Chapter 5
Considering the Art of the Possible: The Budget Constraint
In This Chapter
Checking out how a budget constraint affects consumer choice
Seeing how consumers want to push utility to the maximum
Modeling consumer behavior in two ways
Chapter 4 discusses consumer behavior in terms of what’s called unconstrained optimization, where the only limiting factor is the amount of utility a person achieves. Ah, such freedom!
In the real world, of course, you never quite have the time and money to accomplish all the things you want to. You could always achieve something else if it wasn’t for those pesky constraints, such as buy an item if you weren’t just so short of moolah. So, in reality, satisfying your wants can never be as simple as Chapter 4’s model: In practice you may be unable to consume anything up to your bliss or saturation point where you no longer gain further utility.
Almost nothing in economics works in an unconstrained model. At some level, everything is constrained, if not by money then by time. When you’re deciding how much of something to consume, your utility isn’t the only important aspect; you also need to consider the availability of resources.
Fortunately, economists are very comfortable with the idea of scarcity. They’ve thought about this problem for over a hundred years and have a number of tools for describing constrained choice. In fact, at the heart of microeconomics is a model of constrained optimization, which deals with precisely this conundrum.
To make the model of Chapter 4 more fitting to the real world, this chapter adds another piece of the puzzle: the budget constraint. This chapter describes what the budget constraint does to the level of consumption that an individual consumer is able to choose. We explain what you can tell from the budget constraint and how you can manipulate it to show how people’s choices are affected by changes in the prices of goods that they want to buy or the income they have for buying goods. We also use the indifference curve from Chapter 4 to show how economists model a constrained choice, and show you one important insight of the model — that the ratio between prices is the same as the marginal rate of substitution (MRS — see Chapter 4) between two goods.
Taking It to the Limit: Introducing the Budget Constraint
The key to moving from unconstrained optimization to constrained optimization is the introduction of a budget constraint. This is a method of conceptualizing all the ways that the choice of doing or buying something is held back by the availability of resources, whether in terms of money, time, or something else. It also provides some insights that help economists put a bit more flesh on the utility model, bringing them closer to modeling the real world.
This section presents the budget constraint, and because we expect that your time and effort is constrained, we do our best to keep things simple.
Introducing the budget line
Imagine, as in Chapter 4, that you have two goods x1 and x2 — you can think of them as coffee and tea, or season tickets to see the Yankees and vacation days in Florida. We assume you have a fixed amount of resources, which we call M for now. The two goods have prices p1 and p2, respectively.
The maximum amount you can spend on both goods is M, and so the budget constraint has the following formula:
This equation is known as the budget line.
If you remember some of your high school algebra, you may have picked up that this equation describes a straight line, and that it slopes downward. By doing some rearranging in the equation, you can express the slope of the line by the price ratio of the two goods:
Any bundle of the goods x1 and x2 up to and including the budget line is feasible; anything beyond it is unfeasible and so is ruled out. Figure 5-1 plots the shape of the set of feasible consumption choices.
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Figure 5-1: The budget line, or budget constraint, splits an area into affordable (or feasible) and unaffordable (or unattainable) sets.
Note that points or bundles beyond the budget line — that is, those farther away from the origin than the budget line — are now ruled out. This means that when we re-introduce indifference curves, the highest possible indifference curve that you can be on is the one that is just touching or tangent in one place only to the budget constraint (assuming, that is, that the indifference curves are strictly convex, as in Chapter 2).
We demonstrate this point again in the later section “Getting the Biggest Bang for your Buck.” For the moment, we show you some ways of manipulating the budget constraint and a couple of points you can glean quite simply from doing so.
Shifting the curve when you get a raise
Let’s say one day your boss calls you into her office and announces that you’re going to get a raise. Of course, you’re delighted — but far more importantly, you’ve gained an opportunity to put your microeconomics to use. The way you do that is by understanding that the M (the fixed amount of resources) in the budget constraint is now bigger than it used to be, and so you can use this fact to manipulate the budget constraint to show your new purchasing possibilities.
A budget constraint maps the relative availability of two goods to a fixed amount of resources, which we call M. In the consumer choice model, this means that you take account of an increase in income by moving the budget constraint away from the origin so that the new curve is parallel to the old, as in Figure 5-2.
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Figure 5-2: Representing a change in income by shifting the budget constraint.
You can look at this another way. If your income goes up and prices stay the same, you can afford to buy more goods. Alternatively, if your income stays the same but prices of goods all decline by the same percentage, you can afford more goods as well. Under either scenario your budget constraint shifts up parallel to the old.
A shift in the budget constraint means that some bundles that the consumer desires are now either available where they hadn’t been before (if the change is positive) or ruled out (if the change is negative).
Twisting the curve when the price of one good changes
Changing income shifts your budget constraint up or down, or if all the prices of the goods you’re interested in change at the same rate, your budget constraint shifts up or down in a similar fashion.
But suppose that some prices change more than others. It’s more likely that some prices go up and others stay the same rather than all prices changing by the same percentage. In this case, you need to look again at the formula for the budget constraint to see how this kind of change affects your feasible consumption.
For instance, take the current consumer price index for the U.S. For urban consumers, the price index increased 0.3 percent in June 2015. The food index posted a large increase due in part to a sharp increase in the price of eggs, and the energy index rose for the second straight month as the prices of gasoline, electricity, and natural gas all increased. However, the prices for medical care, household furnishings and operations, used cars and trucks, and apparel all declined.
A price index is based on a very broad sector-based approach to measuring household consumption, and when you look further into the data, you find that the prices of some goods rose and some fell, and so the overall average picture masks these changes in relative prices.
Relative price changes have an important effect on consumer behavior as they cause substitution between goods. Suppose you’re deciding to take a break from work for a nice hot beverage. Suppose further that coffee has become relatively more expensive. Then, under some circumstances (explained more fully in Chapter 6) you may find it worthwhile to substitute some of your consumption of coffee for tea, preserving your overall level of satisfaction.
When the price of one good, say coffee, or p1, increases, and the price of the other good, p2, tea, stays the same, the budget constraint changes. But instead of a shift, the constraint rotates so that it becomes steeper when the price of good one has risen.
Figure 5-3 shows you how we’d take account of a rise in the price of coffee.
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Figure 5-3: A rise in the relative price of coffee draws in and twists the budget constraint.
A rise in the price of any particular good is similar to a fall in income, because it reduces the number of opportunities to consume. However, a rise in the price of one good (relative to another or all others) restricts the choice of bundles to ones in which the more expensive good is more constrained.
Pushing the line: Utility and the budget constraint
The budget constraint divides what is feasible from what is not feasible. We can now use the model of consumer choice and take a look at what a consumer will do to optimize her utility or satisfaction when a constraint exists. To do this, we have to take a look at what happens when we put the indifference curves from Chapter 4 together with the budget constraint in this section.
As described in Chapter 4, a consumer would, up to a point of satiation (and we assume that we’re some way from that point), try to consume so that she’s on the highest possible indifference curve — that is, one farthest away from the origin. Figure 4-2 in Chapter 4 shows this. Each of the indifference curves has the same level of utility at all points along the curve, and the only way to be at a higher level of utility is to be on a higher indifference curve.
Figure 5-4 reintroduces the budget constraint. Okay, now, we look at three indifference curves (and associated consumption bundles on each curve):
- Indifference curve I1: Lies entirely within the budget constraint and therefore is feasible. But would the consumer choose it? The answer is no, because higher levels of utility or satisfaction would be available by consuming right up to the budget line — all the areas above the indifference curve but within the constraint are still affordable, and all yield higher utility than any point on I1.
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Indifference curve I2: Also has points that are inside the constraint —although some are outside it.
Clearly the consumer would prefer points on I2 to those on I1, because they all confer a higher level of utility. But even though some combinations on I2 are unfeasible, the feasible points all lie away from the budget constraint, meaning that utility is available, as long as we restrict ourselves to combinations of x1 and x2 that are away from the extremes. - Indifference curve I3: Has a multitude of unavailable points, but notice also that this has one very important available point — point D — which is exactly on the budget line (mathematically, you say that it lies on a tangent to the line). This point yields higher utility than any point on I1 or I2 and is feasible. Moreover, any other point on I3 yields the same satisfaction as D, but you can’t afford it with your income or M.
