Solved by verified expert:i need a problem set done for my class. as much help as possible would be great. it deals with product maximization, returns to scale single input firms and optimal production.
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Econ 301 Problem Set #1
Due: 11:00pm, February 9, 2018
(See submission instructions in syllabus. Make sure to show your work.)
Problem 1 – Cobb-Douglas and returns to scale (10 points)
Consider the Cobb-Douglas production function F (L, K) = ALα K 1/2 , where α > 0 and A > 0.
Refer to the definitions of returns to scale in the text and in the lecture slides.
1. The Cobb-Douglas function can be either increasing, decreasing or constant returns to scale
depending on the values of the exponents on L and K. Prove your answers to the following
three cases.
(a) For what value(s) of α is F (L, K) decreasing returns to scale?
(b) For what value(s) of α is F (L, K) increasing returns to scale?
(c) For what value(s) of α is F (L, K) constant returns to scale?
Problem 2 – A single input firm (30 points)
Consider a firm that produces a single output with a single input, labor, using 2 different plants.
Denote by L1 the assignment of labor input into plant 1 and
√ by L2 the assignment of labor input
into plant 2. Plant 1’s√production function is F1 (L1 ) = 4 L1 , for L1 ≥ 0. Plant 2’s production
function is F2 (L2 ) = 8 L2 , for L2 ≥ 0.
1. State the average product function of each plant as a function of the labor assignment. Denote
them by AP1 (L1 ) and AP2 (L2 ).
2. State the marginal product function of each plant as a function of the labor assignment.
Denote them by M P1 (L1 ) and M P2 (L2 ).
3. Define total quantity produced for a given labor assignment by Q(L1 , L2 ) = F1 (L1 ) + F2 (L2 ).
Suppose the firm has a total of 100 units of labor available, L = 100. It can freely assign
them across the two plants subject to L1 + L2 = L. In a graph show total output produced
for different choices of L1 ∈ [0, 100] where L2 = L − L1 .
4. For L = 100, find the input assignment, (L∗1 , L∗2 ), that maximizes total output, Use the insight
that M P1 (L∗1 ) = M P2 (L∗2 ).
5. We want to derive the firm’s efficient production function frontier for any total labor input
L ≥ 0. Call it F (L). It is the greatest output that can be produced with L units of workers.
(a) For a given L, find the input assignment L∗1 (L), L∗2 (L) that maximizes total output.
Verify that L∗1 (100), L∗2 (100) matches your answer to question 4.
∗
∗
(b) Define
√ F (L) = F1 (L1 (L)) + F2 (L2 (L)). Show that it can be written in the form F (L) =
A L. What is the value of A?
1
Problem 3 – A single input firm (10 points)
Consider a firm that produces a single output with a single input, labor, using production function
F (L) = 100L − 4L2 , for L ∈ [0, 12.5]. The input price is W = 2.
1. Determine the firm’s cost function C (Q), that is, the lowest cost of producing Q units of
output. State the associated labor choice for a givsn required output, L (Q). Consider only
the range Q ∈ [0, 625] .
2. What is the firm’s marginal cost curve, M C (Q)?
Problem 4 – Adam’s Apples (25 points)
”Adam’s Apples” produces apple cider using labor (L) and apples (A) according to the Cobb-Douglas
production function Q = F (L, A) = 2L1/4 A3/4 . The price of apples is pA = 2 and the price of labor
is W = 10. Adam’s Apples also has a fixed cost for farm buildings, F C = 100.
1. If Adam’s Apples wants to produce 100 gallons of apple cider, Q=100, what is its lowest
achievable input cost? (Roadmap: (1) Determine the optimal input choice (L∗ , A∗ ) by tangency condition, M RT SLA (L∗ , A∗ ) = W/pA and production requirement 100 = F (L∗ , A∗ ).
(2) Variable costs are then found by W L∗ + pA A∗ . (3) Total cost is sum of variable and fixed
costs.)
2. Determine Adam’s Apples’ cost function, C (Q). Verify that C (100) gives you same answer
as in question 1.
3. Determine Adam’s Apples’ Average Cost function, AC(Q), and marginal cost function, M C (Q) .
Problem 5 – Optimal production (25 points)
The Pear Corp produces high end consumer electronics using labor L and capital K according to
production function Q = F (L, K) = 100L1/4 K 1/4 . Let the price of a unit of labor be given by W
and the price of a unit of capital is given by R. The output price is P . Pear Corp has fixed costs
of F C = 5 that are unavoidable in the short run, but avoidable in the long run. Both labor and
capital are fully adjustable in both short and long run. Set input price W = 20 and R = 5.
1. Is the production function increasing, decreasing, or constant returns to scale? Explain you
answer.
2. What is Pear Corp’s marginal rate of technical substitution, M RT S (L, K) for a given input
bundle (L, K)? (MRTS is the absolute value of the slope of the isoquant in a diagram with L
on the horizontal axis and K on the vertical axis).
