Question

1. Assume the following production function: X = 50 N^0.5, where N is labor. Calculate the marginal cost of labor.
2. Assume now that N = 25, and calculate the size of the marginal cost of labor. Explain with words what this number shows.
3. From this point on, ignore that N = 25. Assume now that the production function expands with/includes capital (K), and that the production (as a whole) is given by X = f(N,K). Explain briefly based on the production function what is meant by isoquants.
4. Consider a company that will maximize the production for a given cost. Explain how the company will optimize (their position) in the factor market. (Sorry for the bad English, just skip if you don’t understand what I mean. With optimize, I basically mean what the company will do in the factor market).
5. Explain and show graphically how an increase in the price of labor will affect the demand for the two inputs (factors of production).

Answer 1

NOTE: I am assuming you mean Marginal Product and not Marginal Cost.

a) Now,  X = 50 N^0.5,

Thus, MP = dX/dN = 25X^-0.5

b) For N = 25, MP(25) = 5. This implies that an additional unit of labor will result in increase in output by 5 units.

c) For X = f(N,K) implies that the firm can now carry out production using two inputs - labor and capital. Now, an isoquant represents a particular level of production along which the level of output remains the same for different level of inputs. Thus, using the new production function, the firm can employ several combinations of labor and capital to produce a given/pre-specified level of output.

d) Say the company faces the cost of capital 'r' and cost of capital 'r'. Now, the company can optimize his production iff the Marginal Products of the two factors equals the ratio of the factor prices, i.e., at the point where

MP_N/MP_K = w/r

Basically, at this point the trade-off between the two inputs for the firm equals the market asking trade-off for the two inputs.