Macroeconomics Before the Great Depression

2. Consider the production function Y=K^aL^1-a, where 0<a<1. Does the marginal product of labor increase or decrease when the capital stock increases? Show the effect of an increase in the capital stock on employment and output.

The direct way to approach this question is to differentiate Y with respect to L to get the marginal product of labor:

dY/dL=MP_L=(1-a)K^aL^-a.

Now we want to know how this expression changes when the capital stock increases. To find this, just differentiate again, this time with respect to capital:

d(MP_L)/dK=d(dY/dL)/dK=d²Y/(dLdK)=a(1-a)K^a-1L^-a>0

which is greater than zero because 0<a<1. Thus, the marginal product of labor increases when capital increases. This is equivalent to saying that the demand curve of labor shifting upward at every possible quantity of labor when capital is increased (see the answer to question 1).