ch07Solution manual

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Chapter 7

Costs and Cost Minimization

Solutions to Review Questions

1. A biotechnology firm purchased an inventory of test tubes at a price of $0.50 per tube at some point in the past. It plans to use these tubes to clone snake cells. Explain why the opportunity cost of using these test tubes might not equal the price at which they were acquired.

Acquisition cost and opportunity cost are not necessarily the same. As the text points out, opportunity costs are forward looking. The opportunity cost is the payoff associated with the best of the alternatives that are not chosen. Once the test tubes are purchased, the decision is to use the tubes to clone snake cells or something else. It is possible that someone values the tubes for some purpose at higher (or lower) than $0.50 so that selling the tubes would earn the firm something more (or less) than $0.50 per tube. The opportunity cost then is different than the acquisition cost.

2. You decide to start a business that provides computer consulting advice for students in your residence hall. What would be an example of an explicit cost you would incur in operating this business? What would be an example of an implicit cost you would incur in operating this business?

Since the business is computer consulting, an explicit cost, a cost involving a direct monetary outlay, might be the cost of paper and ink used to advertise your service. An implicit cost, a cost not involving a direct monetary outlay, might be the opportunity cost of your time, e.g., to earn money working at the student fitness center or to study for your own classes.

3. Why does the “sunkness” or “nonsunkness” of a cost depend on the decision being made?

Whether or not a particular cost is sunk or not depends on the decision being made. If the cost does not change as a result of the decision the cost is sunk, while if the cost does change the cost is not sunk.

4. How does an increase in the price of an input affect the slope of an isocost line?

A firm’s total costs are TC = rK + wL, so the equation for a typical isocost line is

Since the slope of the isocost line is given by , if the price of labor increases the isocost line will become steeper and if the price of capital increases the isocost line will become flatter.

Copyright © 2014 John Wiley & Sons, Inc. Chapter 7 - 1


5. Could the solution to the firm’s cost-minimization problem ever occur off the isoquant representing the required level of output?

The solution to the firm’s cost minimization problem must lie on an isoquant. While the firm could produce a given output with a combination of inputs not on the isoquant, say by using more labor and more capital than necessary, a combination such as this would not be efficient and therefore not cost minimizing.

6. Explain why, at an interior optimal solution to the firm’s cost-minimization problem, the additional output that the firm gets from a dollar spent on labor equals the additional output from a dollar spent on capital. Why would this condition not necessarily hold at a corner point optimal solution?

To understand why at an interior optimum the additional output the firm gets from a dollar spent on labor must equal the additional output the firm gets from a dollar spent on capital, assume these were not equal. For example, suppose the firm could get more output from a dollar spent on labor than on a dollar spent on capital. Then the firm could take one dollar away from capital and reallocate it to labor. Since the firm gets more output from a dollar of labor than from a dollar of capital, it will require the firm to spend less than one dollar on labor to offset the decline in output from taking one dollar away from capital. This implies the firm can keep output at the same level but do so at a lower cost. Therefore, if these amounts are not equal the firm is not minimizing cost.This requirement does not necessarily hold at a corner solution. While the firm could potentially reduce cost by reallocating spending to the more productive input, at a corner solution, by definition, the firm is not using one of the inputs. There is no further opportunity to reallocate spending if the firm is spending nothing on one of the inputs, i.e., the firm cannot move to a point where one of the inputs is negative.

7. What is the difference between the expansion path and the input demand curve?

The expansion path traces out the cost minimizing combinations of all inputs as the level of output is increased (expanded) holding the prices of the inputs fixed. An input demand curve traces out a firm’s cost minimizing quantity of one input as the price of that input varies holding the level of output and the prices of the other inputs fixed.

8. In Chapter 5 you learned that, under certain conditions, a good could be a Giffen good: An increase in the price of the good could lead to an increase, rather than a decrease, in the quantity demanded. In the theory of cost minimization, however, we learned that, an increase in the price of an input will never lead to an increase in the quantity of the input used. Explain why there cannot be “Giffen inputs.”

Giffen goods arise when the income effect is so severely negative that it offsets the substitution effect. This can happen because in consumer choice, income was an exogenous variable – therefore, changes in price affect both the relative substitutability of goods (via the tangency condition) as well as the consumer’s purchasing power (via the budget constraint). By contrast,



in the cost minimization problem output is exogenous while the expenditure is the objective function. Thus, a change in an input price affects only the relative substitutability of inputs (via the tangency condition) – there is no corresponding effect on the production constraint, since prices do not appear there. So while there is a “substitution effect” in cost minimization, there is no corresponding “income effect” as in consumer choice. Therefore, increases in input prices will always lead to decreases in the use of that input (except at corner solutions, where there might be no change). So there cannot be a Giffen input.

9. For a given quantity of output, under what conditions would the short-run quantity demanded for a variable input (such as labor) equal the quantity demanded in the long run?

