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6: Individual choice

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    Chapter 6: Individual choice

    In this chapter we will explore:

    6.1

    Rationality

    6.2

    Consumer choice with measurable utility

    6.3

    Consumer choice with ordinal utility

    6.4

    Applications of indifference analysis

    6.1 Rationality

    A critical behavioural assumption in economics is that agents operate in a way that is oriented towards achieving a goal. This can be interpreted to mean that individuals and firms maximize their personal well-being and/or their profits. These players may have other goals in addition: Philanthropy and the well-being of others are consistent with individual optimization.

    If individuals are to achieve their goals then they must act in a manner that will get them to their objective; broadly, they must act in a rational manner. The theory of individual maximization that we will develop in this chapter is based on that premise or assumption. In assuming individuals are rational we need not assume that they have every piece of information available to them that might be relevant for a specific decision or choice. Nor need we assume that they have super computers in their brain when they evaluate alternative possible strategies.

    What we do need to assume, however, is that individuals act in a manner that is consistent with obtaining a given objective. The modern theory of behavioural economics and behavioural psychology examines decision making in a wide range of circumstances and has uncovered many fascinating behaviours – some of which are developed in Application Box 6.1 below.

    We indicated in Chapter 1 that as social scientists, we require a reliable model of behaviour, that is, a way of describing the essentials of choice that is consistent with everyday observations on individual behaviour patterns. In this chapter, our aim is to understand more fully the behavioural forces that drive the demand side of the economy.

    Economists analyze individual decision making using two different, yet complementary, approaches – utility analysis and indifference analysis. We begin by portraying individuals as maximizing their measurable utility (sometimes called cardinal utility); then progress to indifference analysis, where a weaker assumption is made on the ability of individuals to measure their satisfaction. In this second instance we do not assume that individuals can measure their utility numerically, only that they can say if one collection of goods and services yields them greater satisfaction than another group. This ranking of choices corresponds to what is sometimes called ordinal utility – because individuals can order groups of goods and services in ascending order of satisfaction. In each case individuals are perceived as rational maximizers or optimizers: They allocate their income so as to choose the outcome that will make them as well off as possible.

    The second approach to consumer behaviour is frequently omitted in introductory texts. It can be omitted here without interpreting the flow of ideas, although it does yield additional insights into consumer choice and government policy. As in preceding chapters, we begin the analysis with a motivating numerical example.

    Application Box 6.1 Rationality and impulse

    A number of informative and popular books on decision making have appeared recently. Their central theme is that our decision processes should not be viewed solely as a rational computer – operating in one single mode only, and unmoved by our emotions or history. Psychologists now know that our brains have at least two decision modes, and these are developed by economics Nobel Prize winner Daniel Kahneman in his book "Thinking, Fast and Slow". One part of our brain operates in a rational goal-oriented forward-looking manner (the 'slow' part), another is motivated by immediate gratification (the 'fast' part). Decisions that we observe in the world about us reflect these different mechanisms.

    Richard Thaler, a Chicago economist and his law professor colleague Cass Sunstein, have developed a role for public policy in their book entitled "Nudge". They too argue that individuals do not inevitably operate in their own best long-term interests, and as a consequence individuals frequently require a nudge by government to make the long-term choice rather than the short-term choice. For example, when individuals begin a new job, they might be automatically enrolled in the company pension plan and be given the freedom to opt out, rather than not be enrolled and given the choice to opt in. Such policies are deemed to be 'soft paternalism'. They are paternalistic for the obvious reason – another organism is directing, but they are also soft in that they are not binding.

    6.2 Choice with measurable utility

    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
    img181.png

    Figure 6.2 TU from jazz
    img182.png

    Figure 6.3 MU from snowboarding
    img183.png

    Figure 6.4 MU from jazz
    img184.png

    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
    img189.png
    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.

