P denotes the ounces of perfume used and denotes …..., Aunt Joyce’s cross-price elasticity of...

Econ 3070 Prof. Barham

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Problem Set 3 Solutions 1. Aunt Joyce purchases two goods, perfume and lipstick. Her preferences are

represented by the utility function

( ) PLLPU =, ,

where P denotes the ounces of perfume used and L denotes the quantity of lipsticks used. Let PP denote the price of perfume, PL denote the price of lipstick, and I denote Aunt Joyce’s income. a. What is Aunt Joyce’s maximization problem?

MaxL,PU (P,L) = P.L

s.t. PLL+ PpP = I

b. What are the endogenous and exogenous variables? The endogenous variables are: P and L The exogenous variables are: PL, Pp, I

c. Derive her demand for perfume. Your answer should be an equation that

gives P as a function of PP , PL , and I. Determine this by using calculus and maximizing the objective function, do not use the tangency condition.

To find the demand for perfume we need to find the optimal amount of perfume, it will be a function of income and prices.

Step 1: Utility is a function of two variables. Since we don’t know how to maximize

when utility is a function of two variables we need to substitute for one of them. Since we are trying to find the demand for perfume, we will substitute for lipstick, so we are utility will be just a function of perfume.

Rewriting the budget constraint:

L = I ! PPPPL

Now we can substitute this into the utility function

MaxP

U(P)=P*I ! PPPPL

"

#$$

%

&''= P

IPL

"

#$$

%

&''!

PPPL

*P2"

#$$

%

&''


2

Step 2: Now we need to find the value of P that maximizes utility. We know that we

need the value of P where the slope of the utility curve is zero.

!U!P

= gives us the slope of the utility function with respect to P

!U!P

=IPL"PPPL

2P = 0

2PPPLP = I

PL

P= IPL

PL2PP

P = I2PP

Demand function for perfume

d. Derive her demand for lipstick. Your answer should be an equation that gives

L as a function of PP , PL , and I. To find the demand function for lipstick, we can repeat a similar exercise that we did in

part A. Or we can substitute our value of P back into the budget constraint. (Or recognize that the answer has to be symmetric).

PLL+ PPI

2PP

!

"##

$

%&&= I

PLL = I 'I2

PLL =I2

L = I2PL

demand function for lipstick


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d. Is lipstick a normal good? Draw her demand curve for lipstick when I = 200. Label the demand curve D1. Draw her demand curve for lipstick when I = 300 and label this demand curve D2.

A normal good is a good that a consumer purchases more of as income rises. Since Aunt Joyce’s demand for lipstick increases as I increases, lipstick is a normal good. The two demand curves are depicted in the figure below:

e. What can be said about her cross-price elasticity of demand of perfume with respect to the price of lipstick? In part a, we found that Aunt Joyce’s demand for perfume is given by

PPI

P2

= .

Since her demand for perfume does not depend on PL, Aunt Joyce’s cross-price elasticity of demand of perfume with respect to the price of lipstick is zero. That is, a 1% change in the price of lipstick generates a 0% change in the demand for perfume.

2. Ch 5, Problem 5.7 Karl’s preferences over hamburgers (H) and beer (B) are described by the utility function U(H,B)=min(2H,3B). His monthly income is I dollars, and he only buys these two goods out of his income. Denote the price of hamburgers by PH and of beer by PB.

a. Write out the consumer’s maximization problem. Remember this is a case of perfect complements so the indifference curves would be L shaped.

PL

L

D1 D2

10

10

12.25

12.25


4

MaxL,PU (P,L) = min(2H ,3B)

s.t. PHH + PBB = I

c. Draw a graph in H and B space of where the budget constraint and indifference curves must be for the utility maximization. Don’t use real numbers just draw what the basic shape of the curves look like.

b. Derive Karl’s demand curve for beer as a function of the exogenous variables (hint, you can’t maximize this function normally but you know that to be at an optimum 2H=3B). We are maximizing utility at the corner point of the L shaped indifference curve, so at that point Equation 1 from Utility function: 2H=3B of H=3/2B (I rewrote it this way as we are trying to find the demand curve for B. Equation 2: The budget constraint also has to hold PHH + PBB=I We have 2 equations and 2 unknowns. We can substitute equation 1 into equation 2. PH(3/2B) + PBB=I Now I can try to solve for B as a function of price and income which give us our demand curve.

B(32PH +PB ) = I

B(32PH +PB ) = I

B = I32PH +PB

IC

BL

Optimum is at the corner, so 2H=3B


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b. You can answer this just by looking at the demand curve. Because it has a larger

coefficient, the price of hamburgers affects the demand for beer more than the price of beer. A one dollar increase in HP decreases demand for beer more than a one dollar increase in BP .

