# Part 1: Suppose that a cost minimizing firm with two inputs, capital K and labour L recently increased the quantity of each inputs by 10%. As a…

Part 1: Suppose that a cost minimizing firm with two inputs, capital K and labour L recently increased the quantity of each inputs by 10%. As a result, output has increased from 1,000 to 1,200 units. For each of the following statement, indicate whether it is true, false, or uncertain and explain your answer.

a) When output is 1,000 units, there are increasing returns to scale.

b) When output is 1,000 units, long-run average cost is downward sloping.

c) When output is 1,000 units, long-run marginal cost exceeds long-run average cost.

d) When output is 1,000 units, long-run marginal cost is downward sloping.

e) If the firm had increased the quantity of each input by 5%, output would increase from 1,000 to 1100 units.

Part 2: Research suggests that successful performance on exams requires preparation (that is, studying – L) and rest (that is, sleep – S). Neither by itself produces good exam grades, but in the right combination they maximize your exam performance. Suppose that the production technology described above can be captured by the production function q = 40L0.25S0.25, where q is your exam grade, L is the number of hours spent studying, and S is the number of hours spent sleeping. MPL = 10L-0.75S0.25 and MPL = 10L0.25 S-0.25.

a) Prove that this production process has decreasing returns to scale.

b) Set up the cost minimization problem and solve for the conditional studying and sleeping demands as functions of pL(what you are willing to pay to get back an hour of studying) ,pS (what you are willing to pay to get back an hour of sleep), and q (your exam performance).

c) Discuss the demand functions you derived in b). Is sleep and studying normal inputs? What happens to optimal amount sleep and pS increases? What happens to optimal amount of studying as pL decreases?

d) Now, assume that you are always willing to pay \$5 to get back an hour of sleep and \$20 to get back an hour of studying. What is your “optimal production plan”?

e) Derive the cost function and simplify the function as much as you can.

f) Continue with total cost function derived in part e) and derive the average cost. Are marginal and average cost curves for this problem upward or downward sloping? Explain. What is the relationship between MC and AC?

g) What is your “optimal production plan” if you wish to reach the exam performance of 100?

h) What is the cost of this 100- score performance? Now suppose your instructor Lucia clearly tells you that you have to study for 5 hours to get 100-score performance. Reconsider the “short-run” problem where studying is fixed at L .

i) What is your “optimal production plan”? What is your “optimal production plan” if you wish to reach the exam performance of 100?

j) What is the short-run cost function? What is the short-run marginal cost function? Average cost function?

k) Compare cost functions from part j) with cost functions from part f).

Part 3: There are m consumers who consume only two kinds of goods, x (apples) and y (oranges). The preference ordering for each consumer can be represented by the utility function U(x; y) = x 1-a y a : Each consumer has income M. The price of good x is px, that of y, py. The production function for apples (good x) is q = AK0.5L0.5; where q is the quantity of apples produced, K is the amount of land used, L is the amount of labour used and A is a productivity parameter. Let w represent the wage rate and r the rental rate of land. There are n firms currently producing apples. Each currently uses K0 units of land. In the short-run it is prohibitively expensive to change the amount of land utilized. Assume perfect competition.

a) Find the demand curves for apples for a representative consumer.

b) Find the market demand curve for apples.

c) Find the short-run cost function for apples.

d) Find the short-run marginal cost function for apples.

e) Find the supply function of a representative firm producing apples.

f) Find the market supply function for apples.

g) Find expressions for the equilibrium price and quantity of apples.

Part 4: In a perfectly competitive market, the market demand and market supply curves are given by QD = 1000 −10P and QS = 30P. Suppose the government provides a subsidy of \$20 per unit to all sellers in the market.

a) Find the equilibrium quantity demanded and supplied; find the equilibrium market price paid by buyers; find the equilibrium after-subsidy price received by firms.

b) Find the consumer surplus and producer surplus in the absence of the subsidy. What is the net economic benefit in the absence of a subsidy?

c) Find the consumer surplus and producer surplus in the presence of the subsidy. What is the impact of the subsidy on the government budget? What is the net economic benefit under the subsidy program?

d) Does the subsidy result in a deadweight loss? If so, how much is it?

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