    6.3 Choice with ordinal utility

    The budget constraint

    In the preceding section, we assumed that utility is measurable in order to better understand how consumers allocate their budgets, and how this process is reflected in the market demands that are observed. The belief that utility might be measurable is not too extreme in the modern era. Neuroscientists are mapping more and more of the human brain and understanding how it responds to positive and negative stimuli. At the same time, numerous sociological surveys throughout the world ask individuals to rank their happiness on a scale of one to ten, or something similar, with a view to making comparisons between individual-level and group-level happiness – see Application Box 6.2. Nonetheless, not every scientist may be convinced that we should formulate behavioural rules on this basis. Accordingly we now examine the economics of consumer behaviour without this strong assumption. We assume instead that individuals are able to identify (a) different combinations of goods and services that yield equal satisfaction, and (b) combinations of goods and services that yield more satisfaction than other combinations. In contrast to measurable (or cardinal) utility, this concept is called ordinal utility, because it assumes only that consumers can order utility bundles rather than quantify the utility.

    Ordinal utility assumes that individuals can rank commodity bundles in accordance with the level of satisfaction associated with each bundle.

    The budget constraint

    Neal's monthly expenditure limit, or budget constraint, is $200. In addition, he faces a price of $30 for lift tickets and $20 per visit to jazz clubs. Therefore, using S to denote the number of snowboard outings and J the number of jazz club visits, if he spends his entire budget it must be true that the sum of expenditures on each activity exhausts his budget or income (I):

    img191.pngimg192.png
    img193.pngimg192.png
    img194.pngimg195.png

    Since many different combinations of the two goods are affordable, it follows that the budget constraint defines all bundles of goods that the consumer can afford with a given budget.

    The budget constraint defines all bundles of goods that the consumer can afford with a given budget.

    The budget constraint, then, is just what it claims to be—a limit on behaviour. Neal's budget constraint is illustrated in Figure 6.6, where the amount of each good consumed is given on the axes. If he spends all of his $200 income on jazz, he can make exactly ten jazz club visits img196.png. The calculation also applies to visits to Whistler. The intercept value is always obtained by dividing income by the price of the good or activity in question.

    Figure 6.6 The budget line
    img197.png
    FC is the budget constraint and defines the affordable combinations of snowboarding and jazz. F represents all income spent on snowboarding. Thus F=I/Ps. Similarly C=I/Pj. Points above FC are not attainable. The slope = OF/OC =(I/Ps)/(I/Pj)=Pj/Ps=20/30=2/3. The affordable set is 0FC.

    In addition to these affordable extremes, Neal can also afford many other bundles, e.g., (S=2,J=7), or (S=4,J=4), or (S=6,J=1). The set of feasible, or affordable, combinations is bounded by the budget line, and this is illustrated in Figure 6.6.

    The affordable set of goods and services for the consumer is bounded by the budget line from above; the non-affordable set lies strictly above the budget line.

    The slope of the budget line is informative. As illustrated in Chapter 1, it indicates how many snowboard visits must be sacrificed for one additional jazz visit; it defines the consumer's trade-offs. To illustrate: Suppose Neal is initially at point A (J=4,S=4), and moves to point K (J=7,S=2). Clearly, both points are affordable. In making the move, he trades two snowboard outings in order to get three additional jazz club visits, a trade-off of 2/3. This trade-off is the slope of the budget line, which, in Figure 6.6, is AB/BK=–2/3, where the negative sign reflects the downward slope.

    Could it be that this ratio reflects the two prices ($20/$30)? The answer is yes: The slope of the budget line is given by the vertical distance divided by the horizontal distance, OF/OC. The points F and C were obtained by dividing income by the respective price—remember that the jazz intercept is img198.png. Formally, that is I/PJ. The intercept on the snowboard axis is likewise I/PS. Accordingly, the slope of the budget constraint is:

    img199.png

    Since the budget line has a negative slope, it is technically correct to define it with a negative sign. But, as with elasticities, the sign is frequently omitted.