Assuming quantity is fixed, the short-run demand for a variable input would equal its long-run demand if the level of the fixed input in the short run was cost minimizing for the quantity of output being produced in the long run.

Solutions to Problems

7.1. A computer-products retailer purchases laser printers from a manufacturer at a price of $500 per printer. During the year the retailer will try to sell the printers at a price higher than $500 but may not be able to sell all of the printers. At the end of the year, the manufacturer will pay the retailer 30 percent of the original price for any unsold laser printers. No one other than the manufacturer would be willing to buy these unsold printers at the end of the year.a) At the beginning of the year, before the retailer has purchased any printers, what is the opportunity cost of laser printers?b) After the retailer has purchased the laser printers, what is the opportunity cost associated with selling a laser printer to a prospective customer? (Assume that if this customer does not buy the printer, it will be unsold at the end of the year.)c) Suppose that at the end of the year, the retailer still has a large inventory of unsold printers. The retailer has set a retail price of $1,200 per printer. A new line of printers is due out soon, and it is unlikely that many more old printers will be sold at this price. The marketing manager of the retail chain argues that the chain should cut the retail price by $1,000 and sell the laser printers at $200 each. The general manager of the chain strongly disagrees, pointing out that at $200 each, the retailer would “lose” $300 on each printer it sells. Is the general manager’s argument correct?

a) $500b) 30% of $500, or $150c) By not lowering the price and assuming the firm cannot sell any more printers, the best the firm can hope for is the $150 the firm can receive from the manufacturer. If the firm drops the price to $200 and sells the printers on their own they can actually “profit” an additional $50 over their best available alternative.



7.2. A grocery shop is owned by Mr. Moore and has the following statement of revenues and costs:

Revenues $250,000Supplies $25,000Electricity $6,000Employee salaries $75,000Mr. Moore’s salary $80,000

Mr. Moore always has the option of closing down his shop and renting out the land for $100,000. Also, Mr. Moore himself has job offers at a local supermarket at a salary of $95,000 and at a nearby restaurant at $65,000. He can only work one job, though. What are the shop’s accounting costs? What are Mr. Moore’s economic costs? Should Mr. Moore shut down his shop?

The accounting costs are simply the sum: 25,000 + 75,000 + 80,000 + 6,000 = $186,000 and the shop’s accounting profit is $64,000 which means that Mr. Moore’s total gain from this venture is 80,000 + 64,000 = $144,000.The economic costs also include the opportunity cost of the land rental ($100,000) and of Mr. Moore’s next best alternative, which in this case is $95,000. That is, Mr. Moore loses $15,000 by not choosing his next best alternative. Therefore Mr. Moore’s total economic costs are 186,000 + 100,000 + 15,000 = $301,000, which exceeds his revenues by $51,000.If he were to shut down the shop, Mr. Moore would earn 100,000 + 95,000 = $195,000 which is more than the $144,000 he currently earns (by precisely the $51,000 figure from above). Therefore he should shut down the shop.

7.3. Last year the accounting ledger for an owner of a small drugstore showed the following information about her annual receipts and expenditures. She lives in a tax-free country (so don't worry about taxes).Revenues $1,000,000Wages paid to hired labor (other than herself) $300,000Utilities (fuel, telephone, water) $ 20,000Purchases of drugs and other supplies for the store $500,000Wages paid to herself $100,000She pays a competitive wage rate to her workers, and the utilities and drugs and other supplies are all obtained at market prices. She already owns the building, so she has no cash outlay for its use. If she were to close the business, she could avoid all of her expenses, and, of course, would have no revenue. However, she could rent out her building for $200,000. She could also work elsewhere herself. Her two employment alternatives include working at another drugstore, earning wages of $100,000, or working as a freelance consultant, earning $80,000. Determine her accounting profit and her economic profit if she stays in the drug store business. If the two are different, explain the difference between the two values you have calculated.

Her accounting profit equals revenues less all of the expenses reflected in the ledger:$1,000,000 - $300,000 - $20,000 - $500,000 - $100,000 = $80,000.



All of the accounting costs are also economic costs. The first three expense items (wages paid to hired labor, utilities, and purchases of drugs and supplies) are expenses in competitive markets, so the opportunity cost is reflected in the market prices. Further, the wages she pays herself are the same as the opportunity cost of her time, because the most she could earn if she closes her business is $100,000 working at another drugstore.

The economic costs of the business include all of the accounting costs, plus the $200,000 opportunity cost of the building because she could earn that if she exits the drug store business. Her economic profit is her accounting profit ($80,000) less the additional opportunity cost ($200,000) not included in the accounting cost. So her economic profit is actually -$120,000.

We can look at this another way. If she continues to work at the grocery store, she earns an accounting profit of $80,000, plus the salary she pays herself ($100,000). But if she exits the business, her salary working for another drugstore would be $100,000, and she would receive $200,000 rent for the building. She would therefore be better off by $120,000 if she takes the job working for another drugstore.