3. Determine Pear Corp’s variable cost function V C (Q). That is, the lowest possible variable
input expenditure associated with the production of Q ≥ 0 units of output. For short and
long run, respectively, state Pear Corp’s cost function, CSR (Q) and CLR (Q), (That is the
total cost of producing Q ≥ 0 units).
2
Problem 1 – Cobb-Douglas and returns to scale
Consider the Cobb-Douglas production function F (L,K) = ALαK1/2, where α > 0 and A > 0. Refer to the
definitions of returns to scale in the text and in the lecture slides.
1. The Cobb-Douglas function can be either increasing, decreasing or constant returns to scale
depending on the values of the exponents on L and K. Prove your answers to the following three
cases.
(a) For what value(s) of α is F(L,K) decreasing returns to scale?
(b) For what value(s) of α is F(L,K) increasing returns to scale?
(c) For what value(s) of α is F(L,K) constant returns to scale?
Problem 2 – A single input firm
Consider a firm that produces a single output with a single input, labor, using 2 different plants. Denote
by L1 the assignment of labor input into plant 1 and by√ L2 the assignment of labor input
into plant 2. Plant 1’s production function is√
F1 (L1) = 4
L1, for L1 ≥ 0. Plant 2’s production
function is F2(L2) = 8 L2, for L2 ≥ 0.
1. State the average product function of each plant as a function of the labor assignment. Denotethem
by AP1(L1) and AP2(L2).
2. State the marginal product function of each plant as a function of the labor assignment.Denote them
by MP1(L1) and MP2(L2).
3. Define total quantity produced for a given labor assignment by Q(L1,L2) = F1(L1)+F2(L2). Suppose
the firm has a total of 100 units of labor available, L = 100. It can freely assign them across the two
plants subject to L1 + L2 = L. In a graph show total output produced for different choices of L1 ∈
[0,100] where L2 = L − L1.
4. For L = 100, find the input assignment,
.
, that maximizes total output, Use the insight that
5. We want to derive the firm’s efficient production function frontier for any total labor inputL ≥ 0.
Call it F(L). It is the greatest output that can be produced with L units of workers.
(a) For a given L, find the input assignment
that maximizes total output. Verify
that
matches your answer to question 4.
. Show that it can be written in the form F(L) =
(b) Define√
A L. What is the value of A?
1
Problem 3 – A single input firm
Consider a firm that produces a single output with a single input, labor, using production function F (L) =
100L − 4L2, for L ∈ [0,12.5]. The input price is W = 2.
1. Determine the firm’s cost function C (Q), that is, the lowest cost of producing Q units of output.
State the associated labor choice for a givsn required output, L(Q). Consider only the range Q ∈
[0,625].
2. What is the firm’s marginal cost curve, MC (Q)?
Problem 4 – Adam’s Apples
”Adam’s Apples” produces apple cider using labor (L) and apples (A) according to the Cobb-Douglas
production function Q = F (L,A) = 2L1/4A3/4. The price of apples is pA = 2 and the price of labor is W = 10.
Adam’s Apples also has a fixed cost for farm buildings, FC = 100.
1. If Adam’s Apples wants to produce 100 gallons of apple cider, Q=100, what is its lowest achievable
input cost? (Roadmap: (1) Determine the optimal input choice (L∗,A∗) by tangency condition,
MRTSLA (L∗,A∗) = W/pA and production requirement 100 = F (L∗,A∗). (2) Variable costs are then
found by WL∗ + pAA∗. (3) Total cost is sum of variable and fixed costs.)
2. Determine Adam’s Apples’ cost function, C (Q). Verify that C (100) gives you same answer as in
question 1.
3. Determine Adam’s Apples’ Average Cost function, AC(Q), and marginal cost function, MC (Q).
Problem 5 – Optimal production
The Pear Corp produces high end consumer electronics using labor L and capital K according to
production function Q = F (L,K) = 100L1/4K1/4. Let the price of a unit of labor be given by W and the price
of a unit of capital is given by R. The output price is P. Pear Corp has fixed costs of FC = 5 that are
unavoidable in the short run, but avoidable in the long run. Both labor and capital are fully adjustable in
both short and long run. Set input price W = 20 and R = 5.
1. Is the production function increasing, decreasing, or constant returns to scale? Explain youanswer.
2. What is Pear Corp’s marginal rate of technical substitution, MRTS (L,K) for a given input bundle
(L,K)? (MRTS is the absolute value of the slope of the isoquant in a diagram with L on the horizontal
axis and K on the vertical axis).
3. Determine Pear Corp’s variable cost function V C (Q). That is, the lowest possible variable input
expenditure associated with the production of Q ≥ 0 units of output. For short and long run,
respectively, state Pear Corp’s cost function, CSR(Q) and CLR (Q), (That is the total cost of producing
Q ≥ 0 units).
…
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