Question

A firm’s production is represented by the following Cobb-Douglas function: ? = ?^2/3?^1/3. The rental rate, r, of capital is given by $200 and the price of labor is $100.

a) For a given level of output, what should be the ratio of capital to labor in order to minimize costs?

b) How much capital and labor should be used to produce those 300 units?

c) What is the minimum cost of producing 300 units?

d) What is the short run and long run cost of increasing output to 500 units?

e) Does this production function exhibit increasing, decreasing, or constant returns to scale? Answer based on the cost calculations in parts c and d.

Answer #1

A firm’s production is represented by the following Cobb-Douglas
function: ? = ?^2/3?^1/3. The rental rate, r, of capital is given
by $200 and the price of labor is $100.
a) For a given level of output, what should be the ratio of
capital to labor in order to minimize costs?
b) How much capital and labor should be used to produce those
300 units?
c) What is the minimum cost of producing 300 units?
d) What is the short...

33 II) A firm’s production is represented by the following
Cobb-Douglas function: ? = ?^1/3 ?^2/3. The rental rate, r, of
capital is given by $100 and the price of labor is $200.
a. For a given level of output, what should be the ratio of
capital to labor in order to minimize costs?
b. How much capital and labor should be used to produce those
300 units?
c. What is the minimum cost of producing 300 units?
d. What...

Cobb-Douglas Production Function & Cost of
Production
A firm’s production function is given as –
q =
2K0.4N0.6
What kind of returns to scale does this production technology
exhibit? Justify your answer.
Find out the expression for the marginal product of labor.
Find out the expression for the marginal product of
capital.
Find out the expression for MRTS.

A? Cobb-Douglas production function
A. exhibits constant returns to scale.
B. exhibits decreasing returns to scale.
C. exhibits increasing returns to scale.
D. can exhibit? constant, increasing, or decreasing returns to
scale.

2. Suppose a firm is producing 200 widgets. The firm’s
production function is Cobb-
Douglas with decreasing returns to scale. (This means we have
normal, convex
isoquants). The firm uses K’ units of capital and L’ units of
labor. Initially, the input prices
are w’ and r’. However, an exogenous shock in the labor market
causes an increase in
the wage rate, resulting in an increase in input prices from w’
to w’’ where w’<w’’. Using
a graph (of isoquant...

Suppose that a firm has the Cobb-Douglas production function Q =
12K ^ (0.75) L^ (0.25). Because this function exhibits (constant,
decreasing, increasing) returns to scale, the long-run average cost
curve is (upward-sloping, downward-sloping, horizontal), whereas
the long-run total cost curve is upward-sloping, with (an
increasing, a declining, a constant) slope.
Now suppose that the firm’s production function is Q =
KL. Because this function exhibits (constant, decreasing,
increasing) returns to scale, the long-run average cost curve is
(upward-sloping, downward-sloping,...

Given the Cobb-Douglas production function q = 2K 1 4 L 3 4 ,
the marginal product of labor is: 3 2K 1 4 L 1 4 and the marginal
product of capital is: 1 2K 3 4 L 3 4 .
A) What is the marginal rate of technical substitution
(RTS)?
B) If the rental rate of capital (v) is $10 and the wage rate
(w) is $30 what is the necessary condition for cost-minimization?
(Your answer should be...

For each part of this question write down a Cobb-Douglas
production function with the returns to scale called for and
perform a proof for each that shows the production function has the
correct returns to scale.
Constant returns to scale
Decreasing returns to scale
Increasing returns to scale
Increasing returns to scale

6.7 The production function
Q=KaLb where 0≤ a, b≤1 is called a Cobb-Douglas production
function. This function is widely used in economic research. Using
the function, show the following:
a. The production function in Equation 6.7 is a special case of
the Cobb-Douglas.
b. If a+b=1, a doubling of K and L will double q.
c. If a +b < 1, a doubling of K and L will less than double
q.
d. If a +b > 1, a doubling...

Consider the Cobb-Douglas production function F (L, K) =
(A)(L^α)(K^1/2) , where α > 0 and A > 0.
1. The Cobb-Douglas function can be either increasing, decreasing
or constant returns to scale depending on the values of the
exponents on L and K. Prove your answers to the following three
cases.
(a) For what value(s) of α is F(L,K) decreasing returns to
scale?
(b) For what value(s) of α is F(L,K) increasing returns to
scale?
(c) For what value(s)...

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