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Unformatted text preview: CHAPTER 1 9 PROFIT
MAXIMIZATION In the last chapter we discussed ways to describe the technological choices
facing the ﬁrm. In this chapter we describe a model of how the ﬁrm chooses
the amount to produce and the method of production to employ. The
model we will use is the model of proﬁt maximization: the ﬁrm chooses a
production plan so as to maximize its proﬁts. In this chapter we will assume that the ﬁrm faces ﬁxed prices for its in—
puts and outputs. We said earlier that economists call a market where the
individual producers take the prices as outside their control a competitive
market. So in this chapter we want to study the proﬁt—maximization prob
lem of a ﬁrm that faces competitive markets for the factors of production
it uses and the output goods it produces. 19.1 Profits Proﬁts are deﬁned as revenues minus cost. Suppose that the ﬁrm produces
n outputs (yl, . . . , y“) and uses m inputs (x1, . . . ,mm). Let the prices of the
output goods be (131, . . . ,pn) and the prices of the inputs be (101, . . . ,wm). PROFITS 335 The proﬁts the ﬁrm receives, 7r, can be expressed as n m
7T = E piyi — 2 “(Utah‘
i=1 i=1 The ﬁrst term is revenue, and the second term is cost. In the expression for cost we should be sure to include all of the factors
of production used by the ﬁrm, valued at their market price. Usually this
is pretty obvious, but in cases where the ﬁrm is owned and operated by the
same individual, it is possible to forget about some of the factors. For example, if an individual works in his own ﬁrm, then his labor is an
input and it should be counted as part of the costs. His wage rate is simply
the market price of his laboruwhat he would be getting if he sold his labor
on the open market. Similarly, if a farmer owns some land and uses it in
his production, that land should be valued at its market value for purposes
of computing the economic costs. We have seen that economic costs like these are often referred to as op
portunity costs. The name comes from the idea that if you are using
your labor, for example, in one application, you forgo the opportunity of
employing it elsewhere. Therefore those lost wages are part of the cost of
production. Similarly with the land example: the farmer has the oppor—
tunity of renting his land to someone else, but he chooses to forgo that
rental income in favor of renting it to himself. The lost rents are part of
the opportunity cost of his production. The economic deﬁnition of proﬁt requires that we value all inputs and
outputs at their opportunity cost. Proﬁts as determined by accountants do
not necessarily accurately measure economic proﬁts, as they typically use
historical costsewhat a factor was purchased for originallyvrather than
economic costswwhat a factor would cost if purchased now. There are
many variations on the use of the term “proﬁt,” but we will always stick
to the economic deﬁnition. Another confusion that sometimes arises is due to getting time scales
mixed up. We usually think of the factor inputs as being measured in
terms of flows. So many labor hours per week and so many machine hours
per week will produce so much output per week. Then the factor prices will
be measured in units appropriate for the purchase of such ﬂows. Wages are
naturally expressed in terms of dollars per hour. The analog for machines
would be the rental rate—the rate at which you can rent a machine for
the given time period. In many cases there isn’t a very welldeveloped market for the rental of
machines, since ﬁrms will typically buy their capital equipment. In this
case, we have to compute the implicit rental rate by seeing how much it
would cost to buy a machine at the beginning of the period and sell it at
the end of the period. 336 PROFIT MAXIMIZATION (Ch. 19) 19.2 The Organization of Firms In a capitalist economy, ﬁrms are owned by individuals. Firms are only
legal entities; ultimately it is the owners of ﬁrms who are responsible for
the behavior of the ﬁrm, and it is the owners who reap the rewards or pay
the costs of that behavior. Generally speaking, ﬁrms can be organized as proprietorships, partner
ships, or corporations. A proprietorship is a ﬁrm that is owned by a
single individual. A partnership is owned by two or more individuals. A
corporation is usually owned by several individuals as well, but under the
law has an existence separate from that of its owners. Thus a partnership
will last only as long as both partners are alive and agree to maintain its
existence. A corporation can last longer than the lifetimes of any of its
owners. For this reason, most large ﬁrms are organized as corporations. The owners of each of these different types of ﬁrms may have different
goals with respect to managing the operation of the ﬁrm. In a proprietor
ship or a partnership the owners of the ﬁrm usually take a direct role in
actually managing the day—to—day operations of the ﬁrm, so they are in a
position to carry out whatever objectives they have in operating the ﬁrm.
