San Francisco State University
ECON 101
Michael Bar
Theory of the Firm: Production, Costs and Pro…t
1
Introduction
There are millions of businesses and …rms in the world and the U.S., and they are all di¤erent.
Nevertheless, there are some principles of economics, that apply to all …rms. One feature
common to all …rms, is that they all want to maximize pro…t, even non-pro…t organizations
or charity organizations. This chapter focuses on the most fundamental choices that all
businesses must make: how much output to produce and how many inputs to employ in
order to achieve maximum pro…t.
Consider a simple …rm that produces a single product. The pro…t of such …rm, as a
function of the …rm’s output, is given by:
(Q) = R (Q)
T C (Q) = R (Q)
TFC
T V C (Q)
where
Q - …rm’s output (quantity),
(Q) - …rm’s pro…t (as a function of output),
R (Q) - …rm’s Revenue (as a function of output),
T C (Q) - …rm’s Total Cost (as a function of output),
T F C - Total Fixed Cost (cost that are committed in advance, for certain period of
time, and do not depend of the amount of output produced),
T V C (Q) - Total Variable Cost (cost that depend on the amount of output produced).
Before we proceed, we need to explain what do we mean by cost. Economists distinguish
between explicit and implicit costs.
Explicit cost - cost paid in money
Implicit cost - opportunity cost of using the factors of production for other purposes
When calculating pro…t, economists include both types of costs.
Economic Pro…t = Revenue - Explicit cost - Implicit cost
In all the examples below, the cost will include both explicit and implicit cost, and the
resulting pro…t is economic pro…t. For example, say you invest $100,000 to start a business,
and in that year you earn $170,000 in revenue. Your accounting pro…t would be $70,000.
However, say that same year you could have earned an income of $80,000 had you been
employed. Therefore, you have an economic loss of $10,000 (170,000 - 100,000 - 80,000 =
-10,000).
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What is the optimal output of the …rm? In other words, we would like to …nd the output
level that maximizes the …rm’s pro…t. At the optimal output, a small change in output
should not change the pro…t, as illustrated in the next …gure.
Thus, the slope of the pro…t at the optimum is:
(Q)
=
Q
The expression
expression
R(Q)
Q
(Q)
Q
R (Q)
Q
reads "the change in pro…t
T C (Q)
=0
Q
as we increase Q by 1 unit". The
reads "the change in revenue as we increase Q by 1 unit", and this is
change is called marginal revenue, and denoted M R (Q). Finally, the expression T C(Q)
Q
is "the change in total cost as we increase Q by 1 unit", and this change is called marginal
cost. Therefore, at the optimum output, we must have
M R (Q ) = M C (Q )
That it, the marginal revenue is equal to the marginal cost. It is easy to see why M R (Q ) =
M C (Q ) must hold whenever the …rm maximizes it’s pro…t. Suppose that M R (Q ) >
M C (Q ). This means that producing one more unit will bring more revenue than the cost
of producing that unit, and the …rm can increase its pro…t by producing more. If instead,
M R (Q ) < M C (Q ), then the last unit produced at a higher cost than the revenue it
generated. The …rm can then increase pro…t by producing less.
Even if the …rm produces the optimal output, the …rm may still lose money. Thus, we
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also want the pro…t to be positive, in order for the …rm to stay in business.
(Q ) = R (Q ) T C (Q )
(Q )
R (Q ) T C (Q )
=
Q
Q
Q
AR (Q )
AT C (Q )
0
0
However, in the short run, the …rm is committed to pay the …xed cost, even if it does not
operate. Thus, in the short-run, the …rm will operate (stay in business) as long as
R (Q)
AR (Q )
T V C (Q)
AV C (Q )
We can see from the above discussion, that understanding the cost structure of the …rm,
is essential for its most basic decisions: how much to produce, and whether to produce at
all. The next section illustrates how the …rm’s costs are derived from the …rm’s production
process.
2
Production and Cost
We start with describing a simple short-run production of a …rm that produces a single
output. Suppose that some inputs are …xed in the short run, for example, physical capital
(machines, building) and technology, and output can be changed only by varying the labor number of workers. The next table describes a typical short-run relationship between labor
(L, …rst column) and total product of output (T P (L), second column). The third and fourth
columns calculate the associated average product per worker (AP (L) = T PL(L) ) and marginal
product (M P (L) = T PL(L) ).
L
0
1
2
3
4
5
6
7
8
9
10
TP/L ∆TP(L)/∆L
Q = TP( L ) AP( L ) MP( L )
0
5
5
5
15
7.5
10
28
9.333
13
41
10.25
13
52
10.4
11
60
10
8
65
9.286
5
67
8.375
2
68
7.556
1
68.5
6.85
0.5
3
The next …gure plots the graphs of the total product, and the associated average and
marginal products of labor.
Notice that when the number of workers is small, there is increasing marginal return to
labor (M P (L) is increasing), but after 4 workers, there is a deceasing marginal return
to labor (M P (L) is decreasing). When few workers are employed, they may help each other
perform task as a team, so additional worker may contributes to the productivity more than
the one before him. However, with …xed capital, as the number of workers increases, at
some point the contribution of additional worker to output will start decreasing (due to
congestion).
Also notice that as long as the marginal is above the average, the average is increasing,
and whenever the marginal is below the average, the average is decreasing. This is true for
any marginal and average quantities in the world. For example, if your next grade is above
your GPA, the GPA will increase, and if your next grade is below your GPA, the GPA will
decrease.
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3
Short-Run Costs
Suppose that the in the short run, some inputs are …xed. For example, the physical capital,
i.e. the machines and the …rm’s plant (building). The costs associated with …xed factors do
not vary with the level of production, and are called …xed cost, and denoted by T F C - Total
Fixed Cost. The cost of the …rm’s variable inputs are called variable cost, and denoted
T V C (Q) - Total Variable Cost. We write T V C as a function of quantity to emphasize that
variable cost vary with the level of output. In our example above, with the only variable
input being labor, the T V C is the cost of labor. In real business, the variable cost can also
include electricity or other energy, food ingredients for a restaurant, etc.
Thus, the total cost of the …rm is the sum of …xed and variable cost:
T C (Q) = T F C + T V C (Q)
In the above example, lets suppose that the …xed cost is T F C = 50, and labor is paid $10
per hour. Thus, the last 3 columns in the next table, show the cost …gures for the above
example.
L
0
1
2
3
4
5
6
7
8
9
10
Q = TP( L )
0
5
15
28
41
52
60
65
67
68
68.5
AP( L )
MP( L )
5
7.5
9.333
10.25
10.4
10
9.286
8.375
7.556
6.85
5
10
13
13
11
8
5
2
1
0.5
5
TFC
50
50
50
50
50
50
50
50
50
50
50
TVC(Q) TC(Q)
0
50
10
60
20
70
30
80
40
90
50
100
60
110
70
120
80
130
90
140
100
150
These costs are plotted in the next …gure.
3.1
Average and marginal cost
From the total costs above, we can derive the average cost per unit produced:
TFC
Q
T V C (Q)
AV C (Q) =
Q
T C (Q)
T F C + T V C (Q)
AT C (Q) =
=
= AF C (Q) + AV C (Q)
Q
Q
AF C (Q) =
Notice that the Average Fixed Cost is decreasing with the volume of production. As we
showed in the introduction, the importance of the average costs is that they determine
whether the …rm should operate or not. The marginal cost determine the optimal (pro…t
maximizing) output. The marginal cost is given by:
M C (Q) =
T C (Q)
=
Q
T V C (Q)
Q
The last step follows from the fact that …xed cost does not change as output changes. The
next table uses data from the above example, and adds 3 more columns with AT C, AV C,
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and M C.
L
0
1
2
3
4
5
6
7
8
9
10
Q = TP( L )
0
5
15
28
41
52
60
65
67
68
68.5
TP/L ∆TP(L)/∆L
AP( L ) MP( L )
5
7.5
9.333
10.25
10.4
10
9.286
8.375
7.556
6.85
5
10
13
13
11
8
5
2
1
0.5
TFC
50
50
50
50
50
50
50
50
50
50
50
TVC(Q)
0
10
20
30
40
50
60
70
80
90
100
TFC+TVC(Q)
TC(Q)
50
60
70
80
90
100
110
120
130
140
150
TC(Q)/Q
ATC(Q)
TVC(Q)/Q
AVC(Q)
∆TC(Q)/∆Q
MC(Q)
12.00
4.67
2.86
2.20
1.92
1.83
1.85
1.94
2.06
2.19
2.00
1.33
1.07
0.98
0.96
1.00
1.08
1.19
1.32
1.46
2.00
1.00
0.77
0.77
0.91
1.25
2.00
5.00
10.00
20.00
Notice once again, that when the marginal is above the average, the average goes up, and
when the marginal is below average, the average must go down. This is true for marginal
and average product, marginal and average cost, or any marginal and average quantities you
can think of.
