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Above, we have examined the trade-offs that economic actors (e.g., consumers, companies, investors) are willing to make. In this section, we recognize that circumstances almost always impose constraints on the trade-offs that these actors are able to make. In other words, we need to explore how to model the constraints on behavior that are imposed by the fact that we live in a world of scarcity: There is simply not enough of everything to satisfy the needs and desires of everyone at a given time. Consumers must generally purchase goods and services with their limited incomes and at given market prices. Companies, too, must divide their limited input resources in order to produce different products. Investors are not able to choose both high returns and low risk simultaneously. Choices must be made, and here we examine how to represent the set of choices from which to choose.
Previously, we examined what would happen if Warren and Smith were each given an endowment of bread and wine and were allowed to exchange at some pre-determined ratio. Although that circumstance is possible, a more realistic situation would be if Warren or Smith had a given income with which to purchase bread and wine at fixed market prices. Let Warren’s income be given by I, the price he must pay for a slice of bread be P_{B}, and the price he must pay for an ounce of wine be P_{W}. Warren has freedom to spend his income any way he chooses, so long as the expenditure on bread plus the expenditure on wine does not exceed his income per time period. We can represent this income constraint (or budget constraint) with the following expression:
Equation (3)
P_{B}Q_{B} + P_{W}Q_{W} ≤ I
This expression simply constrains Warren to spend, in total, no more than his income. At this stage of our analysis, we are assuming a one-period model. In effect, then, Warren has no reason not to spend all of his income. The weak inequality becomes a strict equality, as shown in Equation 4, because there would be no reason for Warren to save any of his income if there is “no tomorrow.”
Equation (4)
P_{B}Q_{B} + P_{W}Q_{W} = I
From this equation, we see that if Warren were to spend all of his income only on bread, he could buy I/P_{B} slices of bread. Or if he were to confine his expenditure to wine alone, he could buy I/P_{W} ounces of wine. Alternatively, he could spread his income across bread and wine expenditures any way he chooses. Graphically, then, his budget constraint would appear as in Exhibit 6:
Exhibit 6. The Budget ConstraintNote: The budget constraint shows all the combinations of bread and wine that the consumer could purchase with a fixed amount of income, I, paying prices P_{B} and P_{W}, respectively.
A simple algebraic manipulation of Equation 4 yields the budget constraint in the form of an intercept and slope:
Equation (5)
QW=I/PW−((PB/PW)*(QB))
Notice that the slope of the budget constraint is equal to –P_{B} /P_{W}, and it shows the amount of wine that Warren would have to give up if he were to purchase another slice of bread. If the price of bread were to rise, the budget constraint would become steeper, pivoting through the vertical intercept. Alternatively, if the price of wine were to rise, the budget constraint would become less steep, pivoting downward through the horizontal intercept. If income were to rise, the entire budget constraint would shift outward, parallel to the original constraint, as shown in Exhibit 7:
Exhibit 7. Changing Prices and IncomeAs a specific example of a budget constraint, suppose Smith has $60 to spend on bread and wine per month, the price of a slice of bread is $0.50, and the price of an ounce of wine is $0.75. If she spent all of her income on bread, she could buy 120 slices of bread. Or she could buy up to 80 ounces of wine if she chose to buy no bread. Obviously, she can spend half her income on each good, in which case she could buy 60 slices of bread and 40 ounces of wine. The entire set of bundles that Smith could buy with her $60 budget is shown in Exhibit 8:
Exhibit 8. A Specific Example of a Budget ConstraintNote: This exhibit shows Smith’s budget constraint if she has an income of $60 and must pay $0.50 per slice of bread and $0.75 per ounce of wine.
EXAMPLE 4Nigel’s Pub has a total budget of £128 per week to spend on cod and lamb. The price of cod is £16 per kilogram, and the price of lamb is £10 per kilogram.
Calculate Nigel’s budget constraint.
Construct a diagram of Nigel’s budget constraint.
Determine the slope of Nigel’s budget constraint.
The budget constraint is simply that the sum of the expenditure on cod plus the expenditure on lamb be equal to his budget: 128 = 16 Q_{C} + 10 Q_{L}. Rearranging, it can also be written in intercept slope form: Q_{C} = 128/P_{C} – (P_{L}/P_{C}) Q_{L} = 8 – 0.625 Q_{L}.
We can choose to measure either commodity on the vertical axis, so we arbitrarily choose cod. Note that if Nigel spends his entire budget on cod, he could buy 8kg. On the other hand, if he chooses to spend the entire budget on lamb, he could buy 12.8kg. Of course, he could spread his £128 between the two goods in any proportions he chooses, so the budget constraint is drawn as follows:
With quantity of cod measured on the vertical axis, the slope is equal to –(P_{L}/P_{C}) = –10/16 = –0.625. (Note: If we had chosen to measure quantity of lamb on the vertical axis, the slope would be inverted: –(P_{C}/P_{L}) = −1.6.)
Companies face constraints on their production opportunities, just as consumers face limits on the bundles of goods that they can consume. Consider a company that produces two products using the same production capacity. For example, an automobile company might use the same factory to produce either automobiles or light trucks. If so, then the company is constrained by the limited capacity to produce vehicles. If it produces more trucks, it must reduce its production of automobiles; likewise, if it produces more automobiles, it must produce fewer trucks. The company’s production opportunity frontier shows the maximum number of units of one good it can produce, for any given number of the other good that it chooses to manufacture. Such a frontier for the vehicle company might look something like that in Exhibit 9.
Exhibit 9. The Production Opportunity FrontierNote: The production opportunity frontier for a vehicle manufacturer shows the maximum number of autos for any given level of truck production. In this example, the opportunity cost of a truck is 0.8 autos.
There are two important things to notice about this example. First, if the company devoted its entire production facility to the manufacture of automobiles, it could produce 1 million in a year. Alternatively, if it devoted its entire plant to trucks, it could produce 1.25 million a year. Of course, it could devote only part of the year’s production to trucks, in which case it could produce automobiles during the remainder of the year. In this simple example, for every additional truck the company chooses to make, it would have to produce 0.8 fewer cars. That is, the opportunity cost of a truck is 0.8 cars, or the opportunity cost of a car is 1.25 trucks. The opportunity cost of trucks is the negative of the slope of the production opportunity frontier: 1/1.25. And of course, the opportunity cost of an automobile is the inverse of that ratio, or 1.25.
The other thing to notice about this exhibit is that it assumes the opportunity cost of a truck is independent of how many trucks (and cars) the company produces. The production opportunity frontier is linear with a constant slope. Perhaps a more realistic example would be to increase marginal opportunity cost. As more and more trucks are produced, fewer inputs that are particularly well suited to producing truck inputs could be transferred to assist in their manufacture, causing the cost of trucks (in terms of cars) to rise as more trucks are produced. In this event, the production opportunity frontier would become steeper as the company moved its production point away from cars and toward more trucks, resulting in a frontier that would be concave as viewed from the origin.
The investment opportunity set is examined in detail in readings on investments, but it is appropriate to examine it briefly here because we are learning about constraints on behavior. Consider possible investments in which one option might be to invest in an essentially risk-free asset, such as a US Treasury bill. There is virtually no possibility that the US government would default on a 90-day obligation to pay back an investor’s purchase price, plus interest. Alternatively, an investor could put her money into a broadly diversified index of common shares. This investment will necessarily be more risky because of the fact that share prices fluctuate. If investors inherently find risk distasteful, then they will be reluctant to invest in a risky asset unless they expect to receive, on average, a higher rate of return. Hence, it is reasonable to expect that a broadly diversified index of common shares will have an expected return that exceeds that of the risk-free asset, or else no one would hold that portfolio.
Our hypothetical investor could choose to put some of her funds in the risk-free asset and the rest in the common shares index. For each additional dollar invested in the common shares index, she can expect to receive a higher return, though not with certainty; so, she is exposing herself to more risk in the pursuit of a higher return. We can structure her investment opportunities as a frontier that shows the highest expected return consistent with any given level of risk, as shown in Exhibit 10. The investor’s choice of a portfolio on the frontier will depend on her level of risk aversion.
Exhibit 10. The Investment Opportunity FrontierNote: The investment opportunity frontier shows that as the investor chooses to invest a greater proportion of assets in the market portfolio, she can expect a higher return but also higher risk.