The Budget Line

As we have already seen, a consumer’s choices are limited by the budget available. Total spending for goods and services can fall short of the budget constraint but may not exceed it.

Algebraically, we can write the budget constraint for two goods X and Y as:

where P_X and P_Y are the prices of goods X and Y and Q_X and Q_Y are the quantities of goods X and Y chosen. The total income available to spend on the two goods is B, the consumer’s budget. Equation 7.7 states that total expenditures on goods X and Y (the left-hand side of the equation) cannot exceed B.

Suppose a college student, Janet Bain, enjoys skiing and horseback riding. A day spent pursuing either activity costs $50. Suppose she has $250 available to spend on these two activities each semester. Ms. Bain’s budget constraint is illustrated in Figure 7.9 "The Budget Line".

For a consumer who buys only two goods, the budget constraint can be shown with a budget line. A budget lineGraphically shows the combinations of two goods a consumer can buy with a given budget. shows graphically the combinations of two goods a consumer can buy with a given budget.

The budget line shows all the combinations of skiing and horseback riding Ms. Bain can purchase with her budget of $250. She could also spend less than $250, purchasing combinations that lie below and to the left of the budget line in Figure 7.9 "The Budget Line". Combinations above and to the right of the budget line are beyond the reach of her budget.

The vertical intercept of the budget line (point D) is given by the number of days of skiing per month that Ms. Bain could enjoy, if she devoted all of her budget to skiing and none to horseback riding. She has $250, and the price of a day of skiing is $50. If she spent the entire amount on skiing, she could ski 5 days per semester. She would be meeting her budget constraint, since:

$$\$50\times 0+\$50\times \text{5 }=\$250$$

The horizontal intercept of the budget line (point E) is the number of days she could spend horseback riding if she devoted her $250 entirely to that sport. She could purchase 5 days of either skiing or horseback riding per semester. Again, this is within her budget constraint, since:

$$\$50\times \text{5 }+\$50\times 0=\$250$$

Because the budget line is linear, we can compute its slope between any two points. Between points D and E the vertical change is −5 days of skiing; the horizontal change is 5 days of horseback riding. The slope is thus $-5/5=-1$. More generally, we find the slope of the budget line by finding the vertical and horizontal intercepts and then computing the slope between those two points. The vertical intercept of the budget line is found by dividing Ms. Bain’s budget, B, by the price of skiing, the good on the vertical axis (P_S). The horizontal intercept is found by dividing B by the price of horseback riding, the good on the horizontal axis (P_H). The slope is thus:

Simplifying this equation, we obtain

After canceling, Equation 7.9 shows that the slope of a budget line is the negative of the price of the good on the horizontal axis divided by the price of the good on the vertical axis.

Indifference Curves

Suppose Ms. Bain spends 2 days skiing and 3 days horseback riding per semester. She will derive some level of total utility from that combination of the two activities. There are other combinations of the two activities that would yield the same level of total utility. Combinations of two goods that yield equal levels of utility are shown on an indifference curveGraph that shows combinations of two goods that yield equal levels of utility..Limiting the situation to two goods allows us to show the problem graphically. By stating the problem of utility maximization with equations, we could extend the analysis to any number of goods and services. Because all points along an indifference curve generate the same level of utility, economists say that a consumer is indifferent between them.

Figure 7.10 "An Indifference Curve" shows an indifference curve for combinations of skiing and horseback riding that yield the same level of total utility. Point X marks Ms. Bain’s initial combination of 2 days skiing and 3 days horseback riding per semester. The indifference curve shows that she could obtain the same level of utility by moving to point W, skiing for 7 days and going horseback riding for 1 day. She could also get the same level of utility at point Y, skiing just 1 day and spending 5 days horseback riding. Ms. Bain is indifferent among combinations W, X, and Y. We assume that the two goods are divisible, so she is indifferent between any two points along an indifference curve.

Now look at point T in Figure 7.10 "An Indifference Curve". It has the same amount of skiing as point X, but fewer days are spent horseback riding. Ms. Bain would thus prefer point X to point T. Similarly, she prefers X to U. What about a choice between the combinations at point W and point T? Because combinations X and W are equally satisfactory, and because Ms. Bain prefers X to T, she must prefer W to T. In general, any combination of two goods that lies below and to the left of an indifference curve for those goods yields less utility than any combination on the indifference curve. Such combinations are inferior to combinations on the indifference curve.

