{"id":6379,"date":"2018-02-05T15:35:22","date_gmt":"2018-02-05T15:35:22","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-microeconomics\/chapter\/reading-a-tool-for-maximizing-utility\/"},"modified":"2018-06-18T21:30:32","modified_gmt":"2018-06-18T21:30:32","slug":"rules-for-maximizing-utility","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-kbcc-microeconomics\/chapter\/rules-for-maximizing-utility\/","title":{"raw":"Rules for Maximizing Utility","rendered":"Rules for Maximizing Utility"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"list\">\r\n \t<li>Explain why maximizing utility requires that the last unit of each item purchased must have the same marginal utility per dollar<\/li>\r\n \t<li>Calculate the utility-maximizing choice<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe problem of finding <strong>consumer equilibrium<\/strong>, that is, the combination of goods and services that will maximize an individual's total utility, comes down to comparing the trade-offs between one affordable combination (shown by a point on the budget line in Figure 1, below) with all the other affordable combinations.\r\n\r\nMost people approach their utility-maximizing combination of choices in a step-by-step way. This step-by-step approach is based on looking at the tradeoffs, measured in terms of marginal utility, of consuming less of one good and more of another. You can think of this step-by-step approach as the \u201cbiggest bang for the buck\u201d principle. For example, say that Jos\u00e9 starts off thinking about spending all his money on T-shirts and choosing point P, which corresponds to four T-shirts and no movies, as illustrated in Figure 1.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"585\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2042\/2018\/02\/05153520\/CNX_Econ_C06_001.jpg\" alt=\"The points on the graph show how a budget is affected by spending choices. Spending more money at the movies (y-axis) means that Jose' has less money to spend on T-shirts (x-axis).\" width=\"585\" height=\"300\" \/> <strong>Figure 1<\/strong>. <strong>A Choice between Consumption Goods.<\/strong> Jos\u00e9 has income of $56. Movies cost $7 and T-shirts cost $14. The points on the budget constraint line show the combinations of movies and T-shirts that are affordable.[\/caption]\r\n\r\nJos\u00e9 chooses this starting point randomly; he has to start somewhere. Then he considers giving up the last T-shirt, the one that provides him the least marginal utility, and using the money he saves to buy two movies instead. Table 1 tracks the step-by-step series of decisions Jos\u00e9 needs to make (<span class=\"emphasis\"><em>Key<\/em><\/span>: T-shirts cost $14, movies cost $7, and Jose's\u00a0income is $56).\r\n<table id=\"Table_06_04\" summary=\"The table has for columns. Column 1 is labeled \u201cTry,\u201d Column 2 is labeled \u201cWhich Has,\u201d Column 3 is labeled \u201cTotal Utility,\u201d Column 4 is labeled \u201cMarginal Gain and Loss of Utility, Compared with Previous Choice,\u201d Column 5 is labeled \u201cConclusion.\u201d Row 1 shows the following: Column 1 = Choice 1: P. column 2 = 4 t-shirts and 0 movies; column 3 = 81 from 4 \u2013t-shirts + 0 from 0 movies = 81; columns 4 and 5 are blank. Row 2 shows column 1 = Choice 2: Q; column 2 = 3 t-shirts and 2 movies; column 3 = 63 from 3 t-shirts + 31 from 0 movies = 94; column 4 = loss of 18 from 1 less t-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13; column 5 = Q is preferred over P. Row 3 shows the following: column 1 = Choice 3: R, column 2 = 2 t-shirts and 4 movies; column 3 = 43 from 2 t-shirts + 58 from 4 movies = 101; column 4 = loss from 1 less t-shirt, but gain of 27 from two more movies for a net utility gain of 7; column 5 = R is preferred over Q. Row 4 shows the following: Choice 4 : S; column 2 = 1 t-shirt and 6 movies; column 3 = 22 from 1 t-shirt + 81 from 6 movies = 1-3; column 4 = loss of 21 from 1 less t-shirt, but gain of 23 from two more movies, for a net utility gain of 2; column 5 = s is preferred over R. Row 5 shows the following: column 1 = Choice 5: T; column 2 = 0 t-shirts and 8 movies; column 3 = 0 from 0 t-shirts + 100 from 8 movies = 100; column 4 = loss of 22 from 1 less t-shirt, but gain of 19 from two more movies, for a net utility loss of 3; column 5 = s is preferred over T.\"><caption>Table 1. A Step-by-Step Approach to Maximizing Utility<\/caption>\r\n<thead>\r\n<tr>\r\n<th>Try<\/th>\r\n<th>Which Has<\/th>\r\n<th>Total Utility<\/th>\r\n<th>Marginal Gain and Loss of Utility, Compared with Previous Choice<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Choice 1: P<\/td>\r\n<td>4 T-shirts and 0 movies<\/td>\r\n<td>81 from 4 T-shirts + 0 from 0 movies = 81<\/td>\r\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\r\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 2: Q<\/td>\r\n<td>3 T-shirts and 2 movies<\/td>\r\n<td>63 from 3 T-shirts + 31 from 0 movies = 94<\/td>\r\n<td>Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13<\/td>\r\n<td>Q is preferred over P<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 3: R<\/td>\r\n<td>2 T-shirts and 4 movies<\/td>\r\n<td>43 from 2 T-shirts + 58 from 4 movies = 101<\/td>\r\n<td>Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7<\/td>\r\n<td>R is preferred over Q<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 4: S<\/td>\r\n<td>1 T-shirt and 6 movies<\/td>\r\n<td>22 from 1 T-shirt + 81 from 6 movies = 103<\/td>\r\n<td>Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2<\/td>\r\n<td>S is preferred over R<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 5: T<\/td>\r\n<td>0 T-shirts and 8 movies<\/td>\r\n<td>0 from 0 T-shirts + 100 from 8 movies = 100<\/td>\r\n<td>Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3<\/td>\r\n<td>S is preferred over T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>DECISION MAKING BY COMPARING MARGINAL UTILITY<\/h3>\r\nJos\u00e9 could use the following thought process (if he thought in utils) to make his decision regarding how many T-shirts and movies to purchase:\r\n\r\n<strong>Step 1.<\/strong> From Table 1, Jos\u00e9 can see that the marginal utility of the fourth T-shirt is 18. If Jos\u00e9 gives up the fourth T-shirt, then he loses 18 utils.\r\n\r\n<strong>Step 2.<\/strong> Giving up the fourth T-shirt, however, frees up $14 (the price of a T-shirt), allowing Jos\u00e9 to buy the first two movies (at $7 each).\r\n\r\n<strong>Step 3.<\/strong> Jos\u00e9 knows that the marginal utility of the first movie is 16 and the marginal utility of the second movie is 15. Thus, if Jos\u00e9 moves from point P to point Q, he gives up 18 utils (from the T-shirt), but gains 31 utils (from the movies).