Unit 1.1
The market for a certain kind of prescription drug is described by the following supply and demand curves. Use them to solve problems 1-9.
500 50 50 25 D S Q P Q P = − = +
1. Graph the supply and demand curves. 2. Calculate the equilibrium price and quantity
3. Suppose that the government imposes a price floor of $8. a. What is the quantity exchanged?
b. Is there a shortage or a surplus? How many units?
4. Suppose that the government forces companies to sell the drug for free. a. What is the quantity exchanged?
b. Is there a shortage or a surplus? How many units?
5. Suppose that the government passes a law that only 100 units of the drug can be exchanged. a. What price floor would achieve the same objective?
b. What price ceiling would achieve the same objective?
6. Suppose that the government imposes a $3 excise tax to be paid by sellers of the drug. a. What is the equilibrium quantity of the drug traded after the tax is imposed? b. What price is paid by buyers?
c. What is the net price received by sellers (after the tax is paid)?
d. How much of the tax, in percentage terms, is paid by buyers and sellers? 7. Suppose that the government imposes a $3 excise tax to be paid by buyers of the drug.
a. What is the equilibrium quantity of the drug traded after the tax is imposed? b. What is the net price paid by buyers (after the tax is paid)?
c. What price is received by sellers?
d. How much of the tax, in percentage terms, is paid by buyers and sellers? 8. Suppose that the government pays a $6 subsidy to sellers of the drug.
a. What is the equilibrium quantity of the drug traded after the subsidy? b. What price is paid by buyers?
c. What price is received by sellers (including the subsidy)?
d. In percentage terms, how much of the subsidy benefits buyers versus sellers? 9. Suppose that the government imposes a 40% VAT tax to be paid by sellers of the drug.
a. What is the equilibrium quantity of the drug traded after the tax is imposed? b. What price is paid by buyers?
c. What is the net price received by sellers (after the tax is paid)?
1. For each function below, calculate the first partial derivative with respect to x
f ∂ ∂f / x
2 2
ax +bxy+cy
40 xy
a b Ax y
1 a a Ax− y
(
)
(
)
ln 3 ln 2
a x− +b y−
(
)
(
)
ln 3 ln 2
a −x +b −y
(
) (
2)
2 1
x− − y−
(
)
31/3 1/3 x +y
1/ 2 2 2 1
40x y
−
− −
⎛ + ⎞
⎜ ⎟
⎝ ⎠
2. Find the value of x that maximizes f x
( )
=ln( )
x +ln 200 5(
− x)
Unit 1.3
1. For each of the following markets, suppose that an excise tax is charged to sellers. What fraction of the economic incidence of the tax is paid by sellers in each case?
a. ε = −0.732 and η =1.266
b. ε =0 and η =0.455 (perfectly inelastic demand) c. ε = −1.528 and η=0 (perfectly inelastic supply) d. ε = −∞ and η=0.351 (perfectly elastic demand)
2. Consider a market where demand is QD =2500 50− P and supply is QS =150P−500.
a. Calculate the equilibrium price and quantity.
b. Calculate the elasticity of demand and the elasticity of supply at equilibrium. c. Calculate the consumer surplus and producer surplus at equilibrium.
d. At what price is the elasticity of demand ε = −0.5.
3. The elasticity of demand for green peas is ε = −2.58.
a. By how much should producers cut the price in order to sell 25% more green peas? b. By how much should producers cut production in order to raise the price by 25%?
4. According to a recent study, immigration into the US increased the number of workers in the US by 11.0%, reducing workers’ wages by 3.2%.
a. Does this information help you calculate the elasticity of labor demand or the elasticity of labor supply? Explain.
b. Calculate the appropriate elasticity and interpret it in words.
5. Suppose that the demand for beef depends on the price of beef PB, the price of lamb PL, the price of rice PR and consumer income Y according to the function Q=0.3P P P YB−1 L1/3 R2/3 1/ 2. Currently, beef costs AED 16, lamb costs AED 12, rice costs AED 3 and income is AED 10.
a. From the demand function, are beef and rice substitutes or complements? Explain. b. Calculate the cross-price elasticity of demand for beef with the price of rice at the
current prices.
a community, measured by Y. The demand function is 1000 1 20
P= − +Q Y, and the supply
function is 1 1 2 40 P= Q+ Y .
a. Find the equilibrium price P* in terms of Y.
b. Calculate dP*
Unit 2.1
1. Petra’s income is $100. She consumers bottled water, which costs $5 per bottle, and salads, which cost $10 apiece.
a. Draw Petra’s budget constraint and label it B0. Be sure to label the axes.
b. The bottled water company changes its pricing. Now, the first four bottles Petra buys cost $10 per bottle. However, if she buys more than four, each additional bottle costs only $5. Draw her new budget constraint and label it B1.
c. Now say that the bottled water company changes its pricing policy again. To buy any water at all, Petra must join the “water club”. It costs $40 to join, but she gets four bottles of water just by joining. Thereafter, she can purchase water for $5 per bottle. Draw her budget constraint now and label it B2.
2. Clara has odd preferences for tuba lessons. For her first 5 lessons, she can’t play very well, so the tuba is clunky and sounds bad – she dislikes her first 5 lessons. However, after 5 lessons, she can start to play real music – and then she enjoys her tuba lessons. On the other hand, Clara always enjoys ballet lessons. Sketch a few of her indifference curves over ballet lessons and tuba lessons. Use arrows to indicate direction of increasing preference.
