Download PS3

Microeconomics: Problem set 3 (Due on May 25-26 in the TA recitation)
Instructor: Toshihiro Ichida
Page 1 of 6
7.6.
Suppose the production of airframes is characterized by a CES production function
1
1
Q  (L2  K 2 ) 2
The marginal products for this production function are
1
1
-
1
MPL  (L2  K 2 )L 2
1
1
MPK  (L2  K 2 )K
-
1
2
Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the
cost-minimizing combination of labor and capital for an airframe manufacturer that wants to produce
121,000 airframes.
7.7.
Suppose the production of airframes is characterized by a Cobb-Douglas production function
Q = LK
Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the
cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000
airframes.
7.9.
Consider, again, the production function
Q = LK.
Suppose that the price of labor equals w, and the price of capital equals r. Derive expressions for the
input demand curves.
7.10.
Suppose that the firm’s production function is given by
1
3
Q  10KL .
The firm’s capital is fixed at K . What amount of labor will the firm hire to solve its short-run cost
minimization problem?
7.11.
Suppose that the firm uses three inputs to produce its output: capital, K, labor, L, and materials, M.
The firm’s production function is given by
Microeconomics: Problem set 3 (Due on May 25-26 in the TA recitation)
Instructor: Toshihiro Ichida
Page 2 of 6
1
3
1
3
1
3
Q  10 K L M .
The prices of capital, labor, and materials are r = 1, w = 1, and m = 1, respectively.
a) What is the solution to the firm’s long-run cost-minimization problem given that the firm wants
to produce Q units of output?
b) What is the solution to the firm’s short-run cost-minimization problem when the firm wants to
produce Q units of output and capital is fixed at K ?
c) .Verify that when Q = 4 and K = 4, which is the long-run cost-minimizing quantity of capital
when Q = 4, the short-run and long-run quantities of labor and materials are the same.
8.3.
Consider a production function of two inputs, labor and capital, given by
1
1
Q= [L 2 +K 2 ] 2.
The marginal products associated with this production function are as follows:
1
1
1
MPL= [L 2 +K 2 ] L 2 .
1
1
1
MPK= [L 2 +K 2 ] K 2 .
Let w=2 and r=1.
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 of the firm’s long-run total cost curve.
c) Find the equation of the firm’s long-run average cost curve.
d) Find the solution to the firm’s short-run cost-minimization problem when capital is fixed at a
quantity of 9 units（i.e., K = 9）
．
e) Find the short-run total cost curve, and graph it along with the long-run total cost curve.
f) Find the associated short-run average cost curve.
8.4.
Consider a production function of three inputs, labor, capital, and materials, given by
Q = LKM
The marginal products associated with this production function are follows:
MP L = KM
MPK = LM
Microeconomics: Problem set 3 (Due on May 25-26 in the TA recitation)
Instructor: Toshihiro Ichida
Page 3 of 6
MP M = LK
Let w = 5, r = 1, and m = 2, where m is the price per unit of materials.
a) Suppose that the firm is required to produce Q units of output. Show how the cost-minimizing
quantity of capital depends on the quantity Q. Show how the cost-minimizing quantity of
materials depends on the quantity Q.
b) Find the equation of the firm’s long-run total cost curve.
c) Find the equation of the firm’s long-run average cost curve.
d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a
quantity of 50 units (i.e., K =50).
on the quantity Q.
Show how the cost-minimizing quantity of labor depends
Show how the cost-minimizing quantity of materials depends on the
quantity Q.
e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of 50 units
(i.e., K = 50) and graph it along with the long-run total cost curve.
f)
Find the equation of the associated short-run average cost curve.
8.6.
A producer of hard disk drives has a short-run total cost curve given by
STC (Q)  K 
Q2
.
K
Within the same set of axes, sketch a graph of the short-run average cost curves for three different
plant sizes: K  10 , K  20 , and K  30 . Based on this graph, what is the shape of the
long-run average cost curve?
9.2
The bolt industry currently consists of 20 producers, all of whom operate with identical short-run
total cost functions
,
where Q is the annual output of a firm. The corresponding short-run marginal cost curve is
The market demand curve for bolts is
where P is the market price.
a) Assuming that all of the firm’s $16 fixed cost is sunk, what is a firm’s short-run supply curve?
b) What is the short-run market supply curve?
c) Determine the short-run equilibrium price and quantity in this industry.
Microeconomics: Problem set 3 (Due on May 25-26 in the TA recitation)
Instructor: Toshihiro Ichida
Page 4 of 6
9.3.
A market contains a group of identical firms. Each firm acts as a price taker and has a short-run total
cost function.
STC(Q) = 250 + Q2 ,
where Q is the annual output of each firm. The corresponding marginal cost curve is
SMC(Q )= 2Q .
A study reveals that part of the $250 fixed cost is sunk, and each firm will produce if the price
exceeds $20 per unit and will shut down if price is less than $20 per unit. The market demand curve
for the industry is
D(P )= 240 – 1/2 P,
where P is the market price.
a) How much of the fixed cost is sunk for each firm?
