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for optimal consumer choice. \u2022 Solve for an optimal consumption basket, given information about income, prices, and marginal utilities. \u2022 Explain why the optimal consumption basket solves both a utility maximization problem and an expendi- ture minimization problem. \u2022 Explain why the optimal consumption basket could occur at a corner point. c04consumerchoice.qxd 7/14/10 3:16 PM Page 104 \u2022 Illustrate the budget line and optimal consumer choice graphically when one of the goods a consumer can choose is a composite good. \u2022 Describe the concept of revealed preference. \u2022 Employ the concept of revealed preference to determine whether observed choices are consistent with utility maximization. 4.1 THE BUDGET CONSTRAINT 105 Number of households 121,171,000 14,720,000 11,824,000 37,322,000 Average number 2.5 2.2 2.4 3.1 of people in household Age of reference person* 48.8 52.3 46.8 47.0 Percent (reference person) 60 44 59 79 having attended college Income before taxes $ 63,091 $ 24,893 $44,555 $130,455 Income after taxes $60,858 $24,709 $43,628 $ 124,613 Average annual expenditures $49,638 $29,704 $41,083 $ 84,072 Expenditure on selected categories Food $ 6,133 $ 4,071 $ 5,689 $ 9,464 Housing (including $ 16,920 $ 10,994 $ 13,997 $ 27,408 shelter, utilities, supplies, furnishings, and equipment) Apparel and services $ 1,881 $ 1,016 $ 1,517 $ 3,275 Transportation $ 8,758 $ 5,434 $ 7,346 $ 14,362 Health care $ 2,853 $ 2,841 $ 2,800 $ 3,928 Entertainment $ 2,698 $ 1,375 $ 2,029 $ 4,927 All Households Households with Income $20,000\u2013 $29,999 Households with Income $40,000\u2013 $49,999 Households with Income over $70,000 TABLE 4.1 U.S. Average Expenditures by Household, 2007 *Reference person: The first member mentioned by the respondent when asked to \u201cStart with the name of the person or one of the persons who owns or rents the home.\u201d Source: Bureau of Labor Statistics. All data come from Table 2, Income Before Taxes: Average Annual Expenditures and Characteristics, Consumer Expenditure Survey, 2007. The Consumer Expenditure Survey tables are available online at www.bls.gov/cex/tables.htm, August 19, 2009. 4.1 THE BUDGET CONSTRAINT The budget constraint defines the set of baskets that a consumer can purchase with a limited amount of income. Suppose a consumer, Eric, purchases only two types of goods, food and clothing. Let x be the number of units of food he purchases each month and y the number of units of clothing. The price of a unit of food is Px, and the price of a unit of clothing is Py. Finally, to keep matters simple, let\u2019s assume that Eric has a fixed income of I dollars per month. c04consumerchoice.qxd 7/14/10 3:16 PM Page 105 106 CHAPTER 4 CONSUMER CHOICE Eric\u2019s total monthly expenditure on food will be Pxx (the price of a unit of food times the amount of food purchased). Similarly, his total monthly expenditure on clothing will be Py y (the price of a unit of clothing times the number of units of clothing purchased). The budget line indicates all of the combinations of food (x) and clothing ( y) that Eric can purchase if he spends all of his available income on the two goods. It can be expressed as (4.1) Figure 4.1 shows the graph of a budget line for Eric based on the following as- sumptions: Eric has an income of I \ufffd $800 per month, the price of food is Px \ufffd $20 per unit, and the price of clothing is Py \ufffd $40 per unit. If he spends all $800 on food, he will be able to buy, at most, I\ufffdPx \ufffd 800 \ufffd20 \ufffd 40 units of food. So the horizontal intercept of the budget line is at x \ufffd 40. Similarly, if Eric buys only clothing, he will be able to buy at most I \ufffdPy \ufffd 800 \ufffd40 \ufffd 20 units of clothing. So the vertical intercept of the budget line is at y \ufffd 20. As explained in Figure 4.1, Eric\u2019s income permits him to buy any basket on or inside the budget line (baskets A\u2013F ), but he cannot buy a basket outside the budget