3  Budget Constraints

For more information on these topics, see Varian Chapter 2: Budget Constraint.

Objective

I’ll begin this course by developing a theory of the consumer. In particular, economists assume that consumers choose the best bundle of goods they can afford.

In precise terms, what do I mean by can afford? That’s the focus of this chapter.

What do I mean by best bundle? That’s the focus of next chapter.

3.1 The Budget Constraint

Consumption Bundle \(X = (x_1, x_2)\): In the case of two goods, a consumption bundle \(X\) represents the quantities of each good that a consumer chooses to consume. Specifically, \(x_1\) denotes the amount of good 1, and \(x_2\) denotes the amount of good 2 in the bundle.

Practice Question: If you’re shopping for coffee and detergent, you have $20 to spend, and coffee is $5 while detergent is $6, which of these describe bundles that are affordable to you?

\(5 x_{coffee} \leq 20\)

\(6 x_{detergent} \leq 20\)

\(5 x_{coffee} + 6 x_{detergent} \leq 20\)

\(5 x_{coffee} + 6 x_{detergent} \geq 20\)

Practice Question: According to the conditions in the previous question, which of these bundles would not be affordable?

2 coffees and 1 detergent

2 coffees and 2 detergents

0 coffees and 3 detergents

1 coffee and 2 detergents

3.2 Why Limiting to Two Goods Isn’t Restrictive

Considering only two goods at once is less restrictive than it seems. One of the goods can represent everything else the consumer might want to consume.

For example, if we want to study consumer demand for cups of espresso, we can let \(x_1\) measure a person’s consumption of espresso, and \(x_2\) represent everything else the consumer might want to consume. It is convenient to let \(x_2\) stand for the dollars the consumer can use to spend on other goods. Thus, the price of good 2 is automatically 1, since the price of one dollar is one dollar. Therefore, our budget constraint becomes:

\[p_1 x_1 + x_2 \leq m.\]

In this context, \(x_2\) is called a composite good.

Composite good: a theoretical representation that stands for all other goods a consumer might want to consume besides a specific good being studied, often measured in dollars for convenience.

Practice Question: If you’re going shopping and you have $50 to spend, flowers cost $4.99, and you plan on buying other things, which of these represent your budget constraint?

\(4.99 x_{flowers} \geq 50\)

\(4.99 x_{flowers} \leq 50\)

\(4.99 x_{flowers} + x_2 \geq 50\)

\(4.99 x_{flowers} + x_2 \leq 50\)

3.3 Classwork 3

1. We learned that a budget constraint describes a set of bundles \((x_1, x_2)\) that are affordable given \(p_1\) and \(p_2\) and an income \(m\): \[p_1 x_1 + p_2 x_2 \leq m.\] Let the budget line be the equality: \[p_1 x_1 + p_2 x_2 = m\] which refers to bundles that exactly exhaust the consumer’s income \(m\). Let \(x_1 = x\) and \(x_2 = y\) and rearrange the budget line equation to the form \(y = mx + b\). The slope of the budget line is \(\underline{\hspace{2cm}}\) and the y-intercept is \(\underline{\hspace{2cm}}\).

2. Continuing from the previous question, sketch the budget line for \(p_{apple} = 1\), \(p_{banana} = 2\), and \(m = 5\). Note that all the bundles below the budget line are affordable. The x-intercept is \(\underline{\hspace{2cm}}\) and the y-intercept is \(\underline{\hspace{2cm}}\). Note that the significance of the intercepts are that if you spend all your money on a certain good, it is the maximum amount of that good you can afford. What is the x-intercept as a function of \(m\) and \(p_1\)?

3. Write a proof of the fact that increasing the consumer’s income \(m\) will shift out the budget line.

   Hint: start by defining precisely what “shift out” means, in reference to the slope and y-intercept in \(y = mx + b\).

4. If the price of good 2 increases, will the budget line become flatter or steeper? Make sure good 2 is drawn on the y-axis. Will the x and y intercepts change? Sketch what will happen to the budget line in this case.