Question

A monopolist is the sole purchaser of labor in the locality, the supply can be described by:

W = 10 + 0.1L

where W is the daily wage and L is the number of "personal days" work.
The company's production function can be written:

Q = 10L

where Q is output.
The demand for these outputs can be described by:

P = 41 - (Q / 1000)

where P is the output rate.
What is the profit maximizing level of output for the company? How much manpower is then used? Which output price prevailing then?

Answer 1

A Monopolist tends to produce that level of output when his Marginal revenue from producing an additional unit of output is equal to the Marginal cost of producing an additional unit of output i.e., MR = MC and at that point, his profits are maximised.

We have W = 10 + 0.1L, So if L = 1 then W = 10.1, if L = 2 then W = 10.2, if L = 3 then W = 10.3 and so on. From this, we can say that MC of hiring additional labour is 0.1. We could have found it simply by differentiating the wage function w.r.t L which gives 0.1 but I have done it that way to explain in a bit detail. So, at last, we know the MC = 0.1.

Now we have Q = 10L and P = 41 - (Q/1000), Therefore, TR = P*Q = [41 - (Q/1000)]*Q = 41Q - Q²/1000. Now we will differentiate the TR function w.r.t Q to find the MR. So MR = 41 -Q/500. We know that Q = 10L therefore MR = 41 - 10L/500 = 41 - L/50.

Now we equate MR = MC => 41 - L/50 = 0.1 => L/50 = 40.9 => L = 40.9*50 = 2045. At L= 2045, Q = 10*2045) = 20450 units and P = 41 - (20450/1000) = 20.550.

Therefore the profit maximising level of output is 20450 units and 2045 person-days are used. The output price is equal to $20.550.