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Lecture 5: Budget Constraints

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Up to now we’ve discussed a consumer’s preferences over elements of a choice space, and how we could model those preferences using utility functions.

Next we’re going to be thinking about how the problem of constrained optimization: that is, we’re going to assume that an agent cannot, in fact, choose from the entire choice space, but rather from a constrained subset of that choice space.

For example, if you think about your decision of where to go to college (or more broadly, what to do after high school), you weren’t choosing among all colleges, universities, jobs, etc. You applied to a certain number of colleges and universities, and some subset of the ones you applied to accepted you; that became your feasible set from which you made the eventual optimal choice of Stanford University.

In the particular case of a consumer, the feasible set is determined by the amount of money they have available to spend and the prices of goods. Let’s think of a teenager named Jordan, who like many 13-year-olds, has a modest allowance and spends it on things like snacks.

We’ll call Jordan’s feasible set her budget set. 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.

As with many of our models, we will take three approaches: mathematical, visual, and intuitive:

For the remainder of the lecture we’ll look at some more realistic (and less simple) kinds of budget constraints.

The mathematics of the budget constraint

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:

See interactive graph online here.

General budget constraint for $n$ goods, and the “dot product” of two vectors

Let me take a very brief pause here and teach you a little bit of mathematical notation which may be useful to you in some later classes. Recall that a vector is an ordered set of numbers. The main two ways to denote a vector are as a bolded letter (like $\textbf{x}$), or as a letter with a little arrow above it: \(\vec{x} = (x_1,x_2,x_3,..., x_n)\) If you have two vectors of the same dimension $n$ (i.e. the same number of elements), then you can define the dot product of those two vectors as \(\vec{x} \cdot \vec{y} = \sum_{i = 1}^n x_iy_i = x_1y_1 + x_2y_2 + x_3y_3 + \cdots + x_ny_n\) For example, if we have the following two four-dimensional vectors: \(\begin{aligned} \vec{x} &= (x_1,x_2,x_3,x_4) = (1,4,3,8)\\ \vec{y} &= (y_1,y_2,y_3,y_4) = (5,5,3,1) \end{aligned}\) then their dot product is \(\begin{aligned} \vec{x} \cdot \vec{y} &= x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4\\ &= (1 \times 5) + (4 \times 5) + (3 \times 3) + (8 \times 1)\\ &= 5 + 20 + 9 + 8\\ &= 42 \end{aligned}\) With this notation in mind, let’s now think about the budget set for more than 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 $i$. Using our newfound 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!

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 Jordan 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 $€24$ 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 $€24$ 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. (The bar above these variables indicates that it’s the maximum possible amount of each good, not the average.)

To think about the slope, let’s think of what happens as Jordan 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:

See interactive graph online here.

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 Jordan 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. Jordan’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

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:

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

Interpreting the price ratio: the opportunity cost of good 1

So, given that the slope of the budget line is measured in units of good 2 per unit of good 1, what does that mean in real life? In fact, it means something really important: the magnitude of the slope of the budget line represents the opportunity cost of another unit of good 1, in terms of units of good 2.

Remember from Econ 1 what opportunity cost means: when making a decision, the opportunity cost of one choice is the value of the next best use of the resources consumed by that choice. So in this case, if the price of apples is $p_1 = 4$ and the price of bananas is $p_2 = 2$, and Jordan wants another apple, the cost of that apple is four dollars; but the opportunity cost of that apple (assuming she spends money on only apples and bananas) is the quantity of bananas she could have bought with that four dollars; and since each banana costs two dollars, that works out to be two bananas. So: the opportunity cost of one apple is two bananas, and the price ratio is \(\frac{p_1}{p_2} = \frac{4\ \cancel{\text{dollars}}\text{/apple}}{2\ \cancel{\text{dollars}}\text{/banana}} = 2 \frac{\text{ bananas}}{\text{ apple}}\) This will be critically important in the next lecture…so make sure you have it down!

A special case: composite goods

The above analysis treated both good 1 and good 2 as actual goods: apples and bananas. However, there’s a different kind of tradeoff we might think about, which is between one good (e.g. apples) and all other goods one could spend one’s money on. In this case, “good 2” can be “money spent on other goods,” measured in dollars. We call good 2 in this case a “composite good,” because it’s not just one thing like a banana; it’s a “composite” of everything that isn’t an apple.

Because one dollar spent on another good costs one dollar, in this formulation we have $p_2 = 1$; the equation of the budget line is just $p_1x_1 + x_2 = m$; and the magnitude of the slope of the budget line is $p_1$! In other words, if good 2 is measured in dollars, then the opportunity cost of good 1 is just the price you have to pay for it in dollars.

