chapter8 new

Presented by Suong Jian & Liu Yan, MGMT Panel , Guangdong University of Finance.

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Chapter 8 PRODUCTION ANALYSIS AND COMPENSATION POLICY QUESTIONS & ANSWERS Q8.1 Is use of least-cost input combinations a necessary condition for profit maximization?

Is it a sufficient condition? Explain. Q8.1 ANSWER

Employment of least-cost input combinations is a necessary but not sufficient condition for profit maximization. It is necessary because a failure to operate with a least-cost input combination means that costs could be lowered and profits increased at any given output level. It is not a sufficient condition because the cost-minimizing level does not incorporate any information concerning demand relations, and therefore provides no information about the optimal level at which to operate: that is, information concerning demand relations must be added to the analysis to determine how much to produce for profit maximization (an optimal level of output).

In short, employment of a least-cost input combination will result in an optimal production of a target level of output. Conversely, employment of inputs such that MRPi = Pi for each input will result in an optimal production of an optimal level of output.

Q8.2 AOutput per worker is expected to increase by 10% during the next year. Therefore,

wages can also increase by 10% with no harmful effects on employment, output prices, or employer profits.@ Discuss this statement.

Q8.2 ANSWER

This statement is correct so long as the projected increase in output per worker is solely due to an improvement in labor productivity and provided that the demand for output is also expected to rise. Gains in labor productivity are sometimes derived from an improvement in worker skill due to education or experience, elimination of obsolete work rules, labor-saving technical change, and so on. When increases in output per worker can be directly attributed to such gains in labor productivity, a commensurate increase in wages can be justified with no resulting increase in output prices or decrease in employer profits. So long as output demand is growing as fast as the gain in labor productivity, no reduction in employment opportunities will result. However, should output demand be stagnant, an increase in labor


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productivity could reduce employment opportunities by reducing the number of workers required to produce a given level of output.

It is important to recognize that increases in output per worker are sometimes made possible by increased capital investment per worker, improvements in supplier efficiency, and so on. In such instances, increases in output per worker are not directly attributable to gains in labor productivity, and do not imply that a higher wage rate could be justified.

Q8.3 Commission-based and piece-rate-based compensation plans are commonly

employed by businesses. Use the concepts developed in the chapter to explain these phenomena.

Q8.3 ANSWER

Commission-based and piece rate-based compensation plans ensure that the relevant labor cost per unit of output is the same for all units produced (or sold). More productive employees earn greater total compensation, although less productive employees earn the same compensation per unit of output. Using output-oriented labor compensation schemes, employers ensure that the MPL/PL ratio is equal for all employees and that optimal labor input proportions are employed.

Q8.4 AHourly wage rates are an anachronism. Efficiency requires incentive-based pay

tied to performance.@ Discuss this statement. Q8.4 ANSWER

Given that many successful firms use hourly wage rates, it seems rash to dismiss them as an inefficient method for employee compensation. When hourly wages are paid, employees are expected to provide a standard level of effort per hour. Because reprimand or dismissal for substandard performance, or Agoldbricking,@ is always possible, even hourly employees have strong incentives to provide a satisfactory level of performance. In addition, Abeing there@ is often an important component of an employee's service to customers. Thus, hourly input is often synonymous with the hourly output of service.

Q8.5 Explain why the MP/P relation is deficient as the sole mechanism for determining

the optimal level of resource employment. Q8.5 ANSWER

The equality of the MP/P ratio across input factors in a production system is necessary to insure a least-cost input combination for production of any level of output. Satisfying this requirement does not, however, insure that the optimal

Production Analysis and Compensation Policy


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activity level has been selected because it does not reflect the demand for output. An equality of MP/P ratios can be found at activity levels both above and below that activity (output) level at which profits are maximized.

Q8.6 Develop the appropriate relations for determining the optimal quantities of all inputs

to employ in a production system, and explain the underlying rationale. Q8.6 ANSWER

The necessary and sufficient condition for optimal resource employment is that each input must be used at a level such that its price equals its marginal revenue product (PX = MRPX). This relation is derived from the simple notion that marginal costs and marginal revenues for each input must be equal for profit maximization. Symbolically:

Q8.7 Suppose that labor, capital, and energy inputs must be combined in fixed proportions. Does this mean that returns to scale will be constant?

Q8.7 ANSWER

No, the fact that labor, capital, and energy inputs must be combined in fixed proportions does not imply that returns to scale will be constant. In such a situation, returns to scale could just as easily be increasing or diminishing. To judge the nature of returns to scale, we must consider the relation between the increase in output caused by a proportionate increase in all inputs. If output increases faster (slower) than all inputs, returns to scale are increasing (decreasing). Returns to scale are constant when a given increase in all inputs leads to a proportionate increase in output.

X

Q X

X

Marginal cost of Marginal revenue of =

last input unit last input unitTC TR = X X

TR Q = x P Q X = x MR MP

= MRP

∂ ∂∂ ∂∂ ∂∂ ∂

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Q8.8 What is meant by the Apace@ of economic productivity growth, and why is it important to economic welfare?

Q8.8 ANSWER

The pace of productivity growth is the rate of increase in output per unit of input. For example, if the amount of output produced in the economy were to grow by 5% following only a 2% increase in the quantity of inputs employed, then the overall rate of productivity growth would be roughly 3%. When productivity growth is robust in the overall economy, economic welfare per capita rises quickly. When productivity growth is sluggish, economic welfare improves slowly. If productivity growth is robust for individual companies, or within specific industry groups, superior efficiency is suggested and exceptional profitability often ensues. Thus, the rate of productivity growth is important both for managers and investors in individual companies, and for decision makers in the public sector.

Q8.9 Cite some potential ways for increasing productivity growth in the United States. Q8.9 ANSWER

During the late-1990s, annual rises in productivity in nonfarm businesses have averaged 3.1%, a big jump from the 1.4% annual rate common during the early-1990s. This burst in productivity growth is similar in timing and magnitude in many advanced industrial economies. As a result, it cannot be explained by purely domestic factors.

Many economists point to a business cycle effect as partial explanation for the recent burst in productivity growth. During economic booms, such as that experienced during the late-1990s, productivity jumps. However, economists argue such business cycle effects contribute only 0.04 percent to the recent boost in productivity.

Faster growth of inputs, both physical and human capital, is a major cause of faster productivity growth. In the U.S., the capital-labor ratio has grown faster since the early-1990s, but not enough to account for all of the increase in productivity growth. The percentage point contribution from greater capital services (capital deepening) may be as much as 0.38 percent. The rate of increase of human capital, as measured by the average education level and experience of workers, has accelerated since the 1950s and 1960s. Human capital growth is now responsible for roughly one-quarter of total productivity growth during the past 30 years, but relatively flat over the past decade. Thus, although policies to increase investment,



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education, and training, are important, they do not explain the underlying causes of the recent boost in productivity.

From an accounting perspective, a significant share of the recent boost in productivity growth is attributable to the more effective use of worker skills made possible through recent improvements in communications technology. Increasingly, companies have been eager to buy powerful computers and computer software at relatively low prices. Rapid advances in computer hardware and software technology, combined with the widespread adoption of the Internet, have led to an unprecedented boom in communications technology. Benefits from the recent boom in communications technology are evident in every home and workplace, and are broadly reflected in the late-1990s burst in productivity growth.

Q8.10 Explain why company productivity is important to managers, employees, and

investors. Is superior worker productivity a necessary and sufficient condition for above-average compensation?

Q8.10 ANSWER

For managers and other employees, profits and revenues per employee give helpful insight concerning the income potential from employment. When profits and revenues per employee are high, the potential for high wages and growing incomes for exceptional employees can be significant. On the other hand, companies in industries that seldom generate an attractive rate of return rarely have the wherewithal to pay attractive and growing salaries. For example, Microsoft is well known for generous compensation policies that have allowed many Microsoft employees to earn stock-based rewards in excess of one million dollars each. At the same time, total compensation tends to be low for employees of regulated utilities and in other low-profit industries. Similarly, for investors, profits and revenues per employee give valuable insight on the investment potential of various companies and industries. For example, financial services like stock brokerage and specialized insurance are marvelous businesses that hold out the potential for exceptional rates of return for investors. Conversely, garbage collection (environmental and waste) is a tough business where it is extraordinarily difficult to make above-average rates of return.

Finally, it is worth remembering that superior worker productivity is only a necessary and not sufficient condition for above-average compensation. When profits and revenues per employee are high, the potential for high wages and growing incomes for exceptional employees can be significant. However, wages and employee compensation reflect the relative productivity of the marginal employee.

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Only superior employees with hard-to-duplicate contributions can expect to earn above-average compensation.

SELF-TEST PROBLEMS & SOLUTIONS ST8.1 Optimal Input Usage. Medical Testing Labs, Inc., provides routine testing services

for blood banks in the Los Angeles area. Tests are supervised by skilled technicians using equipment produced by two leading competitors in the medical equipment industry. Records for the current year show an average of 27 tests per hour being performed on the Testlogic-1 and 48 tests per hour on a new machine, the Accutest-3. The Testlogic-1 is leased for $18,000 per month, and the Accutest-3 is leased at $32,000 per month. On average, each machine is operated 25 eight-hour days per month.

A. Describe the logic of the rule used to determine an optimal mix of input usage.

B. Does Medical Testing Lab usage reflect an optimal mix of testing equipment?

C. Describe the logic of the rule used to determine an optimal level of input usage.

D. If tests are conducted at a price of $6 each while labor and all other costs are

fixed, should the company lease more machines? ST8.1 SOLUTION A. The rule for an optimal combination of Testlogic-1 (T) and Accutest-3 (A)

equipment is

This rule means that an identical amount of additional output would be produced with an additional dollar expenditure on each input. Alternatively, an equal marginal cost of output is incurred irrespective of which input is used to expand output. Of course, marginal products and equipment prices must both reflect the same relevant time frame, either hours or months.

B. On a per hour basis, the relevant question is

T A

T A

MP MP = P P



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27$18,000 /(25 x 8)

= ? 48$32,000 /(25 x 8)

0.3 = _ 0.3

On a per month basis, the relevant question is

27 x (25 x 8)$18,000

= ? 48 x (25 x 8)$32,000

0.3 = _ 0.3

In both instances, the last dollar spent on each machine increased output by the same 0.3 units, indicating an optimal mix of testing machines.

C. The rule for optimal input employment is MRP = MP Η MRQ = Input Price

This means that the level of input employment is optimal when the marginal sales revenue derived from added input usage is equal to input price, or the marginal cost of employment.

D. For each machine hour, the relevant question is Testlogic-1 MRPT = MPT Η MRQ = ? PT 27 Η $6 = ? $18,000/(25 Η 8) $162 > $90. Accutest-3 MRPA = MPA Η MRQ = ? PA 48 Η $6 = ? $32,000/(25 Η 8)

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$288 > $160.

Or, in per month terms: Testlogic-1 MRPT = MPT Η MRQ = ? PT 27 Η (25 Η 8) Η $6 = ? $18,000 $32,400 > $18,000. Accutest-3 MRPA = MPA Η MRQ = ? PA 48 Η (25 Η 8) Η $6 = ? $32,000 $57,600 > $32,000.

In both cases, each machine returns more than its marginal cost (price) of employment, and expansion would be profitable.

