Question

Consider the following Cobb-Douglas production function: y(K,L) = 2K^(0.4)*L^(0.6), where K denotes the amount of capital and L denotes the amount of labour employed in the production process.

a) Compute the marginal productivity of capital, the marginal productivity of labour, and the MRTS (marginal rate of technical substitution) between capital and labour. Let input prices be r for capital and w for labour. A representative firm seeks to minimize its cost of producing 100 units of output.

b) By applying the Lagrangian method, derive the optimal demands for capital and labour.

c) What is the impact of an increase in w on the optimal demand for labour?

d) If w = r = 1, what is the cost of producing 100 units of output?

Answer #1

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...

2. Consider a Cobb-Douglas production function Q = A . L^a . K^b
. Answer the following in terms of L, K, a, b
(a) What is the marginal product of labour ?
(b) What is the marginal product of capital ?
(c) What is the rate of technical substitution (RTS L for
K)?
(d) From the above what is the relation between K L and RT
SL,K?
(e) What is the relation between ∆ K L ∆RT SL,K (f)...

A cost-minimizing firm has the following production function:
Q=LK+2M. Where L denotes Labor, K denotes Capital, and M denotes
Materials. The prices for the inputs are as follows: w=$4, r=$8,
and m=$1. The firm is operating in the long run. Answer the
following questions as you solve for the total cost of producing
400 units of output. Assume an interior solution (i.e. positive
values of all inputs).
a) Set up constrained optimization problem of the firm:
b) Write out the...

Assuming a Cobb-Douglas production function with constant
returns to scale, then, as L rises with K and A constant, it will
be the case
Group of answer choices
Both the marginal product of labour and the marginal product of
capital will fall
Both the marginal product of labour and the marginal product of
capital will rise
The marginal product of labour will rise and the marginal
product of capital will fall
The marginal product of labour will fall and the...

A firm’s production function is given by Q = 5K1/3 +
10L1/3, where K and L denote quantities of capital and
labor, respectively.
Derive expressions (formulas) for the marginal product of each
input.
Does more of each input increase output?
Does each input exhibit diminishing marginal returns?
Prove.
Derive an expression for the marginal rate of technical
substitution (MRTS) of labor for capital.
Suppose the price of capital, r = 1, and the price of labor, w
= 1. The...

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?...

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...

The production function is Y=K0.5L0.5 where K is capital, L is
labor and Y is output. The price of L is 1 and the price of K is
2.
a) Find the optimal levels of K and L that should be employed to
produce 100 units of output. What is
the cost of producing this level of output?
b) Will the optimal capital-labor ratio change if the price of
labor goes up to 2 and the price of K goes...

In the Cobb-Douglas production function :
the marginal product of labor (L) is equal to β1
the average product of labor (L) is equal to β2
if the amount of labor input (L) is increased by 1 percent,
the output will increase by β1 percent if the amount of Capital
input (K) is increased by 1 percent,
the output will increase by β2 percent
C and D

Suppose a Cobb-Douglas Production function is given by the
following:
P(L,K)=10L0.9K0.1
where L is units of labor, K is units of capital, and
P(L,K)P(L,K) is total units that can be produced with this
labor/capital combination. Suppose each unit of labor costs $400
and each unit of capital costs $1,200. Further suppose a total of
$600,000 is available to be invested in labor and capital
(combined).
A) How many units of labor and capital should be "purchased" to
maximize production subject...

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