Intermediate Microeconomics

Final Exam

This is an open book, open notes exam. You have the entire period (195 minutes) to finish this exam. There arefour (4) questions. You must answer all parts of all questions. Each question has several parts, with the later partsof each question being more difficult. Show your work as much as possible. If I can see the process you used to reachyour answer, you may receive partial credit even if the answer is incorrect. Manage your time well and do not getstuck on any one question. It is strongly suggested that you read the entire exam before starting so that you can leavemore difficult problems for later in the exam. At the end of the exam, you MUST RETURN THIS SHEET with youranswers.

Please answer ALL the following FOUR questions

1. (25 points total) Suppose that a firm’s technology can be expressed as the following production function q =f(K,L) = 1

4K2L2. The price of a unit of labor is denoted w and the price of a unit of capital is denoted r.

First, consider the short-run decisions of this firm when w = 10, r = 5, and capital is fixed such that K = 5.

a) Define and compute the following as functions of the quantity of output q (8 points):

• Short-run fixed costs FCSR (and average fixed costs AFCSR).

• Short-run variable costs V CSR (and average variable costs AV CSR).

• Short-run total costs TCSR (and average total costs ATCSR).

• Short-run marginal costs MCSR.

Solution:

In the short-run, the firm has fixed capital of K = 5. As a result, q = 14 (5)2L2 = 25

4 L2. So for all short-run cost

functions, L(q) = 25

√q.

FCSR = rK̄ = 25 AFCSR = 25q

V CSR = wL(q) = 4√q AV CSR = 4√

q

TCSR = wL(q) + rK̄ = 4√q + 25 ATCSR = 25

q + 4√q

MCSR = 2√q

Now, consider the long-run decisions of this firm when w = 10, r = 5, but capital is now variable.

a) Define an isoquant and draw this firm’s isoquants for q = 2 and q = 4. (2 points)

Solution:

An isoquant is a curve connecting all of the combinations of inputs that produce a fixed level of output.

b) Is this firm’s production technology increasing, constant, or decreasing returns to scale? Explain. (2 points)

Solution:

f(δL, δK) =1

4(δK)2(δL)2 = δ4

1

4K2L2 > δf(L,K), for δ > 1

The production technology is increasing returns to scale.

c) Find the long-run expansion path that indicates the optimal combinations of capital and labor. (5 points)

Solution:

1

The long run expansion path solves MRTS = MPL

MPK= w

r

MPL

MPK=

1/2K2L

1/2KL2=K

L

K

L=

w

r= 2

K = 2L

d) Compute the following as functions of the quantity of output q (8 points):

• Long-run total costs TCLR.

• Long-run average total costs ATCLR.

• Long-run marginal costs MCLR.

Solution:

First, solve for labor and capital as functions of output.

q =1

4(2L)2L2

q = L4

L = q14

K = 2q14

Long run costs are given by

TC = wL(q) + rK(q)

= 10q0.25 + 10q0.25

= 20q0.25

ATC = 20q−0.75

MC = 5q−0.75

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2. (25 points total) Ace Armstrong and Butch B. Bradley are the only two inhabitants on a remote desert island.Unfortunately for both, there is no food or water source on the island itself. Fortunately for them both, two cargocontainers— one full of Hostess snack cakes and one full of Evian spring water— washed up on the island. Eachcargo container 100 units of each good inside. Ace’s initial endowment is 20 units of Hostess cakes and 80 bottlesof water. Butch’s initial endowment is 80 units of Hostess cakes and 20 bottles of water.

Denote units of Hostess cakes as H and bottles of spring water as W . Ace’s utility function over snack cakesand spring water is given as UA = H0.5

A W 0.5A . Butch’s utility function over snack cakes and water is given as

UB = H0.5B W 2

B . They are free to trade with each other, provided that the trades are voluntary, i.e. both gain fromthe exchange. In this environment, answer the following questions:

a) Is the initial allocation Pareto optimal? (5 points)

Solution:

For further reference, we can calculate the equation for the set of Pareto efficient allocations (the “contractcurve”) by equating MRSA with MRSB .

MRSA =WA

HA

MRSB =WB

4HB

WA

HA=

100−WA

400− 4HA

400WA − 4HAWA = 100HA −HAWA

400WA − 3HAWA = 100HA

WA =100HA

400− 3HA

Since 100(20)400−3(20) = 2000

340 6= 80 at the initial allocation HA = 20,WA = 80, it can not be Pareto efficient.

b) Show that Ace having 80 Hostess cakes and 50 bottles of water (with Butch having the remaining goods) is aPareto efficient allocation. (5 points)

Solution:

WA =100(80)

400− 3(80)

= 50

Since 100HA

400−3HA= WA at this allocation, it is Pareto efficient.

c) Call the allocation where Ace and Butch each have half of each good (50 units of Hostess cakes and 50 bottlesof water) the equitable allocation. Would each be willing to give up their endowments to move to the equitableallocation? Is the equitable allocation Pareto optimal? (5 points)

Solution:

WA =100(50)

400− 3(50)

=2

250= 20 6= 50

Since 100HA

400−3HA6= WA at this allocation, it is not Pareto efficient.

UA(50, 50) =√

502 = 50 > UA(20, 80) =√

80(20) = 40

UB(50, 50) = 5052 > UB(80, 20) =

√80(20)2

Both people achieve higher utility by moving to the equitable allocation from their endowments.

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d) Assume that the price of Hostess cakes is pH = 1. Calculate the competitive equilibrium price of water pW suchthat the markets for both Hostess cakes and Evian water clear (i.e. total amount of each good is exactly equalto the sum of quantities demanded by each individual). (10 points)

Solution:

First, calculate the demands of each individual. Taking advantage of the Cobb-Douglas utility functions, we canwrite demand functions as income shares divided by prices.

Ace is endowed with 20 Hostess cakes, each worth pH = 1, and 80 bottles of water, each worth pW . He spends halfon each good, so his demand functions are written

H∗A =1

2(20 + 80pW ) = 10 + 40pW

W ∗A =1

2pW(20 + 80pW ) =

10

pW+ 40

Butch is endowed with 80 Hostess cakes, each worth pH = 1, and 20 bottles of water, each worth pW . He spendsone-fifth on Hostess cakes and four-fifths on water, so his demand functions are written

H∗B =1

5(80 + 20pW ) = 16 + 4pW

W ∗B =4

5pW(80 + 20pW ) =

64

pW+ 16

At a competitive equilibrium, the total supply of each good has to be equal to the total demand.

Market clearing for Hostess cakes:

100 = 10 + 40pW + 16 + 4pW

44pW = 74

pW =37

22= 1.68

Market clearing for water:

100 =10

pW+ 40 +

64

pW+ 16

74

pW= 44

pW =37

22= 1.68

Equilibrium allocation:

H∗A =850

11= 77.27

W ∗A =1700

37= 45.95

H∗B =250

11= 22.73

W ∗B =2000

37= 54.05

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3. (25 points total) Samsung and LG produce liquid crystal display (LCD) panels for televisions sets. Suppose(for now) the panels they produce are identical and Samsung and LG compete in the quantities of display panelsthey produce. Samsung can produce a panel for a marginal cost of $50 per unit and LG can produce a panelfor a marginal cost of $80 per unit. Demand for LCD panels is given by the following inverse demand functionp = 500− 0.01Q.

a) If the two firms must produce their quantities simultaneously, what is the Cournot-Nash equilibrium in thismarket? In other words, what quantity does each firm produce and what is the market price? What is eachfirm’s profit? (5 points)

Solution:

Samsung’s reaction function is given by setting MR = MC:

500− 0.01qL − 0.02qS = 50

0.02qS = 450− 0.01qL

qS = 22500− 0.5qL

LG’s reaction function is given by setting MR = MC:

500− 0.01qS − 0.02qL = 80

0.02qL = 420− 0.01qS

qL = 21000− 0.5qS

Solving for qS and qL:

qL = 21000− 0.5qS

qL = 21000− 0.5(22500− 0.5qL)

0.75qL = 9750

qL = 13000

qS = 16000

p = 210

πS = 2, 560, 000

πL = 1, 690, 000

b) Suppose instead that Samsung chooses its quantity first, followed by LG. What is the Stackelberg equilibriumin this market? In other words, what quantity does each firm produce and what is the market price? What iseach firm’s profit? (5 points)

Solution:

LG’s reaction function is given by setting MR = MC:

500− 0.01qS − 0.02qL = 80

0.02qL = 420− 0.01qS

qL = 21000− 0.5qS

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Samsung’s profit function can now be written in terms of just qS :

π = (500− 0.01qS − 0.01(21000− 0.5qS))qS − 50qS

= (290− 0.005qS)qS − 50qSdπ

dqS= 290− 0.01qS − 50 = 0

0.01qS = 240

qS = 24000

qL = 9000

p = 170

πS = 2, 880, 000

πL = 810, 000

Suppose now Samsung focuses on LCD panels while LG focuses on plasma panels. As a result, the two manufacturersproduce imperfect substitutes. Each firm’s costs are the same as above. Samsung’s demand function is nowqS = 50000− 100pS + 100pL and LG’s demand function is now qL = 50000− 100pL + 100pS .

a) What is LG’s optimal price for any given price Samsung charges pS , i.e. LG’s reaction function? (5 points)

Solution:

LG’s reaction function is given by differentiating the profit function with respect to pL and setting equal to zero:

π = pLqL − 80qL = (50000− 100pL + 100pS)pL − 80(50000− 100pL + 100pS)

dπ

dpL= 50000− 200pL + 100pS + 8000 = 0

200pL = 58000 + 100pS

pL = 290 + 0.5pS

b) If Samsung and LG have to set prices simultaneously, what is the Bertrand-Nash equilibrium in this market?In other words, what price does each firm charge and what quantity does each firm produce? (5 points)

Solution:

Samsung’s reaction function is given by differentiating the profit function with respect to pL and setting equalto zero:

