7
PRODUCTION Chapter ANALYSIS AND COMPENSATION POLICY
QUESTIONS AND ANSWERS
Q7.1
Is use of least-cost input combinations a necessary condition for profit
maximization? Is it a sufficient condition? Explain.
Q7.1
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.
Q7.2
Output per worker is expected to increase by 10 percent during the next year.
Therefore, wages can also increase by 10 percent with no harmful effects on
employment, output prices, or employer profits. Discuss this statement.
Q7.2
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
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
169
Chapter 7
directly attributable to gains in labor productivity, and do not imply that a higher
wage rate could be justified.
Q7.3
Commission-based and piece-rate-based compensation plans are commonly
employed by businesses. Use the concepts developed in the chapter to explain these
phenomena.
Q7.3
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.
Q7.4
Hourly wage rates are an anachronism. Efficiency requires incentive-based pay
tied to performance. Discuss this statement.
Q7.4
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 goldbricking, is always
possible, even hourly employees have strong incentives to provide a satisfactory
level of performance. In addition, being 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.
Q7.5
Explain why the MP/P relation is deficient as the sole mechanism for determining
the optimal level of resource employment.
Q7.5
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
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.
Q7.6
Clarify how profits are maximized and the optimal level of employment is achieved
in a competitive labor market when the price of labor PL = MRPL.
Production Analysis and Compensation Policy
Q7.6
170
ANSWER
The MRPL represents the value created by each additional worker, and represents net
marginal revenue considering all costs except wages. In a competitive labor market,
the price of labor represents the marginal cost of employment. When the market
wage rate PL < MRPL, the firm has an incentive to increase employment to further
expand profits. When the market wage rate PL > MRPL, wages paid at the margin
exceed the amount of additional revenue generated. Therefore, when the market
wage rate PL > MRPL, the firm has an incentive to cut back on employment. An
optimal level of employment is achieved only when the market price of labor PL =
MRPL. At the optimal level of employment, all profitable workers are employed but
no unprofitable workers are employed.
Q7.7
Oregons minimum wage increased from $4.75 in 1996 to $5.50 in 1997, to $6 in
1998, and to $6.50 in 1999. According to a study by the Oregon Center for Public
Policy, the minimum wage increases in Oregon did not harm welfare recipients
opportunities to find work. In fact, a larger percentage of welfare recipients in
Oregon found jobs after the minimum wage increased than before the increases.
Discuss how these facts could be consistent with a downward-sloping demand curve
for unskilled labor.
Q7.7
ANSWER
It is interesting to note that sharp increases in the Oregon state minimum wage
during the late-1990s had little apparent effect on the ability of welfare recipients to
find new higher-pay job opportunities. While perhaps surprising, these facts are
entirely consistent with a downward sloping demand curve for unskilled labor. By
itself, a simple increase in the state minimum wage has the effect of decreasing the
quantity demanded of unskilled labor. This involves an upward movement along the
labor demand curve. However, during the late-1990s, changes in the pace of
economic activity were not stagnant. The late-1990s was a period of robust
economic expansion and rising demand for labor. As sales rise, companies are able
to spend more to attract workers, including minimum wage workers. Increasing
demand for unskilled labor, reflected by a rightward shift in the labor demand curve,
obviously overwhelmed the employment decreasing effect of rising minimum wages
during this period.
Q7.8
Powerful unions like the AFL-CIO are staunch advocates for increasing the federal
minimum wage despite the fact that highly-trained and experienced AFL-CIO
workers tend to earn far more than the minimum wage. Can you give an economic
rationale for the AFL-CIOs position?
171
Q7.8
Chapter 7
ANSWER
Higher Federal minimum wages increase the number of workers willing to work, but
decrease the number of workers employers are willing to hire. Raising wages by
governmental edict creates a surplus of labor, or unemployment. Ample empirical
evidence has led most economists to accept this conclusion. The magnitude of the
unemployment effects of minimum wage increases can be questioned, but the
existence of those effects is clear.
An increase in the Federal minimum wage has the greatest impact on those
with low skills whose normal wage would be less than or near the legally established
minimum. However, as minimum wages are raised secondary effects can increase
the wages of other workers. Most products can be produced in different ways. For
example, crops can be harvested with stoop labor or by skilled workers using
expensive machinery. As a result, unskilled labor often competes directly or
indirectly with semiskilled and skilled labor. Some economists contend that this
explains why the AFL-CIO staunchly supports increasing the minimum wage even
though almost all AFL-CIO workers earn far more than the minimum. Increasing
the minimum wage is sometimes seen as organized labor's attempt to eliminate price
competition from low-price and low-skilled labor.
Q7.9
Cite some ways for increasing productivity growth in the United States.
Q7.9
ANSWER
During the 2000-05 period, the pace of annual increase in productivity in the United
States averaged 3.1 percent, a big jump from the 1.4 percent annual rate common
during the 1973-1995 period. This burst in U.S. productivity growth is greater than
in many advanced industrial economies.
Faster growth of inputs, both physical and human capital, is a major cause of
faster productivity growth. In the United States, the capital-labor ratio has grown
faster since the early-1990s, but not enough to account for all of the recent increase
in productivity growth. The contribution from greater capital services, called capital
deepening, is clearly evident. 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. Therefore, policies to increase
investment, education, and training, are an important contributor to the recent boost
in U.S. productivity.
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
Production Analysis and Compensation Policy
172
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 2000-05 burst
in U.S. productivity growth.
Q7.10
Explain why company productivity is important to managers, employees, and
investors. Is superior worker productivity a necessary and sufficient condition for
above-average compensation?
Q7.10
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 fail to generate attractive rates of return on investment 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.
Only superior employees with hard-to-duplicate contributions can expect to earn
above-average compensation.
SELF-TEST PROBLEMS AND SOLUTIONS
ST7.1
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.
173
Chapter 7
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?
ST7.1
SOLUTION
A.
The rule for an optimal combination of Testlogic-1 (T) and Accutest-3 (A)
equipment is
MPT = MP A
PT
PA
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
27
$18,000 /(25 8)
48
?
= $32,000 /(25 8)
0.3 = 0.3
On a per month basis, the relevant question is
27 (25 8)
$18,000
0.3
?
=
48 (25 8)
$32,000
= 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 MRQ = Input Price
Production Analysis and Compensation Policy
174
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 MRQ ? PT
=
27 $6
? $18,000/(25 8)
=
$162 > $90.
Accutest-3
MRPA = MPA MRQ ? PA
=
48 $6
? $32,000/(25 8)
=
$288 > $160.
Or, in per month terms:
Testlogic-1
MRPT = MPT MRQ ? PT
=
27 (25 8) $6
? $18,000
=
$32,400 > $18,000.
Accutest-3
MRPA = MPA MRQ ? PA
=
48 (25 8) $6
? $32,000
=
$57,600 > $32,000.
175
Chapter 7
In both cases, each machine returns more than its marginal cost (price) of
employment, and expansion would be profitable.
ST7.2
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):
Q = b0 Lb1 K b2 E b3
where
Q = output
L = labor input in worker hours
K = capital input in machine hours
E = energy input in BTUs
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:
b0 = 0.9; b1 = 0.4; b 2 = 0.4; and b3 = 0.2
The standard error estimates for each coefficient are:
b0 = 0.6; b1 = 0.1; b2 = 0.2; b3 = 0.1
A.
Estimate the effect on output of a 1 percent decline in worker hours (holding K
and E constant).
B.
Estimate the effect on output of a 5 percent reduction in machine hours
availability accompanied by a 5 percent decline in energy input (holding L
constant).
C.
Estimate the returns to scale for this production system.
ST7.2
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 Analysis and Compensation Policy
176
production relation. Here a 1 percent decline in L, holding all else equal, will lead to
a 0.4 percent decline in output. Notice that:
Q/Q Q L
=
L/L L Q
=
(b0 b1 L b1 - 1 K b 2 E b 3) L
Q
=
b0 b1 Lb1 K b 2 E b 3
b 0 L b1 K b 2 E b 3
-1 +1
= b1
And because (Q/Q)/( L/L) is the percent change in Q due to a 1 percent change in
L,
Q/Q
L/L
= b1
Q/Q = b1 L/L
= 0.4(-0.01)
= -0.004 or -0.4 percent
B.