Typically, the owners would be interested in maximizing the proﬁts of their
ﬁrm, but, if they have nonproﬁt goals, they can certainly indulge in these
goals instead. In a corporation, the owners of the corporation are often distinct from
the managers of the corporation. Thus there is a separation of ownership
and control. The owners of the corporation must deﬁne an objective for
the managers to follow in their running of the ﬁrm, and then do their
best to see that they actually pursue the goals the owners have in mind.
Again, proﬁt maximization is a common goal. As we’ll see below, this goal,
properly interpreted, is likely to lead the managers of the ﬁrm to choose
actions that are in the interests of the owners of the ﬁrm. 19.3 Profits and Stock Market Value Often the production process that a ﬁrm uses goes on for many periods.
Inputs put in place at time t pay off with a whole ﬂow of services at later
times. For example, a factory building erected by a ﬁrm could last for 50
or 100 years. In this case an input at one point in time helps to produce
output at other times in the future. In this case we have to value a ﬂow of costs and a ﬂow of revenues over
time. As we’ve seen in Chapter 10, the appropriate way to do this is to
use the concept of present value. When people can borrow and lend in
ﬁnancial markets, the interest rate can be used to deﬁne a natural price
of consumption at different times. Firms have access to the same sorts of PROFITS AND STOCK MARKET VALUE 337 ﬁnancial markets, and the interest rate can be used to value investment
decisions in exactly the same way. Consider a world of perfect certainty where a ﬁrm’s ﬂow of future proﬁts
is publicly known. Then the present value of those proﬁts would be the
present value of the ﬁrm. It would be how much someone would be
willing to pay to purchase the ﬁrm. ' As we indicated above, most large ﬁrms are organized as corporations,
which means that they are jointly owned by a number of individuals. The
corporation issues stock certiﬁcates to represent ownership of shares in the
corporation. At certain times the corporation issues dividends on these
shares, which represent a share of the proﬁts of the ﬁrm. The shares of
ownership in the corporation are bought and sold in the stock market.
The price of a share represents the present value of the stream of dividends
that people expect to receive from the corporation. The total stock market
value of a ﬁrm represents the present value of the stream of proﬁts that the
ﬁrm is expected to generate. Thus the objective of the ﬁrm—maximizing
the present value of the stream of proﬁts the ﬁrm generates—could also
be described as the goal of maximizing stock market value. In a world of
certainty, these two goals are the same thing. The owners of the ﬁrm will generally want the ﬁrm to choose production
plans that maximize the stock market value of the ﬁrm, since that will make
the value of the shares they hold as large as possible. We saw in Chapter
10 that whatever an individual’s tastes for consumption at different times,
he or she will always prefer an endowment with a higher present value to
one with a lower present value. By maximizing stock market value, a ﬁrm
makes its shareholders’ budget sets as large as possible, and thereby acts
in the best interests of all of its shareholders. If there is uncertainty about a ﬁrm’s stream of proﬁts, then instructing
managers to maximize proﬁts has no meaning. Should they maximize ex
pected proﬁts? Should they maximize the expected utility of proﬁts? What
attitude toward risky investments should the managers have? It is difﬁ—
cult to assign a meaning to proﬁt maximization when there is uncertainty
present. However, in a world of uncertainty, maximizing stock market value
still has meaning. If the managers of a ﬁrm attempt to make the value of
the ﬁrm’s shares as large as possible then they make the ﬁrm’s owners—the
shareholdersias well—off as possible. Thus maximizing stock market value
gives a well—deﬁned objective function to the ﬁrm in nearly all economic
environments. Despite these remarks about time and uncertainty, we will generally limit
ourselves to the examination of much simpler proﬁt—maximization prob—
lems, namely, those in which there is a single, certain output and a single
period of time. This simple story still generates signiﬁcant insights and
builds the proper intuition to study more general models of ﬁrm behavior.
Most of the ideas that we will examine carry over in a natural way to these
more general models. 338 PROFIT MAXIMIZATION (Ch. 19) 19.4 The Boundaries of the Firm One question that constantly confronts managers of ﬁrms is whether to
“make or buy.” That is, should a ﬁrm make something internally or buy it
from an external supplier? The question is broader than it sounds, as it can
refer not only to physical goods, but also services of one sort or another.
