ECONS 301 - INTERMEDIATE MICROECONOMICS

LECTURE NOTES

Felix Munoz-Garcia1

School of Economic Sciences

Washington State University

This document contains a set of partial lecture notes that are intended to serve as a

starting point when coming to class, so every student can complement them with

additional examples, exercises and applications discussed in class. (Do not quote).

1 103G Hulbert Hall, School of Economic Sciences, Washington State University, Pullman, WA 99164-6210. e-mail: fmunoz@wsu.edu. Tel. +1-509-335-8402.

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EconS 301 – Intermediate Microeconomics

Chapter 2 – Demand and Supply – Lecture notes

In chapter 2 we deal with demand and supply analysis in perfectly competitive markets. Perfectly competitive markets consist of a large number of buyers and sellers. In competitive markets, the transactions of any individual buyer or seller are so small in comparison to the overall volume of the good or the service traded in the market that the buyer or seller essentially “takes” the price set by the market. Therefore, we consider this the perfect competition model as a model of price-taking behavior.

Demand Curve:

Market Demand Curve: A curve that shows us the quantity of good that consumers are willing to buy at different prices.

• For instance, in the graph below, the demand curve shows us the amount of corn a buyer would be willing to buy at different prices.

This illustration brings us to a crucial law in economics, the Law of Demand…

Law of Demand: The inverse relationship between the price of a good and the quantity demanded, when all other factors that influence demand are held fixed. This law is intuitive, as a higher price will make consumers less willing to purchase that good. With almost all goods, the consumer will behave in this manner, treating price and quantity demanded as inversely related.

A demand curve will take the form . . .

Q = a – bP where ‘a’ = vertical intercept, and ‘b’ = slope

Rearranging and solving for P, we get

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P = (a/b) – (Q/b)

This is the Inverse Demand Curve, which is simply the demand curve where P = some function of Q

Example:

Demand: Q = 100 -2P

Inverse Demand: P = 50 – (Q/2)

• The vertical intercept is therefore 50 and represents the Choke Price, or the price at which consumers of the product will not desire any of the good.

• The horizontal intercept is therefore 100, and represents the amount of the good the consumer would want to purchase at a price of 0.

Or more generally…

Demand: Q = a – bP

Inverse Demand: P = (a/b) – (Q/b

Choke Price 

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Supply Curve

Market Supply Curve: A curve that shows us the total quantity of goods that their suppliers are willing to sell at different prices.

Example:

Supply Curve: QS = 0.15 + P

a) Supply of wheat if P = $2 → QS = .15 (2) = 2.15 P = $3 → QS = .15 (3) = 3.15 b) Sketch the supply curve…

QS = 0.15 + P and solving for P, we get P = QS – 0.15 • So the slope = 1 (coefficient of QS) • Intercept = -0.15

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Equilibrium

As you might guess, the market equilibrium in a perfectly competitive market is the intersection of the supply and demand curves. That is, in equilibrium, a perfectly competitive market will set a price and quantity such that there is no excess supply and no excess demand, hence demand equals supply.

Example:

Demand Curve: Qd = 500 – 4P

Supply Curve: QS = -100 + 2P

a) Let us sketch these curves on the same graph with quantity on the horizontal axis and price on the vertical axis. Inverse Demand Curve → P = (500/4) – (Qd/4) Inverse Supply Curve → P = (QS/2) + 50

b) At what price and quantity do you reach equilibrium? QS = Qd

500 – 4P = -100 + 2P 600 = 6P 100 = P

And then take this p=100 and plug it into either the demand or supply curve to find the equilibrium quantity…

QS = 500 – 4(100) = 100

And so, equilibrium occurs at p=100 and Q=100 An Increase in Demand, for any given price

When P=0, Qd=500‐4.0=500 

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An increase in demand as the one depicted about can originate from an increase in income, or in consumer’s preference for the good. For any given price, the quantity that consumers demand has now gone up. You can visually see that by extending a long horizontal dotted line which maintains your focus on a given (fixed price). The point where the dotted line crosses each demand curve represents the quantity demanded.

A decrease in supply, for any given price

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A decrease in supply might originate from an increase in production costs, which lead producers of the good to supply lower amounts of the good at any given price. [Follow similar graphical representation as above]

Example: Market for Aluminum

Qd = 500 – 50P + 10I where P=price and I=income

QS = -400 + 50P

a) Equilibrium when I = 10 Plug in I=10 into Qd to get Qd = 500 – 50P + 10(10) = 600 – 50P

Now equate Qd to QS,

600 – 50P = -400 + 50P 1000 = 100P

10=P→ QS = -400 + 50(10) = 100

b) Equilibrium when I = 5 Plug I = 5 into Qd to get Qd = 500 – 50P +10(5) = 550 – 50P

Now equate Qd to QS,

550 – 50P = -400 + 50P 950 + 100P 9.5 = P→

QS = -400 + 50(9.5) = 75

Summarize…

Supply P=10 

P=9.5 

10075

P 

Q 

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A change in both the demand and supply curve

A decrease in demand and an increase in supply:

• Unambiguously produce a reduction in the equilibrium price; but…

• the effect on the equilibrium quantity is more ambiguous. In the figure, the outward shift in the supply curve dominates the inward shift in the demand curve, producing an overall increase in the equilibrium quantity. Otherwise, the equilibrium quantity would decrease.

Price Elasticity of Demand

The price elasticity of demand measures the sensitivity of the quantity demanded to price changes.

Price Elasticity of Demand: A measure of the rate of percentage change of quantity demanded with respect to price, holding all other determinants of demand constant. Or, it is the percentage change in quantity demanded (Q) brought about by a 1% change in the price.

,% %

∆

∆∆∆

Example:

When P=10, quantity is Q=50

When P=12, quantity is Q=45 (So ∆Q = -5)

,∆∆

52

1050

0.5

Remember, (P/Q) represents the initial price and quantity

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Why is the elasticity of demand always negative? Because (∆Q/∆P) is the slope of the demand curve, which is negative by the law of demand and (P/Q) is always positive because neither P nor Q can ever be negative.

Example

Let’s find the elasticities for different prices along the demand curve Q = 100 – 2(P), first when…

o A) P=40, so Q= 20 o B) P=25, so Q= 50 o C) P=10, so Q= 80

Remember, the demand curve is in the form Q = a – bP, so the elasticity equation is єd = -b (P/Q)

A) Єd = -2 (40/20) = -4 B) Єd = -2 (25/50) = -1 C) Єd = -2 (10/80) = -(1/4)

Vertical Intercept:

єd = -2 (50/0) = -∞

Horizontal Intercept:

Єd = -2 (0/100) = 0

Elasticity of demand in the linear demand curve, Q = a-bP

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Usual mistake: say that the price-elasticity of demand is equal to the slope of the demand curve. NO!

We just saw that a demand curve with a constant slope (-b) can have different price-elasticities of demand, depending on the price at which the elasticity is evaluated.

Why do we make things so difficult? Wouldn’t it be easier to simply talk about the slope of the demand curve, rather than the price-elasticity? The reason we use price elasticity of demand (and not simply the slope of the demand curve) is because by using the former we can produce a unit-free measure of how sensitive is the demand curve to changes in prices. Indeed, note that units cancel out when you use the formula of price-elasticity of demand, but they wouldn’t if you were simply using the slope of the demand curve.

Different Types of Elasticities of Demand

As we have seen, elasticities show us how one component of a demand equation changes with another component. So far we have only compared how quantity demanded varies with price, in order to see how price sensitive certain commodities are. This analysis, however, can be extended to compare more than just quantity with price. As we will show, we can compare changes in quantity with changes in income, or even with the price of other goods.

Constant Elasticity Demand Curve: A demand curve of the form Q = aP-b where ‘a’ and ‘b’ are positive constants. The term –b is the price elasticity of demand along this curve.

• If we take the derivative of Q with respect to P, we can see that the elasticity is simply the –b exponent

Income Elasticity of Demand: The ratio of the percentage change in quantity demanded to the percentage change in income, holding price and all other determinants of demand constant.

,

∆

∆ ,∆∆

Cross-Price Elasticity of Demand: The ratio of the percentage change of the quantity of one good demanded with respect to the percentage change in the price of another good.

∆

∆ ∆∆

• For example, research shows that a 1% increase in the price of the Nissan Seutra cause a 0.454% change in the quantity demanded of the Ford Escort

Coke vs. Pepsi

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Coke Pepsi

Price Elasticity of Demand єQ,P -1.47 -1.55 Cross-Price Elasticity of Demand 0.52 0.64 Income Elasticity of Demand 0.58 1.38

SO… a ∆1% in the price of Coke causes a 1.47% drop in the Qd of coke, but a 0.52% increase in the Qd of pepsi. This shows us that they are substitutes, so as the price of one rises consumers will demand less of that product and more of its substitute.

Price Elasticity of Supply

This is a very similar analysis to price elasticity of demand, except we are seeing how supply reacts to a 1% increase in the price. Simply, this elasticity will tell us how sensitive quantity supplied is to the price of that good.

,∆∆

Fitting Linear Demand Curves

Now that we know how a demand curve is constructed and how to use its various parts to analysis the relationship between quantity demand, quantity supplied, and price, we can work backwards to actually construct a demand curve. More simply, if we know the prevailing market Q and P and єQ,P, we can have all the components we need to construct the actual demand curve equation of Qd = a - bP

1) ЄQ,P = -b(P/Q) →So… b = -єQ,P(Q/P) 2) Q = a – bP → So…

a = Q + bP

a = Q + (-єQ,P(Q/P))P

a = Q + (-єQ,P(Q))

a = (1 -єQ,P) Q

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CHAPTER 3 - Consumer Preferences and Utility

• Bundle (food, clothing)- each bundle/basket is represented by a point on the graph below. There are three assumptions that must hold true in consumer preference theory.

• Consumer preferences: These three preferences must hold true and

1. Complete ability to compare, either A >B or B >A, or A~B (in other words the consumer has to state their preferences. Whether they prefer A over B, or B over A or if they both make them equally happy.)

2. Transitive if A >B and B >C, then A >C (the choices that the consumer makes must be consistent with each other. Explaining the symbols above, the consumer prefers A over B, and B over C, such that the consumer should prefer A over C)

Example of non-transitive preferences: “I would rather have two cookies than a glass of milk. And I would rather have a glass of milk than a slice of pizza. However (puzzlingly) I would rather have a slice of pizza than two cookies.”

3. Nonsatiated (more is better)- a fairly easy concept to understand because a consumer should prefer two hot dogs to one, and three hot dogs to two. As consumers we want more.

• Ranking Systems:

• There are two type of ranking systems that the book references.

Rationality 

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1) Ordinal Ranking: allows us to gather information about the order in which a consumer ranks a bundle. We can know that a consumer prefers basket A to B but not how much more they prefer it.

2) Cardinal Ranking: allows us to answer the intensity at which a consumer prefers one bundle to another. For example, I like basket A twice as much as basket B. It is usually hard for consumers to articulate their intensity but both ranking systems are important to recognize.

• Utility Function: measures the level of satisfaction from consuming different bundles. It represents the consumer’s preferences. The unit of measurement is Utils, which does not have a real life translation. Instead of considering the absolute or numerical value for the level of satisfaction we compare relative values. For example, the consumer receives more utility from consuming an orange than from consuming an apple. Different consumers may have different utility functions. Thus knowing that John receives 5 Utils from consuming an apple and Jill receives 6 Utils does not give us any useful information. If we knew John received 7 Utils from consuming an orange we could say that John would prefer to consume an orange more than an apple. But again we can only compare relative values. We can think of Utils as a consumers level of happiness in consuming one good.

Let’s begin with just one good, u(y)=(y)1/2. This utility function represents consumer preferences that satisfy:

1) Completeness

For any A, B we have u(A) ≥ or ≤ u(B) that indicates either A >B, A <B, or A ~B.

2) Transitive

For example what if:

A=1, B=4, C=5, then u(A) = (1)1/2 = 1

u(B) = (4)1/2 = 2

u(C) = (3)1/2 > 2

Thus if u(A) > u(B) and u(B) > u(C), then we have that u(A) > u(C).

Therefore the preferences are transitive because

if A >B and B >C then A >C

3) Non-satiation (more is better)

Indeed is satisfied since u(C) > u(B) > u(A)

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Marginal utility: rate at which utility changes as consumption increases. Often thought of as, how much better off we would be if we received one more of something. In mathematical terms as you will see below, the marginal utility is the first derivative of the utility function.

MUy= du/dy = change in u/change in y

Each representation is equivalent.

MUy = ∆u/∆y = ∂U(y)/∂y

Example:

If u(y)=(y)1/2 , then ∂u/∂y = ½ y1/2-1 = ½ y-1/2 = ½ (1/y1/2) = 1/(2 y1/2)

MUy (at y=4)=slope of U at y=4 

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Diminishing marginal utility: additional units add less utility (∆U smaller and smaller) but we will assume that ∆U > 0 always. That is, more is always better, although it is not as good as the last unit consumed.

There are three important points in mind when drawing total and marginal utility curves.

1) Total Utility and marginal utility cannot be plotted on the same graph, because the Y-axis variable differs on each graph.

2) The marginal utility function is the slope of the total utility function.

3) The relationship between total and marginal utility holds for other measures in economics that will be discussed later on.

For example: copies of the same DVD movie. When you get the first one you’re very excited and value the DVD movie greatly. When you get a second one you’re not as happy because you already own a copy of the DVD. The property that you receive less utility from the 2nd copy of the DVD than the 1st copy and less utility from the 3rd copy than from the 2nd copy is called diminishing marginal utility.

Is more always better?

More than one good

Example Utility function: U(x,y) = (x×y)1/2 = x1/2×y1/2

If x = 2, y = 8 then we plug into the utility function to find (2×8)1/2 = (16)1/2 = 4 utils.

In the 3D graph below points A, B, and C all represent the same level of utility

x = 4, y = 4 4 utils

x = 8, y = 2 4 utils

x = 2, y = 8 4 utils

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MUx=∆U/∆x │y is held constant = ∂U(x,y)/∂x │y is held constant

MUy=∆U/∆y │x is held constant = ∂U(x,y)/∂y │x is held constant

This notation represents partial derivatives. When we take a derivative with respect to x we must hold y constant. For example if x is apples and y

is oranges. We would take a partial derivative with respect to x in order to find out how much utility increases from consuming one more apple.

We must hold the number of oranges consumed constant because consuming more oranges would also cause utility to increase.

Calculus example: Then

MUx=∂U(x,y)/∂x = ½ x-1/2×y1/2 = 1/(2x1/2) y1/2

MUy=∂U(x,y)/∂y = ½ x1/2×y-1/2 = 1/(2y1/2) x1/2

Learning-by-doing 3.1

U(x,y) = (x×y)1/2

1. Satisfies more is better?

2. Satisfies diminishing marginal utility?

1. More is better can be checked in two different ways:

a) Increase in x increase in U and increase in y increase in U

b) MUx > 0 and MUy > 0 for any x,y > 0

2. Diminishing marginal utility just tells us that the additional utility we get from consuming amount of goods is smaller and smaller. The additional utility we get from consuming additional goods is MUx and MUy. So we notice that:

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MUx is decreasing in x (x is in the denominator)

MUy is decreasing in y (y is in the denominator)

Mathematically we may take a derivative twice to check for diminishing marginal utility.

a) MUxx < 0 and MUyy < 0 for any x,y > 0

Indifference curves: curves connecting consumption bundles that yield the same level of utility.

Properties:

1) When the consumer likes both goods (MUx>0, MUy>0), indifference curves are negatively sloped (Figure)

2) Indifference curves cannot intercept. They cannot intersect because if they did, then a bundle could creat two different levels of utility which would destroy rational thinking.

In the Figure the ICs increase to the Northeast (upper right)

3) Every consumption bundle lies on one and only one IC (Figure)

4) ICs are not thick They are not thick because then it would violate the more is better assumption if a bundle with more of one good and a constant amount of the other would create the same utils.

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MRS (marginal rate of substitution)

The rate at which a consumer will give up one good to get one additional unit of another good, keeping his or her utility level constant. In our earlier example the consumer gets the same utility from 4x and 4y as she does from 8x and 2y. This in the second situation she’d be exactly indifferent between a trade of 4x for 2y. In this situation we would say the MRS of x for y is 4 for 2, or 2 for 1, or just 2.

Graph depicting MRS

Moving along the curve we see if the consumer has a lot of good x she will trade a lot of good x for a single unit of good y. In the same way if the consumer has a lot of good y she will trade a lot of good y for a single unit of good x. This is known as diminishing MRS.

∆y/∆x is the rate at which the consumer is giving up y in order to get an additional unit of x.

MRSx,y = - slope of I.C. = MUx/MUy

Totally differentiating

Diminishing MRS

Initially you are willing to give up many glasses of lemonade (y) for m additional hamburgers (x). When you have few glasses of lemonade and many hamburgers you will not be willing to give up as many (or none) glasses of lemonade in order to get an additional hamburger.

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Totally differentially x and y (since I am lowering y in order to get more of x), we obtain

-MRS=slope of Ind. Curve

MRS= -slope of Ind. Curve

ICs are bowed in toward the origin.

Application 3.2. Demand for attributes in cars

U(x,y) = (x×y)1/2 where x is horsepower

y is gas mileage

MRSx,y represents how a typical consumer will be willing to forgo horsepower x in order to get an additional mile per gallon (y).

In 1969, MRSx,y = 3.79

Δu = MU xΔx + MU yΔy

−MUxΔx = MUyΔy−MUx

MUy

=ΔyΔx

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In 1986, MRSx,y = 0.71

The decrease means that people became more willing to give up horsepower in order to get an additional mile per gallon (y).

Learning-by-doing 3.3

U(x,y) = x×y MUx = ∂U(x,y)/∂x = y MUy = ∂U(x,y)/∂y = x

a)

• Draw the IC correspondingly to U1 = 128

x×y = 128 in many of its combinations

For example, G: x = 8 , y = 16

H: x = 16, y = 8

I: x = 32, y = 4

• Does I.C. intersect either axis?

No, if IC intersects x-axis, then y = 0, U1 = 0 ≠ 128

y-axis, then x = 0, U1 = 0 ≠ 128

• Does IC indicate that MRSx,y is diminishing?

Yes, since IC is bowed in toward the origin.

and connect them  

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Note that

MRSx,y = MUx/MUy = y/x so we more from left to the right (∆x), we get smaller ratios, smaller MRSx,y (diminishing).

b) Similar, but for U2 = 200

Not graded: Learning-by-doing 3.4. Increasing MRSx,y

U = Ax2 + By2 MUx = 2Ax

MUy = 2By

Special Utility Functions-There are certain goods that will create unique utility functions. These functions are important to recognize because they will not always have diminishing MRS, and it is important to check what type of utility function your working with.

Perfect Substitutes: Aquafina versus Dasani bottled water or Kingston versus Scandisk memory sticks. Close substitutes: butter versus margarine or coffee versus black tea.

MRSB,M = MRSM,B = 1 but it can be any constant

Example

MRS = 2Ax/2By = Ax/By, which increases in x 

MRSx,y =MUx

MUy

=2Ax2Ay

=AxBy

, which increases in x

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U = aB + aM, then MUB = a and MUM = a

Hence, MRSB,M = MUB/MUM = a/a = 1

Then , slope of IC = -1

Example: Pancakes and Waffles

U = P + 2W, MUp = 1 and MUW = 2

The consumer is always willing to trade 2 pancakes for 1 waffle.

MRSP,W = MUP/MUW = ½ slope of IC = -1/2

Constant slope of IC- perfect substitutes will always have a linear indifference curve and are the simplest to sketch.

Constant MRS (not diminishing as in previous examples)

Similarly, U = ax + by, MRSx,y = a/b

Perfect compliments left shoe and right shoe (consumer wants them in fixed proportions), or two scoops of peanut butter to one scoop of jelly for the perfect PB&J sandwhich.

Example:

U = 10×min(R,L)

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G R = 2 and L = 2, then U2 = 10×2 = 20

H R = 3 and L = 2, then U2 = 10×2 = 20

(consumption choices should be at the kink)

Slope of IC (MRS) is:

Zero at the flat segment of the IC

Infinity at the vertical segment of the IC

Undefined at the kink of the IC (infinitely many slopes

Cobb-Douglas

U = (x×y)1/2 and U = x×y are examples of Cobb-Douglas utility functions

Generally, U = Axαyβ

if α = β = ½, then U = Ax1/2y1/2 = A(xy)1/2

if α = β = 1 and A =1, then U = xy

Properties:

1) MUx > 0 and MUy > 0. That is, “more is better” is satisfied

MUx = ∂U(x,y)/∂x = Aαxα-1yβ > 0 for any x

MUy = ∂U(x,y)/∂y = Aβxαyβ-1 > 0 for any y

2) Since MUx > 0 and MUy > 0, then IC is downward slopping (recall property 1 of ICs)

[what about the case in which the consumer dislikes some goods, MUx < 0? Review Session]

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3) Diminishing MRSx,y:

MRSx,y = Aαxα-1yβ / Aβxαyβ-1 = α/β × xα-1-α × yβ-(β-1)

= α/β × 1/x × y = α/β × y/x

The movements from left to right (∆x) will shrink this ratio, and reduce MRSx,y Diminishing MRSx,y

Quasi-Linear preferences

Used in economic applications for situations in which the amount of a commodity (such as toothpaste or garlic) doesn’t change very much to income.

ICs are parallel displacements to each other

General form

U(x,y) = ν(x) + by (linear in y, but not generally linear in x)

where ν(x) is a function that increases in x, such as ν(x) = (x)1/2 or ν(x) = x2.

where b > 0.

For example you make $400 dollars and spend $5 on toothpaste and $395 on pizza. With quasi-linear preferences if got a better job and made $500 dollars you would still spend $5 on toothpaste and increase pizza purchases to $495.

