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6.2: Choice with measurable utility

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    Neal loves to pump his way through the high-altitude powder at the Whistler ski and snowboard resort. His student-rate lift-ticket cost is $30 per visit. He also loves to frequent the jazz bars in downtown Vancouver, and each such visit costs him $20. With expensive passions, Neal must allocate his monthly entertainment budget carefully. He has evaluated how much satisfaction, measured in utils, he obtains from each snowboard outing and each jazz club visit. We assume that these utils are measurable, and use the term cardinal utility to denote this. These measurable utility values are listed in columns 2 and 3 of Table 6.1. They define the total utility he gets from various amounts of the two activities.

    Table 6.1 Utils from snowboarding and jazz
    1 2 3 4 5 6 7
    Visit Total Total Marginal Marginal Marginal Marginal
    # snowboard jazz snowboard jazz utils snowboard jazz utils
    utils utils utils utils per $ per $
    1 72 52 72 52 2.4 2.6
    2 132 94 60 42 2.0 2.1
    3 182 128 50 34 1.67 1.7
    4 224 156 42 28 1.4 1.4
    5 260 180 36 24 1.2 1.2
    6 292 201 32 21 1.07 1.05
    7 321 220 29 19 0.97 0.95
    Price of snowboard visit=$30. Price of jazz club visit=$20.

    Cardinal utility is a measurable concept of satisfaction.

    Total utility is a measure of the total satisfaction derived from consuming a given amount of goods and services.

    Neal's total utility from each activity in this example is independent of the amount of the other activity he engages in. These total utilities are plotted in Figures 6.1 and 6.2. Clearly, more of each activity yields more utility, so the additional or marginal utility (MU) of each activity is positive. This positive marginal utility for any amount of the good consumed, no matter how much, reflects the assumption of non-satiation—more is always better. Note, however, that the decreasing slopes of the total utility curves show that total utility is increasing at a diminishing rate. While more is certainly better, each additional visit to Whistler or a jazz club augments Neal's utility by a smaller amount. At the margin, his additional utility declines: He has diminishing marginal utility. The marginal utilities associated with snowboarding and jazz are entered in columns 4 and 5 of Table 6.1. They are the differences in total utility values when consumption increases by one unit. For example, when Neal makes a sixth visit to Whistler his total utility increases from 260 utils to 292 utils. His marginal utility for the sixth unit is therefore 32 utils, as defined in column 4. In light of this example, it should be clear that we can define marginal utility as:

    img179.png (6.1)

    where img180.png denotes the change in the quantity consumed of the good or service in question.

    Marginal utility is the addition to total utility created when one more unit of a good or service is consumed.

    Diminishing marginal utility implies that the addition to total utility from each extra unit of a good or service consumed is declining.

    Figure 6.1 TU from snowboarding
    Figure 6.2 TU from jazz
    Figure 6.3 MU from snowboarding
    Figure 6.4 MU from jazz

    The marginal utilities associated with consuming different amounts of the two goods are plotted in Figures 6.3 and 6.4, using the data from columns 4 and 5 in Table 6.1. These functions are declining, as indicated by their negative slope. It should also be clear that the MU curves can be derived from the TU curves. For example, in figure 6.2, when going from 2 units to 3 units of Jazz, TU increases by 34 units. But 34/1 is the slope of the TU function in this range of consumption – the vertical distance divided by the horizontal distance. Similarly, if jazz consumption increases from 4 units to 5 units the corresponding change in TU is 24 units, again the vertical distance divided by the horizontal distance, and so the slope of the function. In short, the MU is the slope of the TU function.

    Now that Neal has defined his utility schedules, he must consider the price of each activity. Ultimately, when deciding how to allocate his monthly entertainment budget, he must evaluate how much utility he gets from each dollar spent on snowboarding and jazz: What "bang for his buck" does he get? Let us see how he might go about allocating his budget. When he has fully spent his budget in the manner that will yield him greatest utility, we say that he has attained equilibrium, because he will have no incentive to change his expenditure patterns.

    If he boards once, at a cost of $30, he gets 72 utils of satisfaction, which is 2.4 utils per dollar spent (=72/30). One visit to a jazz club would yield him 2.6 utils per dollar (=52/20). Initially, therefore, his dollars give him more utility per dollar when spent on jazz. His MU per dollar spent on each activity is given in the final two columns of the table. These values are obtained by dividing the MU associated with each additional unit by the good's price.

    We will assume that Neal has a budget of $200. He realizes that his initial expenditure should be on a jazz club visit, because he gets more utility per dollar spent there. Having made one such expenditure, he sees that a second jazz outing would yield him 2.1 utils per dollar expended, while a first visit to Whistler would yield him 2.4 utils per dollar. Accordingly, his second activity is a snowboard outing.

    Having made one jazz and one snowboarding visit, he then decides upon a second jazz club visit for the same reason as before—utility value for his money. He continues to allocate his budget in this way until his budget is exhausted. In our example, this occurs when he spends $120 on four snowboarding outings and $80 on four jazz club visits. At this consumer equilibrium, he gets the same utility value per dollar for the last unit of each activity consumed. This is a necessary condition for him to be maximizing his utility, that is, to be in equilibrium.

    Consumer equilibrium occurs when marginal utility per dollar spent on the last unit of each good is equal.

    To be absolutely convinced of this, imagine that Neal had chosen instead to board twice and to visit the jazz clubs seven times; this combination would also exhaust his $200 budget exactly. With such an allocation, he would get 2.0 utils per dollar spent on his marginal (second) snowboard outing, but just 0.95 utils per dollar spent on his marginal (seventh) jazz club visit.1 If, instead, he were to reallocate his budget in favour of snowboarding, he would get 1.67 utils per dollar spent on a third visit to the hills. By reducing the number of jazz visits by one, he would lose 0.95 utils per dollar reallocated. Consequently, the utility gain from a reallocation of his budget towards snowboarding would outweigh the utility loss from allocating fewer dollars to jazz. His initial allocation, therefore, was not an optimum, or equilibrium.