3. Ahmed likes cashews better than almonds and likes almonds better than walnuts. He likes pecans equally as well as dates, but prefers dates to almonds. Assume his preferences are transitive.
a. Does Ahmed prefer pecans or walnuts? b. Does Ahmed prefer dates or cashews?
4. Initially, good X costs $120 and good Y costs $80. If the price of X increases by $18 and the price of Y increases by $12, then describe what the new budget line will look like relative to the original budget line.
1. Jim has an income of $50 and spends it all on apples, bananas and cherries. Suppose that the price of an apple is $1, the price of a banana is $4 and the price of a cherry is $10. The table below shows the marginal utility that Jim obtains for various quantities of the three goods. For example, Jim gets 80 extra units of utility from his first cherry but only 70 units of extra utility from his second cherry.
Determine Jim’s optimal bundle.
Quantity MU Apple MU Banana MU Cherry
1 5 40 80
2 5 40 70
3 4 24 50
4 3 16 20
5 2 8 10
Unit 2.3
1. The diagram below shows part of Lisa’s indifference map over coconuts and carrots.
Initially, Lisa is on budget line B0, where the price of coconuts is $8 and the price of carrots is $2. As illustrated on budget line B1, the price of coconuts drops to $4, while the price of carrots remains at $2. Also shown is budget line B’ -- Lisa’s hypothetical budget constraint after the Hicksian compensation.
a. What is Lisa’s income?
b. What is the total effect corresponding to the price decrease of coconuts? c. What is the income effect corresponding to the price decrease of coconuts? d. What is the substitution effect corresponding to the price decrease of coconuts? e. Are coconuts an inferior or a normal good for Lisa? Be clear about which budget
constraints you compared.
f. Are coconuts a Giffen good for Lisa? Be clear about which budget constraints you compared.
g. Are coconuts and carrots substitutes or complements? h. Sketch Lisa’s Marshallian demand curve for coconuts.
i. Calculate Lisa’s price elasticity of demand for coconuts. Use the midpoint method. j. Calculate Lisa’s income elasticity of demand for coconuts. Use the midpoint method. k. Can you calculate Lisa’s cross-price elasticity of demand for coconuts with the price of
carrots? If so, calculate it. If not, explain what information you need.
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Are weapons bout which b Are weapons
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ketch Tarzan he points tha
good that co pons from $1 he hypothetic
of mass des budget const of mass des ou compared ncome effect ubstitution e n’s Marshall oints that you n’s Hicksian at you used in
osts $1. Tarz 1 to $2, mov cal His optim
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rom the price ng from the d curve for w riving your g urve for weap your graph an
e is $24. The from budget on each budg
or an inferio
for Tarzan?
e increase? B price increa weapons of m
graph and lab pons of mass nd label this
e graph show constraint B get constrain
or good for T
Be clear abo
Be careful ab se? Be caref mass destruct bel this curv s destruction
curve Dh
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B to budget nt is given.
Tarzan? Be c
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3. The diagram below shows a consumer’s preferences over right shoes and left shoes. As shown in the diagram, the goods are perfect complements. The consumer’s income is $10. Initially, the consumer is on budget constraint B0, where the price of both right shoes and left shoes is $1. As shown in the diagram, the price of left shoes increases to $4, moving the consumer to budget constraint B1. The optimal bundle on each budget constraint is shown. The hypothetical budget constraint B’ may be useful in answering some of the questions below.
a. What is the total effect resulting from the price increase of left shoes? b. What is the income effect resulting from the price increase of left shoes? c. What is the substitution effect resulting from the price increase of left shoes? d. Consider your answers in (a)-(c). Why does this make intuitive sense?
g. Sketch the consumer’s Marshallian demand curve for left shoes. Label clearly the points that you used in deriving your graph and label this curve Dm.
1. For the utility function U x x
(
1, 2)
= x1 +x2, sketch in a clearly labeled graph the indifference curves corresponding to U =4 and U =62. Suppose that a consumer with income of Y =1500 faces prices P1 =10 and P2 =20. His utility function is U x x
(
1, 2)
= x1 + x2a. Solve for the optimal bundle.
b. Find the MRS at the optimal bundle and show that it is equal to the price ratio. c. Show that marginal utility per dollar is equal across the two goods at the optimal
bundle.
d. How much utility does the consumer obtain at the optimal bundle? e. Calculate the numerical value of the Lagrange multiplier λ.
f. Suppose that income rises to Y =1501. How much utility does the consumer obtain at the new optimal bundle?
g. Compare your answers to (e) and (f). Give a simple interpretation of the Lagrange multiplier in words.
3. Consider a consumer who buys three goods subject to the budget constraint
1 1 2 2 3 3
P x +P x +P x =Y . Find the demand functions for the following utility functions:
a. U x x x
(
1, 2, 3)
=x x x1 2 3b. U x x x
(
1, 2, 3)
=lnx1+lnx2+lnx3c. Compare your answers to (a) and (b) and explain your results.
4. Consider a consumer who buys 2 goods: x1 and x2. He maximizes the utility function
(
1, 2)
ln(
1 5)
ln( )
2U x x = x − + x subject to the usual budget constraint P x1 1+P x2 2 =Y
a. Find the demand functions for x1 and x2
b. Are the two goods normal or inferior? How do you know?
