Chapter 6 Henderson and Quandt Microeconomics, Questions 6.12-6.13_ Solutions

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6-12 Consider the following markets which are characterized by lagged supply response:

(a) = 40-10p,; = 2+9

(b) = 30-5p₁; = 20 -

Determine the equilibrium price and quantity for each market. Assume an initial price 20 percent

below the equilibrium price for each market, and determine the number of periods necessary for

each price to adjust to within 1 percent of equilibrium.

(a) The market equilibrium price and quantity are determined by equating demand and supply:

Dt = St

40 – 10p = 2 + 9pt-1

Rearranging the terms and substituting t-1 with t-2:

P = 4.15, q = 16.6

To determine the adjustment process, we start with an initial price of:

P₀ = 0.8 * 4.15 = 3.32

And solve for p1, p2, p3, ... until convergence:

P₁ = 4.18, error = 0.96%

P₂ = 4.15, error = 0.00%

So, it takes two periods for the price to adjust to within 1% of equilibrium.

(b) Equating demand and supply:

Dt = St

30 – 5p = 20 – pt-1

P = 5, q = 5

Starting with an initial price of:

P₀ = 0.8 * 5 = 4

And solving for p1, p2, p3, ... until convergence:

P₁ = 4.5, error = 10%

P₂ = 4.75, error = 5%

P₃ = 4.88, error = 2.67%

P₄ = 4.94, error = 0.80%

P₅ = 4.97, error = 0.40%

P₆ = 4.99, error = 0.20%

P₇ = 5.00, error = 0.04%

It takes seven periods for the price to adjust to within 1% of equilibrium.

6-13 A sugar refiner has a strictly concave production function for which labor and raw sugar cane

are the only inputs. His production of refined sugar and purchase of inputs will take place next spring, but he must determine his future production level today. The future prices of refined sugar and labor are known with certainty, but the price of raw sugar will assume one of the values (.....,) with the respective probabilities (v₁, ...,.). Show how you would determine his futures-market raw sugar demand.

The sugar refiner’s problem is to choose the optimal quantity of raw sugar cane, denoted as q, to purchase today, given the prices of labor, refined sugar, and the uncertain price of raw sugar cane. Let p_r denote the uncertain price of raw sugar cane, which can take on n possible values (r_1, ..., r_n), each with a probability of occurrence (v_1, ..., v_n).

The refiner’s production function is strictly concave, implying that the marginal product of raw sugar cane is diminishing. Suppose that the price of refined sugar is denoted as p, and the price of labor is denoted as w. Then, the refiner’s profit-maximization problem can be written as:

Max [π = pq – wL – p_rq]

Where L is the amount of labor input, and q is the amount of raw sugar cane input. The first-order condition for this problem is:

P = p_r[1 – (dQ/dq)/(Q/q)]

Where Q is the quantity of refined sugar produced, and (dQ/dq)/(Q/q) is the elasticity of output with respect to raw sugar cane input. This elasticity depends on the specific form of the production function.

Since the refiner must choose q today based on the uncertain price of raw sugar cane, he will form his demand for raw sugar cane based on his expected profit. The expected profit can be written as:

E(π) = p_qq – wL – E(p_r)q

Where E(p_r) is the expected price of raw sugar cane. The refiner’s optimal quantity of raw sugar cane can then be obtained by maximizing this expected profit with respect to q:

dE(π)/dq = p_q – E(p_r) – w(dQ/dq)/(Q/q) = 0

This condition determines the optimal quantity of raw sugar cane that the refiner should purchase today, given his expectations about the future price of raw sugar cane. The futures-market raw sugar demand is the function that relates this optimal quantity to the expected price of raw sugar cane.

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