    Tastes and indifference

    We now consider how to represent a consumer's tastes in two dimensions, given that he can order, or rank, different consumption bundles, and that he can define a series of different bundles that all yield the same satisfaction. We limit ourselves initially to considering just "goods," and not "bads" such as pollution.

    Figure 6.7 Ranking consumption bundles
    img200.png
    L is preferred to R since more of each good is consumed at L, while points such as V are less preferred than R. Points W and T contain more of one good and less of the other than R. Consequently, we cannot say if they are preferred to R without knowing how the consumer trades the goods off – that is, his preferences.

    Figure 6.7 examines the implications of these assumptions about tastes. Each point shows a consumption bundle of snowboarding and jazz. Let us begin at bundle R. Since more of a good is preferred to less, any point such as L, which lies to the northeast of R, is preferred to R, since L offers more of both goods than R. Conversely, points to the southwest of R offer less of each good than R, and therefore R is preferred to a point such as V.

    Without knowing the consumer's tastes, we cannot be sure at this stage how points in the northwest and southeast regions compare with R. At W or T, the consumer has more of one good and less of the other than at R. Someone who really likes snowboarding might prefer W to R, but a jazz buff might prefer T to R.

    Let us now ask Neal to disclose his tastes, by asking him to define several combinations of snowboarding and jazz that yield him exactly the same degree of satisfaction as the combination at R. Suppose further, for reasons we shall understand shortly, that his answers define a series of points that lie on the beautifully smooth contour UR in Figure 6.8. Since he is indifferent between all points on UR by construction, this contour is an indifference curve.

    Figure 6.8 Indifference curves
    img201.png
    An indifference curve defines a series of consumption bundles, all of which yield the same satisfaction. The slope of an indifference curve is the marginal rate of substitution (MRS) and defines the number of units of the good on the vertical axis that the individual will trade for one unit of the good on the horizontal axis. The MRS declines as we move south-easterly, because the consumer values the good more highly when he has less of it.

    An indifference curve defines combinations of goods and services that yield the same level of satisfaction to the consumer.

    Pursuing this experiment, we could take other points in Figure 6.8, such as L and V, and ask the consumer to define bundles that would yield the same level of satisfaction, or indifference. These combinations would yield additional contours, such as UL and UV in Figure 6.8. This process yields a series of indifference curves that together form an indifference map.

    An indifference map is a set of indifference curves, where curves further from the origin denote a higher level of satisfaction.

    Let us now explore the properties of this map, and thereby understand why the contours have their smooth convex shape. They have four properties. The first three follow from our preceding discussion, and the fourth requires investigation.

    1. Indifference curves further from the origin reflect higher levels of satisfaction.

    2. Indifference curves are negatively sloped. This reflects the fact that if a consumer gets more of one good she should have less of the other in order to remain indifferent between the two combinations.

    3. Indifference curves cannot intersect. If two curves were to intersect at a given point, then we would have two different levels of satisfaction being associated with the same commodity bundle—an impossibility.

    4. Indifference curves are convex when viewed from the origin, reflecting a diminishing marginal rate of substitution.

    The convex shape reflects an important characteristic of preferences: When consumers have a lot of some good, they value a marginal unit of it less than when they have a small amount of that good. More formally, they have a higher marginal valuation at low consumption levels—that first cup of coffee in the morning provides greater satisfaction than the second or third cup.

    Consider the various points on UR, starting at M in Figure 6.8. At M Neal snowboards a lot; at N he boards much less. The convex shape of his indifference map shows that he values a marginal snowboard trip more at N than at M. To see this, consider what happens as he moves along his indifference curve, starting at M. We have chosen the coordinates on UR so that, in moving from M to R, and again from N to H, the additional amount of jazz is the same: CR=FH. From M, if Neal moves to R, he consumes an additional amount of jazz, CR. By definition of the indifference curve, he is willing to give up MC snowboard outings. The ratio MC/CR defines his willingness to substitute one good for the other. This ratio, being a vertical distance divided by a horizontal distance, is the slope of the indifference curve and is called the marginal rate of substitution, MRS.

