Microeconomics problem set (ECO601)

Microeconomic Theory Yong Wang (SITE, UIBE)

Microeconomic Theory, Problem Set 2

Due: Nov 28, 2022

Question 1. Suppose that a firm produces a single output q from two inputs x1 and x2. You ar given

100 monthly observations, and two of these monthly observations are shown in the following table:

Inputs prices Inputs levels Output price Output level

Month w1 w2 x1 x2 p q

3 3 1 40 50 4 60

9 2 2 55 40 4 60

Is there any contradiction between the profit-maximization assumption and these two monthly observa-

tions?

Question 2. Consider the following profit function that has been obtained from a technology that uses

a single input, z:

π(p, w) = p2wα,

where p is the output price, w is the input price, and α is a parameter value.

1. Find the value of α for the profit function to be homogeneity of degree 1 in (p, w).

2. Assuming the value of α for which the profit function satisfies homogeneity of degree 1, check if

the profit function π(p, w) satisfies the following properties: (1) non-decreasing in output price p,

(2) non-increasing in input price w, and (3) convex in prices p and w.

3. Calculate the supply function of the firm, q(p, w), and its demand for input, z(p, w).

Question 3. Suppose that a firm owns two plants, each producing the same good. Every plant j′s

average cost is given by

ACj(qj) = α+ βjqj , ∀qj > 0,

where j ∈ {1, 2} where coefficient βj may differ from plant to plant. Assume that you are asked to

determine the cost-minimizing distribution of aggregate output q = q1 + q2, among the two plants (i.e.,

for a given aggregate output q, how much q1 to produce in plant 1 and how much q2 to produce in plant

2). For simplicity, consider that aggregate output q satisfies q < α/maxj |βj |.

1. If βj > 0 for every plant j, how should output be located among the two plants?

2. If βj < 0 for every plant j, how should output be located among the two plants?

3. If β1 > 0 while β2 < 0, how should output be located among the two plants?

Question 4. Consider a strictly risk-averse decision maker who has an initial wealth of w but who runs

a risk of a loss of D dollars. The probability of the loss is θ. It is possible, however, for the decision

maker to buy some insurance. One unit of insurance costs q dollars and pays 1 dollar if the loss occurs.

What is the optimal level of insurance when q = θ?

Question 5. A farmer uses his own labor, x, to produce wheat, q, with linear production function

q = f(x) = x. At the end of the harvesting season, the farmer sells all production at the given price of

wheat.

The price he receives is a random variable, in particular, has two possible realizations: pH with

probability γ, pL with probability 1− γ, where pH > pL, 0 < γ < 1.

The farmer’s preferences are given by utility function u(wk)−f(x), where wk represents the farmer’s

income when the realization of the random variable is k, where k ∈ {H,L}. Suppose that u(·) is strictly

increasing and concave. Assume that the cost of effort is strictly increasing and convex, and denote by

w0 the farmer’s initial income.

The government has just created a price-guarantee program that ensures that the farmer will receive

a price pG, where pH > pG > pL, for the proportion of his production that he includes in this program at

the beginning of the season, that is, before knowing the particular price for that year. Let λ ∈ [0, 1] denote

the proportion of the farmer’s production that he chooses to include in the price-guarantee program.

1. Set up the farmer’s expected utility maximization problem. For simplicity, assume inner solutions.

Take the first-order conditions with respect to the two choice variables of the farmer: his own

labor, x, and the proportion of his production that he includes in the price-guarantee program, λ.

Characterize the solution (note that the solution will be implicit, since we do not have a functional

form for the farmer’s utility function).

2. Consider now that the price-guarantee program produces zero expected profits for the government.

Explain how your implicit solution for x and λ in above question is affected.

Question 6. Assume that there are only two assets. The first is a riskless asset that pays $1, and the

second pays amounts a and b (b > a) dollars with probabilities of π ∈ (0, 1) and 1 − π, respectively.

Denote the demand for the two assets by ( x1, x2

) .

Consider a risk averse decision-maker with initial wealth $1. Prices of both assets are equal to 1, and

short-selling is prohibited for both assets. Thus budget constraint is

x1 + x2 = 1, x1, x2 ∈ [0, 1]

a. Give a simple condition (involving a and b only) for the demand for the riskless asset to be strictly

positive.

b. Give a simple condition (involving a, π and b only) for the demand for the risky asset to be strictly

positive.

Following suppose that the conditions obtained in (a) and (b) are satisfied.

c. Write down the first order conditions for expected utility maximization in the asset demand prob-

lem.

d. Assume that a < 1. Show by analyzing the first order conditions that dx1 da

< 0.

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