Question

**(10pts)The Classical Model: Cobb-Douglas Production Function**:

(a) (6pts) Using calculus, demonstrate how the percentage of total income attributable to capital is equal to the exponent of capital in the Cobb-Douglas production function. As you work through the proof, explain what each variable represents as if you were explaining this to a fellow student for the first time.

(b) (4pts) Using the
following production function:
Y=10K^{0.25}*L^{0.75}, suppose that capital (K)
increases by 20%.

i) (2pts)
How much will total output increase in terms of percent? (Hint:
Start with the equation for MPk and recall that MP_{k}can
be written as the ΔY/ΔK.)

ii) (2pts) Explain in words what will happen to the rental price of capital and the real wage as a result of the increased amount of capital?

Answer #1

You are given this estimate of a Cobb-Douglas production
function: Q = 10K0.6L0.8 A. Calculate the output elasticities of
capital and labor. (Note: As shown on p. 300, for the Cobb-Douglas
production function Q = 10KaLb the output elasticity of capital is
EK = (%ΔQ/%ΔK) = a and the output elasticity of labor is EL =
(%ΔQ/%ΔL) =
b. B. Using what you found in Part (A), by how much will output
increase if the firm increases capital by 10...

Problem 2
Suppose that an economy’s production function is Cobb- Douglas
with parameter = 0.3.
c. Suppose that a gift of capital from abroad raises the capital
stock by 10 percent. What happens to total output ( in percent)?
The rental price of capital? The real wage?
d. Suppose that a technological advance raises the value of the
parameter A by 10 percent. What happens to total output ( in
percent)? The rental price of capital? The real wage?

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...

1. Using the Cobb-Douglas production function:
Yt =
AtKt1/3Lt2/3
If K = 27, L = 8 A = 2, and α = 1/3, what is the value of Y?
(For K and L, round to the nearest whole number) ______
2. If Y = 300, L = 10, and α = 1/3, what is the marginal product
of labor? ______
3. Using the values for Y and α above, if K = 900, what is the
marginal product of capital?...

Which is/are incorrect about the Cobb-Douglas production
function: Y equals K to the power of alpha L to the power of 1
minus alpha end exponent (0 < alpha < 1 )? All are correct it
increases in both K and L the share of total income that goes to
capital and labor depend on the amount of K and L it exhibits
diminishing marginal returns to both K and L it is constant returns
to scale

Assume that a competitive economy can be described by a
constant-returns-to-scale Cobb-Douglas production function and all
factors of production are fully employed. Holding other factors
constant, including the quantity of capital and technology,
carefully explain how a one-time, 10 percent increase in the
quantity of labor as a result of a special immigration policy, will
change the following: (12 points)
The level of output produced
The real wage of labor
The real rental price of capital
Labor share of total...

1. In previous problem, the given Cobb-Douglas
production function was Q = 6 L½
K½ and the cost function was given as:
C = 3L + 12K. For $384
of total cost, the optimum labor usage was determined to be 64, and
capital of 16.
a. If the cost
function now changes to C = 3L + 18K, it implies that the total
cost will become $480. Compute the new level of total cost for Q =
192. Can...

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...

1. Consider the Cobb-Douglas production function Q = 6 L^½ K^½
and cost function C = 3L + 12K. (For some reason variable "w" is
not provided)
a. Optimize labor usage in the short run if the firm has 9 units
of capital and the product price is $3.
b. Show how you can calculate the short run average total cost
for this level of labor usage?
c. Determine “MP per dollar” for each input and explain what the
comparative...

3. The White Noise Corporation has estimated the following
Cobb-Douglas production function using monthly observations for the
past two years:
ln Q = 2.485 + 0.50 ln K + 0.50 ln L + 0.20 ln N
where Q is the number of units of output, K is the number of
units of capital, L is the number of unit of labor, and N is the
number of units of raw materials. With respect to the above
results, answer the following...

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