Generalizing the CES cost function The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is q 5 f 1k, l2 5 3 1αk2 ρ 1 1βl2 ρ4

γ /ρ

.

a. What is the total-cost function for a firm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is v/α and for a unit of labor input is w/β.

b. If γ 5 1 and α 1 β 5 1, it can be shown that this production function converges to the Cobb—Douglas form q 5 kαl

β

as ρ S 0. What is the total cost function for this particular version of the CES function?

c. The relative labor cost share for a two-input production function is given by wl/vk. Show that this share is constant for the Cobb–Douglas function in part (b). How is the relative labor share affected by the parameters

α and b?

d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in w/v? How is the direction of this effect determined by the elasticity of substitution, σ? How is it affected by the sizes of the parameters α and β?