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Unformatted text preview: CHAPTER 1 8 TECHNOLOGY In this chapter we begin our study of ﬁrm behavior. The ﬁrst thing to do is
to examine the constraints on a ﬁrm’s behavior. When a ﬁrm makes choices
it faces many constraints. These constraints are imposed by its customers,
by its competitors, and by nature. In this chapter we’re going to consider
the latter source of constraints: nature. Nature imposes the constraint that
there are only certain feasible ways to produce outputs from inputs: there
are only certain kinds of technological choices that are possible. Here we
will study how economists describe these technological constraints. If you understand consumer theory, production theory Will be very easy
since the same tools are used. In fact, production theory is much simpler
than consumption theory because the output of a production process is
generally observable, Whereas the “output” of consumption (utility) is not
directly observable. 18.1 Inputs and Outputs Inputs to production are called factors of production. Factors of produc—
tion are often classiﬁed into broad categories such as land, labor, capital, DESCRIBING TECHNOLOGICAL CONSTRAINTS 323 and raw materials. It is pretty apparent what labor, land, and raw mate—
rials mean, but capital may be a new concept. Capital goods are those
inputs to production that are themselves produced goods. Basically capital
goods are machines of one sort or another: tractors, buildings, computers,
or whatever. Sometimes capital is used to describe the money used to start up or
maintain a business. We will always use the term ﬁnancial capital for
this concept and use the term capital goods, or physical capital, for
produced factors of production. We will usually want to think of inputs and outputs as being measured
in flow units: a certain amount of labor per week and a certain number of
machine hours per week will produce a certain amount of output a week. We won’t ﬁnd it necessary to use the classiﬁcations given above very
often. Most of what we want to describe about technology can be done
without reference to the kind of inputs and outputs involvedkjust with
the amounts of inputs and outputs. 18.2 Describing I'echnological Constraints Nature imposes technological constraints on ﬁrms: only certain combi—
nations of inputs are feasible ways to produce a given amount of output,
and the ﬁrm must limit itself to technologically feasible production plans. The easiest way to describe feasible production plans is to list them.
That is, we can list all c0mbinations of inputs and outputs that are tech
nologically feasible. The set of all combinations of inputs and outputs that
comprise a technologically feasible way to produce is called a production
set. Suppose, for example, that we have only one input, measured by 3:, and
one output, measured by 3;. Then a production set might have the shape
indicated in Figure 18.1. To say that some point ($3114) is in the production
set is just to say that it is technologically possible to produce y amount
of output if you have m amount of input. The production set shows the
possible technological choices facing a ﬁrm. As long as the inputs to the ﬁrm are costly it makes sense to limit our
selves to examining the maximum possible output for a given level of input.
This is the boundary of the production set depicted in Figure 18.1. The
function describing the boundary of this set is known as the production
function. It measures the maximum possible output that you can get
from a given amount of input. Of course, the concept of a production function applies equally well if
there are several inputs. If, for example, we consider the case of two inputs,
the production function f(:z:1,:c2) would measure the maximum amount of output y that we could get if we had :61 units of factor 1 and 332 units of
factor 2. 324 TECHNOLOGY (Ch. 18) M y = OUTPUT y: f(x) = production function x = lNPUT A production set. Here is a possible shape for a production
set. M In the two—input case there is a convenient way to depict production
relations known as the isoquant. An isoquant is the set of all possible
combinations of inputs 1 and 2 that are just sufﬁcient to produce a given
amount of output. Isoquants are similar to indifference curves. As we’ve seen earlier, an
indifference curve depicts the different consumption bundles that are just
sufﬁcient to produce a certain level of utility. But there is one important
difference between indifference curves and isoquants. Isoquants are labeled
with the amount of output they can produce, not with a utility level. Thus
the labeling of isoquants is ﬁxed by the technology and doesn’t have the
kind of arbitrary nature that the utility labeling has. 18.3 Examples of Technology Since we already know a lot about indifference curves, it is easy to under—
stand how isoquants work. Let’s consider a few examples of technologies
and their isoquants. Fixed Proportions Suppose that we are producing holes and that the only way to get a hole is
to use one man and one shovel. Extra shovels aren’t worth anything, and
neither are extra men. Thus the total number of holes that you can produce
will be the minimum of the number of men and the number of shovels that
you have. We write the production function as f($1,a:2) = min{:c1,a:2}. EXAMPLES OF TECHNOLOGY 325 Isoquants *1 Fixed preportions. Isoquants for the case of ﬁxed propor—
tions. The isoquants look like those depicted in Figure 18.2. Note that these
isoquants are just like the case of perfect complements in consumer theory. Perfect Substitutes Suppose now that we are producing homework and the inputs are red
pencils and blue pencils. The amount of homework produced depends only
on the total number of pencils, so we write the production function as
f (291 , 3:2) : x1 + :32. The resulting isoquants are just like the case of perfect
substitutes in consumer theory, as depicted in Figure 18.3. Cobb—Douglas If the production function has the form f ($1,332) 2 Axilwg, then we say
that it is a CobbDouglas production function. This is just like the
functional form for Cobb~Douglas preferences that we studied earlier. The
numerical magnitude of the utility function was not important, so we set
A 2 1 and usually set a + b = 1. But the magnitude of the production
function does matter so we have to allow these parameters to take arbitrary
values. The parameter A measures, roughly speaking, the scale of produc—
tion: how much output we would get if we used one unit of each input.