    The marginal rate of substitution is the slope of the indifference curve. It defines the amount of one good the consumer is willing to sacrifice in order to obtain a given increment of the other, while maintaining utility unchanged.

    At N, the consumer is willing to sacrifice the amount NF of boarding to get the same additional amount of jazz. Note that, when he boards less, as at N, he is willing to give up less boarding than when he has a lot of it, as at M, in order to get the same additional amount of jazz. His willingness to substitute diminishes as he moves from M to N: The quantity NF is less than the quantity MC. In order to reflect this taste characteristic, the indifference curve has a diminishing marginal rate of substitution: A flatter slope as we move down along its surface.

    A diminishing marginal rate of substitution reflects a higher marginal value being associated with smaller quantities of any good consumed.

    Optimization

    We are now in a position to examine how the consumer optimizes—how he gets to the highest level of satisfaction possible. The constraint on his behaviour is the affordable set defined in Figure 6.6, the budget line.

    Figure 6.9 displays several of Neal's indifference curves in conjunction with his budget constraint. We propose that he maximizes his utility, or satisfaction, at the point E, on the indifference curve denoted by U3. While points such as F and G are also on the boundary of the affordable set, they do not yield as much satisfaction as E, because E lies on a higher indifference curve. The highest possible level of satisfaction is attained, therefore, when the budget line touches an indifference curve at just a single point—that is, where the constraint is tangent to the indifference curve. E is such a point.

    Figure 6.9 The consumer optimum
    img202.png
    The budget constraint constrains the individual to points on or below HK. The highest level of satisfaction attainable is U3, where the budget constraint just touches, or is just tangent to, it. At this optimum the slope of the budget constraint (–Pj/Ps) equals the MRS.

    This tangency between the budget constraint and an indifference curve requires that the slopes of each be the same at the point of tangency. We have already established that the slope of the budget constraint is the negative of the price ratio (img203.png). The slope of the indifference curve is the marginal rate of substitution MRS. It follows, therefore, that the consumer optimizes where the marginal rate of substitution equals the slope of the price line.

    Optimization requires:

    img204.png(6.3)

    A consumer optimum occurs where the chosen consumption bundle is a point such that the price ratio equals the marginal rate of substitution.

    Notice the resemblance between this condition and the one derived in the first section as Equation 6.2. There we argued that equilibrium requires the ratio of the marginal utilities be same as the ratio of prices. Here we show that the MRS must equal the ratio of prices. In fact, with a little mathematics it can be shown that the MRS is indeed the same as the (negative of the) ratio of the marginal utilities: img205.png. Therefore the two conditions are in essence the same! However, it was not necessary to assume that an individual can actually measure his utility in obtaining the result that the MRS should equal the price ratio in equilibrium. The concept of ordinal utility is sufficient.

    Adjusting to income changes

    Suppose now that Neal's income changes from $200 to $300. How will this affect his consumption decisions? In Figure 6.10, this change is reflected in a parallel outward shift of the budget constraint. Since no price change occurs, the slope remains constant. By recomputing the ratio of income to price for each activity, we find that the new snowboard and jazz intercepts are 10 img206.png and 15 img207.png, respectively. Clearly, the consumer can attain a higher level of satisfaction—at a new tangency to a higher indifference curve—as a result of the size of the affordable set being expanded. In Figure 6.10, the new equilibrium is at E1.

    Figure 6.10 Income and price adjustments
    img208.png
    An income increase shifts the budget constraint from I0 to I1. This enables the consumer to attain a higher indifference curve. A price rise in jazz tickets rotates the budget line I0 inwards around the snowboard intercept to I2. The price rise reflects a lower real value of income and results in a lower equilibrium level of satisfaction.