Same slope, same MRS between goods even if income increases 

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Application 3.4 Pet Rock Fad

 

 

 

 

 

 

 

 

 Pet Rock Fad is not completely over For more practice on derivatives see: APP. A3 and A4, pp. 690-695

Another way to find MRS

∆U = MUx×∆x + MUy×∆y

0 = MUx×∆x + MUy×∆y

-MUx×∆x = MUy×∆y

-MUx/ MUy = ∆y/∆x

-MRS = slope of IC

or

MRS = - slope of IC

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CHAPTER 4 – Consumer Choice Chapter 4 introduces us to consumer choice. That is, we are introduced to how consumers choose between different bundles of goods in order to maximize their utility. As we will see, this requires that we know not only consumer preferences but also the constraints that any consumer is bound by, the budget constraint. Using both preferences and constraints we can deduct which baskets a utility-maximizing will choose in a given situation between various goods. Preferences Preferences tell us how a consumer ranks some bundle compared to another bundle, and the utility function representation tells us how much utility a consumer derives from different bundles. Remember also that from a utility function we can create our indifference map, which will later be crucial in choosing an optimal bundle.

However, the bundles are not equally costly, and some may be unavailable at the current prices and income.

Hence, in order to understand how a consumer chooses among different goods we need to take into account both – consumer preferences and budget constraint. Simply put, given that a consumer faces a certain budget constraint (i.e. he or she only has so many resources by which to attain goods), what bundle is the affordable and utility maximizing? Budget constraint The budget constraint defines the set of baskets that a consumer can purchase with a limited amount of income. Graphically, the budget constraint will be represented by a specific region on the graph where all bundles in that region are affordable. To find the limit (outermost edge) of our budget constraint we must construct a budget line (B.L.). The B.L. indicates all of the combinations of a given set of goods that a consumer can afford to consume if she spends all of her money. x – food, px y – clothing, py I – income Example: Let’s use the example from the book where Eric is faced with the decision of purchasing some combination of food and clothing. Let px=price of food, py=price of clothing, and I=Eric’s income over the period.

Budget Line: pxx + pyy = I where pxx – total expenditure on food pyy – total expenditure on clothing

Combination of x and y that the consumer can buy when spending her entire income

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px = 20 , py = 40 , I = 800

pxx + pyy ≤ I meaning the you can only spend as much money as you have this is called the budget constraint (B.C.). Notice that both the x and y intercepts are simply Eric’s income divided by the price of the good on that respective axis. Each intercept therefore represents how much food or clothing Eric could consume if he were to spend his entire income on that one good. Slope of Budget Line (B.L.) pxx + pyy = I

pyy = I - pxx

y = I/py – (px /py ) x Notice that this equation resembles the general equation of a straight line y = b + m×x, where b = vertical intercept , m = slope Hence, the slope of the B.L. = – (px /py ) and indicates, when moving from left to right along the budget line, how many units of y the consumer must give up to increase his consumption of x in one unit. An easy way to remember the slope is simply as the negative of the price ratio. But why is the slope negative? Because to get more of good x while staying on the same budget line you must give up some of good y. If the slope were positive, you would be getting more of good x by also getting more of good y, which intuitively doesn’t make sense when we are constrained by a budget. Example – (px /py ) = - 20/40 = -1/2, and the consumer must give up ½ units of clothing (y) in order to obtain an additional unit of food (x). This ratio is constant because prices are constant. ▲Income ▲is the symbol for “change in”

Budget line Unattainable at the current prices and income pxx + pyy = 20μ20 + 40μ15 = $1,000 > $ 800

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Unchanged slope since – (px /py ) = - 20/40 = -1/2 Hence, new B.L. is parallel to old B.L., but shifted outward. The set of available choices increases with the increases in Income. Why is the B.L. still parallel after an increase or decrease in income? Remember that the slope of the B.L. is the negative of the price ratio between food and clothing. A shift in income has no effect on the prices of the two goods and thus leaves the slope of the line unchanged. ▲px What if the price of good x increases from 20 to 25?

The consumer must give up 5/8 units of y to obtain one more unit of x. New slope = – (px /py ) = -25/40 = -5/8 since |-5/8 | > |-1/2 | , BL2 is steeper than BL1. In this case where prices are changed, the slope of the B.L. is inevitably changed as well, as shown in the above graph. Now, to stay on the budget line as Eric moves from left to right, his tradeoff between clothing and food is now occurring in a different ration than before the change in the price of food.

4

Good news or Bad News? What if both income and all prices are doubled? Intuitively it would seem that if both prices and income double, the two effects would cancel each other out and Eric would be left no better or worse off than before the change. Mathematically, this proves to be the case… Vertical intercept 2I/2py = I/py

Horizontal intercept 2I/2px = I/px Slope = - 2px/2py = - px/py

PUTTING PREFERENCES AND B.L. TOGETHER: Having now identified what a consumer is constrained by when making a choice between bundles, we can put together both his preferences and constraint to find the optimal bundle. This process is central to intermediate micro and will come up time and time again throughout the course, so let’s learn it well… Optimal choice Consumer choice of a basket of goods that 1) maximizes utility 2) is affordable (in the B.C.)

• No points inside B.L. (utility could be increased) • No points outside B.L. (unaffordable)

Intuitively, an optimal consumption bundle will not only maximize a consumer’s utility, but it will also be affordable. These very simple conditions are what we translate into mathematical condition by which to discover the optimal bundle. Also note that a consumer would not choose any bundle inside or outside of the budget line as his optimal bundle. Any point outside the line would be unaffordable for the consumer, and any point inside the line will not maximize utility because it would leave income to be spent which would further increase his utility (remember that this consumer spends all of his income to maximize his utility). Here is our mathematical expression of the two conditions for optimization… Max U(x,y) reaches the highest I.C. max U(x,y) means x and y are the choice variables x,y x,y subject to pxx + pyy = I that still belongs to the set of affordable baskets(on or inside B.L.)

The 2s cancel out, thus no element of B.L. changes

5

Looking at the above graph, why are other baskets not optimal?

D unaffordable (outside of B.L.) C money unspent could increase our utility by spending it (generally, it could be future consumption of the good, or savings) B lower I.C. than at A (less utility at same cost)

Tangency condition at the optimum, A We refer to the condition for the optimal bundle as a tangency condition. This is because at the optimum bundle A the I.C. and the B.L. are tangent, that is, they have the same slope. Knowing this allows us to simply express that optimum condition as…

Slope of B.L. = slope of I.C. at A – (px /py ) = - MUx/MUy = MRSx,y (remember MRS is the slope of I.C.)

px /py = MUx/MUy MUy / py = MUx / px Extra utility per dollar spent on y = Extra utility per dollar spent on x

Same “BANG FOR THE BUCK” spent across different goods.

• The expression MUy / py = MUx / px tells us that at the optimum bundle A both the marginal utility per dollar of good x and the marginal utility of good y are equal. So, at this bundle each dollar spent on each good result in the same level of utility received from consuming that good. If these marginal utilities per dollar were different this bundle would not be optimal. Why is this the case? Because if one good had a higher MU per dollar than it wouldn’t make sense to consume that bundle. It would make sense to consume more of the good with the higher MU per dollar, thus increasing your utility, and thus changing the bundle.

6

Example U(x,y) = xy MUx = y, MUy = x I = 800, px = 20, py = 40 B.L. 1) pxx + pyy = I 20x + 40y = 800 Tangency Condition 2) MUx/MUy = px /py y/x = 20/40 2y = x

• Plug (2y=x) back into the B.L. equation so that we only have one variable

NOTE: In an optimization problem where the MU’s are both positive, the optimal bundle we find at the point of tangency is referred to as an interior optimum. An interior optimum is simply an optimal basket at which the consumer will be purchasing positive amounts of all commodities. Point B in the graph was not optimal since:

MUy / py = MUx / px y/20 ∫x/40 16/20 ∫8/40 0.8 ∫0.2 In particular, MUy / py > MUx / px ▲x Thus, spending less Money on y and more Money on x increases total utility (different marginal utilities per dollar). USING THE LAGRANGE MULTIPLIER METHOD OF CONSTRAINT MAXIMIZATION IN THE CONSUMER CHOICE PROBLEM Using the Lagrange method, we can mathematically show this tangency condition as the optimal bundle. Here we again see that the optimal bundle is where MUx/Px=MUy/Py. Max U(x,y) x,y subject to pxx + pyy = I £(x,y;l) = U(x,y) + l [I - pxx - pyy] F.O.C.s: ∑£/∑x = ∑U(x,y)/∑x - lpx = 0 MUx / px = l (1) MUx

20(2y) + 40y = 800 80y = 800 y = 10 x = 2y = 2μ10 = 20

7

∑£/∑y = ∑U(x,y)/∑y - lpy = 0 MUy / py = l (2) MUy ∑£/∑l = I - pxx - pyy = 0 (3) • From (1) and (2), we have MUy / py = MUx / px (same “bang for the buck”), and from

(3) we have the B.L. • These are the same two conditions we use to find optimal consumption choice for the

consumer. More on maximization problems: APP.A5 pp.695-697 constrained optimization: APP.A7 pp.701-702 and Lagrange multipliers: APP.A.8 pp. 702-704 Example: Cobb-Douglas Utility Here we use the Lagrange method with a Cobb-Douglas Utility Function. U(x,y) = xay1-a subject to pxx + pyy = I £(x,y;l) = xay1-a + l [I - pxx - pyy] F.O.C.s: ∑£/∑x = axa-1y1-a - lpx = 0 axa-1 y1-a

(1/x) = lpx MUx U(x,y) aU(x,y) / xpx = l (1) ∑£/∑y = (1-a)xay1-a-1 - lpy = 0 (1-a)xay-a (1/y) = lpy MUy U(x,y) (1-a)U(x,y) / ypy = l (2) ∑£/∑l = I - pxx - pyy = 0 pxx + pyy = I (3) From (1) and (2) we have aU(x,y) / xpx = (1-a)U(x,y) / ypy xpx / ypy = a/1-a Or alternatively, px x = (a/1-a ) pyy Substituting this result into (3),

8

(a/1-a ) pyy + pyy = I (1+ a/1-a ) pyy = I (1 - a + a/1-a ) pyy = I y = (1 - a) I/py And similarly for x. Starting with py y = (1-a/a ) pxx , and plugging it into (3), (1-a/a ) pxx + pxx = I (1+ 1-a/a ) pxx = I (a+ 1-a/a ) pxx = I x = (I/px ) a Budget shares The proportion of total income spent on each good. Budget share of x: pxx/I = px (I/ px a) / I = I×a / I = a Budget share of y: pyy/I = py (I/ py (1-a)) / I = I×(1-a) / I = (1-a) Notice from an empirical point of view: if you observe a consumer spending a % of his wealth on x and (1-a) % on y, you know his utility preferences function from consuming these two goods. Expenditure minimization problem (EMP) alternative way to think about optimality

This is simply another way to look at the optimization conditions. Whereas before we were maximizing utility given a certain budget constraint, here we approach optimization by minimizing expenditure subject to a certain utility level. This shows that the answer you reach through the tangency condition is the cheapest bundle at a certain utility level. As you can see with bundle A above, it is the least expensive and in fact the only affordable bundle that reaches the utility level of U2. This approach is mathematically expressed as… Min Expenditure = pxx + pyy (x,y) Subject to U(x,y) = U (x,y) (given level of utility) U(x,y) = xay1-a Budget pxx + pyy

9

Min pxx + pyy x,y subject to U(x,y) = xay1-a ≥Ū £(x,y;l) = pxx + pyy + l [Ū - xay1-a] F.O.C.s: ∑£/∑x = px - l axa-1y1-a = 0 px = laxa-1y1-a

(1/x) = U(x,y) px×x / aU(x,y) = l (1) ∑£/∑y = py - l (1-a)xay1-a-1 = 0 py = l (1-a)xay-a (1/y) U(x,y) py ×y/ (1-a)U(x,y) = l (2) ∑£/∑l = Ū - xay1-a= 0 (3) From (1) and (2) we have px×x / aU(x,y) = py ×y/ (1-a)U(x,y) (1-a)px x = a pyy xpx / ypy = a/1-a Hence, py ×y= px×x (1-a)/a, or y = (px×x/ py) ×((1-a)/a) Substituting this result into (3), we obtain, Ū = xa[(px×x/ py) ×((1-a)/a)]1-a Ū = xa× x1-a [(px/ py) ×((1-a)/a)]1-a Solving for x, x = Ū/ [(px/ py) ×((1-a)/a)]1-a = Ū×[(px/ py) ×((1-a)/a)]a-1 And similarly for y, we start from x = (a/1-a)×( py/ px)×y. Substituting it into (3). We have Ū = [(a/1-a)×( py/ px)y]a y1-a = [(a/1-a)×( py/ px)]a ya y1-a

Solving for y, y = Ū/ [(a/1-a)×( py/ px)]a = Ū×[(a/1-a)×( py/ px)]-a

10

py y = (1-a/a ) pxx , and plugging it into (3), (1-a/a ) pxx + pxx = I (1+ 1-a/a ) pxx = I (a+ 1-a/a ) pxx = I x = (I/px ) a Corner Points Solutions So far we have been dealing with optimal bundles where the consumer purchases positive amounts of all commodities (interior optimum). However, in the real world this is not always the case. Many times a consumer will not consume a good at all, and thus, we need a way to find optimal bundles in these situations. This search will take us to what we refer to as corner point solutions: a solution to the consumer’s optimal choice problem at which some good is not being consumed at all, in which case the optimal basket lies on an axis. Example: U(x,y) = xy + 10x MUx = y + 10 MUy = x I = 10, px = 1, py = 2 1) x + 2y = 10 2) MUx / MUy = px / py (y+10)/x = ½ 2y + 20 = x Attempting to find the optimal bundle via the tangency condition gives us a negative amount consumed of good clothing. Since we can obviously not consume negative clothing, this simply informs us that our optimal bundle is going to be where we consume 0 units of clothing. This is a corner point. Then, consumer wants to increase his consumption of x as much as possible. That is, the consumer spends all his income in x, I/ px = 10/1 = 10 , and no income in y.

(2y+20) + 2y = 10 4y = -10 y = -2.5 (NO!! y cannot be negative, thus we have a corner solution where y = 0) x = 2y + 20 = - 5 + 10 = 5

11

The graph shows our corner solution where all income is spent on good x. In this situation, the MU per dollar of food is always higher than the MU per dollar of clothing, so you will this consumer will continually substitute clothing for more food until he reaches 0 units of clothing (i.e. all his income is spent on food at bundle R). Corner Points with Perfect Substitutes What if a consumer is always willing to substitute clothing for food, but in a constant ratio? This is a case of perfect substitutes. Instead of a curved I.C., the I.C.’s will be straight line, since the slope is constant.

MRSchoc.,vanilla = MUc/MUv = 2 (constant MRS because of Perfect substitutes)

pc/pv > MUc/MUv MUv /pv > MUc /pc and as a consequence the consumer should only buy vanilla ice cream.

Once again, the MU per dollar for both goods is never equal, which will in turn lead the optimal bundle to a corner point where 0 units of the less preferred good is consumed. In other words, the consumer only likes chocolate ice cream twice as much as vanilla ice cream, but vanilla ice cream is three times cheaper than chocolate ice cream. pc/pv = 3

Don’t confuse this Indifference curve with B.L.

[What if price ratio was lower than MRS? Then BL becomes flatter than I.C., and the consumer buys only chocolate]

12

Coupons vs cash subsidies Occasionally governments and other institutions want to increase the consumption of a particular good. They can do this by issuing coupons, which can only be spent on the good of interest, or through cash subsidies, which are lump sum payments that can be spent on any good (but which the issuer hopes will be spent on the good of interest). To isolate the good of interest, say housing, we lump all other goods into a variable we refer to as a composite good. A composite good is simple a good which represents the collective expenditures on every other good except the commodity being considered. Cash subsidies: B.L. moves from KJ to EG Vouchers (coupon): B.L. moves from KJ to KFG

• In the above graph, both methods of either issuing a coupon or a subsidy push the consumer from I.C. U1 to U2. Thus, both achieve the desired effect of raising housing consumption to hB. However, this is not always the case. Notice that if the U2 I.C. was tangent to the segment EF, the coupon would not allow the consumer to move to a higher I.C. while the subsidy would.

Joining a Club You pay a lower price if you are a club member, however you must pay to join the club. Is this trade off worth joining the club? Let’s look at an example…

I = 300 px = $20 py = $1

13

I = 300 – 100 (fee) = $200 px = $10 py = $1 Before joining: I= 300, px = $20 I/ px = 300/20 = 15 units py = $1 I/ py = 300/1 = 300 units The consumer chooses basket A. After joining: I = 300 – 100 (fee) = $200, px = $10 I/ px = 200/10 = 20 units py = $1 I/ py = 200/1 = 200 units Flatter BL, and the consumer selects basket B.

• Basket B is on a higher I.C. than basket A, and therefore this consumer is better off after joining the club.

A similar analysis is applicable for

1) cell phone service (higher monthly subscription charge in order to have lower price per minute in calls)

2) gym, golf club Example: Sprint Cellular in 2006 Plan A: $ 30 monthly charge, 200 minutes, and $0.10 beyond these minutes. Plan B: $60 monthly charge, 1,000 minutes, and $0.10 beyond these minutes. Drawing Plan A: income is $120, but the consumer pays $30 on monthly charge

• 200 first minutes are free, then the slope of BL is – 0.1 • If he didn’t get free minutes, the highest amount of minutes he could

consume is 90/0.1 = 900, but he gets 200 min.for free = 1,100 • MRT is the BLA

14

Drawing Plan B: income is $120, but the consumer pays $60 on monthly charge • 1,000 first minutes are free, then the slope of BL is – 0.1 • If he didn’t get free minutes, the highest amount of minutes he could

consume is 60/0.1 = 600, but he gets 1,000 min.for free = 1,600 • MRT is the BLB

If the consumer were to spend all income on cell phone minutes he would choose Plan B to receive more minutes. Borrowing and Lending So far we have considered the consumer’s income as fixed within a time period. However, this is not always the case in the real world. As we well know as college students, consumers can both borrow and lend money. Borrowing and lending is really a tradeoff in consumption. In borrowing you consuming more today by consuming less tomorrow. In lending you are consuming less today so that you can consume more tomorrow. But how does this affect our optimization problem? Without borrowing and lending consume I1 today and I2 tomorrow (as we have considered thus far) With borrow and lending (allowing for)

• If you don’t consume anything today, then you have I2 + I1(1+r) tomorrow because you lent some consumption today to be paid back with interest (1+r) tomorrow, where r=interest rate earned by loaning the money

• If you don’t consume anything tomorrow, then you can consume: I1 + I2/(1+r) today by borrowing

How to find a slope?

• Our x-axis here will be consumption today while the y-axis will represent consumption tomorrow

• Intuitively, the price of giving up one unit of consumption tomorrow (y axis) in order to gain one more unit of consumption today (x axis) is measured by the opportunity costs of every dollar, (1+r).

• How to prove it more formally? We know that the B.L. is y = a + m×x y = I2 + I1(1+r) + m×x a and, in addition, y = 0 and x = I1 + I2/(1+r) , Then , 0 = I2 + I1(1+r)+ m×( I1 + I2/(1+r)) m = [I2 + I1(1+r)]/-[ I1 + I2/(1+r)] = [I2 + I1(1+r)]/-[ (I1 (1+r)+ I2)/(1+r)] = - (1+r)[I2 + I1(1+r)]/ [I2 + I1(1+r)] = - (1+r)

15

• A consumer with these preferences chooses to : o Borrow today if C1B > I1 o Repay the loan tomorrow if C2B<I2

• How can we draw the ind. curve of an individual that prefers to:

o Save today? o Get the returns of his savings tomorrow?

By drawing ind.curve where the tangency point (optimal consumption) is above the point of no-borrowing or lending (A)

WHAT IF THE INTEREST RATE FOR BORROWING AND LENDING DO NOT COINCIDE? Assume I1 = 10,000 I2 = 13,200

16

Example: Lending, rL = 0.05 Next year he will have 13,200 from tomorrow’s income = 13,200 10,000 (1+0.05) from savings = 10,500 ----------------- 23,700

(Vertical intercept, all of consumer’s income was consumed next year) Borrowing, rB = 0.10 Today I can have : 10,000 from today’s income = 10,000 13,200/1+0.10 = 12,000 = 12,000 -------------- 22,000 Where is the I.C. of a saver? Tangency point at segment AE Where is the I.C. of a borrower? Tangency point at segment AG Quantity Discounts Firms can also incentivize the consumption of a product by offering quantity discounts (i.e. discounts after you purchase over a given number of units). In this case, the budget line for a consumer will have a kink where the discounted price kicks in. Let’s look at an example… py = $1 px = $11 for the first 9 units $5.5 for all additional units

How to find the vertical and horizontal intercepts in this case? Vertical Intercept =

Slope = - px / py = - 5.5/1 = -5.5

Slope = - px / py = - 11/1 = -11

IPy

=$440

$1= 440

IPx

=44011

= 40

17

Horizontal intercept of BL1 (no discounts) = Horizontal intercept of BL2 (with discounts) =

• Notice that this consumer is better off with the quantity discount, and that she is consuming 7 more units than before.

• Frequent flyer programs are another example of quantity discounts. Revealed Preferences We have learned how to find optimal baskets for a consumer if we know her preferences and budget line. But, what if we don’t know the consumer’s preferences? Her choices in different settings will help us infer her preferences. That is, by looking at how she chooses bundles in different situation, we can infer which bundles she prefers to others.

• Note that the dark shaded region represents that the area to the northeast of any

bundle will always be preferred to that bundle. Through revealed preferences we can establish if a consumer weakly prefers a bundle to another (i.e. prefers it at least as much as the other bundle when both choices are affordable), or if he strongly prefers one bundle to another (i.e. prefers the bundle more when both choices are affordable).

