The Marginal Product of Labor If we take a total product of labor function of the form Q = Ax/Rx/E and we take A and K to be fixed in the short run, we can define a marginal product of labor function that tells us how much output increases for each unit increase in labor. The marginal product of labor at say L : 10 will be given by the slope of the total product of labor function at that point. If you know a bit of calculus this is easy to calculate as: MPL(K,L): =-A- (1) 8Q 1 K 8L 2 L There is a neat trick for constant returns to scale CobbDouglas production functions of the form F(K, L) = A' K\" ' LHI . Their marginal product of labor will always be in the form: F(K,L) MPL(K, L) = (1 a)- L 1 With CL = 5 you can confirm this gives us equation (1). For a CRS CobbDouglass production function, the marginal product of labor will be a proportion of the average product of labor F(K7 L) APL(K, L) : , which is simply total output . . . . . 1 dIVIded by labor Input. For this case With a = E you can confirm that this leads us to the same result as equation (1) above with a marginal product of labor equal to one half the average product of labor. Note that we can just as easily derive a formula for the average product of capital F(K, L) APK(K, L) = as well as a marginal product of capital F K L MPK(K,L):d- (') K f : a . A . _ K G) SFM100 Review Suppose we have a production function of the form F(K,L)=A * sqrt(K) * sqrt(L). A and K are fixed. What happens to the marginal product of labor, at any given level of L, as we increase the amount of fixed capital? A The MPL stays the same \\/ The MPL increases Graph 1.a On the top half of a page draw a graph of the total product of labor function for the Cobb-Douglas function above with A = 1 and K : 10 (this should be simple as you can mimic what's drawn in the geogebra app). Draw in indicator lines to indicate L : 10 on the horizontal axis and the corresponding level of output on the vertical axis Graph 1.b On the same graph now draw in the total product of labor graph for the same function afterthe factory increases its capital from K = 10 to K = 20. Draw in the new total product of labor curve and again draw in indicator lines to indicate L=10L=10 on the horizontal axis and the corresponding level of output on the vertical axis