Chapter 21: The Discounted Utility Model


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1 Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal preferences. In fact, it is not only empirically relevant but it also has the important normative property that individuals with such preferences are never dynamically inconsistent. What is meant by this is that the relative evaluation of consumption in period t and in period s remains constant through time for instance, if today an individual regards 1000 in 2015 as equivalent to him or her as 1500 in 2017, they should think the same way tomorrow and in 2003, and so on. This means that it represents the kind of intertemporal preferences that people ought to have if they want to be consistent in their intertemporal decisionmaking. These normative properties are not relevant in the twoperiod world that we have been analysing so far, but become relevant in a morethantwo period world. We shall consider such a world after we have described the model and explored its implications in a twoperiod world. 21.2: The Discounted Utility Model in a TwoPeriod World This is a model of preferences over bundles of consumption (c 1, c 2 ) where c 1 denotes consumption in period 1 and c 2 denotes consumption in period 2. It is simply given as follows: U(c 1, c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) (21.1) Do note carefully that there are two utility functions here: one is U, which is defined over the consumption bundle (c 1, c 2 ) and which represents the utility of that bundle; the other is u, which is defined over a single period s consumption and which represents the utility gained from consuming a particular amount in a particular period. The Discounted Utility Model states that the utility of a consumption bundle (c 1, c 2 ) is given by the utility gained from consuming the amount c 1 (in period 1) plus the utility gained from consuming the amount c 2 (in period 2) divided by (1+ρ). In this model, ρ is a parameter of the model and usually varies from individual to individual. For most individuals the parameter ρ is positive, which means that (1+ ρ) is greater than 1, which means that u(c 2 )/(1+ρ) is smaller than u(c 2 ) and hence that greater weight is attached to a particular level of consumption if it consumed in period 1 rather than in period 2. Why is it called the Discounted Utility Model? Because it discounts the utility gained from consumption in period 2 at the rate ρ. This parameter for a particular individual is called the individual s discount rate. It is called this for the following reason. Consider the present value of the stream of income m 1 in period 1 and m 2 in period 2 when the market interest rate is r. The present value is (see section 20.5) is m 1 + m 2 /(1+r) As we have already noted in chapter 20, as viewed from period 1 the future income m 2 is discounted by the market at the rate r because it will not be received for a period. Now notice the similarity between this expression and that for the Discounted Utility Model in (21.1). In the latter the individual discounts the future utility because it will not be received for a period. He or she discounts it at the rate ρ. We should emphasise that this discount factor is individual specific some individuals have a high value for ρ, some a low value. What it depends upon is how the
2 individual regards the utility of consumption in period 2 relative to the utility of consumption in period 1. It may be the case that the individual regards them equally in which case the parameter ρ takes the value 0 the individual in this case does not discount the future. However for most of us it seems to be the case that we weight the present more highly than the future in which case the value of the parameter ρ is positive. Furthermore, the more the individual regards the present relative to the future the higher is ρ  alternatively we can say the higher the individual regards the present relative to the future, the more the individual discounts the future. We should also point out that, as a parameter representing preferences, the value of ρ is independent of the value of the market discount rate r. The parameter ρ captures the individual s preferences regarding the relative weighting of present and future consumption. To complete the specification of the Discounted Utility Model we also need to specify the function u. This will also be individual specific. Normally we expect it to be a concave function as consumption rises then so does the utility gained from that consumption, but it rises less than proportionately. Or, in more casual terms, for every increase of 1 in consumption there is an increase in utility but these increases get smaller as the amount consumed increases. In what follows we will assume that the function u is the square root function (which is concave for positive consumption 1 ). An alternative is that the u function is the logarithmic function (which is also concave for positive consumption). With this function we get slightly different demand functions but the important property concerning the relationship of r with ρ (which we shall prove shortly) is true for all concave u functions, including the logarithmic function. Of course, in practice, the form of the function depends upon the individual s preferences. 21.4: Indifference Curves with the Discounted Utility Model We have now specified the Discounted Utility Model. In the next section we explore its implications. But first, as we are going to use the framework for analysis developed in chapter 20, we need to look at the properties of the indifference curves in (c 1, c 2 ) space implied by this model. There follows a little bit of mathematics; if you do not like maths, look away until we discuss the implications of the results. An indifference curve in (c 1, c 2 ) space is given, as ever, by U(c 1, c 2 ) = constant If we substitute in the specification of the Discounted Utility Model from (21.1) we get the following equation for an indifference curve in (c 1, c 2 ) space. u(c 1 ) + u(c 2 )/(1+ρ) = constant From this we can find the slope of the indifference curve (a proof is provided in the Mathematical Appendix to this chapter). The slope is given by: slope of indifference curve =  (1+ρ) [du(c 1 )/dc 1 ]/[du(c 2 )/dc 2 ] (21.2) where du(c)/dc denotes the derivative of u(c) with respect to c that is the rate at which utility is increasing with consumption. 1 It is difficult to conceive of negative consumption.
