Decisions with a budged constraint

Posted 2023-02-13 15:15:43

Key points

The budget constraint is the boundary of the opportunity set—all possible combinations of consumption that someone can afford given the prices of goods and the individual’s income.
Opportunity cost measures cost in terms of what must be given up in exchange.
Marginal analysis is the process of comparing the benefits and costs of choosing a little more or a little less of a certain good.
The law of diminishing marginal utility indicates that as a person receives more of a good, the additional—or marginal—utility from each additional unit of the good declines.
Sunk costs are costs that occurred in the past and cannot be recovered; they should be disregarded in making current decisions.
Utility is the satisfaction, usefulness, or value one obtains from consuming goods and services.

Introduction

Most consumers have a limited amount of income to spend on the things they need and want. Alphonso, for example, has $10 in spending money each week that he can use to buy bus tickets for getting to work and the burgers that he eats for lunch. Burgers cost $2 each, and bus tickets are 50 cents each.

The figure above shows Alphonso’s budget constraint—the outer boundary of his opportunity set. The opportunity set identifies all the opportunities for spending within his budget—in this case, bus tickets and burgers. The budget constraint indicates all the combinations of burgers and bus tickets Alphonso can afford before he exhausts his budget, given the prices of the two goods.

The vertical axis in the figure shows burger purchases, and the horizontal axis shows bus ticket purchases. If Alphonso spends all his money on burgers, he can afford five per week—$10 per week divided by $2 per burger equals five burgers per week. But if Alphonso uses all his money on burgers, he will not be able to afford any bus tickets. This choice—zero bus tickets and five burgers—is shown by point A in the figure.

Alternatively, if Alphonso spends all his money on bus tickets, he can afford 20 per week—$10 per week divided by $0.50 per bus ticket equals 20 bus tickets per week. If he does this, however, he will not be able to afford any burgers. This choice—20 bus tickets and zero burgers—is shown by point F.

If Alphonso is like most people, he will choose some combination that includes both bus tickets and burgers. That is, he will choose some combination on the budget constraint that connects points A and F. Every point on or inside the constraint shows a combination of burgers and bus tickets that Alphonso can afford. Any point outside the constraint is not affordable because it would cost more money than Alphonso has in his budget.

The budget constraint shows the tradeoff Alphonso faces in choosing between burgers and bus tickets. Suppose he is currently at point D, where he chooses to buy 12 bus tickets and two burgers. What would it cost Alphonso for one more burger? It would be natural to answer $2, but that’s not the way economists think. Economists think about the true cost of a burger—the number of bus tickets Alphonso will have to sacrifice.

What is opportunity cost?

Economists use the term opportunity cost to indicate what must be given up to obtain something that is desired. The idea behind opportunity cost is that the cost of one item is the lost opportunity to do or consume something else; in short, opportunity cost is the value of the next best alternative.

For Alphonso, the opportunity cost of a burger is the four bus tickets he would have to give up in order to afford another burger. He must decide whether or not to choose the burger depending on whether the value of the burger exceeds the value of the forgone alternative—in this case, bus tickets. Since few if any people have unlimited financial resources, consumers inevitably face tradeoffs in which they have to give up things they desire to get other things they desire more.

A fundamental principle of economics is that every choice has an opportunity cost. If you sleep through your economics class—not recommended, by the way—the opportunity cost is the learning you miss from not attending class. If you spend your income on video games, you cannot spend it on movies. If you choose to marry one person, you give up the opportunity to marry anyone else. In short, opportunity cost is all around us and is part of human existence.

Understanding budget constraints

One way for us to better understand budget constraints is to build an equation. Let's make P and Q the price and quantity of items purchased and start text, B, u, d, g, e, t, end text the amount of income one has to spend.

start text, B, u, d, g, e, t, end text, equals, start text, P, end text, start subscript, 1, end subscript, space, times, space, Q, start subscript, 1, end subscript, space, plus, space, P, start subscript, 2, end subscript, times, space, Q, start subscript, 2, end subscript

We can apply the budget constraint equation to Alphonso's scenario:

Budget $ 10 = = P^{1} \times Q^{1} + P^{2} \times Q^{2} $ 2 \times Q^{burgers} + $ 0.50 \times Q^{b u s t i c k e t s}

Using a little algebra, let's turn this into the equation of a line:

y = b + m x

If we plug in the variables from Alphonso's scenario, we get the following:

