Assignment 3-D, Econ 2101
Consider the following production function
Q = KLb,
where b is a positive constant, with b < 1.
Using calculus, show that the marginal product of labor and the marginal product of
capital are always positive.
Using calculus, and working with repeated partials, determine whether capital and labor
each exhibit diminishing marginal returns.
Using calculus, determine how an increase in capital affects the marginal product of
Does the production function exhibit increasing, decreasing, or constant returns to scale?
Only algebra, and no calculus, is needed to answer this part.
Consider the production function in Question 1 above.
Suppose we are in the short-run and K is
fixed at 1 unit (K = 1), further suppose that b = ½.
Derive the expression for the (short-run) “labor requirement function”–this expression
simply tells us how much labor the firm needs to employ given how much output it
chooses to produce.
It is obtained by manipulating/rearranging the production function
given that K =1 and b = ½.
Letting w denote the wage rate (or price) of labor, write down an expression for the
variable cost function (it must have output Q as its argument).
You will need to make
use of the labor requirement function derived in part (a).
Given the variable cost function, use calculus to derive the marginal cost function and
show that it is the same as w/MPL, where MPL is the marginal product of labor.
Given the variable cost function, derive the average variable cost function.
Sketch the marginal cost and average variable cost functions in a diagram.
A firm produces shirts using the following production function:
Q = K1/2 + L1/2
where K and L are capital and labor, respectively.
(a) Using calculus, derive an expression (formula) for the MRTS of L for K for this production
(b) Suppose labor costs $9 per unit and capital costs $1 per unit.
Suppose the firm wants to
produce 100 shirts.
Calculate the cost-minimizing quantity of labor and of capital the firm should
use to produce 100 shirts.
(c) What is the corresponding (minimum) total cost of producing 100 shirts?