EC 203 - INTERMEDIATE MICROECONOMICS Bo gazi˘ci...

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EC 203 - INTERMEDIATE MICROECONOMICS Bo˘ gazi¸ciUniversity - Department of Economics Fall 2019 Problem Set 2 - Solutions 1. Consider a consumer who wants to consume only two commodities and has an income of $100. Assume the price of good 1 is $10 per unit and the price of good 2 is $20 per unit. Now, inflation causes the price of good 1 to increase to $20 per unit, while the price of good 2 increases to $25 per unit. On the other hand, the consumer also gets a raise of $100 (so her new income is $200). Is she better off or worse off? Solution: Originally, the consumer’s budget constraint is 10x 1 + 20x 2 100. The budget line has horizontal intercept m p 1 = 100 10 = 10 and vertical intercept m p 2 = 100 20 =5. After the change, her budget constraint becomes 20x 1 + 25x 2 200, and so the new budget line has horizontal intercept m p 1 = 200 20 = 10 (the same as before) and vertical intercept m p 2 = 200 25 = 8 (higher than before), and so the consumer’s budget set has grown: she can now afford bundles she previously could not, and she can still afford all bundles she previously could. Therefore she is better off. 2. Suppose prices are P x = 2, P y = 1 and income is m = 100. (a) Show the change in the budget set if price of x decreased and price of y increased such that P 0 x =1 and P 0 y = 2. Solution: The new budget line is shown as the blue line in the figure belove. 100 50 50 100 x y -2 -1/2 (b) Suppose that the government taxes (at the new set of prices P 0 x = 1 and P 0 y = 2) the consumer an extra 0.50 dollar for each unit of x he buys beyond 70 units. That is, no tax is collected for units x< 70, and 0.50 dollar tax for each x 70. Show the new budget set. 1

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Page 1: EC 203 - INTERMEDIATE MICROECONOMICS Bo gazi˘ci Universityweb.boun.edu.tr/muratyilmaz/my/EC203_files/EC203 - Problem Set 2... · Draw his budget set. Solution: Now he has 100 coupons

EC 203 - INTERMEDIATE MICROECONOMICS

Bogazici University - Department of Economics

Fall 2019

Problem Set 2 - Solutions

1. Consider a consumer who wants to consume only two commodities and has an income of $100. Assume

the price of good 1 is $10 per unit and the price of good 2 is $20 per unit. Now, inflation causes the price

of good 1 to increase to $20 per unit, while the price of good 2 increases to $25 per unit. On the other

hand, the consumer also gets a raise of $100 (so her new income is $200). Is she better off or worse off?

Solution: Originally, the consumer’s budget constraint is

10x1 + 20x2 ≤ 100.

The budget line has horizontal intercept mp1

= 10010 = 10 and vertical intercept m

p2= 100

20 = 5. After the

change, her budget constraint becomes

20x1 + 25x2 ≤ 200,

and so the new budget line has horizontal intercept mp1

= 20020 = 10 (the same as before) and vertical

intercept mp2

= 20025 = 8 (higher than before), and so the consumer’s budget set has grown: she can

now afford bundles she previously could not, and she can still afford all bundles she previously could.

Therefore she is better off.

2. Suppose prices are Px = 2, Py = 1 and income is m = 100.

(a) Show the change in the budget set if price of x decreased and price of y increased such that P ′x = 1

and P ′y = 2.

Solution: The new budget line is shown as the blue line in the figure belove.

100

50

50 100

x

y

−2

−1/2

(b) Suppose that the government taxes (at the new set of prices P ′x = 1 and P ′y = 2) the consumer an

extra 0.50 dollar for each unit of x he buys beyond 70 units. That is, no tax is collected for units

x < 70, and 0.50 dollar tax for each x ≥ 70. Show the new budget set.

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Solution: P newx = 1 for the first 70 units, there is no tax. For each additional good x purchased

after 70 units, consumers pay $0.50 per unit tax. There is no tax on the second good. This is shown

in the following figure.

100

50

70 90

x

y

−1/2

−3/4

slope

slope

(c) Suppose that on top of part (b) government subsidizes the consumer with a 0.40 dollar for each unit

of y he buys beyond 40 units. That is, there is no subsidy for y > 40, and consumer pays 0.40 dollar

less for each y ≥ 40. Show the new budget set.

Solution: P newy = 2 for the first 40 units, there is no subsidy. For each additional good y purchased

after 40 units, consumers pays $1.60. This is shown in the following figure.

100

70 90

x

y

−1/2

−3/4

slope

slope

40

20

52.5slope−5/8

3. Consider a consumer who is choosing how many of two goods to buy: Footballs and cricket balls. The

consumer has an income of $20, and the cost of a football is $4 and a cricket ball is $2.

