DEPARTMENT OF ECONOMICS

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1 ON THE DEMAND FOR HYPER-SUBSTITUTES WITH SATIATION by Peter G. Moffatt and David J. Butler DISCUSSION PAPER DEPARTMENT OF ECONOMICS THE UNIVERSITY OF WESTERN AUSTRALIA NEDLANDS, WESTERN AUSTRALIA 697 ISSN ISBN

2 ON THE DEMAND FOR HYPER-SUBSTITUTES WITH SATIATION by Peter G. Moffatt University of East Anglia and David J. Butler Department of Economics The University of Western Australia DISCUSSION PAPER 96-3 January 1996 ISSN ISBN

3 ON THE DEMAND FOR HYPER-SUBSTITUTES WITII SATIATION Peter G. Moffatt University of East Anglia Norwich NR47TJ United Kingdom and David J. Butler University of W estem Australia Nedlands W estem Australia 697 Australia December 1995 ABSTRACT Hyper-substitutes are defined as goods which the consumer has an aversion to consuming in mixtures, violating the axiom of convexity. Beer and wine provide an obvious example. Under the assumption that more is preferred to less, the indifference map for hyper-substitutes consists of indifference curves which are concave to the origin. In this paper it is argued that the axiom of non-satiation should also be relaxed in order for the hyper-substitute model to be internally consistent The indifference map for hyper-substitutes with satiation is constructed and is shown to have very interesting properties. In particular, it is shown that maximisation of the underlying utility function subject to a budget constraint does not automatically result in only one of the two goods being consumed. Thus an explanation is offered of why rational individuals might mix their drinks in spite of having an aversion to mixtures. JEL classifications: A23, Dll

4 1. Introduction In the teaching of elementary consumer theory, it is usually assumed that only two goods are available to the consumer, and that the consumer prefers mixtures of the two goods to extreme quantities of either one of the two goods. It is easily demonstrated that, under the assumption that more is preferred to less, this preference for mixtures gives rise to indifference curves which are convex to the origin. Preference for mixtures is, in fact, embodied in one of the axioms of consumer theory, namely, the axiom of convexity. An interesting variation on the simple model, which is also popular in teaching, is concerned with the situation in which the consumer has an-aversion to consuming mixtures of the two goods (see, for example, Varian, 1993, p.53, question 11, and also p.47). In Varian's example, the two goods are ice-cream and anchovies. Another example, preferred by us, concerns two different types of alcoholic beverage. Sproul (1984, p.25) reports that "I enjoy drinking beer and wine but I prefer not to mix them. I would be just as happy with a bottle of either, or with three-quarters of a bottle of each". The alcohol example is also tackled by Clements and Selvanathan (1991), and in more humorous spirit by Covicks (1974). In deciding on the level of consumption of beer and wine, the consumer is influenced by the prospect of the undesirable consequences of alcohol consumption, which are likely to be worse when a mixture has been consumed, than when a larger total quantity of a single beverage-type has been consumed. This aversion to mixtures amounts to a violation of the axiom of convexity, and, again assuming that more is preferred to less, gives rise to indifference curves which are concave to the origin. For want of a better term, let us refer to such a pair of goods as "hyper-substitutes". This proposed nomenclature can be seen as a natural extension of existing terminology. If indifference curves are L-shaped, we refer to the two goods as perfect complements. If indifference curves slope downwards and are convex to the origin, there is a degree of substitutability between the two goods, and this degree of substitutability rises as the curves become more linear. Eventually, we reach the case of straight line, downward-sloping indifference curves, and in this case we refer to the two goods as perfect substitutes. By continuing this process, and allowing indifference curves to bend the "wrong" way, we move to the case of hyper-substitutes, which are the focus of attention in this paper. Usually, when finding the optimal levels of consumption of hyper-substitutes in the presence of a budget constraint, it is assumed that the utility function is a monotonically increasing function of both of its arguments; that is, the consumer never reaches a state of "satiation" whatever level of consumption is achieved. With this assumption, known as the axiom of nonsatiation, it is easily verified that the solution to the constrained optimisation problem is on one of the two axes, with only one of the two goods being consumed (see Zamagni, 1987, p.143). However, since the prospect of the undesirable consequences of consuming mixtures has been built into the utility function, there is no reason why the undesirable consequences of consuming a considerable quantity of a single good should be ignored. Surely it is the case that consuming a very large amount of alcohol is undesirable, mixture or no mixture, but that mixing large quantities makes the consequences even more undesirable. What is being suggested here is that the undesirable consequences of excess alcohol consumption, per se, should be built in to the utility function, by relaxing the assumption of non-satiation. What is required is a utility function which is decreasing in its arguments beyond a certain level of consumption, the "satiation level". Beyond this satiation level, goods become "bads". The concept of a utility function with satiation is not new (see, for example, Varian, 1993, p.43). Leaving aside, for the moment, the problem of hyper-substitutes, a natural way to relax 1