7.4. A consulting firm has just finished a study for a manufacturer of wine. It has determined that an additional man-hour of labor would increase wine output by 1,000 gallons per day. Adding another machine-hour of fermentation capacity would increase output by 200 gallons per day. The price of a man-hour of labor is $10 per hour. The price of a machine-hour of fermentation capacity is $0.25 per hour. Is there a way for the wine manufacturer to lower its total costs of production and yet keep its output constant? If so, what is it?

At the optimum we must have

In this problem we have

This implies that the firm receives more output per dollar spent on an additional machine hour of fermentation capacity than for an additional hour spent on labor. Therefore, the firm could lower cost while achieving the same level of output by using fewer hours of labor and more hours of fermentation capacity.

7.5. A firm uses two inputs, capital and labor, to produce output. Its production function exhibits a diminishing marginal rate of technical substitution.a) If the price of capital and labor services both increase by the same percentage amount (e.g., 20 percent), what will happen to the cost-minimizing input quantities for a given output level?



b) If the price of capital increases by 20 percent while the price of labor increases by 10 percent, what will happen to the cost-minimizing input quantities for a given output level?

a) If the price of both inputs change by the same percentage amount, the slope of the isocost line will not change. Since we are holding the level of output fixed, the isocost line will be tangent to the isoquant at the same point as prior to the price increase. Therefore, the cost-minimizing quantities of the inputs will not change.b) If the price of capital increases by a larger percentage than the price of labor, then, relatively speaking, the price of labor has become cheaper. The firm will substitute away from capital and add labor until either the tangency condition holds or a corner solution is reached.

7.6. A farmer uses three inputs to produce vegetables: land, capital, and labor. The production function for the farm exhibits diminishing marginal rate of technical substitution.a) In the short run the amount of land is fixed. Suppose the prices of capital and labor both increase by 5 percent. What happens to the cost-minimizing quantities of labor and capital for a given output level? Remember that there are three inputs, one of which is fixed.b) Suppose only the cost of labor goes up by 5 percent. What happens to the cost-minimizing quantity of labor and capital in the short run?

a) The amount of land used in production is fixed in the short-run. Hence, in the short-run the farmer chooses amount of capital and labor. It follows that cost-minimizing quantities of labor and capital have to satisfy equation MPL / MPK = w/r where w and r denote prices of labor and capital. Notice that w/r = (1.05 w)/ (1.05 r). The cost-minimizing quantities of inputs, for each level of output, do not change when prices of both inputs go up by 5% and quantity of land is fixed.b) For a given output level, the cost-minimizing farmer uses more capital and less labor.

7.7. The text discussed the expansion path as a graph that shows the cost-minimizing input quantities as output changes, holding fixed the prices of inputs. What the text didn’t say is that there is a different expansion path for each pair of input prices the firm might face. In other words, how the inputs vary with output depends, in part, on the input prices. Consider, now, the expansion paths associated with two distinct pairs of input prices, (w1, r1) and (w2, r2). Assume that at both pairs of input prices, we have an interior solution to the cost-minimization problem for any positive level of output. Also assume that the firm’s isoquants have no kinks in them and that they exhibit diminishing marginal rate of technical substitution. Could these expansion paths ever cross each other at a point other than the origin (L = 0, K = 0)?

Imagine that two expansion paths did cross at some point. Recall that the expansion path traces out the cost- minimizing combinations of inputs as output increases. Essentially the expansion path traces out all of the tangencies between the isocost lines and isoquants. These tangencies occur at the point where



If the expansion paths cross at some point then the cost minimizing combination of inputs must be identical with both sets of prices. This would require that

and

Unless the input prices are proportional, i.e. unless w1 / r1 = w2 / r2, it is not possible for both of these equations to hold. Therefore, it is not possible for the expansion paths to cross unless the prices are proportional, in which case the two expansion paths will be identical.

7.8. Suppose the production of airframes is characterized by a CES production function: Q = (L½ + K½)2. The marginal products for this production function are MPL = (L½ + K½)L−½ and MPK = (L½+ K½)K−½. 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.

The tangency condition implies

Given that and , this implies

Returning to the production function and assuming yields



Since , . The cost minimizing quantities of capital and labor to produce 121,000 airframes is and .

7.9. Suppose the production of airframes is characterized by a Cobb–Douglas production function: Q = LK. The marginal products for this production function are MPL = K and MPK = L. Suppose 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.

The tangency condition implies

Substituting into the production function yields

Since , . The cost-minimizing quantities of labor and capital to produce 121,000 airframes are and .