1) When facing BL1, consumer chose A instead of any point inside BL1, such as B. Then A ≥B

2) A and C were equally costly, but he chose A. Hence A ≥C 3) Since C lies to the northwest of B, then C >B

Choices:

A >B

440 − 9.115.5

= 62

18

Failing to max utility I= $24 ( px , py ) = ($4, $2) (x1, y1) = (5, 2) A ( px , py ) = ($3, $3) (x2, y2) = (2, 6) B Here the consumer fails to maximize utility. Let’s see why…

BL1 :

1) when facing BL1, the consumer chose A instead of C, in spite of being available (affordable) A≥C

2) Since C is to the northwest of B, C>B BL2 :

1) when facing B in spite of D was available B≥D

2) Since D is to the northwest of A, D>A Contradiction occurs because under both budget lines A and B were affordable. If A were chosen over B under BL1 then it should always be chosen over B whenever both are affordable consumer wasn’t maximizing utility when

24/3=8 24/4=6

24/2=12

24/3=8

A >B

B >A

19

• Choosing A within BL1 • Choosing B within BL2

Alternative approach

BL1 : Consumer prefers basket A. Thus A >B

Cost of basket A, $4×5 + $2×2 = $24 Cost of basket B, $4×2 + $2×6 = $20

BL2 : Consumer prefers basket B. Thus B >A

Cost of basket B, $3×2 + $3×6 = $24 Cost of basket A, $3×5 + $3×2 = $21 Case 1. Any point on B.L. is strongly preferred to points inside B.L.

BL2 C>A since it is to the northwest of A B ≥C since B was chosen when C was affordable Case 2 An example of choices that are inconsistent with utility-maximizing behavior BL2 : From above, we know that B >A since A is inside BL2 BL1: Additionally, A ≥B since both were affordable when facing BL1 Hence, consumer is NOT maximizing utility Case 3: BL1 : the consumer chose A when both A and B were affordable, A ≥B BL2 : the consumer chose B when A wasn’t affordable Case 4 BL1 : chose A but B wasn’t affordable

B >A

Contradiction

We can’t infer anything from these choices because we don’t know which bundle the consumer would choose if both were affordable

20

BL2 : chose B but A wasn’t affordable The theory of revealed preference:

1) Allows us to infer how a consumer ranks one basket of goods over others her preferences

2) We can discover when a consumer is not maximizing his utility

1  

EconS 301 – Intermediate Microeconomics

Chapter 5

Optimal Choice and Demand

It is important to realize that given the consumers preferences, income, and price of all goods we could determine how much they will buy. That is simply one point on their demand curve for that good. We can find more point by repeating the process by changing the price of that one good for example ice cream.

Figure 5.1 Does a good job of explaining how to use indifference curves and budget lines to find the demand curve.

2  

The curve used to find the demand curve is known as the price consumption curve (the set of utility maximizing baskets as the price of one good varies while holding constant income and the prices of the other goods). Effects of a change in Income: We can also change the income of the consumer and observe how their demand shifts based on income. Figure 5.2 displays how we use an income consumption curve to see shifting demands.

From the picture above we can see the Income Consumption Curve is the set of utility maximizing baskets as income varies while prices are held constant! There is one last way of showing how a consumer’s choice of a particular good varies with income and that is to draw an Engle curve (a curve that relates the amount of commodity purchased to the level of income, holding constant the prices of all goods). It is important to remember there are more than one type of good, and an Engle curve can help you distinguish which type of good the consumer is dealing with. A normal good is a good that a consumer purchases more of as income rises and an inferior good is a good that a consumer purchases less of as income rises.

3  

The next page contains algebraically how to find the demand curve as well as the engle curve with visual representation. Finding a demand curve (Learning by doing 5.2): A consumer purchase two goods X and Y where the utility function is represented by

*

*

and

1)

2) plug into 1

2 plug into y2

22

x y

x y

x x x x

y y y y

xx y x

y x

xx

y y

U xy MU y MU x

p x p y I

MU p y p p xMRS yMU p x p p

p x Ip x p I p x I xp p

Ipp Iy

p p

= ⇒ = =

+ =

= = ⇒ = ⇒ =

⎛ ⎞+ = ⇔ = ⇔ =⎜ ⎟⎜ ⎟

⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠= =

The last equations ( )* * x and y represent the demand curve. Given any value for Income and for prices of goods, we can find the quantity of good the consumer will purchase. Demand with corner point solution: Using the same goods from previous example but here we change the utility function such that

( )

( )

( )

10 10 and

Suppose that 100 and 1 then

1) 1 100 100

10 12) 10 plug into 1

100 1010 100 2 10 100

2

100 102

x y

x y

x y y

x xy

y y y

yy y y y

y

y

U xy x MU y MU x

I p p unknow

p x p y x p y

MU p yMRS x y pMU p x p

py p p y p y p y

p

pp

= + ⇒ = + =

= = =

= + = ⇒ + =

+= = ⇒ = ⇒ = +

−+ + = ⇔ + = ⇔ =

−0 only if 100-10 0 100 10 10

100 10 if <10

2

0 otherwise

y y yy

yy

y

p p p

pp

py

≥ ≥ ⇔ ≥ ⇔ ≥

−⎧⎪⇒ = ⎨⎪⎩

Only the consumer demand positive quantities of normal goods.

4  

What happens if >10yp ? The consumer will be at a the corner point because yx

x y

MUMUP P

>

Engel Curve in corner solution: Recall that Engel curve relate the amount of good consumed to the level of income, then the Engel Curve will be the same like axis X, depends where is the corner solution. [Figures] Engel Curve for non corner solution:

When the Engel curve have positive slope then the good will be normal

Normal goods, , ,0Q I Q Ix II x

δε εδ+

⎛ ⎞> ⇔ = ⎜ ⎟⎝ ⎠

When consumer buy less of one good when

income increase then the good will inferior

Inferior goods, , ,0Q I Q Ix II x

δε εδ−

⎛ ⎞< ⇔ = ⎜ ⎟⎝ ⎠

5  

Note that some good must be normal, not all can be inferior. To show this using elasticities,

1 1 2 2 ... n np x p x p x I+ + + = Differentiate w.r.t. I,

( )1 21 2 1 2... 1 since , ,..., ,n

n i ndxdx dxp p p x p p p I

dI dI dI+ + + =  

Multiply and divide by ix I ,

1 2, , ,1 2

1 2

1 1 2 21 2

1 2

1 1 1 2 2 2

1 2

1 , 2 , ,

... 1

... 1

... 1

nQ I Q I Q In

n

n nn

n

n n n

n

Q I Q I n Q I

x I dxx I dx x I dxp p px I dI x I dI x I dI

p x dxp x dx p x dxI I II dI x I dI x I dI xθ θ θε ε ε

θ ε θ ε θ ε

+ + + =

+ + + =

+ + + =

Since 1 2, ,..., 0nθ θ θ ≥ , you cannot have , 0Q Iε < for all goods and still have the expression be equal to one.

6  

Example: Food / all other goods in China

[ ]0,1 , but ?f oε ε∈

We know that: ( )1

1 11

ff o o

food other

θεθε θ ε ε

θ−

+ − = ⇔ =−

If 0fε = , then 11oε θ

=−

, which is >1 since the expenditure share, theta, is between 0 and 1.

If 1fε = , then 1 11o

θεθ

−= =

−, which is the lower bound.

According to the data, food’s share of total expenditure in 1983 was 0.6θ = , and in 20050.37θ = .

Then our upper bound was 1 1 2.51 1 0.6oε θ

= = =− −

in 1983, and 1 1 1.591 1 0.37oε θ

= = =− −

in

2005. Hence you know that in 1983 [ ]1, 2.5oε ∈ , and in 2005 [ ]1,1.59oε ∈ .

7  

Change in the Price of a Good: Substitution Effect and Income Effect Substituion effect: The change in the amount of a good that would be consumed as the price as that good changes holding all other prices and the level of utility constant. Income Effect: The change in the amount of a good that a consumer would buy as purchasing power changes holding all other prices constant. The following figure does a nice job of showing step by step how to find both the substitution and Income effect graphically.

8  

Decomposition Budget Line: After define what good is what, we can observe the effects after the price of one good change. One will be the substitution effect and the other will be the income effect, both conform the decomposition budget line, how? Well the SE will be in the same curve and the IE will shift the curve (up or down).

1) Parallel to the BL2 (since prices are the same) 2) Tangent to IC1 (same utility as initial basket)

Substitution Effect Income Effect Total Effect

Normal + + + Inferior + (>) - + Giffen + (<) - -

Here are some pictures of different possibilities, the one above was x being a normal good. Case 2: X is neither a normal good nor an inferior good-

Case 3: X is an inferior good-

9  

Case 4: X is a giffen good-

IE & SE for a price decrease (Learning by doing 5.4):

( )

1 2

and

Suppose that 72 and the initial 9 then decrese to 4, 1

:1) 9 72

92) 9 plug into 11

9 9 72 18 72 4 plug into y9(4) 36(4,36)

x y

x x y

y

x x

y y

U xy MU y MU x

I p p p

Basket Ax p y

MU p yMRS y xMU p x

x x x xyA

Ba

= ⇒ = =

= = = =

+ =

= = ⇒ = ⇒ =

+ = ⇔ = ⇔ =

= ==

( )

C :1) 4 72

42) 4 plug into 11

4 4 72 8 72 9 plug into y9(4) 36(9,36)

y

x x

y y

sketx p y

MU p yMRS y xMU p x

x x x xyC

+ =

= = ⇒ = ⇒ =

+ = ⇔ = ⇔ =

= ==

Decomposition Basket B:

1) B must give the same utility level as the initial basket A, Hence, from (4,36)A = we know 1 4(36) 144U xy= = = . Then the (x,y) of basket B must satisfy xy=144.

10  

2) BLd must be tangent to IC1. Since the slope of BLd is the same as BL2, then 41

x

y

pp

= .

Then,

d 1slope of BL slope of IC41

4 4

x

y

MUMU

y y xx

=

=

= ⇒ =

3) Then we have,

( )

( )( )

21444 144 4 144 6

4

4 6 24

6, 24

xyx x x x

y x

y

B

= ⎫⇒ = ⇒ = ⇒ =⎬= ⎭

= =

=

Where,

6 4 23

B A

C B

SE x xIE x x

= − = − == − =

Summary: As price of good Y decrease the SE leads us to an increase in consumption of good Y (normal good) from 4 to 6. The IE will measure the consumption from decomposition basket B to basket C Note: When you confront SE you are in the same Utility but when you confront IE you are in the same slope…remember is just a trick IE & SE for a price increase: Do Learning by doing 5.5 yourself. Learning by Doing 5.6 IE & SE for a Quasilinear Utility: Recall that the distinguish characteristic for Quasi-linear utility function is that as we move due north on the indifference map, the marginal rate of substitution of x for y remains the same, so be careful with IE.

11  

( )

( )

1 2

12 and 1

10, 0.5, 0.2, 1

:1) 0.5 10

10.5 1 12) 4

1 1 2

0.5 4 10 8(4,8)

C :1) 0.2 10

10.2 1 12) 25

1 1 5

0.2 25 10

x y

x x y

x x

y y

x x

y y

U x y MU MUx

I p p p

Basket Ax y

MU p xMRS xMU p x

y yA

Basketx y

MU p xMRS xMU p x

y

= + ⇒ = =

= = = =

+ =

= = ⇒ = ⇒ = ⇒ =

+ = ⇒ =

=

+ =

= = ⇒ = ⇒ = ⇒ =

+ = ⇔ 5(25,5)

yC

=

=

Decomposition Basket B:

1) Same utility level as basket A=(4,8), so 1 2 2 4 8 12U x y= + = + = . Thus we know,

2 12x y+ = .

2) BLd must be tangent to IC1 and parallel to BL2.

( )

1 2d

slope of IC slope of BL(and BL )

10.2 1 1 25

1 1 5

2 25 12 225, 2

25 4 2125 25 0

x x

y y

B A

C B

MU p x xMU p x

y yBSE x xIE x x

= ⇒ = ⇒ = ⇒ =

+ = ⇒ =

=

= − = − == − = − =

Notice there is no IE because she or he consumes the same amount of good at B and C.

12  

13  

Compensated Demand Curve: [Figures] In our exercise from last Tuesday, if α=0.6,

( )0.4 0.4

1 20.6, , 1.180.4

y y

x x

p ph p p U U U

p p⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Using ( )1 2, ,h p p U we obtain the minimal expenditure,

( ) ( ) ( )1 2 1 1 1 2 2 2 1 2, , , , , ,E p p U p h p p U p h p p U= + Hence,

( )1 1 2 11

, ,E h p p U qp

δδ

= ≡

Slutsky Equation:

We know that ( ) ( )1 2 1 2 1 2

minimal expenditure to reach

, , , , , ,

U

h p p U D p p E p p U

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

Differentiating w.r.t. p1,

1

1 1 1

11 1

q

h D D Ep p E p

h D D qp p E

δ δ δ δδ δ δ δ

δ δ δδ δ δ

= +

= +

Multiplying all terms by 1

1

pq

and the last by EE

,

1 1

1

,

1 1 11

1 1 1 1 1

*

Q I

p qD EE q E

p p ph D D Eqp q p q E q E

θε

ε ε δδ

δ δ δδ δ δ

= + 

Rearranging, ,

,

*

*Q I

Q ITE SE IE

ε ε ε θ

ε ε ε θ

= +

= −

14  

Examples: 1) Garlic: If 0θ = , then *

TE SE

ε ε= or the TE = SE.

2) If 0.76θ = , , 0.88Q Iε = , 0.76ε = − and * 0.09ε = − , then , 0.6688Q Iε θ = and we have, 0.76 0.09 0.6688− = − −

3) Housing: If 0.4θ = , , 1.38Q Iε = , and 0.6ε = − , then we have

,* 0.6 (0.4)1.38) 0.04Q Iε ε ε θ= + = − + = For an inferior good, ε >0, but , ,0 *Q I Q Iε ε ε ε θ

− −−

+

−

< ⇒ = −

For a giffen good, ε <0 because , 0Q Iε < and ,* Q Iε ε ε θ− −

−

+

+

= −

The Concept of Consumer Surplus: CS= maximum willingness to pay for a good – price paid for good. i.e. how much better off the consumer will be when he purchases the good. The area under a demand curve measure net benefits for a consumer only if the consumer experiences no income effect over the range of price change. Learning by doing: Q=# of gallons of milk purchased at P dollars Q=40-4p —> but we want p to be in the y-axis, the rewrite p=10-Q/4

1) Point A is the crossing of p=10-Q/4 and p=3. Then,

3 10 7 284 4Q Q Q= − ⇔ = ⇔ =

2) CS is the shaded blue (G) triangle

15  

1 (28)(10 3) 982

CS = − =

Consumer Surplus with a Cobb-Douglas utility function:

0.6 0.4 15 20 300x xU x y p p I′= = → = =

1) Find the consumer’s demand curve, 0.6 0.6(300) 180

x x x

Ixp p p

= = =

2) [Figure]

[ ] [ ]20

20

1515

180 180 ln 180 ln 20 ln15 51.79x xx

CS d p pp

∇ = − = − = − − = −∫

3) What if we were assuming linear demand: Area of A=(20-15)9=45 Area of B=(20-15)3=15 Total loss in CS =60

Welfare changes resulting from variations in prices — CV and EV: CV: A measure of how much money a consumer would be willing to give up after a reduction in the price of a good to be just as well off as before the price decrease. If prices increase, then I need more income to maintain the same level of utility. How much money would I need? CV If prices decrease, then I don’t need all my original income to remain at original utility. How much money would I be willing to give up in order remaining at the original utility? CV EV:A measure of how much additional money a consumer would need before a price reduction to be as well off as after the price decrease. If prices are going to decrease, then after the decrease, consumers are going to be better off because they can purchase more. How much income should we give them today, before the price increase, to make them just as well-off as they will be tomorrow after the price decrease? EV If prices are going to increase, then after the increase, consumers are going to be worse off. How much money would we need to take from the consumer today, before the price increase, in order for him to be just as worse-off as he is going to be tomorrow? EV

16  

0K— consumer’s income 0L— income needed to purchase basket B at the new prices of x 0K-0L=KL is the CV

The amount of money that the consumer will be willing to give up, after the price change, in order to maintain the original utility that he had before the price change.

0J— income needed to buy basket E at the old prices of x. 0J-0K=JK is the EV

The amount of money that we need to give to the consumer, before the price change, in order to make him just as well-off as he will be after the price change.

CV ≠ EV, except with quasilinear utilities (CV = EV). CV and EV with no income effect (Based on Learning by Doing 5.6): Recall in this case the IE was zero

1 2

210, 0.5, 0.2, 1x x y

U x yI p p p= += = = =

a) CV= Cost of initial basket A ($10) less cost of decomposition basket B (0.2(25)+2=$7) CV= 10-7=$3

b) EV= Cost of buying basket E at initial prices – cost of buying initial basket A But what is basket E? 1) We know E must reach a utility of U2 =15, hence 2 15x y+ = 2) We also know that at E, the slope of U2 = slope of BL1

10.5 1 1 4

1 1 2x x

x= ⇒ = ⇒ =

Hence, 2 4 15 11 (4,11)y y E+ = ⇒ = ⇒ = Cost of buying E=(0.5)4+11=13 EV=13-10=$3

17  

Hence the CV=EV when the IE is absent (which occurs when we have a quasilinear utility)

What if we measure the change in welfare by using CS? You can easily check that when,

1 2

210, 0.5, 0.2, 1x x y

U x yI p p p= += = = =

The demand for x is, 2

1

x

xp

= . Then,

 

18  

0.20.2

20.5 0.5

1 1 1 1 30.2 0.5x

x x

CS d pp p

⎡ ⎤Δ = = = − =⎢ ⎥

⎣ ⎦∫ which is exactly the same as the CV and EV when we

have a quasilinear utility function. What if IE>0?

1 272, 9, 4, 1x x y

U xyI p p p== = = =

a) CV= cost of buying initial basket A ($72) – cost of buying decomposition basket B ((4)6+24=448) CV=72-48=24

b) EV= cost of buying initial basket E – cost of buying basket A ($72) What is E?

1) We know basket E must reach U2, so 4(36) 324xy = = 2) We also know that at E, the slope of U2 = slope of BL1

( )

9 91

9 324 69(6) 54(6,54)

y y xxx x xyE

= ⇒ =

= ⇔ =

= ==

EV=108-72=36 c) CV ≠ EV What if we measure the change in welfare using CS? Given,

72, 1y

U xyI p== =

It is straight forward to find the demand 72 362 2x x x

Ixp p p

= = =

19  

[ ] [ ]9

9

44

,

36 36 ln 36 ln 9 ln 4 29.20

29.224 different, but is this usual? Not so much since and are low.36

x xx

Q I

CS d p pp

CSCVEV

θ ε

Δ = = = − =

Δ = ⎫⎪= ⎬⎪= ⎭

∫

Application: Automobile export restrictions for Japanese cars in 1984. Prices went up about 20% for Japanese cars, and CV = $14 billion. That is the additional income needed after the price change (due to export restrictions). Since in 1984 there were 2 million new cars bought, the CV per car buyer was $14,000/2 = $8,000. Market Demand: If we have three demand curves, which two are fro two kinds of consumers and the last one the market demand.

15 3 if 5( )

0 if 5

6 2 if 3( )

0 if 3The market demand will be

21 5 if 3 (Both)( ) 15 3 if 3 5 (Only h consumes)

0 if 5 (Neither

h

c

M

p pQ p

p

p pQ p

p

p pQ p p p

p

− <⎧= ⎨ ≥⎩

− <⎧= ⎨ ≥⎩

− <= − ≤ ≤

≥ consumes)

⎧⎪⎨⎪⎩

Graphically we can represent this

20  

21  

Network Externality: Bandwagon effect: A positive network externality that refers to the increase in consumer demand for a good as more consumers buy the good. (ie online games) Snob effect: A negative network externality that refers to the decrease in consumer demand for a good as more consumers buy the good. Perloff: Ch.4 pp. 111-116, Ch.5 pp. 136-141 and 152-164. Tax Revenue and Labor Supply: TaxRevenue ( )where ( ) is the number of hours worked when wage net of taxes is (1- )

whh w

τ ωω ω τ

==

2

positive effect on Tfrom higher rate negative effect on T

from fewer hours worked

( )T dhwh wd

δ ω τδτ ω

−+

= −

For 0Tδδτ

< so a decrease in tax causes an increase in revenue, we need,

2 1( )( )

dh dh wwh wd d h

ω τω τ ω ω

< ⇔ <

Multiply both sides by 1-τ,

( )

elasticity of supply of labor

11( )

1( )

wdhd hdhd h

ω

τττ ω ωτ ω

τ ω ω

−−<

−<

Hence, for T to raise from a small fall in the tax rate, we need

supply, 1

ωτε

τ−

>

Example: 25%τ = if your income is about average $35,000 a year

supply, 1 0.25 3

0.25ωε −> = which is unlikely. (ie Bush, Reagan)

In Sweden 90%τ = , supply, 1 0.9 .111

0.9ωε −> = which is very likely. (ie Japan, Sweden, Kennedy)

22  

Labor Supply:

( )

( )

1

from work from leisure

1 1

1 1

1

( ) (24 )

( ) (24 ) ( ) 1 (24 ) 0

( ) (24 ) ( ) 1 (24 )

( ) (24 ) 1

(24 ) 1

2424

U wh h

FOCsU w wh h wh hh

w wh h wh h

w wh h

hh

h h hh

α α

α α α α

α α α α

δ α αδα α

α αα α

α α αα

−

− − −

− − −

−

= −

= − − − − =

⇔ − = − −

⇔ − = −

⇔ − = −

⇔ − = −⇔ =

If, for example, alpha was one third, then hours worked would be eight, regardless of wage.

