The Geometry of the Budget Line

In a situation with constant prices, the budget constraint is linear; so we call it a budget line. The equation of a budget line with two goods is \(p_1x_1 + p_2x_2 = m\) To analyze the geometry of the budget line, let’s think about its intercepts and slope. Let’s imagine that a consumer has 24 dollars to spend on apples (good 1) and bananas (good 2). If apples cost $p_1 = 4$ dollars per apple, and bananas cost $p_2 = 2$ dollars per banana, then if she spent all 24 dollars on apples she could buy \(\overline x_1 = \frac{24\ \cancel{\text{dollars}}}{4\ \cancel{\text{dollars}}\text{/apple}} = 6 \text{ apples}\) Likewise, if she spent all 24 dollars on bananas, she could buy \(\overline x_2 = \frac{24\ \cancel{\text{dollars}}}{2\ \cancel{\text{dollars}}\text{/banana}} = 12 \text{ bananas}\) These points represent the intercepts of her budget line. To think about the slope, let’s think of what happens as the consumer spends more money on apples and less on bananas. If she buys one more apple, it costs her 4 dollars; since each banana costs her 2 dollars, she has to buy two fewer bananas. Therefore the market prices of $p_1 = $4 dollars per apple and $p_2 = $2 dollars per banana allow her to trade off between those goods at a rate of 2 bananas for every apple; thus the slope of the budget line is 2 bananas per apple.

We can see these illustrated in the following graph. As always, feel free to play around with the parameters to see how the budget line’s slope and intercepts change:

Formulas for the budget line intercepts and slope

Let’s analyze the equations for her slope and intercept for general values of $p_1$, $p_2$, and $m$.

If the consumer spends all her money on good 1 (so $x_2 = 0$), then this becomes $p_1x_1 = m$, so her horizontal intercept is given by \(\overline x_1 = {m \over p_1}\) Likewise, if she spends all her money on good 2, we have $p_2x_2 = m$, so her vertical intercept is \(\overline x_2 = {m \over p_2}\) Finally, if we write the budget line as \(x_2 = {m \over p_2} - {p_1 \over p_2}x_1\) we can see that the slope of the budget line is \(\text{Slope of BL }=-{p_1 \over p_2}\)

Let’s pause a moment take a look at the units of this, just to be sure. We know $\overline x_1$ should be measured in units of good 1, $\overline x_2$ should be measured in units of good 2, and the slope of the budget line should be in units of good 2 per units of good 1. The consumer’s income $m$ is measured in dollars. The price $p_1$ is measured in dollars per unit of good 1, and the price $p_2$ is measured in dollars per unit of good 2. Therefore we can confirm that the equation for the horizontal intercept works out in terms of units: \(\overline x_1 = \frac{m\ \cancel{\text{dollars}}}{p_1\ \cancel{\text{dollars}}\text{/unit of good 1}} = \frac{m}{p_1} \text{ units of good 1}\) \(\overline x_2 = \frac{m\ \cancel{\text{dollars}}}{p_2\ \cancel{\text{dollars}}\text{/unit of good 2}} = \frac{m}{p_2} \text{ units of good 2}\) \(\text{Slope of BL} = -\frac{p_1\ \cancel{\text{dollars}}\text{/unit of good 1}}{p_2\ \cancel{\text{dollars}}\text{/unit of good 2}} = \frac{p_1}{p_2} \frac{\text{ units of good 2}}{\text{ units of good 1}}\) This is very important: the numbers we deal with in this course (almost) always have important units attached to them!

Shifts in the budget line

Since we’re interested in studying comparative statics, an important question to ask is how budget lines shift when prices or income change.

We know that the intercepts of the budget line are $\overline x_1 = m/p_1$ and $\overline x_2 = m/p_2$, and that the magnitude of the slope (in absolute value) is the price ratio $p_1/p_2$.

From these formulas, we can derive the following results:

• An increase in $p_1$ will cause the horizontal intercept to move left without affecting the vertical intercept, so the budget line will become steeper.
• A decrease in $p_1$ will cause the horizontal intercept to move right without affecting the vertical intercept, so the budget line will become flatter.
• An increase in $p_2$ will cause the vertical intercept to move down without affecting the horizontal intercept, so the budget line will become flatter.
• A decrease in $p_2$ will cause the vertical intercept to move up without affecting the horizontal intercept, so the budget line will become steeper.
• An increase in $m$ will cause a parallel shift out of the budget line: the vertical intercept will move up and the horizontal intercept will move right, but the slope will remain unchanged.
• A decrease in $m$ will cause a parallel shift in of the budget line: the vertical intercept will move down and the horizontal intercept will move left, but the slope will remain unchanged.

Again, you can play with the graph above to see these for yourself.

The price ratio as the opportunity cost of good 1

We will often call $p_1/p_2$ the “price ratio” or the “relative price of good 1;” we’ll also call the ratio $p_2/p_1$ the “relative price of good 2.” This notion of a relative price is crucially important when we analyze applications involving exchange. Generally speaking, we’ll be talking about relative prices much more than absolute prices.

In the above example, we showed that if the consumer bought one more apple for $p_1 = 4$ dollars, she had to give up 2 bananas, since they cost $p_2 = 2$ dollars per banana, so the opportunity cost of an apple was two bananas. More generally, if the consumer spent $p_1$ dollars to buy a unit of good 1, she would have to spend $p_1$ fewer dollars on good 2, thereby reducing her consumption of good 2 by $p_1/p_2$ units of good 2. Hence, we can think of the price ratio $p_1/p_2$ as the opportunity cost of good 1 in terms of good 2.

A Special Case: Good 2 as a “composite good”

The choice space can represent a choice between two commodities, like apples and bananas. However, many times we’re interested in how much a consumer wants to spend on one good, versus saving her money for anything else. In cases like this, we might think about “good 1” as a specific good, and good 2 as “money spent on other things.” This is sometimes called a “composite good.”

If good 2 is “money spent on other things,” then “units of good 2” are dollars, and the price of a dollar is just a dollar (that is, $p_2 = 1$. In such a case, the budget line becomes \(p_1x_1 + x_2 = m\) Occasionally, since we’re only concerned with a single good, we’ll let $p$ be the price of that good, and simply write \(px_1 + x_2 = m\) In this case the price ratio is just $p_1$, and is measured in dollars per unit of good 1.