5 non-satiation would be to allow the utility function to attain a global maximum at some point at which consumption of both goods is finite, and to decrease monotonically with increases in consumption of either good beyond this point. Tue consumption bundle at which utility attains this global maximum is known as the "bliss point". Tue indifference curves in such a situation may be concentric circles, or ellipses, around the bliss point, with utility falling the further away from the bliss point, in any direction, the consumer strays. Close examination of such an indifference map reveals that it consists of four distinct regions. In the south-west region, marginal utility is positive for both goods. In the north-east region, both marginal utilities are negative. In the other two regions, north-west and south-east, marginal utility is positive for one of the two goods, and negative for the other. Tue constrained optimisation rule is simple: if the budget line passes through the north-east region, that is, if it lies to the right of the bliss point, the optimal bundle is at the bliss point. Otherwise, the optimal bundle is the one where an indifference curve is tangential to the budget line. We therefore see that relaxing the assumption of non-satiation, under otherwise normal circumstances, gives rise to an indifference map which is straightforward to analyse, albeit non-standard. In this paper, however, we set out to relax the assumption of non-satiation in an already non-standard situation, namely, when the axiom of convexity is violated, due to the two goods being hyper-substitutes. As we shall see, this combination of two non-regular features gives rise to an indifference map of a truly extraordinary nature. Tue indifference map has the following striking features. Firstly, the indifference curves are concave to the origin in a region near the origin, but are convex to the origin in a region further away from the origin. This requires that there be a sharp "satiation ridge" separating the two regions. There are then two bliss points (i.e. maxima of the utility function) where the satiation ridge meets the axes. Finally, there is a saddle point on the ridge at a location where consumption of both goods is positive. The first objective of this paper is to construct an indifference map with these properties and represent it graphically. This will be done in section 2. As already mentioned, the constrained optimisation problem for hyper-substitutes, with the assumption of non-satiation in place, always has a solution on one of the axes, with only one of the goods being consumed. This is hardly surprising given the premise that mixtures are undesirable. However, after the assumption of non-satiation is relaxed, it can be shown, astonishingly, that the solution to the constrained optimisation problem can, under certain circumstances, be at a point where consumption of both goods is positive. Tue second objective of this paper is to demonstrate this phenomenon. This will be done in section 3. Section 4 concludes. 2. The Indifference Map Tue starting point for the analysis is the indifference map for hyper-substitutes with the assumption of non-satiation. As already mentioned, this consists of indifference curves which are all concave to the origin. The objective here is to relax the assumption of non-satiation. Relaxation of non-satiation generally results in the emergence of a point in the indifference map which is preferred to all others, the "bliss point". Here we are starting from the premise that mixtures are less desirable than extreme quantities of one good. It therefore must be the case that for any interior point, there must be a point on an axis which is preferred. It follows that if there is a bliss point, it must be on one of the axes. Furthermore, since both goods are subject to satiation, there must be a bliss point on each axis. Of course, it may be that one of these points gives higher utility than the other. However, it will be assumed in what follows that the utility function is symmetric in the two quantities. This assumption is not costly in terms of 2