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Figure 5-4: The optimal point is on an indifference curve tangent to the budget constraint.
When looking at utility given a budget constraint, the best available point must lie on an indifference curve tangent to the budget constraint, because that’s when the consumer has spent to the last penny available and can get no more satisfaction.
Getting the Biggest Bang for Your Buck
The shape of the budget constraint itself is of great importance to microeconomists. In and of itself, the nature of the constraint provides information that’s useful for looking at consumer behavior.
One of the key ways in which a tax, subsidy, or rationing affects consumers is through lowering (or, better, raising) the budget constraint. Economists manipulate the budget constraint into showing you those kinds of points in a number of ways. This section looks at two.
Economists assume that consumers want to get as much utility as possible for the least possible cost. Changes in relative prices of goods change the budget constraint and therefore the highest level of utility that a consumer can get. So, if anything affects the budget constraint, it also affects how consumers will choose which bundle of goods to consume.
Exploring relative price changes with the numeraire
Sometimes economists prefer to move away from absolute prices — for example, $4.79 for a cheeseburger — to relative prices that express prices for one good in terms of how much something else costs. For instance, the same cheeseburger costs 4.79 times the price of a bag of chips, which costs $1, so a cheeseburger on the market is worth 4.79 bags of chips. Looking at the ratios of prices rather than at the prices themselves can make choice simpler to understand.
A numeraire (pronounced “noomer-air”) price is what you get when you fix the price of one of the goods on the budget line to 1. Doing that can be useful when you want to eliminate the effect of the absolute level of prices and just look at the effects of changes relative to the price of the numeraire good.
To get the numeraire price, start by considering that setting one price equal to 1 and allowing the other prices and income to change around it doesn’t change the properties of the budget set at all. In the original budget line used earlier in this chapter (in section “Taking It to the Limit: Introducing the Budget Constraint”), the formula is as follows:
But if we fix things so that p2 is 1, we still get exactly the same budget line, except with the intercept for x2 (M/p2) now just being M. The key thing is that the relative price of everything else is now expressed holding the numeraire price of x2 at 1.
If everything in the budget set changes by the same rate, the only effect on the consumer is to shift the budget line as a whole in or out. If you multiply the budget line by a number, the optimal choice of the consumer doesn’t change. So if the price of all goods and services in an economy changes by exactly the same amount in a year, inflation is balanced and doesn’t affect consumers’ purchases.
A numeraire good with a price equal to one is one way of fixing our attention on relative prices. If you’re unsure about the change in price of one good relative to another, the numeraire helps you see how the ratio matters. Again, the slope of the budget constraint is the important factor, because it measures relative prices rather than absolute prices. If you use a numeraire price, you can quite simply see what’s happening, because it sets one of the prices at 1, making the role of the price ratio clearer.
Using the budget line to look at taxes and subsidies
In a two-good model, the budget line is a simple straight line whose slope is the ratio of prices. But if, for instance, a tax changes the cost of a good relative to others, that is tantamount to a price change, and you can use the shape of the budget line to think about how to analyze the effect of the tax.
Before doing so, we have to be a bit more specific about the type of tax, because different taxes do different things to the shape of the budget line.
Two types of tax
We want to distinguish two types of tax (or their seemingly positive cousin, subsidies) that affect the constraint:
- Quantity taxes: A tax per unit of something bought. Examples are the tax that government levies on gasoline, expressed per gallon, or the “sin” taxes levied on certain goods, such alcohol and cigarettes per unit. These taxes, also called excise taxes, simply change the price paid for that quantity: If x1 is the quantity of unleaded gasoline, and the quantity tax is τ per unit, the price of a gallon is p1 + τ, and you can treat the imposition of the tax as a price change.
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Ad valorem (“to the value of”) taxes: Instead of being levied on a per unit quantity of the good, an ad valorem tax is levied as a percentage of the purchase price of the good. A common example is the sales tax.
In the U.S., the sales tax varies according to jurisdictions within the country. For example, the sales tax in Chicago is 10.25% — consisting of 6.25% state, 1.25% city, 1.75% county, and 1% for the regional transportation authority. In Baton Rouge, Louisiana, the sales tax is 9%, consisting of 4% state and 5% local rate.