7.10. The processing of payroll for the 10,000 workers in a large firm can either be done using 1 hour of computer time (denoted by K) and no clerks or with 10 hours of clerical time (denoted by L) and no computer time. Computers and clerks are perfect substitutes; for example, the firm could also process its payroll using 1/2 hour of computer time and 5 hours of clerical time.a) Sketch the isoquant that shows all combinations of clerical time and computer time that allows the firm to process the payroll for 10,000 workers.



b) Suppose computer time costs $5 per hour and clerical time costs $7.50 per hour. What are the cost-minimizing choices of L and K? What is the minimized total cost of processing the payroll?c) Suppose the price of clerical time remains at $7.50 per hour. How high would the price of an hour of computer time have to be before the firm would find it worthwhile to use only clerks to process the payroll?

a)

K and L are perfect substitutes, meaning that the production function is linear and the isoquants are straight lines. We can write the production function as Q = 10,000K + 1000L, where Q is the number of workers for whom payroll is processed.

b) If and , the slope of a typical isocost line will be . This is steeper than the isoquant implying that the firm will employ only computer time ( ) to minimize cost. The cost minimizing combination is and . This outcome can be seen in the graph below. The isocost lines are the dashed lines.

The total cost to process the payroll for 10,000 workers will be .

c) The firm will employ clerical time only if MPL / w > MPK / r. Thus we need 0.1 / 7.5 > 1/r or r > 75.



7.11. A firm produces an output with the production function Q = KL, where Q is the number of units of output per hour when the firm uses K machines and hires L workers each hour. The marginal products for this production function are MPK = L and MPL = K. The factor price of K is 4 and the factor price of L is 2. The firm is currently using K = 16 and just enough L to produce Q = 32. How much could the firm save if it were to adjust K and L to produce 32 units in the least costly way possible?

Currently the firm must be using L = Q/K = 32/16 = 2 units of labor. Let the factor prices of capital and labor be, respectively, r and w.Its total expenditure is C = wL + rK = 2(2) + 4(16) = 68.If it were to minimize cost, it would hire L and K so that (1) MPK/r = MPL/w, or L/4 = K/2, or L = 2K and (2) Q = LK.(1) and (2) imply that Q = 2K2, or 32 = 2K2, and thus K = 4 and L = 8.So Q = 32 can be produced efficiently with a cost of C = wL + rK = 2(8) + 4(4) = 32.The firm could save 68 – 32 = 36 by producing efficiently.

7.12. A firm operates with the production function Q = K2L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. The marginal product of capital is 2KL, and the marginal product of labor is K2. The manager has been given a production target: Produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit. The wage rate paid to each worker is $200 day.a) Currently the firm employs at 80 workers per day. What is the firm’s daily total cost if it rents just enough capital to produce at its target?b) Compare the marginal product per dollar sent on K and on L when the firm operates at the input choice in part (a). What does this suggest about the way the firm might change its choice of K and L if it wants to reduce the total cost in meeting its target?c) In the long run, how much K and L should the firm choose if it wants to minimize the cost of producing 8,000 units of output day? What will the total daily cost of production be?

a) Suppose that the firm is operating in the short run, with L = 80. To produce Q = 8,000, how much K will it require? From the production function we observe that 8,000 = K2 (80) => K = 10.The total cost would be C = wL + rK = $200(80) + $400(10) = $2,000 per day.

b) Let’s examine the “bang for the buck” for K and L when K = 10 and L = 80. For capital: MPK / r = 2KL / 400 = 2(10)(80) / 400 = 4For labor: MPL / w = K2 / 200 = 102 / 200 = 0.5So the marginal product per dollar spent on capital exceeds that of labor. The firm would like to rent more capital and hire fewer workers.



c) Because the production function is Cobb-Douglas, we know that it has diminishing MRTSL,K and that the isoquants do not intersect either the K or L axis. Thus the cost reducing basket (K,L) will be interior (with K > 0 and L > 0). To find the optimum, we use the two conditions:(1) Tangency condition: MPK / MPL = r / w => 2KL/K2 = 400 / 200 => K = L(2) Production Requirement: K2L = 8,000Together equations (1) and (2) tell us that K = 20 and L = 20.The total cost would be C = wL + rK = $200(20) + $400(20) = $12,000 per day.

7.13. Consider the production function Q = LK, with marginal products MPL = K and MPK = L. Suppose that the price of labor equals w and the price of capital equals r. Derive expressions for the input demand curves.

From the tangency condition, we get

Substituting into the production function yields

This represents the input demand curve for . Since

we have

This represents the input demand curve for .



7.14. A cost-minimizing firm’s production function is given by Q = LK, where MPL = K and MPK = L. The price of labor services is w and the price of capital services is r. Suppose you know that when w = $4 and r = $2, the firm’s total cost is $160. You are also told that when input prices change such that the wage rate is 8 times the rental rate, the firm adjusts its input combination but leaves total output unchanged. What would the cost-minimizing input combination be after the price changes?

Using the tangency condition, with the original input prices: . So, K = 2L. Also, using

the information on total costs, Combining these two equations, we get (L, K) = (20, 40). Therefore the firm produces 20*40 = 800 units of output.After the prices change, even though we don’t know the numerical values of the input prices, we can still answer the question using the fact that we’re told w = 8r. The tangency condition

implies that so K = 8L. Also, we have . This implies that the optimal input

combination is (L, K) = (10, 80).