(a) Write down the equation for the consumer’s budget constraint and graph it.

Solution: Let F be the amount of footballs and C be the amount of cricket balls. The budget is

4F + 2C = 20, that is 2F + C = 10.

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F

C

10

5

− pF

pC= −2

(b) The government decides that football is evil and needs to be taxed. They introduce a 50% tax on

each football sold. Rewrite and re-graph the budget constraint.

Solution: The new cost of buying a football is $6 after the tax. Here the budget line is 6F+2C = 20,

or 3F + C = 10.

F

C

10

10/3

− pF

pC= −3

(c) A new government is elected that hates all sports. They now tax both footballs and cricket balls at

50%. What does the budget constraint look like now?

Solution: Now the cost of buying a cricket ball is $3. The budget line now is 6F + 3C = 20.

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F

C

20/3

10/3

− pF

pC= −2

(d) Due to a threat of revolt amongst sports fans, the government hands out a subsidy of $10 to the

consumer. What does their new budget constraint look like?

Solution: The budget now is 6F + 3C = 30.

F

C

10

5

(e) Revolution comes, and all taxes and subsidies are abolished. Even better, the consumer finds a new

shop that offers bulk discounts. In this shop, footballs cost $4 each if you buy 3 or less. However,

the cost of any additional football after 3 is $2. What does the budget set look like now?

Solution: PF = $4 for the first 3 units. P discountF = $2 for any additional football after 3 units.

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F

C

10

73

4

− pF

pC= −2

− pF

pC= −1

4. William has $300 to spend on food and clothing. The price of food is $2 per unit and the price of clothing

is $1 per unit.

(a) Write down the equation for William’s budget constraint and graph it.

Solution: Let F denote the amount of food, and C denote the amount of closing. The budget line

is 2F + C = 300.

F

C

300

150

− pF

pC= −2

(b) Suppose that the government gives William 100 free coupons as food stamps (assuming that William

can get free unit of food for each coupon) and strictly enforce a rule which prohibits William from

selling his food stamps to someone. Draw the new budget set.

Solution: For the first 100 units of Food, William can use coupons, and resale is prohibited.

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F

C

300

250

− pF

pC= −2

100

(c) Suppose that the rule prohibiting the food stamps sale is difficult to enforce and William can trade

each food stamp in black market for $0.50. Draw his budget set.

Solution: Now he has 100 coupons for food, but he can sell it in the black market at $0.5 per

coupon.

F

C

300

250

− pF

pC= −2

100

350 − pF

pC= −1/2

(d) Suppose that in addition to 100 units of food stamps, government also gives 50 free coupons for

clothing (each coupon can be used for a free unit of clothing). Assume that trading both coupons

in black market can be prohibited. Draw the budget set.

Solution: He has coupon for both C and F (trading in black market is prohibited).

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F

C

250

− pF

pC= −2

100

350

50

5. Mark has preferences for guns and banjos represented by the utility function u(xg, xb) = xg +√xb.

(a) Write down the equation for an indifference curve, that is, for some utility level u, write down the

number of guns as a function of the number of banjos.

Solution: u = xg + (xb)12 , that is xg = u− (xb)

1/2

(b) Find the marginal rate of substitution of banjos for guns.

Solution: We have MUb = 12x−1/2b and MUg = 1, then we get MRSbg = MUb

MUg= 1

2x−1/2b .

(c) Determine whether these preferences are convex, that is, determine whether there is diminishing

MRS property.

Solution: Convexity of preferences is equivalent to the diminishing MRS property. You can check

whether MRS is decreasing, as xb increases and xg decreases: MRS is actually decreasing since12x−1/2b is decreasing in xb. Alternatively, you can check the second order derivative of the indifference

curve you found:dxg

dxb= −1

2x−1/2b , then

d2xg

dx2b

= 14x−3/2b . Second order derivative is positive, so the

indifference curve has a convex shape.

(d) Show that, for a budget constraint pbxb + pgxg = m, at any point of tangency between budget line

and indifference curves, it must be the case that xb = (1/4)(pgpb

)2.

Solution: Tangency is equivalent to MRSbg = pbpg

. Then 12x−1/2b = pb

pg, that is, x

1/2b =

pg2pb

, which

implies, xb = (1/4)(pgpb

)2.

(e) What is the optimal consumption bundle, if pg = $2 , pb = $1 and m = $8?

Solution: Since this utility function is well behaved (it is monotone, represents convex preferences

since MRS is diminishing, and continuous and differentiable) at an interior solution tangency should

hold. From the tangency condition above, plugging prices, we get x∗b = (1/4)(2/1)2 = (1/4)4 = 1.

