3.3 Budgets
For more information on these topics, see Varian Intermediate Microeconomics Chapter 2: Budget Constraint.
This unit builds the theory of consumer behavior. In the simplest possible terms, economists model consumers choosing the best bundle of goods they can afford. In this assignment, you will explore what it means for a consumer to be able to afford a bundle of goods in mathematical terms.
Math Refresher: Plotting Lines
What is the slope and y-intercept of this line? \(y = 3 x - 2\)
Answer:
Solve this equation for y, then find the slope and y-intercept of this line: \(x + 2 y = 4\)
Answer:
\[\begin{align} \end{align}\]
- Use ggplot to plot the line from the previous question.
Answer:
Find the x-intercept and y-intercept of this line: \(3x + 2 y = 6\) (recall that you can find the x-intercept by setting y = 0, and you can find the y-intercept by setting x = 0).
Answer:
Budget Constraints
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.
- 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?
\(5 x_{coffee} \leq 20\)
\(6 x_{detergent} \leq 20\)
\(5 x_{coffee} + 6 x_{detergent} \leq 20\)
\(5 x_{coffee} + 6 x_{detergent} \geq 20\)
Answer:
- 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
Answer:
A budget constraint describes all bundles \((x_1, x_2)\) that are affordable given prices \(p_1\) and \(p_2\) and income \(m\): \[p_1 x_1 + p_2 x_2 \leq m\] The budget line represents bundles that exactly use up the consumer’s income: \[p_1 x_1 + p_2 x_2 = m\]
For a budget line with \(p_{apple} = 1\), \(p_{banana} = 2\), and \(m = 5\), the x-intercept represents the maximum amount of apples you can afford if you spent all your money on apples (which is ___ apples), and the y-intercept represents the maximum amount of bananas you can buy if you spent all your money on bananas (which is ___ bananas). Solve the budget line \(1 x_1 + 2 x_2 = 5\) for \(x_2\) and sketch the budget line using ggplot.
Answer:
Add another line to your ggplot: what happens if prices stay the same, but your income increases to \(m = 6\)? Your budget line (pivots to be flatter; pivots to be steeper; shifts in; shifts out).
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Why Limiting to Two Goods Isn’t Restrictive
Limiting our analysis to only two goods is less restrictive than it seems because one of the goods can represent everything else the consumer might want to consume.
For example, if we want to study consumer demand for housing, we can let \(x_1\) measure a person’s consumption of housing, 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.
- 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?
- \(4.99 x_{flowers} \geq 50\)
- \(4.99 x_{flowers} \leq 50\)
- \(4.99 x_{flowers} + x_2 \geq 50\)
- \(4.99 x_{flowers} + x_2 \leq 50\)
Answer:
Inflation
- Consider a scenario where all prices in the economy double. How much would your income have to change by to leave you with the same purchasing power (i.e., leave you equally well off) as before the price increase?