The parameters a and b measure how the amount of output responds to 326 TECHNOLOGY (Ch.18) WW x2 \ .quuams Perfect substitutes. Isequants for the case of perfect substi
tutes. X1 changes in the inputs. We’ll examine their impact in more detail later on.
In some of the examples, we will choose to set A : 1 in order to simplify
the calculations. The CobbDouglas isoquants have the same nice, well—behaved shape
that the Cobb~Douglas indifference curves have; as in the case of utility
functions, the CobbDouglas production function is about the simplest ex
ample of wellbehaved isoquants. 18.4 Properties of Technology As in the case of consumers, it is common to assume certain properties
about technology. First we will generally assume that technologies are
monotonic: if you increase the amount of at least one of the inputs, it
should be possible 'to produce at least as much output as you were pro—
ducing originally. This is sometimes referred to as the property of free
disposal: if the ﬁrm can costlessly dispose of any inputs, having extra
inputs around can’t hurt it. Second, we will often assume that the technology is convex. This means
that if you have two ways to produce 3,1 units of output, (391, 332) and (21, 22),
then their weighted average will produce at least y units of output. One argument for convex technologies goes as follows. Suppose that you
have a way to produce 1 unit of output using (11 units of factor 1 and a2 PROPERTIES OF TECHNOLOGY 327 units of factor 2 and that you have another way to produce 1 unit of output
using b1 units of factor 1 and b2 units of factor 2. We call these two ways
to produce output production techniques. Furthermore, let us suppose that you are free to scale the output up by
arbitrary amounts so that (100011, 10002) and (100b1,100b2) will produce
100 units of output. But now note that if you have 25m + 75b1 units of
factor 1 and 25oz + 75b2 units of factor 2 you can still produce 100 units
of output: just produce 25 units of the output using the “a” technique and
75 units of the output using the “b” technique. This is depicted in Figure 18.4. By choosing the level at which you
operate each of the two activities, you can produce a given amount of output
in a variety of different ways. In particular, every input combination along
the line connecting (100021, 100a2) and (100b1, 100b2) will be a feasible way
to produce 100 units of output. (25a1 + 75b1, 25a2 + 75b2) 1 00b2  Isoquant 100aI 100b, X1 Convexity. If you can operate production activities indepen—
dently, then weighted averages of production plans will also be
feasible. Thus the isoquants will have a convex shape. In this kind of technology, where you can scale the production process up
and down easily and where separate production processes don’t interfere
with each other, convexity is a very natural assumption. 328 TECHNOLOGY (Ch. 18) 18.5 The Marginal Product Suppose that we are operating at some point, (3:1, :62), and that we consider
using a little bit more of factor 1 while keeping factor 2 ﬁxed at the level
5172. How much more output will we get per additional unit of factor 1? We
have to look at the change in output per unit change of factor 1: All : fflb‘i + Axum) “ f($1,$2)
A331 A5171 We call this the marginal product of factor 1. The marginal product
of factor 2 is deﬁned in a similar way, and we denote them by MP1(:L"1,2:2)
and M P2 (£01,532), respectively. Sometimes we will be a bit sloppy about the concept of marginal product
and describe it as the extra output we get from having “one” more unit of
factor 1. As long as “one” is small relative to the total amount of factor 1
that we are using, this will be satisfactory. But we should remember that