6 generality of the results. One consequence of this assumption is that utility is equal at the two bliss points. The distance of a bliss point along an axis is interpreted as the consumption level at which satiation sets in, when only the good measured on that axis is being consumed. Of course, if the consumer were forced to consume the two goods in certain fixed proportions, he would be moving along a ray through the origin, and satiation would set in at a certain point along that ray. All such rays through the origin have their own satiation points. Assuming that utility is a continuous function of the two quantities, the satiation points thus obtained can be joined by a continuous curve between the two bliss points. It is therefore seen that satiation in the context of hyper-substitutes is represented by a "satiation-ridge" in the utility function. The ridge is a collection of all the satiation points of rays through the origin. The highest points on the satiation ridge are clearly the two bliss points, which are at either end of the ridge, on the axes. The lowest point on the ridge is at a point near the middle of the ridge. This point has a property very unusual in the context of utility functions: it is a saddle-point This is because, while being the lowest point on the ridge, the fact that it is on the ridge implies that it is the highest point on a ray through the origin. An important consequence of there being a satiation ridge, rather than a unique bliss point, is that there is no region of quantity space in which the marginal utilities for the two goods are of opposite signs. Below of the ridge, both marginal utilities are positive, while above the ridge, both are negative. The north-west and south-east regions, which arise in situations where the bliss point is unique, have been "squeezed out" by the ridge. This has an intuitive justification. If the consumer is consuming only one good, in a moderate quantity, so that the marginal utility of additional consumption of that good is positive, it is reasonable that consumption of a unit of the other good would increase utility, although, because the two goods are hyper-substitutes, not as much as would an additional unit of the consumed good. If, however, the consumer is consuming a very large quantity of only one good, so much that his marginal utility of additional consumption of the good is negative, it is not reasonable that utility could be increased by consuming a unit of the other good; this would surely cause utility to fall. The point here is that if satiation has set in with respect to one of the two goods, it has, necessarily, set in with respect to the other as well. It is not therefore logical to have regions in which the marginal utilities have opposite signs. The presence of a ridge avoids such regions, as desired. We now turn to the shape of indifference curves. Below the ridge, both goods have positive marginal utilities. Since mixtures are undesirable, a convex combination of two points on the same indifference curve must be on a lower indifference curve, that is, one closer to the origin. Indifference curves must therefore be concave to the origin in this region. Above the ridge, both goods have negative marginal utilities. In fact, they are not goods at all in this region; they are "bads". As before, a convex combination of two points on the same indifference curve must be on a lower indifference curve, but here, "lower" means further from the origin, since both goods are "bads". Above the ridge, indifference curves are therefore convex to the origin. As will be seen below, this implies that the ridge must in fact be sharp. Let Xi and x2 be consumption of good 1 and good 2 respectively. A simple utility function which bears all of the features introduced so far is: (1) for xi and x2 both in the range (,15). The utility function (1) is the minimum of two functions. The first function is relevant below the ridge, while the second is relevant above the ridge. A 3- dimensional plot of this utility function is presented in figure la. Comparisons with physical 3

