The Demand for Labor

A profit maximizing condition is that a firm should hire more labor as long as

W< P* MPN

P is the price per unit of output and MPN is the amount of output an additional unit of labor will produce. So P* MPN is the value of the output an additional hour of labor will produce. The wage rate,W , is the cost of an additional hour of labor. So if the cost of an additional hour of labor is less than the value of output that can be produced by that hour, you can increase profits by hiring that hour of labor. The firm should stop hiring when

W= P* MPN

The Cobb—Douglas production function is one commonly used in economics. It is written as

q( K, N) = Na K b where q is the amount of output produced using K units of capital and N units of labor. Here we will let a= b = 0.5 so

q( K, N) = N0.5 K 0.5

Define such a function in Maxima (actually a function of N only since we will hold capital fixed).

Next fix capital by giving it a value Next, plot the production function

The diff(f(x), x ) can be used to compute the marginal product of labor

Next assume that wage and price have certain values

and now find the profit maximizing level of labor hours The result is stored in ans. A decimal representation can be found using the float() function

So the profit maximizing level is approximately 250 hours each day.

Next, let’s see what happens if the wage rate is increased and the profit maximizing level of hours is determined for the higher wage rate.

The profit maximizing number of hours of labor has fallen as the wage rate increases. We can plot the W and P*MPN to see a graphical solution

and W=P*MPN at about 62.5 hours of labor.

Let’s repeat this assuming a further increase in the wage rate (N should decline, right?)

and the profit maximizing level of hours is about 15.625 hours per day. Graphically