In order to efficiently study the structure of many important economic models, it is necessary to first discuss an important class of functions known as homogeneous functions. The interest in these functions arose from a problem in the economic theory of distribution. The development of marginal productivity theory by Marshall and others led to the conclusion that factors of production would be paid the value of their marginal products. (This will be studied in the next and subsequent chapters in more detail.) Roughly speaking, factors would be hired until their contribution to the output of the firm just equaled the cost of acquiring additional units of that factor. Letting y = f{x\, X2) be the firm's production function and letting w, denote the wage of factor x, andp the price of the firm's output, the rule developed was that
pMPi = pft = wt
where f{ = 3//3x,. But this analysis was developed in a "partial equilibrium" framework; that is, each factor was analyzed independently. The question then arose, how is it possible to be sure that the firm was capable of making these payments to both factors? All factor payments had to be derived from the output produced by the firm. Would enough output be produced (or perhaps would too much be produced, leaving the excess unclaimed) to be able to pay each unit of each factor the value of its marginal product?
A theorem developed by the great Swiss mathematician Euler (pronounced "Oiler") came to the rescue of this analysis. (It leads to other problems, but those will be deferred.) It turns out that if the production function exhibits constant returns to scale, then the sum of the factor payments will identically equal total output.
Mathematically, if each factor JC, is paid w, = pft, then the total payment
to all JC,- is wtXi = pfiXt. Total payment to both factors is thus
pf\*\ + Phxi = P(f\xi + fixi)
But, as we shall see, constant-returns-to-scale production functions have the convenient property that, identically,
/i*i + /2-X2 = y = f(xi,x2)
Hence, in this case,
W1X1 + W2X2 = pf 1X1 + pf2X2 = p(/i*i + f2X2) = Py
FUNCTIONS OP SEVERAL VARIABLES 57 How is the feature of constant returns to scale characterized? This means that if each factor is increased by the same proportion, output will increase by a like proportion. Mathematically, a
production function y — f(x\, ... ,xn) exhibits constant returns to scale
if
f ( t x1, . . . , t xn) = t f ( xl, . . . , xn) (3- 30)
Note the identity sign: this proportionality of output and inputs must
hold for all x( 's and all t. If, for example, all inputs are doubled,
output will double, starting at any input combination.
The relation (3-30) is a special case of the more general mathematical notion of homogeneity of functions.
Definition 1. A function f(x\,..., xn) is said to be homogeneous of
degree r if and only if
f ( t xl, . . . , t xn) = trf ( xu. . . , xn) (3- 31)
That is, changing all arguments of the function by the same proportion t results in a change in the value of the function by an amount f', identically. Note again the identity sign—this is not an equation that holds only at one or a few points; the above relation is to hold for a \ l t , x i , . . . , xn. Constant returns to scale is the special case where a production function is homogeneous of degree 1. Homogeneity of degree 1 is often called linear homogeneity.
Example 1. Consider the very famous Cobb-Douglas production
function, y = LaKx~a — f(L,K), where L = labor, K = capital. This
production function is homogeneous of degree 1; i.e., it exhibits constant returns to scale. Suppose labor and capital are changed by some factor t. Then,
f(tL,tK) = {tL)a(tK)x~a =taLatx-aKx-a
Output f(L, K) is affected in exactly the same proportion t as are both inputs.
Consider now another important area in which the notion of homogeneity arises. In the theory of the consumer (also to be discussed later), individuals are presumed to possess demand
functions for the goods and services they consume. If Pi,..., pn
represents the money prices of the goods X\, ..., xn that a person actually consumes, and if M represents the consumer's money income, the ordinary demand curves are representable as
xi= x * ( pl, . . . , pn, M ) , (3-
32)
That is, the quantity consumed of any good x, depends on its price
pt, all other relevant prices, and money income M.
How would we expect the consumer to react to a proportionate change in all prices, with the same proportionate change in his or her money income? Although a formal proof must wait until a later
58 THE STRUCTURE OF ECONOMICS
consumption under these conditions. Economists (for good reason) in general assert that only relative price changes, not absolute price changes, matter in consumers' decisions.