7 objects are always useful in describing 3-d plots, and we see that this one resembles a sheet draped over a slack washing line, parts of the sheet being in contact with the ground. The ridge is the washing line and the two bliss points are at either end of the washing line. In figure 1 b, a contour plot of utility function (1) is presented. The contour map consists of two sets of concentric circles: one set has centre (-5,-5); the other has centre (15,15). The ridge is traced out by the points at which the two sets of circles intersect The ridge is, in fact, the straight line joining the two points (1,) and (,1), and these.two points are the two bliss points. The lowest point on the ridge is at (5,5). This point is the saddle-point, because, while being the lowest point on the ridge, it is the highest point on a 45 ray from the origin. That the ridge is a straight line, when viewed from above, is obvious from (1). The utility function is the minimum of two basin-shaped functions. The ri'dge consists of the set of points at which the two functions are equal, and because of their circular contours, this set of points must be a straight line. A consequence of the shape and position of the satiation ridge in this example is that satiation will always be reached after consuming exactly 1 units of drink. regardless of the composition of the 1 units. In figure lb, we see that along the ridge, the contours display sharp kinks. The sharpness of these kinks, and, indeed, the sharpness of the ridge itself, are due to the requirement that indifference curves be concave to the origin below the ridge, and convex to the origin above the ridge. A consequence of the sharpness of the ridge is that marginal utility is not defined at any point on the ridge, since the utility function is not differentiable at these points. While the utility function (1), being very simple. is useful for the purposes of exposition, it does not embody the full nature of the problem. Firstly, it only bears the required features over a certain range; when consumption of one of the goods rises above 15 units, utility starts to rise again. There is a further problem with (1) which prompts us to seek a different function, namely rhat the bliss ridge implied by (1) is a straight line when viewed from above, as we have seen in figure lb. A straight line ridge is not plausible, for the following reason. If the consumer were constrained to consume a mixture of the two goods in fixed proportions, that is, constrained to be on a particular ray, with positive but finite slope, through the origin, it is sensible to assume that satiation would be reached sooner, in terms of the total quantity consumed, than if just one good were being consumed. The reason for this assumption is exactly the same as the reason why the consumer has an aversion to consuming mixtures in the first place. Extending the argument, it is plausible that the more mixed the bundle became, the sooner satiation would be reached This assumption has the very straightforward implication that the ridge is convex to the origin, when viewed from above. For the reason given in the last paragraph, we are now in search of a function with a ridge which is convex to the origin. The other requirements are as before: below the ridge, the function must be increasing in both its arguments, while above the ridge, it must be decreasing in both; and the indifference curves must be concave to the origin below the ridge, and convex to the origin above the ridge. A function with all of the desired properties may be constructed as follows. First let: (2) and let: 4

8 (3) where: '"--x 1 +-xry +-x 1 x,. 'f (4) The required utility function is given by: (5) The ridge is then given by: 1 2 x 1 +x, =l+-(x 1 -x,), (6) a parabola, convex to the origin, as required. It may be verified that g = f on the ridge. Below the ridge, f < g and above the ridge, g < f. Note that the curves g = est. are the same as the curves \jf = est., i.e. hyperbolae convex to the origin. A feature of the utility function (5) is that it is perfectly symmetric in the two quantities. Using (6), we may calculate the positions of the bliss points to be (1.17, ) and (, 1.17), and that of the saddle point to be (.5,.5). We therefore see that, if constrained to consume the goods in equal quantities, we reach satiation after consuming one unit in total, while, if allowed to consume only one of the goods, we do not reach satiation until we have consumed 1.17 units. The utility function (5) is represented in 3D form in figure 2a, and in contour form in figure 2b. 3. The demand for hyper-substitutes with satiation The point has already been made that with the assumption of non-satiation in place, the constrained optimisation problem for hyper-substitutes always results in a solution on one of the axes, with only one good being consumed. Let us now investigate the constrained optimisation problem with satiation. In section 2, two different utility functions were suggested, the first with a straight-line ridge, the second with a ridge convex to the origin. Consider the first, given in (1) and represented graphically in figures la and lb. Examination of figure lb reveals that at any income level and for any pair of prices of the two goods, the constrained optimum will always be on one of the axes. This is, in fact, the case for any utility function with a straight-line ridge. The situation pertaining to utility function (5) is far more interesting. In figure 3a, we reproduce figure 2b, the contour plot of utility function (5), and superimpose the budget line x 1 + x2 = 1.7. Note that this budget line implies that the two goods have equal prices. Once again, nothing is lost by making this simplifying assumption. Notice that this budget line crosses the ridge in two places. In order to find the constrained optimum, we need to expand a part of this indifference map. This is done in figure 3b, which shows an expansion of the 5