If the pre-tax price of the good is p1, then the post-tax price is (1 + τ) p1 where τ is the ad valorem rate of tax. For the 10.25% sales tax that a consumer pays in Chicago, τ equals 0.1075 (convert the percentage to a decimal), and 1 + τ is 1.1075. So the price of a good is 1.1075 times p.
Again, you can treat the introduction of an ad valorem tax as being tantamount to an increase in the price of the good you’re considering and manipulate the budget constraint to show it. In this case, the constraint would show the bundles of goods that can be consumed when the sale tax on good 1 is included in the post-tax price.
An interesting case to consider is what happens when a tax is only levied on consumption of a good above a certain price. In Massachusetts, the sales tax of 6.25% is not levied on clothing that costs less than $175. Any individual clothing item that is more than $175 is taxable on the amount over the basic exemption. If you buy a $200 coat, $1.56 or 6.25% of the $25 taxable amount would be added to the price. So, the microeconomics question is: How do you look at this aspect using a budget constraint?
The answer’s easy: One slope of the line for purchases goes up to the threshold and then the line bends at that point (see Figure 5-5).
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Figure 5-5: The effect on the budget constraint of a stepped tax.
To make everything easier, think about the tax being levied on a quantity rather than a value tax. Suppose, for argument’s sake, that the first item of clothing does not incur a tax, but the second does. Now, while you’re deciding to buy a first item, the budget constraint is the constraint for x1 up to the point where x1 = 1. Here, the slope of the budget constraint is –p1/p2 as it was earlier (see the section “Taking It to the Limit: Introducing the Budget Constraint”). However, beyond x1 = 1, the slope changes to become –(p1 + τ)/p2. As you can see in Figure 5-5, the budget line is steeper beyond the threshold.
You can do the same type of graphing with subsidies, too. A subsidy, in this case, is just a negative tax, and so instead of adding it to the price you subtract it. Therefore, if good x1 is subsidized, the budget slope is –(p1 – t)/p2. We show this in Figure 5-6.
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Figure 5-6: Showing the effect of a subsidy on the budget constraint.
Rationing also affects the budget line. If a good is rationed, one area of the budget set becomes unavailable at any price — the set is said to be truncated in economics-speak.
To show this, cut a vertical line in above the maximum rationed consumption of good x1. To the left of the line, the budget set behaves as normal. To the right, where the maximum consumption is greater than the rationed amount — call it R for the moment — the set consists of goods that the consumer could afford, but can’t get. We present this example in Figure 5-7.
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Figure 5-7: Rationing truncates the budget constraint.
Putting the Utility Model to Work
Essentially, in the standard choice model, a consumer can optimize in two ways:
- Utility maximization approach: The consumer first decides on a budget constraint and then figures out how much utility she can get for that budget (discussed in the earlier section “Taking It to the Limit: Introducing the Budget Constraint”).
- Cost-minimization approach: The consumer has a target level of utility that she wants to achieve and wants to minimize the cost that she has to pay to get it. As a microeconomist, you treat the amount of utility that she wants to get as fixed and then work out which budget constraint is the lowest possible constraint that allows her to afford a bundle from that utility function.
Ultimately, the two approaches work out to the same thing, and should arrive at the same answer, but from two different directions. Deciding which approach to use is a matter of computational efficiency — meaning you use the version that’s easiest to do in the time available — and which is more efficient depends on how much information you have about the constraint or the consumer’s preferences, that is, indifference curves. In microeconomics, the insight that both approaches achieve the same answer is known as the dual.
What’s most important is the conclusion that you can derive from knowing something about utility (see Chapter 4) and something about the budget constraints and how the two have to be related. The relation between the two that is important in this case is that the highest level of utility possible for a constrained consumer occurs when the indifference curve is tangent to the budget constraint. Therefore the slope of the utility curve and the slope of the budget constraint are equal at that point.
The slope of the indifference curve is the marginal rate of substitution (MRS), as in Chapter 4, and the slope of the budget constraint is the relationship between the two prices (–p1/p2). Given that these have to be equal at the optimal choice for the consumer, you know that the optimizing consumer’s best point occurs when
You need to know this equation for Chapter 6’s full discussion on consumer optimization and for Chapter 9, which looks into the famous supply and demand model.