3. Uncle Bob purchases two goods, tweed sport coats and bow ties. His preferences

are represented by the utility function

( ) 75.025.0, CBCBU = ,

where B denotes the number of bow ties purchased and C denotes the number of sport coats purchased. Let $25 be the price of bow ties and $60 be the price of sport coats. And finally, let I denote Uncle Bob’s income. a. Derive Uncle Bob’s Engel curve for bow ties. Your answer should be an

equation that gives B as a function of I. Uncle Bob’s maximization problem is:

1/ 4 3/ 4

,

B

( , )

. . PB C

C

Max U B C B C

s t B P C I

=

+ =

Step 1: Rewrite budget constraint. B

c

I P BCP−

= sub into utility function


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31 44

3 13 14 44 4

3144

3 14 4

3 11 34 44 4

( ) *

1 3 04 4

1 34 4

1 3 4 4

B

Bc

B B B

c c C

B B

c C

B

c

B B B

c c C

I P BMax U B BP

I P B I P B PU B BB P P P

I P B P BP P

I P BBP

I P B I P B P B BP P P

−−

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞− −∂= + − =⎜ ⎟ ⎜ ⎟

∂ ⎝ ⎠ ⎝ ⎠

⎛ ⎞−⎜ ⎟⎝ ⎠ =

⎛ ⎞−⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞− −=⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

c

B

c c

c B

1 3 4 4I 3

PP I 3P P

I B=P 2P

I B=4*25 100

B B

c C

B B

c C

c

I P B P BP PP B P BP P

B

P

I

⎛ ⎞−=⎜ ⎟

⎝ ⎠

− =

=

=

Therefore, Uncle Bob’s Engel curve for bow ties is given by

100I

B = .

b. Draw Uncle Bob’s Engel curve for bow ties on a graph with B on the

horizontal axis and I on the vertical axis.


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c. Are bow ties a normal good? What can be said about Uncle Bob’s income

elasticity of demand for bow ties? Bow ties are a normal good because the demand for bow ties increases as income increases. Since bow ties are a normal good, Uncle Bob’s income elasticity of demand for bow ties is positive. We can calculate the income elasticity of demand as follows:

1100100

1, =⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛==∈

II

BI

dIdB

IB .

So, Uncle Bob’s income elasticity of demand for bow ties is 1 – a 1% increase in his income leads to a 1% increase in his demand for bow ties.

I

B 1

100

Engel Curve


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4. Ch 5, Problem 5.9

Rick purchases two goods: food and clothing. He has a diminishing marginal rate of substitution of food for clothing. Let x denote the amount of food consumed and y the amount of clothing. Suppose the price of food increases from

1xP to

2xP . On a clearly labeled graph, illustrate the income and substitution

effects of the price change on the consumption of food. Do so for each of the following cases: a. Case 1: Food is a normal good.

Given the increase in the price of x, we expect to see the following effects:

Substitution Effect Income Effect

x ↓ ↓ y ↑ ↓

Because the price of x increased, x became relatively more expensive, and y became relatively less expensive. As a result, Rick substitutes away from x in favor of y. This is represented in the table by a down arrow for x and an up arrow for y in the substitution effect column. Moreover, the increase in the price of x reduced Rick’s purchasing power. Since x and y are both normal goods (x being a normal good is given by the problem, y being a normal good is assumed), the reduction in purchasing power causes Rick to purchase less of both x and y. This is represented in the table by the down arrows in the income effect column. The following diagram gives us a graphical representation of the information presented in the table:

The initial consumption bundle is represented by point A, which lies on the initial budget line BL1. The increase in the price of x causes the budget line to shift

y

x

• • A

B

• C

BL1 BL2


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inwards to BL2. The new consumption bundle is represented by point C. We then construct point B in order to separate the substitution effect from the income effect. The movement from A to B represents the substitution effect. Note that as suggested by the table, the movement from A to B shows x going down and y going up. The movement from B to C represents the income effect. Once again, note that as suggested by the table, the movement from B to C shows both x and y going down.

b. Case 2: The income elasticity of demand for food is zero.

In this case, we expect to see the following effects:


x ↓ ⎯ y ↑ ↓

Once again, because the price of x increased, Rick substitutes away from x in favor of y. This is represented in the table by a down arrow for x and an up arrow for y in the substitution effect column. However, the information in the income effect column has changed. Since the income elasticity of demand for x is zero, the reduction in Rick’s purchasing power has no effect on x. This is represented by the horizontal line for x in the income effect column. (The down arrow for y reflects the fact that we are continuing to assume that y is a normal good.) The following diagram gives us a graphical representation of the information presented in the table:

As before, the initial consumption bundle is represented by point A, and the new consumption bundle is represented by point C. Point B is constructed in order to separate the substitution effect from the income effect.

y

x

•

•

A

B

• C

BL1 BL2


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The movement from A to B represents the substitution effect, and as suggested by the table, we observe x going down and y going up. The movement from B to C represents the income effect. As suggested by the table, we observe no change in x since the income elasticity of demand for x is zero. On the other hand, we do observe y going down since y is assumed to be a normal good.

c. Case 3: Food is an inferior good, but not a Giffen good.

