[Editor’s note: We’re bringing back price theory with our series on Price Theory problems with Professor Bryan Cutsinger. You can view the previous problem and Cutsinger’s solution here and here. Share your proposed solutions in the Comments. Professor Cutsinger will be present in the comments for the next two weeks, and we’ll again post his proposed solution shortly thereafter. May the graphs be ever in your favor, and long live price theory!]

Solution:

This question is one I like to ask my students when I introduce the notion of a budget constraint. As I’ll explain shortly, it highlights an important point in consumer theory–namely, that what influences consumer behavior is their real (i.e., inflation-adjusted) wages and the real prices of the goods they consume.

The simplest way to answer this question is to set up the consumer’s budget constraint. In this case, we have a consumer who uses all her money income to purchase two goods, X and Y. Let’s assume that the prices of X and Y are unaffected by how much of either good she purchases–a reasonable approximation for many consumer goods.

We can express the budget constraint mathematically as:

Here, M denotes her money income, which equals the number of hours she works times her hourly wage, P_x  and P_y denote the prices of the two goods, and X and Y denote the quantities she consumes. [1]

Since the question tells us that she uses her money income to purchase only two goods, we know that whatever combination of X and Y our consumer purchases must satisfy this condition.

Solving the budget constraint for Y will be more useful for our purpose:

The ratio P_x/P_y  is the price of X in terms of Y. It represents the amount of Y our consumer must give up in exchange for an additional unit of X. This ratio is the real price of X. By the same logic, the ratio P_y/P_x is the real price of Y.

The ratio M/P_y is the purchasing power of her income in units of Y. Think of this ratio as her real income (we could also express her real income in units of X).

The question states that her money income doubles along with the prices of X and Y. We can illustrate this change as follows:

Viewed this way, it’s clear that doubling her money income and the dollar prices of the two goods she consumes has no effect on her budget constraint, as the twos will cancel out, yielding the initial budget constraint. 

Since real prices and real income are what influence people’s behavior, doubling the dollar prices of X and Y and her money income will not affect the quantities of these goods she purchases (assuming  this doubling did not affect her preferences for goods X and Y).

We could consider interesting extensions. For example, what happens if prices double but her money income doesn’t. Or, we could consider a case where the prices of the two goods rise by different proportions. These extensions involve changes in real prices and real income, and, unsurprisingly, would result in our consumer changing her behavior.

[1] Note that we could express her money income in hourly terms, in which case, M would just be her wage, or in monthly or annual terms. While it doesn’t matter much which option we pick, it’s crucial that we express the quantities of X and Y she consumes in the same terms. For example, if M denotes her annual income, then X and Y should denote the quantities of these goods she consumes per year.

Bryan Cutsinger is an assistant professor of economics in the College of Business at Florida Atlantic University and a Phil Smith Fellow at the Phil Smith Center for Free Enterprise. He is also a fellow with the Sound Money Project at the American Institute for Economic Research, and a member of the editorial board for the journal Public Choice.