Then, from the budget equation, we get 1 · 1 + 2xg = 8, that is, x∗g = 7/2. Since both x∗b = 1 and

x∗g = 7/2 are positive, we have an interior solution at (x∗b = 1, x∗g = 7/2).

(f) What is the optimal consumption bundle if pg = $5 , pb = $1 and m = $4?

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Solution: From the tangency condition, we get xb = 25/4. However, this together with budget

equation implies a negative amount of guns. Hence the best solution is to buy no guns and buy as

much banjos as possible. Mark spends all income on banjos: x∗b = mpb

= 4, and none spent on guns,

x∗g = 0.

You can also see this through comparing MUbpb

andMUg

pg. We have MUb = 1

2x−1/2b and MUg = 1,

that is, MUbpb

=12x−1/2b1 = 1/2

x1/2b

andMUg

pg= 1/5. When spending all income on banjos, the most he

can buy is xb = 4. With this amount MUbpb

= 1/2

41/2= 1/4 which is larger than 1/5. Thus, he would

like to get more xb, but xb = 4 is the limit. Thus, this is the solution, x∗b = 4, together with no guns,

x∗g = 0.

6. Suppose a consumer has a utility function u(x1, x2) = xa1x1−a2 , for goods x1 and x2.

(a) Find her demand for x1 and x2 in terms of her income, m, prices, p1, p2, and the positive constant

0 < a < 1. (Although we derived this in class, I included this problem here, since it may be useful

for the rest of the problem set.)

Solution: This is a Cobb-Douglas utility function, which is well-behaved (it is increasing in x and y,

that is, it is monotone. It has DMRS, that is, it represents convex preferences. It is also continuous

and differentiable. You should be able to check monotonicity and DMRS). And the budget line is

p1x1 + p2x2 = m. We can directly use the tangency condition:

MRS = MU1MU2

=axa−1

1 x1−a2

(1−a)xa1x

−a2

= a1−a

x2x1

. And the price ratio is simply p1p2

. Thus, we get

a

1− a

x2x1

=p1p2

Then, we have x2 = x11−aa

p1p2

. Now plug this into the budget line and solve for x1.

p1x1 + p2x11− a

a

p1p2

= m

that is, p1x1 + x11−aa p1 = m, which implies x1[p1 + 1−a

a p1] = m, or x1p1[1 + 1−aa ] = m, that is,

x1p1[1a ] = m. Thus, we get

x∗1 = am

p1

Plugging this into the equation x2 = x11−aa

p1p2

, we get x2 = amp1

1−aa

p1p2

, that is,

x∗2 = (1− a)m

p2

(b) What share of her budget does she spend on x1, and what share on x2?

Solution: The income share she is spending on good 1 isp1x∗

1m = a, and the income share she is

spending on good 2 isp2x∗

2m = (1− a).

7. Suppose that the preferences of a typical household for quantities of electricity (E) and gasoline (G) are

given by u(E,G) = aln(E)+(1−a)ln(G), where 0 < a < 1. Suppose the prices of gasoline and electricity

in the units provided are both $1/unit and the household has an income of $100. Suppose in addition, the

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government has chosen to ration electricity by allowing a maximum consumption of 50 units of electricity,

that is, Emax = 50.

(a) If a = 1/4, find the optimal consumption bundle of gasoline and electricity.

Solution: The utility function is a positive monotonic transformation of V (E,G) = EaG1−a. Thus

the optimal choice bundle can be found through E∗ = a mpE

and G∗ = (1−a) mpG

, which imply E∗ = 25,

G∗ = 75 (you should be able to solve this without applying these formulas).

(b) If a = 3/4, find the optimal consumption bundle of gasoline and electricity.

Solution: Calculation gives E = .751001 = 75, but this is not allowed. Thus consumers will consume

as much E as possible, that is the maximum possible: E∗ = 50, and the rest will be spent on gasoline

G∗ = 50.

8. Suppose that a consumer consumes only food (f) and entertainment (e) where pf = $5, pe = $10 and

m = $600. Suppose that the consumer has to have a minimum of 50 units of f , and a maximum of 20

units of e.

(a) Draw the budget set.

Solution:

20

50 80 120

E

F

(b) Suppose the utility function is u(f, e) = f · e. Find the optimal consumption bundle.

Solution: This is a monotonic transformation of u(f, e) = f1/2 · e1/2. By Cobb-Douglas solution

above, e = mpe

12 = 30 and f = m

pf12 = 60. But, e = 30 is not allowed. Thus, the consumer will

consumer as much e as he can, the maximum allowed, that is, e∗ = 20. Then, from the budget

equation we get f∗ = (600− (10 · 20))/5 = 80. You can see the optimal bundle in the graph below.

