Question

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

Answer #1

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...

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 production function q
= K2 + L2.
Does this production function exhibit constant, increasing or
decreasing returns to scale?)
Find an expression for the marginal rate of technical
substitution. Does this production function exhibit diminishing
marginal rate of technical substitution? Explain

Suppose the production function is given by formula Q = KL.
A) Draw the isoquant curve for Q = 128. (Draw K on the vertical
axis and L on the horizontal axis.)
B) Suppose K = 4. How many units of labor should the firm use if
it wants to produce 128 units of output? Label it point X on the
isoquant curve.
C) Suppose K = 8.How many units of labor should the firm use if
it wants to...

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...

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

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. ?

2. A firm has the following linear production function:
q = 5L + 2K
a. Does this firm’s production function exhibit diminishing
returns to labor?
b. Does this production function exhibit diminishing returns to
capital?
c. Graph the isoquant associated with q = 20.
d. What is the firm’s MRTS between K and L?
e. Does this production technology exhibit decreasing, constant,
or increasing returns to scale?

Suppose that a rm has a production function given by Q=
5L^2/5K^1/5 (a) Show that there is a positive and diminishing
marginal product of labour, and a positive and diminishing marginal
production of capital. (b) Show that increasing K leads to an
increase in the marginal product of labor, and increasing L leads
to an increase in the marginal product of capital.

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),...

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