In: Economics

Suppose there are two inputs in the production function, labor and capital, and these two inputs are perfect substitutes. The existing technology permits 5 machines to do the work of 2 workers. So the production function is f(E, K) = 2K + 5E. The firm wants to produce q units of output, where q > 0 is some number. Suppose the price of capital is $10 per machine per hour. What combination of inputs will the firm use if the wage rate is $15 or $25 or $35 per hour?

Suppose there are two inputs in the production function, labor
and capital, which are substitutes. The current wage is $10 per
hour and the current price of capital is $25 per hour.
Given the following information on the marginal product of
labor and the marginal product of capital, find the firm’s
profit-maximizing input mix (i.e. number of workers and units of
capital) in the long-run. Show your work and explain.
L
MPL
K
MPK
1
125
1
130
2
100...

Consider a production function of two inputs, labor and capital,
given by Q = (√L + √K)2. Let w = 2 and r = 1. The marginal products
associated with this production function are as follows:MPL=(√L +
√K)L-1/2MPK=(√L + √K)K-1/2
a) Suppose the firm is required to produce Q units of output.
Show how the cost-minimizing quantity of labor depends on the
quantity Q. Show how the cost-minimizing quantity of capital
depends on the quantity Q.
b) Find the equation...

Capital and labor are the only two inputs for the following
production process. Capital is fixed at 4 units, which costs 50
dollars each unit per day. Workers can be hired for 100 each per
day. Complete the following table and plot the marginal cost (MC),
average total cost (ATC), average variable cost (AVC), average
fixed cost (AFC) on the same graph.
The quantity of labor input
Total output per day
AFC
AVC
ATC
MC
0
0
1
100...

A firm produces a product with labor and capital as inputs. Its
production function is described by Q(L,K) = L^1/2 K^1/2. Let w and
r be the prices of labor and capital, respectively.
a. Derive the firm’s long-run total cost and long-run marginal
cost functions.
b. Assume capital is fixed at 4 units in the short-run and
derive the firm’s short-run total cost and short-run average
variable cost functions.
c. Rewrite your short-run and long-run total cost functions (for
the...

Suppose that production of widgets requires capital and labor.
The production function is constant returns to scale and capital
investment is sunk. There are no other barrier to entry. Is the
investment in sunk capital a barrier to entry? Explain. What will
the market equilibrium be if there are many possible entrants?

Suppose that an economy has a Cobb-Douglas production function
with three inputs. K is capital (the number of machines), L is
labor (the number of workers), and H is human capital (the number
of college degrees among workers). Markets for output and factors
of production are both competitive. The production function is Y =
K^1/3*L^1/3*H^1/3
1. Prove that this technology shows constant returns to
scale.
2. Solve the competitive firm’s profit maximization problem by
deriving the first-order conditions.
3. An...

Consider the following production function using capital (K) and
labor (L) as inputs. Y = 10.K0.5L0.5. The marginal product of labor
is (MPL=) 5.K0.5/L0.5, and marginal product of capital (MPK) =
5.L0.5/K0.5.a. If K = 100 and L=100 what is the level of output
Y?b. If labor increases to 110 while K=100, what is the level of
output?c. If labor increases to 110 while K=100, what is the
marginal product of labor?d. If labor increases to 120 while K=100,
what...

A closed economy has two factors of production: capital and
labor. The production function is known to exhibit constant returns
to scale. The capital stock is about 4 times one yearís real GDP.
Approximately 8% of GDP is used to replace depreciating capital.
Labor income is 70% of real GDP. Real GDP grows at an average rate
of 4% per year. Assume the economy is at a steady state. (a) Is the
capital per e§ective worker lower or larger than...

The production function has two input, labor (L) and capital
(K). The price for L and K are respectively W and V.
q = L + K a linear production function
q = min{aK, bL} which is a Leontief production function
1.Calculate the marginal rate of substitution.
2.Calculate the elasticity of the marginal rate of
substitution.
3.Drive the long run cost function that is a function of input
prices and quantity produced.

If you have two inputs, labor and
capital and you are using them at the least cost combination
point. Describe what happens to the isocost line and the
equilibrium least cost point when the cost of labor decreases.

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