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c16GeneralEquilibriumTheory.qxd 8/16/10 9:14 PM Page 693

APPENDIX: DERIVING THE DEMAND AND SUPPLY CURVES 693
DERIVING THE HOUSEHOLD AND MARKET DEMAND
CURVES FOR ENERGY AND FOOD

We begin by deriving the demand curves for each household type in our economy, and
we then sum these demand curves to derive the market demand curves. To do this, we
use the techniques developed in Chapter 5.
Given the utility function for a white-collar household, the marginal utilities of
energy and food are

1
1 y 2
W
MU x a b
2 x
1
1 x 2
W
MU y a b
2 y
The marginal rate of substitution of energy for food is the ratio of the marginal
W
W
utilities: MRS W MU /MU . Using the above expressions for marginal utility,
x, y
x
y
this ratio reduces to MRS W y/x. Assuming that the household maximizes its
x, y
utility subject to its budget constraint, it will equate the marginal rate of substitu-
tion to the ratio of the prices: MRS W P /P . In addition, the budget constraint
x, y
x
y
is satisfied. Thus, utility maximization gives us two equations in two unknowns, x
and y. First, y/x P /P (which follows from MRS W y/x and MRS W P /P ).
x
x
x, y
y
x, y
y
Second, xP yP I W (which follows from the budget constraint), where I W de-
y
x
notes the household’s income level (which, recall, depends on the input prices, w
and r). When we solve these two equations for x and y (treating P , P , and I W as
x
y
constants), we get x (1/2)(I /P ) and y (1/2)(I /P ). These are a typical white-
x
y
W
W
collar household’s demand curves for energy and food.
Let’s suppose that our economy contains 100 such households. We can find the
aggregate demand curves for energy and food from white-collar households by mul-
tiplying the above expressions by 100. This yields the D W and D W demand curves in
x
y
Figure 16.5: x W 50I /P and y W 50I /P .
W
y
x
W
Let’s now turn to the blue-collar households. Given the utility function for a blue-
collar household, the marginal utilities of energy and food are
1 3
3 y 4 1 x 4
B
B
MU x a b and MU y a b
4 x 4 y
Proceeding in the same way we did for white-collar households. We find that the
demand curves for a typical blue-collar household are x (3/4)(I /P ) and y
B
x
(1/4)(I /P ). Multiplying these by 100 gives us the aggregate demand curves for blue-
B
y
B
B
collar households D B x and D B in Figure 16.5: x 75I /P x and y 25I /P .
y
B
B
y
We can now find the market demand curves for energy and food by horizontally
summing the demand curves for both types of household. Let X be the aggregate
amount of energy demanded in the economy. The market demand curve for energy is
B
thus X x W x , or X (50I /P ) (75I /P ). In Learning-By-Doing Exercise
x
B
W
x
16.2 we expressed this as P (50I W 75I )/X. Similarly, the market demand curve
B
x
B
for food is Y y W y , which we expressed as P (50I W 25I )/Y. Notice
B
y
that these market demand curves depend on the income levels of each individual
household.