\r\n\r\n<strong>Step 4.<\/strong> Gaining 31 utils and losing 18 utils is a net gain of 13. This is just another way of saying that the total utility at Q (94 according to the last column in Table 1) is 13 more than the total utility at P (81).\r\n\r\n<strong>Step 5.<\/strong> So, for Jos\u00e9, it makes sense to give up the fourth T-shirt in order to buy two movies.\r\n\r\n<\/div>\r\nJos\u00e9 clearly prefers point Q to point P. Now repeat this step-by-step process of decision making with marginal utilities. Jos\u00e9 thinks about giving up the third T-shirt and surrendering a marginal utility of 20, in exchange for purchasing two more movies that promise a combined marginal utility of 27. Jos\u00e9 prefers point R to point Q. What if Jos\u00e9 thinks about going beyond R to point S? Giving up the second T-shirt means a marginal utility loss of 21, and the marginal utility gain from the fifth and sixth movies would combine to make a marginal utility gain of 23, so Jos\u00e9 prefers point S to R.\r\n\r\nHowever, if Jos\u00e9 seeks to go beyond point S to point T, he finds that the loss of marginal utility from giving up the first T-shirt is 22, while the marginal utility gain from the last two movies is only a total of 19. If Jos\u00e9 were to choose point T, his utility would fall to 100. Through these stages of thinking about marginal tradeoffs, Jos\u00e9 again concludes that S, with one T-shirt and six movies, is the choice that will provide him with the highest level of total utility. This step-by-step approach will reach the same conclusion regardless of Jos\u00e9's starting point.\r\n\r\nThis approach to finding consumer equilibrium is somewhat tedious. Next, we'll turn to a quicker and more intuitive approach.\r\n<h2>A Rule for Maximizing Utility<\/h2>\r\nThis process of decision making described previously suggests a rule to follow when\u00a0maximizing utility. Since the price of T-shirts is not the same as the price of movies, it's not enough to just compare the marginal utility of T-shirts with the marginal utility of movies. Instead, we need to control for the prices of each product. We can do this by computing and comparing marginal utility per dollar of expenditure for each product.\u00a0<strong>Marginal utility per dollar<\/strong> is the amount of additional utility Jos\u00e9 receives given the price of the product.\r\n<div id=\"eip-255\" style=\"text-align: center;\">[latex]\\begin{array}{rcl}\\text{marginal utility per dollar}&amp; =&amp; \\frac{\\text{marginal utility}}{\\text{price}}\\end{array}[\/latex]<\/div>\r\nFor Jos\u00e9's T-shirts and movies, the marginal utility per dollar is shown in Table 2.\r\n<table class=\"aligncenter\" summary=\"The table has 8 columns and 7 rows of data. Column 1 is \u201cquantity of t-shirts\u201d; Column 2 is \u201ctotal utility\u201d; column 3 is \u201cmarginal utility\u201d; column 4 is \u201cmarginal utility per dollar\u201d; column 5 is \u201cquantity of movies\u201d; column 6 is \u201ctotal utility\u201d; column 7 is \u201cmarginal utility\u201d; column 8 is \u201cmarginal utility per dollar.\u201d Column 1 has the following data: 1, 2, 3, 4, 5, 6, 7. Column 2 has the following data: 22, 43, 63, 81, 97, 111, 123. Column 3 has the following data: 22, 21, 20, 18, 16, 14, 12. Column 4 has the following data: (22\/$14=1.6); (21\/$14=1.5); (20\/$14=1.4); (18\/$14=1.3); (16\/$14=1.1); (14\/$14=1); (12\/$14=1.2). Column 5 has the following data: 1, 2, 3, 4, 5, 6, 7. Column 6 has the following data: 16, 31, 45, 58, 70, 81, 91. Column 7 has the following data: 16, 15, 14, 13, 12, 11, 10. Column 8 has the following data: (16\/$7=2.3); (15\/$7=2.14); (14\/$7=2); (13\/$7=1.9); (12\/$7=1.7); (11\/$7=1.6); (10\/$7=1.4).\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"8\">Table 2. Marginal Utility per Dollar<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th>Quantity of T-Shirts<\/th>\r\n<th>Total Utility<\/th>\r\n<th>Marginal Utility<\/th>\r\n<th>Marginal Utility per Dollar<\/th>\r\n<th>Quantity of Movies<\/th>\r\n<th>Total Utility<\/th>\r\n<th>Marginal Utility<\/th>\r\n<th>Marginal Utility per Dollar<\/th>\r\n<\/tr>\r\n<tr>\r\n<td><strong>1<\/strong><\/td>\r\n<td><strong>22<\/strong><\/td>\r\n<td><strong>22<\/strong><\/td>\r\n<td><strong>22\/$14=1.6<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>16<\/td>\r\n<td>16<\/td>\r\n<td>16\/$7=2.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>43<\/td>\r\n<td>21<\/td>\r\n<td>21\/$14=1.5<\/td>\r\n<td>2<\/td>\r\n<td>31<\/td>\r\n<td>15<\/td>\r\n<td>15\/$7=2.14<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>63<\/td>\r\n<td>20<\/td>\r\n<td>20\/$14=1.4<\/td>\r\n<td>3<\/td>\r\n<td>45<\/td>\r\n<td>14<\/td>\r\n<td>14\/$7=2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>81<\/td>\r\n<td>18<\/td>\r\n<td>18\/$14=1.3<\/td>\r\n<td>4<\/td>\r\n<td>58<\/td>\r\n<td>13<\/td>\r\n<td>13\/$7=1.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>97<\/td>\r\n<td>16<\/td>\r\n<td>16\/$14=1.1<\/td>\r\n<td>5<\/td>\r\n<td>70<\/td>\r\n<td>12<\/td>\r\n<td>12\/$7=1.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>111<\/td>\r\n<td>14<\/td>\r\n<td>14\/$14=1<\/td>\r\n<td><strong>6<\/strong><\/td>\r\n<td><strong>81<\/strong><\/td>\r\n<td><strong>11<\/strong><\/td>\r\n<td><strong>11\/$7=1.6<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7<\/td>\r\n<td>123<\/td>\r\n<td>12<\/td>\r\n<td>12\/$14=1.2<\/td>\r\n<td>7<\/td>\r\n<td>91<\/td>\r\n<td>10<\/td>\r\n<td>10\/$7=1.4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-idp43427504\" class=\"economics workout\">\r\n<div class=\"textbox exercises\">\r\n<h3>A Rule for maximizing Utility<\/h3>\r\n<div id=\"fs-idp43427504\" class=\"economics workout\">\r\n<p id=\"fs-idm16605536\">If a consumer wants to maximize total utility, for every dollar that they spend, they should spend it on the item which yields the greatest marginal utility per dollar of expenditure.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2><strong>Applying the Rule<\/strong><\/h2>\r\nJos\u00e9's first purchase will be a movie. Why? Jos\u00e9's choices are to purchase either a T-shirt or a movie. Table 1 shows that the marginal utility per dollar spent on the first T-shirt is 1.6 compared with 2.5 for the first movie.\u00a0Because the first movie gives Jos\u00e9 more marginal utility per dollar than the first T-shirt, and because the movie is within his budget, he will purchase a movie first.