Question:

Russ buys 5 sirloins per week. True or false: If the price of sirloin rises by $5 apiece, and if Russ’ preferences and income remain constant, he will have $25 a week less to spend on other things.

Solution:

One of the first things I emphasize in my micro principles course is that the behavioral patterns we observe in the real world are shaped by prices. When prices change, so does behavior.

This idea comes straight from consumer theory. In standard models, people maximize utility by consuming each good up to the point where the marginal value of one more unit equals its market price. When the price of a good rises—as in the example we’re considering here—the marginal value at the optimum must also rise. In this case, the marginal value at the new optimum must be $5 higher to match the new price of sirloin.

Because marginal value falls as Russ consumes more sirloin, he can restore the equality between marginal value and price—the condition for his optimum—only by consuming less. Without more information about his income or preferences, we can’t say exactly how much less, but we do know that he will buy fewer sirloins than before.

Thus, the statement in the original question is false: Russ will reduce his consumption of sirloin when its price rises, so it doesn’t necessarily follow that he has $25 less to spend on other things.

 

Once More, With Math

We can also see this result by examining Russ’s budget constraint. Suppose Russ uses his income, M, to purchase sirloin, S, and a composite good we’ll call “all other goods,” Y. His budget constraint is therefore

M=PSS+PYY

Here, PS​ and PY​ denote the price of sirloin and the price of all other goods, respectively.

The question tells us that the price of sirloin rises by $5, so his new budget constraint is

M=(PS+5)S+PYY

Since Russ’s income, M, and the price of other goods, PY,​ remain constant, the maximum amount of “all other goods” he could buy if he purchased no sirloin remains the same at M/PY​. In that sense, the maximum quantity of all other goods he can consume hasn’t changed.

However, the slope of his budget line has changed: sirloin has become relatively more expensive, so the budget line pivots inward around that intercept. This change in relative prices reduces Russ’s feasible combinations of sirloin and other goods, prompting him to move to a new optimum with less sirloin and more of other goods.

In this sense, Russ’s real income has fallen even though his nominal income remains the same. But because he reoptimizes—reallocating his spending between sirloin and other goods when the price changes—it does not follow that he has $25 less each week to spend on other goods.