Indeed, in the broadest interpretation, “make or buy” applies to almost
every decision a ﬁrm makes. Should a company provide its own cafeteria? Janitorial services? Pho
tocopying services? Travel assistance? Obviously, many factors enter into
such decisions. One important consideration is size. A small mom—and—pop
video store with 12 employees is probably not going to provide a cafeteria.
But it might outsource janitorial services, depending on cost, capabilities,
and staffing. Even a large organization, which could easily afford to operate food ser—
vices, may or may not choose to do so, depending on availability of alter
natives. Employees of an organization located in a big city have access to
many places to eat; if the organization is located in a remote area, choices
may be fewer. One critical issue is whether the goods or services in question are exter
nally provided by a monopoly or by a competitive market. By and large,
managers prefer to buy goods and services on a competitive market, if they
are available. The secondbest choice is dealing with an internal monop—
olist. The worse choice of all, in terms of price and quality of service, is
dealing with an external monopolist. Think about photocopying services. The ideal situation is to have dozens
of competitive providers vying for your business; that way you will get
cheap prices and highquality service. If your school is large, or in an urban
area, there may be many photocopying services vying for your business. On
the other hand, small rural schools may have less choice and often higher
prices. The same is true of businesses. A highly competitive environment gives
lots of choices to users. By comparison, an internal photocopying division
may be less attractive. Even if prices are low, the service could be sluggish.
But the least attractive option is surely to have to submit to a single
external provider: An internal monopoly provider may have bad service,
but at least the money stays inside the ﬁrm. As technology changes, what is typically inside the ﬁrm changes. Forty
years ago, rms managed many services themselves. Now they tend to
outsource as much as possible. Food service, photocopying service, and
janitorial services are often provided by external organizations that spe—
cialize in such activities. Such specialization often allows these companies
to provide higher quality and less expensive services to the organizations
that use their services. SHORTRUN PROFIT MAXIMIZATION 339 19.5 Fixed and Variable Factors [n a given time period, it may be very difficult to adjust some of the inputs.
Typically a ﬁrm may have contractual obligations to employ certain inputs
at certain levels. An example of this would be a lease on a building, where
the ﬁrm is legally obligated to purchase a certain amount of space over the
period under examination. We refer to a factor of production that is in
a fixed amount for the ﬁrm as a ﬁxed factor. If a factor can be used in
different amounts, we refer to it as a variable factor. As we saw in Chapter 18, the short run is deﬁned as that period of time
in which there are some ﬁxed factors—factors that can only be used in
ﬁxed amounts. In the long run, on the other hand, the ﬁrm is free to vary
all of the factors of production: all factors are variable factors. There is no rigid boundary between the short run and the long run. The
exact time period involved depends on the problem under examination.
The important thing is that some of the factors of production are ﬁxed in
the short run and variable in the long run. Since all factors are variable in
the long run, a ﬁrm is always free to decide to use zero inputs and produce
zero output—that is, to go out of business. Thus the least proﬁts a ﬁrm
can make in the long run are zero proﬁts. In the short run, the ﬁrm is obligated to employ some factors, even if it
decides to produce zero output. Therefore it is perfectly possible that the
ﬁrm could make negative proﬁts in the short run. By deﬁnition, ﬁxed factors are factors of production that must be paid
for even if the ﬁrm decides to produce zero output: if a ﬁrm has a long—
term lease on a building, it must make its lease payments each period
whether or not it decides to produce anything that period. But there is
another category of factors that only need to be paid for if the ﬁrm decides
to produce a positive amount of output. One example is electricity used
for lighting. If the ﬁrm produces zero output, it doesn’t have to provide
any lighting; but if it produces any positive amount of output, it has to
purchase a ﬁxed amount of electricity to use for lighting. Factors such as these are called quasiﬁxed factors. They are factors of
production that must be used in a ﬁxed amount, independent of the output
of the ﬁrm, as long as the output is positive. The distinction between
ﬁxed factors and quasiﬁxed factors is sometimes useful in analyzing the
economic behavior of the ﬁrm. 19.6 ShortRun Profit Maximization Let’s consider the shortrun proﬁtmaximization problem when input 2 is
ﬁxed at some level 52. Let f ($1,562) be the production function for the
ﬁrm, let p be the price of output, and let 1111 and 1112 be the prices of the 340 PROFIT MAXIMIZATION (Ch. 19) two inputs. Then the proﬁtmaximization problem facing the ﬁrm can be written as HEX Pf($1752) “ 1111331" w252 The condition for the optimal choice of factor 1 is not difficult to determine.