\r\n\r\nJos\u00e9 will continue to purchase the good which gives him the highest marginal utility per dollar until he exhausts the budget. Jos\u00e9 will keep purchasing movies because they give him a greater \"bang for the buck\" until the sixth movie is equivalent to a T-shirt purchase. Jos\u00e9 can afford to purchase that T-shirt. So Jos\u00e9 will choose to purchase six movies and one T-shirt. That combination, six movies and one T-shirt, is his consumer equilibrium\r\n<h2>Another Rule for Maximizing Utility<\/h2>\r\nSince the price of T-shirts is twice as high as the price of movies, to maximize utility the last T-shirt chosen needs to provide exactly twice the marginal utility (MU) of the last movie. If the last T-shirt provides less than twice the marginal utility of the last movie, then the T-shirt is providing less \"bang for the buck\" (i.e., marginal utility per dollar spent) than if the same money were spent on movies. If this is so, Jos\u00e9 should trade the T-shirt for more movies to increase his total utility. Marginal utility per dollar measures the additional utility that Jos\u00e9 will enjoy given what he has to pay for the good.\r\n\r\nIf the last T-shirt provides more than twice the marginal utility of the last movie, then the T-shirt is providing more \"bang for the buck\" or marginal utility per dollar, than if the money were spent on movies. As a result, Jos\u00e9 should buy more T-shirts. Notice that at Jos\u00e9's optimal choice the marginal utility from the first T-shirt, of 22 is exactly twice the marginal utility of the sixth movie, which is 11. At this choice, the marginal utility per dollar is the same for both goods. This is a tell-tale signal that Jos\u00e9 has found the point with highest total utility.\r\n\r\nThis argument can be written as another rule: the utility-maximizing choice between consumption goods occurs where the marginal utility per dollar is the same for both goods, and the consumer has exhausted his or her budget.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{MU_1}{P_1}=\\frac{MU_2}{P_2}[\/latex]<\/p>\r\nA sensible economizer will pay twice as much for something only if, in the marginal comparison, the item confers twice as much utility. Notice that the formula for the table above is\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{22}{14}=\\frac{11}{7}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]1.6=1.6[\/latex]<\/p>\r\nThe following feature provides step-by-step guidance for this concept of utility-maximizing choices.\r\n<div class=\"textbox exercises\">\r\n<h3><strong>Another Rule for Maximizing Utility<\/strong><\/h3>\r\nThe rule, [latex]\\displaystyle\\frac{MU_1}{P_1}=\\frac{MU_2}{P_2}[\/latex], means that the last dollar spent on each good provides exactly the same marginal utility. So:\r\n\r\n<strong>Step 1.<\/strong> If we traded a dollar more of movies for a dollar more of T-shirts, the marginal utility gained from T-shirts would exactly offset the marginal utility lost from fewer movies. In other words, the net gain would be zero.\r\n\r\n<strong>Step 2.<\/strong> Products, however, usually cost more than a dollar, so we cannot trade a dollar's worth of movies. The best we can do is trade two movies for another T-shirt, since in this example T-shirts cost twice what a movie does.\r\n\r\n<strong>Step 3.<\/strong> If we trade two movies for one T-shirt, we would end up at point R (two T-shirts and four movies).\r\n\r\n<strong>Step 4.<\/strong> Choice 4 in Table 3 shows that if we move to point S, we would lose 21 utils from one less T-shirt, but gain 23 utils from two more movies, so we would end up with more total utility at point S.\r\n<table summary=\"The table has for columns. Column 1 is labeled \u201cTry,\u201d Column 2 is labeled \u201cWhich Has,\u201d Column 3 is labeled \u201cTotal Utility,\u201d Column 4 is labeled \u201cMarginal Gain and Loss of Utility, Compared with Previous Choice,\u201d Column 5 is labeled \u201cConclusion.\u201d Row 1 shows the following: Column 1 = Choice 1: P. column 2 = 4 t-shirts and 0 movies; column 3 = 81 from 4 \u2013t-shirts + 0 from 0 movies = 81; columns 4 and 5 are blank. Row 2 shows column 1 = Choice 2: Q; column 2 = 3 t-shirts and 2 movies; column 3 = 63 from 3 t-shirts + 31 from 0 movies = 94; column 4 = loss of 18 from 1 less t-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13; column 5 = Q is preferred over P. Row 3 shows the following: column 1 = Choice 3: R, column 2 = 2 t-shirts and 4 movies; column 3 = 43 from 2 t-shirts + 58 from 4 movies = 101; column 4 = loss from 1 less t-shirt, but gain of 27 from two more movies for a net utility gain of 7; column 5 = R is preferred over Q. Row 4 shows the following: Choice 4 : S; column 2 = 1 t-shirt and 6 movies; column 3 = 22 from 1 t-shirt + 81 from 6 movies = 1-3; column 4 = loss of 21 from 1 less t-shirt, but gain of 23 from two more movies, for a net utility gain of 2; column 5 = s is preferred over R. Row 5 shows the following: column 1 = Choice 5: T; column 2 = 0 t-shirts and 8 movies; column 3 = 0 from 0 t-shirts + 100 from 8 movies = 100; column 4 = loss of 22 from 1 less t-shirt, but gain of 19 from two more movies, for a net utility loss of 3; column 5 = s is preferred over T.\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"5\">Table 3. A Step-by-Step Approach to Maximizing Utility<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th>Try<\/th>\r\n<th>Which Has<\/th>\r\n<th>Total Utility<\/th>\r\n<th>Marginal Gain and Loss of Utility, Compared with Previous Choice<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 1: P<\/td>\r\n<td>4 T-shirts and 0 movies<\/td>\r\n<td>81 from 4 T-shirts + 0 from 0 movies = 81<\/td>\r\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\r\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 2: Q<\/td>\r\n<td>3 T-shirts and 2 movies<\/td>\r\n<td>63 from 3 T-shirts + 31 from 0 movies = 94<\/td>\r\n<td>Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13<\/td>\r\n<td>Q is preferred over P<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 3: R<\/td>\r\n<td>2 T-shirts and 4 movies<\/td>\r\n<td>43 from 2 T-shirts + 58 from 4 movies = 101<\/td>\r\n<td>Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7<\/td>\r\n<td>R is preferred over Q<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 4: S<\/td>\r\n<td>1 T-shirt and 6 movies<\/td>\r\n<td>22 from 1 T-shirt + 81 from 6 movies = 103<\/td>\r\n<td>Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2<\/td>\r\n<td>S is preferred over R<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Choice 5: T<\/td>\r\n<td>0 T-shirts and 8 movies<\/td>\r\n<td>0 from 0 T-shirts + 100 from 8 movies = 100<\/td>\r\n<td>Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3<\/td>\r\n<td>S is preferred over T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn short, the rule shows us the utility-maximizing choice.