In this case, we expect to see the following effects:


x ↓ ↑ y ↑ ↓

Once again, because the price of x increased, Rick substitutes away from x in favor of y. Moreover, the reduction in Rick’s purchasing power reduces his demand for y (a normal good). What is new is that x is an inferior good; that is, the reduction in Rick’s purchasing power causes Rick to purchase more x. This is represented by the up arrow for x in the income effect column. The following diagram gives us a graphical representation of the information presented in the table:

As before, the initial consumption bundle is represented by point A, and the new consumption bundle is represented by point C. Point B is constructed in order to separate the substitution effect from the income effect. The movement from A to B represents the substitution effect, and as suggested by the table, we observe x going down and y going up. The movement from B to C represents the income effect. As suggested by the table, we observe y going up (since y is assumed to be a normal good) and x going down (since x is assumed to be an inferior good).

y

x

•

•

A

B

•

C

BL1 BL2


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The last thing to take note of is that the diagram indicates that x is not a Giffen good. The diagram indicates that the income effect is not strong enough to dominate the substitution effect; that is, the increase in x going from B to C is smaller than the decrease in x going from A to B.

The last thing to take note of is that the diagram indicates that x is a Giffen good. The diagram indicates that the income effect dominates the substitution effect; that is, the increase in x going from B to C is larger than the decrease in x going from A to B.

5. Ch 5, Problem 5.11 ed. 5. Ginger’s Utility function is U(x,y)=x2y. She has income I=240 and faces prices Px=$8 and Py=$2. Part A. The maximization problem is:

2

, ( , )

. . 8 2 240x yMax U x y x y

s t X Y

=

+ =

Part B. Using the budget constraint we rewrite the maximization problem in terms of one variable.

2

2

( ) (120 4 )

240 12 0

* 20* 120 4(20) 40

XMaxU x x x

U x xxxy

= −

∂= − =

∂=

= − =

Her optimal bundle is (x,y)=(20,40) and utility is 16,000 Part C. To be just as well off as before, her utility must be 16,000. We will use this fact later to help us. But first we can just set this up as a usual maximization problem like in part A. I’ll start after I sub in the budget constraint since the steps are just the same as in part A. I will change the price of y to be $8, and will use Px in the place of the price of X


12

2

2

( ) (30 )8

360 08

860 1603

x

x

x

x

P xMaxU x x

P xU xx

P x

= −

∂= − =

∂

= =

Now we can sub PxX into the budget constraint to find Y. 160+8Y=240. So Y* =10. To find out what X is we can use the fact that utility must equal 16,000 U(x,y)=16,000=x210 X*=40. We know PxX=160 and X* is 40, so Px=4 if Ginger is just as well off as before the price change. 6. Ch 5, Problem 5.18 ed 5. The demand function for Kendamas is given by D(P)=16-2P (note that D(P) is just a way of saying writing the demand function where the quantity demanded is a function of P which you are used to seeing as QD. Compute the change in consumer surplus when the price of a widget increases from $1 to $3. First show your results graphically. First lets graph this demand curve it is linear, so the slope is -2 If P=0, Q=16 If Q=0 then P= ? just write the inverse demand curve P=(16-QD)/2 so P=8 If P=1 then QD or D(1) =14 If P=3 then QD or D(1) =10 For price of a widget equal to $1 consumer surplus is

CS$1 = ½ · (8 – 1) · D(1) = ½ · 7 · 14 = 49. When price is equal to $3 consumer surplus is

CS$3 = ½ · (8 – 3) · D(3) = ½ · 5 · 10 = 25.

So the change in consumer surplus is 49-25=24 or Area EBDC


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7. Ch 5, Problem 5.26 ed. 5. Suppose that Bart and Homer are the only people in Springfield who drink 7-UP. Moreover, their inverse demand curve for 7-UP are:

Bart: P=10-4QB Homer: P=25-2QB

Neither one can consume a negative amount. Write down the market demand curve for 7-UP in Springfield, as a function of all possible prices. Bart will only consume when the price is less than 10. To see this, see what the price has

to be if QB is zero. Therefore his demand curve for 7-UP is ,4

10 PQB−

= when P<10

and zero otherwise. Homer will only consume if the price is less than 25 so his demand curve is

,2

25 PQH−

= when P < 25 and zero otherwise.

Therefore the market demand curve for 7-UP as a function of all possible values of price is:

$8

$1

$3

D(P)

Area of ABE triangle CS when P = $3 is 25

D(P) = 16 – 2P

A

C D

B E

Area of ACD triangle CS when P = $1 is 49

P


14

10 if ,4

360

2510 if ,2

2525 if ,0

<−

=

<<−

=

>=

PPQ

PPQ

PQ

M

M

M

P denotes the ounces of perfume used and denotes …..., Aunt Joyce’s cross-price elasticity of...

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