5. Consider a consumer who buys only water W and food F. He maximizes the utility function U W F
(
,)
=ln( )
W +F subject to the usual budget constraint P WW +P FF =Y.a. Find the demand functions for food and water.
b. Say that the price of food is $4 and the price of water is $1. How much food and water will a consumer with an income of $100 buy?
c. Write the equation of the budget constraint for the consumer in part (b).
d. Using the same prices as in (b), how much food and water will a consumer with an income of $200 buy?
1. A consumer buys only housing and food. Housing is a necessity good.
a. If consumers spend 30% of their incomes on food, what is the range of possible income elasticities for food?
b. How would your answer to (a) change if consumers spent 50% of their incomes on food?
c. Use the Slutsky equation, what is largest possible Marshallian elasticity for food in (a)?
Unit 3.3
1. A consumer’s indirect utility function is
(
1 2)
1 2
, , Y
V P P Y
P P
= +
a. Find the Marshallian demand for good 1. b. Find the expenditure function.
c. Use Shephard’s Lemma to find the Hicksian demand function for good 1.
2. A consumer’s utility function is U x x
(
1, 2) (
= x1−5)
x2. The Marshallian demandfunctions are 1 1
1 5 5
2
m Y P
x
P
−
= + and 1
2
2 5 2
m Y P
x
P
− =
a. Find the indirect utility function. b. Find the expenditure function.
c. Find the Hicksian demands in two different ways without solving a Lagrangian. d. Write the Lagrangian that you would solve to find the Hicksian demand curves
directly. You do not need to solve it.
3. A consumer buys three goods. An economist determines that his expenditure function is
(
1, 2, 3,)
1 3 21. For each scoop of ice cream (I) that you eat, you need 2 cherries (C) and 10 scoops of fudge (F). You are not interested in ice cream unless served with these toppings.
a. Write a utility function over I, C and F that represents your preferences. b. If each scoop of ice cream costs $1, each cherry costs $0.50 and each scoop of
fudge costs $0.10, then determine the optimal bundle for any income Y.
2. A consumer’s utility function over apple juice A and orange juice O is
(
,)
15 60 U A O = A+ O.a. How many units of apple juice is the consumer willing to accept in place of one unit of orange juice?
b. If 0.25PA = , 0.5PO= and Y =30, what is the optimal bundle?
3. Consider a consumer who chooses among n goods
{
x x1, 2,...,xn}
. His utility function is(
1, 2,..., n)
1 1 2 2 ... n nU x x x =a x +a x + +a x . Write out the Marshallian demand functions.
4. A consumer’s utility function over x1 and x2 is
(
1, 2)
1 12 ln( )
2 2U x x = x + x . Suppose that
the prices are P1=2 and P2 =1.
a. Find the optimal bundle when Y =4. (Hint: consider the interior solution as well as the possible corner solutions).
Unit 3.5
1. When a tomato costs $4 and a grapefruit costs $6, Abdulla buys 6 tomatoes and 6 grapefruit. When a tomato costs $6 and a grapefruit costs $3, Abdulla buys 10 tomatoes and no grapefruit. Do Abdulla’s choices satisfy WARP?
2. Ronald visits McDonalds locations in several different countries. Prices are different in each country. The table shows how many BigMacs (B) and orders of french fries (F) he buys in each country. Does Ronald’s behavior satisfy WARP? What about SARP?
Country Price of B Price of F Quantity of B Quantity of F
Albania 1 1 5 35
Bulgaria 1 2 35 10
Croatia 1 1 10 15
Denmark 3 1 5 15
Estonia 1 2 10 10
3. A consumer buys
(
x x x1, 2, 3)
at price vector(
P P P1, 2, 3)
. We make three observations of his behavior:At
(
P P P1, 2, 3)
=(1,1, 2), the consumer chooses(
x x x1, 2, 3)
=(5,19, 9) At(
P P P1, 2, 3)
=(1,1,1), the consumer chooses(
x x x1, 2, 3)
=(12,12,12) At(
P P P1, 2, 3)
=(1, 2,1), the consumer chooses(
x x x1, 2, 3)
=(27,11,1)1. Consider a consumer whose utility function over daily income and consumption is given by U Y N
(
,)
= +Y 240 N . He can work as many hours as he wants each day for a wage of w per hour. In addition to labor income, he has nonlabor income of Y0.a. Find the optimal levels of Y and N . Your answers should depend on w and Y0. b. Find the labor supply function. Is the income effect or the substitution effect
stronger for this consumer?
c. Suppose that Y0 rises. Note that this induces only an income effect, and no substitution effect. Is this income effect built into an increase in Y, an increase in
N, or both?
2. Initially, an individual can work as many hours per day as he wants for a wage of w. He chooses to work 7 hours. Suppose now that his boss offers to pay him “time and a half” if he works more than 8 hours, i.e. he earns 1.5w per hour for any hours he works each day beyond 8 hours.
a. Graph the original budget line and the new budget line. b. Will the individual choose to work more than 7 hours?
3. Suppose that leisure was an inferior good rather than a normal good. What then would be the effect of a wage increase on labor supply?