π = pSqS − 50qS = (50000− 100pS + 100pL)pS − 50(50000− 100pS + 100pL)

dπ

dpS= 50000− 200pS + 100pL + 5000 = 0

50000− 200pS + 100pL + 5000 = 0

pS = 275 + 0.5pL

Solving for pS and pL:

pS = 275 + 0.5(290 + 0.5pS)

pS = 420 + 0.25pS

pS = 560

pL = 570

qS = 51000

qL = 49000

c) Suppose instead that Samsung chooses its price first, followed by LG. What are the equilibrium prices andquantities for each firm in the sequential game? (5 points)

Solution:

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Substituting LG’s reaction function into Samsung’s profit function, we differentiate with respect to pS and setequal to zero:

π = pSqS − 50qS = (50000− 100pS + 100(290 + 0.5pS))pS − 50(50000− 100pS + 100(290 + 0.5pS))

dπ

dpS= 50000− 200pS + 29000 + 100pS + 5000− 2500 = 0

100pS = 81500

pS = 815

pL = 697.5

qS = 38250

qL = 61750

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4. (25 points total) Ned has the utility function UN (W ) = W , Arya has the utility function UA(W ) =√W , and

Sansa has the utility function US(W ) = W 2. For each of these individuals, answer the following questions.

a) Is the individual risk averse, risk neutral, or risk seeking? (5 points)

Solution:

Ned:

UN (W ) = W → U ′′N (W ) = 0→ risk neutral

Arya:

UA(W ) =√W → U ′′A(W ) = −0.5(0.5)W−1.5 → risk averse

Sansa:

US(W ) = W 2 → U ′′S (W ) = 2→ risk seeking

b) Suppose each chooses to invest x proportion of their wealth in a risky investment that returns $8 for every$1 invested with probability 1

2 and $0 with probability 12 and (1 − x) proportion of their wealth in a riskless

investment that pays $2 for every $1 invested with certainty. What is the optimal choice of x for each person?(Hint: Remember second-order conditions. If the second derivative is negative, setting the first-derivativeequal to zero provides a maximum. If the second derivative is positive, setting the first-derivative equal to zeroprovides a minimum.)(10 points)

Solution:

Ned:

EUN = 0.5(8x+ 2(1− x)) + 0.5(0x+ 2(1− x)) = 4x+ 1− x+ 1− x= 2x+ 2

dEU

dx= 2 > 0

Ned will always invest everything in the risky asset. Ned is risk neutral and therefore only cares that theexpected value of the risk investment is higher than that of the riskless investment.

Arya:

EUN = 0.5(√

8x+ 2(1− x)) + 0.5(√

0x+ 2(1− x)) = 0.5√

6x+ 2 + 0.5√

2− 2x

dEU

dx= 0.5(0.5)

6√6x+ 2

− 0.5(0.5)2√

2− 2x= 0

6

4√

6x+ 2=

1

2√

2− 2x

12√

2− 2x = 4√

6x+ 2

144(2− 2x) = 16(6x+ 2)

18− 18x = 6x+ 2

24x = 16

x =2

3

Arya is risk averse and therefore will invest only 23 of her wealth in the risky investment.

Sansa:

Sansa is risk-seeking and therefore will invest all of her wealth in the risky investment. Taking first-orderconditions here provides a minimum,

EUN = 0.5((8x+ 2(1− x))2) + 0.5((0x+ 2(1− x))2) = 0.5(6x− 2)2 + 0.5(2− 2x)2

dEU

dx= 6(6x− 2)− 2(2− 2x) = 0

36x− 12 = 4− 4x

40x = 16

x =2

5

But at x = 25 , EU = .8. At x = 1, EU = 8. Even at x = 0, EU = 4. Clearly, x = 2

5 is not a maximum ofexpected utility.

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c) Suppose each individual has $10,000. In addition, each believes that, over the course of the year, with probability0.99 they will suffer no injuries, but with probability 0.01 they will break one of their legs. The loss for eachindividual of breaking a leg (in medical bills, lost wages, etc.) is $10,000. What is the maximum annual premiumeach individual will be willing to pay UPMC to insure themselves against the loss of breaking a leg? (10 points)(Hint: It is best to think of this problem as a 99% chance of keeping $10,000 and a 1% chance of ending up withnothing).

Solution:

The maximum an individual is willing to pay sets the expected utility from going uninsured to the utility forsure of negating the loss and paying the premium.

Ned:

0.99(10000) + 0.01(0) = 10000− P9900 = 10000− PP = 100

Ned is willing to pay up to $100 to insure against the loss. Since he is risk-neutral, this is equal to the fairinsurance price.

Arya:

0.99(√

10000) + 0.01(√

0) =√

10000− P99 =

√10000− P

9801 = 10000− PP = 199

Arya is willing to pay up to $199 to insure against the loss. Since she is risk-averse, this is more than the fairinsurance price.

Sansa:

0.99(100002) + 0.01(02) = (10000− P )2

99000000 = (10000− P )2

9949.87 = 10000− PP = 50.13

Sansa is willing to pay up to $50.13 to insure against the loss. Since she is risk-seeking, this is less than the fairinsurance price.

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