\r\n\r\n<\/div>\r\nThere is another, equivalent way to think about this. The rule can also be expressed as\u00a0<span class=\"emphasis\"><em>the ratio of the prices of the two goods should be equal to the ratio of the marginal utilities.<\/em><\/span> When the price of good 1 is divided by the price of good 2, at the utility-maximizing point this will equal the marginal utility of good 1 divided by the marginal utility of good 2. This rule can be written in algebraic form:\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{P_1}{P_2}=\\frac{MU_1}{MU_2}[\/latex]<\/p>\r\nAlong the budget constraint, the total price of the two goods remains the same, so the ratio of the prices does not change. However, the marginal utility of the two goods changes with the quantities consumed. At the optimal choice of one T-shirt and six movies, point S, the ratio of marginal utility to price for T-shirts (22:14) matches the ratio of marginal utility to price for movies (of 11:7).\r\n<div class=\"textbox examples\">\r\n<h3>Watch It<\/h3>\r\nThis video applies these same concepts to a graph. It demonstrates how there will come a point when the marginal utility per dollar of one good goes down enough so that it makes more sense to buy another good.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/KbW6OiuRa1Y?showinfo=0\" width=\"800\" height=\"470\" frameborder=\"0\"><\/iframe>\r\n\r\n<\/div>\r\n<h2><span class=\"cnx-gentext-section cnx-gentext-t\">Measuring Utility with Numbers<\/span><\/h2>\r\nThis discussion of utility started off with an assumption that it is possible to place numerical values on utility, an assumption that may seem questionable. You can buy a thermometer for measuring temperature at the hardware store, but what store sells an \"utilimometer\" for measuring utility? However, while measuring utility with numbers is a convenient assumption to clarify the explanation, the key assumption is not that utility can be measured by an outside party, but only that individuals can decide which of two alternatives they prefer.\r\n\r\nTo understand this point, think back to the step-by-step process of finding the choice with highest total utility by comparing the marginal utility that is gained and lost from different choices along the budget constraint. As Jos\u00e9 compares each choice along his budget constraint to the previous choice, what matters is not the specific numbers that he places on his utility\u2014or whether he uses any numbers at all\u2014but only that he personally can identify which choices he prefers.\r\n\r\nIn this way, the step-by-step process of choosing the highest level of utility resembles rather closely how many people make consumption decisions. We think about what will make us the happiest; we think about what things cost; we think about buying a little more of one item and giving up a little of something else; we choose what provides us with the greatest level of satisfaction. The vocabulary of comparing the points along a budget constraint and total and marginal utility is just a set of tools for discussing this everyday process in a clear and specific manner. It is welcome news that specific utility numbers are not central to the argument, since a good utilimometer is hard to find. Do not worry\u2014while we cannot measure utils, by the end of the next module, we will have transformed our analysis into something we can measure\u2014demand.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/7956\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/7957\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/7958\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/7959\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nThese questions allow you to get as much practice as you need, as you can click the link at the top of the first question (\u201cTry another version of these questions\u201d) to get a new set of questions. Practice until you feel comfortable doing the questions.\r\n\r\n[ohm_question]155309[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nThese questions allow you to get as much practice as you need, as you can click the link at the top of the first question (\u201cTry another version of these questions\u201d) to get a new set of questions. Practice until you feel comfortable doing the questions.\r\n\r\n[ohm_question sameseed=1]155322-155321[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nClick through the buttons in this simulation to think about applications of utility.\r\n\r\nFirst, you'll imagine that you are desperately searching for food and water on a hot day. Given the heat and your time constraints, you'll need to decide what to obtain first based on its utility. Then, you can see these combinations on a graph and consider what the utility of each choice would be.\r\n\r\nBe sure to choose the option to \"learn more about marginal utility per dollar spent\" so that you can see how marginal-decision making comes into play when choosing to forage for either apples or berries.\r\n\r\n<iframe src=\"https:\/\/www.branchtrack.com\/projects\/i0fyro7x\/embed\" width=\"850\" height=\"500\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<\/div>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/7774\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Glossary<\/h3>\r\n[glossary-page][glossary-term]consumer equilibrium:[\/glossary-term]\r\n[glossary-definition]the combination of goods and services that will maximize an individual's total utility[\/glossary-definition][glossary-term]marginal utility per dollar:[\/glossary-term]\r\n[glossary-definition]the additional satisfaction gained from purchasing a good given the price of the product; MU\/Price[\/glossary-definition]\r\n[\/glossary-page]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"list\">\n<li>Explain why maximizing utility requires that the last unit of each item purchased must have the same marginal utility per dollar<\/li>\n<li>Calculate the utility-maximizing choice<\/li>\n<\/ul>\n<\/div>\n<p>The problem of finding <strong>consumer equilibrium<\/strong>, that is, the combination of goods and services that will maximize an individual&#8217;s total utility, comes down to comparing the trade-offs between one affordable combination (shown by a point on the budget line in Figure 1, below) with all the other affordable combinations.