At the current market price, each firm produces 20 units. What is market price and how many firms
are in this industry?
9.4.
The wood-pallet market contains many identical firms, each with a short-run total cost
function
STC(Q)= 400 + 5Q + Q2 ,
where Q is a firm’s annual output(and all of a firm’s $400 fixed cost is sunk). The corresponding
marginal cost function is
SMC(Q) = 5 + 2Q
The market demand curve for this industry is
D( P )  262.5 
1
P,
2
where P is the market price. Each firm in the industry is currently earning zero economic profit. How
many firms are in this industry, and what is the marker equilibrium price?
9.7.
Propylene is used to make plastic. The propylene industry is perfectly competitive, and each
producer has a long-run marginal cost function given by
MC(Q)=40-12Q+
.
The corresponding long-run average cost function is
AC(Q)=40-6Q+
.
Microeconomics: Problem set 3 (Due on May 25-26 in the TA recitation)
Instructor: Toshihiro Ichida
Page 5 of 6
The market demand curve for propylene is
D(P)=2200-100P.
(a)
What is the long-run equilibrium price in this industry?
(b)
At this price, how much would an individual firm produce?
(c)
How many firms are in the propylene market in a long-run competitive equilibrium?
(d)
Suppose the demand curve shifted so that it is now
D(P)=A-100P.
How large would A have to be so that in the new long-run competitive equilibrium, the number of
propylene firms was twice what it was in the initial long-run equilibrium?
9.10
The long-run average cost for production of hard-disk drives is given by
2
AC(Q) = wr (120 － 20Q + Q )
where Q is the annual output of a firm, w is the wage rate for skilled assembly labor,
and r is the price of capital services. The corresponding long-run marginal cost is
2
MC(Q) = wr (120 － 40Q + 3Q ).
Suppose, further , that the demand for labor for an individual firm is
LQ, w, r  

r 120  20Q 2  Q 3
2 w
.
Let’s suppose throughout this problem that the price of capital services is fixed at r = 1.
a) In a long-run competitive equilibrium, how much output will each firm produce?
b) In a long-run competitive equilibrium, what will be the market price? Note that your
answer will be expressed as a function of w.
c) At a long-run competitive equilibrium, how much skilled labor will each firm demand?
Again, your answer will depend on w.
d) Now, suppose that the market demand curve is given by
D P  
10,000
.
P
What is the market equilibrium quantity as a function of w?
e) What is the long-run equilibrium number of firms as a function of w?
f) Using your answers to parts (c) and (e), what is the overall demand for skilled labor in
this industry as a function of w?
g) Suppose that the supply curve for the skilled labor used in this industry is given by
( w)  50 w
At what value of w does the supply of skilled labor equal the demand for skilled labor?
Microeconomics: Problem set 3 (Due on May 25-26 in the TA recitation)
Instructor: Toshihiro Ichida
Page 6 of 6
h) Using your answer from part (g), go back through parts (b), (d), and (e) to determine
the long-run equilibrium price, market demand, and number of firms in this industry.
i) Repeat the analysis in this problem, now assuming that the market demand curve is given
by
D P  
20,000
P
9.11
Consider an industry in which chief executive officers (CEOs) run firms. There are two types of
CEOs: exceptional and average. There is a fixed supply of 100 exceptional CEOs and an unlimited
supply of average CEOs. Any individual capable of being a CEO in this industry is willing to work
for a salary of $144,000 per year. The long – run total cost of a firm that hires an exceptional CEO at
this salary is TCE(Q) = 144+
1 2
Q , if Q > 0.
2
0 if Q = 0.
where Q is annual output in thousands of units, and total cost is expressed in thousands of dollars per
year. The corresponding long – run marginal cost curve is MCE(Q) = Q, where marginal cost is
expressed as dollars per unit. The long – run total cost for a firm that hires an average CEO for
$144,000 per year is TCA(Q) = 144 + Q2.
The corresponding marginal cost curve is MCA(Q) = 2Q.
The market demand curve in this market is D(P) = 7200-100P, where P is the market price and D(P)
is the market quantity, expressed in thousands of per year.
a) What is the minimum efficient scale for a firm run by an average CEO? What is the minimum
level of long – run average cost for such firms?
b) What is the long – run equilibrium price in this industry, assuming that it consists of firms with
both exceptional and average CEOs?
c) At this price, how much output will a firm with an average CEO produce? How much output
will a firm with an exceptional CEO produce?
d) At this price, how much output will be demanded?
e) Using your answers to parts (c) and (d), how many firms with average CEOs will there be in this
industry at a long – run equilibrium?
f)
What is the economic rent attributable to an exceptional CEO?
g) If firms with exceptional CEOs hire them at the reservation wage of $144,000 per year, how
much economic profit do these firms make?
Assuming that firms bid against each other for the services of exceptional CEOs, what would you
expect their salaries to be in a long – run competitive equilibrium?