If x: is the proﬁtmaximizing choice of factor 1, then the output price times the marginal product of factor 1 should equal the price of factor 1. In symbols, pMP1(a:’{,'372): wl. In other words, the value of the marginal product of a factor should equal
its price. In order to understand this rule, think about the decision to employ a
little more of factor 1. As you add a little more of it, Am, you produce
Ay : M PIAzrl more output that is worth pMPlAacl. But this marginal
output costs wlAccl to produce. If the value of marginal product exceeds
its cost, then proﬁts can be increased by increasing input 1. If the value
of marginal product is less than its cost, then proﬁts can be increased by
decreasing the level of input 1. If the proﬁts of the ﬁrm are as large as possible, then proﬁts should
not increase when we increase or decrease input 1. This means that at a
proﬁt—maximizing choice of inputs and outputs, the value of the marginal
product, pMPﬁwLTg), should equal the factor price, wl. We can derive the same condition graphically. Consider Figure 19.1. The
curved line represents the production function holding factor 2 ﬁxed at T2.
Using 3; to denote the output of the ﬁrm, proﬁts are given by 7r : py — 101231 — 1112552. This expression can be solved for y to express output as a function of $1: 7T ’11} w
y 2 — + in, + ix}. (19.1)
p p p This equation describes isoproﬁt lines. These are just all combinations
of the input goods and the output good that give a constant level of proﬁt,
11'. As 7r varies we get a family of parallel straight lines each with a slope of
101 / p and each having a vertical intercept of 7r/p+ wgfg / p, which measures
the proﬁts plus the ﬁxed costs of the ﬁrm. The ﬁxed costs are ﬁxed, so the only thing that really varies as we move
from one isoproﬁt line to another is the level of proﬁts. Thus higher levels of
proﬁt will be associated with isoproﬁt lines with higher vertical intercepts. The proﬁtmaximization problem is then to ﬁnd the point on the produc—
tion function that has the highest associated isoproﬁt line. Such a point
is illustrated in Figure 19.1. As usual it is characterized by a tangency
condition: the slope of the production function should equal the slope of COMPARATIVE STATICS 341 OUTPUT isoproﬁt lines slope = w1/p
fix}; 532) '
production
a function
Y
r 12;;
p + P x: ' ' " x, Proﬁt maximization. The ﬁrm chooses. the input and output
combination that lies on the highest inoproﬁt line. In this case
the proﬁtmaximizing point is (cf, 3;“). the isoproﬁt line. Since the slope of the production function is the marginal
product, and the slope of the isoproﬁt line is ml /p, this condition can also
be written as MP1 2 3‘1, P which is equivalent to the condition we derived above. 19.7 Comparative Statics We can use the geometry depicted in Figure 19.1 to analyze how a ﬁrm’s
choice of inputs and outputs varies as the prices of inputs and outputs
vary. This gives us one way to analyze the comparative statics of ﬁrm
behavior. For example: how does the optimal choice of factor 1 vary as we vary its
factor price w]? Referring to equation (19.1), which deﬁnes the isoproﬁt
line, we see that increasing in} will make the isoproﬁt line steeper, as shown
in Figure 19.2A. When the isoproﬁt line is steeper, the tangency must occur
further to the left. Thus the optimal level of factor 1 must decrease. This
simply means that as the price of factor 1 increases, the demand for factor 1
must decrease: factor demand curves must slope downward. Similarly, if the output price decreases the isoproﬁt line must become
steeper, as shown in Figure 19.2B. By the same argument as given in the 342 PROFIT MAXIMIZATION (Ch. 19) “Fm—mm f(x1) Low p High p x1 x1
A B Comparative statics. Panel A shows that increasing w; will
reduce the demand for factor 1. Panel B shows that increasing
the price of output will increase the demand for factor 1 and
therefore increase the supply of output. last paragraph the proﬁtmaximizing choice of factor 1 will decrease. If the
amount of factor 1 decreases and the level of factor 2 is ﬁxed in the short
run by assumption, then the supply of output must decrease. This gives us
another comparative statics result: a reduction in the output price must
decrease the supply of output. In other words, the supply function must
slope upwards. Finally, we can ask what will happen if the price of factor 2 changes?
