Question

Consider the following production function Y=z*(a*K + (1-a)*N) where z represents total factor productivity, a is a parameter between 0 and 1, K is the level of capital, and N is labor. We want to check if this function satisfies our basic assumptions about production functions.

1. Does this production function exhibit constant returns to scale? Ex- plain

2. Is the marginal product of labor always positive? Explain

3. Does this function exhibit diminishing marginal product of labor? Ex- plain

4. Does increasing the amount of capital increase the marginal product of labor? Explain

Answer #1

1. Consider the following production function:
Y=F(A,L,K)=A(K^α)(L^(1-α))
where α < 1.
a. Derive the Marginal Product of Labor(MPL).
b. Show that this production function
exhibit diminishing MPL.
c. Derive the Marginal Production of Technology (MPA).
d. Does this production function exhibit diminishing MPA? Prove
or disprove

2. Bridgestone Company has
the following production function for tires: Q = 20 K
0.2 L 0.8, where K represents machine hours
and L represents labor hours. They pay $ 15 per hour to rent their
machines and $ 10 per hour to their workers. They have $ 12,000 to
spend on capital and labor.
A. Does this production function
exhibit constant, increasing, or decreasing returns to scale?
B. Does this production function
exhibit diminishing marginal returns to capital and...

Wheat is produced according to the production function Q = 100
K^0.8 L^0.2
a. Beginning with a capital input of 4 and a labor input of 49,
show that the marginal product of labor and the marginal product of
capital are both decreasing.
b. Does this production function exhibit increasing, decreasing,
or constant returns to scale?
please explain in 4 sentences thank you!

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

Consider a production function for an economy:
Y = 20(L.5K.4N.1)where L is labor, K is capital, and N is land. In
this economy the factors of production are in fixed supply with L =
100, K = 100, and N = 100.
a) What is the level of output in this country?
b) Does this production function exhibit constant returns to scale?
Demonstrate by an example.
c) If the economy is competitive so that factors of production are
paid 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.

Consider the following Cobb-Douglass production
function?≡??(?,?):?=??1/3?2/3
where Y is output, the constant z measures productivity, K is
physical capital, and N is labor. Suppose ?=2, ?=0.16, ?=0.06, and
?=0.02.
a. What are the steady-state (numerical) values of ?, ?, and
??
b. What is the golden-rule (numerical) level of capital per
worker?
c. If the government wants to achieve the golden rule level of
k, should savings increase, decrease or remain unchanged? Solve
for/obtain its (numerical) value. Explain briefly.

The production function for a firm is given by q = L0.75 K0.3
where q denotes output; L and K labor and capital inputs
. (a) Determine marginal product of labor. Show whether or not
the above production function exhibits diminishing marginal
productivity of labor.
(b) Calculate the output (or production) elasticity with respect
to labor.
c) Determine the nature of the Return to Scale as exhibited by
the above production function. Show and explain all
calculations

Consider the following production function:
Y = output = AK1/2N1/2, A = productivity, K = capital, N =
labor.
a) (3 pts.) Suppose that Y = 1331, K =121, and N = 121. Find
A.
b) (4 pts.) Find the marginal product of capital (MPK), measured
as the additional output that arises when the capital stock is
increased by 1 unit. (Start with the values of A, K and N that you
found in part (a).)
c) (4 pts.) Suppose...

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...

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