ST8.2 Production Function Estimation. Washington-Pacific, Inc., manufactures and sells

lumber, plywood, veneer, particle board, medium-density fiberboard, and laminated beams. The company has estimated the following multiplicative production function for basic lumber products in the Pacific Northwest market using monthly production data over the past two and one-half years (30 observations):

where

Q = output

L = labor input in worker hours

K = capital input in machine hours

E = energy input in BTUs

1 2 3b b b0Q = b L K E



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Each of the parameters of this model was estimated by regression analysis using monthly data over a recent three-year period. Coefficient estimation results were as follows:

ˆ0b = 0.9; ˆ1b = 0.4; ˆ2b = 0.4; and ˆ3b = 0.2

The standard error estimates for each coefficient are:

ˆ0bσ = 0.6; ˆ1bσ = 0.1; ˆ2bσ = 0.2; ˆ3bσ = 0.1

A. Estimate the effect on output of a 1% decline in worker hours (holding K and E

constant).

B. Estimate the effect on output of a 5% reduction in machine hours availability accompanied by a 5% decline in energy input (holding L constant).

C. Estimate the returns to scale for this production system.

ST8.2 SOLUTION A. For Cobb-Douglas production functions, calculations of the elasticity of output with

respect to individual inputs can be made by simply referring to the exponents of the production relation. Here a 1% decline in L, holding all else equal, will lead to a 0.4% decline in output. Notice that:

And because (ΜQ/Q)/(ΜL/L) is the percent change in Q due to a 1% change in L,

Q/QL/L

∂∂

= b1

1 2 3

1 2 3

1 2 3

- 1b b b0 1

- 1 + 1b b b0 1

b b b0

1

Q/Q Q L = x L/L L Q

( ) x Lb b L K E = Q

b b L K E = b L K E

= b

∂ ∂∂ ∂

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ΜQ/Q = b1 Η ΜL/L = 0.4(-0.01) = -0.004 or -0.4% B. From part A it is obvious that: ΜQ/Q = b2(ΜK/K) + b3(ΜE/E) = 0.4(-0.05) + 0.2(-0.05) = -0.03 or -3% C. In the case of Cobb-Douglas production functions, returns to scale are determined by

simply summing exponents because:

Here b1 + b2 + b3 = 0.4 + 0.4 + 0.2 = 1 indicating constant returns to scale. This means that a 1% increase in all inputs will lead to a 1% increase in output, and average costs will remain constant as output increases.

PROBLEMS & SOLUTIONS P8.1 Marginal Rate of Technical Substitution. The following production table provides

estimates of the maximum amounts of output possible with different combinations of two input factors, X and Y. (Assume that these are just illustrative points on a spectrum of continuous input combinations.)

Units of Y Used

Estimated Output per Day 5

210

305

360

421

470

4 188

272

324

376

421

3 162

234

282

324

360

1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

b b b0

b b b0

+ + b b b b b b0

+ + b b b

Q = b L K EhQ = (kL (kK (kE) ) )b

= bk L K E = Qk



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2 130 188 234 272 305 1

94

130

162

188

210

1

2

3

4

5

Units of X used

A. Do the two inputs exhibit the characteristics of constant, increasing, or decreasing marginal rates of technical substitution? How do you know?

B. Assuming that output sells for $3 per unit, complete the following tables:

X Fixed at 2 Units

Units of Y Used

Total Product

of Y

Marginal Product

of Y

Average Product

of Y

Marginal Revenue Product

of Y 1

2

3

4

5

Y Fixed at 3 Units

Units of X Used

Total Product

of X

Marginal Product

of X

Average Product

of X

Marginal Revenue Product

of X 1

2

3

4

5

C. Assume that the quantity of X is fixed at 2 units. If output sells for $3 and the

cost of Y is $120 per day, how many units of Y will be employed?

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D. Assume that the company is currently producing 162 units of output per day

using 1 unit of X and 3 units of Y. The daily cost per unit of X is $120 and that of Y is also $120. Would you recommend a change in the present input combination? Why or why not?

E. What is the nature of the returns to scale for this production system if the

optimal input combination requires that X = Y? P8.1 SOLUTION A. The inputs exhibit the characteristic of a decreasing marginal rate of technical

substitution throughout. For decreasing MRTS, the slope of the production isoquants diminishes as one input is increasingly substituted for another. We can also see this point algebraically by holding X or Y constant in the input-output matrix and noting the decline in the relative marginal product of the other input as its usage level grows.

B.

X Fixed at 2 Units Units of

Y Employed

TPY (1)

MPY (2)

APY (3)

MRPY

(4) = $3 Η (2) 1

130

130

130

$390

2

188

58

94

174 3

234

46

78

138

4

272

38

68

114 5

305

33

61

99

Y Fixed at 3 Units Units of

X Employed

TPX (1)

MPX (2)

APX (3)

MRPX

(4) = $3 Η (2) 1

162

162

162

$486

2 234

72

117

216

3

282

48

94

144 4

324

42

81

126

5

360

36

72

108



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C. Y = 3 will be employed. The marginal value of the first three units of Y is greater

than their marginal cost. The marginal value of the fourth unit is only $114 or $6 less than its cost, and hence, the firm would employ no more than 3 units of Y.

D. A change would be in order because the firm could produce 188 units at the same

cost using 2 units of each output: that is, the marginal product to price ratios of the two inputs are not equal at the current input proportions. Relatively less Y, and more X, is needed to provide an optimal combination.

E. The system exhibits constant returns to scale. This is true because a given increase

in both inputs causes an increase in output of the same proportion.

X

Y

Output 1

1

94 Η 1 = 94

2

2

94 Η 2 = 188 3

3

94 Η 3 = 282

4

4

94 Η 4 = 376 5

5

94 Η 5 = 470

P8.2 Production Function Concepts. Indicate whether each of the following statements is

true or false. Explain your answers.

A. Decreasing returns to scale and increasing average costs are indicated when εQ < 1.

B. If the marginal product of capital falls as capital usage grows, the returns to

capital are decreasing.

C. L-shaped isoquants describe production systems in which inputs are perfect substitutes.

D. Marginal revenue product measures the profit earned through expanding input

usage.

E. The marginal rate of technical substitution will be affected by a given percentage increase in the marginal productivity of all inputs.

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P8.2 SOLUTION A. True. When εQ < 1, the percentage change in output is less than a given percentage

change in all inputs. Thus, decreasing returns to scale and increasing average costs are indicated.

