Question

Need as much details as possible. Microeconomics.

A firm produces output using a single input L. The firm’s marginal productivity is increasing in L. The firm currently produces 50 units at the marginal cost of 100. If the firm were to produce 60 units,

a. We cannot determine how the marginal cost changes.

b. The marginal cost would be less than 100.

c. The marginal cost would be greater than 100.

d. The marginal cost would equal 100.

Answer 1

b. The marginal cost would be less than 100.

The firm is producing output using a single input L.The firm’s marginal productivity is rising in L. So we can say that the firm is operating under increasing return to scale. As long as the marginal productivity of the variable input rises, the marginal cost will decrease.

Here the only input is L. Let the price of L is 'w' or wage. So the firm's total cost would be the product of 'w' and 'L'.

Or, Total cost (TC) = w * L ........ (1)

Marginal cost (MC) is the change in total cost due to one unit increase in output (Q).

MC = d(TC) / dQ = TC / Q , where 'd'= '' stands for change

Marginal productivity of a factor is the change in total output due to change in one additional unit of the factor.Here the only factor is 'L'.

Marginal productivity of 'L' (MP_L) = dQ / dL = Q / L

Now differentiating equation (1) i.e TC = w * L , with respect to Q, we get,

d(TC)/dQ = w * dL/dQ , 'w' is fixed in short-run.

Or, MC = w * [ 1/(dQ /dL)]

Now, as  dL / dQ = 1/MP_L,

MC = w * (1 /MP_L) .....(2)

So from equation (2), we see that the marginal cost is the product of the factor price (here, w) of the variable factor and the reciprocal of the marginal productivity of the variable factor (here, MP_L).

When , MP_L rises , (1/ MP_L) falls ,

MC = w * (1 /MP_L) also falls

Now, as the firm's marginal productivity is increasing in L, so its marginal cost is decreasing in L. The firm currently produces 50 units of output at the marginal cost of 100. Now, if the firm were to produce 60 units of output , its marginal cost(MC) would be less than the marginal cost at 50 units of output.As MP_L is gradually rising , so MC is gradually falling with the rise in output.

So,  If the firm were to produce 60 units of output, the marginal cost would be less than 100.

'Q' shows the quantity measured on the horizontal axis, 'MP' shows marginal productivity and 'MC" shows marginal cost , measured on vertical axis.'MP' is the marginal product curve and 'MC' is the marginal cost curve. When MP is maximum, MC is minimum.

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