Question

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

up to 4? Why or why not? Will you still use the same amount of K and L to produce 100 Y?

Answer #1

. Let the production function be Y=AL^1/2*K^1/2 where Y is
output, K is capital, L is labor and A represents the level of
technology.
a. What happens to the marginal product of capital as the level
of capital increases?
b. If L=100, A=5, the savings rate is 1/2 and the depreciation
rate is 1/3, what will the steady-state levels of capital, output
and consumption be?

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...

Suppose that you have estimated the following output function
where L is labor and K is capital: Y = K1/4 L1/2.
-You know that the current price of labor is $10 and capital
cost is $150 per machine (capital). You currently use 81 units of
capital. For #1a, the output (Y) is 100.
1a: How many employees (L) do you need to hire to achieve your
output goal?
b. Given a fixed level of capital (K=81) and a price of...

Suppose that you have estimated the following output function
where L is labor and K is capital: Y = K1/4L1/2
You know that the current price of labor is $10 and capital cost
is $150 per machine (capital).
You currently use 81 units of capital.
For #3a, the Output (Y) is 100
Using calculus, Please show all work in order to understand
where things are coming from. Thank you.
a.How many employees (L) do you need to hire...

A firm produces an output with the production function Q=K*L2,
where Q is the number of units of output per hour when the firm
uses K machines and hires L workers each hour. The marginal product
for this production function are MPk =L2 and MPl = 2KL. The factor
price of K is $1 and the factor price of L is $2 per hour.
a. Draw an isoquant curve for Q= 64, identify at least three
points on this curve....

Consider a production function for an economy:
Y = 20(L.5K.4N.1)where L is labor, K is capital, and N is land. In
this economy the factors of production are in fixed supply with L =
100, K = 100, and N = 100.
a) What is the level of output in this country?
b) Does this production function exhibit constant returns to scale?
Demonstrate by an example.
c) If the economy is competitive so that factors of production are
paid the...

A firm has production function y = f(K,L), where y is output, K
is capital, and L is labour. We have:
a. f(K,L) = K^0.4 + L^0.4
b. f(K,L) = (K^0.4)(L^0.4)
What are the firm's production function degree of
homogeneity?
I know the answer is 0.4 for A and for B it is 0.8.
But I don't know how to get those answers. I know m = degree of
homogeneity. I'm guessing they found A from m = 0.4. For...

An electronics plant’s production function is Q = L 2K, where Q
is its output rate, L is the amount of labour it uses per period,
and K is the amount of capital it uses per period.
(a) Calculate the marginal product of labour (MPL) and the
marginal product of capital (MPK) for this production function.
Hint: MPK = dQ/dK. When taking the derivative with respect to K,
treat L as constant. For example when Q = L 3K2 ,...

Consider a production function for an economy: Y =
20(L^0.5K^0.4N^0.1)
where L is labor, K is capital, and N is land. In this economy
the factors of production are in fixed supply with L = 100, K =
100, and N = 100.
a. What is the level of output in this country?
b. Does this production function exhibit constant returns to
scale. Demonstrate by example.
c. If the economy is competitive so that factors of production
are paid the...

Question 1) Relate to the following information. A firm has
production function F(K,L)=K^0.5L^0.5, and faces a cost of labor of
$5 per unit, and cost of capital of $20 per unit.
A) How much capital should the firm use to minimize cost if it
wants to produce 100 units of output? (Set up a Lagrangian function
where the cost function is the objective function and the
production target is the constraint.)
B) How much labor should the firm use to...

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