Question

A farmer uses three inputs to produce vegetables: land, labor, and capital. The production function for the farm exhibits diminishing marginal rate of technical substitution.

 a. In the short run the amount of land is fixed. Suppose the prices of capital and labor both increase by 5%. What happens to the cost-minimizing quantities of labor and capital for a given level of output? (Remember that there are three inputs, one of which is fixed).

 b. Suppose only the cost of labor goes up by 5%. What happens to the cost-minimizing quantity of labor and capital in the short run?

Answer 1

a. As the prices of both capital and labor will increase by 5 per cent, then the cost minimizing quantities of both capital and labor hired will decrease for a given level of output. This is because as the price of the factor of production increases, the quantity of the factor hired will reduce.

b. When the cost of labor goes up by 5 per cent , then the producer will substitute capital for labor and thus amount of capital hired in the short run will increase and quantity of labor hired in the short run will reduce because wages have increased and the price of capital has remained constant.

Answer 2

2.

a.

In profit maximization equilibrium, we have:

where,

MPL = Marginal product of labor

MPK = Marginal product of capital

w = wage rate

r = rental rate of capital

If both prices of labor and capital rise by 5%, we have:

The condition does not change and hence, equilibrium quantities of both inputs will stay same.

b.

If only cost of labor goes up by 5%, we have:

Since there are diminising returns to factors, MPL must be falling with rise in labor and MPK must be falling with rise in capital.

Now to increase MRTS, ratio of MPL to MPK must be raise in order to match to 1.05w / r.

MPL can be raised by reducing labor and MPK can be decreased by using more capital.

Hence in equilibrium, less labor and more capital will be used.