The Budget Constraint

What money allows

In consumer theory we assume a person has three fixed pieces of information:

• their income (or budget) $M$, that is, the money they can spend;

• the price of each good;

• some tastes, which we cover in the other sections of the chapter.

To draw everything on a plane we work with two goods, $x$ and $y$. It's not that only two exist: the model simplifies, and the second good stands for "everything else" (remember ceteris paribus: everything else is held constant).

With $M = \$100$, corn on the cob ($x$) at $\$20$ and hibiscus water ($y$) at $\$10$ a glass, we can try combinations $(x, y)$:

The equation and the graph

If you buy $x$ corns at $p_x$ and $y$ waters at $p_y$, you spend $p_x \cdot x + p_y \cdot y$. That amount cannot exceed $M$:

The reason is simple: each good "uses up" part of the budget, and the sum of all the parts cannot exceed what you have. If you solve for $y$ you get a linear function $y = \dfrac{M}{p_y} - \dfrac{p_x}{p_y}\,x$: y-intercept $\dfrac{M}{p_y}$ and negative slope.

There are three types of points:

• Below the line (inside the triangle): affordable, with money left over.

• On the line: affordable and they use up exactly the whole budget.

• Above the line: unaffordable, they cost more than $M$.

Opportunity cost and slope

Look at the line above. If you move along it to buy one more corn, you need $\$20$ and you take it from the waters: you give up buying $20 \div 10 = 2$ waters. That sacrifice is the opportunity cost of the corn measured in waters.

The slope of the line says the same thing:

In the example: $\dfrac{20}{10} = 2$ waters per corn. The other way around, one water costs $\dfrac{10}{20} = 0.5$ corns.

When income or prices change

The line is not fixed. With the corn example ($M = \$100$, $p_x = \$20$, $p_y = \$10$):

• Income rises (from $\$100$ to $\$150$): the line shifts outward, parallel to the original. Relative prices don't change, so the slope is the same; there are simply more affordable combinations. If income falls, the line shifts inward.

• The price of one good falls (corn from $\$20$ to $\$10$): the line pivots around the intercept on the other axis. Now $10$ corns are affordable, but $10$ waters are still affordable.

• The price of one good rises: the line pivots inward, with the same intercept for the other good.

Choosing well: marginal utility per peso

Knowing what you can afford doesn't tell you what is best for you to buy. To choose, you compare what you get for each peso. In the section on marginal utility you saw that MU is what one more unit adds (in "utils"). Since goods have different prices, we compare marginal utility per peso:

Intuition: if a peso spent on $y$ gives you more satisfaction than a peso spent on $x$, it pays to shift a peso from $x$ to $y$. You keep doing this until there is no more gain.

Back to the beginning: with $\$100$ at the street market you can only afford the combinations in the region below the line $20x + 10y = 100$. Among them, you choose using the $\dfrac{MU}{p}$ rule.

Summary