4. Consider a housewife who takes N hours of leisure each day. She can also spend time preparing food, and each unit of food F requires PF hours to prepare. Her utility function is U F N
(
,)
=FN1/ 2a. Write the housewife’s time constraint.
b. Fine the housewife’s optimal choices of F and N.
Unit 4.2
1. Amal earns income of $2000 this year and $1100 next year. She can borrow or lend money at an interest rate of 10%. She consumes C1 this year and C2 next year. Her utility function is U C C
(
1, 2)
=C C1 2.a. Write the present value budget constraint. What is the present value of Amal’s income?
b. Write the future value budget constraint. What is the future value of Amal’s income?
c. Find the optimal consumption bundle C1 and C2.
d. Does Amal borrow or save in the first period? How much?
e. Suppose that the interest rate rises to 20%. What happens to Amal’s first period consumption C1? What does this say about the relative magnitudes of the income effect and the substitution effect?
2. An individual lives for 5 years – 4 years of work followed by 1 year of retirement. He earns $25,000 during each year of work but $0 during retirement. The interest rate is 5%. Assume that the individual wants his consumption to be the same in each of the 5 years.
a. What utility function would justify this choice of consumption pattern? b. Write out the intertemporal budget constraint in present value form. c. What level of consumption does he choose?
1. Faisal is driving to Jebel Ali and has two choices. On E-11, he faces probability 1/2 of a $200 speeding fine. On E-311, he faces probability 1/4 of a $300 speeding fine. Faisal’s wealth is W =300 before beginning his trip and his utility function is U W
( )
= W .a. Is Faisal’s expected wealth higher by driving on E-11 or E-311? b. Is Faisal’s expected utility higher by driving on E-11 or E-311? c. Which road will Faisal choose?
2. Suppose that most people will not speed if the expected fine is at least $500. The actual fine for speeding is $800. How high must the probability of being caught be in order to discourage speeding?
3. You have two pieces of jewelry, each worth $1000, that you want to mail to your sister. You can either mail them separately in two packages or together in one package. For any package, the probability that it will be lost in the mail is 0.01. Your utility function is
( )
U W = W , where W is the value of the jewelry that your sister ends up receiving.
a. What is the expected value of the jewelry received if you mail the pieces separately? Together?
b. What is your expected utility if you mail the pieces separately? Together? c. Relate your answers to idiosyncratic and systemic risk.
d. Suppose that your sister begins with wealth of W =500 before receiving the jewelry. If each package costs ω to mail, write down an equation characterizing the highest value of ω for which you would send the jewelry in two packages. You do not need to solve the equation.
4. Michael and Samer are neighbors. Each owns a car valued at $10,000. Michael’s wealth, including the value of his car, is $80,000. Samer’s wealth, including the value of his car, is $20,000. Each has utility function U W
( )
=W0.4, where W designates total wealth. If a car is parked in the street, there is a probability 0.5 that the car will be stolen. If a car is parked in a garage, it will not be stolen.a. What is the largest amount that Michael would be willing to pay for the garage? b. What is the largest amount that Samer would be willing to pay for the garage? c. Compare Michael’s and Samer’s Arrow-Pratt measures of risk aversion and
5. A driver in the UAE faces a 2% probability that his car will be in an accident and will be worth nothing. Consider three drivers with cars that have value AED 100,000 and no other wealth. Abdulla’s utility function is U W
( )
=ln 1(
+W)
. Bedriya’s utility function is( )
100 0.5U W = + W . Ciera’s utility function is U W
( )
=W2.1. A consumer with income of Y =300 and a utility function U =min
{
x x1, 2}
initially faces prices P1=1 and P2 =1. As a result of government policy, the price of good 1 rises to1 2
P = , with the price of good 2 remaining at P2 =1.
a. How much utility does the consumer obtain at the old prices? b. How much utility does the consumer obtain at the new prices?
c. At the new prices, how much additional income would be needed to return the consumer to the old level of utility?
d. At the original prices, how much income would need to be taken away in order for the consumer’s utility to fall by the same amount as it falls as a result of the price increase?
e. Find the CV and the EV for this price increase. f. (bonus) Find the change in consumer surplus.
2. Consider a consumer whose utility function is U =ln
( )
x1 +x2. Throughout this problem, set P2 =1 to normalize.a. Find the Marshallian demand functions. They should depend on Y and P1. b. Find the indirect utility function. It should depend on Y and P1.
c. Find the expenditure function. It should depend on U and P1.
d. For a consumer with income Y =10, find the compensating variation for a price increase of good 1 from P1=1 to P1 =2.
e. Find the equivalent variation for this price increase.
Unit 5.1
1. Consider a firm whose production function is q=1000 min
{
L K, 3}
.a. Sketch a couple of isoquants for this production function.
b. Suppose that capital is fixed at K =5 in the short-run. Sketch the total product curve as L rises.
c. Refer to (b) and sketch the marginal product curve.
2. Let E designate the number of hours spent studying for exams, let P designate the number of hours spent writing papers and let G designate course grade. Noor’s “grade production” function is G=2.5E0.36P0.64.
a. What is Noor’s marginal product of studying for exams? Describe in words what this represents.
b. What is Noor’s MRTS? Describe in words what this represents.
3. Consider firms with the production functions below, operating in the short run with capital fixed at K =10. In each case, decide whether or not the firm will experience diminishing marginal returns to labor in the short run.
a. q=10L+K b. q=L K3/ 4 1/ 4
4. State whether each production function displays increasing, constant or decreasing returns to scale. If it depends on the values of the parameters, specify the range of parameters over which the production function displays various scale properties.
a. q=aL bK+ b. q=min
{
aL bK,}
c. q=
(
Lρ+Kρ)
1/ρd. q=
(
Lρ+Kρ)
φ ρ/5. The production function for the automotive industry has been estimated as 0.27 0.16 0.61
q=L K M , where M represents raw materials.
a. What is the marginal product of raw materials?
b. What kind of returns to scale does the automotive industry feature?
6. After the 2004 tsunami, a Navy Admiral was asked in a radio interview whether doubling the number of helicopters would “produce twice as much relief”. His answer was:
1. Pepsi uses two inputs to produce soda: bottling machines K and workers L. The machine costs $1000 per day to run and each worker is paid $200 per day. At the current level of production, the marginal product of a machine is an additional 200 cases of soda per day and the marginal product of a worker is an additional 50 cases per day. Is this firm minimizing cost? If not, explain how it should adjust its input usage.
2. A music publisher pays $10 to prepare a plate for printing music. After the plate is prepared, the publisher can make as many copies of the music as he wants for $1 per copy. Graph the ATC, AVC and MC functions.
3. Samia works in a flower shop where she produces 10 floral arrangements per hour. She is paid $10 per hour for the first 8 hours she works and then $15 per hour for each
additional hour. Give equations for the TC, ATC, AVC and MC functions.
4. A firm faces the production function q=10L0.32K0.56. Find the short-run total cost function, along with the MC and AVC functions when capital is fixed at K =1.
5. For the production functions below, find the input demand functions and the long-run cost function. Your answers should depend only on q, r and w. (hint: (c) is the only part for which you should be using a Lagrangian.)
a. q= +L K b. q=min
{
L K,}
c. q= L+ Kd. What are the returns to scale in the production functions in (b) and (c)? e. What are the economies of scale in the cost functions in (b) and (c)?
6. A power plant must produce 4 units of electricity every hour during the day (12 hours) and 3 units of electricity every hour at night (12 hours). Let K designate capital and F designate fuel. The hourly production function at the plant is q= KF , where q is units of electricity produced. Each unit of capital costs r per hour and each unit of fuel costs
w per hour. The plant cannot change its level of capital usage between day and night, but it can change its usage of fuel.
a. How much capital should the firm buy in order to minimize its costs of supplying the necessary amount of power?
1. Consider a competitive firm selling Christmas trees. Its total cost function is estimated as
3
7 37
6860
12 27, 000, 000 TC= +⎛⎜T + +t ⎞⎟q+ q
⎝ ⎠ . Here T is the wholesale cost of a tree and t
is the transport cost.
a. Suppose that T =11.5 and t=2. Find the marginal cost function. b. Suppose that T =11.5 and t=2. Find the shut-down price. c. Suppose that T =11.5 and t=2. Find the zero-profit price.
d. Find the seller’s supply function, with price as a function of q, T and t.
e. Where *P is the seller’s supply price, compute P* t
∂
∂ and explain in words what
this represents.
2. The market for lobsters is perfectly competitive. Total cost for a firm that harvests q lobsters is TC=800 0.5+ q2. The market demand curve for lobsters is Q=2000 5− P.
a. Find output by each firm q, market price P and the number of firms in operation when the market is in long-run equilibrium.
b. Now suppose that the government imposes a $450 tax on each firm, raising costs to TC=1250 0.5+ q2. In the short-run, where the number of firms is the same as in (a), find the output by each firm q and the market price P.
c. Calculate the profit or loss earned by each firm. Will there be entry or exit? d. Find the new long run equilibrium output by each firm q, market price P and the
number of firms in operation after the imposition of the tax.
Unit 5.4
1. A monopoly faces demand curve P=10q−1/ 2. The firm faces a cost of 5 for each unit of output it produces.
a. Find the profit-maximizing price for the monopoly to charge.
b. Suppose that the government imposes a $1 excise tax on the firm, raising the cost of each unit to 6. Find the new profit-maximizing price.
c. What seems odd about your answers to (a) and (b)?
2. Two states face the same statewide demand curve for cigarettes: P= −5 0.001Q. There is a perfectly elastic supply of cigarettes at a marginal cost of $2 per unit.
a. In state A, the state maintains a monopoly in the sale of cigarettes. Find the profit-maximizing price and quantity.
b. In state B, the market for cigarettes is competitive. Find the equilibrium price and quantity.
c. Calculate the excise tax on sellers that would need to be imposed in state B so that the price of cigarettes there equals the price in state A.
d. Show that the tax revenue in state B equals the monopoly profit in state A.
3. Apple charges $499 for its Mac mini computer, even though its marginal cost of production is only $258 per computer. Assuming that Apple is charging the profit-maximizing price
a. What is Apple’s Lerner index?
b. What elasticity of demand does Apple face for the Mac mini?
4. The government licenses MedFraud as a monopoly for selling aspirin. The government charges a tax equal to 10% of MedFraud’s profits.
a. Show how the tax affects MedFraud’s profit-maximizing output and price. b. How would your answer to (a) change if the tax were 25%?