Because this is a shortrun analysis, changing the price of factor 2 will not
change the ﬁrm’s choice of factor 2Ain the short run, the level of factor 2
is ﬁxed at E2. Changing the price of factor 2 has no effect on the slope of
the isoproﬁt line. Thus the optimal choice of factor 1 will not change, nor
will the supply of output. All that changes are the proﬁts that the ﬁrm
makes. 19.8 Profit Maximization in the long Run In the long run the ﬁrm is free to choose the level of all inputs. Thus the
long—run proﬁtmaximization problem can be posed as max Pf($1a$2) — 1111561 — 11/2372
931,352 This is basically the same as the short—run problem described above, but
now both factors are free to vary. INVERSE FACTOR DEMAND CURVES 343 The condition describing the optimal choices is essentially the same as
before, but now we have to apply it to each factor. Before we saw that
the value of the marginal product of factor 1 must be equal to its price,
whatever the level of factor 2. The same sort of condition must now hold for each factor choice:
pMP1(mi,w§) = wi pMP2($i,$3) =w2. If the ﬁrm has made the optimal choices of factors 1 and 2, the value of
the marginal product of each factor should equal its price. At the optimal
choice, the firm’s proﬁts cannot increase by changing the level of either
input. The argument is the same as used for the shortwrun proﬁt—maximizing
decisions. If the value of the marginal product of factor 1, for example,
exceeded the price of factor 1, then using a little more of factor 1 would
produce M P1 more output, which would sell for pM P1 dollars. If the value
of this output exceeds the cost of the factor used to produce it, it clearly
pays to expand the use of this factor. These two conditions give us two equations in two unknowns, 933‘ and 30;.
If we know how the marginal products behave as a function of 361 and 302,
we will be able to solve for the optimal choice of each factor as a function
of the prices. The resulting equations are known as the factor demand
curves. 19.9 Inverse Factor Demand Curves The factor demand curves of a ﬁrm measure the relationship between
the price of a factor and the proﬁtmaximizing choice of that factor. We saw
above how to ﬁnd the proﬁtmaximizing choices: for any prices, (p,w1,w2),
we just ﬁnd those factor demands, (22333;), such that the value of the
marginal product of each factor equals its price. The inverse factor demand curve measures the same relationship,
but from a different point of View. It measures what the factor prices must
be for some given quantity of inputs to be demanded. Given the optimal
choice of factor 2, we can draw the relationship between the optimal choice
of factor 1 and its price in a diagram like that depicted in Figure 19.3. This
is simply a graph of the equation PMP1($1,$§) = w1 This curve will be downward sloping by the assumption of diminishing
marginal product. For any level of 271, this curve depicts what the factor
price must be in order to induce the ﬁrm to demand that level of $1, holding
factor 2 ﬁxed at 344 PROFIT MAXIMIZATION (Ch. 19) pMP (x , x“) = price A: marginal
' 1 1 2 productof good 1 X1 The inverse factor. demand'curve. This measures what the _
price of factor l must'be to get .931 unitssd'emanded if’the level
.of the other factor is held ﬁxed at £233. 19.10 Profit Maximization and Returns to Scale There is an important relationship between competitive proﬁt maximiza
tion and returns to scale. Suppose that a firm has chosen a long—run proﬁt
maximizing output 11* 2 f (221‘, m3), which it is producing using input levels
(a a). Then its proﬁts are given by 7r* : py" — 101331“ — 10239;. Suppose that this ﬁrm’s production function exhibits constant returns to
scale and that it is making positive proﬁts in equilibrium. Then consider
what would happen if it doubled the level of its input usage. According to
the constant returns to scale hypothesis, it would double its output level.
What would happen to proﬁts? It is not hard to see that its proﬁts would also double. But this con
tradicts the assumption that its original choice was proﬁt maximizing! We
derived this contradiction by assuming that the original proﬁt level was
positive; if the original level of proﬁts were zero there would be no prob—
lem: two times zero is still zero. This argument shows that the only reasonable longrun level of proﬁts
for a competitive ﬁrm that has constant returns to scale at all levels of
output is a zero level of proﬁts. (Of course if a ﬁrm has negative proﬁts in
the long run, it should go out of business.) REVEALED PROFITABILJTY 345 Most people ﬁnd this to be a surprising statement. Firms are out to
maximize proﬁts aren’t they? How can it be that they can only get zero
proﬁts in the long run? Think about what would happen to a ﬁrm that did try to expand indef
initely. Three things might occur. First, the ﬁrm could get so large that it
could not really operate effectively. This is just saying that the ﬁrm really
doesn’t have constant returns to scale at all levels of output. Eventually,
due to coordination problems, it might enter a region of decreasing returns
to scale. Second, the ﬁrm might get so large that it would totally dominate the
market for its product. In this case there is no reason for it to behave
competitively—to take the price of output as given. Instead, it would
make sense for such a ﬁrm to try to use its size to inﬂuence the market
price. The model of competitive proﬁt maximization would no longer be
a sensible way for the ﬁrm to behave, since it would effectively have no
competitors. We’ll investigate more appropriate models of ﬁrm behavior
in this situation when we discuss monopoly. Third, if one ﬁrm can make positive proﬁts with a constant returns to
scale technology, so can any other ﬁrm with access to the same technology.
