We were talking about the Cobb-Douglas production function. And a Cobb-Douglas production
function shows you how much output you can get for different combinations of two inputs, say,
capital and labor. Suppose our Cobb-Douglas production function is just this: Q equals K to the 0.5 power times L
to the 0.5. And suppose we want to evaluate the marginal product of labor when K is equal to 64 and L is
equal to 100. How would we do that? Let K be equal to 64. Let L be equal to 100.. For these values of K and L, evaluate the marginal product of labor. Find the marginal product of labor. What is the marginal product of labor? Mathematically. What is the marginal product of labor? [Student comment] Whenever you’re talking about a partial derivative, it’s the partial derivative of the dependent variable with respect to the independent variable. [Student comment] It’s partial Q partial L. The marginal product of labor is just partial Q partial L. And what this tells you is the rate of change of output per unit change in labor for given values of K and L. If we were to take partial Q partial L, what is it? What is partial Q partial L? [Student comment] Bring this one-half or 0.5 down. [Student comment] So, it’s 0.5 times K to the 0.5 times L to the 0.5 over – No, actually, let’s do it a slightly different way. What do you do to this exponent right here? When you differentiate with respect to L? [Student comment] So, it becomes L to the minus 0.5. [Student comment] Which we could write as 0.5 times K to the 0.5 divided by L to the 0.5. Because L to the minus
0.5 power is just 1 over L to the 0.5. And, actually, we could simplify this a little bit more. Notice K and L have the same exponent. So, we could just write this as 0.5 times K over L raised to the 0.5. And raising K over L to the 0.5 is the same thing as doing what? Square root. So, it’s the 0.5 times the square root of K over L. Now if you want to evaluate the marginal product of labor at this value of K and L, you just plug
those values in. And, in this case, it might be easier to look at this formulation right here. What is the square root of 64? 8. So, it’s just 0.5 times 8, the square root of 64. What is the square root of 100? [Student comment] 10. So, it’s 4 over 10 or 0.4. So, the marginal product of labor is just 0.4. So, what this means is approximately, if you’re currently using 64 units of capital, 100 units of labor, if you use one more unit of labor, your output will increase by 0.4 units. I say approximately because a partial derivative is really the limit of the difference quotient, delta Q over delta L as L goes to zero. So, a one unit change in labor is not the same as an infinitesimally small change, but it will be approximately right.