What is being asserted here, mathematically? We are asserting homogeneity of degree 0 of the above demand equations, i.e.,
x*(tp\, ..., tpn, tM) = t°x*(pi, ..., pn, M) = x*(pi, . . . ,
pn, M)
The functional value is to be unchanged by proportionate change in all the independent variables; this is precisely homogeneity of degree 0. The demands for goods and services are not to depend on the absolute levels of prices and income.* The theoretical reasons for asserting this proposition will become clearer in later chapters; our purpose here is only to illustrate and motivate the usefulness of the concept of homogeneity of functions.
Consider now the Cobb-Douglas production function again, y =
La Kl~a = f(L, K). The marginal products of labor and capital are,
respectively,
l-a
= fK = ( \ - a)LaK~a = (1 - or) (~
These marginal products exhibit a feature worth noting: They can be written as functions of the ratios of the two inputs. They are independent of the absolute value of either input. Only their proportion to one another counts.
Because of this dependence only on ratios, the marginal products of the Cobb-Douglas function are homogeneous of degree 0:
ftK\l-a (K
MPL(fL, **) = « — =a -
Similarly,
t K\ / K \
— J = ( l - a ) ( — J =M?K(L,K)
If labor and capital are changed, by the same proportion, say they are both doubled, the marginal products of labor and capital will be unaffected. Geometrically, changing each input by the same proportion means moving along a ray out of the origin,
^There was a time, in the macroeconomics literature, when this homogeneity of demand functions was denied, under the name "money illusion." It was asserted that a completely neutral inflation would lead an economy out of depression; that even though people were not in fact richer, a higher money income (together with proportionately higher money prices) would somehow make people "feel" richer, increasing their consumption expenditures. This line of argument has been largely abandoned.
FUNCTIONS OF SEVERAL VARIABLES 59 through the original point. At every point along any such ray, the marginal products of the Cobb-Douglas production function (and others?) are the same.
To what extent, if any, are these results peculiar to the Cobb- Douglas functions; i.e., to what extent do other functions exhibit the
same or similar properties? Consider first any function f(x\, ..., xn) that
is homogeneous of degree 0. By definition,
f { t xutx2, . . . , t xn) = f ( Xi , x2, . . . , xn) Since this holds for any t, let t = l/x\. Then we have
r / N J- I 1 X2 xn \ X2 xn \
f ( xux2, . . . , xn) = f I , — , . . . , — = g — , . . . , —
\ X\ X] J \X\ X\ J
Similarly, we could let t — 1 /xt . What the above shows is that any
function that is homogeneous of degree 0 is representable as a function of the ratios of the independent variables to any one such variable. Hence, that the marginal products of the Cobb-Douglas function were representable as functions of the capital-labor ratios is not peculiar to that production function; it will hold for any marginal product functions that are homogeneous of degree 0.
What, then, are the conditions that the marginal products be homogeneous of degree 0? The answer is given, in a more general form, by the following theorem:
Theorem 1. If f(x\, x2,..., xn) is homogeneous of degree r, then the
first partials /i ,...,/„ are homogeneous of degree r — 1.
Proof. By assumption, f{tx\, ..., txn) = f f{x\, ..., xn). Since this is an
identity, it is valid to differentiate both sides with respect to xt:
df djtXj) _ f df d(tXi) dXj dXj
However, 3(rx,)/3x, = t. Dividing both sides of the identity by t therefore yields
df _ f_x df
But this says that the function f, evaluated at {txx,..., txn) equals f ~'
f, (*,•,..., xn). Hence, f is homogeneous of degree r — 1.
If y = f(x\, ..., xn) is any production function exhibiting
constant returns to scale, the marginal products are homogeneous of degree 0. That is, the marginal products are the same at every point along any ray through the origin. The Cobb-Douglas function is thus only a special case of this theorem.
Homogeneity of any degree implies that the slopes of the level curves of the function are unchanged along any ray through
the origin. This can be shown as follows: Let y = f(x\, ..., xn) be a
production function, for example, that is homogeneous of degree r.