If one ﬁrm wants to expand its output, so would other ﬁrms. But if all ﬁrms
expand their outputs, this will certainly push down the price of output and
lower the proﬁts of all the ﬁrms in the industry. 19.11 Revealed Profitability When a proﬁt~maximizing ﬁrm makes its choice of inputs and outputs
it reveals two things: ﬁrst, that the inputs and outputs used represent a
feasible production plan, and second, that these choices are more proﬁtable
than other feasible choices that the ﬁrm could have made. Let us examine
these points in more detail. Suppose that we observe two choices that the ﬁrm makes at two dif—
ferent sets of prices. At time t, it faces prices (1921031123) and makes
choices (yt, At time s, it faces prices (p3, mi, 1113) and makes choices
(gs, (ti, If the production function of the ﬁrm hasn’t changed between
times 5 and t and if the ﬁrm is a proﬁt maximizer, then we must have p‘y1t — wlmi — HIE“BE Z ptys  10533? ~ 103333 (192) and
it pays — wigs? — Z psyt — u)wa  (19.3)
That is, the proﬁts that the ﬁrm achieved facing the t period prices must be
larger than if they used the 5 period plan and vice versa. If either of these
inequalities were violated, the ﬁrm could not have been a proﬁtmaximizing
ﬁrm (with an unchanging technology). 346 PROFIT MAXIMIZATION (Ch. 19) Thus if we ever observe two time periods where these inequalities are
violated we would know that the ﬁrm was not maximizing proﬁts in at least
one of the two periods. The satisfaction of these inequalities is virtually
an axiom of proﬁtmaximizing behavior, so it might be referred to as the
Weak Axiom of Proﬁt Maximization (WAPM). If the ﬁrm’s choices satisfy WAPM, we can derive a useful comparative
statics statement about the behavior of factor demands and output supplies
when prices change. Transpose the two sides of equation (19.3) to get —psyt + wins) + U)ng 2 —psys + w‘fmi + was; (19.4)
and add equation (19.4) to equation (19.2) to get (29‘ — psly‘ e (mi  wim  (W2  103% 2 (int — if)? — (w: — mini P (w; ~— 1113);"; (19.5) Now rearrange this equation to yield
(Pt ~ p3)(yt r ys)  (wi — wini — mi)  (w; — w§)($§ r 903) Z 0 (196) Finally deﬁne the change in prices, Ap : (pIt — p5), the change in output,
Ag 2 (gt a y”), and so on to ﬁnd ApAy — AwlAml — mass 2 0. (19.7) This equation is our ﬁnal result. It says that the change in the price of
output times the change in output minus the change in each factor price
times the change in that factor must be nonnegative. This equation comes
solely from the deﬁnition of proﬁt maximization. Yet it contains all of the
comparative statics results about proﬁtmaximizing choices! For example, suppose that we consider a situation Where the price of
output changes, but the price of each factor stays constant. If Awl :
Awg = 0, then equation (19.7) reduces to ApAy 2 0. Thus if the price of output goes up, so that Ap > 0, then the change in
output must be nonnegative as well, Ag 2 0. This says that the proﬁt—
maximizing supply curve of a competitive ﬁrm must have a positive (or at
least a zero) slope.