Point Z, with 3 days of skiing and 4 days of horseback riding, provides more of both activities than point X; Z therefore yields a higher level of utility. It is also superior to point W. In general, any combination that lies above and to the right of an indifference curve is preferred to any point on the indifference curve.

We can draw an indifference curve through any combination of two goods. Figure 7.11 "Indifference Curves" shows indifference curves drawn through each of the points we have discussed. Indifference curve A from Figure 7.10 "An Indifference Curve" is inferior to indifference curve B. Ms. Bain prefers all the combinations on indifference curve B to those on curve A, and she regards each of the combinations on indifference curve C as inferior to those on curves A and B.

Although only three indifference curves are shown in Figure 7.11 "Indifference Curves", in principle an infinite number could be drawn. The collection of indifference curves for a consumer constitutes a kind of map illustrating a consumer’s preferences. Different consumers will have different maps. We have good reason to expect the indifference curves for all consumers to have the same basic shape as those shown here: They slope downward, and they become less steep as we travel down and to the right along them.

The slope of an indifference curve shows the rate at which two goods can be exchanged without affecting the consumer’s utility. Figure 7.12 "The Marginal Rate of Substitution" shows indifference curve C from Figure 7.11 "Indifference Curves". Suppose Ms. Bain is at point S, consuming 4 days of skiing and 1 day of horseback riding per semester. Suppose she spends another day horseback riding. This additional day of horseback riding does not affect her utility if she gives up 2 days of skiing, moving to point T. She is thus willing to give up 2 days of skiing for a second day of horseback riding. The curve shows, however, that she would be willing to give up at most 1 day of skiing to obtain a third day of horseback riding (shown by point U).

The maximum amount of one good a consumer would be willing to give up in order to obtain an additional unit of another is called the marginal rate of substitution (MRS)The maximum amount of one good a consumer would be willing to give up in order to obtain an additional unit of another., which is equal to the absolute value of the slope of the indifference curve between two points. Figure 7.12 "The Marginal Rate of Substitution" shows that as Ms. Bain devotes more and more time to horseback riding, the rate at which she is willing to give up days of skiing for additional days of horseback riding—her marginal rate of substitution—diminishes.

The Utility-Maximizing Solution

We assume that each consumer seeks the highest indifference curve possible. The budget line gives the combinations of two goods that the consumer can purchase with a given budget. Utility maximization is therefore a matter of selecting a combination of two goods that satisfies two conditions:

1. The point at which utility is maximized must be within the attainable region defined by the budget line.
2. The point at which utility is maximized must be on the highest indifference curve consistent with condition 1.

Figure 7.13 "The Utility-Maximizing Solution" combines Janet Bain’s budget line from Figure 7.9 "The Budget Line" with her indifference curves from Figure 7.11 "Indifference Curves". Our two conditions for utility maximization are satisfied at point X, where she skis 2 days per semester and spends 3 days horseback riding.

The highest indifference curve possible for a given budget line is tangent to the line; the indifference curve and budget line have the same slope at that point. The absolute value of the slope of the indifference curve shows the MRS between two goods. The absolute value of the slope of the budget line gives the price ratio between the two goods; it is the rate at which one good exchanges for another in the market. At the point of utility maximization, then, the rate at which the consumer is willing to exchange one good for another equals the rate at which the goods can be exchanged in the market. For any two goods X and Y, with good X on the horizontal axis and good Y on the vertical axis,

Utility Maximization and the Marginal Decision Rule

How does the achievement of The Utility Maximizing Solution in Figure 7.13 "The Utility-Maximizing Solution" correspond to the marginal decision rule? That rule says that additional units of an activity should be pursued, if the marginal benefit of the activity exceeds the marginal cost. The observation of that rule would lead a consumer to the highest indifference curve possible for a given budget.

Suppose Ms. Bain has chosen a combination of skiing and horseback riding at point S in Figure 7.14 "Applying the Marginal Decision Rule". She is now on indifference curve C. She is also on her budget line; she is spending all of the budget, $250, available for the purchase of the two goods.

