- Email: [email protected]
A note on the relevance of the q-exponential function in the context of intertemporal choices
ARTICLE IN PRESS
Physica A 364 (2006) 385–388 www.elsevier.com/locate/physa
A note on the relevance of the q-exponential function in the context of intertemporal choices Daniel O. Cajueiro Graduate Programme in Economics, Universidade Cato´lica de Brası´lia, Brazil Received 1 December 2004; received in revised form 15 August 2005 Available online 19 September 2005
Abstract This paper shows that the q-exponential function well known in the deformed algebra inspired in the Tsallis’s nonextensive thermodynamics may be used to model discount functions in intertemporal choices which present the phenomenon known as increasing patience. Moreover, we show that this insight may also be used to provide a measure of the degree of dynamic inconsistency presented in such decisions. r 2005 Elsevier B.V. All rights reserved. Keywords: Behavioral economics; Increasing patience; q-exponential; Tsallis’s thermodynamics
1. Introduction A large part of the behavioral economic literature has been paying attention to a deviation from the expected utility theory known as dynamic inconsistency. Dynamic consistency requires that the ranking of consumption streams be unchanged over time. But, empirical evidence has shown that this is not true in all situations [1–4]. For example, if people are asked to choose between (a1) $1000 in 1 year and (a2) $1050 in 1 year and 1 week or (b1) $1000 today and (b2) $1050 in 1 week, then according to the expected utility theory someone who chooses (a2) in the first situation must choose (b2) in the second situation. However, greater impatience for immediate rewards can make one choose (a2) and (b1). According to [5], the only way for agents to achieve intertemporal consistency is to discount the cash flows from all delayed events by a constant rate-per-time unit. Therefore, in the context of the classical expected theory, the exponential function has played a fundamental role.1 The problem is that with the discounting supported by the exponential function, the phenomena presented in the example above cannot be modelled. For that reason, the hyperbolic discount was introduced [6] as an alternative to the exponential discount model in order to take the phenomena of increasing impatience described above into account. In this paper, I introduce the definition of the q-exponential utility function based on the q-exponential function discount which is an extension of both previous ideas: (1) the concept of the classical utility function based on exponential discount and (2) the concept of the hyperbolic utility function based on the hyperbolic
1
E-mail address: [email protected]. This is obvious, since ex is the solution of differential equation dx=x ¼ c dt, where c is a constant.
0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.08.056
ARTICLE IN PRESS D.O. Cajueiro / Physica A 364 (2006) 385–388
386
discount. Actually, the q-exponential is a very well-known function in the deformed algebra inspired in nonextensive thermodynamics [7] and was very well studied by [8]. This paper is organized as follows. In Section 2, the q-exponential function is introduced. In Section 3, an example of intertemporal choice is presented. Finally, Section 4 presents some conclusions of this work. 2. The q-exponential utility function The q-exponential utility function can be defined as Uðx1 ; x2 ; . . . ; xT Þ ¼
T X
bq ðtÞuðxt Þ ,
(1)
t¼1
where u is an instantaneous utility function, T is a fixed time in the future and bq ðtÞ, the q-exponential discount, is the inverse of the q-exponential function given by bq ðtÞ ¼
1 1 ¼ eq ðatÞ ½1 þ ð1 qÞat1=ð1qÞ
(2)
with q 2 ½0; 1. One should note that when q ! 1 then bq ðtÞ recovers the classical exponential discount. On the other hand, when q ¼ 0 then bq ðtÞ yields the hyperbolic discount [9]. Moreover, the definition of the q-exponential utility function proposed above is quite similar to the definition of a model in this framework that generalizes a large class of hyperbolic functions [10]. q may play a very important role here. It may be defined as a measure of consistency in the intertemporal decisions. If q ! 1 then the intertemporal choices are consistent. On the other hand, if q 2 ½0; 1Þ, then the choices are inconsistent. 3. The buffer-stock problem with q-exponential discount In this paper, the buffer-stock problem2 with q-exponential is solved here in order to provide some insight about the effect of the q-exponential discount in the consumers choices. The buffer-stock problem is a standard problem of a consumer who lives for many periods and chooses optimal current consumption and contingency plans for future consumption to maximize the expected value of his lifetime separable utility function. The only source of uncertainty considered here is the labor income. The markets are considered incomplete in the sense that individuals cannot hedge against this uncertainty by trading contingent claims. Formally, this problem can be summarized in the following way. The consumer chooses a policy fc0 ; c1 ; . . . ; cT g in order to maximize " # T X E0 bq ðtÞuðcðtÞÞ (3) t¼0
subject to xðt þ 1Þ ¼ RðxðtÞ cðtÞÞ þ yðt þ 1Þ ,
(4)
xðtÞ cðtÞX0 ,
(5)
cðtÞX0 ,
(6)
where xðtÞ is the consumer’s wealth in period t (after receiving income and before consuming), R41 is the real return on his savings, cðtÞ is consumption in period t, yðtÞX0 is the wage in period t which is independently and 2
Since the incomes are stationary, independently and identically distributed over time, the name of this problem comes from the idea of a poor farmer in a developing country where assets play the role of buffer stock and the consumer saves and dissaves in order to smooth consumption in face of income uncertainty. This model (without the q-exponential argument) has been useful to explain several puzzles in economics literature. For details, see [11].
