http://www.geocities.com/vrppe
Fuente: Microeconomics: An Integrated Approach, Besanko-Braeutigan
Producción, Costos, Oferta de la empresa
Problems
1.
Consider the short-run production function
Q  25LK
a)
Derive the cost-minimizing quantity of labor as a function of output and
capital. How does the cost-minimizing quantity of labor vary with Q ? with
K?
Answer
To find the cost-minimizing quantity of labor in the short-run, simply solve
the production function for L in terms of output and capital.
Q  25LK
L
Q
25K
Since output is in the numerator, the cost-minimizing quantity of labor will
increase as the quantity of output goes up. Conversely, since the fixed level
of capital is in the denominator, the cost-minimizing quantity of labor will
decrease as the quantity of capital goes up.
b)
Assume the price of labor is w  50 and the rental rate of capital is r  10 .
Derive the short-run total cost function. How does the short-run total cost
vary with Q ? with K ?
Answer
To find the short-run total cost function, plug the cost-minimizing quantity
of labor and the fixed level of capital into the cost function TC  wL  rK .
 Q 
TC  50 
  10 K
 25 K 
2Q
TC 
 10 K
K
1
http://www.geocities.com/vrppe
Since output only shows up in the numerator of the left-hand term, total cost will increase
as the level of output goes up. The fixed level of capital shows up in both terms. Since it is
in the denominator of the left-hand term, as K increases, the left-hand term will fall, i.e.,
the labor costs will fall. At the same time, however, as K increases the right-hand term
will also increase, i.e., capital costs will increase. Therefore, without knowing more about
the current level of K , it is impossible to determine whether total costs will increase or
decrease.
Problems
2.
Suppose a firm has the production function
Q  30 KL
with MPL  30K and MPK  30L .
a)
If the wage rate is $10 per unit of labor and the rental rate of capital is $5 per
unit of capital, how much capital and labor should the firm employ in the
long run to minimize the cost of producing 37,500 units?
Answer
Use the tangency condition to determine the optimal capital-labor ratio.
MPL MPK

w
r
30 K 30 L

10
5
5K  10 L
K  2L
Next, use this result with the production function to determine the optimal
quantities of capital and labor.
Q  30 KL
37,500  30(2 L) L
625  L2
L  25
Finally, since K  2 L , K  25(2)  50.
b)
Using the solution in part a), what will the firm’s long-run total cost be?
2
http://www.geocities.com/vrppe
TC  wL  rK
TC  10(25)  5(50)
TC  50
3.
Suppose a firm has the production function
Q  30 KL.
where MPL  30K and MPK  30L . If the wage rate is w and the rental rate of
capital is r , derive the input demand curves for labor and capital.
Answer
Use the tangency condition to determine the optimal capital-labor ratio.
MPL MPK

w
r
30 K 30 L

w
r
rK  wL
K  wr L
Next, use this result with the production function to determine the optimal
quantities of capital and labor.
Q  30 KL
Q  30( wr L) L
rQ
 L2
30 w
L
rQ
30 w
Finally, since K  wr L we have
K  wr L
K
w rQ
r 30 w
K
wQ
30r
3
http://www.geocities.com/vrppe
Problems
4.
Suppose a firm’s short-run total cost curve is given by
STC  6Q2  5Q  1
with short-run marginal cost SMC  12Q  5 .
a)
What is the equation for the firm’s short-run supply curve?
Answer
First, we find the minimum of average variable cost by setting average
variable cost equal to short-run marginal cost.
6Q  5  12Q  5
Q0
At Q  0 , average variable cost is AVC  6Q  5  6(0)  5  5 . The supply
curve is the short-run marginal cost curve above the minimum point of
average variable cost. Thus,
P 5
P5

s( P)   12

0 P5
0
b)
How many units will the firm supply if market price is P  29 ?
Answer
Plugging P  29 into the supply curve yields s (29)  2 . The firm will
supply two units.
Page Reference: 358-359
5.
Suppose in some market that a typical firm’s short-run total cost curve is given by
STC  10Q2  4Q  100
with short-run marginal cost SMC  20Q  4 . There are 100 identical firms in the
market and all fixed costs are sunk. In addition, market demand is given by
D( P)  100  P .
a)
Derive the equation for the typical firm’s short-run supply curve.
4
http://www.geocities.com/vrppe
Answer
To find the firm’s short-run supply curve, begin by finding the minimum
point of the average variable cost curve by setting average variable cost
equal to short-run marginal cost.
10Q  4  20Q  4
Q0
At Q  0 , average variable cost is AVC  10Q  4  10(0)  4  4 . The
firm’s supply curve is short-run marginal cost above the minimum point of
average variable cost. Thus, supply is given by
P 4
P4

s ( P)   20
0
0P4
b)
What is the short-run equilibrium market price?
Answer
Since there are 100 identical firms with the supply curve given in a), market
supply is given by (for price greater than $4)
S ( P)  100s ( P)
 P4
S ( P)  100 