3 The slope is negative so the indifference curves are downward sloping. Moreover if u is concave then, as we move down and rightwards along an indifference curve, c 1 is rising while c 2 is falling, and hence du(c 1 )/dc 1 is falling while du(c 2 )/dc 2 is rising, and so the magnitude of the slope is falling. From this it follows that if u is concave then the indifference curves are convex. If however u is linear then both du(c 1 )/dc 1 and du(c 2 )/dc 2 are constant and so the slope of the indifference curves are constant that is they are linear. If we continue this line of argument with a convex function u, then we get the following result: If u is concave, linear or convex then the indifference curves in (c 1, c 2 ) space are convex, linear or concave. As we have already argued the concave u function is the empirically more realistic as people s consumption increases, their utility rises, but at a decreasing rate. One final result is of particular importance. If, in (21.2) we put c 1 = c 2 we get that the slope of an indifference curve is (1+ρ). Calling the line c 1 = c 2 the equal consumption line we get the important result that: Along the equal consumption line the slope of every indifference curve of an individual with Discounted Utility Model preferences is equal to (1+ρ). You should remember this. We illustrate in figure (21.1). In this figure I have used the square root utility function for u, and I have put ρ = 0.2. (that is, the individual discounts the future at 20%). In figure 21.1 I draw some of his or her indifference curves in (c 1, c 2 ) space. I have also inserted the equal consumption line. It can be seen that the slope of every indifference curve along this line is 1.2. For an individual with a higher discount rate than 20%, his or her indifference curves are everywhere steeper along the equal consumption line. 21.5: The Implications in a TwoPeriod World You may be able to guess some of the implications. We know that along the equal income line the slope of the indifference curves are all (1+ρ). It follows therefore that to the right of the equal income line the magnitude of the slope is everywhere less than (1+ρ) and to the left of the equal income line the magnitude of the slope is everywhere more than (1+ρ). Why is this important? Because we know that the magnitude of the slope of the budget line is (1+r) and we know that at the optimum point the magnitude of the slope of the budget line is equal to the magnitude of the slope of the indifference curve at the optimum point. (Because the budget line must be tangential to
4 the indifference curve at the optimal point.) So at the optimal point the slope of the indifference curve must be (1+r). There are three cases to consider. The simplest is when r = ρ. In this case the optimum point must lie somewhere along the equal consumption line for we know that the slopes of the indifference curves there are (1+ρ), and if this is equal to (1+r) (because r = ρ) it follows that the optimum point (where the slopes of the indifference curve and the budget line are equal) must be along the equal consumption line. In this case the individual consumes the same in both periods the reason being that the market discounts the future at exactly the same rate as the individual. When r > ρ we can argue that the optimal point must lie to the left of the equal consumption line so that the individual consumes more in the second period than the first. And when r < ρ we can argue that the optimal point must lie to the right of the equal consumption line so that the individual consumes more in the first period than the second. What is crucial is whether the market discounts the future more heavily than the individual. These properties are confirmed in the following example, in which we keep fixed the initial incomes of 40 (in period 1) and 40 (in period 2) and vary the rate of interest from zero (as in figure 21.3) upwards. We assume a value of ρ equal to 0.2 (20%) that is, we use the preferences of (21.1) above. You might like to verify that when the market rate of interest reaches 20% the optimal point is on the equal consumption line. The implied borrowing and lending are pictured in figure (Recall that the rate of return is one plus the rate of interest. The downward sloping line is the net demand for consumption in period 1 and the upward sloping line the net demand for consumption in period 2. Note that they cross the axis at a rate of interest 20%.) If we repeat this exercise for an individual with ρ = 0.4 we get figure Note the similarities and the differences between this and figure 21.4.
5 21.6: The Discounted Utility Model in a ManyPeriod World Although a little outside the scope of this book, it is interesting to consider the extension of the Discounted Utility Model to a manyperiod world. The extension is the natural one and follows the extension of the discounting formula discussed in section Suppose we are in a world which lasts T periods, where T may be finite, infinite or random. Suppose the individual has consumption c 1 in period 1, c 2 in period 2,, c t in period t,, and c T in period T. We consider his or her preferences over consumption bundles over these T periods bundles which we denote by (c 1, c 2,, c t,, c T ). The Discounted Utility Model states that preferences over these bundles are given by U(c 1, c 2,, c t,, c T ) = u(c 1 ) + u(c 2 )/(1+ρ) + + u(c t )/(1+ρ) t1 + +u(c T )/(1+ρ) T1 (21.3) Once again the model is specified by a discount rate ρ and a utility function u. The model is the natural extension of the twoperiod model. The utility of the bundle is the sum of the utilities of consumption in each period discounted at the rate ρ. If ρ is positive the weight attached to future consumption declines with the time from the present: period 1 s consumption is given a weight of 1; period 2 s consumption is weighted by 1/(1+ρ); ; period t s consumption by 1/(1+ρ) t1 ; ; and period T s by 1/(1+ρ) T1. You may find it constructive to compare (21.3) with the formula in section 20.5 giving the present value of an income stream there future incomes are discounted by the market at the rate r because they are received in the future. In the Discounted Utility Model future utilities are discounted by the individual at the rate ρ because they are experienced in the future. Whether this is a good description of actual intertemporal preferences is obviously an empirical issue but it may be instructive to give the normative reason why such preferences might be appropriate. In essence the idea is that an individual with such preferences is dynamically consistent in the sense that they carry out plans that they formulate. In a certain world there is no reason why one should ever want to change plans once formulated, so this property is an appealing one. Let us discuss it in more detail. Suppose we start in period 1 and we know that the income stream is going to be m 1 in period 1, m 2 in period 2,, m t in period t,, m T in period T. Then the individual in period 1 formulates a plan for consumption (c 1, c 2,, c t,, c T ) through time on the basis of maximising the utility (21.3) subject to the income stream and the rate of interest (which we are assuming is fixed and known). Accordingly in period 1 the individual consumes the planned c 1. In period 2 what happens? Does the individual implement the planned consumption c 2? It depends. Suppose the individual replans at this stage. At this stage he or she has a different objective
6 function because period 1 has been and gone. Now only periods 2 through T remain. If we update (21.3) to take account of this the objective function is now U(c 2,, c t,, c T ) = u(c 2 ) + u(c 3 )/(1+ρ) + + u(c t )/(1+ρ) t2 + +u(c T )/(1+ρ) T2 (21.4) Does the maximisation of this, given the remaining stream of income, lead to the same optimal values c 2,, c t,, c T that were the optimal values in the original plan c 1, c 2,, c t,, c T formulated at time period 1? The answer to this question is yes though the reason for this answer may not be obvious. A proof is provided in the Mathematical Appendix, which shows that we can express the utility from period 1 onwards in the following form: U(c 1, c 2,, c t,, c T ) = u(c 1 ) + U(c 2,, c t,, c T )/(1+ρ) (21.5) So the choice of the best values of c 2,, c t,, c T given the optimal value of c 1 from the maximisation of utility as viewed from the first period (with respect to the choice of c 1, c 2,, c T ) leads to the same choice of c 2,, c t,, c T as in the original choice of c 1, c 2,, c t,, c T in the maximisation of utility as viewed from the first period. The individual would not want to change the plans originally made. In essence and this should be clear from (21.5)  the reason is that if we compare any two periods s and t the relative weight attached to the utility of consumption in period s is always the same value (1+ρ) st irrespective of which period we are viewing it from. This means that the individual never wishes to revise any plans that he or she has formulated. (Of course, in a certain world, there are no reasons why one should ever wish to revise any plans.) If, instead, the individual used a varying discount rate through time, then it is possible that the individual might want to revise plans made earlier. In a sense this is because the individual, with a varying discount rate, is not just one individual but several and they have conflicting views what is best to do. This is exactly the kind of person who resolves every morning not to drink in the evening and then ends up doing so. 21.7: Summary Most of this chapter has been concerned with the twoperiod Discounted Utility Model though the final section contained an extension to the manyperiod model in which we provided a normative justification