ARTICLE IN PRESS D.O. Cajueiro / Physica A 364 (2006) 385–388
387
1 0.95 0.9 0.85
fq (t)
0.8
q=0 q=0.2 q=0.4 q=0.6 q=0.8 q→1
0.75 0.7 0.65 0.6 0.55 0.5
0
5
10
15
20
25 Time
30
35
40
45
50
Fig. 1. The q-exponential discount factor for several values of q and a ¼ 0:6.
identically distributed, E t is the expectation operator conditional on information available at time t, uð:Þ is the one period utility function and T is a nonstochastic time of death.3 While Eq. (4) says that whatever the consumer does not spend is saved, Eq. (5) rules out borrowing. Considering the dynamic programming approach and that this problem has only interior solutions, it is easy to show that4 u0 ðc ðtÞÞ ¼ f q ðtÞRE t ½u0 ðc ðt þ 1ÞÞ ,
(7)
where c ðtÞ is the optimal consumption and f q ðtÞ ¼ bq ðt þ 1Þ=bq ðtÞ is the instantaneous discount factor. The real rate of the return R in Eq. (7) means that for each unit of consumption saved in period t, one receives R units of consumption in t þ 1. Eq. (7) expresses the standard marginal condition for an optimum: 1 u0 ðc ðtÞÞ which is the loss in satisfaction if the investor saves another unit of consumption and ðbq ðt þ 1Þ=bq ðtÞÞRE t ½u0 ðc ðt þ 1ÞÞ is the increase in (discounted, expected) utility he obtains from the extra unit of consumption saved at t þ 1. The investor continues to buy or sell the asset until the marginal loss equals the marginal gain. On the other hand, f q ðtÞ (with qa1) takes an interesting phenomena into account here. It is easy to note from Fig. 1, that the patience of the consumer increases over time. While in the short run, he presents a high appetite for consumption, in the long run this appetite is nothing more than a small fraction of the previous one.5 On the other hand, if q ! 1, the discount factor yields a constant.
3
In this paper, it is considered that T is an arbitrary distant date, but finite and known. Since this problem is not time consistent, it is very difficult (may be impossible) to be solved for the infinite horizon case by means of the dynamic programming approach. 4 Details of the dynamic programming approach may be found in [12]. 5 This kind of behavior has been experimentally proved. For details, see [1–4].