From part A it is obvious that:
Q/Q
= b2(K/K) + b3(E/E)
= 0.4(-0.05) + 0.2(-0.05)
= -0.03 or -3 percent
177
C.
Chapter 7
In the case of Cobb-Douglas production functions, returns to scale are determined by
simply summing exponents because:
Q = b0 L b1 K b 2 E b 3
hQ = b0 (kL ) b1 (kK ) b 2 (kE )b 3
= k b1 + b 2 + b 3 b0 L b1 K b 2 E b 3
= k b1 + b 2 + b 3 Q
Here b1 + b2 + b3 = 0.4 + 0.4 + 0.2 = 1 indicating constant returns to scale. This
means that a 1 percent increase in all inputs will lead to a 1 percent increase in
output, and average costs will remain constant as output increases.
PROBLEMS AND SOLUTIONS
P7.1
Marginal Rate of Technical Substitution. The following production table gives
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
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
Total
Marginal
Average
Marginal
Revenue
Production Analysis and Compensation Policy
Units of
Y Used
Product
of Y
178
Product
of Y
Product
of Y
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?
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?
P7.1
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
179
Chapter 7
Units of
Y Employed
TPY
(1)
MPY
(2)
APY
(3)
MRPY
(4) = $3 (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 (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
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
Output
1
1
94 1 = 94
2
2
94 2 = 188
3
3
94 3 = 282
4
4
94 4 = 376
5
P7.2
Y
5
94 5 = 470
Production Function Concepts. Indicate whether each of the following statements
is true or false. Explain your answers.
Production Analysis and Compensation Policy
180
A.
Decreasing returns to scale and increasing average costs are indicated when
Q < 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.
P7.2
SOLUTION
A.
True. When Q < 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.
P7.3
Compensation Policy. Pay for performance means that employee compensation
closely reflects the amount of value derived from each employees 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
181
Chapter 7
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.
P7.3
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.
P7.4
Returns to Scale. Determine whether the following production functions exhibit
constant, increasing, or decreasing returns to scale.
A.
B.
Q = 3L + 10K + 500
C.
Q = 4A + 6B + 8AB
D.
Q = 7L2 + 5LK + 2K2
E.
P7.4
Q = 0.5X + 2Y + 40Z
Q = 10L0.5K0.3
SOLUTION
Production Analysis and Compensation Policy
A.
182
Initially, let X = Y = Z = 100, so output is:
Q = 0.5(100) + 2(100) + 40(100) = 4,250
Increasing all inputs by an arbitrary percentage, say 2 percent, leads to:
Q = 0.5(102) + 2(102) + 40(102) = 4,335
Because a 2 percent increase in all inputs results in a 2 percent 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 percent, leads to:
Q = 3(103) + 10(103) + 500 = 1,839
Because a 3 percent increase in both inputs results in a 2.2 percent 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 percent, leads to:
Q = 4(101) + 6(101) + 8(101)(101) = 82,618
Because a 1 percent increase in both inputs results in a 2 percent 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 percent, leads to:
Q = 7(1022) + 5(102)(102) + 2(1022) = 145,656
183
Chapter 7
Because a 2 percent increase in both inputs results in a 4 percent 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 percent, leads to:
Q = 10(1040.5)(1040.3) = 411
Because a 4 percent increase in both inputs results in a 3.3 percent 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.
P7.5
Optimal Compensation Policy. Caf-Nervosa.com, based in Seattle, Washington, is
a rapidly growing family business that offers a line of distinctive coffee products to
local and coffee regional shops. Assume 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 Frasier 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 Frasier 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. Frasier Crane is considering changing from the current compensation
plan to one based on a 5 percent commission. He sees such a plan as being fairer to
the parties involved and believes it would also provide strong incentives for needed
market expansion.
A.
Calculate Caf-Nervosa.com's salary expense for each employee expressed as
a percentage of the monthly sales generated by that individual.
Production Analysis and Compensation Policy
184
B.
C.
P7.5
Calculate monthly income for each employee under a 5 percent of monthly
sales commission-based system.
Will a commission-based plan result in efficient relative salaries, efficient
salary levels, or both?
SOLUTION
A.
Average
Monthly Sales
(2)
Monthly
Salary
(3)
Commission
(4) = (3)/(2)
Roz Doyle
$160,000
$6,000
3.75 percent
Daphne Moon
100,000
4,500
4.50 percent
Martin Crane
90,000
3,600
4.00 percent
Niles Crane
75,000
2,500
3.33 percent
Sales Staff
(1)
B.
Average
Sales Staff
(1)
Commission
(3) = (2) 0.05
Roz Doyle
$160,000
$8,000
Daphne Moon
100,000
5,000
Martin Crane
90,000
4,500
Niles Crane
C.
Monthly
Sales
(2)
75,000
3,750
The commission-based compensation plan will result in more efficient relative
salaries for sales personnel. Under this plan, Caf-Nervosa.com sales compensation
costs average 5 percent, 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 percent commission may or may not result in an optimal level of
compensation being paid to each employee. If 5 percent 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.
185
P7.6
Chapter 7
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?
P7.6
SOLUTION
A.
No. The rule for an optimal combination of newspaper (N) and magazine (M) ads is:
MP N = MPM
PN
PM
Here, the question is
3,000
$500
1,000
?
= $125
68
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 percent (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
percent from 3,000 to 4,000. To check:
MP N
PN
4,000
$500
=
MPM
PM
? 1,000
=
$125
Production Analysis and Compensation Policy
186
8 =8
P7.7
Marginal Revenue Product of Labor. To better serve customers interested in
buying cars over the Internet, Smart Motors, Inc., hired Nora Jones to respond to
customer inquiries, offer price quotes, and write orders for leads generated by the
companys web site. During last year, Jones averaged 1.5 vehicle sales per week.
On average, these vehicles sold for a retail price of $25,000 and brought the
dealership a profit contribution of $1,000 each.
A.
Estimate Jones= annual (50 workweek) marginal revenue product.
B.
Jones earns a base salary of $60,000 per year, and Smart Motors pays an
additional 28% of this base salary in taxes and various fringe benefits. Is
Jones a profitable employee?
P7.7
SOLUTION
A.
In the long run, Jones= marginal revenue product is the maximum amount Smart
Motors could pay in base salary plus all fringe benefits. It is the amount of added
revenue after all other variable costs that Jones= effort brings to the firm. In this
case, Jones= marginal revenue product is determined by the number of cars sold and
the profit contribution earned on each sale.
MRPL = MPL MRQ
= (Car sales per year) (Profit contribution per unit)
= (1.5 50) ($1,000)
= $75,000
Because Jones is only engaged in the sales function, Jones does not produce cars.
What Jones does produce are car sales, and the added value to the employer of
Jones= sales effort is what determines the amount the employer is willing and able to
pay.
B.
No. In addition to base salary, employers must pay additional taxes and fringe
benefits. All of these costs must be justified by the amount of marginal revenue
product generated to justify employment. In this case, employment costs for Jones
are $76,800 (= $60,000 1.28). A comparison of marginal revenue product figures
with these employment cost data suggests:
MRPL = $75,000 < $76,800 = PL
187
Chapter 7
Therefore, Jones brings in $75,000 per year in additional profit contribution, but
costs Smart Motors $76,800. This means that Jones brings in $1,800 per year less in
net marginal revenues than the marginal cost of employment. At the margin, Joness
employment represents a marginal loss to Smart Motors. Jones is not a profitable
employee.
P7.8
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 30second 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
Sales (units)/ Advertising = 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 production 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 profitmaximizing number of radio ads.
P7.8
SOLUTION
A.
The marginal product of advertising is given by the expression:
MPA = S/A
= $100 - A
Production Analysis and Compensation Policy
B.
