Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006
114 ADVANCED TOPICS IN CONSUMER DEMAND THEORY [CHAP. 5(a) For 1989,P P 1 q 1E ¼ P P 0q 0 ¼ P1 x X1 þ P 1 y Y1 ($5)(6) þ ($4)(6)Px 0X 0 ¼þ Py 0 0Y ($4)(5) þ ($3)(3) ¼ $54 ffi 1:86 or 186%$29P P 1 q 0L ¼ P P 0q 0 ¼ P1 x X0 þ P 1 y Y0¼$29($5)(5) þ ($4)(3)$29¼ $37 ffi 1:28 or 128%$29P P 1 q 1P ¼ P P 0q 1 ¼ $54Px 0X1 þ P ¼ $54y 0Y1 ($4)(6) þ ($3)(6) ¼ $54 ffi 1:29 or 129%$42Since E . L and L . P, the consumer’s standard of living increased between 1984 and 1989.(b) For 1990,E ¼ $44$29ffi 1:52 L ¼$45$29Thus, the standard of living fell between 1977 and 1983.(c) For 1991,E ¼ $52$29ffi 1:79 L ¼$51$29$44ffi 1:55 P ¼ ffi 1:57$28$52ffi 1:76 P ¼ ffi 1:86$28Thus, we cannot say what happened to the standard of living between 1984 and 1991.UTILITY THEORY UNDER UNCERTAINTY5.7 With reference to Fig. 5-3, if the individual’s income is either OA = $30,000 with probability of 0.95 orOB ¼ $5000, (a) what is the expected income of this individual? (b) What is the maximum amount ofinsurance that this individual would be willing to pay?(a) I ¼ ( p)(OA) þ (1 p)(OB)I ¼ (0:95)($30,000) þ (0:05)($5000)I ¼ $28,5000 þ $250I ¼ $28,750(b)We can answer this question by drawing a horizontal line from point C 0 to the left, until it crosses the TU curvein Fig. 5-3. The horizontal distance from the crossing point on the TU curve to the vertical line AA 0 representsthe maximum amount of insurance that this individual is willing to pay. The reason for this is that the utility ofthe certain income with insurance given by the crossing point on the TU curve is the same (i.e., it is of the sameheight) as point C 0 (the expected income without insurance). Utility theory under uncertainty is associatedwith the names of Friedman and Savage—the original investigators.5.8 Suppose that an individual is just willing to accept a gamble to win or lose $1000 if the probability ofwinning is 0.6. Suppose that the utility gained if the individual wins is 100 utils. (a) Is this consumer aninsurer or a gambler? Why? (b) How much utility does one lose if one loses the gamble?(a)The individual is an insurer because this person required better-than-even odds of winning before beingwilling to gamble.
CHAP. 5] ADVANCED TOPICS IN CONSUMER DEMAND THEORY 115(b)Since the individual is just induced to accept the gamble when the probability of winning is 0.60 and wouldgain 100 utils upon winning, we can measure the utility lost in losing the gamble as follows:Expected gain in utility ¼ expected loss in utility(0:6)(100 utils) ¼ (0:4)(utils lost)(0:6)(100 utils)utils lost ¼ ¼ 150 utils0:4Since the individual would gain 100 utils upon winning $1000 and lose 150 utils on losing the $1000,the MU of money decreases, making the individual an insurer. Note that in choices involving risk, theconsumer maximizes expected utility rather than utility. This general method of calculation is oftenreferred to as modern utility theory.5.9 Do the calculations in Problem 5.8(b) give a cardinal measure of utility? Why?The calculations of Problem 5.8(b) (and modern utility theory) do not really give us a cardinal measure ofutility, since the results obtained are arbitrary with regard to both origin and scale. For example, if we assigned200 utils to the winning of $1000, the utility lost in losing the $1000 would have been 300 utils instead of 150utils. Furthermore, 300 utils should only be taken to imply more utility than 150 utils, and not twice as muchutility. Thus, modern utility theory only gives a method of ordering utility in conditions involving risk.A NEW APPROACH TO CONSUMER THEORY—THE DEMAND FOR CHARACTERISTICS5.10 Starting with panel A of Fig. 5-4, draw a figure that shows a hypothetical equilibrium with (a) 33%increase in the consumer’s income and (b) 40% reduction in the price of honey (with no change inthe price of sugar and in the consumer’s income).(a)A 33% increase in the consumer’s income extends ray OA by 33% to OA 0 in Fig. 5-11 or OB to OB 0 . Thebudget frontier is then A 0 B 0 , and equilibrium may take place at C 0 on indifference curve III, with OB characteristicsfrom honey and BC 0 (equals OG) from sugar. See Fig. 5-11.Fig. 5-11 Fig. 5-12(b)A 40% reduction in the price of honey extends ray OB by 40% to OB 00 in Fig. 5-12, so that the budget frontierbecomes AB 00 . Equilibrium may then take place at point C 00 on indifference curve V and is reached with ONcharacteristics from honey and NC 00 (equals OR) from sugar. See Fig. 5-12.
