Homework assignment #1
Lecturer: Dmytro Vikhrov
Due date: April 3, 2012 (before midterm)
Problem 1
If we graph Jana’s indifference curve with avocados on the horizontal and grapefruits on
the vertical axis, then whenever she has more grapefruits than avocados, the slope of the
indifference curve is −2. Whenever she has more avocados than fruits, the slope is −1
2
.
Jana is indifferent between the bundle with 24 avocados and 36 grapefruits and another
bundle with 32 avocados and x grapefruits. Find x and plot Jana’s indifference curve.
Problem 2
Clara’s utility function is U(x1, x2) = (x1 + 2)(x2 + 1). If her marginal rate of
substitution is −4 and she is consuming 14 units of x1, how many x2 is she consuming?
Problem 3
You know that U(x1, x2) = x1 − x−1
2 , prices of goods x1 and x2 are p1 and p2, and w is
the available budget.
a. Find the Marshallian demand functions for both goods and check whether they are
normal or inferior, Giffen or non-Giffen. Are both goods complements or substitutes?
b. Compute elasticities of each good with respect to p1 and p2. Is any of the goods
luxury?
c. Obtain the indirect utility function, derive the cost function and the Hicksian
demand.
d. Setup the cost minimization problem and verify that the Hicksian demand in the
point above is correct. Compute the income and substitution effects for both goods
separately and show that the Slutsky equation holds. How is x1 different from x2.
Draw IE and SE for both goods in separate graphs.
Problem 4
Suppose that the economy is populated by two consumers: A and B with respective
utility functions: UA
(x1, x2) = x
1
4
1 x
3
4
2 and UB
(x1, x2) = x
2
3
1 x
1
3
2 . Endowments are
(eA
1 , eA
2 ) = (1, 1) and (eB
1 , eB
2 ) = (3, 2).
a. Setup the maximization problem of each consumer and solve it.
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b. Setup the market clearing conditions (2 equations, 2 unknowns).
c. Define the competitive equilibrium. Is the price vector unique in this equilibrium?
Given the price vector, is the allocation for both consumers unique?
d. Draw your findings in the Edgeworth box and provide respective analysis.
Problem 5
Consumer’s utility function is U(C, L) = Cγ
γ
− L, where C is consumption and L is
labor.
a. Setup the budget constraint when the time endowment is 1.
b. Derive the optimal labor supply L∗
and show it in a graph.
c. Re-derive L∗
τ when τ is a per-unit tax imposed on the wage. How does L∗
τ compare
to L∗
?
d. Now let us model the government - its objective is to maximize the tax revenue.
Setup the government’s maximization problem.
e. Find the revenue maximizing tax rate. Define the competitive equilibrium here.
Problem 6
You are given the total cost function: TC(q) = Aq3
− Bq2
+ Cq + D.
a. Find MC(q) and AC(q). What restrictions can you impose on the parameters
A, B, C, D? Depict all three cost functions in a graph.
b. Find q∗
MC that minimizes the marginal costs and q∗
AC that minimizes the average
costs. Is q∗
MC equal to q∗
AC? Why or why not?
Problem 7
Assume a company producing two goods under perfect competition. Its total cost
function is TC(q1, q2) = 2q2
1 + q1q2 + 2q2
2.
a. For this structure of the market, what are the choice variables for the company?
Setup the profit function and take the first-order conditions.
b. Derive the profit-maximizing quantities q∗
1 and q∗
2. How can you make sure that
these quantities really maximize the firm’s profit? Show that the profit function is
concave in q∗
1 and q∗
2.
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Now suppose that the same company is a monopolist on both markets. The demand
functions for goods are: q1 = 40 − 2p1 + p2, q2 = 15 + 2p1 − p2.
c. Are the goods substitutes or complements?
c. Setup the monopolist’s profit function and find the equilibrium quantities and prices.
c. Compute the deadweight loss of the monopoly in both markets.
Problem 8
A monopolist faces a linear marginal revenue and a fixed marginal cost,
MR(q) = a − bq, MC(q) = c. There is also a fixed cost of production f.
a. Find the total revenue, demand function, total costs and average cost.
b. Find the profit maximizing output and the maximized profit.
Suppose that the government imposes a per-unit tax on the sale price, τ.
a. Setup the new profit function. Find the optimal quantity sold. What is the effect of
the tax on the sales volume?
b. Can you find the government’s Laffer curve here? Obtain the tax rate that
maximizes the tax revenue.
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