UNIT I:Theory of the Consumer

UNIT I:Theory of the Consumer

• Introduction: What is Microeconomics?• Theory of the Consumer• Individual & Market Demand

6/24

Theory of the Consumer

• Indifference Curves• Utility Functions• Optimization under Constraint• Income & Substitution Effects

How do consumers make optimal choices?

How do they respond to changes in prices and income?

Utility Functions

X

U

Assume 1 Good:

U = 2X

Utility: The total amount of satisfaction one enjoys from a given level of consumption (X,Y)

Utility Functions

X

U

Assume 1 Good:

X

U

U = 2X

MUx = U/X

= 2

Marginal Utility: The amount by which utility increases when consumption (of good X) increases by one unit

MUx = U/X

MUx

Utility Functions

XX

U U

Assume 1 Good:

X

U

U

X

U = 2X

MUx = U/X

= 2

U (X)

We generally assume diminishing marginal utility

Utility Functions

Y

X

U

Now Assume 2 Goods:

U (X)U (Y)

U = f(X,Y)

Utility Functions

Y

X

U

U0 U1 U2 U3

U1

U2

U3

U0

U = f(X,Y)

Indifference curves

Utility Functions

Y

X

U

U0 U1 U2 U3

U1

U2

U3

U0

U = f(X,Y)

XY

Utility FunctionsMarginal Rate of Substitution (MRS): The rate at which a consumer is willing to trade between 2 goods. The amount of Y he is willing to give up for 1 unit of X.

Y Utility = No. of Apples + 2(No. of Oranges) U Along an indifference curve, U = 0

Therefore, MUxX + MUyY = 0- MUxX = MUyY

- (MUx/MUy)X = YY/X = - MUx/MUy

= MRS= slope

X

Generally, this rate will not be constant; it will depend upon the consumer’s endowment.

XY

OptimizationWe assume that a rational consumer will attempt to maximize her utility. But utility increases with consumption of all goods, so utility functions have no maximum -- more is always better!

Y Utility = No. of Apples + 2(No. of Oranges)

U

X

Increasing utility

OptimizationThe optimal consumption bundle places the consumer on the highest feasible indifference curve, given her preferences and the opportunities to trade (her income & the prices she faces).

Y Utility = No. of Apples + 2(No. of Oranges)

U

Y*

X* X

Indifference Curves depict consumer’s “willingness to trade”

Slope = - MRSBudget Constraint depicts “opportunities to trade”

Slope = - Px/Py

At point C, MRS = Px/Py, so consumer can’t improve thru trade.

C

Two Conditions for Optimization under Constraint:

1. PxX + PyY = I Spend entire budget

2. MRSyx = Px/Py Tangency

Optimization

MRSyx = MUx/MUy = Px/Py

=> MUx/Px = MUy/Py

The marginal utility of the last dollar spent on each good should be the same.

Optimization: An ExamplePat divides a monthly income of $1800 between consumption of food (X) and consumption of all other goods (Y). Pat’s preferences can be described by the following utility function:

U = X2Y

If the price of food is $1 and the price of all other goods is $2, find Pat’s optimal consumption bundle.

Pat should choose the combination of food and all other goods that places her on the highest feasible indifference curve, given her income and the prices she faces. This is the point where an indifference curve is tangent to the budget constraint (unless there is a comer solution).


U = X2Y


Since Pat’s utility function is U = X2Y, MUx = 2XY and MUy = X2. MRS = (-)MUx/MUy = (-)2XY/X2 = (-)2Y/X. Setting this equal to the (-)price ratio (Px/Py), we find ½ = 2Y/X, X = 4Y. This is Pat’s optimal ratio of the goods, given prices.


U = X2Y


To find Pat’s optimal bundle, we substitute the optimal ratio into the budget constraint: I = PxX + PyY, 1800 = (1)X + (2)Y,

1800 = (1)4Y + (2)Y = 6Y, so

Y* = 300, X* = 1200.

Y

X

900

Y*=300

600 X*=1200

U = XY

Optimization: An ExampleGraphically:

Maximize: U = X2Y

Subject to: I = PxX + PyY

I = 1800; Px = $1; Py = $2

Y* = 300, X* = 1200.


U = X2Y

Now suppose the price of food rises to $2.

MRS = (-)2Y/X. Setting this equal to the new (-)price ratio (Px/Py), we find 1 = 2Y/X, X = 2Y. Substituting in Pat’s new budget constraint: I = PxX + PyY, 1800 = (2)X + (2)Y,

1800 = (2)2Y + (2)Y = 6Y, so

Y** = 300, X** = 600.

