Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

104 ADVANCED TOPICS IN CONSUMER DEMAND THEORY [CHAP. 5
and to the left and below shaded area LAM. Such an indifference curve must be negatively sloped and convex to the origin.
To locate more closely the indifference curve, we can chip away the lower zone of ignorance by showing that the consumer
can be induced to purchase basket B on NN if P x /P y falls sufficiently, say, to PP. Then, every point on PP is inferior to B,
which is inferior to A. We have, thus, eliminated area NBP from the lower zone of ignorance. On the other hand, budget line
S 0 S 0 through point A shows the same real income as at point A, but since P x /P y is higher than at point A, the consumer will
purchase less of X and more of Y, as at point G. Then all baskets in the shaded area above and to the right of G are preferred
to G, which is preferred to A. Thus, we have eliminated some of the upper zone of ignorance. These processes can be
repeated any number of times so that the location of the indifference curve can be pinpointed more precisely (see Problems
5.3 and 5.4). While not being very practical for actually deriving an indifference curve, the theory of revealed preference can
be used to derive a demand curve easily and without indifference curves (see Problem 5.5).
Fig. 5-2
5.3 INDEX NUMBERS AND CHANGES IN THE STANDARD OF LIVING
A consumer is better off in period 1 than in a base period if that person’s total expenditures or income in
period 1 exceeds the cost of the base period basket of goods in terms of period 1 prices. That is, if
or
P 1 x X1 þ P 1 y Y1 . P 1 x X0 þ P 1 y Y0
X
P 1 q 1 . X P 1 q 0
where P refers to price, q to quantity, the superscripts 1 and 0 refer to period 1 and the base period, respectively,
while P refers to “the sum of.” By the same reasoning, the consumer is better off in the base period if
X
P 0 q 0 . X P 0 q 1
This is because period 1 basket of X and Y was available in the base period but was not chosen.
Dividing both sides of the above inequality for an increase in the standard of living by P P 0 q 0 , we get
P P 1 q
P 1 P P 1
P0 q 0 . q
P 0
P0 q 0
or
E . L
where E is the expenditure index, which measures the ratio of period 1 expenditures to base period expenditures,
while L is the Laspeyres price index, which measures the cost of base period quantities at period
1 relative to base period prices. Thus, if E . L, the consumer’s standard of living increased.