Question
A hat manufacturing firm has the following production function with capital and labor being the inputs:...
A hat manufacturing firm has the following production function with capital and labor being the inputs: Q = min(5L,3K) (it has a fixed-proportions production function). If w is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm’s optimal inputs, long-run total cost curve, average cost curve, and marginal cost curve in terms of the input prices and Q.
b) A firm has the linear production function Q = 2L + 7K. Derive the expression for the optimal inputs and the 1ong-run total cost that the firm incurs, as a function of Q and the factor prices, w and r
Homework Answers
Answer #1
a. Q = min(5L,3K)
This is a fixed proportions production function. So, Q = 5L =
3K
So, optimal inputs are: L = Q/5 and K = Q/3
Long run total cost (TC) = wL + rK = (wQ/5) + (rQ/3)
Average cost (AC) = TC/Q = 
Marginal cost (MC) = 
b. Q = 2L + 7K
MRTS = MPL/MPK = 2/7
When MRTS > w/r, that is, w/r < 2/7 then only L will be
used. So, K = 0
So, Q = 2L + 7(0) = 2L
So, L = Q/2
Optimal combination is L = Q/2 and K = 0
Total cost (TC) = wL + rk = w(Q/2) + r(0) = wQ/2
When MRTS < w/r, that is, w/r > 2/7 then only K will be
used. So, L = 0
So, Q = 2(0) + 7K = 7K
So, K = Q/7
Optimal combination is L = 0 and K = Q/7
Total cost (TC) = wL + rk = w(0) + r(K/7) = rK/7
Not the answer you're looking for?
Similar Questions
a firm produces a product with labor and capital as inputs. The production function is described...
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...
A firm produces a product with labor and capital. Its production function is described by Q...
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...
Consider a firm that used only two inputs, capital (K) and labor (L), to produce output....
Consider a firm that used only two inputs, capital (K) and labor
(L), to produce output. The production function is given by: Q =
60L^(2/3)K^(1/3) .
a.Find the returns to scale of this production function.
b. Derive the Marginal Rate of Technical Substitutions (MRTS)
between capital and labor. Does the law of diminishing MRTS hold?
Why? Derive the equation for a sample isoquant (Q=120) and draw the
isoquant. Be sure to label as many points as you can.
c. Compute...
2. Optimal Inputs Your boss has given you the following production function for labor (L) and...
2. Optimal Inputs
Your boss has given you the following production function for
labor (L) and capital (K) used by your company:
q¯ = K0.2L 0.8
You want to produce q = 100 units for sale and faces prices for
labor of w = 2 and capital of r = 6.
a) What is the marginal rate of technical substitution?
b) What are the optimal amounts of each input used by the firm?
Round to three decimal places as needed....
III. A production process uses two inputs, labor and capital which can be written as: Q...
III. A production process uses two inputs, labor and capital
which can be written as: Q = 5LK, MPL = 5K and MPK = 5L 'w' = $150
and 'r' = $1000. Find the least cost combination of L and K when
output Q = 1000 tons per day. What is the total cost of producing
1000 tons per day using this least cost combination?
A firm’s production function is Q! = min(4L ,5K ). The price of labor is w...
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....
A firm’s production function is Q = min(K , 2L), where Q is the number of...
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,...
A price-taking firm has production function: q41/4K1/4 This is a long-run problem. Labor and capital cost...
A price-taking firm has production function: q41/4K1/4 This is a
long-run problem. Labor and capital cost w and r, respectively.
Show that the long-run Total Cost function is TC-2 (rw Hint:This is
just like the consumption problem where you derive the Hicks demand
functions then the expenditure function
A cost-minimizing firm has the following production function: Q=LK+2M. Where L denotes Labor, K denotes Capital,...
A cost-minimizing firm has the following production function:
Q=LK+2M. Where L denotes Labor, K denotes Capital, and M denotes
Materials. The prices for the inputs are as follows: w=$4, r=$8,
and m=$1. The firm is operating in the long run. Answer the
following questions as you solve for the total cost of producing
400 units of output. Assume an interior solution (i.e. positive
values of all inputs).
a) Set up constrained optimization problem of the firm:
b) Write out the...
A firm has the following production function: q=5LK^0.5+2L^2K-L^3K What is its short-run production function if capital...
A firm has the following production function:
q=5LK^0.5+2L^2K-L^3K
What is its short-run production function if capital is fixed at
K=9?
What are the firm’s marginal product of labour and average
product of labour in the short run?
Show that the firm’s elasticity of output with respect to labour
in the short run is a function of marginal product of labour and
average product of labour. Calculate the short-run elasticity of
output with respect to labour
ADVERTISEMENT
Need Online Homework Help?
Get Answers For Free
Most questions answered within 1 hours.
ADVERTISEMENT
Active Questions
asked 32 minutes ago
asked 45 minutes ago
asked 50 minutes ago
asked 1 hour ago
asked 1 hour ago
asked 1 hour ago
asked 1 hour ago
asked 1 hour ago
asked 1 hour ago
asked 2 hours ago
asked 2 hours ago
asked 2 hours ago