Assignment 3-D, Econ 2101

1

.Consider the following production function

Q = KLb,

where b is a positive constant, with b < 1

.a

. Using calculus, show that the marginal product of labor and the marginal product of

capital are always positive

. b

. Using calculus, and working with repeated partials, determine whether capital and labor

each exhibit diminishing marginal returns

. c

. Using calculus, determine how an increase in capital affects the marginal product of

labor

. d

. Does the production function exhibit increasing, decreasing, or constant returns to scale?

Only algebra, and no calculus, is needed to answer this part

.

2

.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

.a

. 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

. b

. 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)

. c

. 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

. d

. Given the variable cost function, derive the average variable cost function

. e

. Sketch the marginal cost and average variable cost functions in a diagram

.

3

.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

function

.(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?