60 THE
STRUCTURE OF ECONOMICS K
FIGURE 3-5
Invariance of the Slope of Isoquants to a Proportionate Increase in Each Factor. Consider any point (L°, K°). Suppose each input is doubled. If the production function is homogenous of any degree, the slope of the isoquant, — fi / f^, will be the same at (2L°, 2K°) as at (L°, A:0). This property is known as homotheticity. The most general functions that exhibit this property can be
written F(f{x\, ..., xn)), where
f{x\, ..., xn) is homogenous of any degree and F^O. But f i ( t x u . . . , t x n ) = t r - l f j ( x
x , . . . , x n ) f j ( t x l t . . . , t x n ) ~ t r - x f j ( x u . . . , x n ) = f i ( x u . . . , x n ) fj(X\, . . . , Xn) Thus, the slope of any isoquant evaluated along a radial expansion of an initial point is identical to the slope at the original point. In other words, the ratios of the marginal products along any ray from the origin remain unchanged for homogeneous functions. The level curves are thus radial blowups or reductions of each other. This situation is depicted in Fig. 3-5.
The following
describes a related class of production functions. Let y
— /Ui, ..., xn) be
homogeneous of degree r, and let z = F(y), where F'(y) > 0. [F(y) is a monotonic transformation of y.] The function z(xi, ..., xn) is called a homothetic function. It is easy to show that homothetic functions also preserve the property that slopes along a radial blowup remain unchanged, i.e., that the slopes of isoquants z(tx\, ..., txn) are
the same as at z{x\, ..., xn), and this is left to the student as an exercise. It is less than easy to show, but nonetheless true, that this is the most general class of production functions that have this property.^ Example 2. Consider the function z — g(L, K) = F(y), where y = LaKx~a and F{y) = log_y. Then z = =\ogLa Kl~a =cdogL -a ) lo g That is, the original function LaKx~a is transformed by the function "F," in this case "log." We note that F'(y) = \/y > 0, for positive L, K. Now La Kx~a is homogeneous
^See, e.g., F. W. McElroy, "Returns to Scale, Euler's Theorem, and the Form of
Production Functions," Econometrica,
FUNCTIONS OF SEVERAL VARIABLES 61
of degree 1, as noted before, but log(LaKl~a) is not a
homogeneous function: g(tL,tK) = a\ogtL + (l -a)logtK
= a(log/ + logL) + (l -a)(\ogt
+ \ogK) -logt + logVK'~a
^fg(L,K)
However, g(L, K) — a log L+(l—a) log K is homothetic: The slope of a level curve i s -8L _ -a/L _ -a K ~ g ~ 7 ~ d - a ) / K ~ l - a L
As before, —gi/gK is unaffected by changing Kand L by a factor of t; the r's cancel in the expression K/L and, hence, the slope of the
level curves of log La Kl'a are the same along any ray out of the
origin. This function is not homogeneous, but it is homothetic. Suppose that instead of defining homothetic functions as
F{f{x\, ..., xn)), where/ is homogeneous of degree r, that instead we
restrict/to be linearly homogeneous; i.e., homogeneous of degree 1. Though it might not seem so at first, this latter definition is just as general as the first definition; i.e., no functions are left out by so doing. The reason is that any homogeneous function of degree r can be converted to a linear homogeneous function by taking the rth root of f(x\, ..., xn). Then, [f(x\, ..., xn)]l/r can be transformed by some function F. Thus, since we can always consider F to be a composite of two transformations, the first of which takes the rth root of/ and the second, which operates on that, no generality is lost by defining homothetic functions as transformations of linear homogeneous functions.
Example 3. Let y = f(xx, x2) = x\X2. Here, f(x\, x2) is homogeneous
of degree 2. Let
g(xux2) = F(f(xl,x2)) = log(jci*2) = log*i +logx2
This function is homothetic but not homogeneous. How could g(x\,
x2) be constructed out of a linear homogeneous function? Let
g{xx,x2) = 21ogUix2)1/2 Thus,
where 0 means "take square root" and F is log, as before. Then the same function
g(xx,x2) = logx{+logx2 is constructed as
a transformation of the linear homogeneous function (x\X2)l/2.
We now prove the main theorem of this section.
Theorem 2 (Euler's theorem). Suppose f(x{,..., xn) is homogeneous
of degree r. Then
9/ 3/
- X] + ■ ■ • + - -xn =
rf{x\,..., xn)
ax\ oxn
62 THE STRUCTURE OF ECONOMICS