1

EconS 301 – Intermediate Microeconomics Chapter 6

Inputs and Production Function In order for a firm to produce output, either products or services, the firm must use a certain function of inputs to create their output. Inputs (factors of production) are resources, such as labor, capital equipment, and raw materials, that are combined to produce finished goods. A firm must decide how to use these inputs to produce a given output level. For instance, they may use a lot of labor and very little capital (machines, etc.) or they may use a lot of capital and very little labor. This is called a firm’s production function: a mathematical representation that shows the maximum quantity of output a firm can produce given the quantities of inputs that it might employ.

( ) ,Q f L K= The production function tells us the maximum amount of production, Q, for a given amount of inputs L and K. (analogous to utility function). Note: we could include any number of inputs in the production function but L and K are the two inputs most firms face and L and K allow us to develop the main ideas of production theory.

Example: Technical Inefficiency among U.S Manufacturers (63% of efficiency) (39% of inefficiency)

2

( )/ , 0.63Q f K L = but much closer to the frontier if the firm: • faces competition • not a major player in its industry

We can also invert the production function to get L=g(Q), which tells us the minimum amount of labor needed to get a specific quantity of output. This approach is called the Labor Requirements Function. Marginal and Average Products We now want to characterize the productivity of the firm’s labor input. Looking at the production function we can derive two distinct types of productivity for a given input: the average productivity and the marginal productivity of a certain input. In the cases below, we are looking at these two distinctions for labor…

The first is the Average Product of Labor, which tells us the average output per worker, which we write as the APL.

/LAP Q L= “Labor productivity” • Looking above at the Total Product graph, the Average Product of Labor is the

slope of the ray between any point on the curve and the origin (for instance, the ray from point A to the origin).

The second is the Marginal Product of Labor, which tells us the rate at which total output rises as the firm increases its quantity of labor. We write this as MPL.

3

/LMP Q L= Δ Δ (analogous to Marginal Utility in Consumer Choice Theory) • Looking above at the Total Product graph, the Marginal Product of Labor is the

slope of any line tangent to any point on the graph (for instance, the line BC is tangent to L1).

Before we continue we need to focus on “Law of diminishing Marginal Returns”… Relationship between AP and MP: the relationship between MR and AR can be illustrated with the simple example of adding one more test score to a collection of scores. If the added test score (marginal effect) is greater than the current average, it will bring up the overall average. If the added score is lower than current average, it will bring down the average score.

Æaverage grade its marginal effect was positive ∞average grade its marginal effect was negative AP is increasing MPL > APL AP is decreasing MPL < APL AP is flat MPL = APL

2 INPUTS: When we consider a production function with two inputs, the associated graph is three-dimensional. On the two horizontal axis’s are the two inputs, and on the vertical axis is the quantity produced. We refer to this graph as the Total Product Hill. If the production output is ( ) ,Q f K L=

( )L K cte

K L cte

MP Q / L |Slopes

MP ( Q / K) |⎛ = Δ Δ⎜⎜ = Δ Δ⎝

They are the slopes of the hill for labor and capital when the other inputs are fixed.

• We can therefore derive the Total Product Function for one input from a two input production function by holding either K or L constant and following the hill as the other input increases.

4

Example: 24, K and L= Δ (eastward), we move from A to B to C (peak).

Given the green line is our TP when ( ) Q f L= because we were fixing K (in particular, fixing it at K = 24).

( )( )

tan

tan

/ / |

/ / |L K is held cons t

K L is held cons t

MP change in Q change in L Q L

MP change in Q change in K Q K

= = Δ Δ

= = Δ Δ

MPL is the steepness of the green line, when we hold K fixed at 24K = . ISOQUANTS: To show the economic trade-offs in production, it is helpful to flatten the three dimensional Total Product Hill to two dimensions. By taking a fixed level of Q from the Total Product Hill and then dropping those fixed levels to the same level, we can graph a contour plot very similar to the indifference plot we constructed in consumer choice. We call these fixed levels of production given various combinations of two inputs Isoquants. As stated in the book, an isoquant is a curve that shows all the combinations of labor and capital that can produce a given level of output. Level curves, representing all the points of the mountain with the same heights (Q).

Downward slopping: a firm can substitute K for L, keeping Q unchanged. Isoquant as a line in topographical map (Mount Hood) oregon.

5

Example:

( ) ( )( )

1/ 2 1/ 2

1/ 2 2 2

20 400 400 /

, /

Q KL KL KL K L

Generally Q KL Q KL K Q L

= → = → = → =

= => = → =

Economic/Uneconomic Regions But why don’t we include the full circle of the isoquant (think of topographic map) on the actual plot of isoquants? This is because of economic and uneconomic regions. The upward slope and backward bending regions correspond to a situation where an input has a negative marginal product (i.e. diminishing total return), but a normal firm that wants to minimize its production cost will never operate in this regions. In other words, in the upward and backward sloping regions of the isoquant that firm would be producing a certain output level at an unnecessarily high cost. They could produce the same level of output by using far fewer inputs. Marginal Rate of Technical Substitution By measuring the slope of the isoquant, we can see the trade-off a firm must make between inputs in order to maintain a certain level of output. This is referred to as the Marginal Rate of Technical Substitution. Or as stated in the book, the MRTS is the rate at which the quantity of capital can be reduced for every one unit of increase in the quantity of labor, holding the quantity of output constant.

• Measure how steep an Isoquant(IQ) is. • Slope of IQ (analog to MRS being slope of IC)

6

• Rate at which the quantity of one input could or ↑ ↓ for every one unit or ↓ ↑in the quantity of the other input, holding Output constant.

Diminishing MRTSL,K (isoquants are bowed in towards the origin) Example

( ) 2 3 , 0.1 3 – 0.1q f L K LK L K L K= = +

Assume __

10K = , so that ( ) 2 3 2 3,10 0.1 10 3 10 – 0.1 10 30 – f L L L L L L L= × + × × = +

1) 2 / 1 60 3LMP dq qL L L= = + − Where does it reach a max?

( )21 60 3 / 60 – 6 0 60 6 10 d L L dL L L L+ − = = → = → =

2) ( ) ( )2 3 2 ,10 / 30 – / 1 30 LAP f L L L L L L L L= = + = + − Where does it reach a max?

( )2 1 30 / 30 – 2 0 30 2 15 d L L dL L L L+ − = = → = → = Is this point L = 15 the crossing point between APL and MPL ?

2 2

2

1 60 3 1 30 30 2 30 2

15

L L L LL L

LL

+ − = + −

===

Generally, why do APL and MPL cross each other at the max of APL ? If ( ) /LAP f L L= , then it reaches its Max at

7

( ) ( ) ( ) ( )( ) ( )( ) ( )

( )( )

2 2/ ’ 1 / ’ / – / 0

’ – ’ / 0

,

( , /

L

L L

L L

dAP dL f L L f L L f L L f L L

f L f L L L MP AP L

MP APQ f K L

Q f K L

⎡ ⎤= × − × = =⎣ ⎦⇔ × = − × =

=

=

Δ = ∂ ∂ ( )

,

) ( , / )

0

/ / /

K L

K L

L K

L K

L K L K

K K f K L L LQ MP K MP L

MP K MP LMP L MP K

MP MP K L slope of IQMP MP MRTS

Δ + ∂ ∂ Δ

Δ = Δ + Δ= Δ + Δ

− Δ = Δ= − Δ Δ ==

Intuitively, the MPL and the APL will always cross at the maximum of the APL because, at that point, the MPL is not putting any upward or downward pressure on the APL. Remember the relationship between the APL and MPL. If the MP is above or below the AP, it will force the AP up or down, respectively. So they will cross where they two are equal. Example Learning-by-Doing 6.1 The production function is

,

The marginal products are

Where /

L

K

L K

Q KL

MP KMP L

MRTS K L

=

==

=

Diminishes as L increases and K falls as we move along an Isoquant so MRTS of Labor for Capital is diminishing, why? Because MRTS is a measure of slope of tangency of one point on Isoquant and compared with the slope of other points in the same Isoquant (see points B and D). When L increases the MRTS will reduce. (Slope of point B is more vertical than slope in point D) MRTS is analogous to MRS from consumer theory.

8

Example MRTS between low and high tech workers º 6 in US ELASTICITY OF SUBSTITUTION

( )( )( )L,K

% K / L

% MRTSσ

Δ=

Δ

This elasticity shows how easy it is to substitute L for K. Where K/L is the slope of the rays from origin to a given point on the isoquant MRTS is the slope of the isoquant at the given point. Example

9

( )( )( ) ( ) ( ) ( )

L,K

% K / L 1- 4 / 4 / 1- 4 / 4 - 0.75 / -0.75 1

% MRTSσ

Δ⎡ ⎤ ⎡ ⎤= = = =⎣ ⎦ ⎣ ⎦Δ

If s is close to zero, then MRTS changes drastically, as in Figure 6.11 (a) If s is large, then K/L changes drastically while MRTS is almost constant, as in Figure 6.11 (b)

Application Industries like Chemicals, Motor vehicles, Food…etc.

10

SPECIAL PRODUCTION FUNCTIONS

1) Linear: MRTS is constant • The MRTS is constant because the lines are straight. That is, the two

inputs in this production process can always be substituted in perfect ratios. Think of a manufacturing process that requires oil or gas. That firm can always substitute a certain amount of oil for a certain amount of gas to achieve the same level of output.

Example: 20 10Q H L= + (always additive). This is an example of High vs. Low productivity computers.

( ),

200 20 1010 ½

10 ½ 1 / 2 L H

H LH L

H L equation of isoquantMRTS for all L

= += +

= −

= −

Case for Linear: Perfect Substitutes

As example before, two low capacity workers are as productive as a one high capacity worker ( ), 1 / 2 L HMRTS for all L= − .Since the denominator is zero, the elasticity is infinity, which implies that it is perfectly easy for the firm to substitute between these two inputs while maintaining the same level of output.

( )( )( )

( )( )L,

% / L % / L

0 % MRTS H

H Hσ

Δ Δ= = = ∞

Δ

If L = 0 then 200=20H H =10

If H = 0 then Q=10L 200=10L L=20

Since the slope of the Isoquant is constant

11

2) Fixed proportion or Perfect Complement In the case of Fixed Proportions, think of the Perfect Complements in Consumer Choice. Here, adding more of one input given the other input is fixed will not change the total Output. That is the reason you are going to use the minimum value (going to the corner point). As the inputs are combined in fixed proportions what happens with the elasticity of substitution? Well, is clear that will be zero, why? Because the numerator of σ is zero (not change in the ratio K

L ).

Example:

{ } { } min / 2, min , 2 Q L K L K= = i.e. 2 units of labor for each unit of K. Note that in the equation every L value is twice the quantity of every K value, thus the K value is multiplied by 2.

{ }{ }

3 min 6 / 2, 3

3 min 10 / 2, 3

=

=

s = 0 since MRTS goes from ¶ to 0. Application→Elasticity of substitution in German industry Chemicals 0.37 Iron 0.50 Motor vehicles 0.10 low substitutes between L and K Food 0.66 high substitutes between L and K

12

3) Cobb- Douglas This production function is an intermediate between the previous ones. The elasticity of substitution for a Cobb-Douglas production function is always equal to 1 and the isoquants are downward sloping curves bowed in toward the origin. The general form for a Cobb-Douglas is…

] [1 1,

/ [ / ] /L K L K

Q AL KMRTS MP MP AL K AL K K L

α β

α β α βα β α β− −

=

= − − = − − = −

(decreasing in L, therefore flatter isoquant) (FROM CHAPTER 6 APPENDIX)

Rearranging,

, / /L KMRTS K Lβ α− = (1)

Hence,

( )( )

L,K

L,K

K / L ( / ) MRTS

K / L / MRTS ( / )

β α

β α

Δ = − Δ

Δ Δ = − (2)

We also know from (1)

( ),

,

( / ) /

/ / /L K

L K

MRTS K L

MRTS K L

α β

α β

− =

= − (3)

Hence,

( )( )( )

( ) ( )( )( )

( )( ) ( )( )L,K , ,

, ,

% K / L / / /

% MRTS /

/ / / / ( / ) ( / ) 1L K L K

L K L K

K L K LMRTS MRTS

K L MRTS MRTS K L

σ

β α α β

Δ Δ= = =

Δ Δ

= Δ Δ × = = − × − =

This means that the elasticity of substitution along a Cobb-Douglas production function is always equal to 1. Example: if a = b = 0.5

1) ( ) ( ), 12,3 0.5 12 / 0.5 3 4K L = ⇒ − × × = −

2) ( ) ( )K,L 6,6 0.5 6 / 0.5 6 1= ⇒ − × × = −

3) ( ) ( ), 3,12 0.5 3 / 0.5 12 1 / 4K L = ⇒ − × × = −

13

3) CES All the previous ones are derived from it/the other three production functions above are special cases of the constant elasticity production function.

1/ 1/ / 1 [ ]Q aL bKσ σ σ σ σ σ− − −= + where s is the elasticity s = ¶ (linear) substitutes s = 0 (fixed proportions) Compliments s = 1 Cobb-Douglas

Return to Scale Now that we know how to define how a firm substitutes one input for another, we will now see how output changes given a certain increase in all inputs. Remember that from our analysis of the marginal product of inputs we can figure out whether output will increase given an increase in inputs, but what we do not know is by HOW MUCH output will increase. Returns to Scale tell us the percentage increase in output given a certain percentage increase in inputs. This is generally expressed as…

( )( )

% quantity of output.

% quantity of ALL inputR S

Δ=

Δ

Here, this firm increases or scales up production by λ. That is, they multiply both inputs L and K by λ. The resulting change in output will be measured by Ø.

( ), ( , )

% Q % in all inputs

Q f L KQ f L Kφ λ λ

φλ

=

=Δ ←Δ ←

Then, if f > l increasing (i.e. output rises by a higher percentage than the percentage increase in inputs) Fig.(A)

14

f = l cte (i.e. output rises by the same percentage as the percentage increase in inputs) Fig.(B) f < l decreasing (i.e. output rises by a smaller percentage than the percentage increase in inputs) Fig.(C)

Example: Let 1 2&Q Q two different Output levels, both have the same quantity of inputs.

( )( )1

2 1

Q AL K

Q A L K AL K Q

α β

α β α β α β α βλ λ λ λ+ +

=

= = =

If la+b >l , (a+b >1) then Q2 > Q1 increasing if la+b =l , (a+b =1) then Q2 = Q1 cte if la+b <l , (a+b <1) then Q2 < Q1 decreasing Difference between diminish marginal returns and returns to scale Fig.6.19

15

More examples on Returns to Scale Example Decreasing a+b Tobacco 0.51 example:

Doubling inputs leads to 0.51

2 1 1 12 1.42Q Q Q Qα βλ += = =

Food 0.91 Transportation equipment 0.92 Constant a+b Apparel and textiles 1.01 Furniture 1.02 Electronics 1.02

Increasing a+b

Paper products 1.09 Petroleum and coal 1.18 Primary metal 1.24 example: doubling inputs lead to

1.242 1 1 12 2.36Q Q Q Qα βλ += = =

But why is the understanding of Returns to Scale important for economic decision

making? Because if a firm exhibits increasing Returns to Scale, there are cost advantages to large scale production. That is, the firm will be able to produce at a lower cost per unit. On the contrary, a firm can turn away from large scale production to save unduly harsh costs at large levels of production. That is, they could have a lower cost per unit at a smaller scale of production.

16

Technological progress Here, the production function can shift over time, why? Because a firm can achieve equal or more out of output from a given combination of inputs (less or equal quantity of inputs). This can occur because of increased skill by managers from experience, the payoff of previous investment in research and develop, and from new technology. Essentially, this means that as time goes on the firm can produce more with less. Categories of Technological Progress 1) Neutral combination of L and K are fixed (shift inward of the isoquant ) 

Neutral Technological Progess decreases the amount of inputs needed to achieve a certain level of production without affecting the firm’s marginal rate of technical substitution. Graphically, the isoquants will shift inward while maintaining the same slope.

( i.e. shift in parallel along the ray ____

0A )

2) Labor-Saving ÆÆMPK relative to ÆMPL (i.e. MPK increase faster than MPL) and

the Isoquant shift inward too • Intuitively this means that the capital becomes more effective and

therefore less labor is needed to maintain a given level of production.

• flatter isoquant • MRTS goes down for any given ray from the origin • Since MRTS = MPL / MPK , then MPK ÆÆ more than ÆMPL

17

3) Capital-Saving ÆÆMPL relative to ÆMPK and Isoquant shift inward too. Here, labor becomes more effective and thus this firm will require less capital to maintain a certain level of output. Thus the curve becomes steeper, showing that less capital can maintain the same production levels. This type of progress could occur from increased skill of workers or increased education of workers.

• MRTS is steeper • MRTS goes up • ÆÆMPL more than ÆMPK

Example 6.4 In this example, we want to see if the change in the production function represents technological progress, and if so, what kind of technological progress.

( ) ( )

( ) ( )( ) ( )

0.5 0.51 2

0.5 0.5

0.5 0.5

where over time this change to

10.5 0.5

0.5

K K

L L

Q KL Q L K

LMP MP LK K

KMP MP KL

= =

= =

= =

a) Tech.progress? Yes!! 1 2 < 0Q Q for any K and L > b) capital savings since

( )( )

( )( )

1 2

0.50.5

L,K L,K0.5 0.5

0.5 K 2KMRTS MRTS L 10.5 0.5

Q Q

K KLLL LK K

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= = < = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

18

The isoquant becomes steeper after the tech. progress, as in Fig 6.22

1

EconS 301 – Cost Theory Chapter 7

Before we get into using cost concepts for decision making, we need to define some important concepts. The first two are explicit and implicit costs. Explicit Costs: Costs that involve a direct monetary outlay. Implicit Costs: Costs that do not involve outlays of cash. Another important economic concept that should be familiar is the idea of opportunity costs. Opportunity Costs: The value of the next best alternative that is forgone when another alternative is chosen Ex: You are giving up potential salary while going to college so trading the potential salary for an education. Closely related to explicit and implicit costs are economic and accounting costs. Economic Costs: The sum of the firm’s explicit costs and implicit costs. Accounting Costs: The total of explicit costs that have been incurred in the past. There are two more important costs that any firm must recognize when making economic decisions. Those are sunk and nonsunk costs. Sunk Costs: Costs that have already been incurred and cannot be recovered. Nonsunk Costs: Costs that are incurred only if a particular decision is made. The Cost Minimization Problem: The main idea behind the cost minimization problem is finding the input combination that minimizes a firm’s total cost of producing a particular level of output. There are two important time scales to remember when dealing with these types of problems. Long Run: The period of time that is long enough for the firm to vary the quantities of all its inputs as much as it desires. Short Run: The period of time in which at least one of the firm’s quantities cannot be changed. In the utility maximizing problems there was a budget line, in cost minizing problems we will be using an Isocost line. Isocost line: The set of combinations of labor and capital that yield the same total cost for the firm. To find the tangency point it is very similar to the utility maximizing problem. Slope of isoquant=slope of isocost line -MRTSL,K = -w/r

Below is a graphical representation:

2

Corner Point Here, the optimal solution doesn’t have a tangency between an isocost line and an isoquant curve. The next example (perfect substitution) uses the follow information: Production function: 10 2 where 10 and 2Price of labor: 5 per unitPrice of capital: 2 per unitFirm wish produce: 200 units

= += =

===

L K

Q L KMP MPwrQ

Using the previous figure we observed that the optimal combination is a corner solution. Why? Because

,10 5 >2 2

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

LL K

K

MP wMRTSMP r

So there is no point that can satisfy =L

K

MP wMP r

. This tells us that we cannot have an

interior solution. The next question will be where and what is my solution?

10 2 5

2 12

⎛ ⎞ ⎛ ⎞= = > = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

L KMP MPw r

The marginal product per dollar of labor exceeds the marginal product per dollar of capital ( )2 1> , then the firm will substitute labor per capital until it uses no capital (K=0). Another fact here is that the slope of Isoquant is bigger that the slope of Isocost (5 > 2.5). Then, the quantity of labor satisfy 10 2 will be:= +Q L K

( ) 200200 10 2 0 200=10 =2010

= + = ⇒ ⇒ = ⇒L K L L L .

3

Comparative Strategies of Changes in Input Prices

Δw

An increase in the price of labor Δw makes the slopes of the isocost steeper for the same Q=100, then the cost-minimizing amount of labor must go down and the cost-minimizing quantity of capital must go up. Comparative Strategies of Changes in Output

For the first case, we hold the input prices fixed and increased the quantity of output, this move the firm to Isoquants to North East. When this quantity of output increases, the cost-minimizing quantities of L and K also increase. Expansion Path: A line that connects the cost-minimizing input combinations as the quantity of output, Q varies, holding input prices constant.

4

Normal output: An input whose cost-minimizing quantity increases as the firm produces more output. Inferior output: An input whose cost-minimizing quantity decreases as the firm produces more output. The second case show us when the quantity of output increase but this change decrease the cost minimizing quantity of labor and increase the cost minimizing of capital. Example: Burke Mills Revisited. Labor Demand Curve : A curve that shows how the firm’s cost minimizing quantity of labor varies with the price of labor. How can we fix it? First, this curve “generally” is downward slopping and will be affected by changes in price of labor or by increase in output or mixed both. This curve will show how the firm’s cost-minimizing amount of labor varies as price or output varies.