for it. In the twoperiod case we showed the following. The Discounted Utility model states that the intertemporal utility function is given by U(c 1, c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) where ρ is the individual's discount rate. With the Discounted Utility Model the slope of all indifference curves along the equalconsumption line are (1+ρ) Now we recalled that the perfect capital market budget constraint has a slope of (1+r) where r is the market interest rate. Combining these two results we showed that: An individual with Discounted Utility preferences would consume the same in both periods if his or her ρ equalled the market r... and that he or she will consume more in period 1 than in period 2 if r < ρ.. and that he or she will consume more in period 2 than in period 1 if r > ρ
7 We then briefly extended the model to a manyperiod world and showed that one attractive normative property of the model is that an agent with such preferences is never dynamically inconsistent. 21.8: At what rate do you discount the future? The Discounted Utility Model incorporates the notion that individuals do not value consumption in all periods equally. In general, it would appear to be the case that individuals care more about the present than the future, though the extent to which they do this differs from individual to individual. Here you are invited to try and discover how the extent to which you do this. The model in a Tperiod world takes the form of equation (21.3) and in a 2period world, the form of equation (21.1). This latter says that a twoperiod bundle of consumption (c 1,c 2 ) is valued by the function U(c 1, c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) where u(c) is the utility gained from consuming c in some period and ρ is the individual s discount rate. Both of these are specific to the individual. Here we will concentrate on discovering your value of ρ, assuming, of course, that your preferences are in accordance with the Discounted Utility Model 2. The method used is simple we exploit the implications of the equation above, particularly those results that we have already derived concerning the slope of indifference curves. If we can find two points on the same indifference curve, then we can use these results to try to infer the value of ρ. Specifically, let us take equation (21.2): slope of indifference curve =  (1+ρ) [du(c 1 )/dc 1 ]/[du(c 2 )/dc 2 ] If we do not know the individual s utility function, then we start at a position where c 1 and c 2 are equal. This enables us to derive the result stated in the text: Along the equal consumption line the slope of every indifference curve of an individual with Discounted Utility Model preferences is equal to (1+ρ). It follows that if we start at a point where c 1 = c 2 = c and we find a nearby point about which the individual feels the same (that is, is on the same indifference curve) then we can infer the value of ρ. To fix ideas suppose that the point (c  a, c + b) is such a point that is, the individual feels indifferent between the twoperiod consumption bundle (c, c) and the twoperiod consumption bundle (c  a, c + b). Then the slope of the indifference curve is approximately equal to b/a. This is an estimate of (1+ ρ). Hence we have that an estimate of ρ is b/a 1 = (b a)/a. If the individual puts more weight on present than on future consumption, then b will be bigger than a (because to compensate the individual for consuming a less today he or she will require more than a next period), and hence the estimate of ρ is positive. Obviously the value of ρ depends upon the length of the period we are considering. Let us assume here that the period is of length one year. You can now try and implement the above ideas and 2 If they are not, then there is not a ρ to discover.
8 hence find your yearly discount rate. You have to do the following introspection. Suppose you start from a position where you are consuming the same both this year and next year: for example, you are consuming 5000 each year. Now suppose someone suggests that your consumption this year will fall by say 100 (to 4900) but that you will be given some extra consumption next year to compensate. You should ask yourself: what is the minimum compensation I would require? This is quite a difficult introspection, but you should attempt it. Try and narrow it down. Would 1 compensation be enough? (Probably not.) Would 1000 compensation be enough? (Probably more than enough.) Would 50 be enough? And so on. The minimum compensation that you require gives an estimate of your discount rate ρ. Suppose this minimum compensation is 120. Then we have that a = 100 and b = 120, so that your ρ is 0.2 ( = ( )/100). It should be clear that we can derive the following table of examples. Minimum Compensation required for a decrease in period 1 consumption of 100 Implied value of the discount rate ρ Note carefully that we are talking about changes in consumption and not about changes in money income. If it were the latter and there was a perfect capital market in which you could freely borrow and lend at the constant rate of interest r, then the answer to the question would have