Derive a formula showing how the elasticity of labor demand

Question

Economics 713: Assignment 7-

Q1. There is a fixed total X of some good to be divided between two jealous individuals. If person 1 gets X₁ and 2 gets X₂, the utilities are

U₁ = X₁ - kX²₂ and U₂ = X₂ - kX²₁

where k is a positive constant.

a. How would you judge the efficiency of alternative allocations (X₁, X₂)?

b. Suppose a social planner's objective is to maximize the sum of utilities. What is the optimal allocation? Interpret your result.

Q2. The Hicks-Marshall Rules

There is an old piece of economic analysis, going back at least to Marshall, of what determines the elasticity of demand for a factor of production. When the real wage rate rises, for example, producers are more inclined to use a capital-intensive technology, and consumers also tend to substitute away from goods (such as services) which are labor intensive. The following quaint summary is from Pigou:

i. "The demand for anything is likely to be more elastic, the more readily substitutes for the thing can be obtained."

ii. "The demand for anything is likely to be less elastic, the less important is the part played by the cost of that thing in the total cost of some other thing, in the production of which it is employed."

iii. "The demand for anything is likely to be more elastic, the more elastic is the supply of co-operant agents of production."

iv. "The demand for anything is likely to be more elastic, the more elastic is the demand for any further thing which it contributes to produce.

Consider a good which is produced using labor and capital and a CES technology, with constant returns to scale. Everyone has free access to this technology, and so in equilibrium the good must sell at average cost, and each firm which produces it just breaks even, as long as they minimize the cost of production. The total quantity produced is determined by the market demand curve (i.e. it is the total quantity demanded when price equals average cost).

Suppose the demand curve in the product market has constant elasticity, η. Suppose there is an upward sloping market supply curve of capital, with constant elasticity, e, so that when the industry demand curve for capital shifts out because of a wage increase, the price of capital must rise.

a. Derive a formula showing how the elasticity of labor demand, λ, depends on labor share, s, the elasticity of substitution, σ, and the demand elasticity η, assuming that the supply curve of capital is infinitely elastic.

b. Use your formula to determine whether the rules given above are correct, ignoring rule iii.

c. ("Optional") Extend your formula to cover an arbitrary value for e, and re-check all of the rules, including iii.

Q3. 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 Ai, 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.