Daniel l. Rubinfeld

134 Part 2 Producers, Consumers, and Competitive Markets
Thus elasticity increases in magnitude as the price increases (and the quantity
demanded falls),
There is no reason to expect elasticities of demand to be . constal:t.
Nevertheless, \\'e often find the isoelastic dellland Clll"Ue, in which the pnce ~lastl~itv
and the income elasticity are constant, useful to work with. When \vntten 111
it~ log-linear form, the isoelastic demand curve appears as follows:
10g(Q) = a - b 10g(P) + c 10g(I) (4.6)
where log ( ) is the logarithmic function and a, b, and c are the c.onsta~1ts. in the
demand equation. The appeal of the log-lin,ear demand relatIonshIp IS. that
the slope of the line - b is the price elasticity ot de~'nand and the constan.t L. IS the
income elasticity.E Using the data in Table 4.5, tor example, we obtamed the
regression line
10g(Q) = - 0.81 - 0.24log(P) + 1.46log(I)
This relationship tells us that the price elasticity of deman.d.fo~ raspberries is
_ 0.24 (that is, demand is inelastic) and that the income elastIcIty IS 1.46.
We have seen that it can be useful to distinguish between goods that are complements
and goods that are substihltes. Suppose that P 2 represents the price of
a second aood-one which is belie\'ed to be related to the product we are studying,
We c;n then write the demand function in the following form:
10g(Q) = a -
b 10g(P) + b 2 log(P2) + C 10g(I)
rather than before the acquisition. \A/hy Because after the acquisition the lost
sales from consumers who 'would switch a,·vay from Grape Nuts would be
recovered to the extent that they s\yitched to Shredded Wheat.
TI1e extent to which a price increase will cause consumers to switch is given
(in part) by the price elasticity of demand for Grape Nuts. Other things being
equal, the higher the demand elasticity, the greater the loss of sales associated
with a price increase. The more likely, too, that the price increase will be
lmprofitable.
The substitutability of Grape Nuts and Shredded Wheat can be measured by
the cross-price elasticity of demand for Grape Nuts with respect to the price of
Shredded Wheat. The relevant elasticities were calculated using ,",reekly data
obtained from the supermarket scanning of household purchases for 10 cities
over a three-year period. One of the estimated isoelastic demand equations
appeared in the following log-linear form:
10g(QcrJ = 1.998 - 2.08510g(PGl'J + 0.6210g(I) + 0.14log(Psw)
where QCN is the amOlmt (in pounds) of Grape Nuts sold weekly, PCN the price
per pound of Grape Nuts, I real personal income, and Psw the price per pound
of Shredded Wheat Spoon Size.
TI1e demand for Grape Nuts is elastic (at current prices), with a price elasticity
of about - L The income elasticity is 0.62: In other words, increases in
income lead to increases in cereal purchases, but at less than a 1-for-1 rate.
Finally, the cross-price elasticity is 0.14. TIus figure is consistent with the fact
that although the two cereals are substitutes (the quantity demanded of Grape
Nuts increases in response to an increase in the price of Shredded Wheat), they
are not very close substitutes.
&
Chapter 4 Individual and Market Demand 135
When b 2
, the cross-price elasticity, is positive, the tvw goods are substitutes;
when be is negative, the two goods are complements.
e Post Cereals Division of Kraft General Foods acquired the Shredded
Wheat cereals of Nabisco in 1995. The acquisition raised the legal and economic
question of whether Post vvould raise the pri,ce of its best-selling brand,
Grape Nuts, or the price of Nabisco's most successtul brand, Shredded Wheat
Spoon SizeY One important issue in a lawsuit brought,by the state of ~ew
York was whether the two brands \·vere close substihltes tor one another. It so,
it would be more profitable for Post to increase the price of Grape Nuts after
12The naturalloaarithmic function with base c has the property that ~(log(Q)) ~Q/Q foran~
chanae in 10g(QtSimilarl\', ~(log(P)) = ~P/P for any change in 10g(P) It follows that ~(log(Q)) .-,
~Q/Q _ b~ ~(log(P))J ' - bPP/P) Therefore, (~.Q/Q)~(~P/P) =, -:- b,.whIch ,IS the pnce ela~tlClt)
of demand By a similar argument, the income elastiClty ot demand LIS gl\en b} (~Q/Q)/(~L I),
13 State of New York v. Kraft Gwera/ Foods. IIlC, 926 F Supp. 321, 356 (S.D. . .N.Y 1995)
, .
1. Individual consumers' demand curves for a commodity
can be derived from information about their tastes
for all goods and services and from their budget constraints.
2. Engel curves, which describe the relationship
between the quantity of a good consumed and
income, can be useful for discussions of how consumer
expenditures vary with income.
3. Two goods are substitutes if an increase in the price of
one leads to an increase in the quantity demanded of
the other. In contrast, two goods are complements if
an increase in the price of one leads to a decrease in
the quantity demanded of the other.
4. The effect of a price change on the quantity demanded
of a good can be broken into two parts: a
substitution effect, in which satisfaction remains constant
while price changes, and an income effect, in
Which the price remains constant while satisfaction
changes. Because the income effect can be positive or
negative, a price change can have a small or a large
effect on quantity demanded. In the unusual case of a
so-called Giffen good, the quantity demanded may
move in the same direction as the price change,
thereby generating an upward-sloping individual
demand curve.
5. The market demand curve is the horizontal summation
of the individual demand curves of all consumers
in the market for a good. It can be used to calculate
how much people value the consumption of particular
goods and services.
6. Demand is price inelastic when a 1-percent increase in
price leads to a less than 1-percent decrease in quantity
demanded, thereby increasing the consumer's
expenditure. Demand is price elastic when a 1-percent
increase in price leads to a more than 1-percent
decrease in quantity demanded, thereby decreasing