Y

X

900

Y**=300

X**= 600 12001200

U = XY

Optimization: An ExampleGraphically:

Now: U = X2Y

I = 1800; Px’ = $2; Py = $2

Y* = 300, X* = 600.

Y

X

900

Y**=300

X**= 600 1200 1200

U = XY

Graphically:Because the relative price of food has increased, Pat will consume less food (and more of all other goods). This the substitution effect. But because Pat is now relatively poorer (her purchasing power has decreased), she will consume less of both goods. This is the income effect.

Income & Substitution Effects

S

S

Y

X

900

Y**=300

X**= 600 1200 1200

U = XY

Graphically:But because Pat is now relatively poorer (her purchasing power has decreased), she will consume less of both goods. This is the income effect.

In this case, the 2 effects are equal and opposite for Y, additive for X.


Individual & Market Demand

• Income & Substitution Effects (from last time)• Normal, Inferior, and Giffen Goods• Consumer Demand• Price Elasticity of Demand• Next Time: The Theory of the Firm

Individual & Market Demand

We have seen how consumers make optimal choices. A rational consumer will attempt to maximize utility subject to market conditions (relative prices) and income. That is, given I, Px, Py, she chooses X and Y to maximize U.

Now, we want to ask, how do changes in prices effect these consumption decisions? X = f(Px).

We will see that changes in prices affect quantities through two causal channels: Income and substitution effects.

Y

XNow his wage rises to$12/hr for the first 40 hrs/wk; it remains $8/hr above 40

hrs/wk.

1200

960

800

50 60 100 1200

Bullwinkle Moose faces a choice between leisure (X) and income (Y). He can work up to 100 hours a week at a wage of $8/hr. Initially, he chooses to work 50 hrs/wk.


Draw his new budget constraint.

Y


hrs/wk.

1200

960

800

50 60 100 1200



Will he work more than, less than, or equal to 50 hrs/wk?

What is the income effect?

His purchasing power is greater, so he will consume

more leisure, work less.

Y


hrs/wk.

1200

960

800

50 60 100 1200



Will he work more than, less than, or equal to 50 hrs/wk?

What is the substitution effect?

At 50 hrs/wk., the new wage rate is the same as the old ($8/hr).

=> no substitution effect!

Px/Py = 8

Pat divides a monthly income of $1800 between consumption of food (X) and consumption of all other goods (Y). Pat’s preferences can be described by the following utility function:

U = X2Y

Originally, the price of food is $1 and the price of all other goods is $2. Then the price of food rises to $2.

Because the relative price of food has increased, Pat will consume less food (and more of all other goods). This the substitution effect. But because Pat is now relatively poorer (her purchasing power has decreased), she will consume less of both goods. This is the income effect..


Y

X

900

Y**=300

X**= 600 1200 1200

Because the relative price of food has increased, Pat will consume less food (and more of all other goods). This the substitution effect.


S

S

Y

X

900

Y**=300

X**= 600 1200 1200

U = XYBut because Pat is now relatively poorer (her purchasing power has decreased), she will consume less of both goods. This is the income effect.

In this case, the 2 effects are equal and opposite for Y, additive for X.


Y

X

900

Y**=300

X**= 600 1200 1200

The move from A to B is the substitution effect;B to C is the income effect.

B is a point on the original indifference curve, tangent to

the new budget constraint, indicating the bundle the

consumer would choose at the new prices.


AB

C

S

I

I

S

Y

X

900

Y**=300

X**= 600 1200 1200

U = X2Y

We are looking for a point on the indifference curve that includes

Y = 300, X = 1200, for which MRS = 1 (the new price ratio):

At point B, MRS = 2Y/X = 1=> X = 2Y.

Also, Ua = Ub = 432,000,000

U = X2Y4Y3 = 432,000,000

Y3 = 108,000,000Yb = 476; Xb = 952


AB

C

S

I

I

S

Y

X

900

Y**=300

X**= 600 1200 1200

U = X2Y4Y3 = 432,000,000

Y3 = 108,000,000Yb = 476; Xb = 952

So the substitution effect is a decrease in X of 248 and an

increase in Y of 176.

The income effect is a decrease in X of 352

and a decrease in Yof 176.


AB

C

S

I

I

S

Y

X

900

Y**=300

X**= 600 1200 1200

How much would Pat be willing to pay to avoid this price increase?