Similarly, if the price of output and of factor 2 remain constant, equation
(19.7) becomes
“A1121 AfL‘l 2 0, which is to say
Awlel S REVEALED PROFITABILITY 347 Thus if the price of factor 1 goes up, so that Awl > 0, then equation
(19.7) implies that the demand for factor 1 will go down (or at worst stay
the same), so that A331 S 0. This means that the factor demand curve
must be a decreasing function of the factor price: factor demand curves
have a negative slope. The simple inequality in WAPM, and its implication in equation (19.7),
give us strong observable restrictions about how a ﬁrm will behave. It
is natural to ask Whether these are all of the restrictions that the model
of proﬁt maximization imposes on ﬁrm behavior. Said another way, if we
observe a ﬁrm’s choices, and these choices satisfy WAPM, can we construct
an estimate of the technology for which the observed choices are profit
maximizing choices? It turns out that the answer is yes. Figure 19.4 shows
how to construct such a technology. lsoprofit line
for period 5 lsoproﬁt line
for period t "t/Pt “s/Ps X1 Construction of a possible technology. If the observed
choices are maximal proﬁt choices at each set of prices, then we
can estimate the shape of the technology that generated those
choices by using the isoproﬁt lines. In order to illustrate the argument graphically, we suppose that there
is one input and one output. Suppose that we are given an observed
choice in period t and in period 3, which we indicate by (pt,w‘i,yt,wtl)
and (pﬂwf, yiwf). In each period we can calculate the proﬁts 7T5 and 7m
and plot all the combinations of y and 1101 that yield these proﬁts. That is, we plot the two isoproﬁt lines it t
7r1t=py—w1$1 Figure
19.5 348 PROFIT MAXIMIZATION (Ch. 19) and
173 = psy — with. The points above the isoproﬁt line for period t have higher proﬁts than
7rt at period t prices, and the points above the isoproﬁt line for period 3
have higher proﬁts than 71's at period 3 prices. WAPM requires that the
choice in period it must lie below the period 3 isoproﬁt line and that the
choice in period 3 must lie below the period t isoproﬁt line. If this condition is satisﬁed, it is not hard to generate a technology for
which (ytﬂttl) and (yﬂx‘i) are proﬁtmaximizing choices. Just take the
shaded area beneath the two lines. These are all of the choices that yield
lower proﬁts than the observed choices at both sets of prices. The proof that this technology will generate the observed choices as
proﬁt—maximizing choices is clear geometrically. At the prices (pt, w’i), the
choice (garb is on the highest isoproﬁt line possible, and the same goes
for the period 3 choice. Thus, when the observed choices satisfy WAPM, we can “reconstruct”
an estimate of a technology that might have generated the observations.
In this sense, any observed choices consistent with WAPM could be proﬁt—
maxirnizing choices. As we observe more choices that the ﬁrm makes, we get
a tighter estimate of the production function, as illustrated in Figure 19.5. This estimate of the production function can be used to forecast ﬁrm
behavior in other environments or for other uses in economic analysis. ___________________...___._—_————————— lsoprofit
lines X
Estimating the technology. As we observe more choices we get a tighter estimate of the production function. ____________________.___————————— SUMMARY 349 EXAMPLE: How Do Farmers React to Price Supports? The US. government currently spends between $40 and $60 billion a year
in aid to farmers. A large fraction of this amount is used to subsidize
the production of various products including milk, wheat, corn, soybeans,
and cotton. Occasionally, attempts are made to reduce or eliminate these
subsidies. The effect of elimination of these subsidies would be to reduce
the price of the product received by the farmers. Farmers sometimes argue that eliminating the subsidies to milk, for ex
ample, would not reduce the total supply of milk, since dairy farmers would
choose to increase their herds and their supply of milk so as to keep their
standard of living constant. If farmers are behaving so as to maximize proﬁts, this is impossible. As
we’ve seen above, the logic of proﬁt maximization requires that a decrease
in the price of an output leads to a reduction in its supply: if Ap is negative,
then Ay must be negative as well. It is certainly possible that small family farms have goals other than sim
ple maximization of proﬁts, but larger “agribusiness” farms are more likely
to be proﬁt maximizers. Thus the perverse response to the elimination of
subsidies alluded to above could only occur on a limited scale, if at all. 19.12 Cost Minimization If a ﬁrm is maximizing proﬁts and if it chooses to supply some output 3;,
then it must be minimizing the cost of producing y. If this were not so, then
there would be some cheaper way of producing y units of output, which
would mean that the ﬁrm was not maximizing proﬁts in the ﬁrst place. This simple observation turns out to be quite useful in examining ﬁrm
behavior. It turns out to be convenient to break the proﬁtmaximization
problem into two stages: ﬁrst we ﬁgure out how to minimize the costs of
producing any desired level of output 3;, then we ﬁgure out which level of
output is indeed a proﬁt—maximizing level of output. We begin this task in
the next chapter. Summary 1. Proﬁts are the difference between revenues and costs. In this deﬁnition
it is important that all costs be measured using the appropriate market
prices. 2. Fixed factors are factors whose amount is independent of the level of
output; variable factors are factors whose amount used changes as the level
of output changes. 350 PROFIT MAXIMIZATION (Ch. 19) 3. In the short run, some factors must be used in predetermined amounts.
