AE6101 Exercises
Question 1 Derive the cost function c (y), profit function π (p) and supply function y (p) for the following technologies
whose production functions y = f (x) are given by
i. f1 (x) = p2x1 + x2
ii. f2 (x) = min{ax1 + bx2, 2×1 + x2}, for a > b > 1.
iii. f3 (x) = xα
1 · (2×1 + x2)β for 0 < α, β < 1.
Question 2. An individual with initial wealth w = 100 risks a loss L = 75. The individual’s utility-of money function
is v(c) = pc and the probability of loss is 1 − q. The individual’s utility depends on the monetary outcome and the
probability q according to U = (1 − q)pc1 + qpc2 − 5q2, where c1 denotes the individual’s wealth after a loss has
occurred and c2 denotes the individual’s wealth in the absence of a loss.
(a) Suppose that q = 1/2, and that the individual can obtain an insurance payment K in case of a loss by paying
premium K/2. How much insurance (i.e. what K) will the individual purchase?
(b) Suppose instead that the price of insurance is 3/4 dollar per dollar of insured loss. How much insurance (i.e.
what K) will the individual purchase?
(c) Now, assume that there is a moral-hazard problem in that the individual can choose q optimally given the
preferences above. What q will the individual choose if:
i) The individual has no insurance.
ii) The individual has bought insurance according to question (a) above.
Question 3. There are two consumers A and B with the following utility functions and endowments:
uA (x1, x2) = a ln x1 + (1 − a) ln x2 and eA = (20, 10) ,
uB (x1, x2) = min ((1 − a) x1, ax2) and eB = (10, 20) ,
where 0 < a [removed] 0, for i = 1, 2, and the monopolist
has some constantmarginal cost of c > 0. Under what conditionswill themonopolist choose not to price discriminate?
(Assume interior solutions.)
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Question 9 A firm has two factories for which costs are given by: C1 (q1) = 10q2
1 and C2 (q2) = 20q2
2, respectively.
The firm faces the following demand curve: P = 700 − 5Q, where Q is total output, i.e. Q = q1 + q2.
(a). On a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves,
and the total marginal cost curve (i.e., the marginal cost of producing Q = q1 + q2). Indicate the profit-maximizing
output for each factory, total output, and price.
(b) Calculate the values of q1, q2, Q and P that maximize profit.
(c) Suppose labor costs increase in Factory 1 but not in Factory 2. How should the firm adjust the following (i.e.,
raise, lower, or leave unchanged): Output in Factory 1? Output in Factory 2? Total output? Price?
Question 10. In a quantity-competed duopoly, Firm X is a price-taker and Firm Y behaves as Cournot best-responder.
The cost functions for two firms are respectively Cx(q) = cq2/2, c 1, and Cy(q) = q2/2? The inverse market
demand is given by p(x + y) = 100 − (x + y). Denote i , i = x, y as the respective profits of two duopolistic firms.
(a) Evaluate the impacts of c on equilibrium profits and their difference.
(b) Verify that there exists a c such that x y for 1 c c. Briefly comment why this possibility can exist.
(c) For c = 1 (identical cost), shown in x-y plane the regions that x y and x < y and verify that the Walrasian
reaction function of X lies in the regions of x y.
(d) Applying (c) to comment that if Firm Y knows in advance the Walrasian reaction function of X and acts as a
Stackelberg-alike leader, the conclusion in (b) will still hold for some range of c.
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Question 1 Derive the cost function c (y), profit function π (p) and supply function y (p) for the following technologies
whose production functions y = f (x) are given by
i. f1 (x) = p2x1 + x2
ii. f2 (x) = min{ax1 + bx2, 2×1 + x2}, for a > b > 1.
iii. f3 (x) = xα
1 · (2×1 + x2)β for 0 < α, β < 1.
Question 2. An individual with initial wealth w = 100 risks a loss L = 75. The individual’s utility-of money function
is v(c) = pc and the probability of loss is 1 − q. The individual’s utility depends on the monetary outcome and the
probability q according to U = (1 − q)pc1 + qpc2 − 5q2, where c1 denotes the individual’s wealth after a loss has
occurred and c2 denotes the individual’s wealth in the absence of a loss.
(a) Suppose that q = 1/2, and that the individual can obtain an insurance payment K in case of a loss by paying
premium K/2. How much insurance (i.e. what K) will the individual purchase?
(b) Suppose instead that the price of insurance is 3/4 dollar per dollar of insured loss. How much insurance (i.e.
what K) will the individual purchase?
(c) Now, assume that there is a moral-hazard problem in that the individual can choose q optimally given the
preferences above. What q will the individual choose if:
i) The individual has no insurance.
ii) The individual has bought insurance according to question (a) above.
Question 3. There are two consumers A and B with the following utility functions and endowments:
uA (x1, x2) = a ln x1 + (1 − a) ln x2 and eA = (20, 10) ,
uB (x1, x2) = min ((1 − a) x1, ax2) and eB = (10, 20) ,
where 0 < a [removed] 0, for i = 1, 2, and the monopolist
has some constantmarginal cost of c > 0. Under what conditionswill themonopolist choose not to price discriminate?
(Assume interior solutions.)
2
Question 9 A firm has two factories for which costs are given by: C1 (q1) = 10q2
1 and C2 (q2) = 20q2
2, respectively.
The firm faces the following demand curve: P = 700 − 5Q, where Q is total output, i.e. Q = q1 + q2.
(a). On a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves,
and the total marginal cost curve (i.e., the marginal cost of producing Q = q1 + q2). Indicate the profit-maximizing
output for each factory, total output, and price.
(b) Calculate the values of q1, q2, Q and P that maximize profit.
(c) Suppose labor costs increase in Factory 1 but not in Factory 2. How should the firm adjust the following (i.e.,
raise, lower, or leave unchanged): Output in Factory 1? Output in Factory 2? Total output? Price?
Question 10. In a quantity-competed duopoly, Firm X is a price-taker and Firm Y behaves as Cournot best-responder.
The cost functions for two firms are respectively Cx(q) = cq2/2, c 1, and Cy(q) = q2/2? The inverse market
demand is given by p(x + y) = 100 − (x + y). Denote i , i = x, y as the respective profits of two duopolistic firms.
(a) Evaluate the impacts of c on equilibrium profits and their difference.
(b) Verify that there exists a c such that x y for 1 c c. Briefly comment why this possibility can exist.
(c) For c = 1 (identical cost), shown in x-y plane the regions that x y and x < y and verify that the Walrasian
reaction function of X lies in the regions of x y.
(d) Applying (c) to comment that if Firm Y knows in advance the Walrasian reaction function of X and acts as a
Stackelberg-alike leader, the conclusion in (b) will still hold for some range of c.
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