B. True. Returns to the capital input factor are decreasing when the marginal product of

capital falls as capital usage grows. C. False. L-shaped production isoquants reflect a perfect complementary relation

among inputs. D. False. Marginal revenue product is the revenue generated by expanding input usage

and represents the maximum that could be paid to expand usage. Because MRP is calculated before input costs (wages in the case of labor, for example), it does not measure the increase in profit earned through expansion.

E. False. The marginal rate of technical substitution is measured by the relative

marginal productivity of input factors. This relation is unaffected by a commensurate increase in the marginal productivity of all inputs.

P8.3 Compensation Policy. APay for performance@ means that employee compensation

closely reflects the amount of value derived from each employee=s effort. In economic terms, the value derived from employee effort is measured by net marginal revenue product. It is the amount of profit generated by the employee, before accounting for employment costs. Holding all else equal, indicate whether each of the following factors would be responsible for increasing or decreasing the amount of money available for employee merit-based pay.

A. Government mandates for employer-provided health insurance

B. Rising productivity due to better worker training

C. Rising employer sales due to falling imports

D. Falling prices for industry output

E. Rising prevalence of uniform employee stock options.



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P8.3 SOLUTION A. Decreasing. Government mandates for employer-provided health insurance increase

the costs of employment with no offsetting benefit in terms of increasing worker productivity and thereby decrease the funds available for merit-based pay.

B. Increasing. Rising productivity due to better worker training increases the profits

gained through expanding employment and increases the pool of funds available for merit-based pay.

C. Increasing. As imports fall, domestic output and employer sales revenue rise,

holding all else equal. Thus, output demand and MRQ would rise and increase the pool of available funds for merit-based pay.

D. Decreasing. As output prices fall, so too does MRQ and the MRP of workers. This

reduces the pool of funds available for merit-based pay. E. Decreasing. A rising prevalence of uniform employee stock options increases the

costs of employment with no offsetting gain in worker productivity. This reduces the pool of funds available for merit-based pay.

P8.4 Returns to Scale. Determine whether the following production functions exhibit

constant, increasing, or decreasing returns to scale.

A. Q = 0.5X + 2Y + 40Z

B. Q = 3L + 10K + 500

C. Q = 4A + 6B + 8AB

D. Q = 7L2 + 5LK + 2K2

E. Q = 10L0.5K0.3 P8.4 SOLUTION A. Initially, let X = Y = Z = 100, so output is: Q = 0.5(100) + 2(100) + 40(100) = 4,250

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Increasing all inputs by an arbitrary percentage, say 2%, leads to: Q = 0.5(102) + 2(102) + 40(102) = 4,335

Because a 2% increase in all inputs results in a 2% increase in output (Q2/Q1 = 4,335/4,250 = 1.02), the output elasticity is 1 and the production system exhibits constant returns to scale.

B. Initially, let L = K = 100, so output is: Q = 3(100) + 10(100) + 500 = 1,800

Increasing both inputs by an arbitrary percentage, say 3%, leads to: Q = 3(103) + 10(103) + 500 = 1,839

Because a 3% increase in both inputs results in a 2.2% increase in output (Q2/Q1 = 1,839/1,800 = 1.022), the output elasticity is less than 1 and the production system exhibits diminishing returns to scale.

C. Initially, let A = B = 100, so output is: Q = 4(100) + 6(100) + 8(100)(100) = 81,000

Increasing both inputs by an arbitrary percentage, say, 1%, leads to: Q = 4(101) + 6(101) + 8(101)(101) = 82,618

Because a 1% increase in both inputs results in a 2% increase in output (Q2/Q1 = 82,618/81,000 = 1.02), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale.

D. Initially, let L = K = 100, so output is: Q = 7(1002) + 5(100)(100) + 2(1002) = 140,000

Increasing both inputs by an arbitrary percentage, say, 2%, leads to: Q = 7(1022) + 5(102)(102) + 2(1022) = 145,656



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Because a 2% increase in both inputs results in a 4% increase in output (Q2/Q1 = 145,656/140,000 = 1.04), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale.

E. Initially, let L = K = 100, so output is: Q = 10(1000.5)(1000.3) = 398

Increasing both inputs by an arbitrary percentage, say, 4%, leads to: Q = 10(1040.5)(1040.3) = 411

Because a 4% increase in both inputs results in a 3.3% increase in output (Q2/Q1 = 411/398 = 1.033), the output elasticity is less than 1 and the production system exhibits decreasing returns to scale.

P8.5 Optimal Compensation Policy. Café-Nervosa.com, based in Seattle, Washington, is

a rapidly growing family business that offers a line of distinctive coffee products to local and regional coffee shops. Founder and president Frasier Crane is reviewing the company's sales force compensation plan. Currently, the company pays its three experienced sales staff members a salary based on years of service, past contributions to the company, and so on. Niles Crane, a new sales trainee and brother of Fraiser Crane, is paid a more modest salary. Monthly sales and salary data for each employee are as follows:

Sales Staff

Average Monthly

Sales

Monthly Salary

Roz Doyle

$160,000

$6,000 Daphne Moon

100,000

4,500

Martin Crane

90,000

3,600 Niles Crane

75,000

2,500

Niles Crane has shown great promise during the past year, and Fraiser Crane believes that a substantial raise is clearly justified. At the same time, some adjustment to the compensation paid to other sales personnel also seems appropriate. Fraiser Crane is considering changing from the current compensation plan to one based on a 5% commission. He sees such a plan as being fairer to the parties

Chapter 8


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involved and believes it would also provide strong incentives for needed market expansion.

A. Calculate Café-Nervosa.com's salary expense for each employee expressed as

a percentage of the monthly sales generated by that individual.

B. Calculate monthly income for each employee under a 5% of monthly sales commission-based system.

C. Will a commission-based plan result in efficient relative salaries, efficient

salary levels, or both? P8.5 SOLUTION A.

