Risk Vulnerability: a graphical interpretation

Transcription

1 Risk Vulnerability: a graphical interpretation Louis Eeckhoudt a, Béatrice Rey b,1 a IÉSEG School of Management, 3 rue de la Digue, Lille, France, and CORE, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium, louis.eeckhoudt@fucam.ac.be b Université Lyon 1, Lyon, F-69007, France ; Institut de Sciences Financières et d Assurance, 50 Avenue Tony Garnier, Lyon, France, beatrice.rey-fournier@univ-lyon1.fr Abstract The note gives a graphical interpretation of the concept of risk vulnerability. It shows that in a specific intertemporal context the assumption of risk vulnerability adds to prudence what the assumption of decreasing absolute risk aversion adds to risk aversion. Keywords: Decreasing absolute risk aversion; Prudence; Risk vulnerability JEL Classification Numbers: D81 1. Introduction The concept of risk vulnerability introduced by Gollier and Pratt (1996) turned out to be very important for the analysis of decisions in the presence of multiple risks. The definition of risk vulnerability is very intuitive. Using Gollier s terminology (2001) preferences exhibit risk vulnerability if the presence of an exogenous background risk with a nonpositive mean, namely an unfair risk, raises the aversion to any other independent risk. 1 Corresponding author. Tel.: ; Fax. : adress: beatrice.rey-fournier@univ-lyon1.fr 1

2 However when one tries to express the concept in terms of properties of the utility function matters are much less simple. As Gollier (2001) indicates the necessary and sufficient condition for risk vulnerability is rather complex. The purpose of the present note is to give a geometrical interpretation of the concept that can be easily understood and remembered. We basically show that in a specific intertemporal context the assumption of risk vulnerability adds to prudence what the assumption of decreasing absolute risk aversion (DARA) adds to risk aversion. Our tool is an analysis of properties of indifference curves in a simple two-period model of intertemporal choice. Our paper is organized as follows. In section 2 we present the basic elements of the model and we illustrate in that framework the relationship between risk aversion and the DARA assumption. Our main result is in section 3 where risk vulnerability is illustrated in geometrical terms by a natural extension of the results obtained in section 2. A short conclusion is presented in section The model In his influential work on capital theory Boehm-Bawerk (1884) 2 discussed the concept of positive time preference. Summarizing his point of view, Olson and Bailey (1981) claim that positive time preference arises from two causes: diminishing marginal utility of consumption at any given time and/or discounting of future versus present utility. In a two period framework where C 1 and C 2 denote present and future consumptions and where u is instantaneous utility, the consumer s utility U is U = u(c 1 ) + βu(c 2 ) (1) where β (with 0 β 1) is the discounting factor applied to the second period utility u. As is well known in that framework the marginal rate of substitution (MRS) between C 1 and C 2 is given by MRS = dc 2 = u (C 1 ) (2) dc 1 βu (C 2 ) In order to minimize notation and to focus on the effect of diminishing marginal utility we assume from now on that β = 1. The MRS is equal to unity whenever C 1 = C 2, i.e. along a 45 o line in the (C 1, C 2 ) space and concavity of the instantaneous utility function 2 In 1959 the twelve-hundred pages of Capital and Interest were translated into English by Hans Sennholz and George Huncke and published as a single volume. 2

3 implies that for any point below the 45 o line the MRS will be smaller than unity 3 : MRS = u (C) u (C k) < 1 k > 0 Less formally, when future is less favorable (a certain loss is added to the future wealth), a risk averse individual is an individual which will demand an increase in future consumption smaller than the reduction in present consumption. We are going to show that DARA assumption means that the MRS increases when C increases. To illustrate the DARA assumption in the present framework consider two different values of C 1 (C 1a and C 1b ) as in figure 1. These values induce corresponding values of C 2 (C 2a and C 2b ) on the 45 o line. Now take two new values of C 2 which are below the 45 o line at a constant distance k from C 2a and C 2b. They are denoted respectively C 2a and C 2b. C2 C2b C 2b k b C2a C 2a k a 45Â C1a C1b C1 Figure 1. 3 Concavity of u in our intertemporal environment is the equivalent of risk aversion in models of risky choices. This analogy is often done and all our results could be obtained also in a contingent states model with two states. 3