ARTICLE IN PRESS D.O. Cajueiro / Physica A 364 (2006) 385–388
388
4. Conclusions This paper has introduced a very interesting connection between the Tsallis’s thermodynamic statistics and behavioral economics6 by means of the q-exponential function. Moreover, it has been shown that the q-exponential function can be used to take the phenomena known as increasing patience into account and the parameter q can be seen as a measure of consistency in intertemporal choices. Acknowledgements The author would like to express his gratitude to one anonymous referee who made several important suggestions that helped the improvement of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
6
J. Cairnes, M. Van der Pol, J. Econ. Psychology 21 (2000) 191–205. L.R. Keller, E. Strazzera, Examining predictive accuracy among discounting models, J. Risk Uncertainty 24 (2) (2002) 143–160. K.N. Kirby, N.N. Marakovic, Organ. Behav. Human Decision Process. 64 (1995) 22–30. M. Van der Pol, J. Cairnes, J. Econ. Behav. Organ. 49 (2002) 79–96. P. Samuelson, Rev. Econ. Stud. 4 (1937) 155–161. J.E. Mazur, An adjustment procedure for studying delayed reinforcement, in: Quantitative Analysis of Behavior: The Effect to Delay and Intervening Events on Reforcement Value, Erlbaum, Hillsdale, 1987. C. Tsallis, Quı´ mica Nova 17 (1994) 468. E.P. Borges, Physica A 340 (2004) 95–101. C.M. Harvey, Manage. Sci. 32 (1986) 1123–1139. G.F. Loewnstein, D. Prelec, Q. J. Econ. 107 (1992) 573–597. S.P. Zeldes, Q. J. Econ. 104 (1989) 275–298. M.L. Puterman, Markov Decision Processes, Wiley, New York, 1994. C. Tsallis, Chaos, Solitons and Fractals 6 (1995) 539. L. Borland, Phys. Rev. Lett. 89 (2002) 98701. C. Tsallis, C. Anteneodo, L. Borland, R. Osorio, Physica A 324 (2003) 89. I. Matsuba, H. Takahashi, Physica A 319 (2003) 458–468. E.P. Borges, Physica A 334 (2004) 255–266. E.P. Borges, Manifestac- o˜es dinaˆmicas e termodinaˆmicas de sistemas na˜o-extensivos, Tese de Doutorado, Centro Brasileiro de Pesquisas Fı´ sicas, 2004. L. Borland, J.P. Bouchaud, Quant. Finance 4 (2004) 499–514. R. Osorio, L. Borland, C. Tsallis, Nonextensive entropy: interdisciplinary applications, in: C. Tsallis, M. Gell-mann (Eds.), Proceedings Volume in the Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press, Oxford, 2004.
Other interesting connections between the Tsallis’s thermodynamics and economics or finance may be found in [13–20].
Physica A 364 (2006) 385–388 www.elsevier.com/locate/physa
A note on the relevance of the q-exponential function in the context of intertemporal choices Daniel O. Cajueiro Graduate Programme in Economics, Universidade Cato´lica de Brası´lia, Brazil Received 1 December 2004; received in revised form 15 August 2005 Available online 19 September 2005
Abstract This paper shows that the q-exponential function well known in the deformed algebra inspired in the Tsallis’s nonextensive thermodynamics may be used to model discount functions in intertemporal choices which present the phenomenon known as increasing patience. Moreover, we show that this insight may also be used to provide a measure of the degree of dynamic inconsistency presented in such decisions. r 2005 Elsevier B.V. All rights reserved. Keywords: Behavioral economics; Increasing patience; q-exponential; Tsallis’s thermodynamics
1. Introduction A large part of the behavioral economic literature has been paying attention to a deviation from the expected utility theory known as dynamic inconsistency. Dynamic consistency requires that the ranking of consumption streams be unchanged over time. But, empirical evidence has shown that this is not true in all situations [1–4]. For example, if people are asked to choose between (a1) $1000 in 1 year and (a2) $1050 in 1 year and 1 week or (b1) $1000 today and (b2) $1050 in 1 week, then according to the expected utility theory someone who chooses (a2) in the first situation must choose (b2) in the second situation. However, greater impatience for immediate rewards can make one choose (a2) and (b1). According to [5], the only way for agents to achieve intertemporal consistency is to discount the cash flows from all delayed events by a constant rate-per-time unit. Therefore, in the context of the classical expected theory, the exponential function has played a fundamental role.1 The problem is that with the discounting supported by the exponential function, the phenomena presented in the example above cannot be modelled. For that reason, the hyperbolic discount was introduced [6] as an alternative to the exponential discount model in order to take the phenomena of increasing impatience described above into account. In this paper, I introduce the definition of the q-exponential utility function based on the q-exponential function discount which is an extension of both previous ideas: (1) the concept of the classical utility function based on exponential discount and (2) the concept of the hyperbolic utility function based on the hyperbolic
1
E-mail address: [email protected]. This is obvious, since ex is the solution of differential equation dx=x ¼ c dt, where c is a constant.