to be 100(1+r). We would not learn anything about your discount rate only about the rate of interest in the perfect capital market! Note the assumptions carefully: (1) we start along the equal consumption line (so that we do not have to worry about your utility function, which is difficult to infer); (2) we consider a small reduction in period 1 consumption (otherwise we are moving around the indifference curve and its slope may change); (3) you tell us honestly the minimum amount of compensation in terms of period 2 consumption you require. If you do all this, you can find your discount rate. If you are interested, you could explore the implications for a manyperiod world. In the extension to T periods, as given in equation (21.3), you will see that the Discounted Utility Model assumes that the same discount rate is used throughout. Obviously this is a strong assumption, but one that can be tested. You can try one such test yourself. Suppose you have found that the minimum compensation you require (under the assumptions listed above) is 120. Then, starting from an equal consumption point, you regard having 100 less today as being compensatable with an extra 120 in one year s time. Now do the same exercise, but now ask what is the minimum compensation in two years time for having 100 less today. Denote again by b this minimum compensation, but do remember that this will be consumed in two years time. Repeating the argument that we used above it follows that b/a is an estimate of (1+ ρ) 2. Thus to be consistent with the Discounted Utility Model and with your previously derived estimate of ρ (which is 0.2), it must be the case that the minimum compensation you require in two years time is 144. (So that b/a = (1+ ρ) 2.) At first glance you may this odd or, at least inconsistent with your introspection. Let us discover why the Discounted Utility Model makes this prediction. We begin with your first introspection you needed 120 in one year s time to compensate you for having 100 less today, that is, for each
9 1 less today you needed 1.20 in compensation in one year s time. If this story applies not only to consumption deferred for one year from today, it should also apply to consumption deferred for one year from next year. So, if you are to be compensated for having 100 less today, but will receive the compensation in two years time, you can argue as follows: I would need 120 more in one year s time, and to defer each of these 120 for a further year, I will need a compensation of 1.20 for each of those 120 that is a compensation of 1.2 times 120 = 144 in two year s time. Was your introspection consistent with this? You may be interested to know that there have been many experimental tests of the Discounted Model, and particularly its central assumption that the discount rate is constant 3. The great strength of experimental economics is its central tenet that participants should be given appropriate incentives to behave in such a way that their behaviour reveals their preferences. In many areas it is easy to give appropriate incentives as we will see in the chapter on Game Theory but in the area of intertemporal choice it is more difficult. We have already noted some strong assumptions that underlie the inferences we have made. These are difficult to enforce in the laboratory. The greatest problem, however, is that correct incentives in intertemporal choice experiments necessary involve the passage of time. It may be difficult to ensure that participants and experimenters are still around after that passage of time Mathematical Appendix We first derive the proposition concerning the slope of the indifference curves implied by the Discounted Utility Model. As stated in the text, an indifference curve in (c 1, c 2 ) space is given by U(c 1, c 2 ) = constant If we substitute in the specification of the Discounted Utility Model from (21.1) we get the following equation for an indifference curve in (c 1, c 2 ) space. u(c 1 ) + u(c 2 )/(1+ρ) = constant To find the slope of the indifference curve we differentiate this totally, thus getting u'(c 1 ) dc 1 + u'(c 2 ) dc 2 /(1+ρ) = 0 where u'(c) denotes the derivative of u(c) with respect to c. From this we get the slope of an indifference curve This is equation (21.2) of the text. We now derive equation (21.5). Suppose c 1 *, c 2 *,, c t *,, c T * maximise dc 2 /dc 1 =  (1+ρ) u'(c 1 )/u'(c 2 ) 3 Many of these studies suggest that the discount rate is not constant.
10 U(c 1, c 2,, c t,, c T ) = u(c 1 ) + u(c 2 )/(1+ρ) + + u(c t )/(1+ρ) t1 + +u(c T )/(1+ρ) T1 given an income stream m 1, m 2,, m t,, m T, and hence subject to the intertemporal budget constraint m 1 + m 2 /(1+r) + + m T1 /(1+r) T2 + m T /(1+r) T1 = c 1 + c 2 /(1+r) + + c T1 /(1+r) T2 + c T /(1+r) T1 then it must be the case that c * 2,, c * t,, c * T maximise U(c 2,, c t,, c T ) = u(c 2 ) + u(c 3 )/(1+ρ) + + u(c t )/(1+ρ) t2 + +u(c T )/(1+ρ) T2 subject to the constraint m 1 + m 2 /(1+r) + + m T1 /(1+r) T2 + m T /(1+r) T1 = c * 1 + c 2 /(1+r) + + c T1 /(1+r) T2 + c T /(1+r) T1 because U(c 1, c 2,, c t,, c T ) = u(c 1 ) + U(c 2,, c t,, c T )/(1+ρ).
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