Economics 713: Assignments

Many assignments will be drawn from a collection that is available in either pdf (Acrobat) or postscript form. Many of the questions are at an intermediate level (but you won't be told which ones until it is too late). In most cases, it is a good idea to draw diagrams (if you find yourself differentiating a Lagrangean you are probably lost).

• Ghostview (to read and print postscript files)
• Acrobat Reader (to read and print PDF files)

Assignment 1: Problems 2, 12, 19
Due: 12 noon, Monday, March 23 (in TA mailbox)

• Answers (Antonovics and Krasnokutskaya)

Assignment 2: Problems 11, 16, 72, 73
Due: 12 noon, Monday, March 30 (in TA mailbox)

• Answer for Problem 11

Assignment 3: Problems 6, 55, 71, 74
Due: 5pm, Monday, April 6 (in TA mailbox)

• Answers (Antonovics)

Assignment 4: Problems 10, 57, 21C2 (Mas-Colell, page 812), 21D6 (Mas-Colell, page 813)
Due: 5pm, Monday, April 13 (10 and 21C2 in Antonovics's mailbox; 57 and 21D6 in Krasnokutskaya's mailbox)

Assignment 5: Problems 27, 75, 84 (below), 13C5 (Mas-Colell, page 475),
Due: 5pm, Monday, April 20 (27 and 13C5 to Antonovics; 75 and 84 to Krasnokutskaya)

75. A firm has a large accumulated inventory of a storable good. There are no competing sellers of this good, and there is a linear relationship between the quantity sold in each period and the price that the firm sets. Inventory holding costs are negligible, but the cost of production is higher than any buyer would ever pay. The firm can borrow and lend freely at a fixed discount rate, and acts to maximize the present discounted value of profits. How will the firm set prices?

84. Show from first principles that the Condorcet Paradox is inconsistent with single-peaked preferences.

Assignment 6: Problems 76, 77, 85 (below) , 23B2 (Mas-Colell, page 918),
Due: 5pm, Monday, April 27 (76 and 23B2 to Antonovics; 77 and 85 to Krasnokutskaya)

85. Suppose that both shoes and computers are produced using two factors, skilled and unskilled labor. Both technologies are of the CES form, with the same elasticity of substitution between skilled and unskilled labor, but with different factor intensities.

• Assuming that the two product prices are given, and that both products are produced in a competitive economy, show explicitly (and succinctly) how the factor prices are determined, given the product prices, the factor intensity parameters, and the elasticity of substitution.
• Using your result, what explanations can you suggest for recent trends showing that skilled wages have risen while unskilled wages have fallen?

Assignment 7: Problems 58, 59, 86 (below) 13C4 (Mas-Colell, page 475)
Due: 5pm, Monday, May 4 (58 and 13C4 to Antonovics; 59 and 86 to Krasnokutskaya)

86. [Brock, Jan 98 Prelim] Consider production of two products with separable demands D₁(q₁), D₂(q₂) with total cost function C(q₁,q₂) = F + c₁q₁ + c₂q₂ where c_i is constant marginal cost of product i = 1,2 and F > 0 is fixed cost which must be borne if a positive amount of either good is produced. Define consumer benefit, B_i(q_i) to be the area under the demand curve from zero to q_i. Let R_i(q_i) = D_i(q_i)q_i denote revenue from sales of product i. Examine three problems:

(1) (Social Welfare Maximum) Max B₁(q₁) + B₂(q₂)-C(q₁,q₂)

(2) (Second Best) Max B₁(q₁) + B₂(q₂) - C(q₁,q₂), s.t. R₁(q₁) + R₂(q₂) > C(q₁,q₂)

(3) (Monopoly) Max R₁(q₁) + R₂(q₂) - C(q₁,q₂).

a. Use the Kuhn-Tucker Theorem to write out the first order necessary conditions for all three problems. Is fixed cost covered by the solution to (1)? Why or why not?
b. "Ramsey" numbers for product i=1,2 are defined by R_i.= (p_i-c_i.)e_i/ p_i , where e_i is elasticity of demand. Compare Ramsey numbers for problems (1)-(3), assuming interior solutions for all products in all three cases.
c. Assume linear demands D_i(q_i) = A_i - M_i q_i. Locate sufficient conditions for shut down to be socially optimal. Can there be a case where shut down is not socially optimal but it is Second Best optimal to shut down? How can you "fix" this social problem if it is possible?
d. Assume linear demands as in (c). Draw a pair of diagrams side by side with the demand curve for each product and the constant marginal cost line on each. On each demand curve tick off the vertical intercept, call it A_i, tick off the price, call it p_i, and tick off the marginal cost, call it c_i. Show for problem (2), for interior optima, that the ratio of the line segment, A_i - p_i, to the line segment, A_i - c_i, is equated across the two goods i=1,2.