### Production and costs: Cost-Minimizing input combination

```PRODUCTION AND COSTS:
COST-MINIMIZING INPUT
COMBINATION
AP Economics
Mr. Bordelon
Alternative Input Combinations
• A firm can choose among a number of alternative
combinations of inputs that will produce a given level of
output.
• Joslyn Farms
• To produce the optimal quantity of wheat, they could choose to
have a relatively capital-intensive operation by investing in tractors
and hiring little labor.
• To produce the optimal quantity of wheat, they could choose to
have a relatively labor-intensive operation by hiring more workers
to do the planting and harvesting by hand.
Alternative Input Combinations
Joslyn Farms
• To produce the optimal quantity of wheat, they could choose to
have a relatively capital-intensive operation by investing in tractors
and hiring little labor.
• To produce the optimal quantity of wheat, they could choose to
have a relatively labor-intensive operation by hiring more workers
to do the planting and harvesting by hand.
• The same amount of wheat can be produced using many
different combinations of capital and labor. Kyle and Ryan
must determine which combination of inputs will maximize
their profits.
Substitutes and Complements
• Capital and labor can serve as both.
• Substitutes. If the price of one gets to be too expensive, the
Joslyns will make a choice for the substitute to maintain a certain
level of wheat production.
• AP: Tractors for farm workers. ATMs for bank tellers.
• Complements. If more of one increases the marginal product of
the other.
• The Joslyns’ farm workers are more productive with tractors, and each
tractor requires a worker.
• Office workers are more productive when they use faster computers.
• Doctors are more productive with modern X-ray machines.
• The quantity and quality of capital available affect the marginal product
of labor, and thus the demand for labor.
Cost Minimization
• Firms determine the combination of inputs that maximize
profits by finding the input combination that costs the
least—the cost-minimizing input combination.
• Publix can alternate self-checkout stations (capital) and
cashiers (labor) to check out customers shopping at the
store.
• If Publix puts in 20 self-checkout stations, Publix will need to hire 1
cashier to monitor every 5 stations.
• Trained cashiers are faster, so Publix could check out the same
number of customers using 10 cashiers and 10 self-checkout
stations.
Cost Minimization
• Publix can alternate self-checkout stations (capital) and
cashiers (labor) to check out customers shopping at the
store.
• If Publix puts in 20 self-checkout stations, Publix will need to hire 1
cashier to monitor every 5 stations.
• Trained cashiers are faster, so Publix could check out the same
number of customers using 10 cashiers and 10 self-checkout
stations.
Capital
Labor
Rental rate =
\$1,000/month
Wage rate =
\$1,600/month
a.
20
4
b.
10
10
Cost Minimization
a.
cost of capital
cost of labor
20 x \$1,000 =
\$20,000
4 x \$1,600 =
\$6,400
TOTAL
b.
\$26,400
cost of capital
10 x \$1,000
\$10,000
cost of labor
10 x \$1,600
\$16,000
TOTAL
Publix should choose option B, the lower cost combination.
\$26,000
Cost Minimization
• When firms must choose between alternative
combinations of inputs, they evaluate the cost of each
combination and select the one that minimizes the cost of
production. This can be done by calculating the total cost
of each alternative combination of inputs.
• However, because the number of possible combinations
can be very large, it is more practical to use marginal
analysis to find the cost-minimizing level of output.
Cost-Minimization Rule
an input is the marginal product (MP) of that input. Firms
want to receive the highest possible marginal product
from each dollar spent on inputs.
• Cost-minimization rule. Firms adjust their hiring of
inputs until the MP per dollar is equal for all inputs.
• For labor and capital:
MPL
MPK
=
Wage Re ntalRate
Cost-Minimization Rule
MPL
MPK
>
Wage RentalRate
• Assume MPL is 20 and MPK is 100.
• If wage is \$10 and rental is \$100, then MPL per dollar is
20/\$10 = 2 units of output per dollar for labor and
100/\$100 = 1 unit of output per dollar for capital.
• Publix is getting more output for its money by hiring labor,
so it should hire more labor and less capital. Why?
Cost-Minimization Rule
•
•
•
•
•
MPL
MPK
>
Wage RentalRate
Assume MPL is 20 and MPK is 100.
If wage is \$10 and rental is \$100, then MPL per dollar is
20/\$10 = 2 units of output per dollar for labor and
100/\$100 = 1 unit of output per dollar for capital.
Publix is getting more output for its money by hiring labor,
so it should hire more labor and less capital.
Diminishing returns! As Publix hires more labor, MPL
decreases and as it hires less capital, MPK increases.
Publix will continue to substitute labor for capital until both
are equal.
Cost-Minimization Rule
•
•
•
•
•
MPL
MPK
<
Wage RentalRate
Assume MPL is 20 and MPK is 100.
If wage is \$10 and rental is \$25, then MPL per dollar is
20/\$10 = 2 units of output per dollar for labor and
100/\$25 = 4 unit of output per dollar for capital.