$ 10 = $ 2 \times Q^{burgers} + $ 0.50 \times Q^{b u s t i c k e t s}

Next, we simplify the equation by multiplying both sides of the equation by two:

2 \times 1020 = = 2 \times 2 \times Q^{burgers} + 2 \times 0.5 \times Q^{b u s t i c k e t s} 4 \times Q^{burgers} + 1 \times Q^{b u s t i c k e t s}

Then we subtract one bus ticket from both sides:

20 - Q^{bus tickets} = 4 \times Q^{burgers}

Next, we divide each side by four to yield the answer:

5 - 0.25 \times Q^{bus tickets} Q^{burgers} = or = Q^{burgers} 5 - 0.25 \times Q^{bus tickets}

Notice that this equation fits Alphonso's budget constraint figure above. The vertical intercept is five and the slope is –0.25, just as the equation says. If you plug 20 bus tickets into the equation, you get 0 burgers. If you plug other numbers of bus tickets into the equation, you get the results shown in the table below, which are also the points on Alphonso’s budget constraint.

Point	Quantity of burgers at $2	Quantity of bus tickets at 50 cents
A	5	0
B	4	4
C	3	8
D	2	12
E	1	16
F	0	20

Notice that the slope of a budget constraint always shows the opportunity cost of the good which is on the horizontal axis. For Alphonso, the slope is −0.25, indicating that for every four bus tickets he buys, Alphonso must give up one burger.

There are two important observations here. First, the algebraic sign of the slope is negative, which means that the only way to get more of one good is to give up some of the other. Second, the slope is defined as the price of whatever is on the horizontal axis in the graph—in this case, bus tickets—divided by the price of whatever is on the vertical axis—in this case, burgers. So, in our scenario, the slope is dollar sign, 0, point, 50, slash, dollar sign, 2, equals, 0, point, 25. If you want to determine the opportunity cost quickly, just divide the two prices.

Identifying opportunity cost

In many cases, it is reasonable to refer to the opportunity cost as the price. If your cousin buys a new bicycle for $300, then $300 measures the amount of other spending opportunities, or other consumption, that he has given up. For practical purposes, there may be no special need to identify the specific alternative product or products that could have been bought with that $300, but sometimes the price as measured in dollars may not accurately capture the true opportunity cost. This problem can loom especially large when costs of time are involved.

For example, consider a boss who decides that all employees will attend a two-day retreat to build team spirit. The out-of-pocket monetary cost of the event may involve hiring an outside consulting firm to run the retreat as well as room and board for all participants. But an opportunity cost exists as well: during the two days of the retreat, none of the employees are doing any other work.

Attending college is another case where the opportunity cost exceeds the monetary cost. The out-of-pocket costs of attending college include tuition, books, room and board, and other expenses. But in addition, during the hours that a student is attending class and studying, it is impossible for them to work at a paying job. Thus, college imposes both an out-of-pocket cost and an opportunity cost of lost earnings.

In some cases, recognizing opportunity cost can alter behavior. Imagine, for example, that you spend $8 on lunch every day at work. You may know perfectly well that bringing a lunch from home would cost only $3 a day. So, the opportunity cost of buying lunch at the restaurant is $5 each day—the $8 buying lunch costs minus the $3 your lunch from home would cost.

Five dollars each day does not seem to be that much; but, if you add up the cost over a year—250 days a year times $5 per day equals $1,250—it's actually equivalent to a decent vacation. If the opportunity cost were described as “a nice vacation” instead of “$5 a day”, you might make different choices.

Marginal decision-making and diminishing marginal utility

The budget constraint framework helps to emphasize that most choices in the real world are not about getting all of one thing or all of another—choosing a point at one end of the budget constraint or all the way at the other end. Instead, most choices involve marginal analysis, comparing the benefits and costs of choosing a little more or a little less of a certain good.

People desire goods and services for the satisfaction or utility those goods and services provide. Utility is subjective, but that doesn't make it any less real.

Economists typically assume that the more of some good one consumes—for example, slices of pizza—the more utility one obtains. At the same time, the utility a person receives from consuming the first unit of a good is typically more than the utility received from consuming the fifth or the 10th unit of that same good.

