Question

Suppose a firm’s production function is given by Q = 2K^1/2 * L^1/2 , where K is capital used and L is labour used in the production.

(a) Does this production function exhibit increasing returns to scale, constant returns to scale or decreasing returns to scale?

(b) Suppose the price of capital is r = 1 and the price of labour is w = 4. If a firm wants to produce 16 chairs, what combination of capital and labor will it use to minimize costs?

(c) Now, suppose in addition to paying for capital and labor the firm will also have to pay fixed lump-sum tax of 20 to the government. Would this affect your answers for part (a)? Why or why not? (HINT: What kind of cost is this additional tax?)

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Answer #1

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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?

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 ,...

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...

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.

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?...

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...

(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...

The production of sunglasses is characterized by the production
function Q(L,K)= 4L1/2K 1/2 . Suppose that the price of labor is
$10 per unit and the price of capital is $90 per unit. In the
short-run, capital is fixed at 2,500. The firm must produce 36,000
sunglasses. How much money is it sacrificing by not having the
ability to choose its level of capital optimally? That is, how much
more does it cost to produce 36,000 sunglasses the short-run
compared...

Cobb-Douglas Production Function & Cost of
Production
A firm’s production function is given as –
q =
2K0.4N0.6
What kind of returns to scale does this production technology
exhibit? Justify your answer.
Find out the expression for the marginal product of labor.
Find out the expression for the marginal product of
capital.
Find out the expression for MRTS.

Suppose that you are given the
following production function:
Q =
100K0.6L0.4
For each of the following
production functions, determine whether returns to scale
are decreasing, constant, or increasing when capital and labor
inputs are increased from K = L =
1 to K = L = 2.
a. Q =
25K0.5L0.5
b. Q =
2K + 3L + 4KL

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