a marginal product is a rate: the extra amount of output per unit of extra
input. The concept of marginal product is just like the concept of marginal
utility that we described in our discussion of consumer theory, except for
the ordinal nature of utility. Here, we are discussing physical output: the
marginal product of a factor is a speciﬁc number, which can, in principle,
be observed. 18.6 The Technical Rate of Substitution Suppose that we are operating at some point (3:1,:r2) and that we consider
giving up a little bit of factor 1 and using just enough more of factor 2 to
produce the same amount of output 3/. How much extra of factor 2, A122,
do we need if we are going to give up a little bit of factor 1, A331? This
is just the slope of the isoquant; we refer to it as the technical rate of
substitution (TRS), and denote it by TRS(3:1,$2). The technical rate of substitution measures the tradeoff between two
inputs in production. It measures the rate at which the ﬁrm will have to
substitute one input for another in order to keep output constant. To derive a formula for the TRS, we can use the same idea that we used
to determine the slope of the indifference curve. Consider a change in our
use of factors 1 and 2 that keeps output ﬁxed. Then we have Ay 1" MP1(.’L'1,CL'2)AJL'1 + MP2<IE1,$2)ACL‘2 I O, which we can solve to get A332 ... _MP1(931,$2)
ATI m MP2(331,$2)‘ Note the similarity with the deﬁnition of the marginal rate of substitution. TRS(IL‘1,.’L‘2) : DIMlNlSHlNG TECHNICAL RATE OF SUBSTITUTlON 329 18.7 Diminishing Marginal Product Suppose that we have certain amounts of factors 1 and 2 and we consider
adding more of factor 1 while holding factor 2 ﬁxed at a given level. What
might happen to the marginal product of factor 1? ’ As long as we have a monotonic technology, we know that the total
output will go up as we increase the amount of factor 1. But it is natural
to expect that it will go up at a decreasing rate. Let’s consider a speciﬁc
example, the case of farming. One man on one acre of land might produce 100 bushels of corn. If we
add another man and keep the same amount of land, we might get 200
bushels of corn, so in this case the marginal product of an extra worker
is 100. Now keep adding workers to this acre of land. Each worker may
produce more output, but eventually the extra amount of corn produced
by an extra worker will be less than 100 bushels. After 4 or 5 people are
added the additional output per worker will drop to 90, 80, 70 . . . or even
fewer bushels of com. If we get hundreds of workers crowded together on
this one acre of land, an extra worker may even cause output to go down!
As in the making of broth, extra cooks can make things worse. Thus we would typically expect that the marginal product of a factor
will diminish as we get more and more of that factor. This is called the
law of diminishing marginal product. It isn’t really a “law”; it’s just
a common feature of most kinds of production processes. It is important to emphasize that the law of diminishing marginal prod—
uct applies only when all other inputs are being held ﬁxed. In the farming
example, we considered changing only the labor input, holding the land
and raw materials ﬁxed. 18.8 Diminishing Technical Rate of Substitution Another closely related assumption about technology is that of diminish
ing technical rate of substitution. This says that as we increase the
amount of factor 1, and adjust factor 2 so as to stay on the same isoquant,
the technical rate of substitution declines. Roughly speaking, the assump—
tion of diminishing TRS means that the slope of an isoquant must decrease
in absolute value as we move along the isoquant in the direction of increas
ing 1:1, and it must increase as we move in the direction of increasing 3:2.
