7.3 The Budget Set

On Chuck’s desert island, his feasible set was determined by his available resources and the technology that could transform those resources into goods.

We’ll define a consumer as someone with money who wants to buy goods to make themselves happy. Their feasible set is determined by their resources — the amount of money they have available to spend — and the prices of goods. If you think about it, buying a candy bar for two dollars is just like a production function that “transforms” two dollar bills into one candy bar.

For our prototypical consumer, let’s consider (young) Jordan Sanders, played by Marsai Martin in the film $Little$. In this twist on the classic movie $Big$ (do I have a Tom Hanks thing? Doesn’t everybody?), Jordan is a tech CEO who wakes up one day as a thirteen-year-old. Like many 13-year-olds, we’ll assume she has a modest allowance and spends it on things like snacks and drinks.

We’ll call Jordan’s feasible set budget set. Just as a PPF divides the choice space into those goods which are possible to produce and those which are impossible, Jordan’s budget set divides her choice space into those combinations which are “affordable” and those which are “unaffordable” with a given amount of money.

The budget constraint for two goods

Let’s think about our usual “good 1 - good 2” space, in which we are discussing possible combinations of two goods. In particular, let’s let “good 1” be apples and “good 2” be bananas. If Jordan 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 Jordan has a certain amount of money, $m$, to spend on apples and bananas(Note: We will often refer to $m$ as her “income,” though it’s probably more accurate to think of it as the amount of money she has decided to devote to these goods, rather than her entire income.), 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!