Trading from an Endowment

In the previous analysis, we assumed that you started with some amount of money, and could convert that money into goods by buying them.

However, there’s another kind of budget constraint that is perhaps more realistic: suppose you start, not with money, but with some bundle of goods; we call this an “endowment” of goods. You might then trade some of one of the goods for more of the other; or, if no one is willing to trade with you, you might sell some of your endowment of one good, and use the money to buy some more of the other.

For example, suppose that instead of starting with an income of 24 dollars, Jordan had endowment of 4 apples and 4 bananas: that is, she’s starting from a bundle within good 1 - good 2 space. We’ll call this point $E$ for “endowment,” and denote the quantities of goods 1 and 2 in that bundle as $e_1$ and $e_2$.

Let’s assume that Jordan can buy or sell apples and bananas at market prices. For example, suppose again that prices are $p_1 = 4$ and $p_2 = 2$. If Jordan wanted to consume more than $e_2 = 4$ bananas, she could sell some of her apples to buy more bananas, as shown below:

See interactive graph online here.

Geometry of the endowment budget line

As in the case with exogenous income, we can draw Jordan’s budget line; you can add it to the diagram above by checking the box. How do we get the equation of this budget line?

Using the logic from the example above, we can say that if Jordan sells some amount $\Delta x_1$ of good 1, she will earn $p_1 \times \Delta x_1$ from the sale; if she uses the proceeds to buy good 2, the amount of good 2 she can buy is therefore \(\Delta x_2 = \frac{p_1 \times \Delta x_1}{p_2}\) Since $\Delta x_1 = e_1 - x_1$ and $\Delta x_2 = x_2 - e_2$, we can write this equation as \((x_2 - e_2) = \frac{p_1 \times (e_1 - x_1)}{p_2}\) Collecting the $x$ terms on the left-hand side and the $e$ terms on the right hand side, we can write this as \(p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2\) If we compare this to the budget line with income from above, we can see that the left-hand side is the same. We can interpret the right-hand side as the “monetary value” (or “liquidation value”) of the endowment: \(\hat m = p_1e_1 + p_2e_2\) In other words, one way of thinking about the endowment budget constraint is that Jordan could sell all her endowment for $\hat m$ dollars, and then go back and spend the money on goods 1 and 2 as usual.

How can we interpret the endpoints of the budget line? Mathematically they’re given by \(\begin{aligned}\overline x_1 &= {\hat m \over p_1} = e_1 + {p_2e_2 \over p_1}\\ \\ \overline x_2 &= {\hat m \over p_2} = e_2 + {p_1e_1 \over p_2} \end{aligned}\) In the context of trading from an endowment, each of these represents the total amount of a good Jordan could afford if she sold all her other goods and used the proceeds to buy that good. For example, the first expression for the maximum amount of good 1 says that if Jordan sold all her good 2, she could get $p_2e_2$ for it, and use the proceeds to buy $p_2e_2/p_1$ units of good 1; so her final consumption would be her initial endowment $e_1$ plus $(p_2/p_1)e_2$.

Effect of changes in prices

In the budget line with exogenous income, an increase in the price of either good would lead to a reduction in the size of the budget set, and a decrease would lead to an increase in the size of the budget set. This is because income was fixed, and didn’t change due to a change in prices. With an endowment, the situation is a little more complicated.

We just saw that the monetary value of Jordan’s endowment was $p_1e_1 + p_2e_2$. An increase in the price of a good therefore leads to an increase in the monetary value of Jordan’s endowment; to the extent that Jordan has a lot of good 1 which she might want to sell, this is good for her. However, it also means that buying additional good 1 will be more expensive than it was before. It’s also possible that Jordan doesn’t own any good 1, in the which case this is unquestionably bad for her.

Geometrically, one way of thinking about the endowment budget line is that it’s a line that passes through the endowment point and has a slope equal to the price ratio. Therefore:

The graph below summarizes all of this. The diagram in its initial state shows the situation with $p_1 = 4$ and $p_1 = 2$. If you change the prices, a green dotted line shows this original budget line, so you can see the effect of the change:

See interactive graph online here.

A few things to try:

Kinked constraints

Up until now, we’ve been dealing with a linear budget constraint because prices were constant throughout the problem: Jordan could buy as many apples or bananas at a single, constant price for each.

However, in the real world, there are many instances of non-linear pricing, which result in a kinked budget constraint (i.e., a constraint whose slope varies, instead of being a straight line). These are particularly important in economic applications, and they can help us add a dose of realism to an otherwise pretty simple model.

Different prices for buying and selling

One kind of kinked budget constraint involves trading from an endowment. In the above example, we considered the case in which Jordan can buy or sell goods at a constant price; and in particular, we’ve been assuming that the price she could get by selling some the apples from her endowment is the same as the price she’d have to pay for additional apples.