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Page 10: EC 203 - INTERMEDIATE MICROECONOMICS Bo gazi˘ci Universityweb.boun.edu.tr/muratyilmaz/my/EC203_files/EC203 - Problem Set 2... · Draw his budget set. Solution: Now he has 100 coupons

20

50 80 120

E

F

9. Suppose that a consumer only consumes good 1 and good 2 under the following prices and income:

p1 = $2, p2 = $1 and m = $100. Consumer’s preferences can be represented by the utility function,

u(x1, x2) = 3x1 + x2.

(a) Find the optimal bundle.

Solution: This is a perfect substitutes case. MU1 = 3 and P1 = 2, and MU2 = 1 and P2 = 1. Thus

MRS12 = MU1MU2

= 31 = 3 > P1

P2= 2

1 = 2, that is, MU1p1

> MU2p2

. Thus, this consumer will spend all

income on good 1, which means x∗1 = 50, x∗2 = 0.

(b) Now suppose that the consumer receives a coupon of $20 which can be spent only in good 2. Draw

the new budget constraint and find the new optimal consumption bundle.

Solution: The budget set is drawn below. Since $20 coupon can only be spent on good 2, he will

use the coupon to buy good 2, and use all income to buy good 1, as the prices did not change, that

is MU1p1

> MU2p2

is still the case. Thus we get, x∗1 = 50, x∗2 = 20 as the optimal consumption bundle.

120

20

50 x1

x2

10. Pamir spends most of his time in Just Coffee shop. Pamir has $12 a week to spend on coffee and muffins.

Just Coffee sells muffins for $2 each and coffee for $1.2 per cup. Pamir consumes xc cups of coffee per

week and xm muffins per week. His utility function for coffee and muffins is u(xc, xm) = xkcxkm, where

k > 0.

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(a) Find Pamir’s optimal consumption bundle. Does it depend on k?

Solution:

A positive monotonic transformation of u(xc, xm) = xkcxkm gives u(xc, xm) = x

1/2c x

1/2m . Then, using

the solution we found for the Cobb-Douglas utility function, we get

x∗c =1

2

m

pc=

1

2

12

1.2= 5

x∗m =1

2

m

pm=

1

2

12

2= 3

The solution does not depend on k!

(b) Now Just Coffee has introduced a frequent-buyer card: For every five cups of coffee purchased at

the regular price of $1.2 per cup, Pamir receives a free sixth cup. Draw Pamir’s new budget set.

Solution:

6

3

5 6 11 12

xm

xc

(c) With frequent-buyer card, find the new optimal consumption bundle.

Solution: (5,3) is the optimal point without the frequent-buyer card. With the frequent-buyer

card, the upper portion of the budget line (the portion above (5,3)) cannot be optimal because, all

those bundles were affordable but Pamir chose (5,3), thus they all are on a lower indifference curve

than (5,3) is on. Thus, Pamir has two possibilities for optimal choice: it might be at the kink (6,3),

or it might be in the portion of the budget line to the right of the kink. Consider the modified m:

Pamir now has m = 1.2 · 6 + 3 · 2 = 13.2. The Cobb-Douglas solution implies xc = 12mPc

= 1213.21.2 5.5,

and xm = 12

mPm

= 1213.22 = 3.3, however this is not feasible as this bundle lies above the flat portion

of the budget set. Thus the optimal point should be at the kink (6, 3). Then x∗c = 6 and x∗m = 3.

11. A consumer can buy two goods: good 1 denoted by x1 and good 2 denoted by x2. Her utility function is

given by u(x1, x2) = x1x2x1+x2

, and p1 and p2 are the prices of good 1 and good 2, respectively, and m is the

consumer’s income level.

(a) Is this utility function well-behaved? (Hint: It is continuous and differentiable. How about mono-

tonicity and DMRS?)

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Solution: Both marginal utilities are positive: MU1 =x22

(x1+x2)2, and MU2 =

x21

(x1+x2)2. Thus,

u(x1, x2) is increasing in both x1 and x2. It is monotone. For DMRS, find MRS first: using

MU1 =x22

(x1+x2)2and MU2 =

x21

(x1+x2)2we get MRS12 =

x22

x21

= (x2x1

)2. This is decreasing as x1

increases and x2 decreases, implying DMRS. Thus this is well-behaved.

(b) Solve for her demand for x1 and x2 both as a function of p1, p2 and m, that is, x1(p1, p2,m) and

x2(p1, p2,m).

Solution: Since u(·) is well-behaved, at an interior solution tangency should hold. Equating MRS

to price ratio gives: MRS12 = (x2x1

)2 = p1p2

. Then x2 = x1√

p1p2

. Plugging this into the budget line,

p1x1 + p2x2 = m, and solving for x1 we get,

x∗1(p1, p2,m) =m

p1 +√p1p2

Then solving for x2, we get

x∗2(p1, p2,m) =m

p2 +√p1p2

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