9 region of figure 3a contained in the superimposed rectangle. The same budget line as drawn in figure 3a is drawn in figure 3b. The highest indifference curve attainable subject to this budget constraint is the one which touches the budget line. In figures 3a and 3b, this indifference curve is the fourth nearest to the bliss point. What is very interesting is that the point at which this indifference curve touches the budget line is not on an axis. It is an interior point. It is, in fact, a point on the ridge. This means that an individual faced with this budget constraint will consume positive quantities of both goods. The optimising bundle is, in fact (.99,.161), and because of the symmetry of the problem, an alternative solution is (.161,.99). The value of utility at the optimum is.49. The highest utility attainable on an ax.is is only.45. The numerical example just given is one of many possibilities. There is a range of income levels for which this phenomenon is observed, these income levels being slightly lower than the income level required to attain one of the bliss points. The phenomenon could also be observed when the prices of the two goods differ. A necessary condition for the phenomenon to be observed is that the budget line should cross the ridge in two places. 4. Conclusion Violations of two axioms of consumer theory have been considered: convexity and nonsatiation. It is the presence of hyper-substitutes which gives rise to a violation of convexity. Hyper-substitutes and satiation, considered separately, are straightforward to analyse. In this paper, it has been suggested that the demand for hyper-substitutes should not be analysed while continuing to assume non-satiation. To do so is internally inconsistent Simultaneous violation of both axioms is therefore necessary. This simultaneous violation has also been shown to be very interesting. The indifference map arising when both axioms are violated has been shown to possess very unusual properties. Because of its complicated nature and its relevance to a real situation, the construction of the indifference map would provide a useful exercise for advanced undergraduates. Textbooks rarely do more than scratch the surface of problems like this. The demand for hyper-substitutes has also been analysed. The most interesting outcome occurs when income is not quite sufficient to attain the bliss points. Under certain circumstances, the highest point attainable is a point on the satiation ridge at which consumption of both goods is positive. The explanation for this phenomenon lies in the shape of the ridge. Because satiation is reached sooner when a mixture is consumed than when extreme quantities are consumed, the ridge is convex to the origin. Although the utility attained at satiation when a mixture is consumed is obviously not as high is that attained at the axis bliss points, it may be that the budget constraint allows only the central portion of the ridge to be attained, and that a point on this portion of the ridge gives higher utility than any ax.is point which is attainable. This phenomenon has been illustrated in figures 3a and 3b. What makes this phenomenon surprising is the way optimising behaviour appears to contradict the assumptions initially imposed on preferences. The utility function (5) is constructed from the premise that mixtures are undesirable. Everywhere on the indifference map, the curvature of indifference curves bears out this premise. However, it has been shown that, in spite of this built-in aversion to mixtures, the constrained maximisation of (5) can under certain circumstances result in positive consumption of both goods. Do we have an explanation here of why rational individuals mix their drinks? 6

10 Fig. 1 a: 3 plot of utility function (1) () Fig. 1 b: contour plot of utility function (1)..-- C\I x I.!) x1

11 Fig. 2a: 3 plot of utility function (5) Fig. 2b: contour plot of utility function (5) C\J T""" T""" co co C\J x "'1" ci C\J ci x1

12 Q Fig. 3a: Utility function (5) with budget constraint C\J x co xi Fig. 3b: Portion of fig. 3a expanded C\J LO..- C\J..- x LO ci ci \ x1

13 References Clements, K. and S. Selvanathan, 1991, 'Tue Economic Determinants of Alcohol Consumption", Australian J oumal of Agricultural Economics, Covicks,., 1974, 'Tue quantity theory of drink", Australian Economic Papers, Sproul, M., 1984, Instructor's Manual for Price Theory and Applications, 3rd edition, by Jack Hirschleifer, Prentice Hall. Varian, H., 1993, Intermediate Microeconomics; 3rd edition, Norton. Za.magni, S., 1987, Microeconomic Theory: an Introduction, Basil Blackwell. 1