line, such as G. To buy G he would need to spend $1,000, which is more than his monthly income. These two sets of baskets\u2014those Eric can buy and those he cannot buy\u2014exemplify what is meant by the budget constraint. Since the budget constraint permits a consumer to buy baskets both on and inside the budget line, the equation for the budget constraint is somewhat different from Px x \ufffd Py y \ufffd I budget constraint The set of baskets that a consumer can purchase with a limited amount of income. budget line The set of baskets that a consumer can purchase when spending all of his or her available income. FIGURE 4.1 Budget Line The line connecting baskets A and E is Eric\u2019s budget line when he has an income of I \ufffd $800 per month, the price of food is Px \ufffd $20 per unit, and the price of clothing is Py \ufffd $40 per unit. The equation of the budget line is Px x \ufffd Py y \ufffd I (i.e., 20x \ufffd 40y \ufffd 800). Eric can buy any basket on or inside the budget line\u2014baskets A\u2013F (note that basket F would cost him only $600). However, he cannot buy a basket outside the budget line, such as basket G, which would cost him $1,000, more than his monthly income. A B C G I Px = 40 D E F 100 5 10 15 20 20 30 Budget line: Income = $800 per month Slope = \u2013 40 x, units of food y, u n its o f c lo th in g I Py = 20 = \u2013 Px Py 1 2 c04consumerchoice.qxd 6/18/10 5:31 PM Page 106 4.1 THE BUDGET CONSTRAINT 107 equation (4.1) for the budget line. The budget constraint can be expressed as: (4.1a) What does the slope of the budget line tell us? The slope of the budget line is \ufffdy/\ufffdx. If Eric is currently spending his entire income on basket B in Figure 4.1\u2014that is, consuming 10 units of food (x) and 15 units of clothing ( y)\u2014and he wants to move to basket C, he must give up 5 units of clothing (\ufffdy \ufffd \ufffd5) in order to gain 10 units of food (\ufffdx \ufffd 10). We can see that, in general, since food is half as expensive as cloth- ing, Eric must give up 1\ufffd2 unit of clothing for each additional unit of food, and the slope of the budget line reflects this (\ufffdy \ufffd\ufffdx \ufffd \ufffd5\ufffd10 \ufffd \ufffd1\ufffd2). Thus, the slope of the budget line tells us how many units of the good on the vertical axis a consumer must give up to obtain an additional unit of the good on the horizontal axis. Note that the slope of the budget line is \ufffdPx\ufffdPy.1 If the price of good x is three times the price of good y, the consumer must give up 3 units of y to get 1 more unit of x, and the slope is \ufffd3. If the prices are equal, the slope of the budget line is \ufffd1\u2014the consumer can always get 1 more unit of x by giving up 1 unit of y. HOW DOES A CHANGE IN INCOME AFFECT THE BUDGET LINE? As we have shown, the location of the budget line depends on the level of income and on the prices of the goods the consumer purchases. As you might expect, when income rises, the set of choices available to the consumer will increase. Let\u2019s see how the budget line changes as income varies. In the example just discussed, suppose Eric\u2019s income rises from I1 \ufffd $800 per month to I2 \ufffd $1,000 per month, with the prices Px \ufffd $20 and Py \ufffd $40 unchanged. As shown in Figure 4.2, if Eric buys only clothing, he can now purchase I2\ufffdPy \ufffd 1000\ufffd40 \ufffd 25 units of clothing, corresponding to the vertical intercept of the new budget line. The extra $200 of income allows him to buy an extra 5 units of y, since Py \ufffd $40. If he buys only food, he could purchase I2\ufffdPx \ufffd 1000\ufffd20 \ufffd 50 units, correspon- ding to the horizontal intercept on the new budget line. With the extra $200 of in- come he can buy an extra 10 units of x, since Px \ufffd $20. With his increased income of $1,000, he can now buy basket G, which had formerly been outside his budget line. The slopes of the two budget lines are the same because the prices of food and clothing are unchanged (\ufffdy\ufffd\ufffdx \ufffd \ufffdPx