0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.08.056
ARTICLE IN PRESS D.O. Cajueiro / Physica A 364 (2006) 385–388
386
discount. Actually, the q-exponential is a very well-known function in the deformed algebra inspired in nonextensive thermodynamics [7] and was very well studied by [8]. This paper is organized as follows. In Section 2, the q-exponential function is introduced. In Section 3, an example of intertemporal choice is presented. Finally, Section 4 presents some conclusions of this work. 2. The q-exponential utility function The q-exponential utility function can be defined as Uðx1 ; x2 ; . . . ; xT Þ ¼
T X
bq ðtÞuðxt Þ ,
(1)
t¼1
where u is an instantaneous utility function, T is a fixed time in the future and bq ðtÞ, the q-exponential discount, is the inverse of the q-exponential function given by bq ðtÞ ¼
1 1 ¼ eq ðatÞ ½1 þ ð1 qÞat1=ð1qÞ
(2)
with q 2 ½0; 1. One should note that when q ! 1 then bq ðtÞ recovers the classical exponential discount. On the other hand, when q ¼ 0 then bq ðtÞ yields the hyperbolic discount [9]. Moreover, the definition of the q-exponential utility function proposed above is quite similar to the definition of a model in this framework that generalizes a large class of hyperbolic functions [10]. q may play a very important role here. It may be defined as a measure of consistency in the intertemporal decisions. If q ! 1 then the intertemporal choices are consistent. On the other hand, if q 2 ½0; 1Þ, then the choices are inconsistent. 3. The buffer-stock problem with q-exponential discount In this paper, the buffer-stock problem2 with q-exponential is solved here in order to provide some insight about the effect of the q-exponential discount in the consumers choices. The buffer-stock problem is a standard problem of a consumer who lives for many periods and chooses optimal current consumption and contingency plans for future consumption to maximize the expected value of his lifetime separable utility function. The only source of uncertainty considered here is the labor income. The markets are considered incomplete in the sense that individuals cannot hedge against this uncertainty by trading contingent claims. Formally, this problem can be summarized in the following way. The consumer chooses a policy fc0 ; c1 ; . . . ; cT g in order to maximize " # T X E0 bq ðtÞuðcðtÞÞ (3) t¼0
subject to xðt þ 1Þ ¼ RðxðtÞ cðtÞÞ þ yðt þ 1Þ ,
(4)
xðtÞ cðtÞX0 ,
(5)
cðtÞX0 ,
(6)
where xðtÞ is the consumer’s wealth in period t (after receiving income and before consuming), R41 is the real return on his savings, cðtÞ is consumption in period t, yðtÞX0 is the wage in period t which is independently and 2
Since the incomes are stationary, independently and identically distributed over time, the name of this problem comes from the idea of a poor farmer in a developing country where assets play the role of buffer stock and the consumer saves and dissaves in order to smooth consumption in face of income uncertainty. This model (without the q-exponential argument) has been useful to explain several puzzles in economics literature. For details, see [11].
ARTICLE IN PRESS D.O. Cajueiro / Physica A 364 (2006) 385–388
387
1 0.95 0.9 0.85
fq (t)
0.8
q=0 q=0.2 q=0.4 q=0.6 q=0.8 q→1
0.75 0.7 0.65 0.6 0.55 0.5
0
5
10
15
20
25 Time
30
35
40
45
50
Fig. 1. The q-exponential discount factor for several values of q and a ¼ 0:6.
