Question

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. At what output level is K = 8 a cost-minimizing choice of capital in the long run?

Answer #1

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...

Suppose that a firm's fixed proportion production function is
given by q = min(2k, 4L), and that the rental rates for capital and
labor are given by v = 1, w = 3.
A) Calculate the firm's long-run total, average, and marginal
cost curves.
B) Graph these curves.
C) Suppose that k is fixed at 10 in the short run. Calculate the
firm's short-run total, average, and marginal cost curves and graph
them.
D) Now suppose in the long run...

A firm produces a product with labor and capital. Its production
function is described by Q = min(L, K). Let w and r be the prices
of labor and capital, respectively.
a) Find the equation for the firm’s long-run total cost curve as
a function of quantity Q and input prices, w and r.
b) Find the solution to the firm’s short-run cost minimization
problem when capital is fixed at a quantity of 5 units (i.e., K =
5). Derive...

A firm’s production function is Q = min(K , 2L), where Q
is the number of units of output produced using K units of capital
and L units of labor. The factor prices are w = 4 (for labor) and r
= 1 (for capital). On an optimal choice diagram with L on the
horizontal axis and K on the vertical axis, draw the isoquant for Q
= 12, indicate the optimal choices of K and L on that isoquant,...

Firm B’s production function is q = min {8L,
10K} where L is the quantity of labor and
K is the quantity of capital used to produce output
q. Let PL and PK
denote price of labor and price of capital, respectively. Derive
Firm B’s long-run total cost function. Show your work.

a. A cost minimizing firm’s production is given by Q=L1/2K1/2.
Suppose the desired output is Q=10. Let w=12 and r=4. What is this
firm’s cost minimizing combination of K & L? What it the total
cost of producing this output?
b. Suppose the firm wishes to increase its output to Q=12. In
the short run, the firm’s K is fixed at the amount found in (a),
but L is variable. How much labor will the firm use? What will the...

a firm produces a product with labor and capital as inputs. The
production function is described by Q=LK. the marginal products
associated with this production function are MPL=K and MPK=L. let
w=1 and r=1 be the prices of labor and capital, respectively
a) find the equation for the firms long-run total cost curve
curve as a function of quantity Q
b) solve the firms short-run cost-minimization problem when
capital is fixed at a quantity of 5 units (ie.,K=5). derive the...

3. A firm’s production function is Q=min(K, 3L ). Input prices a
re as follows: w=$ 2 and r=$1.
On the optimal choice diagram below, draw the isoquant for Q=12.
Calculate the optimal choice of K and L for this level of output as
well as the total cost.
Then, draw in (with a dotted line) the isocost line consistent
with your Total Cost value.
It won't let me copy the graph template but it is a simple graph
with...

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...

Given the production function Q=K2L2 and
the price of capital and labor as r=4 and w=8, respectively, if the
goal is Q=2500 find the level of capital (K) and labor (L) to
minimize cost.

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