Y

X

900

Y**=300

X**= 600 1200 1200

To calculate this amount, start by finding the minimum

income Pat needs to purchase a bundle on the new indifference curve.


Y

X

900

Y**=300

X**= 600 1200 1200

The difference between the market price of this bundle

and her income ( = 1800) is the amount she’d be willing

to pay to avoid the price increase. We call this the

equivalent variation measure of utility loss.


Normal & Inferior Goods

X

Y

Income-Expansion Path

Normal Good

For most goods, the quantity consumed will increase as income increases.

We call these normal goods. Y = f(X)

optimal ratio


XX

Y Income

Engels Curve

Normal Good


X = f(I)


X

Y

Inferior Good

For some goods, consumption will decrease at higher levels of income (e.g., hamburger).

We call these inferior goods.



XX

Y Income

Engels Curve

Inferior Good



X

Y

Normal Good

BA

B

A

X

Y

SS

Px = 1Px = 2

Px increases from $1 to $2.

The movement from A to B is the

substitution effect.


X

Y

Normal Good

BA

Inferior Good

B

A

For both normal and inferior goods, the substitution effect is negative: consumption will increase as price decreases.

X

Y

SS

Px = 1Px = 2

Px increases from $1 to $2.


X

Y

Normal Good

BA

C

Inferior Good

B

A

C

For normal goods the income effect is positive, and for inferior goods it is negative.

II

Y

X


X

Y

Giffen Good

BA

C

C

A

For some inferior goods, the income effect is so large it outweighs the substitution effect (eg., ?).

S

I

Px

2

1

X

Px = 1

Px = 2

… giving rise to a upward sloping

demand curve.

Do any of these cases violate the assumptions of well-behaved preferences that we look at last time?

No. Well-behaved preferences can give rise to all sorts of demand curves (depending on income and prices).


Y

X

900

Y**=300

400 1200

Consumer DemandU = X2YI = 1800; Py = 2

Px*** = $3 Y*** = 300, X*** = 400.

Consumer Demand

XX

Y

:

Px

Px = 3 2 1 400 600 1200

Demand Curve

3

2

1

X = f(Px)

400 3

600 2

1200 1

U = X2YI = 1800; Py = 2

Find the equation for the demand

curve.

Consumer Demand

XX

Y

U = X2YI = 1800; Py = 2

Px

Px = 3 2 1 400 600 1200

Demand Curve

3

2

1

X = f(Px)

400 3

600 2

1200 1

MUx = 2XY; MUy = X2

MRS = 2Y/X = Px/Py = Px/2

=> Y = (1/4)PxX

I = PxX + PyY

1800 = PxX + (2)(1/4)PxX

= (3/2)PxX

X = 1200/Px

Solve for Y & substitute

Consumer Demand

XX

Y

:

Px

Px = 3 2 1 400 600 1200

Demand Curve

3

2

1

Price-Consumption Curve

U = X2YI = 1800; Py = 2

In this case, consumption of Y is

unaffected by changes in Px. Cross-price

elasticity is zero.

Consumer Demand

XX

Y

Px

X

Px

Price-Consumption Curve Demand Curve

3

2

1

Or, cross-price elasticity can be

positive ...

… with a smaller response in demand.

Consumer Demand

XX

Y

:

Px

Px = 3 2 1 400 600 1200

Demand Curve

3

2

1

Price-Consumption Curve

U = X2YI = 1800; Py = 2

Px

X

Price Elasticity of Demand

Price Elasticity of Demand (Ep) Measures how sensitive quantity demanded is to changes in price.

Demand Equation: Qd = a – bP

Ep = (%Q)/(%P) = Q/Q)/(P/P) = Q/P(P/Q) = -b(P/Q)

Ep < 1 Inelastic: Total expenditure increases as price increases.

Ep > 1 Elastic: Total expenditure decreases as price increases.

Ep = 1 Unit Elastic: Total expenditure doesn’t change

Non-Price Determinants of Demand

What determines consumer demand?

• Preferences • Income• Prices of Related Goods

– Substitutes– Complements

Determinants of Price Elasticity

What determines price elasticity of demand?

• Substitutes (+)• Budget share

– Normal (+)– Inferior ( -)

• Short v long run (+) ex. Oil• Network Effects• Bandwagon and Snob Effects

normal goods have higher elasticities, because income effect reinforces substitution effect.

Next Time

6/29 Theory of the Firm

Pindyck, Ch 6.

Besanko ,Chs 6-7.

or

Varian, Chs 18, 20-21.

UNIT I:Theory of the Consumer

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