Question

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 labor?

C. Derive the optimal bundle of capital and labor to achieve the desired level of output. What is the maximum amount of output that they can produce with this $ 12,000 in expenditures?

Answer #1

A. Production function exhibits constant returns to scale.

Since the production function is a Cobb Douglas production function.

a+b=1 as 0.2+0.8=1, So the production function is a constant returns to scale.

B. Marginal product of capital= 0.2*20* k^-0.8 * L^0.8

Marginal product of labor= 0.8*20* K^0.2* L^-0.2

Marginal product diminishes as Capital increases and marginal product diminishes as labor increases.

So this function exhibits diminishing Returns in Capital and labor.

c. MRTS= 4k/L

At equilibrium , MRTS= r/w=15/10=1.5

L=6K

12000= wk+rL

12000=15k+10*6k

12000=75k

k*= 160

L*= 960

Q=20* (160)^0.2 (960)^0.8

Q= 13419.1

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!

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,
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The production function for the Roundtree Laser Company is:
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raw materials used per week
a) Does the production function exhibit decreasing returns to
scale?
b) Does the production function exhibit diminishing marginal
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Suppose a firm’s long-run production function is given by
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Derive expressions (formulas) for the marginal product of each
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Does each input exhibit diminishing marginal returns?
Prove.
Derive an expression for the marginal rate of technical
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Suppose the price of capital, r = 1, and the price of labor, w
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a) (4) Find the marginal product of labor and capital.
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c) (2) Does this production function exhibit increasing,
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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

Joe’s coffee house operates under the production function Q =2√?
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b. Does labor exhibit diminishing marginal returns when capital
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A firm produces output according to the production function.
Q=sqrt(L*K) The
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(a) Does this production function have increasing, decreasing, or
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(b) Does this production function have increasing, decreasing or
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(d) Find the firm's long-run total cost function...

An electronics plant’s production function is Q = L 2K, where Q
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