188
The rule for determining the optimal amount of a resource to employ is:
MRPA = PA
The logic of this rule can be best understood by simply dissecting the above
relations:
MRPA = PA
MPA MRQ = PA
Q TR
A Q
=
TC
A
TR
A
=
TC
A
TR = TC
Inflow = Outflow
C.
The optimal advertising level is found where:
MRPA = PA
MPA MRQ = PA
(100 - A) $2 = $100
200 - 2A = 100
-2A = -100
A = 50
P7.9
Net Marginal Revenue. Crane, Poole & Schmidt, LLC is a successful Boston-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
189
Chapter 7
percent of revenues generated by partners and 60 percent 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.
P7.9
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 MRQ
= (Billable hours per year) (Net marginal revenue per hour)
= (25 50) ($250 0.5)
= $156,250
MRPA = MPA MRQ
= (Billable hours per year) (Net marginal revenue per hour)
= (35 50) ($125 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
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, a $175,000
Production Analysis and Compensation Policy
190
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 might seek to marginally reduce the number of
partners. Alternatively, some small increase in the number of associates could 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 = PP, no further
change in staffing would be necessary. On the other hand, some adjustment
(reduction) in the number of partners may be required.
P7.10
Production Function Estimation. Consider the following Cobb-Douglas production
function for bus service in a typical metropolitan area:
Q = b0 Lb1 k b2 Fb3 ,
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:
b0 = 1.2; b1 = 0.28; b 2 = 0.63; and b3 = 0.12
The standard error estimates for each coefficient are:
b0 = 0.4; b1 = 0.15; b2 = 0.12; b3 = 0.07
A.
B.
Estimate the effect on output of a 3 percent reduction in fuel availability
accompanied by a 4 percent decline in bus transit hours (holding L constant).
C.
P7.10
Estimate the effect on output of a 4 percent decline in worker hours (holding K
and F constant).
Estimate the returns to scale for this production system.
SOLUTION
191
A.
Chapter 7
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 percent decline in L, holding all else equal, will lead to
a 1.12 percent decline in output because:
Q/Q Q L
=
L/L L Q
=
(b0 b1 L b1 - 1 k b 2 Fb 3) L
Q
-1 + 1
b0 b1 L b1 k b 2 Fb 3
b0 L b1 K b 2 Fb 3
= b1
Because (Q/Q)/( L/L) is the percent change in Q due to a 1 percent change in L,
=
Q/Q
L/L
= b1
Q/Q = b1 L/L
= 0.28(-0.04)
= -0.0112 or -1.12 percent
B.
From part A it is obvious that:
Q/Q = b2( K/K) + b3( F/F)
= 0.63(-0.04) + 0.12(-0.03)
= -0.0288 or -2.88 percent
C.
In the case of Cobb-Douglas production functions, returns to scale are determined by
simply summing exponents because:
Q = b0 L b1 K b 2 Fb 3
hQ = b0 (kL ) b1 (kK ) b 2 (kF )b 3
= k b1 + b 2 + b 3 b0 L b1 K b 2 Fb 3
= k b1 + b 2 + b 3 Q
Production Analysis and Compensation Policy
192
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 percent increase in all inputs will lead to a 1.03
percent increase in output, and average costs will fall slightly as output increases.
193
Chapter 7
CASE STUDY FOR CHAPTER 7
Worker Productivity Among Giant U.S. Corporations
Traditional measures of firm productivity tend to focus on profit margins, the rate of return on
stockholders 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
companys ability to manufacture and distribute distinctive products. When profit margins are
high, its 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,
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
stockholders 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 firms ability to collect money owed by customers. High total asset turnover, defied as
sales divided by total assets, documents the firms 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 percent range. Even standout performers, like Toyota, struggle to
earn 6 percent on sales. Meanwhile, even mediocre banks commonly report profit margins in
the 15-20 percent 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 stockholders 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 shareholders equity also falls, and can unfairly inflate the ROE measure. For these reasons,
some analysts look to the growth in net income as a 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, stockholders 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 firms efficient use of human
resources. In the services-based economy of the new millennium, the most telling indicator of a
companys 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
Production Analysis and Compensation Policy
194
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.