<\/p>\n<p>Most people approach their utility-maximizing combination of choices in a step-by-step way. This step-by-step approach is based on looking at the tradeoffs, measured in terms of marginal utility, of consuming less of one good and more of another. You can think of this step-by-step approach as the \u201cbiggest bang for the buck\u201d principle. For example, say that Jos\u00e9 starts off thinking about spending all his money on T-shirts and choosing point P, which corresponds to four T-shirts and no movies, as illustrated in Figure 1.<\/p>\n<div style=\"width: 595px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2042\/2018\/02\/05153520\/CNX_Econ_C06_001.jpg\" alt=\"The points on the graph show how a budget is affected by spending choices. Spending more money at the movies (y-axis) means that Jose' has less money to spend on T-shirts (x-axis).\" width=\"585\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1<\/strong>. <strong>A Choice between Consumption Goods.<\/strong> Jos\u00e9 has income of $56. Movies cost $7 and T-shirts cost $14. The points on the budget constraint line show the combinations of movies and T-shirts that are affordable.<\/p>\n<\/div>\n<p>Jos\u00e9 chooses this starting point randomly; he has to start somewhere. Then he considers giving up the last T-shirt, the one that provides him the least marginal utility, and using the money he saves to buy two movies instead. Table 1 tracks the step-by-step series of decisions Jos\u00e9 needs to make (<span class=\"emphasis\"><em>Key<\/em><\/span>: T-shirts cost $14, movies cost $7, and Jose&#8217;s\u00a0income is $56).<\/p>\n<table id=\"Table_06_04\" summary=\"The table has for columns. Column 1 is labeled \u201cTry,\u201d Column 2 is labeled \u201cWhich Has,\u201d Column 3 is labeled \u201cTotal Utility,\u201d Column 4 is labeled \u201cMarginal Gain and Loss of Utility, Compared with Previous Choice,\u201d Column 5 is labeled \u201cConclusion.\u201d Row 1 shows the following: Column 1 = Choice 1: P. column 2 = 4 t-shirts and 0 movies; column 3 = 81 from 4 \u2013t-shirts + 0 from 0 movies = 81; columns 4 and 5 are blank. Row 2 shows column 1 = Choice 2: Q; column 2 = 3 t-shirts and 2 movies; column 3 = 63 from 3 t-shirts + 31 from 0 movies = 94; column 4 = loss of 18 from 1 less t-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13; column 5 = Q is preferred over P. Row 3 shows the following: column 1 = Choice 3: R, column 2 = 2 t-shirts and 4 movies; column 3 = 43 from 2 t-shirts + 58 from 4 movies = 101; column 4 = loss from 1 less t-shirt, but gain of 27 from two more movies for a net utility gain of 7; column 5 = R is preferred over Q. Row 4 shows the following: Choice 4 : S; column 2 = 1 t-shirt and 6 movies; column 3 = 22 from 1 t-shirt + 81 from 6 movies = 1-3; column 4 = loss of 21 from 1 less t-shirt, but gain of 23 from two more movies, for a net utility gain of 2; column 5 = s is preferred over R. Row 5 shows the following: column 1 = Choice 5: T; column 2 = 0 t-shirts and 8 movies; column 3 = 0 from 0 t-shirts + 100 from 8 movies = 100; column 4 = loss of 22 from 1 less t-shirt, but gain of 19 from two more movies, for a net utility loss of 3; column 5 = s is preferred over T.\">\n<caption>Table 1. A Step-by-Step Approach to Maximizing Utility<\/caption>\n<thead>\n<tr>\n<th>Try<\/th>\n<th>Which Has<\/th>\n<th>Total Utility<\/th>\n<th>Marginal Gain and Loss of Utility, Compared with Previous Choice<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Choice 1: P<\/td>\n<td>4 T-shirts and 0 movies<\/td>\n<td>81 from 4 T-shirts + 0 from 0 movies = 81<\/td>\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\n<\/tr>\n<tr>\n<td>Choice 2: Q<\/td>\n<td>3 T-shirts and 2 movies<\/td>\n<td>63 from 3 T-shirts + 31 from 0 movies = 94<\/td>\n<td>Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13<\/td>\n<td>Q is preferred over P<\/td>\n<\/tr>\n<tr>\n<td>Choice 3: R<\/td>\n<td>2 T-shirts and 4 movies<\/td>\n<td>43 from 2 T-shirts + 58 from 4 movies = 101<\/td>\n<td>Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7<\/td>\n<td>R is preferred over Q<\/td>\n<\/tr>\n<tr>\n<td>Choice 4: S<\/td>\n<td>1 T-shirt and 6 movies<\/td>\n<td>22 from 1 T-shirt + 81 from 6 movies = 103<\/td>\n<td>Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2<\/td>\n<td>S is preferred over R<\/td>\n<\/tr>\n<tr>\n<td>Choice 5: T<\/td>\n<td>0 T-shirts and 8 movies<\/td>\n<td>0 from 0 T-shirts + 100 from 8 movies = 100<\/td>\n<td>Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3<\/td>\n<td>S is preferred over T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>DECISION MAKING BY COMPARING MARGINAL UTILITY<\/h3>\n<p>Jos\u00e9 could use the following thought process (if he thought in utils) to make his decision regarding how many T-shirts and movies to purchase:<\/p>\n<p><strong>Step 1.<\/strong> From Table 1, Jos\u00e9 can see that the marginal utility of the fourth T-shirt is 18. If Jos\u00e9 gives up the fourth T-shirt, then he loses 18 utils.<\/p>\n<p><strong>Step 2.<\/strong> Giving up the fourth T-shirt, however, frees up $14 (the price of a T-shirt), allowing Jos\u00e9 to buy the first two movies (at $7 each).<\/p>\n<p><strong>Step 3.<\/strong> Jos\u00e9 knows that the marginal utility of the first movie is 16 and the marginal utility of the second movie is 15. Thus, if Jos\u00e9 moves from point P to point Q, he gives up 18 utils (from the T-shirt), but gains 31 utils (from the movies).<\/p>\n<p><strong>Step 4.<\/strong> Gaining 31 utils and losing 18 utils is a net gain of 13. This is just another way of saying that the total utility at Q (94 according to the last column in Table 1) is 13 more than the total utility at P (81).<\/p>\n<p><strong>Step 5.<\/strong> So, for Jos\u00e9, it makes sense to give up the fourth T-shirt in order to buy two movies.<\/p>\n<\/div>\n<p>Jos\u00e9 clearly prefers point Q to point P. Now repeat this step-by-step process of decision making with marginal utilities. Jos\u00e9 thinks about giving up the third T-shirt and surrendering a marginal utility of 20, in exchange for purchasing two more movies that promise a combined marginal utility of 27. Jos\u00e9 prefers point R to point Q. What if Jos\u00e9 thinks about going beyond R to point S? Giving up the second T-shirt means a marginal utility loss of 21, and the marginal utility gain from the fifth and sixth movies would combine to make a marginal utility gain of 23, so Jos\u00e9 prefers point S to R.<\/p>\n<p>However, if Jos\u00e9 seeks to go beyond point S to point T, he finds that the loss of marginal utility from giving up the first T-shirt is 22, while the marginal utility gain from the last two movies is only a total of 19. If Jos\u00e9 were to choose point T, his utility would fall to 100. Through these stages of thinking about marginal tradeoffs, Jos\u00e9 again concludes that S, with one T-shirt and six movies, is the choice that will provide him with the highest level of total utility. This step-by-step approach will reach the same conclusion regardless of Jos\u00e9&#8217;s starting point.