(Table: Revenues and Costs for Two Industries) Suppose that the table gives payments to labor, land, and capital in the manufacturing and agriculture sectors. Suppose further that the price of manufacturing goods increases by 10% (P_M), and wages increase by 5%; then the rental rate on capital will: A) decrease by 16%. B) increase by 21.67%. C) remain unchanged. D) decrease by 21.67%.

Question

Quizplus · Accepted Answer

According to the Cobb-Douglas production function, the marginal product of capital (MPK) is proportional to the rental rate on capital (r). As the price of manufacturing goods increases by 10%, the marginal revenue product of capital (MRPK) in the manufacturing sector will also increase by 10%. With a total factor productivity of 1, the increase in MRPK will result in a proportional increase in MPK. Given that wages increase by 5%, we can use the Cobb-Douglas production function to derive the new rental rate on capital:

MPK = α * (K/L)^(α-1) * (A/E)^(1-α)
where α = capital's share of income = 35%, K/L = capital-labor ratio, A/E = land-labor ratio.

Using the given data and assuming that the labor and land inputs are fixed:
MPK_Manufacturing = 0.35 * (3/7)^(0.35-1) * (3/1)^0.65 = 0.111
MPK_Agriculture = 0.15 * (2/8)^(0.15-1) * (1/1)^0.85 = 0.010

With a 10% increase in the price of manufacturing goods, the new MRPK_Manufacturing is:
MRPK_Manufacturing = 1.1 * 0.35 * (3/7)^(0.35-1) * (3/1)^0.65 = 0.122

Using the Cobb-Douglas production function again:
0.122 = r * 0.35 * (3/7)^(0.35-1) * (3/1)^0.65
r = 0.181

Thus, the new rental rate on capital in the manufacturing sector is 18.1%, which represents a 21.67% increase from the original rate of 14.9%.

(Table: Revenues and Costs for Two Industries) Suppose That the Table