# The Budget Set

For the purposes of this analysis, a “consumer” is someone with money who wants to buy goods to make themselves happy. The amount of money, and the prices they have to pay for goods, determine which bundles are *feasible* (or *affordable*) and which are not.

Because we’re assuming the feasible set is determined by the consumer’s “budget,” we’ll call it a **budget set**. A consumer’s budget set divides her choice space into those combinations which are “affordable” and those which are “unaffordable” for a given amount of money.

## The budget constraint for two goods

The simplest consumer choice problem involves buying quantities of two goods. We can analyze this choice in “good 1 - good 2” space. To make things concrete, let’s suppose “good 1” are apples and “good 2” are bananas. If a consumer can buy apples at $p_1$ dollars per apple, and bananas at price $p_2$ per banana, then every possible consumption bundle $X = (x_1,x_2)$ has an associated **cost**:
\(\text{Cost of bundle }(x_1,x_2) = p_1x_1 + p_2x_2\)
If the consumer has a certain amount of money, $m$, to spend on apples and bananas, then the set of combinations she can afford is the set of those combinations that cost less than that amount of money:
\(\text{Budget set: }\{(x_1,x_2)\ |\ p_1x_1 + p_2x_2 \le m\}\)
We can plot this budget set in good 1 - good 2 space. You can drag the bundle $X$ around this space to see which bundles are affordable, and also change the prices $p_1$ and $p_2$ and the amount of money $m$ to see how her budget set changes:

## General Budget Constraint for $n$ Goods

Of course, the notion of a budget set extends well beyond just buying two goods. If $n$ goods may be bought at prices $p_1, p_2, …, p_n$, then the general budget constraint may be written as \(p_1x_1 + p_2x_2 + \cdots + p_nx_n \le m\) where $p_i$ is the price of good $i$, and $x_i$ is the quantity of good $i$. More generally, using vector notation, we can write this constraint as a dot product: \(\vec{p} \cdot \vec{x} \le m\) where $\vec{p} = [p_1,p_2,…,p_n]$ is a vector of the prices of $n$ goods, and $\vec{x} = [x_1,x_2,…,x_n]$ is a vector of quantities of $n$ goods.

Note that each of the terms in this constraint is measured in dollars: for each good $i$, the term $p_ix_i$ represents the expenditure on good $i$:

$$\left[p_i \ \frac{\text{dollars}}{\cancel{\text{units of good $i$}}} \right] \times \left[x_i \ \cancel{\text{units of good $i$}}\right] = p_ix_i \text{ dollars}$$

Again, paying close attention to the units of numbers will be important throughout these models!