    Adjusting to price changes

    Next, consider the impact of a price change from the initial equilibrium E0 in Figure 6.10. Suppose that jazz now costs more. This reduces the purchasing power of the given budget of $200. The new jazz intercept is therefore reduced. The budget constraint becomes steeper and rotates around the snowboard intercept H, which is unchanged because its price is constant. The new equilibrium is at E2, which reflects a lower level of satisfaction because the affordable set has been reduced by the price increase. As explained in Section 6.2, E0 and E2 define points on the demand curve for jazz (J0 and J2): They reflect the consumer response to a change in the price of jazz with all other things held constant. In contrast, the price increase for jazz shifts the demand curve for snowboarding: As far as the demand curve for snowboarding is concerned, a change in the price of jazz is one of those things other than own-price that determine its position.

    Philanthropy

    Individuals in the foregoing analysis aim to maximize their utility, given that they have a fixed budget. Note that this behavioural assumption does not rule out the possibility that these same individuals may be philanthropic – that is, they get utility from the act of giving to their favourite charity or the United Way or Centre-aide. To see this suppose that donations give utility to the individual in question – she gets a 'warm glow' feeling as a result of giving, which is to say she gets utility from the activity. There is no reason why we cannot put charitable donations on one axis and some other good or combination of goods on the remaining axis. At equilibrium, the marginal utility per dollar of contributions to charity should equal the marginal utility per dollar of expenditure on other goods; or, stated in terms of ordinal utility, the marginal rate of substitution between philanthropy and any other good should equal the ratio of their prices. Evidently the price of a dollar of charitable donations is one dollar.

    6.4 Applications of indifference analysis

    Price impacts: Complements and substitutes

    The nature of complements and substitutes, defined in Chapter 4, can be further understood with the help of Figure 6.10. The new equilibrium E2 has been drawn so that the increase in the price of jazz results in more snowboarding—the quantity of S increases to S2 from S0. These goods are substitutes in this picture, because snowboarding increases in response to an increase in the price of jazz. If the new equilibrium E2 were at a point yielding a lower level of S than S0, we would conclude that they were complements.

    Cross-price elasticities

    Continuing with the same price increase in jazz, we could compute the percentage change in the quantity of snowboarding demanded as a result of the percentage change in the jazz price. In this example, the result would be a positive elasticity value, because the quantity change in snowboarding and the price change in jazz are both in the same direction, each being positive.

    Income impacts: Normal and inferior goods

    We know from Chapter 4 that the quantity demanded of a normal good increases in response to an income increase, whereas the quantity demanded of an inferior good declines. Clearly, both jazz and boarding are normal goods, as illustrated in Figure 6.10, because more of each one is demanded in response to the income increase from img209.png to img210.png. It would challenge the imagination to think that either of these goods might be inferior. But if J were to denote junky (inferior) goods and S super goods, we could envisage an equilibrium img34.png to the northwest of img145.png in response to an income increase, along the constraint img210.png; less J and more S would be consumed in response to the income increase.

    Policy: Income transfers and price subsidies

    Government policies that improve the purchasing power of low-income households come in two main forms: Pure income transfers and price subsidies. Social Assistance payments ("welfare") or Employment Insurance benefits, for example, provide an increase in income to the needy. Subsidies, on the other hand, enable individuals to purchase particular goods or services at a lower price—for example, rent or daycare subsidies.

    In contrast to taxes, which reduce the purchasing power of the consumer, subsidies and income transfers increase purchasing power. The impact of an income transfer, compared with a pure price subsidy, can be analyzed using Figures 6.11 and 6.12.

    Figure 6.11 Income transfer
    img211.png
    An increase in income due to a government transfer shifts the budget constraint from I1 to I2. This parallel shift increases the quantity consumed of the target good (daycare) and other goods, unless one is inferior.

    In Figure 6.11, an income transfer increases income from I1 to I2. The new equilibrium at E2 reflects an increase in utility, and an increase in the consumption of both daycare and other goods.

