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The Problem of Divisibility

24 April, 2015 - 15:27

If we are to apply the marginal decision rule to utility maximization, goods must be divisible; that is, itmust be possible to buy them in any amount. Otherwise we cannot meaningfully speak of spending $1 more or $1 less on them. Strictly speaking, however, few goods are completely divisible.

Even a small purchase, such as an ice cream bar, fails the strict test of being divisible; grocers generallyfrown on requests to purchase one-half of a $2 ice cream bar if the consumer wants to spend an additional dollar on ice cream. Can a consumer buy a little more movie admission, to say nothing of a little more car?

In the case of a car, we can think of the quantity as depending on characteristics of the car itself. Acar with a compact disc player could be regarded as containing “more car” than one that has only a cassette player. Stretching the concept of quantity in this manner does not entirely solve the problem. It is still difficult to imagine that one could purchase “more car” by spending $1 more.

Remember, though, that we are dealing with a model. In the real world, consumers may not be able to satisfy Equation 7.3 precisely. The model predicts, however, that they will come as close to doing so as possible.

KEY TAKEAWAYS

  • The utility of a good or service is determined by how much satisfaction a particular consumer obtains from it. Utility is not a quality inherent in the good or service itself.
  • Total utility is a conceptual measure of the number of units of utility a consumer gains from consuming a good, service, or activity. Marginal utility is the increase in total utility obtained by consuming one more unit of a good, service, or activity.
  • As a consumer consumes more and more of a good or service, its marginal utility falls.
  • Utility maximization requires seeking the greatest total utility from a given budget.
  • Utility is maximized when total outlays equal the budget available and when the ratios of marginal utilityto price are equal for all goods and services a consumer consumes; this is the utility-maximizing condition.

TRY IT!

A college student, Ramón Juárez, often purchases candy bars or bags of potato chips between classes; he to limit his spending on these snacks to $8 per week. A bag of chips costs $0.75 and a candy bar costs $0.50 from the vending machines on campus. He has been purchasing an average of 6 bags of chips and 7 candybars each week. Mr. Juárez is a careful maximizer of utility, and he estimates that the marginal utility of an additional bag of chips during a week is 6. In your answers use B to denote candy bars and C to denote potato chips.

  1. How much is he spending on snacks? How does this amount compare to his budget constraint?
  2. What is the marginal utility of an additional candy bar during the week?

Case in Point: Changing Lanes and Raising Utility

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In preparation for sitting in the slow, crowded lanes for single-occupancy-vehicles, T. J. Zane used to stop athis favorite coffee kiosk to buy a $2 cup of coffee as he headed off to work on Interstate 15 in the San Diego area. Since 1996, an experiment in road pricing has caused him and others to change their ways—and to raisetheir total utility.

Before 1996, only car-poolers could use the specially marked high-occupancy-vehicles lanes. With those lanesnearly empty, traffic authorities decided to allow drivers of single-occupancy-vehicles to use those lanes, so long as they paid a price. Now, electronic signs tell drivers how much it will cost them to drive on the special lanes. The price is recalculated every 6 minutes depending on the traffic. On one morning during rush hour, it varied from $1.25 at 7:10 a.m., to $1.50 at 7:16 a.m., to $2.25 at 7:22 a.m., and to $2.50 at 7:28 a.m. The increasing tolls over those few minutes caused some drivers to opt out and the toll fell back to $1.75 and then increasedto $2 a few minutes later. Drivers do not have to stop to pay the toll since radio transmitters read their FasTrak transponders and charge them accordingly.

When first instituted, these lanes were nicknamed the “Lexus lanes,” on the assumption that only wealthydrivers would use them. Indeed, while the more affluent do tend to use them heavily, surveys have discovered that they are actually used by drivers of all income levels.

Mr. Zane, a driver of a 1997 Volkswagen Jetta, is one commuter who chooses to use the new option.

He explains his decision by asking, “Isn’t it worth a couple of dollars to spend an extra half-hour with your family?” He continues, “That’s what I used to spend on a cup of coffee at Starbucks. Now I’ve started bringing my own coffee and using the money for the toll.”

We can explain his decision using the model of utility-maximizing behavior; Mr. Zane’s out-of-pocket commutingbudget constraint is about $2. His comment tells us that he realized that the marginal utility of spending an additional 30 minutes with his family divided by the $2 toll was higher than the marginal utility of the store boughtcoffee divided by its $2 price. By reallocating his $2 commuting budget, the gain in utility of having more time at home exceeds the loss in utility from not sipping premium coffee on the way to work.

From this one change in behavior, we do not know whether or not he is actually maximizing his utility, but his decision and explanation are certainly consistent with that goal.

Source: John Tierney, “The Autonomist Manifesto (Or, How I learned to Stop Worrying and Love the Road),” New York Times Magazine, September 26, 2004, 57–65.

ANSWERS TO TRY IT ! PROBLEMS

  1. He is spending $4.50 (= $0.75 × 6) on potato chips and $3.50 (= $0.50 × 7) on candy bars, for a total of $8. His budget constraint is $8.
  2. In order for the ratios of marginal utility to price to be equal, the marginal utility of a candy bar must be 4.Let the marginal utility and price of candy bars be MUB and PB, respectively, and the marginal utility and price of a bag of potato chips be MUC and PC, respectively. Then we want\frac{MUC}{PC}=\frac{MUB}{PB}

We know that PC is $0.75 and PB equals $0.50. We are told that MUC is 6. Thus\frac{6}{0.75}=\frac{MUB}{0.50}Solving the equation for MUB, we find that it must equal 4.