<\/p>\n<p>This approach to finding consumer equilibrium is somewhat tedious. Next, we&#8217;ll turn to a quicker and more intuitive approach.<\/p>\n<h2>A Rule for Maximizing Utility<\/h2>\n<p>This process of decision making described previously suggests a rule to follow when\u00a0maximizing utility. Since the price of T-shirts is not the same as the price of movies, it&#8217;s not enough to just compare the marginal utility of T-shirts with the marginal utility of movies. Instead, we need to control for the prices of each product. We can do this by computing and comparing marginal utility per dollar of expenditure for each product.\u00a0<strong>Marginal utility per dollar<\/strong> is the amount of additional utility Jos\u00e9 receives given the price of the product.<\/p>\n<div id=\"eip-255\" style=\"text-align: center;\">[latex]\\begin{array}{rcl}\\text{marginal utility per dollar}& =& \\frac{\\text{marginal utility}}{\\text{price}}\\end{array}[\/latex]<\/div>\n<p>For Jos\u00e9&#8217;s T-shirts and movies, the marginal utility per dollar is shown in Table 2.<\/p>\n<table class=\"aligncenter\" summary=\"The table has 8 columns and 7 rows of data. Column 1 is \u201cquantity of t-shirts\u201d; Column 2 is \u201ctotal utility\u201d; column 3 is \u201cmarginal utility\u201d; column 4 is \u201cmarginal utility per dollar\u201d; column 5 is \u201cquantity of movies\u201d; column 6 is \u201ctotal utility\u201d; column 7 is \u201cmarginal utility\u201d; column 8 is \u201cmarginal utility per dollar.\u201d Column 1 has the following data: 1, 2, 3, 4, 5, 6, 7. Column 2 has the following data: 22, 43, 63, 81, 97, 111, 123. Column 3 has the following data: 22, 21, 20, 18, 16, 14, 12. Column 4 has the following data: (22\/$14=1.6); (21\/$14=1.5); (20\/$14=1.4); (18\/$14=1.3); (16\/$14=1.1); (14\/$14=1); (12\/$14=1.2). Column 5 has the following data: 1, 2, 3, 4, 5, 6, 7. Column 6 has the following data: 16, 31, 45, 58, 70, 81, 91. Column 7 has the following data: 16, 15, 14, 13, 12, 11, 10. Column 8 has the following data: (16\/$7=2.3); (15\/$7=2.14); (14\/$7=2); (13\/$7=1.9); (12\/$7=1.7); (11\/$7=1.6); (10\/$7=1.4).\">\n<thead>\n<tr>\n<th colspan=\"8\">Table 2. Marginal Utility per Dollar<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Quantity of T-Shirts<\/th>\n<th>Total Utility<\/th>\n<th>Marginal Utility<\/th>\n<th>Marginal Utility per Dollar<\/th>\n<th>Quantity of Movies<\/th>\n<th>Total Utility<\/th>\n<th>Marginal Utility<\/th>\n<th>Marginal Utility per Dollar<\/th>\n<\/tr>\n<tr>\n<td><strong>1<\/strong><\/td>\n<td><strong>22<\/strong><\/td>\n<td><strong>22<\/strong><\/td>\n<td><strong>22\/$14=1.6<\/strong><\/td>\n<td>1<\/td>\n<td>16<\/td>\n<td>16<\/td>\n<td>16\/$7=2.3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>43<\/td>\n<td>21<\/td>\n<td>21\/$14=1.5<\/td>\n<td>2<\/td>\n<td>31<\/td>\n<td>15<\/td>\n<td>15\/$7=2.14<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>63<\/td>\n<td>20<\/td>\n<td>20\/$14=1.4<\/td>\n<td>3<\/td>\n<td>45<\/td>\n<td>14<\/td>\n<td>14\/$7=2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>81<\/td>\n<td>18<\/td>\n<td>18\/$14=1.3<\/td>\n<td>4<\/td>\n<td>58<\/td>\n<td>13<\/td>\n<td>13\/$7=1.9<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>97<\/td>\n<td>16<\/td>\n<td>16\/$14=1.1<\/td>\n<td>5<\/td>\n<td>70<\/td>\n<td>12<\/td>\n<td>12\/$7=1.7<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>111<\/td>\n<td>14<\/td>\n<td>14\/$14=1<\/td>\n<td><strong>6<\/strong><\/td>\n<td><strong>81<\/strong><\/td>\n<td><strong>11<\/strong><\/td>\n<td><strong>11\/$7=1.6<\/strong><\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>123<\/td>\n<td>12<\/td>\n<td>12\/$14=1.2<\/td>\n<td>7<\/td>\n<td>91<\/td>\n<td>10<\/td>\n<td>10\/$7=1.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-idp43427504\" class=\"economics workout\">\n<div class=\"textbox exercises\">\n<h3>A Rule for maximizing Utility<\/h3>\n<div id=\"fs-idp43427504\" class=\"economics workout\">\n<p id=\"fs-idm16605536\">If a consumer wants to maximize total utility, for every dollar that they spend, they should spend it on the item which yields the greatest marginal utility per dollar of expenditure.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><strong>Applying the Rule<\/strong><\/h2>\n<p>Jos\u00e9&#8217;s first purchase will be a movie. Why? Jos\u00e9&#8217;s choices are to purchase either a T-shirt or a movie. Table 1 shows that the marginal utility per dollar spent on the first T-shirt is 1.6 compared with 2.5 for the first movie.\u00a0Because the first movie gives Jos\u00e9 more marginal utility per dollar than the first T-shirt, and because the movie is within his budget, he will purchase a movie first.<\/p>\n<p>Jos\u00e9 will continue to purchase the good which gives him the highest marginal utility per dollar until he exhausts the budget. Jos\u00e9 will keep purchasing movies because they give him a greater &#8220;bang for the buck&#8221; until the sixth movie is equivalent to a T-shirt purchase. Jos\u00e9 can afford to purchase that T-shirt. So Jos\u00e9 will choose to purchase six movies and one T-shirt. That combination, six movies and one T-shirt, is his consumer equilibrium<\/p>\n<h2>Another Rule for Maximizing Utility<\/h2>\n<p>Since the price of T-shirts is twice as high as the price of movies, to maximize utility the last T-shirt chosen needs to provide exactly twice the marginal utility (MU) of the last movie. If the last T-shirt provides less than twice the marginal utility of the last movie, then the T-shirt is providing less &#8220;bang for the buck&#8221; (i.e., marginal utility per dollar spent) than if the same money were spent on movies. If this is so, Jos\u00e9 should trade the T-shirt for more movies to increase his total utility. Marginal utility per dollar measures the additional utility that Jos\u00e9 will enjoy given what he has to pay for the good.<\/p>\n<p>If the last T-shirt provides more than twice the marginal utility of the last movie, then the T-shirt is providing more &#8220;bang for the buck&#8221; or marginal utility per dollar, than if the money were spent on movies. As a result, Jos\u00e9 should buy more T-shirts. Notice that at Jos\u00e9&#8217;s optimal choice the marginal utility from the first T-shirt, of 22 is exactly twice the marginal utility of the sixth movie, which is 11. At this choice, the marginal utility per dollar is the same for both goods. This is a tell-tale signal that Jos\u00e9 has found the point with highest total utility.