    Suppose now that a government program administrator decides that, while helping this individual to purchase more daycare accords with the intent of the transfer, she does not intend that government money should be used to purchase other goods. She therefore decides that a daycare subsidy program might better meet this objective than a pure income transfer.

    A daycare subsidy reduces the price of daycare and therefore rotates the budget constraint outwards around the intercept on the vertical axis. At the equilibrium in Figure 6.12, purchases of other goods change very little, and therefore most of the additional purchasing power is allocated to daycare.

    Figure 6.12 Price subsidy
    img212.png
    A subsidy to the targeted good, by reducing its price, rotates the budget constraint from I1 to I2. This induces the consumer to direct expenditure more towards daycare and less towards other goods than an income transfer that does not change the relative prices.

    Let us take the example one stage further. From the initial equilibrium img34.png in Figure 6.12, suppose that, instead of a subsidy that took the individual to img213.png, we gave an income transfer that enabled the consumer to purchase the combination img213.png. Such a transfer is represented in Figure 6.13 by a parallel outward shift of the budget constraint from img210.png to img214.png, going through the point img213.png. We now have a subsidy policy and an alternative income transfer policy, each permitting the same consumption bundle (img213.png). The interesting aspect of this pair of possibilities is that the income transfer will enable the consumer to attain a higher level of satisfaction—for example, at point img138.png —and will also induce her to consume more of the good on the vertical axis. The higher level of satisfaction comes about because the consumer has more latitude in allocating the additional real income.

    Application Box 6.3 Daycare subsidies in Quebec

    The Quebec provincial government subsidizes daycare heavily. In the public-sector network called the "Centres de la petite enfance", families can place their children in daycare for less than $10 per day, while families that use the private sector are permitted a generous tax allowance for their daycare costs. This policy is designed to enable households to limit the share of their income expended on daycare. It is described in Figure 6.13.

    The consequences of strong subsidization are not negligible: Excess demand, to such an extent that children are frequently placed on waiting lists for daycare places long before their parents intend to use the service. Annual subsidy costs amount to almost $2 billion per year. At the same time, it has been estimated that the policy has enabled many more parents to enter the workforce than otherwise would have.

    Figure 6.13 Subsidy-transfer comparison
    img215.png
    A price subsidy to the targeted good induces the individual to move from E1 to E2, facing a budget constraint I2. An income transfer that permits him to consume E2 is given by img216.png; but it also permits him to attain a higher level of satisfaction, denoted by img217.png on the indifference curve U3.

    The price of giving

    Imagine now that the good on the horizontal axis is charitable donations, rather than daycare, and the government decides that for every dollar given the individual will see a reduction in their income tax of 50 cents. This is equivalent to cutting the 'price' of donations in half, because a donation of one dollar now costs the individual half of that amount. Graphically the budget constraint rotates outward with the vertical intercept unchanged. Since donations now cost less the individual has increased spending power as a result of the price reduction for donations. The price reduction is designed to increase the attractiveness of donations to the utility maximizing consumer.

    Key Terms

    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.

    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.

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

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

    Ordinal utility assumes that individuals can rank commodity bundles in accordance with the level of satisfaction associated with each bundle.

    Budget constraint defines all bundles of goods that the consumer can afford with a given budget.

    Affordable set of goods and services for the consumer is bounded by the budget line from above; the non-affordable set lies strictly above the budget line.

    Indifference curve defines combinations of goods and services that yield the same level of satisfaction to the consumer.

    Indifference map is a set of indifference curves, where curves further from the origin denote a higher level of satisfaction.

    Marginal rate of substitution is the slope of the indifference curve. It defines the amount of one good the consumer is willing to sacrifice in order to obtain a given increment of the other, while maintaining utility unchanged.

    Diminishing marginal rate of substitution reflects a higher marginal value being associated with smaller quantities of any good consumed.

    Consumer optimum occurs where the chosen consumption bundle is a point such that the price ratio equals the marginal rate of substitution.