When Alphonso chooses between burgers and bus tickets, for example, the first few bus rides that he chooses might provide him with a great deal of utility—perhaps they help him get to a job interview or a doctor’s appointment. But later bus rides might provide much less utility—they may only serve to kill time on a rainy day. Similarly, the first burger that Alphonso chooses to buy may be on a day when he missed breakfast and is ravenously hungry. However, if Alphonso has a burger every single day, the last few burgers may taste pretty boring.

It is a common pattern for consumption of the first few units of any good to bring a higher level of utility to a person than consumption of later units. Economists refer to this pattern—described succinctly, "as a person receives more of a good, the additional, or marginal, utility from each additional unit of the good declines"—as the law of diminishing marginal utility. You could describe this law in more simple terms as "The first slice of pizza brings more satisfaction than the sixth."

The law of diminishing marginal utility explains why people and societies rarely make all-or-nothing choices. You would probably not say, “My favorite food is ice cream, so I will eat nothing but ice cream from now on.” Even though your favorite food has a high level of utility, if you chose to eat it exclusively, the additional or marginal utility from those last few servings would not be very high. Similarly, most workers would not say: “I enjoy leisure, so I’ll never work.” Instead, workers recognize that even though some leisure is very nice, a combination of all leisure and no income is not so attractive. The budget constraint framework suggests that when people make choices in a world of scarcity, they will use marginal analysis and think about whether they would prefer a little more or a little less.

Sunk costs

In the budget constraint framework, all decisions involve what will happen next—what quantities of goods will you consume, how many hours will you work, or how much will you save. These decisions do not look back to past choices. Thus, the budget constraint framework assumes that sunk costs—costs that were incurred in the past and cannot be recovered—should not affect the current decision.

Consider the case of Selena, who pays $8 to see a movie; after watching the film for 30 minutes, she knows that it is truly terrible. Should she stay and watch the rest of the movie because she paid for the ticket, or should she leave? The money she spent is a sunk cost, and unless the theater manager is feeling kindly, Selena will not get a refund. But, staying in the movie still means paying an opportunity cost in time. Her choice is whether to spend the next 90 minutes suffering through a cinematic disaster or to do something—anything—else. The lesson of sunk costs is to forget about the money and time that is irretrievably gone and instead to focus on the marginal costs and benefits of current and future options.

For people and firms alike, dealing with sunk costs can be frustrating. It often means admitting an earlier error in judgment. Many firms, for example, find it hard to give up on a new product that is doing poorly because they spent so much money in creating and launching the product. But the lesson of sunk costs is to ignore them and make decisions based on what will happen in the future.

From a model with two goods to one of many goods

The budget constraint diagram we used to examine Alphonso's situation containing just two goods is not realistic. After all, in a modern economy people choose from thousands of goods. We can, however, think about a model with many goods by extending the ideas we've discussed here.

Instead of drawing just one budget constraint showing the tradeoff between two goods, you can draw multiple budget constraints showing the possible tradeoffs between many different pairs of goods. Or, in more advanced classes in economics, you would use mathematical equations that include many possible goods and services that can be purchased together with their quantities and prices to show how the total spending on all goods and services is limited to the overall budget available.

It's important to remember, though, that the graph above with two goods clearly illustrates that every choice has an opportunity cost, which is an idea that carries over to the real world.

Key Concepts and Summary

Economists see the real world as one of scarcity—a world in which people’s desires exceed what is possible. Economic behavior involves tradeoffs in which individuals, firms, and society must give up something that they desire to obtain things that they desire more.

Individuals must choose which quantities and combinations of goods and services to consume. The budget constraint, which is the outer boundary of the opportunity set, illustrates the range of choices available. The slope of the budget constraint is determined by the relative price of the choices. Choices beyond the budget constraint are not affordable.

Opportunity cost measures cost by what is given up in exchange. Sometimes opportunity cost can be measured in money, but it is often useful to consider time costs as well or to measure opportunity cost in terms of the actual resources that must be given up.

Most economic decisions and tradeoffs are not all or nothing. Instead, they involve marginal analysis, which means they are about decisions on the margin—involving a little more or a little less. The law of diminishing marginal utility points out that as a person receives more of something, whether it is a specific good or another resource, the additional marginal gains tend to become smaller. Because sunk costs occurred in the past and cannot be recovered, they should be disregarded in making current decisions.

Self-check question

Suppose Alphonso’s town raised the price of bus tickets to $1 per trip, the price of burgers stayed at $2, and Alphonso's budget remained $10 per week. Draw Alphonso’s new budget constraint.

What happens to the opportunity cost of bus tickets?