Sales Staff

(1)

Average

Monthly Sales (2)

Monthly Salary

(3)

Commission (4) = (3)/(2)

Roz Doyle

$160,000

$6,000

3.75% Daphne Moon

100,000

4,500

4.50%

Martin Crane

90,000

3,600

4.00% Niles Crane

75,000

2,500

3.33%

B.

Average Sales Staff

(1)

Monthly

Sales (2)

Commission (3) = (2) Η 0.05

Roz Doyle

$160,000

$8,000 Daphne Moon

100,000

5,000

Martin Crane

90,000

4,500 Niles Crane

75,000

3,750



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C. The commission-based compensation plan will result in more efficient relative salaries for sales personnel. Under this plan, Café-Nervosa.com sales compensation costs average 5%, irrespective of which member of the sales staff generates a given dollar of sales. Each employee is treated equally under this plan in the sense that all are paid the same rate for generating business.

Although a commission-based plan will result in an efficient relative salary structure, a 5% commission may or may not result in an optimal level of compensation being paid to each employee. If 5% of sales represents the net marginal revenue (marginal revenue minus all costs except sales expenses) generated by the sales staff, then optimal levels of compensation would be generated under such a commission-based plan. However, if net marginal revenues are different than this rate, some adjustment to the commission rate would be appropriate.

P8.6 Optimal Input Mix. The First National Bank received 3,000 inquiries following the

latest advertisement describing its 30-month IRA accounts in the Boston World, a local newspaper. The most recent ad in a similar advertising campaign in Massachusetts Business, a regional business magazine, generated 1,000 inquiries. Each newspaper ad costs $500, whereas each magazine ad costs $125.

A. Assuming that additional ads would generate similar response rates, is the

bank running an optimal mix of newspaper and magazine ads? Why or why not?

B. Holding all else equal, how many inquiries must a newspaper ad attract for the

current advertising mix to be optimal? P8.6 SOLUTION A. No. The rule for an optimal combination of newspaper (N) and magazine (M) ads is:

Here, the question is

3,000$500

= ? 1,000$125

6 8

N M

N M

MP MP = P P

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In other words, the last dollar spent on newspaper ads attracted six inquiries, while the last dollar spent on magazine ads attracted eight inquiries. Therefore, the current ad combination is not optimal. More magazine ads and/or fewer newspaper ads should be run.

B. Currently, magazine ads return 33% (eight versus six) more inquiries per advertising

dollar than do newspaper ads. Therefore, in order for the current ad mix to be optimal, inquiries generated by newspaper ads would have to increase by 33% from 3,000 to 4,000. To check:

N

N

MPP

= M

M

MPP

4,000$500

= ? 1,000$125

8 = _ 8 P8.7 Optimal Input Level. The Route 66 Truck Stop, Inc., sells gasoline to both self-

service and full-service customers. Those who pump their own gas benefit from the lower self-service price of $1.80 per gallon. Full-service customers enjoy the service of an attendant, but they pay a higher price of $1.90 per gallon. The company has observed the following relation between the number of attendants employed per day and full-service output:

Route 66 Truck Stop, Inc. Number of

Attendants per day

Full-Service Output

(gallons) 0

0

1

2,000 2

3,800

3

5,400 4

6,800

5

8,000

A. Construct a table showing the net marginal revenue product derived from attendant employment.



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B. How many attendants would Route 66 employ at a daily wage rate of $160 (including wages and benefits)?

C. What is the highest daily wage rate Route 66 would pay to hire four attendants

per day? P8.7 SOLUTION A. Because Route 66 operates in a perfectly competitive industry, the 104 price

premium for full-service versus self-service gasoline is stable. Thus, the net marginal revenue product of attendant labor (sometimes referred to as the value of marginal product) is:

Route 66 Truck Stop, Inc.

Number of Attendants

per Day (1)

Full Service

Output (gallons)

(2)

Marginal Product of Labor

(3)

Net Marginal

Revenue Product of Labor

(4) = (3) Η 104

0

0

--

-- 1

2,000

2,000

$200

2

3,800

1,800

180 3

5,400

1,600

160

4

6,800

1,400

140 5

8,000

1,200

120

B. From the table above, it becomes clear that employment of three attendants could be

justified at a daily wage cost of $160 because MRPA=3 = $160. Employment of a fourth attendant could not be justified because MRPA=4 = $140 < $160.

C. From the table above, the marginal revenue product of a fourth attendant MRPA=4 =

$140. Thus, $140 is the highest daily wage cost Route 66 would be willing to pay to hire a staff of four attendants.

P8.8 Optimal Input Level. Ticket Services, Inc., offers ticket promotion and handling

services for concerts and sporting events. The Sherman Oaks, California, branch office makes heavy use of spot radio advertising on WHAM-AM, with each 30-second

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ad costing $100. During the past year, the following relation between advertising and ticket sales per event has been observed:

Sales (units) = 5,000 + 100A - 0.5A2 ΜSales (units)/ΜAdvertising = 100 - A

Here, A represents a 30-second radio spot ad, and sales are measured in numbers of tickets.

Rachel Green, manager for the Sherman Oaks office, has been asked to recommend an appropriate level of advertising. In thinking about this problem, Green noted its resemblance to the optimal resource employment problem studied in a managerial economics course. The advertising/sales relation could be thought of as a production function, with advertising as an input and sales as the output. The problem is to determine the profit-maximizing level of employment for the input, advertising, in this Aproduction@ system. Green recognized that a measure of output value was needed to solve the problem. After reflection, Green determined that the value of output is $2 per ticket, the net marginal revenue earned by Ticket Services (price minus all marginal costs except advertising).

A. Continuing with Green's production analogy, what is the marginal product of

advertising?

B. What is the rule for determining the optimal amount of a resource to employ in a production system? Explain the logic underlying this rule.