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 7.5 gives interesting perspective on employee productivity by showing revenue per
employee and profits per employee for the 30 giant corporations that together comprise the Dow
Jones Industrial Average.
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 7.6 reveals
the following (t statistics in parentheses):
Profit/Emp.= $1,269.016 + 0.220 Ind. Profit/Emp. + 0.084 Rev./Emp. + 0.004 Ass./Emp.
(0.17)
(2.33)
(8.22)
(1.20)
R2 = 89.7 percent, F statistic = 75.48
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
revenues generated on a per employee basis. Differences in employee productivity
195
Chapter 7
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 DJIA, profit per
employee appears to be sensitive to firm-specific factors and industry-specific
factors.
On an overall basis, the simple model estimated here explains a very large
proportion (89.7 percent) of the variation in profits per employee for the 30 giant
firms found within the DJIA. This is a statistically significant explanation (F =
75.49) 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. It is not surprising in this simple model
that industry effects offer some marginal explanation of profits per employee (t =
2.34) when revenue per employee (t = 8.22) and total assets per employee (t = 1.19)
are constrained. In this example, the marginal effects of industry profits per
employee and firm revenues per employee overwhelm the marginal effects of total
assets per employee. 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. At a minimum, results reported here suggest the
importance of industry-related and firm-specific effects on employee productivity.
Production Analysis and Compensation Policy
196
APPENDIX 7A: A CONSTRAINED OPTIMIZATION APPROACH TO DEVELOPING
THE OPTIMAL INPUT COMBINATION RELATIONSHIPS
PROBLEM
7A.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 = P XX + 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
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 , the Lagrangian multiplier, in this problem?
7A.1
SOLUTION
A.
Optimal input proportions are found by solving the following relation:
MPX
PX
=
MPY
PY
3+Y
3
=
5+X
6
197
Chapter 7
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 + ($1,000 - $3X - $6Y).
(1)
LQ/X = 3 + Y - 3 = 0
(2)
LQ/Y = 5 + X - 6 = 0
(3)
LQ/ = 1,000 - 3X - 6Y = 0
To solve, take 2 times (1) minus (2):
2 (1)
minus (2)
6 + 2Y - 6 = 0
- (5 + X - 6 = 0)
1 + 2Y - X = 0
Then, multiplying this result by 3 and adding (3) yields the following:
3 (above)
plus (3)
3 + 6Y - 3X
1,000 - 6Y - 3X
1,003 - 6X
=0
=0
=0
X = 167
Then, from (3):
(3)
1,000 - 3(167) - 6Y
6Y
=0
= 499
Y = 83
Production Analysis and Compensation Policy
198
And, solving for using (1):
(1)
3 + 83 - 3 = 0
3 = 86
= 28.7
Part A above showed that X and Y should be combined in the ratio
X
? 2Y + 1
=
and
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 in Part B. This implies that the marginal
cost of one output unit is
MC =
D.
TC 1
= = $0.035 (or 3.5 cents per unit)
Q
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 + (14,777 - 3X - 5Y - XY)
(1)
(2)
(3)
LTC/ X = 3 - 3 - Y
LTC/ Y = 6 - 5 - X
LTC/ = 14,777 - 3X - 5Y - XY
To solve, take 2 times (1) minus (2):
2 (1)
minus (2)
6 - 6 - 2Y = 0
-(6 - 5 - X = 0)
- + X - 2Y = 0
=0
=0
=0
199
Chapter 7
(- 1 + X - 2Y) = 0
In order for (-1 + X - 2Y) = 0, either or (-1 + X - 2Y) must equal zero. Because
costs are an increasing function of output, 0, and therefore (-1 + X - 2Y) = 0.
This implies
X = 2Y + 1 or Y = (X - 1)/2
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
=
- b b2 - 4 ac
2a
=
12 144 - 4(-2)(14,774)
-4
=
Y
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
Production Analysis and Compensation Policy
200
= 2(83) + 1
= 167
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, , in this problem measures the marginal cost of
producing one additional unit of output.
3 - 3 - Y = 0
3 - 3 - (83)
=0
86
=3
= 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 Q = 1/ = 28.7 units, which was measured by in
part B.)