Economists are usually concerned with functions of more than just one variable because there is almost always more than a single cause of an economic outcome.
To see the effects of many causes, economists must work with functions of several variables. A two-variable function might be written in functional notation as
Y¼fðX,ZÞ (1A:17)
This equation shows thatY’s values depend on the values of two independent variables,XandZ. For example, an individual’s weight (Y) depends not only on calories eaten (X) but also on how much the individual exercises (Z). Increases inX increase Y, but increases in Zdecrease Y. The functional notation in Equation 1A.17 hints at the possibility that there might be trade-offs between eating and exercise. In Chapter 2, we start to explore such trade-offs because they are central to the choices that both individuals and firms make. The next example provides a first step in this process.
Trade-offs and Contour Lines: An Example
As an illustration of how many variable functions can show trade-offs, consider the function
Y¼ ffiffiffiffiffiffiffiffiffiffi
XZ
p ¼X0:5Z0:5; X 0; Z0: (1A:18) Choosing to look at this function is, of course, no accident—it will turn out that this function (or a slight generalization of it) will be used throughout this book
whenever we need to illustrate trade-offs in a simple context.6Here, however, we will look only at some of the function’s mathematical properties. Table 1A.2 shows a few values ofXandZtogether with the result- ing value forY predicted by this function. Two inter- esting facts about the function are shown in the table.
First, notice that if we holdXconstant at, say,X¼2, increasingZalso also increasesY. For example, increas- ingZfrom 1 to 2 increases the value ofYfrom 1.414 to 2. Increasing Z further, to 3, increases Y further to 2.449. But the sizes of these increases get smaller asZ continues to increase further. In economic terms, this shows that the marginal gains from further Z are decreasing for this function if we hold X constant.
Hence, if we were concerned about the cost of Z, we might be careful in buying more of it and instead think about increasing Xto achieve gains in Y. This is pre- cisely the sort of intuition that will guide our discus- sions of trade-offs in households’ and firms’ optimizing behavior.
Contour Lines
A second fact that is illustrated by the calculations in Table 1A.2 is that a number of different combinations ofXandZyield the same value forY. For example,Y¼2 forX¼1,Z¼4, or forX¼2,Z¼2, or forX¼4,Z¼1. Indeed, it seems there are probably an infinite number of combinations ofXandZthat would yield a value ofY¼2. Studying all of these combinations would appear to be a valuable way of learning about trade-offs betweenXandZ.
There are two ways in which we might make progress in examining such trade-offs. The first approach is algebraic—if we setY¼2, we can solve Equa- tion 1A.18 for the kind of relationship that Xand Z must have to yield this outcome
Y ¼2¼ ffiffiffiffiffiffiffiffiffiffi
XZ p
or 4¼XZ orX¼4
Z: (1A:19)
All of the combinations we just illustrated satisfy this relationship, as do many others. In fact, Equation 1A.19 shows precisely how we have to change the values ofXandZto keepYat 2.
Another way to see the trade-offs in a multivariable function is to graph its contour lines. These show the various combinations ofXandZthat yield a given value ofY. The term ‘‘contour lines’’ is borrowed from mapmakers who also use
T A B L E 1 A . 2
Values ofX,Z, a n dYT ha t S a t i s f y th e R e la t i o n sh i p
Y¼ ffiffiffiffiffiffiffiffiffiffiffi
X·Z p
X Z Y
1 1 1.000
1 2 1.414
1 3 1.732
1 4 2.000
2 1 1.414
2 2 2.000
2 3 2.449
2 4 2.828
3 1 1.732
3 2 2.449
3 3 3.000
3 4 3.464
4 1 2.000
4 2 2.828
4 3 3.464
4 4 4.000
6Formally, this function is a particular form of the ‘‘Cobb-Douglas’’ function that we will use to examine the choices of both consumers and firms.
Contour lines
Lines in two dimensions that show the sets of values of the independent variables that yield the same value for the dependent variable.
such lines to show altitude on a two-dimensional map. For example, a contour labeled ‘‘1,500 feet’’ shows the locations on the map that are precisely 1,500 feet above sea level. Similarly, a contour labeledY¼2 shows all those combinations of XandZthat yield a value of 2 for the dependent variableY. Three such contour lines are shown in Figure 1A.5, forY¼1,Y¼2, andY¼3. In this particular case, the contour lines are hyperbolas, as can be seen from Equation 1A.19, which represents the contour line forY¼2.
The slope of these contour lines shows howX andZcan be traded off against one another while still keepingYconstant. In later chapters, we will examine such slopes in much more detail because they will tell us quite a bit about how households and firms behave. For the moment, the most important fact to note is that the slope of the contour lines is constantly changing—that is, the terms at which X and Z can be changed while
F I G U R E 1 A . 5
Co nto ur L in es forY¼ ffiffiffiffiffiffiffiffiffiffiffi X ·Z p
Z
9
4 3 2 1
0 1 2 3 4 9
Y ⴝ 3
Y ⴝ 2
Y ⴝ 1 X Contour lines for the function Y¼ ffiffiffiffiffiffiffiffiffiffiffi
X ·Z
p are rectangular hyperbolas. They can be represented by makingYequal to various supplied values (here, Y¼1,Y¼2,Y¼3) and then graphing the relationship between the independent variablesXandZ
M i c r o Q u i z 1 A . 4
Figure 1A.5 shows three contour lines for the functionY¼ ffiffiffiffiffiffiffiffiffiffiffi
X ·Z
p . How do these lines com- pare to the following contour lines?
1. Contour lines for Y¼9, 4, and 1 for the functionY¼ ffiffiffiffiffiffiffiffiffiffiffi
X·Z p
2. Contour lines forY¼81, 16, and 1 for the functionY¼X2ÆZ2
holding Y constant, changes as we move along any contour line. This fact is important enough to warrant giving it special emphasis.