2 1Δ= ←⎯⎯ =ww w

When increasing output to Q=200 the shift to Labor demand curve will be

• Shift to the right is labor demand normal • Shift to the left if is an inferior

5

Labor Demand Algebraically Suppose that the production function is 50=Q LK we want to find the demand curves for labor and capital: First, tangency condition

( )( )( )( )( )( )

0.5

0.5

0.5 500.5 50

−

−

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

→ → →

L

K

L

K

MP wMP r

dQLK KMP KdL

dQMP LLK LdK

Then, ⎛ ⎞= → =⎜ ⎟⎝ ⎠

K w rL KL r w

, this equation represent our expansion path.

Second, replace into the production function, to find the demand curve for capital first

1 1 12 2 2250

50 50 50⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⇒ = ⇒ = ⇒ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

r Q r Q r Q wQ K K K K Kw w w r

Now we will replace the previous result into our expansion path equation to find our demand curve for labor

1 10.52 2

0.550 50 50⎛ ⎞ ⎛ ⎞= ⇒ = ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

r Q w r w Q Q rL L Lw r r w w

Price of inputs:

• is decreasing in is increasing in ⎧

= ⎨⎩

rK

w

• is decreasing in is increasing in ⎧

= ⎨⎩

wL

r

6

Output: Note that if↑Q , then the demand for both K and L increase. Hence, both K and L are normal inputs.

7

Price Elasticity of Demand for Labor: The percentage change in the cost-minimizing quantity of capital with respect to a 1 percent change in the price of capital. Represent the percentage change in the cost-minimizing quantity of labor with respect to one percent change in the price of labor.

,

*100%.

*100%ε

Δ ΔΔ

= = =Δ Δ ΔL w

L LL wL L

w w w Lw w

It really depends on the substitution between two inputs, ,σ K L

Low σ implies that a change in w has almost no effect in L, which is the first graph. High σ implies that the same change in w induces a great change in L, the right graph.

8

Short-run Costs There exist 3 types where the components of total cost differ from each other.

1. Variable and Nonsunk 0 0

& ⎧ = = ⎫

⇒ ⇒⎨ ⎬Δ → Δ ⎭⎩

If Q then CostsLabor Materials

Q Costs

2. Fixed and Nonsunk 0 0

⎧ = = ⎫

⇒ ⇒⎨ ⎬Δ → ⎭⎩

If Q then Costsheating

Q not change Costs

3. Fixed and Sunk

0 0

⎧Δ → ⎫⇒ ⇒⎨ ⎬= > ⎭⎩

Q not change CostsMortgage payments

If Q then Costs

Cost Minimization Short-run Total Variable Cost: The sum of the expenditures on variable inputs, such as labor and materials, at the short run cost-minimizing input combinations. Total Fixed Costs: The cost of fixed inputs; it does not vary with output.

_

K

0 Q Isoquant

Example:

Suppose that the production function is 50=Q LK but __

K (i.e. capital is fixed) where the only unknown is L. Solving for L:

( )2

2__ __ __2 2__50 50 2500

2500

⎛ ⎞= ⇒ = ⇒ = ⇒ =⎜ ⎟⎜ ⎟

⎝ ⎠

QQ L K Q L K Q L K LK

The last result represents the demand for labor in the short run when the capital amount is fixed.

9

Other example is “Learning by doing” which is considering more than one variable input __

= + +TC wL mM r K (Solve this exercise for yourself) But, What happen if we reduce the price of labor (w) from $2 to $1 (i.e. 50% drop) Then we are going to increase in labor.

5 4.6 8% increase in labor in (a)4.6

5 2.2 127% increase in labor in (e)2.2

−⎧ =⎪⎪⎨ −⎪ =⎪⎩

if

f

Empirical evidence

Industries: textile, paper, chemical, metals in Alabama 1976-1991

1

EconS 301 Chapter 8 Long-Run Total Cost The long run total cost curve shows the total cost of a firm’s optimal choice combinations for labor and capital as the firm’s total output increases. So, each point on the long run total cost curve represents one optimal basket at a specific level of output. Note that the total cost curve will always be zero when Q=0 because in the long run a firm is free to vary all of its inputs.

0 0Δ → Δ= → =

Q TCQ TC

The upper graph shows a change in output and the minimized total cost increase. The below graph shows the long run total cost curve. Remember that the total cost curve is just the aggregation of the total costs of optimal bundles as output (isoquants) increases. Example: Suppose that the production function is 50=Q LK

We know from chapter 7 that

1 12 2

& 50 50

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Q r Q wL Kw r

and

2

( ) ( ) ( )

( )

( ) ( )

1 12 2 0.5 0.5 0.5

0.5

0.5

250 50 50 50 50

25If we are told that 25 & 100, then:

25*100 50 225 25

⎛ ⎞ ⎛ ⎞= + ⇒ + ⇒ + ⇒⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

= =

⎛ ⎞= ⇒ ⇒⎜ ⎟⎝ ⎠

Q r Q w Q Q QTC wL rK w r rw rw rww r

QTC rw

w rQ QTC Q Q

(The L and K that we used to plug into our total cost function are the cost-minimization equations for both inputs L and K. The cost minimization equation for either input can generally be stated as K=TC/r – (w/r)L for capital or L=TC/w – (w/r)K for labor). What happens when just one input price change?

0Q

• Δr forces 1C to pivot to left (flatter) since K is more expensive, but TC remain

constant between 1C and 2C because the isoquant doesn’t change.

• However 0Q output level must remain unchanged, then 2C shifts outward up to 3C , so this firm incurs a higher total cost to maintain the Q0 level of output.

Hence Δr forces a ( )ΔTC Q

3

Example: Trucking firms

Then impliesΔ ⇒r

When Δ = Δr w then KL

doesn’t change because the tradeoff between the two inputs

doesn’t change, but ( )Δ = Δ = ΔTC Q r w

Long-run Average and Marginal Cost

( ) ( ) = =TC Q dTC Q

LRAC LRMCQ dQ

The LRAC is equivalent to the slope of any ray from the origin to a point on the TC curve. The LRMC is equal to the slope of a line tangent to the TC curve. The LRMC tells us how the total cost changes as a firm increases output by one unit.

4

50

50

Slope of =30 30

Slope of tangent at =10 10=

=

⇒ =

⇒ =Q

Q

OA AC

A MC

RELATIONSHIPAC > MC producing a marginal and

reduces ACAC<MC producing a marginal and

raises AC

Note: the MC curve will always cross the AC curve at the minimum of the AC curve because of the relationship between the marginal of anything and the average of anything that we discussed in detail with consumer choice. Application: US Universities Q=Students 195 U.S. universities from 1990 to 1991 The study divides universities in 4 categories, according to the site of grant proposal. Here we only show the group of 66 units with the largest graduated programs.

( )AC Q decreases until Q=30,000 undergraduates students. The effect of an additional student is positive!! ↓ AC .

25,000 →WSU students Example: Given the production function ( )50 and =25 100 we have 2= = =Q LK w r TC Q Q

5

2 2

2

= = =

= =

TC QACQ QdTCMCdQ

Economies of Scale Here, with economies and diseconomies of scale, we want to see whether a firm should expand their production based on the extra cost of doing so. Put differently, is the extra cost worth the extra production?

Economies

dise

cono

mies

Economies of scale: They arise from specialization, but also from indivisible inputs: an input that is available only in a certain minimum size. Its quantity cannot be scaled down as the firm’s output goes to zero, or where the cost of producing a very small amount of output is very similar to the cost of producing a very high output level (satellites for providing cell phone service). Example: minimum-scale packing line for breakfast cereal is equally costly for any firm producing from 0 to 14 million pounds of cereal / year Diseconomies of scale: usually arise from managerial diseconomies: a given %ΔQ forces the firm to largely increase its spending on managers by more than this percentage. Minimum efficient scale: minimum Q for which ( )AC Q attains its minimum point (on the above graph), the leftward most point on the straight line of the AC.

6

High for breakfst cereal and cane sugarLow for bread⎧

= ⎨⎩

MSEQ

Output Elasticity to Total Cost

It’s the percentage change in total cost per 1 percent change in output (the same concept behind all elasticities).

,1. .ε

ΔΔ

= = = =Δ ΔTC Q

TCTC Q MCTC MCQ Q TC AC AC

Q

, 1

Economies of scaleExamples:Iron, StelaElectricityGas

ε <TC Q , 1

Diseconomies of scaleExamples:Textile,Cement

ε >TC Q

Short run TC One or more inputs are fixed at given level. Relate this to LRTC where all inputs can vary so the firm is left with complete freedom in choosing between inputs.

7

TFC

TFC

TFC

In the short run, we can split the cost between the cost that varies (Total Variable Cost) and the cost that is fixed, Total Fixed Cost (hence short run). This is depicted in the graph above where the slope of the Short Run Cost Curve is determined by the TVC, but is vertically higher than the TVC by precisely the amount of the TFC. Example: Short run ( )TC Q

Suppose that the production function is 50=Q LK but __

K (i.e. capital is fixed) where 25 and 100= =w r .

Solving for L:

( )2

2__ __ __2 2__50 50 2500

2500

⎛ ⎞= ⇒ = ⇒ = ⇒ =⎜ ⎟⎜ ⎟

⎝ ⎠

QQ L K Q L K Q L K LK

This is the cost miniminizing amount of labor. Plugging it into the TC

( )__2 2__ __

__ __10025 100

2500 100= + ⇒ + ⇒ +

TFCTVC

Q Q KSTC Q wL r K KK K

Short-run Total Cost vs. Long-run Total Cost When the firm is free to vary the quantity of capital in the long run, it can attain lower total cost than it can when its capital is fixed.

8

1Q

2Q

Short-run Average Cost (Notice that the MC’s still intersect the AC curves at their minimum in the short run)

( )STC Q VC FC VC FCSAC AVC AFCQ Q Q Q

+= = = + = +

9

Long-run Average Cost as an Envelope Curve ( )( ). .i e LRAC Q

(Note that the AC is the summation of the minimums of the SAC curves)

Example: Production function is 50=Q LK a) What is the SAC for a fixed

__

K when 25 and 100w r= = ? From the previous examples we know that

( )

( )

2 __

__

__

__

100100

Thus,

100

100

QSTC Q KK

Q KSAC QQK

= +

= +

b) Sketch ( )SAC Q for { }__

1,2,4K =

( )

( )

( )

__

__

__

1001100

2002200

4004400

QK SAC QQ

QK SAC QQ

QK SAC QQ

= ⇒ = +

= ⇒ = +

= ⇒ = +

10

Application: Railroad Cost An increase in K (track mileage) can benefit the railroad company and reduce the cost

11

Economies of Scope Here, the firm produces two products, 1 2&Q Q and the total cost of one single firm that produce these two goods is less than the total cost of producing those quantities in two single product firms. In essence, there are cost advantages to merging the production of two products to one single firm.

( ) ( ) ( )1 2 1 2, ,0 0,TC Q Q TC Q TC Q< + Then, the variety is better than specialization.

( ) ( ) ( ) ( )1 2 1 2

21

Additional cost of producing Q when Addfirm was only producing Q

0 , ,0 0, 0,0TC Q Q TC Q TC Q TC⇔ − < −

2itional cost of producing Q whenfirm was not producing anything

Example:

Cost of Coca Cola toadd a new product of Cost to a new company toCherry Cola to its live start producing Cherry Colaof products

⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟ < ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎝ ⎠

Example: Cost of adding one more channel to a satellite < Cost of the satelliteShares Eurotunel between Cars and Train < Two Tun

XXXXXnels

Economies of Experience We have two processes. One is a dynamic process which produces a reduction in costs that results from accumulated experience. The other process is static like economies of scale. The economies of experience are described by the experience curve which show a relationship between average variable cost and cumulative production volume. How does accumulated experience affect costs?

( )( )AVC N

( ) [ ]. where >0 and 1,0BAVC N A N A B= ∈ − Where A is the AVC of the first out and B represent the “experience elasticity”

( )( )

1. .

1% in is the % in

BB

dAVC N N NBANdN AVC N AN

N B AVC

−=

Δ ⇒ Δ

12

Economies of Experience vs. Economies of Scale What is the relationship between economies of experience and economies of scale?

( )

( )

Typical for the production of new products, after some results or years the production process gets more efficient.

Typical fro mature industries

N AVC Experience

Q AVC Scale

↑ ⇒↓ Δ

↑ ⇒↓ Δ

Econ of Scale, no experience: mature industries Econ of experience, no scale: handmade products e.g. watches Slope of experience curve: How much does the average variable cost go down as a percentage of an initial level when cumulative output N doubles.

( )( )

( )22 2Slope of experience curve 2B B B

BB B

A NAVC N NAVC N AN N

= = = =

13

The smaller the slope the “steeper” the experience curve the more rapidly AVC goes down as cumulative output increases. Slope of 100% only occurs if B=0 (no Econ of experience)

1

EconS 301 Chapter 9 Perfectly Competitive Markets

What are we going to do in this chapter?

• Characteristics of P.C. markets • Determining P in P.C. markets • Economic surplus and welfare analysis

Characteristics of P.C. markets

1. Fragmented many buyers and sellers (each one is to small to affect the prices)⇒ 2. Undifferentiated products-products that consumers perceive as being identical. 3. Perfect information about prices-full awareness by consumers of the prices charged by all

sellers in the market. 4. Equal access to resources (technology, inputs)

But what are the implications of this characteristics?, see the next point How PCM work

1. Price takers⇒ this come from of the industry is Fragmented 2. Law of one price⇒ from the 2 and 3 characteristic of P.C 3. Free entry in the market⇒From the last characteristic of P.C

Now, what I’m going to do with this theory? Well, know we need to find how this P.C. Market is going to facilitate the allocation of resources and the creation of economic value.

Economic profit vs Accounting profit

Economic Profit = Sales Revenue – Economic Costs (including opportunity costs) Accounting Profit = Sales Revenue – Accounting Costs

Firm Market

2

Q that maximizes Economic Profits Marginal Revenue: The rate at which total revenue changes with respect to output.

– TR TCπ = Where TR = Q×P and given price taking behavior P constant

& TR TCMR MCQ Q

Δ Δ= =

Δ Δ

Where MR > MC (P > MC), we have ▲Q ▲ π MR < MC (P < MC), we have ▼Q ▲ π But when both are equal, what happen?

A price taking firm maximizes π when P = MC. MC must be increasing.

(i.e. the price taking firm Max. profit when produce some Q at which MC equal market price) in roses example this Max profit was in 300. But, is Q = 60 another profit maximizing Q? No! The π is minimized. So, the question is how could I know when is Max or Min my profit? Easy fellas, first when P=MC and when MC is decreasing then we have a Profit Maximization.

3

Now we are going to construct a supply curve for a firm in the short-run. Well, first we need a Price, so I need made assumptions to find this price. Determination of market price in the short-run Short-run: K is fixed for all firms. Number of firms is also fixed in the market. Case: when all fixed costs are sunk Sunk fixed costs: A fixed cost that the firm cannot avoid if it shuts down and produces zero output. Nonsunk fixed costs: A fixed cost that must be incurred for a firm to produce any output but that does not have to be incurred if the firm produces no output.

1. The profit-maximizing condition P = MC makes the supply decisions of the firm (Supply

Curve) to coincide with the SMC curve (red line).

2. If P < AVC, then we have that, apart from the Sunk Fixed Cost, the firm is making losses for TFC+[Q(AVC – P)], the shaded rectangle.

3. This explains Supply Curve concentrated at Q=0 for all p < min AVC.

4. Note, finally, that there are points where p є (AVC, SAC) where the firm is not compensated for its fixed costs, although it is for its variable costs.

5. Short Run supply curve: The supply curve that shows how the firm’s profit-maximizing

output decision changes as the market price changes, assuming that the firm cannot adjust all of its inputs.

4

Example: STC = 100 + 20Q + Q2 , then MC = 20 + 2Q FC TVC

a) AVC = (20Q + Q2)/Q = 20 + Q b) Min AVC

We know that this minimum occurs where MC = AVC. Then, 20 + 2Q = 20 + Q

2Q = Q Q = 0

Then min AVC = 20 + 0 = 20. c)

a. Short-run Supply Curve • For p < 20, we have Q = 0 • For p ≥ 20, we have p = MC. That is p = 20 + 2Q. Then p – 20 = 2Q

Supply (P) = Case: determining Supply with Sunk Costs

In the previous case (all fixed cost are sunk) you could only avoid losing money by Q = 0 when

p AVC< since there is no fixed costs which were non-sunk (all your FC were sunk). Now, you can set Q = 0 and avoid paying FC since some of them are non-sunk.

p/2 – 10 = Q

0 if p < 20 p/2 – 10 if p ≥ 20

5

In this case you will never produce when p ANSC< Shut down price: The price below which a firm supplies zero output in the short run. Example STC = 100 + 20Q + Q2 SMC = 20 + 2Q

a) Assume SFC = 36 and NSFC = 64. What is ANSC? NSC = TC – Sunk Costs = NSFC + Sunk Fixed Costs + VC – Sunk Costs = NSFC + VC Then ANSC = NSFC/Q + VC/Q = 64/Q + (20Q + Q2)/Q = 64/Q + 20 + Q

Min ANSC

It happens where ANSC = MC 64/Q + 20 + Q = 20 + 2Q 64/Q = Q 64 = Q2 Q = 64^(1/2) = 8 Then min ANSC = 64/8 + 20 + 8 = $36

b) Supply Curve

If p < 36 then Q = 0 If p ≥ 36 then Q is determined from p = MC, p = 20 + 2Q. Hence Q = p/2 – 10.

Supply (P) =

0 if p < 36 p/2 – 10 if p ≥ 36

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Now, we need to find the supply curve for the entire industry (recall that previous was just only one price-taking firm). Short-Run Market Supply

Horizontal summation of supply curves. Be careful: if we are adding up Supply 1, 2 for each price, we want Supply 1, 2 in terms of

20 2p Q= + . Secondly, we will differentiate for different price intervals. Suggestion: exercise 9.10. Don’t worries is easier that you expect, but don’t forget the number of types of firm per each supply curve. Short-run market supply curve: The supply curve that shows the quantity supplied in the aggregate by all firms in the market for each possible market price when the number of firms in the industry is fixed. Competitive Equilibrium in the Short-run Market Demand = Market Supply

Example D(p) = 60 – p STC = 0.1 + 150Q2 N=300 SMC = 300Q and AVC = 150Q

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Where from SMC the N=300 are the number of firms.

a) Firm’s supply Min AVC occurs at Q = 0, what implies min AVC = 0. Then, for any p > 0 we have p = MC, p = 300Q, Q = p/300. Firm Supply curve is Q = p/300 for all p > 0.

b) Market Supply 300 × p/300 = p

c) Competitive market

p = 60 – p 2p = 60 Competitive Statics

▲N makes ↓ p and ↑ Q

Increases in Demand

Very elastic supply Very inelastic supply Long-run

p* = $30 Q* = 30

Application ST. Valentine

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Supply curve in the long-run should be use the long-run cost functions.

Similar to short-run, but now there are no fixed or sunk costs: now all costs are avoidable, since the size of the plant, K, act like variable. Long-run P.C. markets Long-run competitive equilibrium: The market price and quantity at which supply equals demand, established firms have no incentive to exit the industry, and prospective firms have no incentive to enter the industry. At the price at which Market Demand = Market Supply we have that

1) No entry, no exit 2) Each firm maximizes long-run π changing Q and K decisions 3) Economics profits is zero, since p = min AC 4) Market Demand = Market Supply

Example

AC(Q) = 40 – Q + 0.01Q2 MC(Q) = 40 – 2Q + 0.03Q2 D(p) = 25,000 – 1,000p

P.C.

We know that in a P.C. market the following 3 conditions must hold: p* = MC(Q*) = 40 – 2Q* + 0.03(Q*)2 (profit maximizing)

p* = AC(Q*) = 40 – Q* + 0.01(Q*)2 (zero profit) D(p*) = n*×Q* 25,000 – 1,000p* = nQ* (market demand = market supply)

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1) From the first two equations

40 – 2Q* + 0.03(Q*)2 = 40 – Q* + 0.01(Q*)2 0.02 (Q*)2 = 2Q* - Q*

0.02 (Q*)2 = Q* 0.02 Q* = 1

Individual supply in equilibrium

2) p* = 40 – 2Q* + 0.03(Q*)2 = 40 - 2×50 + 0.03(50)2 = 50 equilibrium price 3) plugging p* into the Demand function

D(p) = 25,000 – 1,000×15 = 10,000 4) Then, 10,000 = n×50 n = 10,000/50 = 200 firms in equilibrium

Long-run Supply Curve

1) If shift from D0 to D1 from p0 to p1

2) Large profits, since p1 > SRAC in the LHS 3) This induces the entrance of firms, from SS0 to SS1 4) Equilibrium, in the long run, happens to be at

p0 = min AC No entry or exit of additional firms

5) Long-run supply curve is a flat line at p = min AC

But how the long-run supply curve will work with respect to the cost? There is the case of constant cost industry: ΔQ from new firms doesn’t change costs. Increasing cost-industry: an industry in which increases in industry output increases the prices of inputs Decreasing cost industry: An industry in which increases in industry output decrease the prices of some or all inputs. Long-Run Supply when we have increasing costs ΔQ from entering firms ↑ p inputs (▲costs)

Q* = 1/0.02 = 50

Application of n*

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Long-run supply curve with decreasing costs ΔQ from entering firms ↓p inputs (▼costs). Example: an industry which uses chips extensively.

After understand how the supply curve works; now we need to learn how firms and input owners profit from their activities in perfectly competitive markets, for that reason will use the concept of Economic rent and Producer surplus. Economic Rent: The economic return that is attributable to extraordinarily productive inputs whose supply is scarce. Reservation value: The return that the owner of an input could get by deploying the input in its best alternative use outside the industry. Maximum amount that a firm is willing to pay for services of an input – input’s reservation value where reservation value return of the best possible alternative Economic rent≠ Economic profit

11

1) AC` and MC` in left-hand side due to extraordinary master grower who receives a low salary as regular master grower

2) MC and AC in right-hand side since they use run-of-the-mill master grower who receives a low salary as regular master grower, their costs are higher.

