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

CHAP. 4] CONSUMER DEMAND THEORY 81
(b)
If this consumer spent all of his income on commodity Y, he could purchase 16 units of it. If he spent all of his
income on commodity X, he could purchase 8 units of it. Joining these two points by a straight line, we get this
consumer’s budget constraint line. The budget constraint line gives us all the different combinations of X and
Y that the consumer could buy. Thus he could buy 16Y and 0X, 14Y and 1X, 12Y and 2X, ..., 0Y and 8X.
Note that for each two units of Y that the consumer gives up he can purchase one additional unit of X. The
slope of this budget line has a value of –2 and remains constant. Also to be noted is that all points on the
budget line indicate that the consumer is spending all of his income on X and Y. That is,
P x Q x þ P y Q y ¼ M ¼ $16.
4.18 Given a consumer’s money income (M ), P y and P x ,(a) indicate the quantity of Y the consumer could
purchase if she spent all of her income on Y, (b) indicate the quantity of X the consumer could purchase
if she spent all of her income on X, (c) find the slope of the budget constraint line in terms of P x and P y ,
and (d ) find the general equation of the budget constraint line.
(a) Q y0 ¼ M P y
, when Q x ¼ 0
(b) Q x0 ¼ M P x
, when Q y ¼ 0
(c)
slope ¼ DY
DX ¼
Q y0
Q x0
¼
M=P y
M=P x
¼
M P y
P x
M ¼
P x
P y
Fig. 4-18
(d ) The general equation of a straight line can be written as y ¼ a þ bx, where a ¼ y-intercept or the value of y
when x ¼ 0 and b ¼ slope of the line. From the answer to part (a) we know that a ¼ M/P y and from the
answer to part (c) we know that b ¼ 2P x /P y .
Therefore, the general equation of the budget constraint line is
Q y ¼ M P y
P x
P y
Q x
By multiplying each term of the previous equation by P y and then rearranging the terms, we get an equivalent
way of expressing the equation of the budget constraint line. That is,
(P y ) Q y ¼ M
P x
Q x gives P y Q y ¼ M P x Q x
P y P y
By transposing the last term (2P x Q x ) to the left of the equal sign, we get P x Q x þ P y Q y ¼ M.
4.19 (a) Find the specific equation of the budget constraint line in Problem 4.17. (b) Show an equivalent way
of expressing the specific equation of the budget constraint line in part (a).
(a) In Problem 4.17, the y-intercept (a) ¼ M/P y ¼ $16/$1 ¼ 16. The slope of the budget line (b) ¼2P x /
P y ¼ 22/1 ¼ 2. Therefore, the specific equation of the budget line in Problem 4.17 is given by
Q y ¼ 16 2 2Q x . By substituting various values for Q x into this equation, we get the corresponding values
for Q y . Thus, when Q x ¼ 0, Q y ¼ 16; when Q x ¼ 1, Q y ¼ 14; when Q x ¼ 2, Q y ¼ 12; ...; when Q x ¼ 8,
Q y ¼ 0.
(b)
Another way to write the Problem 4.17 budget line equation is
($2)(Q x ) þ ($1)(Q y ) ¼ $16
By substituting various quantities of one commodity into this equation, we get the corresponding quantities
of the other commodity that the consumer must purchase if he is to remain on his budget line. For
example, if Q x ¼ 2, the consumer must purchase 12 units of Y if he is to remain on his budget line (i.e., if
he is to spend all of his income of $16 on X and Y).