Consider a Cobb-Douglas production function: f(l, k) = Alα k1−α , where A is the total factor of productivity (a constant greater than 1), 0 < α < 1, lrepresents labor, and k represents capital. The following sub-questions will guide you through showing that the elasticity of substitution is constant. a) Find the marginal product of labor. Verify that this production function exhibits diminishing marginal productivity of labor. b) Find the marginal product of capital. Verify that this production function exhibits diminishing marginal productivity of capital. c) Find the marginal rate of technical substitution. Write your answer as MRT S = . . . d) In part (C), you should’ve found the MRTS as a function of the input ratio, k l . Take the
Consider a Cobb-Douglas production function:
f(l, k) = Alα k1−α
,
where A is the total factor of productivity (a constant greater than 1), 0 < α < 1, lrepresents
labor, and k represents capital. The following sub-questions will guide you through showing that
the elasticity of substitution is constant.
a) Find the marginal product of labor. Verify that this production function exhibits diminishing
marginal productivity of labor.
b) Find the marginal product of capital. Verify that this production function exhibits diminishing
marginal productivity of capital.
c) Find the
d) In part (C), you should’ve found the MRTS as a function of the input ratio, k
l
. Take the
absolute value of both sides and solve for the input ratio, so that the expression gives the
input ratio as a function of MRTS (i.e. kl = . . .). Take the log of both sides, then take the
derivative with respect to the log of MRTS. Is the elasticity of substitution constant? (Hint: it may or may not help to substitute for the log of MRTS using a variable of your choosing).
2). Suppose there’s a firm with the following Cobb-Douglas production function:
f(`, k) = l0.5k0.5,
where ` represents labor, and k represents capital such that give wage w and rent r, the total cost is wl + rk.
a) What sort of returns to scale does this production function exhibit?
b) We typically think that the firm treats capital as a constant in the short-run because it’s difficult to adjust the level of machinery used in production on a regular (e.g. monthly) basis.
For the rest of this problem, let’s suppose the firm is in the short-run so that capital is a
constant ¯k. Find the amount of labor ¯l required to produce some level of output ¯y. This is
not a profit maximization or cost minimization problem.
c) What is the short-run total cost to produce ¯y? Plug in what you found for part (B) so that
the short-run total cost depends only on w, r, ¯y, and ¯k.
d) Find the short-run average cost using the short-run total cost from part (C).
e) Find the short-run marginal cost using the short-run total cost from part (C).
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