### A.7 Choosing Inputs

```Readings
Baye 6th edition or 7th edition, Chapter 5
BA 445 Lesson A.7 Choosing Inputs
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Overview
Overview
BA 445 Lesson A.7 Choosing Inputs
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Overview
Profit Maximization through choosing each input into production balances the
value of the marginal product of each input with the marginal cost of each input.
Cost Minimization balances the marginal product per dollar invested into an input
across all inputs. Cost minimization makes the most of the resources devoted to
charity.
Cost Measures including total cost and marginal cost are alternatives to
productivity measures to help firms chose output to maximize profit. They also
determine when it is best to shut down production.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Profit Maximization
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Overview
Profit Maximization through choosing each input into
production balances the value of the marginal product of
each input with the marginal cost of each input.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Production Functions allow detailed analysis
• Q = F(K,L) covers the simplest case of a single output
and two inputs.
 Q is quantity of output produced.
 F is a function relating two inputs to a single output.
 L is labor input.
• Labor is rented and not consumed (except for gladiators).

K is capital input.
• Capital is everything that is not labor.
• Capital can be disaggregated to, say, machines and land,
making the function Q = F(K1,K2,L).
• “Capital” is often rented, like machines or land.
• F(K,L) is the maximum amount of output that can be
produced with K units of capital and L units of labor.

Maximum output implies a type of efficient production.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Short-Run vs. Long-Run Decisions
• Some capital or labor is fixed in the short-run.
 Office space (capital) may not be completely
adjustable in the short-run.
 Labor with contracts is not completely adjustable in
the short-run.
• Unless stated otherwise in a homework or exam
problem, for the rest of the course just consider an
extreme division of the short-run from the long-run:
 labor is variable in both the short-run and long-run.
 capital is fixed in the short-run but variable in the
long-run.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Total Product: measures maximum output produced.
• Example: Cobb-Douglas Production Function:
Q = F(K,L) = K.5 L.5
 K is fixed at 16 units in the short-run.
 Short-run Cobb-Douglass production function:
Q = (16).5 L.5 = 4 L.5
 Total Product when 100 units of labor are used?
Q = 4 (100).5 = 4(10) = 40 units
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Average Product of an Input: measures output
produced per unit of input. (It is a common productivity
measure, but it is less useful than marginal product.)
• Average Product of Labor: APL = Q/L.
 Measures the output of an “average” worker.
.5 .5
 Example: Q = F(K,L) = K L
• If the inputs are K = 16 and L = 16, then the
average product of labor is APL = [(16) 0.5(16)0.5]/16
= 1.
• Average Product of Capital: APK = Q/K.
 Measures the output of an “average” unit of capital.
.5 .5
 Example: Q = F(K,L) = K L
• If the inputs are K = 16 and L = 16, then the
average product of capital is APK =
[(16)0.5(16)0.5]/16 = 1.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Marginal Product on an Input: measures the change
in total output caused by the last unit of an input.
• Marginal Product of Labor: MPL = DQ/DL
 Measures output produced by the last worker.
.5 .5
 Example: Q = F(K,L) = K L
• At inputs K = 16 and L = 16, the marginal product
of labor is MPL = .5(16) -0.5(16)0.5 = .5
• Marginal Product of Capital: MPK = DQ/DK
 Measures output produced by the last unit of capital.
• Marginal Products are the essential measures of input
productivity for profit maximization and cost
minimization.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Negative Marginal Productivity should obviously be
avoided. Optimal labor use is also short of that which
maximizes output since labor has positive cost (even if
it is volunteer labor).
Q
Increasing
Marginal
Productivity
Diminishing
Marginal
Productivity
Negative
Marginal
Productivity
Q=F(K,L)
MP
BA 445 Lesson A.7 Choosing Inputs
AP
L
11
Profit Maximization
Guiding the Production Process requires two areas
of management:
• Use the best technology so that, for any input levels
(K,L) of capital and labor, the firm produces maximal
output, F(K,L).
• Employ the right level of inputs.
BA 445 Lesson A.7 Choosing Inputs
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Profit Maximization
Employing the right level of inputs is simplest when
output sells at a competitive market price p and the inputs
are chosen to maximize profit. Then, a manager will vary
inputs and
• hire more or less labor (which is variable in both the short
run and long run) until the value of the marginal product
of labor VMPL = P x MPL equals the wage, VMPL = w.
– For example, if output sells for \$3 per unit and
employing an extra hour of labor produces 4 units
of output, extra labor earns \$12. So, if the wage is
less than \$12, you should hire extra labor.
– Profits are maximized when VMPL = w.
• hire more or less capital (in the long run) until the value of
the marginal product of capital VMPK = P x MPK equals
the rental rate, VMPK = r.
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Cost Minimization
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Overview
Cost Minimization balances the marginal product per dollar
invested into an input across all inputs. Cost minimization
makes the most of the resources devoted to charity.
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Everyone Should Minimize Cost
•
•
•
Firms in all market environments

Monopoly (1 firm with restricted entry)

Duopoly (2 firms with restricted entry)

Oligopoly (a few firms with restricted entry)

Monopolistic competition (a few firms but with free entry)

Perfect Competition (many firms with free entry)
Charities

Toys for Tots should minimize the cost of getting toys to tots
• They should often solicit cash over donated toys or labor. (Instead
of having 1000 people each buying 1 toy for \$20, people could each
donate \$20 and Toys for Tots could buy more than 1000 toys for
\$20,000 by getting a quantity discount.)
Donors

Perfectly altruistic people focused on helping others (not on their own
feelings of satisfaction) should donate money rather than toys or labor,
and so minimize their own cost of giving. (See
http://faculty.pepperdine.edu/jburke2/giving.pdf)
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Employing the right level of inputs is complex when
either firms are in non-competitive markets or charities are
not choosing inputs to maximize profit. In those cases,
guiding the production process requires
1) choosing the right level of output. (For this lesson, we
leave aside that decision.)
2) choosing the right level of inputs that minimize the cost
of producing the chosen output.
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Cost Minimization
To minimize the cost of producing output, the marginal
product per dollar spent should equal for all inputs:
MPL/w = MPK/r
• Example:







Suppose MPL = 12 (an extra unit of labor yields 12 more units of output)
Suppose w = \$4 (an extra dollar to labor can hire 1/w = 1/4 more hours of labor)
Suppose MPK = 15 (an extra unit of labor yields 12 more units of output)
Suppose r = \$3 (an extra dollar to capital can hire 1/r = 1/3 more hours of capital)
Then, MPL/w = 3 (an extra dollar to labor yields 3 more units of output)
Then, MPK/r = 5 (an extra dollar to capital yields 5 more units of output)
Therefore, you can produce the same output at lower cost by hiring less labor
and more capital. That cost savings continues until you restore the costminimization equality
MPL/w = MPK/r
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
The Input Substitution Effect measures changes in the
quantity demanded of inputs caused by changes in input
rental rates. The effect holds output constant and so is not
a complete analysis.
Consider the input substitution effect of an increase in the
rental rate of input X:
 There is a decrease in the quantity demanded of Input X
 There is an increase in the demand for Input Y if the two
inputs are substitutes (like Wood and Labor in building
houses).
 If there are only two inputs (that is, if the production
process uses only two inputs, like Capital and Labor),
then they must be substitutes.
 There is a decrease for Input Y if the two are
complements (like Manuel Labor and Shovels).
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Example 1: Compute the effects of a U.S. crackdown
on illegal Mexican labor
Step 1: There are various ways to crack down on illegal labor:
• decrease the supply of illegal labor able to cross the border.
• decrease the supply of illegal labor wanting to cross the border by
increasing penalties if they are caught.
• decrease the demand for illegal labor by increasing penalties if
they are caught.
Any one of those crackdowns increases the rental rate of illegal labor.
Step 2: The input substitution effect increases the demand for input
substitutes (like unskilled U.S. labor) and decreases the demand for
input complements (like shovels and interpreters). Since the input
substitution effect holds output constant, this is not a complete analysis.
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization
Example 2: Compute the effects of creating jobs for
handicapped workers. Is it good to create such jobs?
• In general, the input substitution effect change in the
rental rate of an input X is a change in the quantity of input
X demanded in the opposite direction.
• In particular, one way to create jobs for handicapped
workers is to decrease the cost of hiring handicapped
workers. That decreased cost increases the quantity
demanded for handicapped workers. But is that a good
idea?
• Consider a specific example:
BA 445 Lesson A.7 Choosing Inputs
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Cost Minimization







Suppose handicapped Mr. H values his time at \$8 per
hour.
Suppose In-N-Out and all other fast-food suppliers value
Mr. H’s time at \$4 per hour for up to 40 hours per week.
(The value of Mr. H’s marginal product of labor is \$4.)
Will Mr. H have a job with In-N-Out?
Now suppose the government subsidizes Mr. H’s
employment \$6 per hour. Will Mr. H now have a job with
In-N-Out?
How much does that \$6-per-hour subsidy cost of
government each week?
Can you reallocate that subsidy money to make Mr. H
happier?
Was the subsidy a good idea?
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
Cost Measures
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
Overview
Cost Measures including total cost and marginal cost are
alternatives to productivity measures to help firms chose
output to maximize profit. They also determine when it is
best to shut down production.
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
Preview
• Cost functions are an alternative to production functions
in formulating production technology. They have some
good and bad features.
• Bad: They do not contain as much information as
production functions.
• Good: Cost functions contain enough information to
help determine profit-maximizing output under
various market conditions (monopoly, oligopoly, …)
discussed later.
• Good: Cost functions help accounting for cost,
revenue, and profit.
BA 445 Lesson A.7 Choosing Inputs
25
Cost Measures
Short-Run vs. Long-Run Decisions
• For the rest of this lesson, just consider the extreme
division of the short-run and the long-run:
 labor is variable in both the short-run and long-run.
 capital is fixed in the short-run but variable in the
long-run.
BA 445 Lesson A.7 Choosing Inputs
26
Cost Measures
Types of Costs
• Short-Run

Fixed costs (FC) is a fixed amount owed for any positive output.
• Since capital is fixed in the short-run, FC equals the cost of
capital.

Sunk costs is the part of fixed cost that is owed even if output is
zero.
• For example, suppose you lease a railroad car for \$10,000
for a month, but can recoup \$6,000 if your output is zero.
• Then, FC = \$10,000, and sunk cost = \$4,000.

Short-run total costs (C or TC)
• Short-run variable costs (VC), defined by VC = C – FC.
• Long-Run

Define FC, Sunk Cost, TC, and VC as in the short run.

Since all capital is variable, Sunk Cost = 0.
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
Graphing Short Run Costs
C(Q): Minimum total cost of
producing alternative levels of
output:
C(Q) = VC(Q) + FC
\$
C(Q) = VC + FC
VC(Q)
VC(Q): Costs that vary with
output.
FC: Costs that do not vary
with output, as long as output
is positive.
Sunk Cost: Cost if output is
zero.
FC
Sunk Cost
0
BA 445 Lesson A.7 Choosing Inputs
Q
28
Cost Measures
Simplifying Assumption
In principle, the entire cost
function C(Q) could be
different between the short
run and the long run. But to
simplify the rest of the course,
assume the only difference
between the short run (SR)
and long run (LR) is C(0), the
cost of 0. And assume SR
sunk cost = fixed cost
• LR C(0) = 0
• SR C(0) = Fixed cost.
\$
Short run
C(Q) = VC + FC
VC(Q)
FC
0
BA 445 Lesson A.7 Choosing Inputs
Q
29
Cost Measures
Marginal Cost of Production: measures the change
in total cost caused by the last unit of output.
• Marginal Cost of output: MC = DC/DQ
• Marginal Cost will be the essential measure of
production cost to determine output that maximizes
profit.
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
A Graph of Marginal Cost
• Marginal cost curves often
initially decrease with output Q
because of specialization of
labor.
\$
• Example: Workers at the Malibu
subway work best when there are
several customers (Q midsize)
because the workers specialize.
• Marginal cost curves
eventually increase with output
when inputs crowd.
•Example: Workers at the Malibu
subway work become less
productive and costs increase if Q
is so large that workers are
crowded.