An exchange of two days of skiing for one day of horseback riding would leave her at point T, and she would be as well off as she is at point S. Her marginal rate of substitution between points S and T is 2; her indifference curve is steeper than the budget line at point S. The fact that her indifference curve is steeper than her budget line tells us that the rate at which she is willing to exchange the two goods differs from the rate the market asks. She would be willing to give up as many as 2 days of skiing to gain an extra day of horseback riding; the market demands that she give up only one. The marginal decision rule says that if an additional unit of an activity yields greater benefit than its cost, it should be pursued. If the benefit to Ms. Bain of one more day of horseback riding equals the benefit of 2 days of skiing, yet she can get it by giving up only 1 day of skiing, then the benefit of that extra day of horseback riding is clearly greater than the cost.

Because the market asks that she give up less than she is willing to give up for an additional day of horseback riding, she will make the exchange. Beginning at point S, she will exchange a day of skiing for a day of horseback riding. That moves her along her budget line to point D. Recall that we can draw an indifference curve through any point; she is now on indifference curve E. It is above and to the right of indifference curve C, so Ms. Bain is clearly better off. And that should come as no surprise. When she was at point S, she was willing to give up 2 days of skiing to get an extra day of horseback riding. The market asked her to give up only one; she got her extra day of riding at a bargain! Her move along her budget line from point S to point D suggests a very important principle. If a consumer’s indifference curve intersects the budget line, then it will always be possible for the consumer to make exchanges along the budget line that move to a higher indifference curve. Ms. Bain’s new indifference curve at point D also intersects her budget line; she’s still willing to give up more skiing than the market asks for additional riding. She will make another exchange and move along her budget line to point X, at which she attains the highest indifference curve possible with her budget. Point X is on indifference curve A, which is tangent to the budget line.

Having reached point X, Ms. Bain clearly would not give up still more days of skiing for additional days of riding. Beyond point X, her indifference curve is flatter than the budget line—her marginal rate of substitution is less than the absolute value of the slope of the budget line. That means that the rate at which she would be willing to exchange skiing for horseback riding is less than the market asks. She cannot make herself better off than she is at point X by further rearranging her consumption. Point X, where the rate at which she is willing to exchange one good for another equals the rate the market asks, gives her the maximum utility possible.

Utility Maximization and Demand

Figure 7.14 "Applying the Marginal Decision Rule" showed Janet Bain’s utility-maximizing solution for skiing and horseback riding. She achieved it by selecting a point at which an indifference curve was tangent to her budget line. A change in the price of one of the goods, however, will shift her budget line. By observing what happens to the quantity of the good demanded, we can derive Ms. Bain’s demand curve.

Panel (a) of Figure 7.15 "Utility Maximization and Demand" shows the original solution at point X, where Ms. Bain has $250 to spend and the price of a day of either skiing or horseback riding is $50. Now suppose the price of horseback riding falls by half, to $25. That changes the horizontal intercept of the budget line; if she spends all of her money on horseback riding, she can now ride 10 days per semester. Another way to think about the new budget line is to remember that its slope is equal to the negative of the price of the good on the horizontal axis divided by the price of the good on the vertical axis. When the price of horseback riding (the good on the horizontal axis) goes down, the budget line becomes flatter. Ms. Bain picks a new utility-maximizing solution at point Z.

The solution at Z involves an increase in the number of days Ms. Bain spends horseback riding. Notice that only the price of horseback riding has changed; all other features of the utility-maximizing solution remain the same. Ms. Bain’s budget and the price of skiing are unchanged; this is reflected in the fact that the vertical intercept of the budget line remains fixed. Ms. Bain’s preferences are unchanged; they are reflected by her indifference curves. Because all other factors in the solution are unchanged, we can determine two points on Ms. Bain’s demand curve for horseback riding from her indifference curve diagram. At a price of $50, she maximized utility at point X, spending 3 days horseback riding per semester. When the price falls to $25, she maximizes utility at point Z, riding 4 days per semester. Those points are plotted as points X′ and Z′ on her demand curve for horseback riding in Panel (b) of Figure 7.15 "Utility Maximization and Demand".