Question

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 with production function (B) given above. As part of your answer, calculate three input bundles on this isoquant.

b For each of production functions (A) and (B) given above, do the following steps.

(i) Calculate the marginal product of labor MPL(L, K) = ∂F(L, K) / ∂L.

(ii) Calculate the marginal product of capital MPK(L, K) = ∂F(L, K) / ∂K.

(iii) Calculate the absolute value of the technical rate of substitution as the ratio of marginal products and simplify as far as possible: |TRS(L, K)| = MPL(L, K) / MPK(L, K).

(iv) Determine whether the absolute value of the technical rate of substitution is constant, diminishing, or increasing in labor along an isoquant. For most production functions, if we move along an isoquant by increasing labor, we must also decrease capital to keep output fixed. Justify your determination. You may recall we did a similar exercise using arrows to determine the shape of indifference curves.

Answer #1

a (i) The isoquant for F(L, K) = 20
L^{2} + 20 K^{2} are drawn below:

As, 20 L^{2} + 20
K^{2} = 2000 => L^{2} + K^{2} = 100

If L = 5 , then K^{2} = 100
– 25 = 75 => K = 8.66

If L= 6 then, K^{2} = 100 –
36 = 64 => K = 8

If L = 7 then, K^{2} = 100 –
49 = 51 => K= 7.14

Figure 1

(ii)

Figure 2

Q = F(L, K) = L^1/2 + K^1/2]^2= 10

If L = 1 , then K_{1} =
10^(1/2) – 1^(1/2) = 3.16 – 1 = 2.16

If L= 6 then, K_{2} =
10^(1/2) – 2^(1/2) = 3.16- 1.414 => 1.746

If L = 7 then, K^{2} =
10^(1/2) – 3^(1/2)=> K= 1.428

B) For the function F(L, K) = 20L^2 + 20K^2

MPL(L, K) = ∂F(L, K) / ∂L = ∂( 20L^2 + 20K^2) / ∂L = 40L

MPK = ∂F(L, K) / ∂K = 40K

For the function F(L, K) = [L^1/2 + K^1/2]^2

MPL = ∂( L^1/2 +
K^1/2]^2) / ∂L = 2( L^1/2 + K^1/2] * 1/2 L ^{1/2 -1} =
(L^1/2 + K^1/2) / L^1/2 = L + (K/L)^1/2

MPK = ∂( L^1/2 + K^1/2]^2) / ∂K = 2(
L^1/2 + K^1/2] * 1/2 K ^{1/2 -1} = K + (L/K)^1/2

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...

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),...

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,...

Consider the following function:
q = 9LK + 6L^2 - (1/3)L^3
Given the following expression for the marginal productivity of
each input:
MPL = 9K + 12L - L^2 and MPK = 9L
Assuming Capital is plotted on the vertical axis and labor is
plotted on the horizontal axis, determine the value of the marginal
rate of technical substitution when K=20 and L =10. (Round your
answer up to two decimal places and include the proper sign.)
MRTS= ___________

For each of the following production functions: a)
Q(K,L) = 2K + 3L b) Q(K,L) = K^0.5L^0.25 c)Q(K,L) = LK +
L
Write an equation and graph the isoquant for Q =
100. ?
Find the marginal rate of technical substitution and
discuss how M RT SLK ?changes as the firm uses
more L, holding output constant. ?

(a) Show that the following Cobb-Douglas production function,
f(K,L) = KαL1−α, has constant returns to scale.
(b) Derive the marginal products of labor and capital. Show
that you the MPL is decreasing on L and that the MPK is decreasing
in K.

Consider a firm which has the following production function
Q=f(L,K)=4?LK
(MPL=2?(K/L) and MPK=2?(L/K).
(a) If the wage w= $4 and the rent of capital r=$1, what is the
least expensive way to produce 16 units of output? (That is, what
is the cost-minimizing input bundle (combination) given that
Q=16?)
(b) What is the minimum cost of producing 16 units?
(c) Show that for any level of output Q, the minimum cost of
producing Q is $Q.

Suppose Cool T-Shirts Co produces T-shirts and employs labor (L)
and capital (K) in production. Suppose production function for Cool
T-Shirts Co is Q=K*L, and Cool T-Shirts Co wants to produce Q=625.
Suppose marginal product of labor (MPL) and marginal product of
capital (MPK) are as follows: MPL=K and MPK=L. Suppose Cool
T-Shirts Co pays workers $10 per hour (w=$10) and interest rate on
capital is $250 (r=250). What is the cost-minimizing input
combination if Cool T-Shirts Co wants to...

(2) Consider the production function f(L, K) = 2K √ L. The
marginal products of labor and capital for this function are given
by MPL = K √ L , MPK = 2√ L. Prices of inputs are w = 1 per hour of
labor and r = 4 per machine hour. For the following questions
suppose that the firm currently uses K = 2 machine hours, and that
this can’t be changed in the short–run.
(e) What is the...

For each of the following production
functions,
• Write an equation and graph the isoquant for Q =
100.
• Find the marginal rate of technical substitution and
discuss how MRTSLK changes as the ?rm uses more L, holding output
constant.
(c) Q(K,L) = LK + L

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 6 minutes ago

asked 23 minutes ago

asked 33 minutes ago

asked 35 minutes ago

asked 38 minutes ago

asked 38 minutes ago

asked 46 minutes ago

asked 48 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago