Firm Theory: production
functions, cost curves and
profit maximization
1
Remarks
 Switching
gears: “Theory of the Firm”
 Embarking on an analysis of the firm.
 Note:
– There are lots of different types of firms.
– There are lots of ways to organize
entrepreneurial activity.
– There are lots of firm objectives.
2
What We Assume
 For
our analysis we assume that:
– we have an owner manager,
– who has a business,
– with the primary and only objective to
maximize economic profit.
3
What’s a Business?





A business is an organization producing goods or
services, also called a firm.
A business, or firm, is assumed to maximize its profits.
Examples of businesses: Microsoft, Kinko’s, the Campus
Store (a business within Cornell University).
Examples of organizations that are not businesses in this
sense: Cornell University as a whole, US Department of
Defense.
Organizations that produce goods and services but are
legally prohibited from using the profit motive are excluded
from our formal model of a firm.
4
What’s a Market?
 A collection
of buyers and sellers
organized for the purpose of
exchanging goods and services for
money.
 Markets can be global, national,
regional, or local depending upon the
item being bought and sold.
5
Profit Maximization
 profit
= total revenue - total cost
 total revenue:
– determined by the level and nature of
competition in your market
 total
cost:
– costs are determined by factor market
prices and the firm’s technology or
production function
6
Economic Profits
 Economic
profits are the difference
between total revenue and total costs.
 Economic total costs include the
opportunity costs of all inputs to the
production process–in particular, the
opportunity costs of the owner’s time
and physical capital (equipment and
space).
 Whenever we talk about profit we mean
economic profit.
7
Accounting Profits
 Accounting
profits are defined as total sales
revenue (the same as total revenue in the
economic profits definition) minus operating
costs (costs of goods sold + administrative
and sales costs for those who know some
accounting).
 Accounting Profits = Sales Revenue Accounting Costs
8
Production & Cost Structures
 There
are lots of ways to describe
production and costs.
 You need to understand them all.
 For example:
– total, fixed and variable concepts
– average and marginal concepts
– long run and short run concepts
– all related to each other
9
Jonathan’s New York State
Apple Farm
 The
farm is a business
organized to grow and
sell apples.
 The owner/proprietor,
Jonathan, tries to
maximize his profits
from the business.
10
The U.S. Apple Market






Americans consume 19 lbs. of apples per person annually, for a
total consumption of 5 billion lbs.
More than 6 billion lbs. of apples are grown in the US each year
(mostly in Washington, New York and Michigan).
Only about 234 million lbs. are imported, while more than 1.3
billion lbs. are exported.
There are about 1,700 apple farms in the State of New York,
with an average of 96 acres of orchards per farm and about
$92,000/year in revenue from apple sales.
80% of the New York apple farms are individual or family
owned.
Markets like the US apple market are classic examples of
competitive product markets.
11
Production Functions




The production function shows the input
requirements for each level of production.
For some businesses the production function is
relatively simple--a few processes with little
substitution.
For some businesses the production function
involves thousands of different processes and
millions of substitution possibilities.
The production function is the economists summary
of the input requirements for each level of
production.
12
Jonathan’s Apple Farm
Production Function
Jonathan's Apple Farm Production Function
Apples
Land
Labor Proprietor's
(tons/year)
(acres)
(hired) time (hours)
0
100
0
1,100
50
100
2,500
1,100
100
100
3,700
1,100
150
100
5,000
1,100
200
100
6,800
1,100
250
100
10,000
1,100
300
100
15,000
1,100
350
100
27,000
1,100

The table describes Jonathan’s inputs for the annual
production of apples shown in the first column.
13
Fixed Factors
 A fixed
factor is one that does not vary
as the quantity produced increases or
decreases.
 Some factors are fixed in the short run
(managerial time).
 Some factors are fixed in the medium
run (cultivated acreage).
 No factors are fixed in the long run.
14
Jonathan’s Fixed Factors
 Jonathan
has two fixed factors
– His cultivated acreage (100 acres)
– His own managerial time (1,100
hours)
15
Variable Factors
 A variable
factor is one that must be
increased in order to increase output.
 The classic variable factor is labor.
 Variable factors usually exhibit
diminishing marginal productivity--the
amount of extra product generated by
each additional unit of the input, holding
other inputs constant, declines.
16
Jonathan’s Variable Factors
 Jonathan
must vary his labor input to
increase his production of apples.
 At first this variation is modest going from 50
tons/year to 100 tons/year requires an
additional 1,200 hours
 Going from 200 to 250 tons/year requires an
additional 3,200 hours.
 Jonathan cannot increase the size of his
farm, his acreage is fixed.
17
Jonathan’s Marginal Product
of Labor


The graph shows how the
marginal product of labor rises,
then falls for Jonathan’s apple
farm. Marginal product is in red.
The marginal product is the extra
amount of apple production
(pounds/hour worked) that can be
produced by an extra hour of
work.
Average product (in blue) is the
ratio of output to labor used.
Average product of labor is
usually called “labor productivity”
in the business press.
Marginal Product of Labor
0.045
0.040
Tons of
apples/hour

0.035
Marginal Product
of Labor (midpoint
formula)
0.030
0.025
0.020
Average Product
of Labor
0.015
0.010
0.005
0.000
0
5,000
10,000 15,000
Hours of Labor
18
The Average/Marginal
Relation
 For
product curves or any other
average/marginal pair of curves:
– If marginal product is above average product, then
average product is rising.
– If marginal product is below average product, then
average product is falling.
– Therefore, marginal product equals average
product when average product is at a critical
value, in this case a maximum.
19
From Production Curves to
Cost Curves
 By
combining the production function
and the factor prices, we produce the
business’s cost curves.
 The total cost curve is the amount spent
on all fixed and variable costs to
produce the indicated output.
 The average cost curve is the ratio of
total costs to units produced.
20
Marginal Cost
 The
most important cost concept is
marginal cost.
 Marginal cost measures the amount by
which costs increase as output
increases by one unit.
21
7 Short Run Cost Curves
 Total
values
– fc = fixed costs = PK K where K is fixed
– vc = variable costs = PLL*(x)
– srtc = short run total costs = fc+vc
 Average
values
– afc = average fixed cost = fc/x
– avc = average variable cost = vc/x
– sratc=short run average total cost=srtc/x=afc+avc
 Marginal
value
– srmc=short run marginal cost = srtc/x = vc/x
22
Short Run Cost Curves
 Short
run cost curves get their shape from
the marginal productivity of the variable factor
(except the fixed costs, of course).
 If capital is held constant (short run) then the
marginal product of labor gives the short run
cost curves their shape.
 The levels of cost curves are determined by
factor market prices along with technology.
23
Jonathan’s Input Prices
Prices
Labor's time
Owner's time
Rent




$8.00 per hour
$12.00 per hour
$124.00 per acre
Each of the entries in this table represents a price
that Jonathan must pay for an input.
Notice that he “pays” for his managerial time
because his next best alternative is to earn $12/hour.
He must pay rent for his land.
Jonathan is a “wage taker” as well as “price-taker.”
24
Jonathan’s Cost Curves
Jonathan's Apple Farm Costs
Apples
(tons/year)
0
50
100
150
200
250
300
350

Land
12,400
12,400
12,400
12,400
12,400
12,400
12,400
12,400
Hired
Labor
0
20,000
29,600
40,000
54,400
80,000
120,000
216,000
Proprietor's
Average
time
Total Cost
Cost
13,200
25,600
13,200
45,600
912
13,200
55,200
552
13,200
65,600
437
13,200
80,000
400
13,200
105,600
422
13,200
145,600
485
13,200
241,600
690
Marginal
Cost
(midpoint
formula)
296
200
248
400
656
1,360
When Jonathan faces the technology and input prices shown previously,
the table shows his cost structure.
25
Graph of Jonathan’s Cost
Curves


The marginal cost of
each ton of apples is
shown as the red line.
The average total
cost curve is shown
as the blue line.
Notice that the
marginal cost =
average cost at
minimum average
cost.
Jonathan's Cost Functions
$/ton

1,000
900
800
700
600
500
400
300
200
100
0
0
100
200
300
400
Apples (tons/year)
Average Cost
Marginal Cost (midpoint formula)
26
Long Run Cost Curves





Describing the long run cost curves requires a
description of all of the technological possibilities for
operating in the industry.
Now both labor and capital are considered variable.
There are three long run cost curves for the firm:
– long run total cost = lrtc = PLL*(x) + PKK*(x)
– long run average total cost = lratc = lrtc/x
– long run marginal cost = lrmc = lrtc/x
The most important one for us is the lratc curve.
Choosing L*(x) and K*(x) will depend on the prices of capital and
labor relative to the marginal products of capital and labor and
what x you expect to produce. The “bang/buck” condition would
have to be met. That is, at L* and K*, the (mpL/PL )= (mpK/PK)
27
Example: 3 Potential
Technologies





Suppose there
are three
different ways
for a company,
System-fixer, to
do business.
Firm sizes A, B
and C illustrate
the possibilities.
Firm A is small,
using only $80
in fixed costs.
Firm B uses
twice the
capital.
Firm C uses
three times the
capital.
Firm A (Capital investment A)
Quantity
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Long Run Cost Curves
Firm B (Capital Investment B)
Firm C (Capital Investment C)
Firm C Marginal
Firm B Marginal
Firm A Marginal
Cost
Average
Cost
Average
Cost
Average
Total (midpoint
Total
Total (midpoint
Total
Total (midpoint
Total
formula)
Cost
formula) Quantity Costs
Cost
formula) Quantity Costs
Cost
Costs
240
0
160
0
80
7.50
250 250.00
1
1.50
161 161.00
1
15.00
100 100.00
6.00
255 127.50
2
2.50
81.50
163
2
10.50
55.00
110
6.50
87.33
262
3
3.00
55.33
166
3
13.50
40.33
121
6.00
67.00
268
4
4.00
42.25
169
4
22.00
34.25
137
6.00
54.80
274
5
5.50
34.80
174
5
33.00
33.00
165
6.50
46.67
280
6
7.00
30.00
180
6
46.50
33.83
203
7.50
41.00
287
7
14.00
26.86
188
7
65.00
36.86
258
8.50
36.88
295
8
26.00
26.00
208
8
90.00
41.63
333
10.00
33.78
304
9
50.00
26.67
240
9
122.50
48.67
438
13.00
31.50
315
10
84.00
30.80
308
10
57.80
578
30.00
30.00
330
11
125.00
37.09
408
11
52.50
31.25
375
12
46.50
558
12
80.00
33.46
435
13
13
140.00
38.21
535
14
14
47.67
715
15
15
28
Question
 What
is the best technology for our
system-fixer firm?
29
Answer
 It
depends on how much System-fixer
expects to produce and sell in the
market.
30
The Firm’s Long Run Average
Total Cost Curve



The firm’s long run average
total cost curve consists of the
minimum of the three curves
illustrated on the right.
System-fixer’s long run average
total cost curve is size A’s (blue)
until 6 units, size B’s (red) from
6 to 10 units and size C’s
(brown) from 11 units onward.
The shape of the firm’s lratc
curve will be determined by how
the technology behaves.
The lratc is the outer envelope
of the possible sratc curves.
Long Run Average Total Cost
60.00
50.00
40.00
Cost (dollars/installation)

30.00
20.00
Firm A Average Total Cost
Minimum Average Total Cost
Firm B Average Total Cost
Minimum Average Total Cost
Firm C Average Total Cost
Minimum Average Total Cost
10.00
0.00
0
1
2
3
4
5
6
7
8
9
10
Quantity (installations/week)
11
12
13
14
15
31
The Firm’s Long Run Average
Total Cost Curve and MES





MES = minimum efficient scale
MES = the level of output at the
minimum of the firm’s lratc
curve.
When there are “economies of
scale,” the lratc is declining.
When there are “diseconomies
of scale,” the lratc is increasing.
At MES all the economies of
scale are exhausted.
In real world empirical analysis,
economists often observe a
more “L-shaped” lratc curve.
Long Run Average Total Cost
60.00
50.00
40.00
Cost (dollars/installation)

30.00
20.00
Firm A Average Total Cost
Minimum Average Total Cost
Firm B Average Total Cost
Minimum Average Total Cost
Firm C Average Total Cost
Minimum Average Total Cost
10.00
0.00
0
1
2
3
4
5
6
7
8
9
10
Quantity (installations/week)
11
12
13
14
15
32
Profit Maximization
 Profit
() = total revenue(tr) - total cost(tc).
 Profit depends on the firm’s output level (x).
 So…  (x) = tr(x) - tc(x)
 Define
– marginal revenue (mr) = tr/x
– marginal cost (mc) = tc/x
33
Profit Maximization
 General
rules for profit maximization:
 If x* maximizes  , then
– mr = mc at x*
– x* is a profit max and not a profit min
– at x* it’s worth operating
 If using a short-run perspective use short run
cost curves.
 If using a long-run perspective use
long run cost curves.
34
Running at a Profit
 When
the market price exceeds the
firm’s average total cost at x*, the firm is
running at a profit.
 Great.
35
Running at a Loss
 When
the market price is less than the
firm’s average total cost at x*, the
business is running at a loss.
 What should the firm do in the short
run?
– produce x* if the firm takes in enough in
revenue to cover its variable costs.
– shut down and play dead if its revenues at
x* don’t even cover variable costs.
36
Running at a Loss
 What
should the firm do in the long run?
– It should plan to go out of business if it
does not expect either the market price to
rise or its costs to fall or both.
37
Sunk Costs and Avoidable
Fixed Costs





In the real world the shut down rule is slightly different.
When the firm actually operates, x>0, it makes sense to
only talk about fixed and variable costs.
However, when the firm considers shutting down in the
short run (when profits are negative at the profit
maximizing output level) then they have to take another
look at their fixed costs.
Fixed costs=sunk costs + avoidable fixed costs
Now change the shutdown rule to be:
– produce x* if the firm takes in enough in revenue to cover its
variable costs + avoidable fixed costs, otherwise shut down.
38