Question

A firm’s production function is Q(L,K) = K^1/2 + L. The firm faces a price of labor, w, and a price of capital services, r.

a. Derive the long-run input demand functions for L and K, assuming an interior solution. If the firm must produce 100 units of output, what must be true of the relative price of labor in terms of capital (i.e. w/r) in order for the firm to use a positive amount of labor? Graphically depict this firm’s expansion path.

b. In general, explain what an expansion path represents and how it compares to an input demand function.

Answer #1

A firm’s production function is Q! = min(4L ,5K ). The price of
labor is w and the price of capital is r.
a) Derive the demand function of labor and capital respectively.
How does the demand of capital change with the price of
capital?
b) Derive the long-run total cost function. Write down the
equation of the long-run expansion path.
c) Suppose capital is fixed at K = 8 in the short run. Derive
the short-run total cost function....

2. A firm has production function Q = k^1/2L^1/2 and faces a
wage for the labor input w = 1 and a rental price of capital r =
9
a. The policy of the Federal Reserve brings the rental price of
capital to r = 4 Graph the change of the cost minimizing equlibrium
explaining the type of substitution that is happening.
b. Compute the new cost function. Suppose a monopoly and show
graphically if after this change in the...

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...

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...

A firm’s production process is represented by y= L^2/3 K^1/3.
The price of Labor, w is $2 and the price of capital, r, is
$27.
(a) Write down the firm’s cost minimization problem
(b) What is the firm’s MRTS?
(c) What are the firm’s cost minimizing levels of labor and
capital (these will both be functions of y)?
(d) What is the firm’s cost curve (ie, derive C(y))?
(e) If the firm chooses output y= 450, what are the firms...

Suppose a firm has a production function given by q = 3L +
K.
The firm can purchase labor, L at a price w = 24, and capital, K
at a price of r = 5.
What is the firm’s total cost function?

Firm A’s production function and cost line are given by:Q=Q(L,K)=2 L^(1/2) K^(1/2) (The production function)?L+?=30000 (The cost line of firm A):?L is the amount of labor hired.?K is the amount of capital hired.??p_L or the price of labor is 1 dollar per
unit.??p_K or the price of capital is 1 dollars per
unit.C or cost (think of it as the firm’s budget) is 30000
dollars.How much labor and capital should this firm optimally hire?

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...

2. Assume that a manufacturer faces a Cobb-Douglas production
function, q=40K^0.5L^0.5
where q is output per period, L is labor, K is capital. The
market price of labor (w) is $50 per unit and the price of capital
(r) is $200 per unit.
a. Specify and illustrate graphically the short-run MPl and APl
for L = 5 to 30 units (assume that the level of capital is 25; use
increments of 5 units of labor). Is this firm operating in...

A firm produces good Q using inputs L & K. The firm’s
production function is X = 20L^0.5 + 11K. The
price of K is $P_K a unit and the price of L is $P_L a unit, and in
the short‐run, the capital input is
fixed at 3 units.
a. If the firm needs an output of X_1 in the short‐run, what is the
firm’s total cost and marginal
cost of production?
b. What is the firm’s fixed cost and...

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