5. The demand curve for sugar in Florida is P=1.787 0.0004641− Q. The supply curve is 0.4896 0.00020165
P= − + Q.
a. Calculate the equilibrium price when the market is competitive.
b. Calculate the equilibrium price when the market is a monopoly, assuming that the supply curve is the marginal cost function.
c. What is the price in the competitive case if the government imposes an excise tax of t=0.01 per unit of sugar?
d. What is the price in the monopoly case if the government imposes an excise tax of 0.01
t= per unit of sugar?
6. The demand curve in a market is P=100−Q and the marginal cost of producing each unit of output is MC=20. Calculate market output Q, consumer surplus, producer surplus and deadweight loss when the market is
a. Perfectly competitive b. A single price monopoly
c. A perfectly price discriminating monopoly
Unit 6.1
1. Consider the Edgeworth Box diagram below in answering this question. Peters and Walters are prisoners. Peters received a shipment containing only chocolate and Walters received a shipment containing only cigarettes.
a. Which point represents the endowment?
b. Suppose that Peters is a nonsmoker and Walters is allergic to chocolate. Describe the contract curve.
2. Suppose that Anna considers one apple to trade off for three bananas. Bob considers an apple to trade off for four bananas.
a. Are Anna and Bob on the contract curve?
b. If Anna gives Bob an apple in exchange for 3 bananas, who is better off? c. If Anna gives Bob an apple in exchange for 3.5 bananas, who is better off?
3. Bert starts with 10 food and 10 clothing. Ernie starts with 10 food and 20 clothing.
a. Represent the initial allocation in an Edgeworth box.
b. Bert regards the two goods as 1-for-1 substitutes, while Ernie regards the two goods as 1-for-1 complements. In your Edgeworth Box, shade the allocations that are Pareto superior to the initial allocation.
4. Anna and Bob live in a pure exchange economy where the goods are x1 and x2. Anna has normally shaped indifference curves and values both goods. Bob, however, is completely indifferent over any possible allocation. Both consumers start with 0.5 units of each good.
a. Identify all Pareto efficient allocations.
b. Consider the allocation where Anna gets 0.9 units of each good and Bob gets 0.1 units of each good. Could this be a competitive equilibrium?
c. Is the allocation in (b) efficient?
1. Consider a pure exchange economy with three goods: x1, x2 and x3. Consumer A’s utility function is UA =min
{
x x1, 2}
. He starts with one unit of x1 and none of the other goods. Consumer B’s utility function is UB =min{
x x2, 3}
. He starts with one unit of x2 and none of the other goods. Consumer C’s utility function is UC =min{
x x1, 3}
. He starts with one unit of x3 and none of the other goods. Find the general equilibrium prices and allocations.2. Anna and Bob live in a pure exchange economy where the goods are x1 and x2. Anna’s utility function is UA =ln
( )
x1 +ln( )
x2 . Bob’s utility function is UB =ln( )
x1 +2 ln( )
x2 . Anna’s endowment is(
e e1A, 2A)
=(
18, 4)
. Bob’s endowment is(
e e1B, 2B)
=( )
3, 6 .a. Characterize the set of Pareto efficient allocations (the contract curve). b. Characterize the core of this economy.
c. Suppose that Anna makes a one time take-it-or-leave-it offer to Bob and no further trade is possible. Show how you would find the allocation that Anna would propose. (Once you set up the problem, you do not need to go through the calculation).
d. If the price of good 1 is P1=1, find the competitive equilibrium price P2. e. Find the equilibrium allocation.
1. The table below shows the profits to Toyota (player 1) and to GM (player 2) depending upon whether each enters or does not enter the market for electric automobiles.
a. Does either firm have a dominant strategy? b. What is the Nash Equilibrium?
c. Suppose that the US government offers to pay GM a subsidy of 50 if it enters the market. What is the Nash Equilibrium now?
Enter Not Enter
Enter 10, -40 250, 0
Not Enter 0, 200 0, 0
2. Consider the game below.
a. Are there any pure strategy equilibria? b. Find the mixed strategy equilibrium.
Y Z A 0,2 7,0 B 2,1 6,6
3. Firms 1 and 2 each produced a movie that can be released on July 4 or July 18. The table below shows the profit to each firm depending upon when the movies are released.
a. Find the Nash Equilibrium.
b. Which release dates maximize the total profit earned by the firms?
c. What is the highest price that firm 2 would be willing to pay in order to acquire the distribution rights to firm 1’s film?
d. What is the lowest price that firm 1 would accept to sell the rights to its film? e. If firm 2 ends up purchasing the distribution rights to firm 1’s film, on what dates
will it release the films?
July 4 July 18
July 4 50,50 80,35
July 18 30,90 20,20
5. Consider the game below.
a. List the inequalities that must hold if (T,L) is a Nash Equilibrium.
b. List the inequalities that must hold if (T,L) is a dominant strategy equilibrium.
L C R T a,b c,d e,f M g,h i,j k,l
B m,n o,p q,r
6. This problem demonstrates a surprising result about mixed strategy equilibria. You can interpret the payoffs below as success probabilities.
a. Find the mixed strategy equilibrium of the game below.
Y Z
A 90,10 20,80
B 30,70 60,40
b. Now suppose that B becomes a better option against Z for player 1 (and
conversely worse for player 2). Find the mixed strategy equilibrium of the new game below.