C. Using the rule for optimal resource employment, determine the profit-

maximizing number of radio ads. P8.8 SOLUTION A. The marginal product of advertising is given by the expression: MPA = ΜS/ΜA = $100 - A B. The rule for determining the optimal amount of a resource to employ is: MRPA = PA



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The logic of this rule can be best understood by simply dissecting the above relations: MRPA = PA MPA Η MRQ = PA

Q TR x A Q

∂ ∂∂ ∂

= TCA

∂∂

TRA

∂∂

= TCA

∂∂

ΜTR = ΜTC Inflow = Outflow C. The optimal advertising level is found where: MRPA = PA MPA Η MRQ = PA (100 - A) Η $2 = $100 200 - 2A = 100 -2A = -100 A = 50 P8.9 Net Marginal Revenue. Will Truman & Associates, LLC is a successful Manhattan-

based law firm. Worker productivity at the firm is measured in billable hours, which vary between partners and associates.

Partner time is billed to clients at a rate of $250 per hour, whereas associate time is billed at a rate of $125 per hour. On average, each partner generates 25 billable hours per 40-hour workweek, with 15 hours spent on promotion, administrative, and supervisory responsibilities. Associates generate an average of 35 billable hours per 40-hour workweek and spend 5 hours per week in administrative and training meetings. Variable overhead costs average 50% of

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revenues generated by partners and, given supervisory requirements, 60% of revenues generated by associates.

A. Calculate the annual (50 workweek) net marginal revenue product of partners

and associates.

B. If partners earn $175,000 and associates earn $70,000 per year, does the company have an optimal combination of partners and associates? If not, why not? Make your answer explicit and support any recommendations for change.

P8.9 SOLUTION A. The annual marginal revenue product calculation for partners (P) and associates (A)

identifies the amount of net revenue generated per employee. MRPP = MPP Η MRQ = (Billable hours per year) Η (Net marginal revenue per hour) = (25 Η 50) Η ($250 Η 0.5) = $156,250 MRPA = MPA Η MRQ = (Billable hours per year) Η (Net marginal revenue per hour) = (35 Η 50) Η ($125 Η 0.4) = $87,500

Here it is important to note that each marginal hour of effort by partners brings to the firm $250 in revenue plus $125 of variable costs, for a net marginal revenue (value) for partner output of $125 per hour. Similarly, the net marginal revenue of associate output is $50 per hour. Both net marginal revenue figures reflect the marginal value of each service output.

B. A comparison of marginal revenue product figures with salary data suggests: MRPP = $156,250 < $175,000 = PP



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MRPA = $87,500 > $70,000 = PA

Therefore, partners bring in $18,750 per year less in net marginal revenues than their salary, whereas associates bring in a surplus of $17,500. At the margin, therefore, a $175,000 salary for each partner represents a marginal loss of $18,750 to the firm, whereas a $70,000 salary for associates represents a marginal profit of $17,500.

Holding all else equal, the firm would seek to marginally reduce the number of partners. Alternatively, some small increase in the number of associates would be warranted. Either move would help shift the firm to a position where MRPL = PL, and profits would be maximized.

As a first step, the firm might expand the number of associates until such a point as MRPA = $70,000 = PA. Then, a reevaluation of MRPP should be made to see if it has increased, as seems likely. If the new MRPP = $175,000 = P, no further change in staffing would be necessary. On the other hand, some adjustment (reduction) in the number of partners may be required.

P8.10 Production Function Estimation. Consider the following Cobb-Douglas production

function for bus service in a typical metropolitan area:

where

Q = output in millions of passenger miles,

L = labor input in worker hours,

K = capital input in bus transit hours, and

F = fuel input in gallons.

Each of the parameters of this model was estimated by regression analysis using monthly data over a recent three-year period. Results obtained were as follows:

ˆ0b = 1.2; ˆ1b = 0.28; ˆ2b = 0.63; and ˆ3b = 0.12

The standard error estimates for each coefficient are:

ˆ0bσ = 0.4; ˆ1bσ = 0.15; ˆ2bσ = 0.12; ˆ3bσ = 0.07

1 2 3b b b0Q = ,b L k F

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A. Estimate the effect on output of a 4% decline in worker hours (holding K and F

constant).

B. Estimate the effect on output of a 3% reduction in fuel availability accompanied by a 4% decline in bus transit hours (holding L constant).

C. Estimate the returns to scale for this production system.

P8.10 SOLUTION A. For Cobb-Douglas production functions, calculations of the elasticity of output with

respect to individual inputs can be made by simply referring to the exponents of the production relation. Here a 4% decline in L, holding all else equal, will lead to a 1.12% decline in output because:

Because (ΜQ/Q)/(ΜL/L) is the percent change in Q due to a 1% change in L,

Q/QL/L

∂∂

= b1

ΜQ/Q = b1 Η ΜL/L = 0.28(-0.04) = -0.0112 or -1.12% B. From part A it is obvious that: ΜQ/Q = b2(ΜK/K) + b3(ΜF/F)

1 2 3

1 2 3

1 2 3

- 1b b b0 1

- 1 + 1b b b0 1

b b b0

1

Q/Q Q L = x L/L L Q

( ) x Lb b L k F = Q

b b L k F = b L K F

= b

∂ ∂∂ ∂



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= 0.63(-0.04) + 0.12(-0.03) = -0.0288 or -2.88% C. In the case of Cobb-Douglas production functions, returns to scale are determined by

simply summing exponents because:

Here b1 + b2 + b3 = 0.28 + 0.63 + 0.12 = 1.03 > 1.0 indicating slight increasing returns to scale. This means that a 1% increase in all inputs will lead to a 1.03% increase in output, and average costs will fall slightly as output increases.

APPENDIX 8A: A CONSTRAINED OPTIMIZATION APPROACH TO DEVELOPING THE OPTIMAL INPUT COMBINATION RELATIONSHIPS PROBLEM & SOLUTION 8A.1 Assume that a firm produces its product in a system described in the following

production function and price data: Q = 3X + 5Y + XY PX = $3 PY = $6

Here, X and Y are two variable input factors employed in the production of Q.