Also notice from the last table, that marginal product of labor and marginal cost are
moving in opposite directions. When M P (L) is increasing, we have M C (Q) decreasing,
and vice versa. To see why this has to be the case, compare the output produced by workers
number 9 and 8. The 8th worker produces 2 units, and is paid $10, so each unit produced by
the 8th worker cost $5 (see the above table). The 9th worker is also paid $10, but produces
only 1 unit. So the cost of each unit produced by the 9th worker is $10.
You can use Excel to plot the graphs of AT C, AV C, and M C in the above table. The
average and marginal costs curves, typically look like in the next …gure.
In the next section we will illustrate how these curves can be used to determine the
optimal output of a competitive …rm, and determine whether the …rm operates or not (i.e.
to stay in business or not).
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4
Pro…t of Perfectly Competitive Firm
A competitive …rm takes the market price as given (a price taker). Thus, the revenue of such
a …rm is:
R (Q) = P Q
where P is the market price. The marginal and the average revenue of a competitive …rm is
simply the market price:
M R = AR = P
The pro…t of a competitive …rm is therefore:
(Q) = P Q
T C (Q)
The optimal output, if the …rm operates, is given by:
P = M C (Q )
The …rm operates in the short run if:
P
min (AV C)
P
min (AT C)
The …rm operates in the long run is:
Thus, the section of the M C curve, which is above the min (AV C), is the …rm’s short run
supply curve of the …rm. At any given price, M C gives the optimal output that the …rm
wants to sell.
The next …gure illustrates these conditions graphically.
Break-even
point
Costs, price
MC
ATC
min(ATC)
AVC
min(AVC)
Shutdown
point
Q
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In the short run, the …rm is committed to pay its T F C, and therefore will operate if the
price is above min (AV C). This way, the …rm covers all of the variable cost, and some of
…xed cost. If the …rm shuts down, it still has to pay the …xed cost. The shutdown point
indicates the min (AV C), so that if the price is below that point, the …rm will shut down,
and will not operate even in the short run.
In the long run, the …rm must make positive pro…t. Thus, the …rm will operate if the
price is above min (AT C). The point of min (AT C) is called break-even point, because if
the price is at the level of min (AT C), the …rm has zero economic pro…t (breaks even).
Example 1 Suppose the Total Fixed Cost
given in the next table.
Q
0
1
2
3
4
5
6
7
8
9
10
and Total Variable Cost of a competitive …rm are
TFC
100
100
100
100
100
100
100
100
100
100
100
TVC(Q)
0
90
170
240
300
370
450
540
650
780
930
1. (a) Find the …rm’s Total Cost, Average Total Cost, Average Variable Cost, and Mar-
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ginal Cost, and plot the graph of AT C, AV C, and M C.
Q
0
1
2
3
4
5
6
7
8
9
10
TFC
100
100
100
100
100
100
100
100
100
100
100
TFC+TVC(Q)
TVC(Q)
TC(Q)
0
100
90
190
170
270
240
340
300
400
370
470
450
550
540
640
650
750
780
880
930
1030
TC(Q)/Q
ATC(Q)
TVC(Q)/Q
AVC(Q)
∆TC(Q)/∆Q
MC(Q)
190.00
135.00
113.33
100.00
94.00
91.67
91.43
93.75
97.78
103.00
90.00
85.00
80.00
75.00
74.00
75.00
77.14
81.25
86.67
93.00
90
80
70
60
70
80
90
110
130
150
(b) What is the minimal market price at which the …rm will operate in the short run?
That is, …nd the shutdown point.
min (AV C) = 74
Thus, the …rm will shut down if the market price falls below $74 per unit.
(c) What is the minimal market price at which the …rm will operate in the long run?
That is, …nd the break-even point.
min (AT C) = $91:43
Thus, the …rm will operate in the long run, if the market price is at least $91.43 per
unit.
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(d) Find the …rm’s optimal output in the short run, and the …rm’s pro…t, for the
following market prices: $70, $74, $80, $91.43, $130.
P
70
74
80
91.43
130
Q
0
5
6
7
9
R
0
370
480
640
1170
TC
100
470
550
640
880
π(Q)
-100
-100
-70
0
290
(e) Illustrate the economic pro…t graphically, when the market price is $130.
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