4 The MRS at point a (C 1a, C 2a) is given by MRS = u (C 1a ) u (C 2a) = u (C 1a ) u (C 1a k) (with C 2a = C 1a ) (3) and it is smaller than unity under risk aversion. Similarly at point b the MRS is given by u (C 1b ) u (C 1b k) < 1 and naturally the question arises: how does the MRS behave when one moves from a to b? It turns out that the answer to this question depends upon the behavior of the absolute risk aversion function. Indeed in general dmrs = u (C 2 )u (C 1 )dc 1 u (C 1 )u (C 2 )dc 2 u (C 2 ) 2. (4) In the special case of a marginal move on a paralell to the 45 o line in the direction of b (i.e. with dc 2 = dc 1 ) one has dmrs = u (C 1 k)u (C 1 ) u (C 1 )u (C 1 k). (5) dc 1 (dc2 =dc 1 ) u (C 1 k) 2 As a result ( dmrs ) ( u (C 1 ) sign = sign dc 1 (dc2 =dc 1 ) u (C 1 ) u (C 1 k) ). (6) u (C 1 k) It follows that the MRS between C 1 and C 2 increases when one moves from a to b if absolute risk aversion is decreasing in C. This result is easy to interpret. When one becomes richer (because of a rightward move along the 45 o line) the loss of k has under DARA a lower impact on the MRS between C 1 and C 2 essentially because the loss k becomes relatively less important. Under DARA and in Boehm-Bawerk s terminology the lower subjective value of present goods due to the loss of a quantity k of future goods becomes less important when the individual is richer (through a right move along the 45 o line). In geometrical terms under DARA indifference curves become more sloping when one moves to the right. 3. Prudence and Risk vulnerability Instead of considering a sure loss applied to points on the 45 o line, we now assume that for all points on this line a zero mean risk ɛ is added to C 2 4. As a result the intertemporal utility is now U = u(c 1 ) + E[u(C 2 + ɛ)] (7) 4 Notice the similarity with the concept of pain apportionment in the unidimensional model of Eeckhoudt and Schlesinger (2006) where two pains are also introduced, a sure loss and a zero mean risk. 4

5 and the marginal rate of substitution becomes MRS = dc 2 dc 1 = u (C 1 ) E[u (C 2 + ɛ)] Under prudence (u > 0) the MRS evaluated at any C 1 = C 2 becomes smaller than unity. As a result the presence of ɛ reduces the time preference in favor of current consumption. This result is illustrated in figure 2. where the indiffrence curves d dd and e ee are those obtained in the absence of the risk ɛ. Their slope (in absolute value) at d and at e is equal to unity as it results from (2) with β = 1. (8) C2 e E d C2b e D E C2a d e D 45 C1a C1b d C1 Figure 2. The indifference curves D dd and E ee are obtained when the risk ɛ is added to any C 2. On the 45 o line their slopes (in absolute value) at d and e is smaller than unity under prudence (u > 0). Of course again a natural question arises: how does the MRS behave when one moves from d to e? We show that this behavior is determined by the assumption of risk vulnerability. Indeed we now have dmrs = E[u (C 2 + ɛ)]u (C 1 )dc 1 u (C 1 )E[u (C 2 + ɛ)]dc 2 E[u (C 2 + ɛ)] 2 (9) 5

6 so that whenever C 1 = C 2 and dmrs = E[u (C 1 + ɛ)]u (C 1 ) u (C 1 )E[u (C 1 + ɛ)] (10) dc 1 (dc2 =dc 1 ) E[u (C 1 + ɛ)] 2 ( dmrs ) ( u (C 1 ) sign = sign dc 1 (dc2 =dc 1 ) u (C 1 ) E[u (C 1 + ɛ)] ). (11) E[u (C 1 + ɛ) If risk vulnerability prevails, this sign is so that when u is risk vulnerable and when the decision maker is prudent an increase in wealth along the 45 o line reduces the impact that a risk ɛ has on the agent s rate of time preference. A move to the right along the 45 o line brings the indifference curve under the presence of risk closer to that prevailing under certainty. Under risk vulnerability the change in the rate of time preference brought about by the introduction of a zero-mean risk for a prudent agent tends to vanish when one moves rightward along the 45 o line. This is exactly the same property as that observed for the introduction of a sure loss when the decision maker is risk averse and exhibits the DARA property. 4. Conclusion The concept of risk vulnerabilty is important for the analysis of choices in the presence of multiple risks. However its implications for the shape of the utility function are not easily interpreted, which limits its use and diffusion. In this note by using a simple model of intertemporal preferences we have given a geometric interpretation of the notion of risk vulnerability that hopefully can be easily remembered. Besides it was shown that risk vulnerability is related to prudence in the same way as DARA is linked to risk aversion. This proximity between the assumptions of DARA and risk vulnerability becomes besides quite transparent when one compares (6) and (11). References Boehm-Bawerk, E., 1959, Capital and interest. (South Holland Libertarian Press.) Eeckhoudt, L. and H. Schlesinger, 2006, Putting risk in its proper place, American Economic Review 96, Gollier, C. and J.W. Pratt, 1996, Risk vulnerability and the tempering effect of a background risk, Econometrica 64, Gollier, C., 2001, The Economics of Risk and Time. (MIT Press). Olson, M. and M-J. Bailey, 1981, Positive time preference. Journal of Political Economy 89,