A. What are the optimal input proportions for X and Y in this production system? Is this combination rate constant regardless of the output level?

B. It is possible to express the cost function associated with the use of X and Y in

the production of Q as Cost = PXX + PYY or Cost = $3X + $6Y. Use the Lagrangian technique to determine the maximum output that the firm can produce operating under a $1,000 budget constraint for X and Y. Show that

1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

b b b0

b b b0

+ + b b b b b b0

+ + b b b

Q = b L K FhQ = (kL (kK (kF) ) )b

= bk L K F = Qk

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the inputs used to produce that level of output meet the optimality conditions derived in Part A.

C. What is the additional output that could be obtained from a marginal increase

in the budget? D. Assume that the firm is interested in minimizing the cost of producing 14,777

units of output. Use the Lagrangian method to determine what optimal quantities of X and Y to employ. What will be the cost of producing that output level? How would you interpret λ, the Lagrangian multiplier, in this problem?

8A.1 SOLUTION A. Optimal input proportions are found by solving the following relation:

MPXPX

= MPYPY

3 + Y3

= 5 + X6

18 + 6Y = 15 + 3X X = 2Y + 1 or Y = (X - 1)/2

Because Q appears in neither of the above expressions, these optimal unit proportions are invariant with respect to the output level.

B. The optimization problem faced by the firm can be written

Maximize Q = 3X + 5Y + XY,

Subject to $3X + $6Y = $1,000,

which suggests the following Lagrangian expression:

LQ = 3X + 5Y + XY + λ($1,000 - $3X - $6Y).

(1) ΜLQ/ΜX = 3 + Y - 3λ = 0



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(2) ΜLQ/ΜY = 5 + X - 6λ = 0

(3) ΜLQ/Μλ = 1,000 - 3X - 6Y = 0

To solve, take 2 times (1) minus (2): 2 Η (1) 6 + 2Y - 6λ = 0 minus (2) - (5 + X - 6λ = 0) 1 + 2Y - X = 0.

Then, multiplying this result by 3 and adding (3) yields the following: 3 Η (above) 3 + 6Y - 3X = 0 plus (3) 1,000 - 6Y - 3X = 0 1,003 - 6X = 0 X = 167.

Then, from (3): (3) 1,000 - 3(167) - 6Y = 0 6Y = 499 Y = 83.

And, solving for λ using (1): (1) 3 + 83 - 3λ = 0 3λ = 86 λ = 28.7.

Part A above showed that X and Y should be combined in the ratio X = ? 2Y + 1

and

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167 = _ 2(83) + 1.

Thus, X and Y are combined in optimal proportions. C. The incremental output obtainable from an additional $1 expenditure on X and Y is

28.7 units as determined by the value of λ in Part B. This implies that the marginal cost of one output unit is

MC = TC 1 = Q

∂∂ λ

= $0.035 (or 3.5 cents per unit).

D. The alternative optimization problem for the firm can be written as Minimize TC = $3X + $6Y Subject to 14,777 = 3X + 5Y + XY,

which suggests the following Lagrangian expression:

LTC = $3X + $6Y + λ(14,777 - 3X - 5Y - XY).

(1) ΜLTC/ΜX = 3 - 3λ - λY = 0 (2) ΜLTC/ΜY = 6 - 5λ - λX = 0 (3) ΜLTC/Μλ = 14,777 - 3X - 5Y - XY = 0

To solve, take 2 times (1) minus (2):

2 Η (1) 6 - 6λ - 2λY = 0 minus (2) -(6 - 5λ - λX = 0) -λ + λX - 2λY = 0 λ(- 1 + X - 2Y) = 0

In order for λ(-1 + X - 2Y) = 0, either λ or (-1 + X - 2Y) must equal zero. Because costs are an increasing function of output, λ 0, and therefore (-1 + X - 2Y) = 0. This implies

X = 2Y + 1 or Y = (X - 1)/2



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Substituting 2Y + 1 for X in (3) yields: (3) 14,777 - 3(2Y + 1) - 5Y - (2Y + 1)Y = 0 14,777 - 6Y - 3 - 5Y - 2Y2 - Y = 0 14,774 - 12Y - 2Y2 = 0

or -2Y2 - 12Y + 14,774 = 0,

which is a quadratic equation of the form aY2 + bY + c = 0,

where a = -2, b = -12, and c = 14,774. Its two roots can be obtained from the quadratic formula

Y = 2-b - 4acb

2a±

= 12 144 - 4(-2)(14,774)

-4±

= 12 344-4±

= 83 or -89

Although both 83 and -89 are mathematically feasible, negative output is not economically feasible. Therefore, Y = 83 is the correct answer, and

X = 2Y + 1 = 2(83) + 1 = 167

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TC = $3(167) + $6(83) = $501 + $498 . $1,000.

This answer is identical with part A because, Q = 3X + 5Y + XY = 3(167) + 5(83) + (167)(83) = 14,777

Maximizing output subject to a given budget constraint is equivalent to minimizing cost subject to a given output constraint.

The Lagrangian multiplier, λ, in this problem measures the marginal cost of producing one additional unit of output.

3 - 3λ - λY = 0 3 - 3λ - λ(83) = 0 86λ = 3 λ = 0.035

The marginal cost of an additional unit of output at the 14,777 unit production level is 3.5 cents. This is identical to our finding in part C. (Note: The increase in output per dollar expended is ΜQ = 1/λ = 28.7 units, which was measured by λ in part B.)

CASE STUDY FOR CHAPTER 8 Employee Productivity Among the Largest S&P 500 Firms Traditional measures of firm productivity tend to focus on profit margins, the rate of return on stockholder=s equity, or related measures like total asset turnover, inventory turnover, or receivables turnover. Profit margin is net income divided by sales and is a useful measure of a company=s ability to manufacture and distribute distinctive products. When profit margins are high, it=s a good sign that customer purchase decisions are being driven by unique product characteristics or product quality rather than by low prices. When profit margins are high,



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companies are also able to withstand periods of fluctuating costs or weak product demand without devastating consequences for net income. While high profit margins have the potential to attract new competitors, they also act as credible evidence that a firm offers a hard-to-imitate combination of attractive goods and services.