3) If P.C. p* = 0.25, then firm 1 profits are larger 4) Competition for such a valuable (talented) extraordinary master grower makes firm 2

offer up to 70,000 to 105,000, as well for Firm 1. Finally, the economic rent will be 105-70 thousand=35 thousand which is entirely captured by the extraordinary master grower. In other words, both are going to offer until observe π1 = π2 = 0

Producer Surplus: A measure of monetary benefit that produces derive from producing a good at a particular price.

Difference between what a producer actually receive for selling one unit and the minimum price he must receive in order to be willing to supply that unit.

• Both in the case of institutional firms and market supply • Note that with the usual supply curves

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Example Market supply Q = 60p p = 2.5 with p = 2.5, Q = 60 × 2.5 = 150

Area:

(A) = ½ × 150,000 × 2.5 = 187,500 (B) = (4 – 2.5) × 150,000 = 225,000 (C) = ½ (4 – 2.5)×(240-150) = 67,500

Increase in Production Supply with p = $4, then Q=60×4 = 240 Producer surplus in the long-run Here the Producer surplus in the long run will be equal to zero, why? Because in the long-run the producer surplus=economic profit which, as we know, in the long run is zero.

(B) + (C) = 292,500

13

How can we have (+) PS if it is supposed to be zero profits in the LR? This PS reflects the economic rent captured by the owners of those inputs which are scarce. Application Roses and Valentine’s Day

Constant cost Increasing cost

1

EconS 301 Chapter 11 Monopoly

Unlike firms who face perfect competition and therefore have no choice over what quantity and price to set for their product, monopolies set the market price for their good. What keeps them from setting an infinitely high price? Monopolists must still take into account a downward sloping market demand curve, as consumer’s will buy less as the price rises. Recall: Perfect competition one firm has no consequence on the market price (price taker) Monopolist one firm has total impact on price (set the market price of his product) Then, why the monopolist does not set price = ¶? Because he faces the market demand curve (i.e. higher prices reduces demand and vice verse). So, how (where on graph) does the monopolist maximize his profit? MR=MC Monopolist profit max condition where MR equals Marginal Revenue and MC equals the Marginal Cost Note that since P>(MC=MR) , Profits >0 So monopolists can make a positive profit, unlike perfect competition firms.

2

Why MR is below Demand?

1) Firstly, the marginal revenue is below the demand curve because the second part of the marginal revenue equation (highlighted region), MR = ∑(pμq)/∑q = p + qμ∑p/∑q, is negative because the change in P is negative. Since this negative number is subtracted from P, MR is below P.

2) Change in monopolist revenue ª III – I, where area I is the revenue lost from decreasing the price of some of the sold product and area III is the revenue gained from selling more units than before.

Area I = Qμ “p = Qμ (-∆p) = - $6(loss) Area III = ∆Qμp = $21 Then, ∆TR = III – I = ∆Qμp +Qμ∆p = $15 Therefore, we can derive the MR by simply deriving the TR equation with respect to Q… MR = ∆TR/∆Q = [∆Qμp +Qμ∆p] / ∆Q = p + Qμ(∆p/∆Q) At Q=0, MR= p (i.e. MR = Demand) Q>0, since ∆p/∆Q <0, then we have that [p + Qμ(∆p/∆Q)] < p

MR<Demand (the same conclusion we came to up above)

3

Average Revenue vs. Marginal Revenue

Why is this so important in monopoly theory? Because TR PQAR PQ Q

= = = ,but in

monopoly theory the firm can decide the price of the product, then this price is determined by the demand curve, well what does this means? It means that

( ) ( )AR Q P Q=

Example

p = a – bQ is our market demand curve. AR = p = a – bQ (remember that AR equals P in the case of a monopoly) MR = p + Qμ(∆p/∆Q) = a – bQ + Q (-b) = a – 2bQ P ∆p/∆Q Alternatively, TR = pQ = (a-bQ)Q = aQ – bQ2 MR = ∆TR/∆Q = a – 2bQ So graphically this implies that the MR has the…

• Same vertical intercept as Demand • Twice the slope of Demand

When the average of a variable is falling, it must be because additional units are below the average.

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Example Another example, where the market demand is given by… p= 12 – Q MC = Q

1) MR TR = pQ = (12-Q) = 12Q – Q2 MR = 12 – 2Q 2) MR=MC (This is the profit maximizing condition for a monopoly) 12 – 2Q = Q 12=3Q Qm = 4 p = 12-4 = 8 So, this monopoly will set its price at 8 and sell a quantity of 4 units.

Lesson: p>MC for the last unit supplied, indeed 12-4 >4 8>4 So therefore this firm has positive profits of B+E = $12 CS>0, area A = ½(12-8)4 = $8, note that these consumers still have a positive

consumer surplus, contrary to what we may have suspected under a monopoly The monopolist doesn’t have a supply curve Perfect competition Firm observes a market price (exogenous and fixed) and determines how much to produce, q. There is a specific relationship between p and q where one forces a certain shift in the other. Monopolist Firm determines simultaneously the market output and price without taking into account anybody else. For different demand curves a monopolist can produce the same output at different prices:

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No unique correspondence between p and Q, so no supply curve. Hence, no unique supply of produced output can be depicted for the monopolist, as the monopoly is the determiner of both price and quantity. Monopoly Supply Conclusions:

• MR < P • Since AR = P, therefore MR<AR • Since AR curve coincides with demand curve, MR curve must be below

demand curve Price Elasticity and Mark-up Prices

Elastic Inelastic pA –MC small pB – MC large These graphs show us that the price elasticity of demand plays an important role in determining by how much a monopolist can raise his price over marginal cost. In graph a, the demand is elastic and thus this monopolist can only set price slightly higher than marginal cost. In graph b, the monopolist can set his price much higher than marginal cost. Then, what is the relation between price elasticity and marginal cost? MR = p + Q (∆p/∆Q) factoring the p out, we get… MR = p [1+(Q/p) μ(∆p/∆Q) ]= p (1+1/eQ,p) And since the monopolist set MC = MR, then MC = p(1+1/eQ,p)

Existence/Non-existence of close substitutes

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MC – p = pμ(1/eQ,p) (MC-p)/p = 1/eQ,p Inverse Elasticity Pricing Rule (IEPR) The left side of the IEPR equation is the monopolist’s optimal price markup expressed in percentage terms. So for this monopolist to optimize the price she can charge, she would use the IEPR. The monopolist’s optimal mark-up of p above MC, as a % of price, is equal to minus the inverse of the price elasticity of demand. So… Æ |eQ,p | ∞|1/eQ,p | (close to 0) ∞ (p-MC)

• The higher the price elasticity of demand, the smaller the price markup Example Demand is… Q=aμP-b So price elasticity of demand equals… eQ,p = (P/Q) μ(∆Q/∆P) = (P/Q) μ(-baP-b-1) = P/aP-b(-b)(aP-b)P-1 = P(-b) P-1= -b Here, we consider a case where MC = 50 and the demand equations are given below…

a) p* if Demand is Q = 100p-2, then eQ,p=-2 Using IEPR, (p-MC)/p = - 1/-2 (p-50)/p = ½ 2p-100 = p, p = 100 b) p* if Demand is Q = 100p-5, then eQ,p=-5 Using IEPR, (p-MC)/p = - 1/-5 (p-50)/p =1/5 5p-250 = p, 4p = 250 p=62.50 Note that when demand is more elastic (as in -5) we have that the monopolist markup is smaller. Application Chewing gum (low elasticity, high mark-up) impulse purchase items Baby goods (high elasticity, low mark-up) non-impulse purchase items

Example MC = $50 p=100 – Q/2 Q = 200 -2p p-MC/p = - 1/eQ,p eQ,p = (∑q/∑p)μ(p/q) = - 2μ(p/200-2p) = -2p/(200-2p) Then,

(p-MC)/p = - 1/eQ,p

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p-50/p = -1/[-2p/200-2p] = (200 – 2p)/ 2p Multiplying both sides by 2p, 2pμ (p-50/p) = 2pμ [200-2p /2p] 2p-100 = 200-2p 4p=300 p=$75 Q = 200 - 2μ75 = 50 units supplied at a price of $75

So to find the profit maximizing price we first find the Price Elasticity of Demand and then set up the IEPR and solve for P. We then plug in that P to the demand function and solve for Q. Elastic Region

The monopolist always produces in the elastic region of the Demand curve. Why? Because MC is positive then from the relation MC= MR = p (1+1/eQ,p) the only way that will be positive is if [ )Q,p 1,ε ∈ − ∞ , means demand is price elastic. That is, the monopolist could always decrease his production (raise his price) and increase profits. Graphically, the I region is larger (which is the profit lost if quantity increases from B to A) than the II region (profit gained from increasing Q from B to A).

Going from A to B we have an ∆TR = I – II >0 Lerner Index of market power Learner Index is a measure of the Market Power. In the real world, firms can be in between perfect competition and monopoly. For instance, Coca Cola is not a monopoly because brands like Pepsi are important competitors, but it also faces a

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downward sloping demand curve and thus has some market power (i.e. not a price taker). The learner index helps us measure that market power. p-MC/p 0 for perfect competition, where p=MC 1 (100%) for monopolists Collusive Cereal 65% 41% Lerner index will be - HIGH for inelastic Demands LOW for elastic Demands (existence of close substitues) Comparative Statics How a shift in demand curve affects monopolist profit maximization? Or, how a shift in MC affects profit maximization? 1) ∆Demand will lead to Æp and ↑Q as long as MC constant or increasing ∆Demand will lead to ∞p and ÆQ as long as MC decreasing So, quantity follows shifts in demand and price changes are dependent on whether MC is increasing or decreasing. 2) ∆MC will lead to Æp and ∞Q

3) “TR as a consequence of ∆MC (e.g., taxes)

if we observe this for a group of firms, they are acting as a monopolist (collusion) if we don’t, then they are not acting collusively.

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Example with Linear Demand

p = a-bQ MC=c MR = a-2bq = c = MC q=(a-c)/2b p = a-b[(a-c)/2b] = a – (a-c)/2 = (2a-a+c)/2 = (a+c)/2 Monopoly pricing as the MIDPOINT RULE, in between the “choke price”, a, and the MC, c (or the vertical intercept of MC, but the MC is a straight line here).

• For monopolies facing linear demand curves and constant marginal cost, the optimal price will always be the midpoint between the choke price (intercept of demand curve) and the MC curve.

Multiplant Monopoly When there exists two plants for the same monopolist you need to answer: How much to produce in overall, well where MR=MC total in each plant

High MC firm produces less. The MC total is a horizontal sum of the individuals MC. To profit maximize, a monopoly will never want to produce quantities in plants when their respective marginal costs are different because they could reallocate production to the plant with a lower MC and reduce total cost. THUS, to maximize profits a multiplant monopolist will always produce at quantities where the plants have the same MC.

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Example p = 120-3Q MR = 120 -6Q MC1 = 10+20Q1 MC2 = 60+5Q2

1) MCT (We first solve for MCT by summing separate MC’s) MCT is the horizontal sum of MC1 and MC2

Firstly, (MC1-10)/20 = Q1 and (MC2-60)/5 = Q2 Adding them up Q1+ Q2 = (MCT-10)/20+(MCT-60)/5= 0.25MCT-2.5 2) Q=0.25MCT-2.5 Solve for MCT = 50+4Q 3) MR = MCT 120-6Q = 50+4Q 70=10Q Q=7 Total units produced by monopoly p = 120 – 3Q = 120-21 = 99 4) Q1,Q2 (No we solve for the quantity produced in each firm) MCT(Q=7) = 50+4μ7 = 78 From inverted MC1 and MC2 curves,

Q1=(78-10)/20 = 3.4 Q1=(78-60)/5 = 3.6

Example. Output choice with two markets. Same price (later on we will analyze how to price discriminate). First, we want to aggregate the individual market demands so that we can determine the aggregate marginal revenue for this firm. We then set aggregate MR equal to the MC to find optimal price. We then find Q by using the aggregate demand curve. Business Q1(p) = 180-p Vacation Q2(p) = 120-p MC(Q) = 30 where p e [120,180], demand is Q=180-p, so inverse demand is p = 180 – Q, MR=180 – 2Q when p<120, aggregate demand is Q=(120-p)+(180-p) = 300-2p Hence, inverse demand is p=150-1/2 Q MR = 150-Q 1) Can p>120? Let’s assume that’s the case Then MR = MC implies 180-2Q = 30 Q=75 p=180-75 = 105 X p is NOT >120 2) Can p<120? Let’s assume that’s the case Then MR=MC implies 150-Q=30 Q=120 p=150-1/2μ120 = 90 YES! p<120 Q1(p) = 180-90 = 90 and Q2(p) = 120-90 = 30

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CARTEL Cooperation by individual entities in order to max joint profits, example OPEC Analysis: same as the one in a multiplant monopoly. That is, we can determine the optimal price and production for the cartel by the same process we used for multiplant monopolists. QT, Q1, Q2

• OPEC production is not consistent with this • So, “A monopolist could have done better”. OPEC is not allocating

production as to equate MC’s of each member and could thus make more profit.

Welfare analysis of Monopoly How is the consumer benefitted or harmed in the monopolistic market versus a perfectly competitive market? Related to deadweight loss (net economic benefit of Perfect Competition and Monopoly are different)

Deadweight loss may be understated because of the existence of rent-seeking activities Upper-bound of DWL of monopoly = (B+E+H) + (F+G)

Deadweight loss of monopoly

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Why do monopoly markets exist? 1) Because they are a natural monopoly (decreasing AC that makes one firm has

smaller costs that many firms). That is, it makes more sense in regards to cost for one firm to supply the market versus many firms. Total cost for one firm is less than the total cost of many firms.

TCone firm = $1μ9,000 = 9,000

TC two firms = [$1.20μ4,500] μ2= $10,800

2) Barriers to entry: • Structural (e.g., cost advantage over other firms) • Legal (e.g., government regulation protecting the market, patents, ect. • Strategic (e.g., regulation of fighting any new comer, incumbent purposefully

attempts to differ other firms from entering) Exercises: Chapters 6, 7, 8, 9, 11, not about 10 Chapter 6: 6.16, 6.11, 6.8, 6.20 Chapter 7: 7.7, 7.8, 7.17 Chapter 8: 8.10, 8.14 Chapter 9: 9.6, 9.10, 9.18, 9.25 Chapter 11: 11.7, 11.12, 11.16 Effects of a specific tax on monopoly - Perloff pp.376-378 The monopolist must pay t per unit produced, so that TC(Q) = c(Q)+ tQ p(Q)= TR(Q)- TC(Q) p(Q)= TR(Q)-c(Q)- tQ FOCs: (derive with respect to Q) [∑TR(Q)/ ∑Q] –[ ∑c(Q)/ ∑Q] - t = 0 SOCs: (second derivative) [∑2TR(Q)/ ∑Q2 ]–[ ∑2c(Q)/ ∑Q2 ]≤ 0, which we know that holds from our standard analysis of monopolist If the monopolist sells Q(t) as a fraction of the tax, what is the effect of rising the tax? 1) ∑TR(Q(t))/ ∑Q(t) - ∑c(Q(t))/ ∑Q(t) - t = 0 2) Differentiate with respect to (t), [∑2TR(Q(t))/ ∑Q(t)2 ]μ[ ∑Q(t)/∑t] - [∑2c(Q(t))/ ∑Q(t)2]μ[ ∑Q(t)/∑t]-1= 0,

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Or alternatively, [∑2TR(Q)/ ∑Q2 - ∑2c(Q)/ ∑Q2] ∑Q/∑t = 1 ∑Q/∑t = 1/ [∑2TR(Q)/ ∑Q2 - ∑2c(Q)/ ∑Q2] from SOC it’s negative Hence, ∑Q/∑t <0, and as Æt ∞Q. So taxes reduce quantity produced.

Tax incidence on consumers Can the imposition of a $1 tax to a monopolist lead to an increase in the price consumers pay of more than $1? YES

1) Inverse demand function p(Q) = Q1/e TR(Q) = pQ = Qμ Q1/e= Q1+1/e MR(Q) = [1+1/e]μ Q1/e 2) MC(Q)=c, but after the tax they become c+t 3) Making MR=MC we obtain [1+1/e]μ Q1/e= c+t Solving for Q we have Q = [m+t]/[1+1/e]μe and plugging back this into the inverse demand function, p(Q)= Q1/e p = [m+t]/[1+1/e] 4) How does p change in t? Let’s derive it so that dp/dt equals…

dp/dt = 1/[1+1/e] But we know that e<-1 for the monopolist, e.g., -2 dp/dt = 1/[1-0.5] = 1/0.5 = 2. So the imposition of the tax increases price by more than the tax. Example Let’s look at an example where a tax is imposed on a monopolist. What happens to price and quantity after tax? p = 24-Q MR=24-2Q MC=2Q, where t =8

1) No tax MR=MC 24-2Q = 2Q Q=6 P=24-6=$18 2) Tax MR=MC 24-2Q = 2Q + 8 Q=4 P = 24 -4 = $20

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∆p=$2 and the consumer is clearly worse off as demand has not changed, but quantity produced has gone down and price has increased. Tax incidence out of $8 of the tax $2 are passed on the consumer $6 are absorbed by the monopolist Monopsony Definition: A market of a single buyer and many firms (sellers). Example: Major League Baseball in 70’s where one team was the buyer of many players (sellers). To profit maximize, a monopsonist. To profit maximize, this firm will want to find a level of production where MRPL=MEL (Marginal revenue product of labor = marginal expenditure of labor). Example: Production function Q=f(L) Firm in perfect competition is the market of coal, p is given. However, firm is the only buyer of labor in this region so it is a monopsonist. TR = pμf(L) Marginal Rev.Product of Labor MRPL : additional revenue that the firm gets when it employs an additional unit of labor. MRPL = pμf”(L) = pμ∆Q/∆L (draw it) Supply of labor is given by w(L) (draw it) Marginal Expenditure on Labor: the rate at which a firm’s total cost goes up, per unit of labor, as it hires more labor. MEL = ∆TC/∆L where ∆TC = I+II, where I = wμ∆L(extra cost from hiring more workers) and

( )II w L= Δ (extra cost that comes from raise the wage for all initial workers) Hence, MEL = [wμ∆L+∆wμL]/∆L = w+(∆w/∆L) μL And since ∆L ∆w, then… MEL = [w+(∆w/∆L) μL] >w = Supply curve MEL lies above supply.

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Profits = pμf(L) – wμL, where pμf(L) is the revenue from labor and wμL =TC from labor monopolist maximizes its profits when MRPL=MEL if L>3,000 the firm does not maximize profits since MRPL<MEL if L<3,000 the firm does maximize profits since MRPL>MEL Example Q=5L (demand), p=$10 and w=2+2L 1) We want to find L such that MRPL=MEL 2) MEL = w+(∆w/∆L) μL= (2+2L)+2L = 2+4L MRPL = pμ∆Q/∆L = $10μ5 = 50 3) MRPL=MEL 2+4L = 50 4L= 48 L=48/4 = 12 units of labor “purchased” Hence, w=2+2L =2+2μ12=2+24=$26 wage for workers. DWL of Monopsony

1  

EconS 301 – Intermediate Microeconomics CHAPTER 12 CAPTURING SURPLUS

Monopolist would like to capture CS and DWL In order to capture all of this surplus, the producer will try to charge different prices (i.e., price discriminate among consumers). Price Discrimination: The practice of charging consumers different prices for the same good or service. There are 3 different degrees of price discrimination. 1st degree 1p = willingness to payi (reservation price: the consumers maximum willingness to pay for that unit.) 2nd degree Different prices to different amounts of units bought: quality desired

$10 0 10 units$5 10-20 units

pp= ==

3rd degree Different prices to different consumer groups who have different demand curves

1 1

2 2

D MR MCD MR MC

→ =→ =

Example Airlines $500 per business class ticket $200 per economy class ticket Requirements

• Some market power to determine prices (not price taken). Downward sloping demand curve

• Some info. about reservation prices • No arbitrage (cannot resell the good to a new consumer)

1st degree price discrimination Demand curve as a willingness-to-pay schedule

E F

G H J

P

DWL

MC

Q

P CS

TR D Q 

P

2  

Uniform monopoly proxy 1st degree CS E + F Zero PS G + H (E + F) + (G + H) + J

DWL J Zero

It is ideal (perfect) for the producer!!

Check? of thee Requirements: 1) Same kind of marked power (Demand B downward energy) 2) Perfect information about willingness to pay, imperfect information is better than none. 3) No arbitrage otherwise p MC can sell to the consumers with the ??? willingness to

pay. Ex. P = 20 – Q MC = 2 a) Monopolist MR = 20 – 2Q = 2 = MC → 18 = 2Q → Q = 9 Price → p = 20 – 9 = 11. Producer Surplus = TR – TC = pQ – MC · Q = 11.9 – 2.9 = 81 b) 1st Degree Dem = MC →20 – Q = 2 → 18 = Q Price → p = 20 – 18 = 2 Producer Surplus = TR – TC = ½ (20-2) · 18 = 162 (↑ wrt

Monopoly) MR = Dem. Curve in 1st degree price discrimination

• With each additional unit sold, the producer gets p from that corner. • However, he doesn’t need to reduce the price of all the previous units.

Application College Education 1) Downward sloppy demand curve (↑Q⇒↓p) 2) Info about willingness to pay? 3) No arbitrage: I can’t sell you my education It is difficult to extract info about your family’s willingness to pay for your undergrad education. But if they ask your family may data about their finances when you apply for financial aid based on the financial need of your faculty, colleges can get a very good approximation of .2.