Y Z
A 90,10 20,80
B 30,70 65,35
1. For the game shown below:
a. Find the SPE outcome.
b. Give the complete SPE strategy.
2. For the game shown below:
a. Write out the reduced normal form. b. Find all Nash Equilibria.
3. For the game shown below:
a. Find the SPE outcome.
b. Could player 2 improve his payoff by committing to a certain strategy in
advance? If so, what strategy should he commit to and what are the new payoffs?
4. Two proposals, A and B, are being debated by politicians in the US. Both, one, or neither of the proposals may become law. The table below shows the payoffs to Congress and to the President depending upon which proposals are passed.
Laws Passed Congress President
A only 4 1
B only 1 4
A and B 3 3
Neither 2 2
a. Suppose that congress first decides which of the four options to implement. If any bill(s) is/are passed, the president can either sign the proposal or veto it. If he vetoes, then neither proposal is implemented. Draw the game tree and find the SPE outcome of this game.
b. Are there other Nash Equilibria in (a) that are not subgame perfect? c. Now suppose that the rules of the game are changed in one respect. If the
congress passes both A and B, then the president can choose to sign A only, sign B only, sign both or veto (this is called a “line item veto”). Draw the game tree and find the SPE outcome of the game now.
two inequalities.
a. If the game is played simultaneously and (B,R) is the Nash Equilibrium of this game, what must be true of X and Y?
b. If the game is played sequentially, with player 1 moving first, and (B,R) is the SPE outcome of this game, what must be true of X and Y?
L R
T 4,8 0,0
B 8,20 X,Y
6. Consider the game shown below.
a. What is the Nash Equilibrium if choices are made simultaneously? b. What is the SPE outcome if player 1 chooses first?
c. What is the SPE outcome if player 2 chooses first?
L R
T 30,30 50,35
B 40,60 20,20
7. Consider a market with two firms where the market demand function is P=60 2− Q. Firm 1 has a marginal cost of production MC1=4, and firm 2’s marginal cost of production is MC2 =20. There are no fixed costs. Fill in the table below.
Model q1 q2 Q P Π1 Π2 Π + Π1 2
Cournot
Stackelberg (firm 1 moves first)
8. There are two entrepreneurs (players 1 and 2) who are working on a joint project, and a venture capitalist (player 3) who is a potential investor in the project. First, player 1 decides whether to devote high or low effort to preliminary work on the project. Player 2 observes this choice and then decides whether to exert high or low effort himself. They then make a presentation to the venture capitalist, who decides whether or not to invest.
The payoffs are as follows. Each entrepreneur obtains 5 if the venture capitalist invests and 0 otherwise. In addition, choosing high effort costs an entrepreneur 1, while choosing low effort is free. Investing costs the venture capitalist 2, but if he invests he gains 3 for each entrepreneur who chose high effort. If the venture capitalist does not invest, his payoff is 0.
1. There are three factories in a town, all of which emit 5 tons of pollution. They could reduce their emissions, but at a cost. The table below shows the marginal abatement cost for each ton by which the firm could reduce its emissions. Suppose that the government wants to reduce total emissions from 15 tons to 6 tons.
a. Suppose that the government orders each firm to reduce its pollution by 3 tons. What is the total abatement cost?
b. Suppose instead that the government issues 2 pollution permits to each firm. Who will buy permits and who will sell them?
c. What is the range of equilibrium prices for the permits?
d. What is the total abatement cost under the pollution permit system?
Tons of
Reduction MC to Firm A MC to Firm B MC to firm C
1 100 200 600 2 300 300 700 3 500 400 800 4 700 500 900
5 900 600 1000
2. Firm Alpha can leave its windows uncovered or pay to install shutters on its windows. Whether the shutters are installed has an effect on Firm Beta’s profit, as illustrated in the table below.
a. Does the shutter installation create a positive or a negative externality? Explain. b. Is it efficient for the shutters to be installed? Explain.
c. Suppose that Firm Alpha has the property rights. What will happen if the firms negotiate?
d. Suppose that Firm Beta has the property rights. What will happen if the firms negotiate?
e. Relate your answers above to the Coase Theorem.
Firm Alpha Profit
Firm Beta Profit
Alpha does not install shutters $800 $200
Alpha installs shutters $700 $500
4. The production of drums creates an externality. The graph below shows the marginal private cost (MSC) and marginal social cost (MSC) associated with the production of drums. Also shown are the demand (MB) curve and the single price monopolist’s marginal revenue (MR) curve.
a. Is the externality positive or negative? Explain. b. What is the socially optimal level of output?
c. Suppose that drums are produced by a single-price monopoly. i. What is the monopolist’s profit-maximizing level of output? ii. What price will the monopolist set?
iii. To induce social efficiency, should the government tax or subsidize the monopoly firm?
iv. What is the dollar value of the necessary tax or subsidy?
d. Suppose that drums are produced in a competitive market in long-run equilibrium. i. How much output will be produced?
ii. What price will firms charge?
iii. To induce social efficiency, should the government tax or subsidize the firms?
1. Tarzan and Jane live alone in the jungle and have trained Cheetah to patrol the perimeter of their camp and to harvest fruits. Cheetah can collect 3 pounds of fruit per hour, but Tarzan and Jane would each be willing to give up one hour of patrolling for 2 more pounds of fruit. If Cheetah currently spends 10 hours patrolling and 14 hours picking fruit, is the current allocation of his time efficient? If not, should he patrol more or less?