Return on equity (ROE), defined as net income divided by the accounting book value of stockholder=s equity, is an attractive measure of firm performance because it reflects the effects of both operating and financial leverage. When ROE is high, the company is able to generate an attractive rate of return on the amount of money entrusted to the firm by shareholders in the form of common stock purchases and retained earnings. High profit margins give rise to high ROE, as do rapid turnover in inventory, receivables and total assets. Rapid inventory turnover reduces the risk of profit-sapping product closeouts where slow-moving goods are marked down for quick sale. Rapid receivables turnover eases any concern that investors might have in terms of the firm=s ability to collect money owed by customers. High total asset turnover, defied as sales divided by total assets, documents the firm=s ability to generate a significant amount of business from its fixed plant and equipment.

Despite these obvious advantages, each of these traditional firm performance measures suffers certain shortcomings. Profit margins are strongly influenced by industry-related factors that might obscure superior firm productivity when firms from different industries are compared. For example, the automobile industry is huge and net profit margins for mediocre performers are commonly in the 2.5-3% range. Even standout performers, like Toyota, struggle to earn 6% on sales. Meanwhile, even mediocre banks commonly report profit margins in the 15-20% range. Similarly, and despite obvious advantages, ROE suffers as a performance measure because steep losses can greatly diminish retained earnings, decimate the book value of stockholder=s equity, and cause ROE to soar. When companies buy back their shares in the open market at prices that greatly exceed accounting book values, the book value of shareholder=s equity also falls, and can unfairly inflate the ROE measure. For these reasons, some analysts look to the accounting rate of return as a more simple and less easily distorted measure of accounting profit performance.

However, the biggest problem with corporate performance measures based upon profit rates tied to sales, stockholder=s equity or assets has nothing to do with measurement problems tied to irregular profit and loss patterns or corporate restructuring. The biggest problem with traditional corporate profit measures is that they fail to reflect the firm=s efficient use of its most precious resources: human resources. In the services-based economy of the new millennium, the most telling indicator of a company=s ability to compete is its ability to attract, train, and motivate a capable workforce. In economics, the term human capital is used to describe the investment made in workers and top management that make them more efficient and more profitable employees. Employee training and education are two of the most reliable tools that companies can use to keep an edge on the competition. However, determining an efficient amount of worker training and education is more tricky than it might seem at first.

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In a competitive labor market, employees can expect to command a wage rate equal to the amount they could compel in their next-best employment opportunity. At least in part, this opportunity cost reflects employee productivity created by better worker training and education. Because dissatisfied workers can be quick to jump ship, employers must be careful to maintain a productive work environment that makes happy employees want to stay and contribute to the firm that paid for their education and training. Employers need capable and well-trained employees, but no employer wants to be guilty of training workers that end up working for the competition! All successful firms are efficient in terms of constantly improving employee productivity, and then motivating satisfied and capable employees to perform. In light of the importance placed upon capable and well-motivated employees, an attractive alternative means for measuring corporate productivity is in terms of profits and revenues per employee. Table 8.6 here

Table 8.6 gives interesting perspective on employee productivity by showing revenue per employee and profits per employee for the largest 30 companies in the S&P 500 Index, when these corporate giants are ranked by the market capitalization of common stock. A. What firm-specific and industry-specific factors might be used to explain differences

among giant corporations in the amount of revenue per employee and profit per employee?

B. A multiple regression analysis based upon the data contained in Table 8.6 reveals the following (t statistics in parentheses):

Profit/Emp.= $17,267.679 + 0.052 Ind. Profit/Emp. + 0.083 Rev./Emp. + 0.006 Ass./Emp.

(1.26) (0.16) (5.19) (3.86)

R2 = 98.0%, F statistic = 417.12

Interpret these results. Is profit per employee more sensitive to industry-specific or firm-specific factors for this sample of giant corporations?

CASE STUDY SOLUTION A. Firm-to-firm variation in the amount of profits per employee is sure to depend upon

both firm-specific and industry-specific factors. For example, at the industry level of aggregation, the amount of capital employed per worker tends to be very high in financial services like banking and credit services. As a result, even the most mediocre banks in terms of relative efficiency report high profits and revenues per employee. However, the most efficient financial service companies will have better trained workers and motivate them most efficiently. Thus, it is important to control for industry effects when considering firm productivity as measured by profits and



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revenues generated on a per employee basis. Differences in employee productivity will be affected by differences in total assets per employee, worker education and training, the effectiveness of incentive compensation plans, the amount of employee stock ownership, and so on.

B. Based upon this sample of giant corporations taken from the S&P 500, profit per

employee appears to be more sensitive to firm-specific factors than to industry-specific factors.

On an overall basis, the simple model estimated here explains a very large proportion (98.0%) of the variation in profits per employee for the 30 largest firms found within the S&P 500, when ranked according to market capitalization. This is a statistically significant explanation (F = 417.12) of a meaningful share of the total amount of variation in profits per employee.

Clearly, the firm-by-firm variation in profits per employee cannot be simply explained as a byproduct of industry effects. What is surprising in this simple model is that industry effects offer no marginal explanation of profits per employee when revenue per employee (t = 5.19) and total assets per employee (t = 3.86) are constrained. In other words, the marginal effects of revenues per employee and total assets per employee overwhelm the marginal effects of industry-specific factors. It is also interesting that both revenues per employee and total assets per employee have marginal influences on the amount of total profits generated on a per employee basis. Of course, the strong influence of revenues per employee on profits per employee stems from the fact that both are commonly employed measures of employee productivity. However, at a minimum, results reported here suggest that profits per employee reflect the amount of revenue generated per employee and the amount of total assets each employee has at their disposal.

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