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Consumer Choice Review for AP Economics (page 2)

By — McGraw-Hill Professional
Updated on Mar 4, 2011

Constrained Utility Maximization

Now we require Joe to pay a price Pc for additional cups of coffee. With a fixed daily income and a price that must be paid, this individual is now a constrained utility maximizer. Joe must ask himself: "Does the next cup of coffee provide at least $Pc worth of additional happiness?" If Joe answers "Yes" to this question for the first three cups of coffee, he maximizes his utility by stopping at three cups. If his answer is "No" to the fourth cup, he does not consume it.

Does this sound familiar? It should, as it is another example of how a consumer never does something if the marginal benefit (in this case, utility) gained is exceeded by the marginal cost incurred.

  • When required to pay a price, the utility maximizing consumer stops consuming when: MB = P.

Demand Curve Revisited

Using the logic outlined above as an example, what would happen if the price of coffee fell? If Joe was facing a new lower price, you should expect that Joe would rationally increase his daily consumption of cups of coffee. Have you heard this behavior described before? Sure! It's the Law of Demand and it has a tight connection to the Law of Diminishing Marginal Utility.

Imagine you are a consumer who has already paid for, and consumed the first pint of ice cream this week. Would you pay the same price for the second pint of ice cream? Doubtful, because the second pint does not provide the same marginal utility as the first. In order to entice you to purchase and consume additional pints of Cherry Garcia ice cream, the price must fall to compensate you for your falling marginal utility.

This Law of Diminishing Marginal Utility is the backbone of the Law of Demand. To convert the relationship between marginal utility and quantity consumed at any price, we might ask you how much you are willing to pay to consume successive pints of ice cream. Because of diminishing marginal utility, you offer to pay less for additional units. Thus, we can then construct your monthly demand curve for ice cream. Figure 7.18 illustrates how diminishing marginal utility from consumption of a good can be converted to a demand curve for that good.

Consumer Choice

Constrained Utility Maximization, Two Goods

Economists see a consumer, constrained by income and prices, as living within a budget constraint. In a simple case where one good is consumed, the consumer maximizes utility by buying units of good X up to the point where the marginal utility of the last unit of good X is equal to the price. Most consumers allocate limited income between many goods and services, each with a price that must be paid. To see how a consumer maximizes utility in this situation, we consider a two-good case where, in addition to daily cups of coffee, Joe also purchases scones. We start with a "rule" and then proceed to solve a couple of problems.

"Learn the definitions first. This will make the logic much more obvious."

—David, AP Student

Utility Maximizing Rule

Given limited income, consumers maximize utility when they buy amounts of goods X and Y so that the marginal utility per dollar spent is equal for both goods. Another way to think about it is that they seek the most "bang for their bucks." Mathematically, this utility maximizing rule is expressed:

    MUx /Px = MUy /Py or MUx/MUy = Px /Py

If the consumer has used all income and the above ratios are equal, they are said to be in equilibrium. Under this condition, no other combination of X and Y provides more total utility.

Example:

Joe has daily income of $20, each cup of coffee costs Pc = $2 and each scone costs Ps = $4. Table 7.6 provides us with Joe's marginal utility received in the consumption of each good.

  • It is very important to remember that consuming more of one good causes the marginal utility to fall, but the total utility to rise.

In order to maximize Joe's utility, he seeks a combination of coffee and scones so that MUc/$2 = MUs/$4 and spends exactly his income of $20. Another way to solve this problem is to rearrange these ratios so that:

    MUc/MUs = $2/$4 = .5

There are several combinations of coffee and scones in Table 7.6 where the ratio of marginal utilities is one-half. For example, Joe could consume one cup of coffee (MU = 10) and three scones (MU = 20) for a total utility of 84 (10+30+24+20). But this combination would only spend a total of $14, and surely Joe would be happier if he used all of his income.

  • To find the total utility of consuming two cups of coffee, sum up the marginal utility of the first and the second. Do the same for scones to calculate total utility.

Another possibility is to consume two cups of coffee (MU = 8) with four scones (MU = 16). This does indeed spend exactly $20. The total utility of 108 confirms that Joe is happier with this combination of coffee and scones. There exists one other combination of goods that satisfies our rule: four cups of coffee (MU = 4) with six scones (MU = 8) expends too much money ($32) for Joe's income.

So according to our rule, Joe's utility maximizing decision would be to use his income of $20 to consume two cups of coffee and four scones. What if he decided to experiment and reallocate his consumption, while still spending only $20 on coffee and scones? For example, four cups of coffee (MU = 4) and three scones (MU = 20) fails our rule, but still would spend $20. On closer inspection, this is a poor decision because total utility falls to 102.

Example:

Now the price of a cup of coffee falls to $1. Joe needs to reexamine his utility maximizing combination of coffee and scones.

    MUc/MUs = $1/$4 = .25

Again, there are three possibilities, but only one uses exactly $20 of income. If Joe buys four cups of coffee (MU = 4) and four scones (MU = 16), he spends exactly his income and receives total utility of 118. The combination of three cups of coffee and two scones does not use all of the income and the combination of five cups of coffee and six scones exceeds the income constraint.

Connection Back to Demand Curves

Joe, as a utility maximizing consumer, chooses two cups of coffee at a price of $2 and four cups of coffee at a price of $1. This sounds familiar! What Joe has done, simply by responding in a utility maximizing way, is illustrate the Law of Demand. The two combinations of price and quantity demanded are two points on Joe's coffee demand curve. By connecting these points, we trace out his demand curve (Figure 7.19).

Consumer Choice

Individual and Market Demand Curves

We can take the individual decisions made by consumers like Joe and expand the analysis to build a market demand curve for coffee and other goods. This process is called horizontal summation. At every price, we would simply add the quantity demanded for all individual consumers.

  • Utility maximizing behavior of individuals creates individual demand curves.
  • Summing the quantity demanded by individuals at each price creates market demand curves.

Review questions for this study guide can be found at:

Elasticity, Microeconomic Policy, and Consumer Theory Review Questions for AP Economics

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