3  

Second Degree Price Discrimination → Quantity Discount Block tariff ≠ two part tariff → a form of 2nd degree price discrimination A form of 2nd degree price discrimination in which we consider 0 → Q - pays one price for units consumed in the first block of output (up to Q ) Q → - pays a different price for any additional units consumed on the second block Optimal block pricing MC = 2 P – 20 – Q 1) Blue segment twice slop of demand curve Starts at P1 ⇒Crosses MC always at the half point between

Q1 and 18. Then 12

18 6 18 24. . 122 2 2

QQ e g+ +⎛ ⎞= = =⎜ ⎟⎝ ⎠

where to set Q1.

After that everything is OK. 2) Constructing PS Receive from 1st block = P1 Q1 and from Demand, (20 – Q1)Q1 Receive from 2nd block = P2 (Q2 – Q1) and from Demand, (20 – Q2) (Q2 – Q1) Total cost = MC · Q2 = 2Q2 Therefore, PS = (20 – Q1)Q1 + (20 – Q2)(Q2 – Q1) – 2 Q2 = 20Q1 – Q1

2 + 20Q2 – 20Q1 – Q22 + Q2Q1 – 2Q2

3) Optimal Q1 and Q2

$11 for each of the first 9 units

$8 for all additional units

(in this case, the consumer only buys 3 more units)

MC

Q

P

D 

9 12

11

8

1st block 2nd block

MC

Q

P

D Q1 Q2 18

P1

P2

2

4  

1

1 21

2 1 1 22

2 2

20 2 20 0

20 2 2 0 2 18

20 2 2 18 20 0Q

PS Q QQPS Q Q Q QQ

Q Q

∂= − − + =

∂∂

= − + − = → = −∂

⎛ ⎞⎜ ⎟− − − + =⎜ ⎟⎝ ⎠

20⇔ 24 36 20Q− + − 2 2

2

0 30 36

12

Q

Q

+ = ⇔ − = −

⇔ =

Pay

1 2

1

2 2

2 18 24 18 6

20 20 6 14 1st block ($4 for all units up to 6 units)

20 20 12 8 2nd block ($8 for all additonal units in excess of 6)

Q Q

P Q

P Q

= − = − =

= − = − =

= − = − =

Subscription and Usage Charges Example: Telephone company charges $20 on subscription fee $0.10 per call (usage charge) How can a producer use Sub. + Usage charges in order to capture more surplus P = MC F = Consumer Surplus It is really so easy? No, different consumers may have different demand curves for the good. If the producer sets F = CS high demand consumers then all low demand consumers will refrain.

MC

Q

P

D 0.05

CS

$20 of subscription fee

5  

As we will see, in the observations, the producer will prefer to offer a new of phone plans. Each consumer picks the one he likes the best. Nonlinear Outlay Schedule: An expenditure schedule in which the average outlay(expenditure) changes with the number of units purchased.

Third Degree Price Discrimination Different prices to different consumer groups who have different demand curves. Ex.

1 1

2 2

3814 0.25

10

P QP QMC

= −= −=

1)

1 1

1 1 1

1

1 1

38 21 38 2 2 38 1 28

14

38 38 14 24

MR QMC MR Q Q Q Q

Q

P Q

= −= → − → = − =

=

= − = − =

MC

Q

P

MR2 D2

Q2

P2

Q

P

MR1 D1

Q1

P1

MC

MC=MR1=MR2

6  

2)

2 2

2 2 2

2

2

14 0.510 14 0.5 0.5 14 10 4

8

14 0.25 8 14 2 12

MR QMC MR Q Q

Q

P

= −= → − − → = − −

=

= − = − =i

Ex.

,

,

1.15 (Business travelers)

1.52 (Vacation travelers)

MC=10

B B

V V

Q P

Q P

ε

ε

= −

= −

1) IEPR for Business travelers ,

1 1101.15

B BQ P

p MC p pP E−

= − ⇔ − = i

2) IEPR for Vacation travelers ,

1 1101.52

V VQ P

p MC p pp ε−

= − ⇔ − = i

B Vp p> Is it so easy to change different prices to different consumers? The answer will need to truthfully reveal the demand for the good with the consumer demanding high do so? Solution: Screening commonly used in practice in order to do 3rd degree price discrimination. Screening: A process for sorting consumers based on a consumer characteristic that (1) the firm can see (such as age or status) and (2) is strongly related to a consumer characteristic that the firm cannot see but would like to observe (such as willingness to pay or elasticity of demand.) ⇒ Sort the consumers according to a perfectly observable character age student - - - This characteristic must be tightly correlated with the willingness to pay for the good. (which the price couldn’t observe). Application: Screening in the travel industry. Perfectly reasonable characteristic: a vacation traveler books the ticket months in advance a business traveler books the ticket but one day in advance (or same day) or

7  

a vacation traveler doesn’t care about staying at the destination over Saturday night a business traveler cares then high prices to couples who book their flight in the last week didn’t stay Saturday night low prices to couples who book their flight months in advance stay Saturday night Other applications: Day/night phone call prices New/old i-pod or computer Coupon and rebates as a screening device TYING (TIE-IN-SALES) A sales practice that allows a customer to buy one product (the tying product) only if that consumer also agrees to buy another product (the tied product). A firm with a patent in a photocopy machine would like to price discriminate, by setting

highnumber low numberof photocopies of photocopiesconsumed consumed

P P>

How to know how many copies is the firm doing? By tying the photocopies with the purchase of paper. BUNDLING (type of tie-in-sales): a type of tie-in sale in which a firm requires customers who buy one of its products also to simultaneously buy another of it products. The firm requires customers who buy one of its products also to simultaneously by other of its products. 3 types of bundling: Option 1: No bundling. The manufacturer does not bundle any goods. Option 2: Pure bundling. The manufacturer only offers bundled goods.

Option 3: Mixed bundling. The manufacturer offers customers different prices for bundled and non-bundled goods.

Examples: Computer and monitor Disney World ticket and all rides at the park

True Δ∇p

8  

Reservation Price

Computer Monitor Comp & Monitor negatively correlated demands

Customer 1 1,200 600 1,800

Customer 2 1,500 400 1,900

MC 1,000 200 1,300 No bundling Bundling Computer: 1,200→ 200 profit · 2 = 400 Comp & Monitor: 1,800 →500 · 2 = 1,000

profit 1,500 →500 profit 1,900 →600 Monitor: 400 →100 profit · 2 = 200 600 →300 profit Clearly: Profits Bundling > Profits No Bundling Take profits = 500 + 300 = 800 (1,000) (800)

9  

Positive (Twohead of Negative) Correlation in Demands

Reservation Price Computer Monitor Comp & Monitor

Customer 1 1,200 400 1,600

Customer 2 1,500 600 2,100

MC 1,000 300 1,300 No bundling Bundling Computer: 1,200→ 200 profit · 2 = 400 Comp & Monitor: 1,600 →300 · 2 = 600 profit 1,500 →500 profit 2,100 →800 Monitor: 400 →100 profit · 2 = 200 600 →300 profit Therefore, bundling does not increase the firms

profits when ??? are positively correlated. Take profits = 500 + 300 = 800 No Mixed Bundling ADVERTISING → non-price strategy to capture surplus. ↑ Advertising ⇒↑Demand, but it is costly to what extent it is profitable to spend on advertising 1) Demand0 and MR0 2) MC constant and AC0 3) P0 and Q0 4) Profitsinitial = I+II 5) Advertising shifts D1 and MR1 6) MC constant, but AC1 shifts upwards

MC

Q

P

P1

Q0 Q1

P1

P0

AC1

MR0 MR1

AC0

D0

II I

III

Adv

10  

7) P1 and Q1 8) Profitsfixed = I+II If I+III – A > I + III III – A > II then you will keep on advertising In other words III II

A AMR MCA> +

You will stop when MRA = MCA For a monopolist to be profit ????, we need MRQ = MCA to be spending the right amount on A, we need MRA = MCA It can be shown that these two conditions lead to

,

,

Q A

Q P

APQ

εε

= −

Ratio of negative of the ratio of the advertising Expenditures to sales elasticity of demand to the own price elasticity of demand

Ex. What APQ

be in two markets with the same ,Q Pε

With different ,Q Aε higher in market 1 Lower in market 2

↑ APQ

should be longer in the first market, since consumer are more sensible to

advertising

Ex. ,

,

1.5

0.1Q P

Q A

ε

ε

= −

=

a) Interpretation of , 0.1

1% in 0.1% in quantity demandedQ A

A

ε =

Δ ⇒ Δ

b) IEPR: ,

1 1 21.5 3

3 3 2 3

Q Q Q

Q P

Q p Q

p MC p MC p MCP P P

p MC p MC

ε− − −

= ⇔ = − ⇔ =

⇔ − = ⇔ =

11  

Advertising-to-paks ratio:

0.1 0.0671.5

APQ

= − =−

Advertising should be a 6.7% of your sales revenues.

1

 

EconS 301 – Intermediate Microeconomics Chapter 13. Market Structure and Competition

Chapter 13 explores different types of market structures. Markets differ on two important dimensions: the number of firms and the nature of product of differentiation. We will analyze competitive markets (many sellers), oligopoly markets (few sellers), and monopoly markets (just one seller). OLIGOPOLY MARKET - A small number of firms sell products that have virtually the same attributes, performance characteristics, image, and price. Think of the U.S. salt industry where Morton Salt, Cargill, and IMC sell virtually the same product at the same price. Oligopoly Cournot Bertrand no product differentiation (homogenous) Stackelberg Dominant firm markets Oligopoly with differentiated products differentiated products

Many firms Monopolistic competition Oligopoly – Cournot- refers to a homogenous products oligopoly. In the Cournot model, firms act as quantity takers because they set the quantity they produce by first making assumptions about what the other firm in the market will produce. The market price adjusts to equilibrium after each firm sets the quantity it will produce. 1) N-2 2) Firms compete in quantities 3) They both simultaneously submit their quantities Residual demand:

  The residual demand curve illustrates the relationship between the market price and a firm’s quantity when rival firms hold their outputs fixed. This demand curve is simply the market demand curve shifted inward by what the exact amount the rival produces. Since that my rival has already sold 50 units, the demand I face has been reduced. Taking into account this reduced demand I act as a monopolist MR50 = MC. That is, I set my Marginal Revenue curve (twice the slope of the residual demand curve) equal to the Marginal Cost (a constant). Profit inventory quantity given that my rival produced 50 40

2

 

60 5 ;

Best response function: q1(q2) A firm’s best response is the quantity where MRresidual=MC. The best response, or reaction function, function is the function that gives us a firm’s best response given any quantity the rival firm produces.

(Reaction function) Equilibrium in Cournot – Occurs where each firm’s output is the best response to the other firm’s output. Thus, neither firm has any after-the-fact reason to regret its output choice. This occurs where the reaction functions cross.

Ex. Market demand 1 2100 10010

= − → = − −=

p Q p q qMC

a) optimal q1 where q2 = 50. Firm 1’s residual demand is 1 1100 50 50p q q= − − = −

MR associated to demand is 1MR 50 2q= −

1 1 150 2 10 40 2 20q q q− = ⇔ − ⇔ =

b) optimal q1 for any arbitrary q2 Firm 1’s residual demand is ( )2 1100p q q= − −

MR =( )

( )

2 1

2 22 1 1

100 2 $109090 2 45

2 2

q qq qq q q

− − =

−− = ⇔ = = −

c) optimal q2 for any arbitrary q1

Given ???

3

 

Practice, but by symmetry 12 45

2qq = −

d) Cournot equilibrium is where Best Response functions cross each other…

21

1 21

2

452 : Th symmetry diopotis

452

qqTrick q q q

qq

⎫= − ⎪⎪→ = =⎬⎪= −⎪⎭

345 452 2

30

100 30 30 40

qq q

q

p

= − → =

=

= − − =

e) What are the monopoly price and Q?

1

2

22.5MR 100 2 $10 90 2 45

22.5100 100 45 55

qQ Q Q

qp Q

== − = ⇔ = → =

== − = − =

Monopoly vs Cournot Pmonop>Pcournot Qmonop<Q(q1+q2)cournot Profitsmonop>Profits1+Profits2

RevenuesReviews of my competitors

qp

Δ↑

∇ ⇒∇

Cournot with N firms Here, we want to analyze the Cournot model with any number of firms N firms, , MC $p a bQ c= − = Residual demand for firm 1 ( ) ( )1 1p a b q X p a bX bg→ = − + → = − −

( ) 1

1 1

MR 21

2 2 2

a bX bq ca bX c a cq q X

b b

= − − =

− − −⇔ = → = −

By symmetry

( )

( )

1 2 2 3 1

1

1 1

... , what makes that ... 1

1 12 2

N N

N

q q q X q q q N q

a cq N qb

−

= = = = + + + = −

−= − −

Manipulating…

Overall production

2 3 ... nq q q+ + +

4

 

( )

( )

( )

1 1

1 1

1 11

1

1 12 2

2 12 2

12 2

11

a cq N qb

q N q a cb

q Nq a c a cq Nb b

a cqN b

−+ − =

+ − −=

+ − −= → + =

−=

+ (A firms optimal quantity given N firms) Market quantity

11

1 1 1

a c a cQ NN b b

N a c a Np a b c cN b N N

− −∞= ⎯⎯→+

− ∞= − = + ⎯⎯→+ + +

i

i

Note: As the number of firms in a Cournot model increases to infinity, the firm and industry profits are zero, like a perfectly competitive market. Note that Cournot markets do not maximize industry profit because the firm’s do not act collusively, but individually. This self-interest erodes industry profits. IEPR in the Cournot model

1 1

MC 1 MC 1 1

market up ( market power)Q P Q P

p pp p N

N

ε ε− −

= − → = −

↑ ⇒↓ ↓

i

Therefore, the more firms there are, the less market power each firm has. Note that this IEPR is the same

as the Monopoly IEPR, except (1/N) is added to the right-hand side.

Bertrand Model Competition in prices, and not competition in quantity as in the Cournot model. Let us assume that firm 2 sets a price p=$40, what price will firm 1 set?

5

 

The process can be infinitely repeated until p=MC. That is, each firm can set a lower and lower price until they cannot set a lower price.

So, the Bertrand model even with N=2 firms we have competitive industry prices Cournot vs. Bertrand The Cournot and Bertrand models show drastically different conclusions within oligopolistic markets. 1) Capacity constraints: Cournot capacity is set firstly, then competition (LR capacity competition) Bertrand enough capacity to satisfy all market demand if necessary (SR price comp.) 2) Firm’s beliefs about the reaction of its competitor: Cournot model competitors cannot adjust their production very much

Bertrand model all my competitors have enough production capacity to steal my customers.

Stackelberg Sequential competition in quantities where one firm acts as a quantity leader, choosing its quantity first ,with other firms acting as followers. 1) Leader and Follower 2) The leader measures profits taken as given the follower’s BR (Reaction) function Ex. We first find the followers BR function, and then plug that into the leaders BR function.

Follower 12 2 45

2qF q→ = −

Leader F1’s residual demand:

1 11 2 1

1 11

1

100 100 45 552 2

55 10MR 55 2 552

45 leaderMC $10

q qp q q q

q qqq

⎛ ⎞= − − = − − − = −⎜ ⎟⎝ ⎠

⎫ − == − = − ⎪⎬ =⎪= ⎭

i

Follower:

25545 22.5 Follower2

q = − =

Price:

Conclusions:

Stackelberg Cournot1 1Stackelberg Cournot2 2Stackelberg Cournot

Stackelberg Cournot1 1Stackelberg Cournot2 2

100 55 22.5

Profits Profits

Profits Profits

pq q

q q

Q Q

= − −

>

<

>

>

>

6

 

Dominant Firm Markets Composition:

1) One dominate firm with a large market store dominates the market (has a significant market share compared to others)

2) Many small firms (Market competitive fringe)

1) DM and SFringe, and MC for everybody is equal. 2) Residual demand for the dominant firm = DM-SFringe When DM=SFringe⇒DR=0 When SFringe = 0 ⇒ DR converges with DM 3) MRR associated to DR MRR = MC determines the equilibrium Q for the dominant form 4) Market price: from DR, not from DM 5) Profit = ( ) ( )MC 50 25 50 $1, 250Rp Q− = − = 6) Fringe firms supply an output of 25 units when p = $50 Product Differentiation Vertical differentiation: two products with differences in their quality Duracell vs. Store-brand batteries

Horizontal differentiation: two products with differences in some attributes, a matter of substitutability. For instance, some consumers may consider Diet Pepsi a poor substitute for Diet Coke, and therefore will still purchase Diet Coke even if Diet Pepsi is a lower price. Firms selling horizontally differentiated products exhibit downward sloping demand curves.

7

 

Weak HD: firm demand is very flat and therefore is very sensitive to its own prices rival’s prices p⇒ q along D0 price ⇒ q price the leftward shift in the demand curve Strong HD: firm demand is much less sensitive to own prices rival’s prices p⇒ slight q D0 price ⇒ slight leftward shift in the demand curve (  q). Bertrand Competition with HD EXAMPLE: 1) Given HD, the demand functions are different for coke and pepsi Coke 1 1 264 4 2Q p p= − + each firm wants to produce ???

Pepsi 2 2 150Q p p= − + Its profits given pure competition Procedure 1st) Find Coke’s profit maximizing price for any arbitrary price of Pepsi, p2 BRF coke p1(p2) 2nd) Substitute one BRF into another, and find given prices

Ex. 1

2

MC $5MC $4

==

a) What is Coca-Cola profit maximizing price when p2 = $8?

( )

1 1 1 1 1 1

1 1 1

1

2

64 4 2.8 64 16 4 48 4 20 0.25

MR 20 0.50 $5 15 0.50 30

Price: 20 0.25 30 $12.50$8

Q p p p p q

q q q

pp

= − + = − − = − ⇔ = −

= − = ⇒ = ⇒ =

= − =

=

b) What is Coca-Cola profit maximizing price for any arbitrary p2?

8

 

1 11 1 2 1 2 1 1

2 11

64 264 4 2 2 64 44

162 4

p QQ p p Q p p p

p Q p

+ −= − + ⇒ − − = − ⇒ =

⇒ + − =

2 1

2 1 21

MR 16 2 5 MC2 4

14 222 2 4

p Q

p Q pQ

⎛ ⎞= + − = =⎜ ⎟⎝ ⎠

+ = ⇔ = +

Substituting back in the demand function, we obtain

2

2 21 1

224 116 10.5

2 4 4

pQp pp p

+ ←⎛ ⎞+ − ⎯⎯⎯⎯→ = +⎜ ⎟⎝ ⎠

Coca-Cola Price Reaction ???

Doing the same for Pepsi, we obtain 12 7

10pp = +

Then,

( ) ( )( ) ( )

221

2 21

2

1

1

2

10.510.544 7 $8.26

10710

8.2610.5 $12.564

64 4 12.56 2 8.26 30.28 units

50 5 8.26 12.56 21.26 units

pppp p

pp

p

Q

Q

⎛ ⎞⎫ += + ⎜ ⎟⎪⎪ ⎝ ⎠= + → =⎬⎪= +⎪⎭

= + =

= − + =

= − + = Application: pricing the Channel Tunnel (Real-world example) £ 87 channel £ 150 ferries

9

 

Page 1  

EconS 301 – Intermediate Microeconomics

Chapter 14: Game Theory

Definitions

Game Theory: The branch of microeconomics concerned with the analysis of optimal decision making in competitive situations.

Strategy: a detailed plan of action under any possible state that the player might face (know as a complete contingent plan).

Nash Equilibrium (NE): a situation in which each player is choosing his Best Response given the strategy chosen by the other player.

Best Responses for the Honda and Toyota game:

Column Row Fix column in Build (meaning consider only the upper left box and the lower left box) Will Honda Build? Yes, because 16>15

( )HBR B B= (meaning Honda’s best response given that Toyota Builds is to Build also)

3. Fix row in Build Will Toyota Build? Yes, because 16>15

( )TBR B B=

Fix column in Not Build Will Honda Build? Yes, because 20>18

( )HBR NB B=

4.Fix row in Not Build Will Toyota Build? Yes, because 20>18

( )TBR NB B=

Page 2  

Row Player Column Player (Honda) (Toyota)

NE ( Build , Build )

Prisoner’s Dilemma

A game where there exists a tension between the self-interests of the player and the collective interest. The players’ strategies do not result in the outcome that is best for everyone.

The story: Ron and David steal a car and drive it off a cliff destroying most of the evidence. The police have a suspicion that they committed the crime but can’t prove it. Nonetheless they arrest both Ron and David and interview them in separate rooms. If neither of them confess to the crime a judge will give them both a year in jail on suspicion. The police make a deal with Ron by saying “if you tell us David did it and he keeps quite we will let you free and put him in jail for 10 years.” They make the same deal with David; if he confesses and Ron keeps quite he will be completely free and Ron will receive 10 years in jail. Lastly if they both turn each other in then the judge will have evidence and will convict them both to 5 years in jail. Let’s examine the interesting equilibrium of this game.

We find that, even though Ron and David would be much better off if they both said nothing to the police, such an outcome is not a Nash Equilibrium.

Ron ( )( )

NE ( , )David

( )( )

R

R

D

D

BR C CBR NC C

C C

BR C CBR NC C

⎫⎪= ⎪⎪= ⎪⎬⎪⎪=⎪

= ⎪⎭

Page 3  

Dominant Strategy: A strategy that is better than any other a player might choose, no matter what strategy the other player follows.

Dominated Strategy: A strategy such that the player has another strategy that gives a higher payoff no matter what the other player does.

Summary:

• Whenever both players have a dominant strategy, those strategies will constitute the Nash Equilibrium in the game.

• If just one player has a dominant strategy, that strategy will be the players Nash equilibrium strategy. We can find the other players Nash equilibrium strategy by identifying that players best response to the first player’s dominant strategy.

• If neither player has a dominant strategy, but both have dominated strategies, we can often deduce the Nash equilibrium by eliminating the dominated strategies and thereby simplifying the analysis of the game.