 20 
S ( P)  5P  20
Setting market supply equal to market demand implies
S ( P)  D( P)
5 P  20  100  P
6 P  120
P  20
5
http://www.geocities.com/vrppe
Mercado
Problems
6.
Suppose market demand is given by Qd  50  2P and market supply is given by
Q s  3P .
a)
With no tax, what is the market equilibrium price and quantity?
Answer
To find the equilibrium, set quantity demanded equal to quantity supplied.
50  2 P  3P
5P  50
P  10
b)
Now suppose the government imposes an excise tax of $10 per unit. What
will the new equilibrium quantity be? What price will the buyer pay? What
price will the seller receive?
Answer
With an excise tax, two conditions must hold. First, the market must clear
so that Q d  Q s . Second, there is a tax wedge of $10, so P d  P s  10 .
Together, these conditions imply
50  2( P s  10)  3P s
5 P s  30
Ps  6
This implies P d  P s  10  16 and the equilibrium quantity will be
(substituting into demand) Q  50  2(16)  18 .
c)
At the equilibrium calculated in part b), what will the government’s total tax
receipts be?
6
http://www.geocities.com/vrppe
Answer
To calculate tax receipts, take the amount of tax, $10, and multiply by the
equilibrium quantity, 18. Tax receipts will be $180.
d)
Who bears the greater burden of the tax, consumers or producers? What
does this tell you about the relative elasticities of supply and demand?
Answer
Consumers pay $6 more than without the tax and sellers receive $4 less than
without the tax. Thus, the consumers carry a greater burden of the tax. This
implies that demand is relatively inelastic when compared with supply.
7.
In the market for compact discs, market demand is given by Q d  300  10 P and
quantity supplied is given by Q s  25  15P where quantity is given in millions of
compact discs sold per year.
a)
With no government subsidy, what is the market equilibrium price and
quantity?
Answer
To find the market equilibrium, set quantity demanded equal to quantity
supplied.
300  10 P  25  15P
25P  325
P  13
At this price, the equilibrium quantity will be Q  300  10(13)  170 .
Consumers will purchase 170 million compact discs.
b)
Suppose the government decided to support this “high-tech” industry by
providing a $10 subsidy per compact disc. What would the new equilibrium
quantity be? What price would consumers pay? What price would sellers
receive?
Answer
In equilibrium, two conditions must be satisfied. First, markets must clear
so that Q d  Q s . Second, there will be a wedge between the price that
consumers pay and the price that sellers receive; in particular, P d  P s  10 .
Together these imply
7
http://www.geocities.com/vrppe
300  10( P s  10)  25  15 P s
25P s  425
P s  17
This is the price sellers will receive. Consumers will pay P d  17  10  7 ,
and the equilibrium quantity will be Q  230 , or 230 million compact discs
per year.
c)
What will this subsidy cost the government?
Answer
To determine the total cost, take the amount of the subsidy, $10, and
multiply it by the market quantity, 230 million. The total cost will be $2,300
million.
8.
In a particular market, Q d  200  15P and Q s  10  5P where Q is given in
thousands of units.
a)
With no government intervention, what are the equilibrium price and
quantity?
Answer
Setting Q d  Q s yields
200  15P  10  5P
20 P  190
P  9.5
At this price, the equilibrium quantity is Q  200  15(9.5)  57.5 , or 57,500
units.
b)
Now suppose the government sets a quota of 50,000 units. How will this
affect the price consumers pay and sellers receive?
Answer
If the government sets a quota of 50,000 units, we can find price consumers
pay and sellers receive by plugging 50,000 into the demand equation and
solving for P . This gives
50  200  15P
P  10
8
http://www.geocities.com/vrppe
In this case, consumers will pay and sellers will receive $10 per unit.
c)
With this quota, what is the amount of the deadweight loss?
Answer
40.0
Qs
35.0
Deadweight Loss
Price
30.0
25.0
20.0
15.0
10.0
5.0
Qd
0.0
0.0
50.0
100.0
150.0
200.0
250.0
Quantity
As the figure above shows, the deadweight loss is the difference between the
price on the demand curve and the price on the supply curve multiplied by
the difference between the quota and the equilibrium quantity. In part b) we
determined the price on the demand curve is $10. Substituting 50 into the
supply curve implies a price of $8 on the supply curve. Thus, the
deadweight loss is 0.5(10-8)(57.5-50) = 7.5. Since Q is given in thousands,
the deadweight loss is $7,500.
9.
In a particular U.S. market, quantity demanded is given by Qd  500  20P and
quantity supplied is given by Q s  50  30 P . The world price for this product is $6,
and an unlimited amount can be purchased at this price.
a)
With imports prohibited by the U.S., what is the domestic equilibrium price
and quantity?
Answer
With imports prohibited, simply set Q d  Q s . This yields
500  20 P  50  30 P
50 P  450
P9
9
http://www.geocities.com/vrppe
At this price, the domestic quantity sold will be Q  500  20(9)  320 .
b)
Now suppose the government opens up the U.S. economy to free trade.
How will this affect the equilibrium market price, the quantity supplied by
domestic suppliers, the quantity purchased by domestic consumers, and the
level of imports?
Answer
When the government opens up the economy to free trade, the equilibrium
price will fall to the world price, $6. At this price, domestic consumers will
demand Qd  500  20(6)  380 units and domestic suppliers will provide
Q s  50  30(6)  230 units. Imports will be the difference, or 150 units.
10