7.15. Ajax, Inc. assembles gadgets. It can make each gadget either by hand or with a special gadget-making machine. Each gadget can be assembled in 15 minutes by a worker or in 5 minutes by the machine. The firm can also assemble some of the gadgets by hand and some with machines. Both types of work are perfect substitutes, and they are the only inputs necessary to produce the gadgets.a) It costs the firm $30 per hour to use the machine and $10 per hour to hire a worker. The firm wants to produce 120 gadgets. What are the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph.b) What are the cost-minimizing input quantities if it costs the firm $30 per hour to use the machine, and $10 per hour to hire a worker? Illustrate your answer with a graph. c) Write down the equation of the firm’s production function for the firm. Let G be the number of gadgets assembled, M the number of hour the machines are used, and L the number of hours of labor.

a) Isoquants for the production function are straight lines. At the given input prices slope of an isoquant is equal to the ratio of the input prices. Hence, all positive input quantities (measured in work hours) such that 4L + 12M = 120 are cost-minimizing.



b) When one hour of the machine’s work costs $20 cost-minimizing firm does not use manual work at all. The cost-minimizing quantity of the machine’s work necessary to produce 120 widgets is equal to M = 120/12 = 10 hours. The firm spends $200. (Note that if the firm were to use only manual labor, the cost would be $300 = 30 hours x $10 per hour).

c) G = 4L + 12M

7.16. A construction company has two types of employees: skilled and unskilled. A skilled employee can build 1 yard of a brick wall in one hour. An unskilled employee needs twice


L

M

10hours

30 hours

Cost-minimizing inputs when pm = $30, pw = $10.

L

M

10hours

30 hours

Cost-minimizing inputs when pm = $20, pw = $10.

Isoquant lineG = 120

Isocost lineTotal Cost = $200


as much time to build the same wall. The hourly wage of a skilled employee is $15. The hourly wage of an unskilled employee is $8.a) Write down a production function with labor. The inputs are the number of hours of skilled workers, LS, the number of hours worked by unskilled employees, LU, and the output is the number of yards of brick wall, Q.b) The firm needs to build 100 yards of a wall. Sketch the isoquant that shows all combinations of skilled and unskilled labor that result in building 100 yards of the wall.c) What is the cost-minimizing way to build 100 yards of a wall? Illustrate your answer on the graph in part (b).

a) The production function is Q = LS + ½ LU where LS denotes hours worked by skilled workers and LU denotes hours worked by unskilled workers. Both types of labor are perfect substitutes.

b) The isoquant is a straight line.

c) MPLs/ws = 1/15; MPLu/wu = 0.5/8 = 1/16. Thus, the “bang for the buck” is higher for skilled labor, and the firm will use only skilled labor. Note that the total cost of building 100 yards with skilled labor is (100 hours)($15/ hour) = $1500. The total cost of building 100 yards with unskilled labor is (200 hours)($8/ hour) = $1600.

The isocost line representing a $1500 expenditure is drawn as a dotted line in the graph in (b). The isocost line is more steeply sloped than the isoquant in the graph because the marginal rate of technical substitution of unskilled labor for unskilled labor is equal to ½, while the ratio of input prices is equal to 8/15.


LU

LS

100h

200h

Isoquant representing 100 yards of wall

Isocost line representing $1500 expenditure

175h


7.17. A paint manufacturing company has a production function Q = K + √L. For this production function MPK = 1 and MPL = 1/(2√L). The firm faces a price of labor w that equals $1 per unit and a price of capital services r that equals $50 per unit.a) Verify that the firm’s cost-minimizing input combination to produce Q = 10 involves no use of capital.b) What must the price of capital fall to in order for the firm to use a positive amount of capital, keeping Q at 10 and w at 1?c) What must Q increase to for the firm to use a positive amount of capital, keeping w at 1 and r at 50?

a) First, note that this production function has diminishing MRSL,K. The tangency condition would imply that or L = 625. Substituting this back into the production function we see that K = 10 – 25 = –15. Since the firm cannot use a negative amount of capital, the tangency condition is not valid in this case. Looking at the corner with K = 0, since Q = 10 the firm requires L = Q2 = 100 units of labor. At this point, MPL / w = (1/20)/1 = 0.05 > MPK / r = 1/50 = 0.02. Since the marginal product per dollar is higher for labor, the firm will use only labor and no capital.

b) The firm will use a positive amount of capital when , or Thus L =

0.25r2. From the production constraint K = = 10 – 0.5r. So if K > 0 then we must have 10 – 0.5r > 0, or r < 20.

c) Again, using the tangency condition we must have Therefore, since r = 50, L = 625. From the production constraint, the input demand for capital is K = = Q – 25. So if K > 0 then we must have Q > 25.

7.18. A researcher claims to have estimated input demand curves in an industry in which the production technology involves two inputs, capital and labor. The input demand curves he claims to have estimated are L = wr2Q and K = w2rQ. Are these valid input demand curves? In other words, could they have come from a firm that minimizes its costs?

