Appendix to Chapter 3 Consumer Preferences and Demand 123unit of X. As X becomes more abundant and Y more scarce, X’s relative value
diminishes and Y’s relative value increases.THE BUDGET CONSTRAINT Having described her preferences, next we deter-
mine the consumer’s alternatives. The amount of goods she can purchase
depends on her available income and the goods’ prices. Suppose the consumer
sets aside $20 each week to spend on the two goods. The price of good X is $4
per unit, and the price of Y is $2 per unit. Then she is able to buy any quantities
of the goods (call these quantities X and Y) as long as she does not exceed her
income. If she spends the entire $20, her purchases must satisfy[3A.1]This equation’s left side expresses the total amount the consumer spends on
the goods. The right side is her available income. According to the equation,
her spending just exhausts her available income.^2 This equation is called the
consumer’s budget constraint.Figure 3A.2 depicts the graph of this constraint.
For instance, the consumer could purchase 5 units of X and no units of Y
(point A), 10 units of Y and no units of X (point C), 3 units of X and 4 units of
Y (point B), or any other combination along the budget line shown. Note that
bundles of goods to the northeast of the budget line are infeasible; they cost
more than the $20 that the consumer has to spend.OPTIMAL CONSUMPTION We are now ready to combine the consumer’s
indifference curves with her budget constraint to determine her optimal pur-
chase quantities of the goods. Figure 3A.3 shows that the consumer’s optimal
combination of goods lies at point B, 3 units of X and 4 units of Y. Bundle B is
optimal precisely because it lies on the consumer’s “highest” attainable indif-
ference curve while satisfying the budget constraint. (Check that all other bun-
dles along the budget line lie on lower indifference curves.)
Observe that, at point B, the indifference curve is tangent to the budget
line. This means that at B the slope of the indifference curve is exactly equal
to the slope of the budget line. Let’s consider each slope in turn. The slope of
the budget line (the “rise over the run”) is 2. This slope can be obtained from
the graph directly or found by rearranging the budget equation in the form
Y 10 2X. As a result, Y/X 2. More generally, we can write the budget
equation in the form:where PXand PYdenote the goods’ prices and I is the consumer’s income.
Rearranging the budget equation, we find Y I/PY(PX/PY)X. Therefore,PXXPYYI,
4X2Y20.
(^2) Because both goods are valuable to the consumer, she will never spend lessthan her allotted
income on the goods. To do so would unnecessarily reduce her level of welfare.
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