Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006
104 ADVANCED TOPICS IN CONSUMER DEMAND THEORY [CHAP. 5and 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 consumercan 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 lineS 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 willpurchase 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 preferredto G, which is preferred to A. Thus, we have eliminated some of the upper zone of ignorance. These processes can berepeated any number of times so that the location of the indifference curve can be pinpointed more precisely (see Problems5.3 and 5.4). While not being very practical for actually deriving an indifference curve, the theory of revealed preference canbe used to derive a demand curve easily and without indifference curves (see Problem 5.5).Fig. 5-25.3 INDEX NUMBERS AND CHANGES IN THE STANDARD OF LIVINGA consumer is better off in period 1 than in a base period if that person’s total expenditures or income inperiod 1 exceeds the cost of the base period basket of goods in terms of period 1 prices. That is, iforP 1 x X1 þ P 1 y Y1 . P 1 x X0 þ P 1 y Y0XP 1 q 1 . X P 1 q 0where 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 ifXP 0 q 0 . X P 0 q 1This 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 getP P 1 qP 1 P P 1P0 q 0 . qP 0P0 q 0orE . Lwhere 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 period1 relative to base period prices. Thus, if E . L, the consumer’s standard of living increased.
CHAP. 5] ADVANCED TOPICS IN CONSUMER DEMAND THEORY 105On the other hand, dividing both sides of the inequality for a decrease in the standard of living by P P 1 q 1 ,we getP P 0 q 0PP1 q 1 . P P 0 q 1PP1 q 11E . 1 PE , Pwhere P is the Paasche price index, which measures the cost of purchasing period 1 quantities at period 1prices relative to base period prices.Thus, if E . L and E . P, the consumer’s standard of living rose; if E , L and E , P, it fell; and ifP . E . L or L . E . P, we cannot say whether it rose, fell, or stayed the same.EXAMPLE 3. Using the data in Table 5.1,Table 5.1Period P x X P y Y0 (base) $2 3 $4 61 4 5 5 7we find thatE ¼P P 1 qP 1P0 q 0 ¼ P1 x X1 þ P 1 y Y1 ($4)(5) þ ($5)(7)¼P 0 x X0 þ P 0 yY0 ($2)(3) þ ($4)(6) ¼ $55$30ffi 1:83L ¼P ¼P P 1 qP 0P0 q 0 ¼ P1 x X0 þ P 1 y Y0$30¼($4)(3) þ ($5)(6)$30P P 1 q 1PP0 q 1 ¼ $55P 0 x X1 þ P 0 y Y1 ¼ $55¼ $42 ffi 1:40$30($2)(3) þ ($4)(7) ¼ $55 ffi 1:45$38Since E . L and E . P, the standard of living increased. The Laspeyres index (L) gives an upward bias of the increase inthe cost of living. The Paasche index (P) gives a downward bias. Because L uses base period quantities, L becomes availablesooner than P. The consumer price index (CPI), published monthly by the Bureau of Labor Statistics is an L index fora “typical” urban family of four. CPI measures the change in the purchasing power of the dollar and is used for inflationadjustment in wage contracts, pensions, welfare payments, and so on. Consumer tastes and product quality are assumedto be constant. Other price indices are the wholesale price index (WPI) and the GNP deflator.5.4 UTILITY THEORY UNDER UNCERTAINTYTraditional economic theory implicitly assumed a riskless world. However, most economic choicesinvolve risk or uncertainty. For example, an individual may decide to become a lawyer or to go into business,where incomes can be either very high or only modest. Similarly, a homeowner may purchase insurance againstthe small chance of a heavy loss through fire and also purchase a lottery ticket offering a small chance of a largewin. Traditional economic theory could not explain choices involving risk because of its strict adherence to theprinciple of diminishing marginal utility. Such an apparently conflicting behavior as the same individual purchasinginsurance and also gambling can be rationalized by a total utility curve that first rises at a decreasingrate (so that MU declines) and then at an increasing rate (so that MU rises).
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CHAP. 5] ADVANCED TOPICS IN CONSUMER DEMAND THEORY 105
On the other hand, dividing both sides of the inequality for a decrease in the standard of living by P P 1 q 1 ,
we get
P P 0 q 0
P
P1 q 1 . P P 0 q 1
P
P1 q 1
1
E . 1 P
E , P
where P is the Paasche price index, which measures the cost of purchasing period 1 quantities at period 1
prices relative to base period prices.
Thus, if E . L and E . P, the consumer’s standard of living rose; if E , L and E , P, it fell; and if
P . E . L or L . E . P, we cannot say whether it rose, fell, or stayed the same.
EXAMPLE 3. Using the data in Table 5.1,
Table 5.1
Period P x X P y Y
0 (base) $2 3 $4 6
1 4 5 5 7
we find that
E ¼
P P 1 q
P 1
P0 q 0 ¼ P1 x X1 þ P 1 y Y1 ($4)(5) þ ($5)(7)
¼
P 0 x X0 þ P 0 yY0 ($2)(3) þ ($4)(6) ¼ $55
$30
ffi 1:83
L ¼
P ¼
P P 1 q
P 0
P0 q 0 ¼ P1 x X0 þ P 1 y Y0
$30
¼
($4)(3) þ ($5)(6)
$30
P P 1 q 1
P
P0 q 1 ¼ $55
P 0 x X1 þ P 0 y Y1 ¼ $55
¼ $42 ffi 1:40
$30
($2)(3) þ ($4)(7) ¼ $55 ffi 1:45
$38
Since E . L and E . P, the standard of living increased. The Laspeyres index (L) gives an upward bias of the increase in
the cost of living. The Paasche index (P) gives a downward bias. Because L uses base period quantities, L becomes available
sooner than P. The consumer price index (CPI), published monthly by the Bureau of Labor Statistics is an L index for
a “typical” urban family of four. CPI measures the change in the purchasing power of the dollar and is used for inflation
adjustment in wage contracts, pensions, welfare payments, and so on. Consumer tastes and product quality are assumed
to be constant. Other price indices are the wholesale price index (WPI) and the GNP deflator.
5.4 UTILITY THEORY UNDER UNCERTAINTY
Traditional economic theory implicitly assumed a riskless world. However, most economic choices
involve risk or uncertainty. For example, an individual may decide to become a lawyer or to go into business,
where incomes can be either very high or only modest. Similarly, a homeowner may purchase insurance against
the small chance of a heavy loss through fire and also purchase a lottery ticket offering a small chance of a large
win. Traditional economic theory could not explain choices involving risk because of its strict adherence to the
principle of diminishing marginal utility. Such an apparently conflicting behavior as the same individual purchasing
insurance and also gambling can be rationalized by a total utility curve that first rises at a decreasing
rate (so that MU declines) and then at an increasing rate (so that MU rises).
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