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

76 CONSUMER DEMAND THEORY [CHAP. 4
(a)
(b)
(c)
The MU curve is shown in Fig. 4-12. In this case, the TU function is a smooth or continuous curve. The MU
corresponding to each point on the TU curve is given by the slope of the TU curve (or the slope of the tangent
to the TU curve) at that point. Thus at point A, the slope of the TU curve is equal to 6. This corresponds to point
A 0 on the MU curve. At point B, the slope of the TU curve or the MU is equal to 20. This gives point B 0 .AtC,
D, and E, the slope of the TU curve is 7, 0, and 24, respectively. Thus we get points, C 0 , D 0 , and E 0 on the MU
curve. By joining points O, A 0 , B 0 , C 0 , D 0 , and E 0 we get the MU curve corresponding to the given TU curve.
As long as the TU curve faces up (from O to B), the TU is increasing at an increasing rate and the MU is rising.
At point B, the TU curve changes direction (from facing up to facing down). At this point, the slope of the TU
curve (the MU ) is maximum. (In mathematics, B is called an inflection point.) Past point B, the TU curve faces
down. That is, TU is increasing at a decreasing rate and MU is falling. At point D, the TU is maximum so the
slope of the TU curve, or the MU, is zero. Past point D, the TU curve begins to fall so that the MU is negative.
In textbooks, the TU function is usually given as a smooth curve and may or may not have a range over which
it is increasing at an increasing rate. However, the economically relevant range of the TU curve is the portion
over which the TU is rising at a declining rate (in the previous diagram, from point B to point D). This corresponds
to the positive but declining range of the MU curve (i.e., from point B 0 to D 0 ). The reason for this will
be discussed in Problem 4.9(e).
CONSUMER EQUILIBRIUM
4.7 (a) What constraints or limitations does the consumer face in seeking to maximize the total utility from
personal expenditures? (b) Express mathematically the condition for consumer equilibrium. (c) Explain
the meaning of your answers to part (b).
(a)
(b)
In seeking to maximize the total utility from personal expenditures, the consumer faces an income and price
constraint or limitation. That is, the consumer has a given and limited income over a specific period of time
and faces given and fixed prices of the commodities he or she seeks to purchase (i.e., the individual consumer
is too small to affect market prices). Thus, given individual income and price constraints, the rational consumer
seeks to maximize the total utility from expenditures.
The condition for consumer equilibrium can be expressed mathematically, as follows
MU x
P x
¼ MU y
P y
¼ ...
subject to the constraint that
P x Q x þ P y Q y þ¼M (the individual’s money income)
(c)
The above two expressions mean that the marginal utility of the last dollar spent on X must be equal to the
marginal utility of the last dollar spent on Y and so on for all commodities purchased, subject to the constraint
that the amount of money spent on X (P x Q x ) plus the amount of money spent on Y (P y Q y ) plus the amount of
money spent on all other commodities purchased by this individual exactly equals the individual’s money
income (if we assume that the total income is spent; i.e., if we assume that nothing is saved).
4.8 Table 4.9 gives an individual’s marginal utility schedule for commodity X and commodity Y. Suppose
that X and Y are the only commodities available, the price of X and the price of Y are $1, and the individual’s
income is $8 per time period and is all spent. (a) Indicate how this individual should spend her
income in order to maximize her total utility. (b) What is the total amount of utility received by the individual
when in equilibrium? (c) State mathematically the equilibrium condition for the consumer.
Table 4.9
(1) Q 1 2 3 4 5 6 7 8 Total
(2) MU x 11 10 9 8 7 6 5 4 60
(3) MU y 19 17 15 13 12 10 8 6 100