2. There are 1000 residents of a town, each of whose marginal benefit (willingness to pay) for having x miles of roads is P=100 5− x. Roads are built competitively at a marginal cost of MC=2000 per mile. Roads are a public good.
a. What is the efficient level of roadway construction?
b. If residents decide on road building independently, how many miles of roadway will be built?
3. Anna and Bob are assigned to write a joint paper in the next 24 hours. Let tA denote the time that Anna spends writing and let tB denote the time that Bob spends writing. The grade is determined by the amount of time spent by both. They get utility from a higher grade, but also get utility from leisure. Specifically, the utility functions are:
(
)
(
)
ln ln 24
Anna
A B A
U = t +t + −t
(
)
(
)
ln ln 24
Bob
A B B
U = t +t + −t
a. Find the Nash equilibrium levels of tA and tB.
b. What levels of tA and tB would have maximized their total utility? c. What sort of market failure does this exercise illustrate?
4. There are n fishermen on a lake. Fisherman 1 catches x1 fish, fisherman 2 catches x2 fish, etc… Let X = + +x1 … xn denote the total fish catch. When X fish are caught, the price of each fish is e−X. Thus, fisherman 1’s profit is x e1 −X, fisherman 2’s profit is
2 X
x e− , etc…
a. If the fishermen act independently, how many fish will each catch?
b. Calculate the profit for each fisherman. Your answer will be a function of n. c. Suppose instead that each fisherman catches 1n fish. Calculate the profit for each
fisherman in this case.
Unit 8.3
1. Explain why first year depreciation of new cars is so high. Indeed, even if you sell a new car a few days after you bought it, the price is much less than the initial sale price.
2. Suppose that half of the population is healthy and the other half is unhealthy. Both types have probability 0.4 of getting sick, but a healthy person pays $1000 of medical expenses if he gets sick, while an unhealthy person pays $10,000 of medical expenses if he gets sick. The insurance company cannot tell whether a customer is healthy. Suppose that all customers start with wealth of $30,000 and that their utility functions are U W
( )
= Wa. What is the actuarially fair price for the insurance if everyone participates? b. If the company charges the price in (a), who will choose to participate?
c. After considering the consumer adjustments in (b), what is the new actuarially fair price for the insurance?
3. Consider the following job signaling game. Low ability workers produce output worth wl regardless of whether they have a college degree. High ability workers produce output worth whnd if they do not have a degree, but produce output worth whd if they have a degree, where whd >whnd. High ability workers can obtain a degree at cost ch. Low ability workers can also obtain a college degree at cost cl. It is more difficult for low ability workers to obtain degrees, meaning that cl >ch. A proportion θ of workers are high ability and a proportion 1−θ are low ability. All markets are competitive.
a. Consider a separating equilibrium where only high ability workers get degrees. Write the inequalities that must hold for this equilibrium to exist.
b. Consider a pooling equilibrium where no workers obtain degrees. Write an equation for the wage w that workers are paid in this equilibrium.
c. Write the inequality that must hold for the equilibrium in (b) to exist. d. Consider a pooling equilibrium where all workers obtain degrees. Write an
equation for the wage w that workers are paid in this equilibrium. e. Write the inequalities that must hold for the equilibrium in (d) to exist.
4. Suppose that stupid workers generate $6 of profit for a firm, while smart workers generate $A of profit for a firm, where A>6. Firms cannot distinguish smart workers from stupid workers, but can observe their education levels. Any worker can acquire as many years of education as he wants. Getting e years of education costs a smart worker
e, but it costs a stupid worker Be, where B>1.
a. Solve for e*, the minimum years of education that smart workers must get in order to differentiate themselves from stupid workers.
1. A firm hires a consultant who may be competent or incompetent. After hired, the
consultant can choose whether to exert high effort or low effort. The firm cannot observe any of this directly, but can observe the revenue that the consultant generates. The correspondence between the consultant’s behavior and the firm’s revenue is shown below. Let w1 indicate the wage that the worker is paid when revenue is 1, with w2 and
3
w defined similarly. It costs the consultant 0.5 to exert high effort, while low effort is free. His best outside option pays 0.
a. Write the incentive compatibility and individual rationality constraints if the firm wants to induce both types to put in high effort.
b. Using (a), solve for the values of w1, w2 and w3 in the optimal contract. c. How would your answers change if the firm were almost certain that the
consultant was competent?
High effort Low effort
Competent 3 2
Incompetent 2 1
2. Explain what the difference would be in problem 1 if the revenue correspondence were as shown below. Is there still a moral hazard here?
High effort Low effort
Competent 3 2.5
Incompetent 1.5 1
3. A firm’s revenue is given by R=10e e− 2, where e is the worker’s level of effort. Putting in high effort is costly, and so the worker’s objective is to maximize w e− , where w is the wage. Find the worker’s level of effort and the firm’s profit (revenue net wage cost) for each of the following compensation schemes
a. w= −R 12.5 b. w=R/ 2
c. w=2 if R≥9 with w=0 otherwise
d. Give some economic intuition comparing your answers to (a)-(c), explaining the incentives created by the principal-agent contracting arrangement.
