Question

Bonus Question. Suppose the production function for a firrm is Q(K,L) = K1/2L1/2, so the marginal...

Bonus Question. Suppose the production function for a firrm is Q(K,L) = K1/2L1/2, so the marginal product of labor is MPL = 1 2 K1/2L−1/2 and the marginal product of capital is MPK = 1 2 K−1/2L1/2.

a) Find the equation of the isoquant for Q = 1. That is, when Q = 1, find L as a function of K or K as a function of L to obtain an equation for the isoquant.

b) Find K1, K2, L3, and L4 so that all four of the following points are on the isoquant for Q(K,L) = 1: (K1,1) (K2, 1/4) (9,L3) (16,L4)

c) Find the marginal product of labor and marginal product of capital for all four points. You may want to first nd the capital to labor ratios, K/L.

d) Find the marginal rate of technical substitution for all four points.

e) Use the definition of the elasticity of substitution to calculate σ for points 1 and 2.

f) Repeat part (e) for points 3 and 4. Comparing your answer for (e) and (f), can we say that the production function is a CES function

Homework Answers

Answer #1

The production function is , and the MPL is or .

(a) For Q=1, the isoquant will be or or .

(b) All the stated points would satisfy the isoquant equation . In other words, for any point on isoquant, would satisfy the equation or .

As (K1,L1=1) satisfies the isoquant, we have or or .

As (K2,L2=1/4) satisfies the isoquant, we have or or .

As (K3=9,L3) satisfies the isoquant, we have or or .

As (K4=16,L4) satisfies the isoquant, we have or or .

(c) The marginal product of labor would be as or . The values would be as below.

  • For (K1,L1), it would be .
  • For (K2,L2), it would be .
  • For (K3,L3), it would be .
  • For (K4,L4), it would be .

The marginal product of capital would be as or . The values would be as below.

  • For (K1,L1), it would be .
  • For (K2,L2), it would be .
  • For (K3,L3), it would be .
  • For (K4,L4), it would be .

(d) The MRTS would be or or . The values would be as below.

  • For (K1,L1), it would be .
  • For (K2,L2), it would be .
  • For (K3,L3), it would be .
  • For (K4,L4), it would be .

(e) The definition of elasticity of substitution is that it is the ratio of rate of change in capital labor ratio and rate of change in MRTS.

The change in capital labor ratio is (similar to midpoint elasticity formula) or or 1.7647. The change in MRTS is or or 1.7647. Hence, the elasticity of substitution is 1.7647/1.7647 or 1.

(f) The change in capital labor ratio is or or 1.0386. The change in MRTS is or or 1.0386. The elasticity of substitution is hence 1.0386/1.0386 or 1.

Thus, YES : We can definitely say that the production function is a (type of) CES function, since changing the capital labor combinations did not change the elasticity of substitution.

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
A firm has the production function: Q = L 1 2 K 1 2 Find the...
A firm has the production function: Q = L 1 2 K 1 2 Find the marginal product of labor (MPL), marginal product of capital (MPK), and marginal rate of technical substitution (MRTS). Note: Finding the MRTS is analogous to finding the MRS from a utility function: MRTS=-MPL/MPK. Be sure to simplify your answer as we did with MRS. A firm has the production function: Q = L 1 2 K 3 4 Find the marginal product of labor (MPL),...
Suppose a firm’s production function is given by Q = L1/2*K1/2. The Marginal Product of Labor...
Suppose a firm’s production function is given by Q = L1/2*K1/2. The Marginal Product of Labor and the Marginal Product of Capital are given by: MPL = (K^1/2)/2L^1/2 & MPK = (L^1/2)/2K^1/2) a) (12 points) If the price of labor is w = 48, and the price of capital is r = 12, how much labor and capital should the firm hire in order to minimize the cost of production if the firm wants to produce output Q = 10?...
Consider the production function Q = f(L,K) = 10KL / K+L. The marginal products of labor...
Consider the production function Q = f(L,K) = 10KL / K+L. The marginal products of labor and capital for this function are given by MPL = 10K^2 / (K +L)^2, MPK = 10L^2 / (K +L)^2. (a) In the short run, assume that capital is fixed at K = 4. What is the production function for the firm (quantity as a function of labor only)? What are the average and marginal products of labor? Draw APL and MPL on one...
Suppose a competitive firm’s production function is Y= 20 L1/2 K1/3. L is Labor , K...
Suppose a competitive firm’s production function is Y= 20 L1/2 K1/3. L is Labor , K is capital and Y is output. a) (4) Find the marginal product of labor and capital. b) (4) What is Marginal Rate of technical Substitution of Labor for Capital? c) (2) Does this production function exhibit increasing, decreasing or constant returns to scale? Show your work.
2. Consider the following production functions, to be used in this week’s assignment: (A) F(L, K)...
2. Consider the following production functions, to be used in this week’s assignment: (A) F(L, K) = 20L^2 + 20K^2 (B) F(L, K) = [L^1/2 + K^1/2]^2 a (i) Neatly draw the Q = 2,000 isoquant for a firm with production function (A) given above, putting L on the horizontal axis and K on the vertical axis. As part of your answer, calculate three input bundles on this isoquant. (ii) Neatly draw the Q = 10 isoquant for a firm...
A firm uses two inputs, capital K and labor L, to produce output Q that can...
A firm uses two inputs, capital K and labor L, to produce output Q that can be sold at a price of $10. The production function is given by Q = F(K, L) = K1/2L1/2 In the short run, capital is fixed at 4 units and the wage rate is $5, 1. What type of production function is F(K, L) = K1/2L1/2 ? 2. Determine the marginal product of labor MPL as a function of labor L. 3. Determine the...
4. output Q according to the production function Q = 6K1/3L1/2, where K = capital and...
4. output Q according to the production function Q = 6K1/3L1/2, where K = capital and L =labor. A. Calculate the marginal product of capital. Calculate the marginal product of labor. Calculate the marginal rate of technical substitution A technological advance occurs which changes the production function to Q = 2KL. D. Calculate the new marginal product of capital. E. Calculate the new marginal product of labor. F. Calculate the new marginal rate of technical substitution for Lazy J Enterprises....
suppose the production function is given by the equation q = L(100K)^(1/2). graph the isoquants corresponding...
suppose the production function is given by the equation q = L(100K)^(1/2). graph the isoquants corresponding to q = 200, q = 400, and q = 500 (give 3 points on the isoquant graph). do these isoquants exhibit diminishing marginal rate of technical substitution? prove your answer and show all work
Suppose a firm’s production function is given by Q = L 1/2 , K 1/2. a)...
Suppose a firm’s production function is given by Q = L 1/2 , K 1/2. a)   Suppose the firm has a fixed cost FC=6, the price of labor is w = 64 and the price of capital is r = 4. Derive the firm’s total cost function, TC(Q). b)   What is the firm’s marginal cost? c)   Graph the firm’s isoquant for Q = 20 units of output. On the same graph, sketch the firm’s isocost line associated with the total...
A firm produces output according to the production function. Q=sqrt(L*K) The associated marginal products are MPL...
A firm produces output according to the production function. Q=sqrt(L*K) The associated marginal products are MPL = .5*sqrt(K/L) and MPK = .5*sqrt(L/K) (a) Does this production function have increasing, decreasing, or constant marginal returns to labor? (b) Does this production function have increasing, decreasing or constant returns to scale? (c) Find the firm's short-run total cost function when K=16. The price of labor is w and the price of capital is r. (d) Find the firm's long-run total cost function...