• Unless a homework or exam problem
explicitly states otherwise, draw the
marginal cost curves U-shaped, as in
the graph on the right.
MC
Decreasing
productivity
of inputs
because of
crowding
Specializatio
n of Labor
BA 445 Lesson A.7 Choosing Inputs
Q
31
Cost Measures
Average Total Cost of Production: measures the
total cost per unit of output.
• Average Total Cost: ATC = C/Q
• Average Total Cost will help compute profit.
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
A Graph of Average Total
Cost
When on the same graph,
marginal cost (MC) and average
total cost (ATC) relate to each
other just as the grade points
you earn on your next class
average.
• MC < ATC implies ATC
decreases (if your next
grade is lower than your
average, then your average
decreases).
• MC > ATC implies ATC is
increases
• Thus, ATC decreases until
it intersects MC.
\$
BA 445 Lesson A.7 Choosing Inputs
MC
ATC
Q
33
Cost Measures
Average Variable Cost of Production: measures the
variable cost per unit of output.
• Average Variable Cost: AVC = VC/Q
• Average Variable Cost will help compute profit.
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
A Graph of Average
Variable Cost
When on the same graph,
marginal cost (MC) and average
variable cost (AVC) relate to
each other like MC and ATC
• MC < ATC implies ATC
decreases (if your next
grade is lower than your
average, then your average
decreases).
• MC > ATC implies ATC is
increases
• Thus, ATC decreases until
it intersects MC.
\$
MC
ATC
AVC
Q
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
Summary
Average Total Cost
ATC = AVC + AFC
ATC = C(Q)/Q
\$
MC
ATC
AVC
Average Variable Cost
AVC = VC(Q)/Q
Marginal Cost
MC = DC/DQ
Q
BA 445 Lesson A.7 Choosing Inputs
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Cost Measures
Recovering Fixed Cost
Q0(ATC-AVC)
\$
= Q0 AFC
= Q0(FC/ Q0)
MC
ATC
AVC
= FC
ATC
AFC
Fixed Cost
AVC
Q0
BA 445 Lesson A.7 Choosing Inputs
Q
37
Cost Measures
Recovering Variable Cost
\$
Q0AVC
MC
ATC
= Q0[VC(Q0)/ Q0]
AVC
= VC(Q0)
AVC
Variable Cost
Minimum of AVC
Q0
BA 445 Lesson A.7 Choosing Inputs
Q
38
Cost Measures
Recovering Total Cost
Q0ATC
\$
MC
= Q0[C(Q0)/ Q0]
ATC
AVC
= C(Q0)
ATC
Minimum of ATC
Total Cost
Q0
BA 445 Lesson A.7 Choosing Inputs
Q
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Cost Measures






Total Cost: C(Q) = 10 + Q + Q2
Variable cost function:
VC(Q) = Q + Q2
Variable cost of producing 2 units:
VC(2) = 2 + (2)2 = 6
Fixed costs (all of which are sunk):
FC = 10
Marginal cost function:
MC(Q) = 1 + 2Q
Marginal cost of producing 2 units:
MC(2) = 1 + 2(2) = 5
BA 445 Lesson A.7 Choosing Inputs
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Summary
Summary
BA 445 Lesson A.7 Choosing Inputs
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Summary
Summary of Choosing the Right Level of Inputs
1) For managers in perfectly-competitive, for-profit
industries, to maximize profits (including minimize
costs), must use inputs such that the value of marginal
of each input equals the price the firm must pay to
employ the input.
• For example, w = VMPL = P x MPL
2) For any industry and for charities, to minimize cost of
producing any level of output, the marginal product per
dollar spent should equal for all inputs:
• For example, MPL/w = MPK/r
 Note: Equation 2) does not determine the level of output, and
Equation 1) does determine output but only in the special case of
perfectly-competitive, for-profit industries.
BA 445 Lesson A.7 Choosing Inputs
42
Review Questions
Review Questions
 You should try to answer some of the review
questions (see the online syllabus) before the next
class.
 You will not turn in your answers, but students may
request to discuss their answers to begin the next class.
 Your upcoming Exam 1 and cumulative Final Exam
will contain some similar questions, so you should
eventually consider every review question before taking