No, these are not valid input demand curves. In both cases the quantity of the input is positively related to the input’s price. Such upward-sloping input demand curves cannot exist.

7.19. A manufacturing firm’s production function is Q = KL + K + L. For this production function, MPL = K + 1 and MPK = L + 1. Suppose that the price r of capital services is equal to 1, and let w denote the price of labor services. If the firm is required to produce 5 units of output, for what values of w would a cost-minimizing firm usea) only labor?b) only capital?c) both labor and capital?



If K = 0, then the firm must hire L = 5 units of labor. For this to be optimal, it must be that MPL / w > MPK / r, or 1/w > 6. In other words, w < 1/6.If L = 0, then the firm must hire K = 5 units of capital. For this to be optimal, it must be that MPL / w < MPK / r, or 6/w > 1. In other words, w > 6.For the firm to use both capital and labor, it must be that 1/6 < w < 6. To see why, notice that the indifference curves will have diminishing MRTSL,K. In particular, MRTSL,K = 6 where the Q = 5 indifference curve intersects the K-axis (where L = 0). Diminishing MRTSL,K implies that the Q = 5 indifference curve will gradually flatten out until it intersects the L-axis (where K = 0), at which point MRTSL,K = 1/6.

7.20. Suppose a production function is given by Q = min(L, K)—that is, the inputs are perfect complements. Draw a graph of the demand curve for labor when the firm wants to produce 10 units of output (Q = 10).

The input demand curves will be vertical lines, representing the fact that the demand by firms for such inputs is inelastic. If the firm’s production function is then, holding fixed the quantity of production and the price of capital, if the wage rate were to increase it would not change the firm’s requirement for labor. Therefore, the demand for each input is independent of price and the demand curves are vertical lines.

7.21. 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, and calculate the total cost.

The isoquant Q = 12 is shown for this Leontief technology. To produce Q = 12, the firm will need at least K = 12 and L = 6. This will cost the firm C = wL + rK = 4(6) + 1(12) = 36. The isocost line representing an expenditure of 36 is drawn. The optimal basket of inputs is A.



7.22. Suppose a production function is given by Q = K + L—that is, the inputs are perfect substitutes. For this production function, MPL = 1 and MPK = 1. Draw a graph of the demand curve for labor when the firm wants to produce 10 units of output and the price of capital services is $1 per unit (Q = 10 and r = 1).

Recall that with a linear production function we are usually going to get corner point solutions. In this case, the firm will employ only labor and no capital if labor is cheap enough or,

i.e. if Similarly it will use just capital if the rental rate is low enough i.e.

. If the firm uses only labor, it will use units regardless of the price, and similarly it

will use units of capital if it uses any capital at all. The input demand curve for labor for a

given price, r, of capital, is shown below.

7.23. Suppose a production function is given by Q = 10K + 2L. The factor price of labor is 1. Draw the demand curve for capital when the firm is required to produce Q = 80.

With this production function the firm views K and L as perfect substitutes.The firm will be at a corner point with K = 0 when MPK/r < MPL/w, or when 10/r <2/1, or when r > 5.The firm will be at a corner point with L = 0 when MPK/r > MPL/w, or when 10/r >2/1, or when r < 5. When the firm needs to produce Q = 80, how much capital will it need? The production function shows that 80 = 10K, or K = 8 units.When r = 5, the firm might use any combination of K and L along the isoquant 80 = 10K + 2L. The firm might therefore use any K such that 0 < K < 8.The graph of the demand for labor is as shown.


L

w

br/a

Q/a


7.24. Consider the production function Q = K + √L. For this production function, MPL = 1/(2√L) and MPK = 1. Derive the input demand curves for L and K, as a function of the input prices w (price of labor services) and r (price of capital services). Show that at an interior optimum (with K > 0 and L > 0) the amount of L demanded does not depend on Q. What does this imply about the expansion path?

The tangency condition implies that . Clearly the demand curve for L is

not a function of the level of output, Q. Therefore, as the level of output changes, the amount of labor is constant. Therefore, if we were to graph isoquants with labor on the horizontal axis, the expansion path for labor would just be a straight, vertical line.The demand curve for capital can be derived by substituting the demand curve for labor into the

production function. That is, .

7.25. A firm has the production function Q = LK. For this production function, MPL = K and MPK = L. The firm initially faces input prices w = $1 and r = $1 and is required to produce Q = 100 units. Later the price of labor w goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the firm’s price elasticity of demand for labor over this range of prices.

Using the tangency condition, initially , implying that K = L. Since KL = 100, we get K =

L = 10.Under the new prices, the tangency condition implies that K=4L. This means that the optimal input combination is (L, K) = (5, 20).The percent change in price is (4 – 1)*100 = 300%. While the percent change in the demand for labor is [(5 – 10)/10]*100 = –50%. Therefore the price elasticity of demand over this range of prices is –50/300 = –1/6.

7.26 A bicycle is assembled out of a bicycle frame and two wheels.a) Write down a production function of a firm that produces bicycles out of frames and wheels. No assembly is required by the firm, so labor is not an input in this case. Sketch the


8

K

r Demand for Capital(Heavy curve)

5


isoquant that shows all combinations of frames and wheels that result in producing 100 bicycles.b) Suppose that initially the price of a frame is $100 and the price of a wheel is $50. On the graph you drew for part (a), show the choices of frames and wheels that minimize the cost of producing 100 bicycles, and draw the isocost line through the optimal basket. Then repeat the exercise if the price of a frame rises to $200, while the price of a wheel remains $50.

a) The production function is Q = min(F, ½ W), where F denotes the number of frames and W denotes the number of wheels.

b) To produce 100 bicycles in the least costly manner, the firm always needs to choose basket A, with 200 wheels and 100 frames.Initially, when the price of a frame is $100 and the price of a wheel is $50, the isocost line is the lighter one shown in the graph; all points on the isocost line indicate an expenditure of $20,000.Later, when the price of a frame is $200 and the price of a wheel is $50, the isocost line is the lighter one shown in the graph; all points on the isocost line indicate an expenditure of $30,000.

7.27. Suppose that the firm’s production function is given by Q = 10KL1/3. The firm’s capital is fixed at K. What amount of labor will the firm hire to solve its short-run cost-minimization problem?

With just two inputs, there is no tangency condition to worry about in the short run. To find the short-run cost-minimizing quantity of labor, we need only solve the production function for in terms of and :

This gives us:


frames

200

100100 bicycle isoquant

400

200

600

Initial Isocost Line:$20,000

Final Isocost Line:$30,000

A

wheels


This is the cost-minimizing quantity of labor in the short run.

7.28. A plant’s production function is Q = 2KL + K . For this production function, MPK = 2L + 1 and MPL = 2K. The price of labor services w is $4 and of capital services r is $5 per unit.a) In the short run, the plant’s capital is fixed at K = 9. Find the amount of labor it must employ to produce Q = 45 units of output.b) How much money is the firm sacrificing by not having the ability to choose its level of capital optimally?

a) Since , we get which implies that L = 36/18 = 2. Therefore the firm’s total cost with this input combination is 4(2) + 5(9) = $53.

b) If the firm could operate optimally, it would choose labor and capital to satisfy the

tangency condition: , implying that Also, Combining

these two conditions, = 4.24 and L = 4.8. Now the firm’s expenditure would be 4(4.24) + 5(4.8) = $41 approximately. Therefore the firm loses about $12 because of its constraint on capital.

7.29. 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 Q = K1/3L1/3M1/3. For this production function, the marginal products of capital, labor, and materials are MPK = 1/3K−2/3L1/3M1/3, MPL = 1/3K1/3L−2/3M1/3, and MPM = 1/3K1/3L1/3M−2/3. 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) When Q = 4, the long-run cost-minimizing quantity of capital is 4. If capital is fixed at K = 4 in the short run, show that the short-run and long-run cost-minimizing quantities of labor and materials are the same.

a) Here we have two tangency conditions and the requirement that , , and produce units of output.



This is a system of three equations in three unknowns. The solution to this system gives us the long-run cost-minimizing input combination:

b) The tangency condition is

which implies

To find the short-run cost-minimizing quantity of labor, we plug this back into the production function and solve for in terms of and .

which when we solve for gives us the short-run cost-minimizing quantity of labor

Since , the short-run cost-minimizing quantity of materials is

c) Plugging into the expressions for the long-run cost-minimizing quantities of labor and materials gives us



Plugging and into the expressions for the short-run cost-minimizing quantities of labor and materials gives us

7.30. Consider the production function in Learning-By-Doing Exercise 7.6: Q = √L + √K + √M. For this production function, the marginal products of labor, capital, and materials are MPL = 1/(2√L), MPK = 1/(2√K), and MPM = 1/(2√M). Suppose that the input prices of labor, capital, and materials are w = 1, r = 1, and m = 1, respectively.a) Given that the firm wants to produce Q units of output, what is the solution to the firm’s long-run cost minimization problem?b) Given that the firm wants to produce Q units of output, what is the solution to the firm’s short-run cost minimization problem when K = 4? Will the firm want to use positive quantities of labor and materials for all levels of Q?(c) Given that the firm wants to produce 12 units of output, what is the solution to the firm’s short-run cost minimization problem when K = 4 and L = 9? Will the firm want to use a positive quantity of materials for all levels of Q?

a) With three inputs, we need two tangency conditions to ensure that the marginal product per dollar spent is equal across all inputs. (We could write down a third tangency condition, but it would be redundant.) Equating the “bang for the buck” between labor and capital implies

or L = K. Similarly, equating the “bang for the buck” between labor and materials implies or L = M. Then using the production constraint to find the input demand for labor yields or L = (1/3)Q2. Since L = M = K from the tangency conditions, we also have K = (1/3)Q2 and M = (1/3)Q2.

b) First, note that with K = 4, the firm can produce up to Q = = 2 units of output without hiring any labor or materials. To produce more than Q = 2, the firm still balances the marginal product per dollar spent on labor and materials; in part (a), we saw this implied L = M. Substituting this and K = 4 into the production constraint, we have Q = which yields L = (1/4)(Q – 2)2 as the input demand for labor. Then L = M implies that the input demand for materials is M = (1/4)(Q – 2)2. Therefore, the input demand functions are



c) Again, with K = 4 and L = 9, the firm can produce up to Q = 5 units of output without hiring any materials. Should it desire to produce greater levels of output, it can hire materials according to Q = , or M = (Q – 5)2. Therefore, the input demand for materials is

7.31. Acme, Inc. has just completed a study of its production process for gadgets. It uses labor and capital to produce gadgets. It has determined that 1 more unit of labor would increase output by 200 gadgets. However, an additional unit of capital would increase output by 150 gadgets. If the current price of capital is $10 and the current price of labor is $25, is the firm employing the optimal input bundle for its current output? Why or why not? If not, which input’s usage should be increased?

The information in the problem tells us that MPL = 200 and MPK = 150 while w = 25 and r = 10. So MPL/w = 8 < MPK/r = 15. Thus Acme could maintain its current level of output while reducing costs by employing more capital and less labor. So it is not employing the optimal input bundle.

7.32. A firm operates with a technology that is characterized by a diminishing marginal rate of technical substitution of labor for capital. It is currently producing 32 units of output using 4 units of capital and 5 units of labor. At that operating point the marginal product of labor is 4 and the marginal product of capital is 2. The rental price of a unit of capital is 2 when the wage rate is 1. Is the firm minimizing its total long-run cost of producing the 32 units of output? If so, how do you know? If not, show why not and indicate whether the firm should be using (i) more capital and less labor, or (ii) less capital and more labor to produce an output of 32.

We have MPL/w = 4/1 = 4 > MPK/r = 2/2 = 1. Thus the firm cannot be minimizing its long-run total cost. By employing more labor and less capital, it could maintain 32 units of output while lowering total costs.

7.33. Suppose that in a given production process a blueprint (B) can be produced using either an hour of computer time (C) or 4 hours of a manual draftsman’s time (D). (You may assume C and D are perfect substitutes. Thus, for example, the firm could also produce a blueprint using 0.5 hour of C and 2 hours of D.)a) Write down the production function corresponding to this process (i.e., express B as a function of C and D).b) Suppose the price of computer time (pc) is 10 and the wage rate for a manual draftsman (pD) is 5. The firm has to produce 15 blueprints. What are the cost minimizing choices of C and D? On a graph with C on the horizontal axis and D on the vertical axis, illustrate your answer showing the 15-blueprint isoquant and isocost lines.



a) Computers are four times as productive as draftsmen; an alternative way of saying this is that MPC = 4MPD. Since C and D are perfect substitutes, we know the production function has the form B = aC + bD, where a and b are positive constants. Thus we can write the production function as B = C + (1/4)D. Note that this is consistent with generating one blueprint (B = 1) from the following combinations of inputs: (C, D) = (1, 0), (C, D) = (0, 4), and (C, D) = (0.5, 2).

b) Notice that . That is, the marginal product per dollar spent on computer time is always higher than the marginal product per dollar spent on draftsman time. So the optimal input combination involves D = 0 and C = 15. The graph below illustrates the (dotted) isocost lines with slope = –pC / pD = –2, along with the (solid) B = 15 isoquant with slope = –MPC / MPD = –4.

7.34. This problem will enable you to apply a revealed preference argument to see if a firm is minimizing the total cost of production. The firm produces output with a technology characterized by a diminishing marginal rate of technical substitution of labor for capital. It is required to produce a specified amount of output, which does not change in this problem. When faced with input prices w1 and r1, the firm chooses the basket of inputs at point A on the graph below, and it incurs the total cost on the isocost line IC1. When the factor prices change to w2 and r2 the firm’s choice of inputs is at basket B, on isocost line IC2. Basket A lies on the intersection of the two isocost lines. Are these choices consistent with cost minimizing behavior?


A

IC1

B

IC2

L

K


Since the firm’s production remains unchanged, it must be producing the same level of output at both points A and B. That is, the isoquant through A also passes through B. Now, suppose that the firm is minimizing costs at point B. Then the isoquant through B is tangent to isocost line IC2 . Since we are told that the MRTS is diminishing, there is no way the isoquant passing through B can also pass through A. You can see this easily from the graph. And you can also reach this conclusion using the property that a line tangent to a curve does not intersect the curve at any point other than the point of tangency. Similarly, if the firm were minimizing costs at A, then isocost line IC1 would be tangent to the isoquant; but then it would be impossible for isocost line IC2 also to be tangent. Thus, it is not possible for both A and B to be cost minimizing input combinations.


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