<\/p>\n<p>This argument can be written as another rule: the utility-maximizing choice between consumption goods occurs where the marginal utility per dollar is the same for both goods, and the consumer has exhausted his or her budget.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{MU_1}{P_1}=\\frac{MU_2}{P_2}[\/latex]<\/p>\n<p>A sensible economizer will pay twice as much for something only if, in the marginal comparison, the item confers twice as much utility. Notice that the formula for the table above is<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{22}{14}=\\frac{11}{7}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]1.6=1.6[\/latex]<\/p>\n<p>The following feature provides step-by-step guidance for this concept of utility-maximizing choices.<\/p>\n<div class=\"textbox exercises\">\n<h3><strong>Another Rule for Maximizing Utility<\/strong><\/h3>\n<p>The rule, [latex]\\displaystyle\\frac{MU_1}{P_1}=\\frac{MU_2}{P_2}[\/latex], means that the last dollar spent on each good provides exactly the same marginal utility. So:<\/p>\n<p><strong>Step 1.<\/strong> If we traded a dollar more of movies for a dollar more of T-shirts, the marginal utility gained from T-shirts would exactly offset the marginal utility lost from fewer movies. In other words, the net gain would be zero.<\/p>\n<p><strong>Step 2.<\/strong> Products, however, usually cost more than a dollar, so we cannot trade a dollar&#8217;s worth of movies. The best we can do is trade two movies for another T-shirt, since in this example T-shirts cost twice what a movie does.<\/p>\n<p><strong>Step 3.<\/strong> If we trade two movies for one T-shirt, we would end up at point R (two T-shirts and four movies).<\/p>\n<p><strong>Step 4.<\/strong> Choice 4 in Table 3 shows that if we move to point S, we would lose 21 utils from one less T-shirt, but gain 23 utils from two more movies, so we would end up with more total utility at point S.<\/p>\n<table summary=\"The table has for columns. Column 1 is labeled \u201cTry,\u201d Column 2 is labeled \u201cWhich Has,\u201d Column 3 is labeled \u201cTotal Utility,\u201d Column 4 is labeled \u201cMarginal Gain and Loss of Utility, Compared with Previous Choice,\u201d Column 5 is labeled \u201cConclusion.\u201d Row 1 shows the following: Column 1 = Choice 1: P. column 2 = 4 t-shirts and 0 movies; column 3 = 81 from 4 \u2013t-shirts + 0 from 0 movies = 81; columns 4 and 5 are blank. Row 2 shows column 1 = Choice 2: Q; column 2 = 3 t-shirts and 2 movies; column 3 = 63 from 3 t-shirts + 31 from 0 movies = 94; column 4 = loss of 18 from 1 less t-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13; column 5 = Q is preferred over P. Row 3 shows the following: column 1 = Choice 3: R, column 2 = 2 t-shirts and 4 movies; column 3 = 43 from 2 t-shirts + 58 from 4 movies = 101; column 4 = loss from 1 less t-shirt, but gain of 27 from two more movies for a net utility gain of 7; column 5 = R is preferred over Q. Row 4 shows the following: Choice 4 : S; column 2 = 1 t-shirt and 6 movies; column 3 = 22 from 1 t-shirt + 81 from 6 movies = 1-3; column 4 = loss of 21 from 1 less t-shirt, but gain of 23 from two more movies, for a net utility gain of 2; column 5 = s is preferred over R. Row 5 shows the following: column 1 = Choice 5: T; column 2 = 0 t-shirts and 8 movies; column 3 = 0 from 0 t-shirts + 100 from 8 movies = 100; column 4 = loss of 22 from 1 less t-shirt, but gain of 19 from two more movies, for a net utility loss of 3; column 5 = s is preferred over T.\">\n<thead>\n<tr>\n<th colspan=\"5\">Table 3. A Step-by-Step Approach to Maximizing Utility<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Try<\/th>\n<th>Which Has<\/th>\n<th>Total Utility<\/th>\n<th>Marginal Gain and Loss of Utility, Compared with Previous Choice<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<tr>\n<td>Choice 1: P<\/td>\n<td>4 T-shirts and 0 movies<\/td>\n<td>81 from 4 T-shirts + 0 from 0 movies = 81<\/td>\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\n<td>\u00a0\u00a0\u00a0\u00a0\u00a0\u2013<\/td>\n<\/tr>\n<tr>\n<td>Choice 2: Q<\/td>\n<td>3 T-shirts and 2 movies<\/td>\n<td>63 from 3 T-shirts + 31 from 0 movies = 94<\/td>\n<td>Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13<\/td>\n<td>Q is preferred over P<\/td>\n<\/tr>\n<tr>\n<td>Choice 3: R<\/td>\n<td>2 T-shirts and 4 movies<\/td>\n<td>43 from 2 T-shirts + 58 from 4 movies = 101<\/td>\n<td>Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7<\/td>\n<td>R is preferred over Q<\/td>\n<\/tr>\n<tr>\n<td>Choice 4: S<\/td>\n<td>1 T-shirt and 6 movies<\/td>\n<td>22 from 1 T-shirt + 81 from 6 movies = 103<\/td>\n<td>Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2<\/td>\n<td>S is preferred over R<\/td>\n<\/tr>\n<tr>\n<td>Choice 5: T<\/td>\n<td>0 T-shirts and 8 movies<\/td>\n<td>0 from 0 T-shirts + 100 from 8 movies = 100<\/td>\n<td>Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3<\/td>\n<td>S is preferred over T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In short, the rule shows us the utility-maximizing choice.<\/p>\n<\/div>\n<p>There is another, equivalent way to think about this. The rule can also be expressed as\u00a0<span class=\"emphasis\"><em>the ratio of the prices of the two goods should be equal to the ratio of the marginal utilities.<\/em><\/span> When the price of good 1 is divided by the price of good 2, at the utility-maximizing point this will equal the marginal utility of good 1 divided by the marginal utility of good 2. This rule can be written in algebraic form:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{P_1}{P_2}=\\frac{MU_1}{MU_2}[\/latex]<\/p>\n<p>Along the budget constraint, the total price of the two goods remains the same, so the ratio of the prices does not change. However, the marginal utility of the two goods changes with the quantities consumed. At the optimal choice of one T-shirt and six movies, point S, the ratio of marginal utility to price for T-shirts (22:14) matches the ratio of marginal utility to price for movies (of 11:7).<\/p>\n<div class=\"textbox examples\">\n<h3>Watch It<\/h3>\n<p>This video applies these same concepts to a graph. It demonstrates how there will come a point when the marginal utility per dollar of one good goes down enough so that it makes more sense to buy another good.