Learning By Doing 14.1

Games with more than one Nash Equilibrium

Game of Chicken. Two players drive their cars toward each other. If they both swerve to avoid an accident they walk away unharmed and with no good story to tell. If one player doesn’t swerve while the other does the swerving player suffers embarrassment while the other player gets to tell a good story. If neither player swerves the cars crash and both players suffer greatly.

A similar result is found in games modeling the Cold War between the USA and the USSR. The game can also be used to model an industry with room for just one firm to make a profit.

Page 4  

Luke ( )( ) Two NEs ( , ) and ( , )

Slick the players do not cordinate( )( )

L

L

S

S

BR Swerve StayBR Stay Swerve Stay Swerve Swerve Stay

BR Swerve StayBR Stay Swerve

⎫⎪= ⎪⎪= ⎪⎬⎪⎪=⎪

= ⎪⎭

Example

The game of chicken between XM and Sirius satellite radio.

Bank Run Game

The equilibriums of this game are coordination. Both players withdraw or both players don’t withdraw.

Mixed Strategies

Pure strategy: a specific choice of a strategy, with 100% probability.

Mixed strategy: the choice among two or more strategies according to a specified probability.

Page 5  

1999 Final match of the Women’s World Cup

The game was between the USA and China. The game was tied 0-0 so the winner would be decided by penalty kicks. p is the probability that the Chinese Goalie plays dive right. q is the probability that the US Kicker plays aim right.

Since there is no overlap of the two country’s players best responses there is no Pure Strategy Nash Equilibrium (PSNE).

However there still may be Mixed Strategy Nash Equilibrium (MSNE). Let’s check:

12

(aim right) (aim left) where is Expected Utility0( ) 10(1 ) 10 0(1 )10 10 1010 20

the US Kicker is indifferent between playing aim right or aim left when the

USA USAEU EU EUp p p p

p pp

p

=+ − = − + −

− = −==

Chinese Goalie plays dive right with 50% probability

12

(Dive right) (Dive left)0 ( 10)(1 ) 10 0(1 )

10 10 1020 10

the Chinese Goalie is indifferent between playing dive right or dive left when the USA Kicker pl

China ChinaEU EUq q q q

q qq

q

=+ − − = − + −

− + = −=

=ays aim right with 50% probability

Now the Best Responses of each player do overlap and we have an equilibrium.

Page 6  

1 1 1 12 2 2 2

Chinese Goalie US Kicker

: ( , )MSNE R L R L+ +

Another popular example of a MSNE is the Battle of the Sexes game. The couple gets an extra benefit if they go somewhere together, but the man wants to go to one place and the woman wants to go to a different place.

Repeated Prisoner’s Dilemma

In the Grim Trigger Strategy the players cooperate until either player is caught cheating and then neither player will ever cooperate again. If a player is caught cheating in the first period, then players receive, the lower, not cooperating payoff in every remaining period.

Page 7  

Tit-for-tat: A strategy in which you do to your opponent in this period what your opponent did to you in the last period.

Cooperation is more likely if:

• Beliefs from cheating are small • The game is repeated enough periods of time • We care about the future periods (we are not too impatient) • Cheating is easy to detect

Sequential Move Games A game in which one player takes an action before the other player.

Game tree: a diagram that shows the different strategies of each player and when these strategies become available to play.

Sequential-moves games: Games in which one player (the first mover) takes an action before another player (the second mover). The second mover observes the action taken by the first mover before deciding what action it should take.

Game tree: A diagram that shows the different strategies that each player can follow in a game and the order in which those strategies get chosen.

Backward induction: A procedure for solving a sequential-move game by starting at the end of the game tree and finding the optimal decision for the player at each decision point.

Example

Honda decides whether to build a large factory (BL), build a small factory (BS), or to not build a factory (NB). Toyota then observes Honda’s choice and makes its own choice about what factory to build.

Page 8  

We may use backwards induction to find that the equilibrium of this sequential game is (BL, NB).

Entry to an industry

Say for example Quiznos is thinking of opening a sandwich shop on campus to compete with Subway. In order to fight against Quiznos Subway may drop its prices drastically. Lower prices would hurt Subway but they might do it because it would also hurt Quiznos. We’ll call this action Fight. Subway could also choose to accommodate Quiznos by keeping their prices the same and losing some business to Quiznos. We’ll call this action Accommodate. Of course Quiznos would rather open a sandwich shop if Subway would accommodate them. Subway receives its greatest payoff if Quiznos does not enter (open a shop).

02⎛ ⎞⎜ ⎟⎝ ⎠

31

−⎛ ⎞⎜ ⎟−⎝ ⎠

21⎛ ⎞⎜ ⎟⎝ ⎠

We may also represent the game in Normal Form:

In the normal form game gives us the two equilibriums:

(Out, Fight if In) (In, Accom if In)

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Using the game tree we narrow this down to the unique equilibrium that survives backwards induction: (In, Accom if In)

Example

201

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

156

−⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

312

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

544

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

01

7

⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠

01

7

⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠

Entry Game Consider the game where you decide on what size factory to build. Then your competitor decides whether to start a price war or accommodate you.

We may use backwards induction to find that the equilibrium of this sequential game is (small, accom).

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EconS301 – Intermediate Microeconomics

Chapter 15 – Risk and Uncertainty

This chapter deals with the role of risk and uncertainty in an economy. For instance, say you were to make a decision over whether or not to buy stock in a certain company. You know that this stock has gone both up and down in the past, and has had various payoffs. However, you cannot predict perfectly what the stock might be worth tomorrow, a month, or a year down the road. Because of this uncertainty, buying the stock caries a certain risk, the risk of losing money if the value of the stock drops. However, there is also a chance that this stock’s return will be positive and you will earn money. Analyzing these types of decisions is crucial in economics, and so, we will deal with it here.

LOTTERY

Lottery – A lottery is simply any event in which the outcome is uncertain, much like our example above or the outcome of a college football game. Probability is the likelihood that any specific outcome will occur (i.e. the likelihood of earning a positive return on the stock you purchased). Furthermore, the probability distribution is a graph that depicts all possible outcomes and their associated probabilities, as depicted below.

• The probability of any particular outcome has to be between 0 and 1 • The sum of the probabilities of all possible outcomes must equal 1

Where do probabilities and prob. distributions come from? Some probabilities result from the laws of nature, as in flipping a coin. We know that there is exactly a 50% chance of flipping heads and a 50% chance of flipping tails. This is referred to as Objective Probability.

On the other hand, deducing probabilities can be much harder (as in buying stock), as there is no clear probability associated with a certain event. These are referred to as Subjective Probabilities: probabilities that reflect subjective beliefs about risky events.

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Expected Value – the Expected Value is the measure of the average payoff that the lottery will generate. That is, it is the weighted average of possible outcome, meaning that it takes into account the probability of each potential outcome in calculating the average.

For instance, using our example from the probability distribution above…

EV =

EV = .30(120) + .40(100) + .30(80) = 100

But, what if some lotteries have the same EV, but one is much more volatile than the other? How do we describe this occurrence? We use a measure of variability called variance.

Each of these probability distributions has an EV = 100, but the one on the left is clearly more risky, as the probabilities are distributed more evenly, versus the graph on the right where we can be quite sure the payoff will be 100.

Variance – The sum of the probability-weighted squared deviations of the possible outcomes of the lottery. That is, the variance gives us a measure of how the potential outcomes (taking into account their associated probabilities) deviate from the expected value we calculated above.

From the example above…

For the internet company…

Var = …

Var = 120 + 0 + 120

Variance = 240

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For the public utilities company…

EV = 100

Var = same process as above…

Var = 40 + 0 + 40

Variance = 80

As we can see, this affirms what the graph shows us. The internet company is a riskier investment because the chance of receiving a payoff different than the EV is greater, that is, the variance is greater than the variance for the public utilities company.

NOTE: Another common measure of riskiness (chance of payoff different than the EV) is the standard deviation. It simply the square root of the variance.

√

Utility Functions and Risk Preferences

Now that we know how to describe risk and uncertainty through the method above, we can now look at how a decision-maker might evaluate and compare alternates whose payoffs have different probability distributions and therefore different degrees of risk.

Let’s consider an example . . .

Say a graduate is offered two jobs upon graduation. One job is from a large and well-established corporation who guarantees the graduate a salary of $54,000 for the coming year. The other offer is from a new start-up company who has been operating at a loss and therefore offers the graduate a $4000 token salary. However, the company also offers the possibility of a $100,000 bonus if the company earns a profit in the ensuing year. What offer should/will this consumer take?

1 54000 $54,000

.50 $4,000 .50 $104,000 $54,000

But, even though these two options have the same EV, the graduate will probably not look at them the same way, because of risk. We can evaluate this more formally by using a utility function…

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• Increasing (more is better) in income • Diminishing marginal utility

That is, even if EVwork 1 = EVwork 2, this shows that EUwork 1 = EUwork 2. That is, your expected utility at the established company is higher than your utility at the start-up company.

Expected Utility – it is simply the expected value of the utility levels that the decision maker receives from the payoffs in the lottery.

Risk Preferences

Risk Adverse – A characteristic of a decision maker who prefers a sure thing to a lottery of equal expected value. That is, the decision maker above is clearly risk adverse. This is represented in the utility function.

EX) √100

5  

EU of internet stock:

.3√100 80 .4√100 100 .3√100 120 99.7

EU of Public Utility

.1√100 80 .8√100 100 .1√100 120 99.9

, then the risk adverse agent will prefer the Public Utility

Risk Neutral – A characteristic of a decision maker who compares lotteries according to their expected value and is therefore indifferent between a sure thing and a lottery with the same expected value.

• This is characterized by a linear utility function , o so 1

EX) a=0, b=100

EU of internet stock

.3 100 80 .4 100 100 .3 100 120 10,000

EU of Public Utility

.1 100 80 .8 100 100 .1 100 120 10,000

,

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Risk Loving - A characteristic of a decision maker who prefers a lottery to a sure thing that is equal to the expected value of the lottery. In the job offer example, a risk-loving person would prefer the start-up company to the established company.

• This is characterized by increasing marginal utility

TRY similar example from above two…

Bearing and Eliminating Risk

Now that we can describe lotteries and can calculate expected utilities to determine a consumer’s preferences amongst different lotteries, we will now move into analyzing when an agent will choose to bear risk and when they will choose to eliminate it. This analysis is based on the assumption that most people are risk-adverse, at least for important decision like whether or not to insure a car or whether to purchase a home, but also, that people still choose to bear risk.

An agent might chose to take the riskier offer if the expected payoff from the gamble is sufficiently larger than that of the sure thing. The graph below illustrates this...

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The exact difference in payoff that would invoke this agent to decide on the riskier choice is referred to as the risk premium – or the necessary difference between the expected value of the lottery and the payoff of a sure thing to make the decision maker indifferent between the lottery and the sure thing.

In a lottery of two payoffs, I1 and I2, with the probabilities p and (1-p), respectively, we can find the risk premium using the following formula:

1 1

EX) A) √ , Find the risk premium associated with the risky start-up company.

1

Where…I1 = 104,000 I2 = 4,000 p=.5 EV=54,000

. 5 104,000 .5 4,000 54,000

192.87 54,000

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37,199 54,000

16,801

B) If the wage offer of the start-up company changes to I1 = 0, I2 = 108,000, then what is the RP?

.5 0 .5 108,000 54,000

. 5√0 .5 108,000 54,000

164.32 54,000

27,000 54,000

27,000

So, as the variance of the lottery increases, the RP also increases. This makes sense, as the riskier the option the higher you would need the payoff to entice you to choose that option.

Fairly Priced Insurance

The logic of risk aversion also sheds light on the circumstances under which a risk-averse person would chose to eliminate risk by buying insurance. When the insurance policy has a premium equal to the expected value of the promised insurance payment, we call this Fairly Priced Insurance.

Let’s consider an example where a person has to decide whether or not to purchase insurance.

Insurance Premium = 500

Insurance Coverage (if accident occurs) = $10,000 (full coverage)

Prob of accident = .05

Prob of no accident = .95

500 = .05(10,000) + .95(0)

500=500

Insurance:

No accident: 50,000-500 = 49,500 So 49,500 is a sure thing, no matter what happens

Accident: 50,000-500 -10,000 + 10,000 = 49,500

Without Insurance:

No Accident: 50,000-500 = 49,500 EV = .5(40,000) + .95(50,000) = 49,500

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Accident: 50,000-500-10,000 = 40,000

, And so, a risk averse person will therefore pick the sure thing (buying insurance) instead of picking the lottery.

Practice this with Learning-by-Doing 15.4

Asymmetric Information

Often in economics, one party in a deal knows more than the other party, and therefore has bartering power. This is referred to as Asymmetric Information – and is basically a situation in which one party knows more about its own actions or characteristics than another party. For instance, you know more about your driving than a car insurance company. For this, car insurance companies charge a premium to safeguard against their lack of information (or think of health insurance deductibles). There are a few types of asymmetric information.

Hidden Action: Moral Hazard

Suppose your car is fully insured. How careful would you be? Probably not as careful as you would be without insurance. That is, insurance in a way incentivized less careful driving. This is a form of moral hazard – or otherwise, a phenomenon whereby an insured party exercises less care than he or she would in the absence of insurance. The insurance company cannot perfectly monitor the actions of their customers (hidden action problem), and therefore charges a deductible to induce customers to be more cautious. That is, a deductible places some of the responsibility of an accident on the driver, and therefore helps to reverse the moral hazard of insurance.

Hidden Information: Adverse Selection

Another reason insurance companies do not provide full coverage is adverse selection – a phenomenon whereby an increase in the premium increases the overall riskiness of the pool of individuals who buy an insurance policy. That is, the higher the premium the insurance company charges, the more likely that only people truly in need of insurance (unhealthy) will buy the policy, and it is exactly these people who will cost the insurance company the most money (obviously something the companies do not desire).

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Decision Trees

Let’s now use a tool called a Decision Tree to analyze how a decision maker might choose a plan of action in the face of risk. A Decision Tree is a diagram that describes the options available to a decision maker as well as the risky events that can occur at each point in time.

In a decision tree, you always start from the end (far right hand options) and work your way back to the beginning.

Auctions

Auctions are a large part of the economic landscape (i.e. governments auctioning off their air waves, eBay, etc.). Auctions typically involve relatively few decision makers who make decision under uncertainty.

Private Values vs Common Values

Private value auctions are auctions where each buyer has his own personal valuation of the auctioned item. You know the value of the item to yourself, but not the value of the item to others (art, antiques). On the contrary, common value auctions are a situation where the item being auctioned has the same intrinsic value to all buyers, but no buyer knows exactly what that value is (oil leases, U.S. Treasury Bills).

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Types of Auctions: Private value Auctions

English Auction – an auction in which participants cry out their bids and each participant can increase his or her bid until the auction ends with the highest bidder winning the object being sold.

First-Price Sealed Bid Auctions – an auction in which each bidder submits one bid, not knowing the other bids. The highest bidder wins the object but pays a price equal to his or her bid.

Second-Priced Sealed-Bid Auction – An auction in which each bidder submits one bid, not knowing the other bids. The highest bidder wins the object but pays an amount equal to the second-highest bid.

Dutch Descending Auction – An auction in which the seller of the object announces a price which is then lowered until a buyer announces a desire to buy the item at that price.

Common Value Auctions

When bidders have common values, a complication arises that does not occur when bidders have private values, the winner’s curse: The winning bidder might bid an amount that exceeds the item’s intrinsic value.

As in the example below, the winning bid is 100 but the intrinsic value of item is only 80.

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EconS 301 – Intermediate Microeconomics Chapter 17 ­ Externalities and Public Goods 

 There is more than one type of market. Some markets have externalities and markets with public goods. These markets are unlikely to allocate resources efficiently. Externality: The effect that an action of any decision maker has on the well-being of other consumers or producers, beyond the effects transmitted by changes in prices. Public Good: A good, such as national defense, that has two defining features: first, one person’s consumption does not reduces the quantity that can be consumed by any other person; second, all consumers have access to the good.  Externalities Vaccination Positive  Consumption Bandwagon Effect  Positive  Consumption Development of laser technology 

Positive  Production 

Manufacturer Polluting in Rive 

Negative  Production 

Highway congestion  Negative  Consumption Computer Network  Negative  Consumption Snob Effect  Negative  Consumption  Negative Externality:

  Equilibrium

(price=p1) Social Optimum (price=p*)

Difference

Consumer Surplus A+B+G+K A -B-G-K Private Producer Surplus

E+F+R+H+N B+E+F+R+H+G B+G-N

-Cost of Externality

-R-H-N-G-K-M -R-G-H M+N+K (external savings cost)

Net Social Benefit A+B+E+F-M A+B+E+F M Deadweight Loss M Zero M

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Is Zero pollution socially efficient? No. At Q=0 Mg Benefit> Mg Cost Q* is efficient amount of production and pollution There are two main ways to reduce pollution externalities: 1) Emission standards A governmental limit on the amount of pollution that can be emitted, since MPC and MSC may be difficult to calculate the government may set an optimal Q* and allow firms to decide if they really want to pollute or not. Tradable emissions standards: low cost firms sell them to high cost firms ( i.e. cap and trade system) 2) Emission fees Tax imposed on pollution the firm releases to the environment.

Equilibrium (with tax) Consumer Surplus A Private Producer Surplus F+R -Cost of Externality -R-H-G Government receipts from emissions tax B+G+E+H Net Social Benefit A+B+E+F Learning By Doing 17.2 Inverse Demand Curve: Pd=24-Q Inverse Supply Curve: MPC=2+Q Industry emits one unit of pollutant for each ton of chemical it produces. As long as fewer than 2 million units of pollutant emitted each year, the external cost is zero. But when pollution exceeds 2 million units the marginal external cost is positive.

3  

MEC: =0, when Q ≤ 2−2 + Q whenQ > 2

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Suppose the government wants to use an emission fee of $T that will induce the market to produce the economically efficient amount of the chemical.

a) Construct a graph and table comparing the equilibria with and without the emission fee.

No Emissions Fee Emissions Fee of $6 a unit Consumer Surplus AJH (60.5 mil) ABM (32 mil) Private Producer Surplus FJH (60.5 mil) FEN (32 mil) -Cost of Externality -VLH=-GUI (-40.5 mil) -VNM=-GKR (-18 mil) Gov’t Earnings from fee Zero ENMB (48 mil) Net Social Benefit AMVF-MLH (80.5 mil) AMVF (94 mil)

In order to find the values above you have to remember your geometric equations. Triangle: 1/2Base*Height Square: Base*Height Example AJH .5{(24-13)(11)}=60.5 Repeat for all other sections of the graph above.

b) Why is the sum of CS + PPS – EC+GOVT+DWL the same without or without the fee?

   The figures in the table show that the sum is going to be 94 million both ways because the figure represents the potential net benefit in the market whether or not there is a fee. When there is no fee the market performs inefficiently because of the negative externality

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and there is a DWL. When there is a fee the market performs efficiently and the entire potential is captured.  Common Property A resource, such as a public park, a highway, or the internet, that anyone can access.

Q1= no congestion Peak: Too much Q5………B-E is the toll Off Peak: Too much Q3…..M-N is the toll Positive Externalities and Under Production

Equilibrium Social Optimum Difference Private Consumer Surplus

B+E+F B+E+F+G+K+L G+K+L

Producer Surplus G+R F+G+R+J+M F+J+M Benefit from Externality

A+H+J A+H+J+M+N+T M+N+T

Gov’t Cost Zero -F-G-J-K-L-M-T -F-G-J-K-L-M-T Net Social Beneifts A+B+E+F+G+H+J+R A+B+E+F+G+H+J+M+N+R M+N Solution: Subsidy

5  

Property Rights and the Coase Theorem Property Rights: The exclusive control over the use of an asset or resource. Why are we not applying property rights and allowing firms to work? Coase Theorem: Regardless of how property rights are assigned with an externality, the allocation of resources will be efficient when the parties can costlessly bargain with each other. Example: Fishery vs. Steel Firm (who owns the river) SPECIAL CLASS ON ENVIRONMENTAL ECONOMICS NEXT FALL – EconS 330 Public Goods Examples: National Defense, Clean Air, Your neighbors Internet Connection (unprotected). Two Features of a Public Good: Non-Rival: When consumption of a good by one person does not reduce the quantity that can be consumed by others. National Defense (yes) Highway (not so much) Clothing (not at all) Non-Exclusive: A good that, once produced, is accessible to all consumers; no one can be excluded from consuming such a good after it is produced. National Defense (yes) Clean air (yes) Highway (not so much) Rival Non-Rival

Excludable Clothing/Food Pay-TV Channels Non-Excludable Hunting National Defense/Clean Air

Note: Not every publicly provided good is a truly public good.

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Efficient Provision of a Public Good - Learning by doing 17.4

if MC= 400 it is efficient to provide zero of the public good if MC= 240 it is efficient to provide Q=30 of the public good if MC= 50 it is efficient to provide Q=150 of the public good Example: P1=100-Q Individual 1 P2=200-Q Individual 2 a) MC =240 Efficient amount of Q?  Start by adding the equations vertically MSB= P1+P2= (100-Q) + (200-Q) = 300-2Q

MSB =300 − 2Q for Q ≤100200 − Q for Q >1000 Q > 200

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

 Set MSB=MC 300-2Q=240

60=2Q Q=30 b) MC=50 MC=MSB 200-Q=50 150=Q

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c) MC=400 300-2Q=400 2Q=-100 Q=-50? The marginal social benefit of this public good is not high enough for any of it to be provided so the Q=0. Free Rider A consumer who doesn’t pay for a non-excludable good, expecting that others pay. Examples: PBS (100 Million viewers, 4 million contributors) National Public Radio (22 million listeners, 3 million contributors) Additional Examples that can be worked out: 17.1, 17.6, 17,16