AE6101 Exercises Question 1 Derive the cost function c (y), profit function π (p) and supply function y (p) for the…

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.)
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.
3

 

Business & Finance homework help

Save your time - order a paper!

Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlines

Order Paper Now