    Exercises for Chapter 6

    EXERCISE 6.1

    In the example given in Table 6.1, suppose Neal experiences a small increase in income. Will he allocate it to snowboarding or jazz? [Hint: At the existing equilibrium, which activity will yield the higher MU for an additional dollar spent on it?]

    EXERCISE 6.2

    Suppose that utility depends on the square root of the amount of good X consumed: img218.png.

    1. In a spreadsheet enter the values 1... 16 as the X column (col A), and in the adjoining column (B) compute the value of utility corresponding to each quantity of X. To do this use the 'SQRT' command. For example, the entry in cell B3 will be of the form '=SQRT(A3)'.

    2. In the third column enter the marginal utility (MU) associated with each value of X – the change in utility in going from one value of X to the next.

    3. Use the 'graph' tool to map the relationship between U and X.

    4. Use the graph tool to map the relationship between MU and X.

    EXERCISE 6.3

    Instead of the square-root utility function in Exercise 6.2, suppose that utility takes the form U=x2.

    1. Follow the same procedure as in the previous question – graph the utility function.

    2. Why is this utility function not consistent with our beliefs on utility?

    EXERCISE 6.4

    1. Plot the utility function U=2X, following the same procedure as in the previous questions.

    2. Next plot the marginal utility values in a graph. What do we notice about the behaviour of the MU?

    EXERCISE 6.5

    Let us see if we can draw a utility function for beer. In this instance the individual may reach a point where he takes too much.

    1. If the utility function is of the form U=6XX2, plot the utility values for X values in the range img222.png, using either a spreadsheet or manual calculations.

    2. At how many units of X (beer) is the individual's utility maximized?

    3. At how many beers does the utility become negative?

    EXERCISE 6.6

    Cappuccinos, C, cost $3 each, and music downloads of your favourite artist, M, cost $1 each from your iTunes store. Income is $24.

    1. Draw the budget line, with cappuccinos on the vertical axis, and music on the horizontal axis, and compute the values of the intercepts.

    2. What is the slope of the budget constraint, and what is the opportunity cost of 1 cappuccino?

    3. Are the following combinations of goods in the affordable set: (4C and 9M), (6C and 2M), (3C and 15M)?

    4. Which combination(s) above lie inside the affordable set, and which lie on the boundary?

    EXERCISE 6.7

    George spends his income on gasoline and "other goods."

    1. First, draw a budget constraint, with gasoline on the horizontal axis.

    2. Suppose now that, in response to a gasoline shortage in the economy, the government imposes a ration on each individual that limits the purchase of gasoline to an amount less than the gasoline intercept of the budget constraint. Draw the new effective budget constraint.

    EXERCISE 6.8

    Suppose that you are told that the indifference curves defining the trade-off for two goods took the form of straight lines. Which of the four properties outlines in Section 6.3 would such indifference curves violate?

    EXERCISE 6.9

    Draw an indifference map with several indifference curves and several budget constraints corresponding to different possible levels of income. Note that these budget constraints should all be parallel because only income changes, not prices. Now find some optimizing (tangency) points. Join all of these points. You have just constructed what is called an income-consumption curve. Can you understand why it is called an income-consumption curve?

    EXERCISE 6.10

    Draw an indifference map again, in conjunction with a set of budget constraints. This time the budget constraints should each have a different price of good X and the same price for good Y.

    1. Draw in the resulting equilibria or tangencies and join up all of these points. You have just constructed a price-consumption curve for good X. Can you understand why the curve is so called?

    2. Now repeat part (a), but keep the price of X constant and permit the price of Y to vary. The resulting set of equilibrium points will form a price consumption curve for good Y.

    EXERCISE 6.11

    Suppose that movies are a normal good, but public transport is inferior. Draw an indifference map with a budget constraint and initial equilibrium. Now let income increase and draw a plausible new equilibrium, noting that one of the goods is inferior.


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