Question

- Answer the following questions about the producer’s production
function: Q = 2K
^{1/2}L^{2}

- Does the production function display increasing, constant, or decreasing returns to scale? [Prove your answer by increasing all inputs by a factor of c in your analysis.]

- Find MP
_{L}if capital is fixed at K_{0}=9 and determine whether the production process follows the law of diminishing returns (LDR) to labor.

- If input prices are r=5 and w=4 for capital and labor,
respectively, and suppose MP
_{K}=40 and the firm is currently using 3 units of labor (L=3). Is the firm in part b (above) minimizing its costs? If not, how could it improve its use of inputs in attempting to minimize costs? Explain.

Answer #1

Ans. Production function, Q = 2K0.5L2

a) To check for returns to scale, suppose both the inputs are increased by factor c, then new output,

Q’ = 2(cK)0.5(cL)2 = c2.5 Q

As Q’ > cQ, so, the production function exhibits increasing return to scale.

b) MPL = dQ/dL = 4K0.5L

At K = 9

MPL = 4*3L = 12L

Differentiating MPL by L we get,

dMPL/dL = 12 > 0

Therefore, the production function doesn’t exhibit diminishing returns to labour.

c) MPL at L = 3 is 12*3 = 36

MPK = 40

r = 5 and w = 4

At cost minimizing level,

MPL/MPK = w/r

But here, MPL/MPK = 9/10 > w/r =4/5

So, firm is not minimizing its cost. To minimise cost capital should be used less which will increase MPK making the required ratios equal.

*Please don’t forget to hit the thumbs up button, if you find the answer helpful.

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 ,...

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 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...

3. Consider the production function, Q = [L0.5 +
K0.5] 2 . The marginal products are given as
follows: MPL = [L0.5 + K0.5] L-0.5
and MPK = [L0.5 + K0.5] K-0.5 and
w = 2, r = 1.
A). what is the value of lambda
B). Does this production function exhibit increasing, decreasing
or constant returns to scale?
C).Determine the cost minimizing value of L
D).Determine the cost minimizing value of K
E).Determine the total cost function
F).Determine the...

(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.

Suppose that a firm's production function is Q =
10L1/2K1/2. The cost of a unit of labor (i.e.
the wage) is $20 and the cost of a unit of capital is $80.
a. What are the cost minimizing levels of capital and labor if
the firm wishes to produce 140 units of output?
b. Illustrate your answer for part (a) on a well-labeled diagram
that shows the firm's production isoquant and isocost equation.

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 for widgets is given by
q = kl -0.8k2- 0.2l2,
where q represents the annual quantity
of widgets produced, k represents annual capital input, and l
represents annual labor input.
Suppose k = 10; graph the total and average productivity of
labor curves. At what level of labor input does this average
productivity reach maximum? How many widgets are produced at that
point?
Again, assuming that k = 10, graph the MPL curve. At
what...

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...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 9 minutes ago

asked 11 minutes ago

asked 28 minutes ago

asked 41 minutes ago

asked 44 minutes ago

asked 46 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago