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Microeconomics Theory And Applications With Calculus 3rd Edition Jeffrey M. Perloff - Solutions

What can you say about Laura’s economies of scope if her time is valued at $5 an hour and her production possibility frontier is PPF1 in Figure 7.10?6. Challenge
A firm’s average cost is AC = αqβ, where α 7 0.How can you interpret α? (Hint: Suppose that q = 1.) What sign must β have if this cost function reflects learning by doing? What happens to average cost as q increases? Draw the average cost curve as a function of output for particular values
A U-shaped long-run average cost curve is the envelope of U-shaped short-run average cost curves. On what part of the curve (downward sloping, flat, or upward sloping) does a short-run curve touch the long-run curve? (Hint: Your answer should depend on where the two curves touch on the long-run
California’s State Board of Equalization imposed a higher tax on “alcopops,” flavored beers containing more than 0.5% alcohol-based flavorings, such as vanilla extract (Guy L. Smith, “On Regulation of ‘Alcopops,’ ” San Francisco Chronicle, April 10, 2009). Such beers are taxed as
A water heater manufacturer produces q water heaters per day, q, using L workers and S square feet of sheet metal per day, using a constant elasticity of substitution production function, q = (L-2 + S-2/40)-0.5. The hourly wage rate is$20, and the price per square foot of sheet metal is 50¢.a.
In Solved Problem 7.6, Equation 7.26 gives the longrun cost function of a firm with a constant-returnsto-scale Cobb-Douglas production function. Show how, for a given output level, cost changes as the wage, w, increases. Explain why. M
A production function is homogeneous of degreeγ and involves three inputs, L, K, and M (materials). The corresponding factor prices are w, r, and e.Derive the long-run cost curve. M
Suppose that your firm’s production function has constant returns to scale. What is the long-run expansion path?
The Bouncing Ball Ping Pong Company sells table tennis sets, which include two paddles and one net. What is the firm’s long-run expansion path if it incurs no costs other than what it pays for paddles and nets, which it buys at market prices?How does its expansion path depend on the relative
For a Cobb-Douglas production function, how does the expansion path change if the wage increases while the rental rate of capital stays the same?(Hint: See Solved Problem 7.5.) M
Derive the long-run cost function for the constant elasticity of substitution production function q = (Lρ + Kρ)1/ρ. (Hint: See Solved Problem 7.4.) M
Replace the production function in Solved Problem 7.4 with a Cobb-Douglas q = ALa Kb, and use calculus to find the cost minimizing capital-labor ratio. M
A firm has a Cobb-Douglas production function, Q = ALa Kb, where a + b 6 1. What properties does its cost function have? (Hint: Compare this cost function to that of the Japanese beer manufacturer.) M
The all-American baseball is made using cork from Portugal, rubber from Malaysia, yarn from Australia, and leather from France, and it is stitched(108 stitches exactly) by workers in Costa Rica. To assemble a baseball takes one unit of each of these inputs. Ultimately, the finished product must be
Suppose that the government subsidizes the cost of workers by paying for 25% of the wage (the rate offered by the U.S. government in the late 1970s under the New Jobs Tax Credit program). What effect does this subsidy have on the firm’s choice of labor and capital to produce a given level of
You have 60 minutes to complete an exam with two questions. You want to maximize your score.Toward the end of the exam, the more time you spend on either question, the fewer extra points per minute you get for that question. How should you allocate your time between the two questions?(Hint: Think
In 2003, Circuit City Stores, Inc., replaced skilled sales representatives who earned up to $54,000 per year with relatively unskilled workers who earned $14 to $18 per hour (Carlos Tejada and Gary McWilliams, “New Recipe for Cost Savings:Replace Highly Paid Workers,” Wall Street Journal, June
A bottling company uses two inputs to produce bottles of the soft drink Sludge: bottling machines, K, and workers, L. The isoquants have the usual smooth shape. The machine costs $1,000 per day to run, and the workers earn $200 per day. At the current level of production, the marginal product of
What is the long-run cost function if the production function is q = L + K? M
What is the effect of a lump-sum franchise tax l on the quantity at which a firm’s after-tax average cost curve reaches its minimum, given that the firm’s before-tax average cost curve is U-shaped?3. Long-Run Costs
The estimated short-run cost function of a Japanese beer manufacturer is C(q) = 0.55q1.67 + 800/q (see the application Short-Run Cost Curves for a Japanese Beer Manufacturer). At what positive quantity does the average cost function reach its minimum?If a $400 lump-sum tax is applied to the firm,
A firm has two plants that produce identical output. The cost functions are C1 = 10q - 4q2 + q3 and C2 = 10q - 2q2 + q3.a. At what output level does the average cost curve of each plant reach its minimum?b. If the firm wants to produce four units of output, how much should it produce in each plant?
A glass manufacturer’s production function is q = 10L0.5K0.5 (Hsieh, 1995). Suppose that its wage, w, is $1 per hour and the rental cost of capital, r, is $4.a. Draw an accurate figure showing how the glass firm minimizes its cost of production.b. What is the equation of the (long-run) expansion
A U.S. chemical firm has a production function of q = 10L0.32K0.56 (Hsieh, 1995). It faces factor prices of w = 10 and r = 20. What are its shortrun marginal and average variable cost curves? M
In 1796, Gottfried Christoph Härtel, a German music publisher, calculated the cost of printing music using an engraved plate technology and used these estimated cost functions to make production decisions. Härtel figured that the fixed cost of printing a musical page—the cost of engraving the
Gail works in a flower shop, where she produces 10 floral arrangements per hour. She is paid $10 an hour for the first eight hours she works and $15 an hour for each additional hour. What is the firm’s cost function? What are its AC, AVC, and MC functions? Draw the AC, AVC, and MC curves. M
A firm builds wooden shipping crates. How does the cost of producing a 1-cubic-foot crate (each side is 1 foot square) compare to the cost of building an 8-cubic-foot crate if wood costs $1 per square foot and the firm has no labor or other costs? More generally, how does cost vary with volume?
The only variable input a janitorial service firm uses to clean offices is workers who are paid a wage, w, of $8 an hour. Each worker can clean four offices in an hour. Use math to determine the variable cost, the average variable cost, and the marginal cost of cleaning one more office. Draw a
A firm’s cost curve is C = F + 10q - bq2 + q3, where b 7 0.a. For what values of b are cost, average cost, and average variable cost positive? (From now on, assume that all these measures of cost are positive at every output level.)b. What is the shape of the AC curve? At what output level is the
Give the formulas for and plot AFC, MC, AVC, and AC if the cost function isa. C = 10 + 10q,b. C = 10 + q2,c. C = 10 + 10q - 4q2 + q3. (Hint: See Solved Problem 7.2.) M
A firm’s short-run cost function is C(q) =200q - 6q2 + 0.3q3 + 400. Determine the fixed cost, F; the variable cost function, AVC; the average cost, AC; the marginal cost, MC; and the average fixed-cost, AFC. (Hint: See Solved Problem 7.2.) M
“There are certain fixed costs when you own a plane,” former tennis star Andre Agassi explained,“so the more you fly it, the more economic sense it makes. . . . The first flight after I bought it, I took some friends to Palm Springs for lunch.” (Ostler, Scott, “Andre Even Flies like a
Many corporations allow CEOs to use their firm’s corporate jet for personal travel. The Internal Revenue Service (IRS) requires that the firm report personal use of its corporate jet as taxable executive income, and the Securities and Exchange Commission (SEC) requires that publicly traded
You have a ticket to go to a concert by one of your favorite groups, the Hives, which you cannot resell.However, you can buy a ticket for $30 to attend a talk by Steven Colbert, at the same time as the concert. You are willing to pay up to $90 to hear Colbert. Given that there are no other costs
For the CES production function q = (aLρ +[1 - a]Kρ)d/ρ, does 0APL/0L have an unambiguous sign? M
During recessions, American firms lay off a larger proportion of their workers than Japanese firms do.(It has been claimed that Japanese firms continue to produce at high levels and store the output or sell it at relatively low prices during recessions.) Assuming that the production function
If a firm lays off workers during a recession, how will the firm’s marginal product of labor change?
Firm 1 and Firm 2 use the same type of production function, but Firm 1 is only 90% as productive as Firm 2. That is, the production function of Firm 2 is q2 = f(L, K), and the production function of Firm 1 is q1 = 0.9f(L, K). At a particular level of inputs, how does the marginal product of labor
Does it follow that, because we observe that the average product of labor is higher for Firm 1 than for Firm 2, Firm 1 is more productive in the sense that it can produce more output from a given amount of inputs? Why or why not?
In a manufacturing plant, workers use a specialized machine to produce belts. A new labor-saving machine is invented. With the new machine, the firm can use fewer workers and still produce the same number of belts as it did using the old machine.In the long run, both labor and capital (the machine)
Until the mid-eighteenth century, when spinning became mechanized, cotton was an expensive and relatively unimportant textile (Virginia Postrel, “What Separates Rich Nations from Poor Nations?” New York Times, January 1, 2004).Where it used to take an Indian hand-spinner 50,000 hours to
Prove Euler’s theorem that, if f(L, K) is homogeneous of degree γ (see Exercise 5.7), then L(0f/0L) + K(0f/0K) = γf(L, K). Given this result, what can you conclude if a production function has constant returns to scale? Express your results in terms of the marginal products of labor and
Show that with a constant returns to scale production function, the MRTS between labor and capital depends only on the K/L ratio and not on the scale of production. (Hint: Use your result from Exercise 5.7.) M
A production function is said to be homogeneous of degree γ if f(xL, xK) = xγf(L, K), where x is a positive constant. That is, the production function has the same returns to scale for every combination of inputs. For such a production function, show that the marginal product of labor and
Is it possible that a firm’s production function exhibits increasing returns to scale while exhibiting diminishing marginal productivity of each of its inputs? To answer this question, calculate the marginal productivities of capital and labor for the production of electronics and equipment,
As asserted in the comment to Solved Problem 6.5, show that γ is a scale elasticity. M
The production function for the automotive and parts industry is q = L0.27K0.16M0.61, where M is energy and materials (based loosely on Klein, 2003). What kind of returns to scale does this production function exhibit? What is the marginal product of energy and materials? (See Solved Problem 6.5) M
Under what conditions do the following production functions exhibit decreasing, constant, or increasing returns to scale?a. q = L + K, a linear production function,b. q = ALa Kb, a general Cobb-Douglas production function,c. q = L + La Kb + K,d. q = (aLρ + [1 - a]Kρ)d/ρ, a CES production
Show in a diagram that a production function can have diminishing marginal returns to a factor and constant returns to scale.
To speed relief to isolated South Asian communities that were devastated by the December 2004 tsunami, the U.S. Navy doubled the number of helicopters from 45 to 90 soon after the first ship arrived. Navy Admiral Thomas Fargo, head of the U.S. Pacific Command, was asked if doubling the number of
What is the elasticity of substitution, σ, of the CES production function q = (aLρ + bKρ)d/ρ? (See Solved Problem 6.4.) M 5. Returns to Scale
What is the MRTS of the CES production function q = (aLρ + bKρ)d/ρ? (See Solved Problem 6.3.) M
Show that the CES production function q =(aLρ + bKρ)1/ρ can be written as q = B(ρ)[cLρ +(1 -c) * Kρ]1/ρ. M
By studying, Will can produce a higher grade, GW, on an upcoming economics exam. His production function depends on the number of hours he studies marginal analysis problems, A, and the number of hours he studies supply and demand problems, R. Specifically, GW = 2.5A0.36R0.64. His roommate
Michelle’s business produces ceramic cups using labor, clay, and a kiln. She can manufacture 25 cups a day with one worker and 35 cups with two workers. Does her production process illustrate diminishing returns to scale or diminishing marginal returns to scale? Give a plausible explanation for
Draw a circle in a diagram with labor services on one axis and capital services on the other. This circle represents all the combinations of labor and capital that produce 100 units of output. Now, draw the isoquant for 100 units of output. (Hint: Remember that the isoquant includes only the
Alfred’s Print Shop can use any one of three fixedproportion technologies. Each involves one printer and one worker. Describe the possible shapes of the firm’s isoquant. (Hint: Review the discussion in the application “A Semiconductor Integrated Circuit Isoquant.”)
Mark launders his white clothes using the production function q = B + 0.5G, where B is the number of cups of Clorox bleach and G is the number of cups of generic bleach that is half as potent. Draw an isoquant. What are the marginal products of B and G? What is the marginal rate of technical
At L = 4 and K = 4, the marginal product of labor is 2 and the marginal product of capital is 3. What is the marginal rate of technical substitution? M
Why might we expect the law of diminishing marginal product to hold?
The production function at Ginko’s Copy Shop is q = 1,000 * min(L, 3K), where q is the number of copies per hour, q, L is the number of workers, and K is the number of copy machines. As an example, if L = 4 and K = 1, then min(L, 3K) = 3, and q = 3,000.a. Draw the isoquants for this production
What is the production function if L and K are perfect substitutes and each unit of q requires 1 unit of L or 1 unit of K (or a combination of these inputs that equals 1)? M
To produce a recorded CD, q = 1, a firm uses one blank disc, D = 1, and the services of a recording machine, M = 1, for one hour. Draw an isoquant for this production process. Explain the reason for its shape.
Suppose that a firm has a fixed-proportions production function in which 1 unit of output is produced using one worker and 2 units of capital. If the firm has an extra worker and no more capital, it still can produce only 1 unit of output. Similarly, 1 more unit of capital produces no extra
What are the differences between an isoquant and an indifference curve?
If the Cobb-Douglas production function is q = L0.75K0.25, and K = 16, what is the elasticity of output with respect to labor? (See Solved Problem 6.2.) M 4. Long-Run Production: Two Variable Inputs
Suppose that the Cobb-Douglas production function is q = L0.75K0.25.a. What is the average product of labor, holding capital fixed?b. What is the marginal product of labor?c. What are the APL and MPL when K = 16? (See Solved Problem 6.1.) M
In the short run, a firm cannot vary its capital, K = 2, but it can vary its labor, L. It produces output q. Explain why the firm will or will not experience diminishing marginal returns to labor in the short run if its production function is q = 10L + K.(See Solved Problem 6.1.) M
Each extra worker produces an extra unit of output, up to six workers. After six, no additional output is produced. Draw the total product of labor, average product of labor, and marginal product of labor curves.
If each extra worker produces an extra unit of output, how do the total product of labor, the average product of labor, and the marginal product of labor vary with the number of workers?
With respect to production functions, how long is the short run?3. Short-Run Production: One Variable and One Fixed Input
What types of organization allow owners of a firm to obtain the advantages of limited liability?2. Production
What types of firms would not normally try to maximize profit?
Remy views ice cream and fudge sauce as perfect complements. Is it possible that either of these goods or both of them are Giffin goods? (Hint: See Solved Problem 4.4.)
Philip’s quasilinear utility function is U = 4q1 0.5+ q2. His budget for these goods is Y = 10. Originally, the prices are p1 = p2 = 1.However, the price of the first good rises to p1 = 2.Discuss the substitution, income, and total effect on the demand for q1. M
Are relatively more high-quality navel oranges sold in California or in New York? Why? (Hint: See Solved Problem 4.3.)
Pat eats eggs and toast for breakfast and insists on having three pieces of toast for every two eggs he eats. Derive his utility function. If the price of eggs increases but we compensate Pat to make him just as “happy” as he was before the price change, what happens to his consumption of eggs?
Sally’s utility function is U(q1, q2) = 4q1 0.5+ q2.Derive her Engel curve. M 3. Effects of a Price Increase
Ryan has a constant elasticity of substitution utility function U = q 1ρ + q 2ρ . Derive his Engel curve. M
Derive the income elasticity of demand for individuals with (a) Cobb-Douglas, (b) perfect substitutes, and (c) perfect complements utility functions. M
Given the estimated Cobb-Douglas utility function in Exercise 1.7, U = q1 0.6q2 0.4, for CDs, q1, and DVDs, q2, derive a typical consumer’s Engel curve for movie DVDs. Illustrate in a figure. M
According to the U.S. Consumer Expenditure Survey for 2008, Americans with incomes below$20,000 spend about 39% of their income on housing. What are the limits on their income elasticities of housing if all other goods are collectively normal? Given that they spend about 0.2% on books and other
Guerdon always puts half a sliced banana, q1, on his bowl of cereal, q2—the two goods are perfect complements. What is his utility function? Derive his demand curve for bananas graphically and mathematically. (Hint: See Solved Problem 4.1.) M
Have your folks given you cash or promised to leave you money after they’re gone? If so, they may think of such gifts as a good. They decide whether to spend their money on fun, food, drink, cars, or give money to you. Hmmm. Altonji and Villanueva (2007) estimated that, for every extra dollar of
In 2005, a typical U.S. owner of a home theater(a television and a DVD player) bought 12 music CDs (q1) per year and 6 Top-20 movie DVDs (q2)per year. The average price of a CD was about p1 = $15, the average price of a DVD was roughly p2 = $20, and the typical consumer spent $300 on these
Draw a figure to illustrate the Application “Quitting Smoking.” That is, show why as the price of cell phones drops, less tobacco is consumed. (Hint:Draw a figure like panel a of Figure 4.2 with cell phones on the horizontal axis and tobacco on the vertical axis. However, unlike in Figure 4.2,
If Philip’s utility function is U = 2q1 0.5+ q2, what are his demand functions for the two goods? M
David consumes two things: gasoline (G) and bread(B).David’s utility function is U(q1, q2) = 10q1 0.25q2 0.75.a. Derive David’s demand curve for gasoline.b. If the price of gasoline rises, how much does David reduce his consumption of gasoline, 0q1/0p1?c. For David, how does 0q1/0p1 depend on
Derive Ryan’s demand curve for q1, given his CES utility function is U = q1ρ + q2ρ. M
How would your answer to Exercise 1.1 change if U = ln(q1 + q2) so that consumers have diminishing marginal utility of diamonds? M
Manufactured diamonds have become as big and virtually indistinguishable from the best natural diamonds (Dan Mitchell, “Fake Gems, Genuine Appeal,” New York Times, June 21, 2008). Suppose consumers change from believing that manufactured diamonds, q1, were imperfect substitutes for natural
Salvo and Huse (2012) found that roughly onefifth of flexible-fuel (cars that can run on a mix of ethanol and gasoline) car owners choose gasoline when the price of gas is 20% above that of ethanol(in energy-adjusted terms) and, similarly, one-fifth of motorists choose ethanol when ethanol is
Einav et al. (2012) found that people who live in high sales tax locations are much more likely than other consumers to purchase goods over the Internet because Internet purchases are generally exempt from the sales tax if the firm is located in another state. They found that a 1% increase in a
In previous years, gasoline was less expensive in the United States than in Canada, but now, due to a change in taxes, gasoline costs less in Canada than in the United States. How will the gasoline-purchasing behavior of a Canadian who lives near the border and can easily buy gasoline in either
Illustrate the logic of the endowment effect using a kinked indifference curve. Let the angle be greater than 90°. Suppose that the prices change, so the slope of the budget line through the endowment changes.a. Use the diagram to explain why an individual whose endowment point is at the kink will
Given that Kip’s utility function is U(qc, qm) = qc 0.5+ qm 0.5, what is his expenditure function? (Hint: See Solved Problem 3.8.) M 5. Behavioral Economics
Wolf’s utility function is U = aq1 0.5+ q2. For given prices and income, show how whether he has an interior or corner solution depends ona. M
Ann’s utility function is U = q1q2/(q1 + q2). Solve for her optimal values of q1 and q2 as a function of p1, p2, and Y. M
Vasco likes spare ribs, q1, and fried chicken, q2. His utility function is U = 10q1 2q2. His weekly income is $90, which he spends on ribs and chicken only.a. If he pays $10 for a slab of ribs and $5 for a chicken, what is his optimal consumption bundle? Show his budget line, indifference curve,
David’s utility function is U = q1 + 2q2. Describe his optimal bundle in terms of the prices of q1 and q2. M

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