This means that the isoquants will have the same sort of convex shape that
wellbehaved indifference curves have. The assumptions of a diminishing technical rate of substitution and di
minishing marginal product are closely related but are not exactly the
same. Diminishing marginal product is an assumption about how the mar
ginal product changes as we increase the amount of one factor, holding the 330 TECHNOLOGY (Ch. 18) other factor ﬁxed. Diminishing TRS is about how the ratio of the marginal
productsﬂthe slope of the isoquantfchanges as we increase the amount
of one factor and reduce the amount of the other factor so as to stay on the
same isoquant. 18.9 The Long Run and the Short Run Let us return now to the original idea of a technology as being just a list
of the feasible production plans. We may want to distinguish between the
production plans that are immediately feasible and those that are eventually
feasible. In the short run, there will be some factors of production that are ﬁxed
at predetermined levels. Our farmer described above might only consider
production plans that involve a ﬁxed amount of land, if that is all he has
access to. It may be true that if he had more land, he could produce more
corn, but in the short run he is stuck with the amount of land that he has. On the other hand, in the long run the farmer is free to purchase more
land, or to sell some of the land he now owns. He can adjust the level of
the land input so as to maximize his proﬁts. The economist’s distinction between the long run and the short run is
this: in the short run there is at least one factor of production that is ﬁxed:
a ﬁxed amount of land, a ﬁxed plant size, a ﬁxed number of machines, or
whatever. In the long run, all the factors of production can be varied. There is no speciﬁc time interval implied here. What is the long run and
what is the short run depends on what kinds of choices we are examining.
In the short run at least some factors are ﬁxed at given levels, but in the
long run the amount used of these factors can be changed. Let’s suppose that factor 2, say, is ﬁxed at E2 in the short run. Then the
relevant production function for the short run is f (931 , E2). We can plot the
functional relation between output and 51:1 in a diagram like Figure 18.5. Note that we have drawn the short—run production function as getting
ﬂatter and ﬂatter as the amount of factor 1 increases. This is just the law
of diminishing marginal product in action again. Of course, it can easily
happen that there is an initial regiOn of increasing marginal returns where
the marginal product of factor 1 increases as we add more of it. In the case
of the farmer adding labor, it might be that the ﬁrst few workers added
increase output more and more because they would be able to divide up
jobs efﬁciently, and so on. But given the ﬁxed amount of land, eventually
the marginal product of labor will decline. 18.10 Returns to Scale Now let’s consider a different kind of experiment. Instead of increasing the
amount of one input while holding the other input ﬁxed, let’s increase the RETURNS TO SCALE 331 W X1 Production function. This is a possible shape for a shortrun
production function. amount of all inputs to the production function. In other words, let’s scale
the amount of all inputs up by some constant factor: for example, use twice
as much of both factor 1 and factor 2. If we use twice as much of each input, how much output will we get?
The most likely outcome is that we will get twice as much output. This is
called the case of constant returns to scale, In terms of the production
function, this means that two times as much of each input gives two times as
much output. In the case of two inputs we can express this mathematically
by 2f(l‘1,1‘2) = f(233‘1,21132). In general, if we scale all of the inputs up by some amount it, constant
returns to scale implies that we should get 75 times as much output: 151007317132): “75331715559 We say that this is the likely outcome for the following reason: it should
typically be possible for the ﬁrm to replicate what it was doing before. If
the ﬁrm has twice as much of each input, it can just set up two plants side
by side and thereby get twice as much output. With three times as much
of each input, it can set up three plants, and so on. Note that it is perfectly possible for a technology to exhibit constant re
turns to scale and diminishing marginal product to each factor. Returns
to scale describes what happens when you increase all inputs, while di
minishing marginal product describes what happens when you increase one
of the inputs and hold the others ﬁxed. 332 TECHNOLOGY (Ch. 18) Constant returns to scale is the most “natural” case because of the repli
cation argument, but that isn’t to say that other things might not happen.
For example, it could happen that if we scale up both inputs by some fac
tor t, we get more than t times as much output. This is called the case of
increasing returns to scale. Mathematically, increasing returns to scale
means that f(tl'1,t.’L‘2) > tf($1,.’L‘2). for all t > 1. What would be an example of a technology that had increasing returns
to scale? One nice example is that of an oil pipeline. If we double the
diameter of a pipe, we use twice as much materials, but the cross section
of the pipe goes up by a factor of 4. Thus we will likely be able to pump
more than twice as much oil through it. (Of course, we can’t push this example too far. If we keep doubling the
diameter of the pipe, it will eventually collapse of its own weight. Increasing
returns to scale usually just applies over some range of output.) The other case to consider is that of decreasing returns to scale,
where f(t.r1,t:r2) < tf(:cl,:v2) for all t > 1. This case is somewhat peculiar. If we get less than twice as much output
from having twice as much of each input, we must be doing something
wrong. After all,...
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