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/KbW6OiuRa1Y?showinfo=0\" width=\"800\" height=\"470\" frameborder=\"0\"><\/iframe><\/p>\n<\/div>\n<h2><span class=\"cnx-gentext-section cnx-gentext-t\">Measuring Utility with Numbers<\/span><\/h2>\n<p>This discussion of utility started off with an assumption that it is possible to place numerical values on utility, an assumption that may seem questionable. You can buy a thermometer for measuring temperature at the hardware store, but what store sells an &#8220;utilimometer&#8221; for measuring utility? However, while measuring utility with numbers is a convenient assumption to clarify the explanation, the key assumption is not that utility can be measured by an outside party, but only that individuals can decide which of two alternatives they prefer.<\/p>\n<p>To understand this point, think back to the step-by-step process of finding the choice with highest total utility by comparing the marginal utility that is gained and lost from different choices along the budget constraint. As Jos\u00e9 compares each choice along his budget constraint to the previous choice, what matters is not the specific numbers that he places on his utility\u2014or whether he uses any numbers at all\u2014but only that he personally can identify which choices he prefers.<\/p>\n<p>In this way, the step-by-step process of choosing the highest level of utility resembles rather closely how many people make consumption decisions. We think about what will make us the happiest; we think about what things cost; we think about buying a little more of one item and giving up a little of something else; we choose what provides us with the greatest level of satisfaction. The vocabulary of comparing the points along a budget constraint and total and marginal utility is just a set of tools for discussing this everyday process in a clear and specific manner. It is welcome news that specific utility numbers are not central to the argument, since a good utilimometer is hard to find. Do not worry\u2014while we cannot measure utils, by the end of the next module, we will have transformed our analysis into something we can measure\u2014demand.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>\t<iframe id=\"lumen_assessment_7956\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=7956&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_7956\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_7957\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=7957&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_7957\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_7958\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=7958&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_7958\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_7959\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=7959&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_7959\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>These questions allow you to get as much practice as you need, as you can click the link at the top of the first question (\u201cTry another version of these questions\u201d) to get a new set of questions. Practice until you feel comfortable doing the questions.<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm155309\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=155309&theme=oea&iframe_resize_id=ohm155309&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>These questions allow you to get as much practice as you need, as you can click the link at the top of the first question (\u201cTry another version of these questions\u201d) to get a new set of questions. Practice until you feel comfortable doing the questions.<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm155322\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=155322-155321&theme=oea&iframe_resize_id=ohm155322&sameseed=1&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Click through the buttons in this simulation to think about applications of utility.<\/p>\n<p>First, you&#8217;ll imagine that you are desperately searching for food and water on a hot day. Given the heat and your time constraints, you&#8217;ll need to decide what to obtain first based on its utility. Then, you can see these combinations on a graph and consider what the utility of each choice would be.<\/p>\n<p>Be sure to choose the option to &#8220;learn more about marginal utility per dollar spent&#8221; so that you can see how marginal-decision making comes into play when choosing to forage for either apples or berries.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.branchtrack.com\/projects\/i0fyro7x\/embed\" width=\"850\" height=\"500\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>\t<iframe id=\"lumen_assessment_7774\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=7774&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_7774\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Glossary<\/h3>\n<div class=\"titlepage\">\n<dl>\n<dt>consumer equilibrium:<\/dt>\n<dd>the combination of goods and services that will maximize an individual&#8217;s total utility<\/dd>\n<dt>marginal utility per dollar:<\/dt>\n<dd>the additional satisfaction gained from purchasing a good given the price of the product; MU\/Price<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-6379\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Maximizing Utlilty. <strong>Authored by<\/strong>: Clark Aldrich for Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Consumption Choices. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: Rice University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/6i8iXmBj@10.31:98vKjzCh@10\/Consumption-Choices\">http:\/\/cnx.org\/contents\/6i8iXmBj@10.31:98vKjzCh@10\/Consumption-Choices<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/bc498e1f-efe9-43a0-8dea-d3569ad09a82@4.4<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Equalizing Marginal Utility per Dollar Spent. <strong>Provided by<\/strong>: Khan Academy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=KbW6OiuRa1Y\">https:\/\/www.youtube.com\/watch?v=KbW6OiuRa1Y<\/a>. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":29,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Consumption Choices\",\"author\":\"OpenStax College\",\"organization\":\"Rice University\",\"url\":\"http:\/\/cnx.org\/contents\/6i8iXmBj@10.31:98vKjzCh@10\/Consumption-Choices\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/bc498e1f-efe9-43a0-8dea-d3569ad09a82@4.4\"},{\"type\":\"copyrighted_video\",\"description\":\"Equalizing Marginal Utility per Dollar Spent\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/www.youtube.com\/watch?v=KbW6OiuRa1Y\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"original\",\"description\":\"Maximizing Utlilty\",\"author\":\"Clark Aldrich for Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"e907eb09-d6f8-4138-834c-2a57a6cf7038, af8b3fa4-af22-4e68-a89c-803558700542, 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