Publix is getting more output for its money by hiring
capital, so it should hire more capital and less labor.
Diminishing returns! As Publix hires more capital, MPK
decreases and as it hires less capital, MPL increases.
Publix will continue to substitute capital for labor until both
are equal.
Cost-Minimization Rule
• Similar to optimal consumption rule, which has consumers
maximize their utility by choosing the combination of
goods so that MU\$ is equal for all goods.
Review
• Firms combine inputs, like labor, capital and land, to
produce output and minimize costs.
• Construction. Carpenters use tools to build houses, but
there are different combinations of labor and capital that
will get the same house built. One man with a nail gun
could be more productive than several men with hammers
and nails, and the firm must decide if that more
expensive, but more productive, nail gun is a better
choice than several men with inexpensive hammers.
Review
• Substitutes. Two factors of production that can do
essentially the same work.
• An ATM machine dispenses cash, accepts deposits and allows you
to transfer money. The ability to perform these banking tasks
makes the machine a substitute for a bank teller.
• A Caterpillar backhoe with one driver can dig holes and ditches. A
team of men with shovels (or a team of students with spoons) can
also dig holes and ditches. These are also substitutes in
production.
• Two types of labor can also be substitutable. An American
computer programmer and a Korean programmer could do the
same work. A group of union autoworkers in Indiana could be
considered substitutable with a group of non-union workers in
Tennessee.
Review
• Complements. Two factors of production that must be
combined to produce output. The presence of one factor
increases the marginal product of the other.
• The Caterpillar backhoe and the driver are complements. Not much
digging gets done if they aren’t combined in production.
• A team of pilots and a 747 passenger jet are complements.
• An 18-wheeler and a truck driver are complements.
Review
• Firms will choose the combination of inputs that can
produce the output at the lowest cost—least-cost
combination of inputs.
• Orange City needs to dig a 100-foot drainage ditch and
the city hires Dana’s Ditch Diggers for the job. DDD has
been experimenting with two combinations of labor and
capital that can each get the ditch dug in the same
amount of time.
• Combo 1: Rented backhoe and skilled driver.
• Combo 2: 10 unskilled workers each with a shovel.
Review
Cost of Labor
Cost of
Capital
Total Cost of
Producing
100 ft of
ditches
Combo 1
1 driver =
\$500
1 backhoe =
\$2,500
\$3,000
Combo 2
10 workers = 10 shovels =
\$1,000
\$250
\$1,250
• Ditch diggers are preferred and DDD will choose Combo
2.
• What if DDD underestimated the productivity of the
backhoe and driver and discovered Combo 1 could
actually produce a drainage ditch 300 ft long in the same
amount of time as 10 men and 100 ft?
Review
Cost of Labor
Cost of
Capital
Total Cost of
Producing
300 ft of
ditches
Combo 1
1 driver =
\$500
1 backhoe =
\$2,500
\$3,000
Combo 2
10 workers = 10 shovels =
\$1,000
\$250
\$3,750
• In this case, Dana’s getting a backhoe.
• Firms need to consider the prices of labor and capital, as
well as productivity before choosing the combination of
inputs that produces output at lowest possible cost.
Review
MPL
MPK
=
Wage Re ntalRate
• Suppose DDD has hired labor to the point where MPL=50 and
•
•
•
•
capital to the point where MPK=40.
MPL/w = 50 units per dollar
MPK/r = 20 units per dollar, so we know that MPL/w > MPK/r
If the firm takes \$2 away from hiring a unit of capital, it could
hire 2 more units of labor. Total costs would remain the same.
Lost production from one less unit of capital = approximately 20
units. Gained production from two more units of labor = a little
less than 50 units. So DDD would see more production, at the
same cost.
DDD would continue to hire more labor (which causes MPL to
decline), and less capital (which causes MPK to rise), until
output can no longer rise.
Review
MPL
MPK
=
Wage Re ntalRate
• Suppose DDD has hired labor to the point where MPL=10 and
•
•
•
•
capital to the point where MPK=60.
MPL/w = 10 units per dollar
MPK/r = 30 units per dollar, so we know that MPL/w < MPK/r
If DDD takes \$2 away from hiring two units of labor, it could
hire one more units of labor. Total costs would remain the
same.
Lost production from two fewer units of labor = approximately
10 units. Gained production from one more units of capital =
about 30 units. So DDD would see more production, at the
same cost. Great deal!
DDD would continue to hire more capital (which causes MPK to
decline), and less labor (which causes MPL to rise), until output
can no longer rise.
Review
MPL
MPK
=
Wage Re ntalRate
• Anytime the marginal product per dollar is not equal, the
firm can reshuffle employment of labor and capital to
increase output while keeping costs unchanged. Once the
firm has found the least-cost combination of labor and
capital, the firm has found the combination that produces
that output at the lowest possible cost.
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