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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!

# Nonlinear (“Kinked”) Budget Constraints

A budget constraint is linear if good can always be bought at constant prices: if you can buy as many units of good 1 as you like for $p_1$ each, and as many units of good 2 as you like for $p_2$ each.

However, there are many examples in real life of prices changing depending on how many units you buy. Let’s add a bit of realism (and complexity) to the budget constraint model by thinking of a few real-world examples of nonlinear pricing.

## 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 electriticity, 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:

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:

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.

## Quantity Discounts

At many grocery stores, you can pay a lower price if you buy goods in bulk. This can cause a discontinuity in a budget constraint.

Suppose a store sells mac and cheese for 2 dollars per box, but has a “10 for 10” deal where if you buy 10 or more, the price is 1 dollar per box. Suppose you have $m = 40$ dollars to spend on mac and cheese (good 1) and other things (good 2, as above). What does your budget set look like?

One way to visualize this is to think about what the budget set would look like if $p_1 = 1$ or $p_2 = 2$; the actual budget set would follow the case of $p_1 = 2$ up until $\hat x_1 = 10$, and then follow the case of $p_1 = 1$ after that. Use the radio button in the following graph to toggle between these three cases:

One question here might be whether the point (8, 30) is really not in the consumer’s budget set. It’s true that you can’t buy exactly 8 boxes of mac and cheese and still have 30 dollars left over; but you could buy 10 boxes and have 30 left over. How do we think about it?

The answer to this question lies in the idea of free disposal. If disposing of goods is free, then if you want to have 8 boxes of mac and cheese, you should just buy 10 and toss 2 of them; so the bundle (8, 30) would be in your budget set. However, if it’s not free to dispose of goods, then you’d have to consider the cost of disposal in your analysis. (Much has been made of the economic assumption of free disposal being directly or indirectly responsible for environmental degredation…but that’s a topic for later.)

There are several cases in which you can pay a flat fee in order to pay a lower (or zero) price for a good.

For example, Walmart sells organic maple syrup at about 80 cents an ounce, and Costco sells it at about 40 cents an ounce. However, a Costco membership costs about 60 dollars.

Suppose you have $m = 160$ dollars. If you don’t buy the Costco membership, your budget line is given by $$0.8x_1 + x_2 = 160$$ If you do buy the Costco membership for $c = 60$, you only have $m - c = 100$ dollars left, but each ounce of organic maple syrup costs only 40 cents; so your budget set is given by $$0.4x_1 + x_2 = 100$$ Visually, we can see that buying the membership lowers the y-intercept of your budget set (because you have less money to spend overall) but reduces the slope of the budget line (because you pay less per unit for good 1):

Should you buy the Costco membership to support your organic maple syrup needs? Obviously, if you’re just going to buy a little bit, it’s not worth it to pay a 60-dollar membership fee. But if you’re going to buy a lot, it’s definitely worth it. In particular, the amount of money you have left over to spend on other things ($x_2$) is $100 - 0.4x_1$ if you buy the membership, and $160 - 0.8x_1$ if you don’t; the value of $x_1$ where these are equal is \begin{aligned} 100 - 0.4x_1 &= 160 - 0.8x_1\\ 0.4x_1 &= 60\\ x_1 &= 150 \end{aligned} So if you’re planning on buying more than 150 ounces of maple syrup, it’s worth it to buy the membership.

When faced with the option of paying for a reduced price for good 1, we can think of kinked budget constraint in which the consumer makes the optimal choice for each value of $x_1$. However, it’s important to realize that once the decision to buy or not to buy has been made, the budget set is a simple straight line.

One last thing to note is that the “reduced price” that you pay may, in fact, be zero! For example, many museums, amusement parks, and other attractions offer both single tickets and annual memberships which allow you to visit as many times as you like. In that case, the effective price of a visit is reduced to zero. To see what this looks like, move $p_C$ to zero in the diagram above.

## General rules

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:

• How much good 2 could I get if I didn’t get any good 1? How much good 1 could I get if I didn’t get any good 2?
• If a quantity is specified (as in the electricity or bulk discount examples), what does the budget set look like to the left of that quantity? To the right?
• If a dollar amount is specified (as in the gift card example), how does that translate into quantities? (For example, a 25-dollar gift card corresponds to five lattes if lattes are 5 dollars each.)
• Once you draw the budget constraint, ask yourself if the slopes make sense. (And remember that a horizontal/vertical slope means one of the prices is zero!)
Copyright (c) Christopher Makler / econgraphs.org