Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006
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CHAP. 4] CONSUMER DEMAND THEORY 77
(a)
(b)
ðcÞ
With continuously decreasing MU, overall TU can be maximized by maximizing the utility received from
spending a dollar at the time. Thus, this individual should spend the first dollar of her income to purchase
the first unit of Y. From this she receives 19 utils. If she spent this first dollar to purchase the first unit of X,
she would receive only 11 utils. The individual should spend her second, third, fourth, and fifth dollars to purchase
the second, third, fourth, and fifth units of Y. From these she received 17, 15, 13, and 12 utils, respectively.
The individual should spend her sixth dollar to purchase the first unit of X (from which she receives 11
utils) rather than the sixth unit of Y (from which she would receive only 10 utils). Her seventh and eighth
dollars should be spent on purchasing the sixth unit of Y and the second unit of X. Both give her 10 utils of
utility. The individual cannot go on purchasing more units of X or Y because her income is exhausted.
When this individual spends her income to purchase 2 units of X and 6 units of Y, her total utility is 107 utils
(see Table 4.9). This represents the maximum utility this individual can receive from the expenditure. If this
individual spent her income in any other way, the total utility would be less. For example, if she gave up the
second unit of X to buy the seventh unit of Y, she would lose 10 utils and gain only 8 utils (see Table 4.9). Similarly,
if she gave up the sixth unit of Y to purchase the third unit of X, she would lose 10 utils and gain only 9
utils. If she spent all of her income to purchase 8 units of X, she would receive only 60 units of utility [see row
(2) of the table]. If she bought 8 units of Y, her total utility would be 100 utils [see row (3) of the table].
MU x
P x
¼ MU y
P y
¼ 10
$1
and P x Q x þ P y Q y ¼ ($1)(2) þ ($1)(6) ¼ $8
4.9 Table 4.10 gives an individual’s marginal utility schedule for commodity X and commodity Y. Suppose
that the price of X and the price of Y are $2, that the individual has $20 of income per time period, and that
he spends it all on X and Y. (a) State the equilibrium condition for this individual, (b) If “commodity” Y is
savings, how would the equilibrium condition be affected? (c) Suppose that the MU of the fourth unit of Y
was 7 utils rather than 8. What effect would this have on the equilibrium condition? (d ) Suppose that the
MU x increased continuously as the individual consumed more of X [while the MU schedule for Y
remained unchanged as indicated in row (3) of Table 4.10]. How should the consumer rearrange his expenditures
to maximize his total utility? (e) Over what range of the MU function do consumers operate?
Table 4.10
(1) Q 1 2 3 4 5 6 7 8 9 10 11
(2) MU x 16 14 11 10 9 8 7 6 5 3 1
(3) MU y 15 13 12 8 6 5 4 3 2 1 0
(a)
MU x
P x
¼ MU y
P y
¼ 8
$2
P x Q x þ P y Q y ¼ ($2)(6) þ ($2)(4) ¼ $20
(b) If Y referred to savings rather than to a consumption good and the MU y schedule of row (3) represented the
utility received by this individual from saving part of his income, the equilibrium condition for this consumer
would remain completely unchanged. In order to maximize the total utility from his income, this consumer
should spend $12 of his income to purchase 6 units of commodity X and save the remaining $8.
(c) If the MU of the fourth unit of Y was 7 rather than 8, in order to be in equilibrium this individual should buy a
little more than 6 units of X and a little less than 4 units of Y. If the consumer could not purchase fractions of a
unit of X and Y, then the individual could continue to buy 6 units of X (6X) and 4 units of Y (4Y), but now the
equilibrium condition would hold only approximately and not precisely.
(d ) If the MU x [row (2) of the table] had been rising continuously rather than falling, this individual should spend
all of his income to buy 10 units of X in order to maximize his total utility.
(e) Since in the real world we observe that consumers spend their income on many commodities rather than on a
single one, consumers operate over the falling portion of the MU function. In addition, the MU function also
becomes irrelevant past the saturation point, because the consumer would not be willing to get more of this
commodity even if it were free. Thus, the relevant portion of the MU function is its positive but falling portion.