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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!

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!

Next: The Geometry of the Budget Line
Copyright (c) Christopher Makler /