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114 ADVANCED TOPICS IN CONSUMER DEMAND THEORY [CHAP. 5
(a) For 1989,
P P 1 q 1
E ¼ P P 0
q 0 ¼ P1 x X1 þ P 1 y Y1 ($5)(6) þ ($4)(6)
Px 0X 0 ¼
þ Py 0 0
Y ($4)(5) þ ($3)(3) ¼ $54 ffi 1:86 or 186%
$29
P P 1 q 0
L ¼ P P 0
q 0 ¼ P1 x X0 þ P 1 y Y0
¼
$29
($5)(5) þ ($4)(3)
$29
¼ $37 ffi 1:28 or 128%
$29
P P 1 q 1
P ¼ P P 0
q 1 ¼ $54
Px 0X1 þ P ¼ $54
y 0Y1 ($4)(6) þ ($3)(6) ¼ $54 ffi 1:29 or 129%
$42
Since E . L and L . P, the consumer’s standard of living increased between 1984 and 1989.
(b) For 1990,
E ¼ $44
$29
ffi 1:52 L ¼
$45
$29
Thus, the standard of living fell between 1977 and 1983.
(c) For 1991,
E ¼ $52
$29
ffi 1:79 L ¼
$51
$29
$44
ffi 1:55 P ¼ ffi 1:57
$28
$52
ffi 1:76 P ¼ ffi 1:86
$28
Thus, we cannot say what happened to the standard of living between 1984 and 1991.
UTILITY THEORY UNDER UNCERTAINTY
5.7 With reference to Fig. 5-3, if the individual’s income is either OA = $30,000 with probability of 0.95 or
OB ¼ $5000, (a) what is the expected income of this individual? (b) What is the maximum amount of
insurance that this individual would be willing to pay?
(a) I ¼ ( p)(OA) þ (1 p)(OB)
I ¼ (0:95)($30,000) þ (0:05)($5000)
I ¼ $28,5000 þ $250
I ¼ $28,750
(b)
We can answer this question by drawing a horizontal line from point C 0 to the left, until it crosses the TU curve
in Fig. 5-3. The horizontal distance from the crossing point on the TU curve to the vertical line AA 0 represents
the maximum amount of insurance that this individual is willing to pay. The reason for this is that the utility of
the certain income with insurance given by the crossing point on the TU curve is the same (i.e., it is of the same
height) as point C 0 (the expected income without insurance). Utility theory under uncertainty is associated
with the names of Friedman and Savage—the original investigators.
5.8 Suppose that an individual is just willing to accept a gamble to win or lose $1000 if the probability of
winning is 0.6. Suppose that the utility gained if the individual wins is 100 utils. (a) Is this consumer an
insurer or a gambler? Why? (b) How much utility does one lose if one loses the gamble?
(a)
The individual is an insurer because this person required better-than-even odds of winning before being
willing to gamble.
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