However, there are lots of economic applications in which the price you get when you go to sell something is (perhaps much) less than the price you’d have to pay for more of it. This can happen, for example, if there are transaction fees: for example, if you try to sell a ticket to an event on StubHub or TicketMaster, the platform itself takes a cut of the deal; so the seller gets a lower price than the buyer pays.

How does this play out in this model? Let’s investigate the example of buying and selling tickets, so let’s have “good 1” be tickets and “good 2” be money. Let’s say you are a season ticket holder to a sports team, so you have 40 tickets; let’s also assume you have €1200 in the bank. Therefore, your endowment is $(40\text{ tickets}, €1200)$. You can choose to sell your tickets online for $p^\text{sell} = €25$ each, but it would cost you $p^\text{buy} = €60$ per ticket to buy additional tickets. How can we construct your budget constraint?

Plotting these all together, we can see a kinked budget set:

See interactive graph online here.

The slope to the left of the endowment represents the price of a ticket when you go to sell it; and the slope to the right represents the price when you want to buy another one. Since they’re different, the budget constraint has a “kink” at the endowment.

Electricity Rates

Electricity is a necessity, but overuse of electricity can cause environmental problems and strain the grid. For this reason, many utilities adopt a “two-tier” system, in which you pay a low rate at first, and then a higher rate beyond a certain level.

For example, suppose a utility charges 10 cents per kilowatt-hour (kWh) for the first 300kWh per day, and 20 cents for every kilowatt-hour after that. The cost of $x_1$ kWh of electricity under such a scheme would be \(c(x_1) = \begin{cases} 0.10x_1 & \text{ if } x_1 \le 300\\ 0.10\times 300 + 0.20 \times (x_1 - 300) & \text{ if } x_1 > 300 \end{cases}\) The second line says that the cost if you use more than 300kWh per month is $0.10 \times 300 = 30$ for the first 300kWh, plus $0.20 \times (x_1 - 300)$ for the remainder, where $x_1 - 300$ is the amount by which your monthly usage exceeds 300kWh.

Suppose you have $m = 120$ to spend on electricity and other goods. For the first 300 units of electricity, you pay 10 cents per kWh; so your budget constraint initially has a slope of $-0.1$, from $x_1 = 0$ to $x_1 = 300$. Beyond that point, you have to pay 20 cents per kWh; so your budget constraint has a slope of $-0.2$ beyond that point. If you spend all 120 dollars on electricity, you can buy the first 300kWh for 30 dollars, and the remaining 90 dollars buys you 90/0.2 = 450kWh, for a total of 750 kWh:

See interactive graph online here.

You can play with the four variables (the initial low price $p_\text{low}$, the second price $p_\text{high}$, your income $m$, and the threshold $\hat x$) to see how the budget constraint is affected.

Should you give gift cards or cash?

A gift card is like cash, but cash that can only be spent on one particular item. Lots of people give these as presents; hence the name “gift card.” But would it be better to just give someone cash? Economists love to say so, and our analysis of budget sets can help to explain why.

Suppose someone has $m = 100$ in cash. You could either give them 25 dollars in cash, or a gift card worth 25 dollars only at Starbucks. Let’s look at their budget set in either case.

Suppose that the only thing they buy at Starbucks are lattes which cost 5 dollars each; call these “good 1,” and as we did before, we’ll let “good 2” be “money spent on other things.”

If you gave them 25 dollars in cash, they would now have $m = 125$, so their budget set would be given by the equation \(5x_1 + x_2 = 125\) On the other hand, if you gave them a 25-dollar gift card, the most they could spend on other goods is 100 dollars. With the gift card in hand, they could buy up to five lattes without giving up any other goods; therefore, their budget set starts out horizontal (with a price ratio of 0), up to the point $(5, 100)$. Beyond that, each additional latte costs the same amount, so the budget set is the same as before. For example, if they bought six lattes, they could use the gift card plus 5 dollars of their own money, leaving 95 dollars left over.

Which is better? In the following diagram, you can toggle between the two options. Notice that the budget set with the cash gift is larger than the budget set with the gift card:

See interactive graph online here.

Notice that the budget set is the same beyond five lattes: so it doesn’t matter whether you give cash or a gift card if they’d choose to consume at least five Starbucks lattes either way. But if they’d spend the money on something else, giving them the gift card reduces their feasible set.

General rules for kinked budget constraints

As should be clear by now, there is no “one rule” for creating budget constraints that are more complicated than a simple line. You need to think about what happens at each point a consumer faces a different price.

Some things you might ask yourself are:

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