    Only when the utility per dollar expended on each activity is equal at the margin will Neal be optimizing. When that condition holds, a reallocation would be of no benefit to him, because the gains from one more dollar on boarding would be exactly offset by the loss from one dollar less spent on jazz. Therefore, we can write the equilibrium condition as

    img185.png (6.2)

    While this example has just two goods, in the more general case of many goods, this same condition must hold for all pairs of goods on which the consumer allocates his or her budget.

    From utility to demand

    Utility theory is a useful way of analyzing how a consumer makes choices. But in the real world we do not observe a consumer's utility, either total or marginal. Instead, his or her behaviour in the marketplace is observed through the demand curve. How are utility and demand related?

    Demand functions relate the quantity of a good consumed to the price of that good, other things being equal. So let us trace out the effects of a price change on demand, with the help of this utility framework. We will introduce a simplification here: Goods are divisible, or that they come in small packages relative to income. Think, for example, of kilometres driven per year, or liters of gasoline purchased. Conceptualizing things in this way enables us to imagine more easily experiments in which small amounts of a budget are allocated one dollar at a time. In contrast, in the snowboard/jazz example, we had to reallocate the budget in lumps of $30 or $20 at a time because we could not "fractionalize" these goods.

    The effects of a price change on a consumer's demand can be seen through the condition that describes his or her equilibrium. If income is allocated to, say, three goods img186.png, such that MUa/Pa=MUb/Pb=MUc/Pc, and the price of, say, good b falls, the consumer must reallocate the budget so that once again the MUs per dollar spent are all equated. How does he do this? Clearly, if he purchases more or less of any one good, the MU changes. If the price of good b falls, then the consumer initially gets more utility from good b for the last dollar he spends on it (the denominator in the expression MUb/Pb falls, and consequently the value of the ratio rises to a value greater than the values for goods a and c).

    The consumer responds to this, in the first instance, by buying more of the cheaper good. He obtains more total utility as a consequence, and in the process will get less utility at the margin from that good. In essence, the numerator in the expression then falls, in order to realign it with the lower price. This equality also provides an underpinning for what is called the law of demand: More of a good is demanded at a lower price. If the price of any good falls, then, in order for the equilibrium condition to be re-established, the MU of that good must be driven down also. Since MU declines when more is purchased, this establishes that demand curves must slope downwards.

    The law of demand states that, other things being equal, more of a good is demanded the lower is its price.

    However, the effects of a price decline are normally more widespread than this, because the quantities of other goods consumed may also change. As explained in earlier chapters, the decline in the price of good b will lead the consumer to purchase more units of complementary goods and fewer units of goods that are substitutes. So the whole budget allocation process must be redetermined in response to any price change. But at the end of the day, a new equilibrium must be one where the marginal utility per dollar spent on each good is equal.

    Applying the theory

    The demand curves developed in Chapter 3 can be related to the foregoing utility analysis. In our example, Neal purchased four lift tickets at Whistler when the price was $30. We can think of this combination as one point on his demand curve, where the "other things kept constant" are the price of jazz, his income, his tastes, etc.

    Suppose now that the price of a lift ticket increased to $40. How could we find another point on his demand curve corresponding to this price, using the information in Table 6.1? The marginal utility per dollar associated with each visit to Whistler could be recomputed by dividing the values in column 4 by 40 rather than 30, yielding a new column 6. We would then determine a new allocation of his budget between the two goods that would maximize utility. After such a calculation we would find that he makes three visits to Whistler and four jazz-club visits. Thus, the combination img187.png is another point on his demand curve. Note that this allocation exactly exhausts his $200 budget.

    By setting the price equal to $20, this exercise could be performed again, and the outcome will be a quantity demanded of lift tickets equal to seven (plus three jazz club visits). Thus, the combination img188.png is another point on his demand curve. Figure 6.5 plots a demand curve going through these three points.

    By repeating this exercise for many different prices, the demand curve is established. We have now linked the demand curve to utility theory.

    Figure 6.5 Utility to demand
    When img190.png, the consumer finds the quantity such that MU/P is equal for all purchases. The corresponding quantity purchased is 4 tickets. At prices of $40 and $20 the equilibrium condition implies quantities of 3 and 7 respectively.
    Application Box 6.2 Individual and Collective Utility

    The example developed in the text is not far removed from what economists do in practice. From a philosophical standpoint, economists are supposed to be interested in the well-being of the citizens who make up an economy or a country. To determine how 'well-off' citizens may be, social scientists frequently carry out surveys on how 'content' or 'happy' people are in their every-day lives. For example, the Earth Institute at Columbia University regularly produces a 'World Happiness Report'. The report is based upon responses to survey questions in numerous economies. One of the measures it uses to compare utility levels is the Cantril ladder. This is an 11-point scale running from 0 to 10, with the lowest value signifying the worst possible life, and 10 the highest possible quality of life. In reporting their findings, the researchers are essentially claiming that some economies have, on average, more contented or happier, people than others. Utility can be considered in exactly this way: A higher reported value on the Cantril ladder suggests higher utility.

    A slightly different measure of well-being across economies is given by the United Nations Human Development Index. In this case, countries score high by having a high level of income, good health (as measured by life expectancy), and high levels of education, as measured by the number of years of education completed or envisaged.

    In practice, social scientists are very comfortable using utility-based concepts to describe the economic circumstances of individuals in different economies.

    This page titled 6.2: Choice with measurable utility is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Curtis and Ian Irvine (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.