identically distributed, E t is the expectation operator conditional on information available at time t, uð:Þ is the one period utility function and T is a nonstochastic time of death.3 While Eq. (4) says that whatever the consumer does not spend is saved, Eq. (5) rules out borrowing. Considering the dynamic programming approach and that this problem has only interior solutions, it is easy to show that4 u0 ðc ðtÞÞ ¼ f q ðtÞRE t ½u0 ðc ðt þ 1ÞÞ ,
(7)
where c ðtÞ is the optimal consumption and f q ðtÞ ¼ bq ðt þ 1Þ=bq ðtÞ is the instantaneous discount factor. The real rate of the return R in Eq. (7) means that for each unit of consumption saved in period t, one receives R units of consumption in t þ 1. Eq. (7) expresses the standard marginal condition for an optimum: 1 u0 ðc ðtÞÞ which is the loss in satisfaction if the investor saves another unit of consumption and ðbq ðt þ 1Þ=bq ðtÞÞRE t ½u0 ðc ðt þ 1ÞÞ is the increase in (discounted, expected) utility he obtains from the extra unit of consumption saved at t þ 1. The investor continues to buy or sell the asset until the marginal loss equals the marginal gain. On the other hand, f q ðtÞ (with qa1) takes an interesting phenomena into account here. It is easy to note from Fig. 1, that the patience of the consumer increases over time. While in the short run, he presents a high appetite for consumption, in the long run this appetite is nothing more than a small fraction of the previous one.5 On the other hand, if q ! 1, the discount factor yields a constant.
3
In this paper, it is considered that T is an arbitrary distant date, but finite and known. Since this problem is not time consistent, it is very difficult (may be impossible) to be solved for the infinite horizon case by means of the dynamic programming approach. 4 Details of the dynamic programming approach may be found in [12]. 5 This kind of behavior has been experimentally proved. For details, see [1–4].
ARTICLE IN PRESS D.O. Cajueiro / Physica A 364 (2006) 385–388
388
4. Conclusions This paper has introduced a very interesting connection between the Tsallis’s thermodynamic statistics and behavioral economics6 by means of the q-exponential function. Moreover, it has been shown that the q-exponential function can be used to take the phenomena known as increasing patience into account and the parameter q can be seen as a measure of consistency in intertemporal choices. Acknowledgements The author would like to express his gratitude to one anonymous referee who made several important suggestions that helped the improvement of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
6
J. Cairnes, M. Van der Pol, J. Econ. Psychology 21 (2000) 191–205. L.R. Keller, E. Strazzera, Examining predictive accuracy among discounting models, J. Risk Uncertainty 24 (2) (2002) 143–160. K.N. Kirby, N.N. Marakovic, Organ. Behav. Human Decision Process. 64 (1995) 22–30. M. Van der Pol, J. Cairnes, J. Econ. Behav. Organ. 49 (2002) 79–96. P. Samuelson, Rev. Econ. Stud. 4 (1937) 155–161. J.E. Mazur, An adjustment procedure for studying delayed reinforcement, in: Quantitative Analysis of Behavior: The Effect to Delay and Intervening Events on Reforcement Value, Erlbaum, Hillsdale, 1987. C. Tsallis, Quı´ mica Nova 17 (1994) 468. E.P. Borges, Physica A 340 (2004) 95–101. C.M. Harvey, Manage. Sci. 32 (1986) 1123–1139. G.F. Loewnstein, D. Prelec, Q. J. Econ. 107 (1992) 573–597. S.P. Zeldes, Q. J. Econ. 104 (1989) 275–298. M.L. Puterman, Markov Decision Processes, Wiley, New York, 1994. C. Tsallis, Chaos, Solitons and Fractals 6 (1995) 539. L. Borland, Phys. Rev. Lett. 89 (2002) 98701. C. Tsallis, C. Anteneodo, L. Borland, R. Osorio, Physica A 324 (2003) 89. I. Matsuba, H. Takahashi, Physica A 319 (2003) 458–468. E.P. Borges, Physica A 334 (2004) 255–266. E.P. Borges, Manifestac- o˜es dinaˆmicas e termodinaˆmicas de sistemas na˜o-extensivos, Tese de Doutorado, Centro Brasileiro de Pesquisas Fı´ sicas, 2004. L. Borland, J.P. Bouchaud, Quant. Finance 4 (2004) 499–514. R. Osorio, L. Borland, C. Tsallis, Nonextensive entropy: interdisciplinary applications, in: C. Tsallis, M. Gell-mann (Eds.), Proceedings Volume in the Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press, Oxford, 2004.
Other interesting connections between the Tsallis’s thermodynamics and economics or finance may be found in [13–20].