In the long run, all factors are free to vary. 4. If the ﬁrm is maximizing proﬁts, then the value of the marginal product
of each factor that it is free to vary must equal its factor price. 5. The logic of proﬁt maximization implies that the supply function of a
competitive ﬁrm must be an increasing function of the price of output and
that each factor demand function must be a decreasing function of its price. 6. If a competitive ﬁrm exhibits constant returns to scale, then its long—run
maximum proﬁts must be zero. REVIEW QUESTIONS 1. In the short run, if the price of the ﬁxed factor is increased, what will
happen to proﬁts? 2. If a ﬁrm had everywhere increasing returns to scale, what would happen
to its proﬁts if prices remained ﬁxed and if it doubled its scale of operation? 3. If a ﬁrm had decreasing returns to scale at all levels of output and it
divided up into two equalsize smaller ﬁrms, What would happen to its
overall proﬁts’.7 4. A gardener exclaims: “For only $1 in seeds I’ve grown over $20 in pro
duce!” Besides the fact that most of the produce is in the form of zucchini,
what other observations would a cynical economist make about this situa—
tion? 5. Is maximizing a ﬁrm’s proﬁts always identical to maximizing the ﬁrm’s
stock market value? 6. If pMP1 > wl, then should the ﬁrm increase or decrease the amount of
factor 1 in order to increase proﬁts? 7. Suppose a ﬁrm is’ maximizing proﬁts in the short run with variable factor
371 and ﬁxed factor 332. If the price of 122 goes down, what happens to the
ﬁrm’s use of 351? What happens to the ﬁrm’s level of proﬁts? 8. A proﬁt—maximizing competitive ﬁrm that is making positive proﬁts
in longrun equilibrium (may/ may not) have a technology with constant
returns to scale. APPENDlX 35! APPENDIX The proﬁtmaximization problem of the ﬁrm is max pf($17 $2) — mixi — M2232,
9111552 which has ﬁrstorder conditions 6131 —_ W1 2 0
paf(m11$2) ﬁwg :0
8532 These are just the same as the marginal product conditions given in the text. Let’s see how proﬁtmaximizing behaviOI looks using the CobbDouglas produc—
tion function. Suppose the CobbDouglas function is given by f(a:1,a:2) : 3:31:65. Then the
two ﬁrst—order conditions beCOme (1—1 h _
pea/:1 $2 — wl — 0 a b—l
pbmlmg — wg = 0. Multiply the first equation by :01 and the second equation by $2 to get b pantme — wlml = 0
a b pbwlmg — wga'g = 0. Using y = $32123 to denote the level of output of this ﬁrm we can rewrite these
expressions as pay 2 1.01331 pby I ’UJQLL‘z.
Solving for $1 and 332 we have
* _ (1179
$1 —— ——'
U11
,. I)
I2 = 22
W2 This gives us the demands for the two factors as a function of the optimal output
choice. But we still have to solve for the optimal choice of output. Inserting the
optimal factor demands into the CobbDouglas production function, we have the expression
(29)” (any 2 y
W1 W2 1 Factoring out the 3; gives (webm
’u}1 1.02 352 PROFIT MAXIMIZATION (Ch. 19) Or _ pa #5 pb 1——23‘5
y  (a) (5;) '
This gives us the supply functiOn of the CobbDouglas ﬁrm. Along with the
factor demand functions derived above it gives us a complete solution to the
proﬁtmaximization problem.
Note that when the ﬁrm exhibits constant returns to scale—when a + b = 1—
this supply function is not well deﬁned. As long as the output and input prices are consistent with zero proﬁts, a ﬁrm with a. Cobb—Douglas technology is indifferent
about its level of supply. ...
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This note was uploaded on 12/04/2009 for the course OPS 111 taught by Professor English during the Spring '09 term at Open Uni..
 Spring '09
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