# American Institute of Mathematical Sciences

January  2011, 7(1): 139-156. doi: 10.3934/jimo.2011.7.139

## Optimal consumption and investment under irrational beliefs

 1 School of Business and Management, Hong Kong University of Science and Technology, Hong Kong, China 2 School of Economics and Management, Tsinghua University, Beijing, China

Received  March 2010 Revised  October 2010 Published  January 2011

In this paper, we study how irrationality affects the investor's consumption and investment decisions. We build a continuous-time financial model, where an irrational investor determines his consumption and investment according to an exogenous price process. The main results are as follows. First, compared with a rational investor, an optimistic irrational investor tends to consume more, while a pessimistic irrational investor tends to consume less. Second, the more irrational the investor, the more volatile his consumption. Third, the extremely irrational investor can get more ex ante expected utility than his rational counterpart, no matter he is optimistic or pessimistic.
Citation: Lei Sun, Lihong Zhang. Optimal consumption and investment under irrational beliefs. Journal of Industrial and Management Optimization, 2011, 7 (1) : 139-156. doi: 10.3934/jimo.2011.7.139
##### References:
 [1] Armen A. Alchian, Uncertainty, evolution, and economic theory, Journal of Political Economy, 58 (1950), 211-221. doi: 10.1086/256940. [2] Fisher Black, Noise, Journal of Finance, 41 (1986), 529-543. doi: 10.2307/2328481. [3] Fisher Black and Myron Scholes, The valuation of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [4] Lawrence Blume and David Easley, If you are so smart, why aren't you rich? Belief selection in complete and incomplete markets, Econometrica, 74 (2006), 929-966. doi: 10.1111/j.1468-0262.2006.00691.x. [5] J. Bradford De Long, Andrei Shleifer, Lawrence H. Summers and Robert J. Waldman, The survival of noise traders in financial markets, Journal of Business, 64 (1991), 1-19. [6] Yuri Fedyk and Johan Walden, High-speed natural selection in financial markets with large state spaces, Working paper, Hass School of Business, UC Berkeley, 2007. [7] Milton Friedman, "Essays in Positive Economics," University of Chicago Press, Chicago, 1953. [8] Lars P. Hansen and Scott Richard, The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models, Econometrica, 55 (1987), 587-613. doi: 10.2307/1913601. [9] John M. Harrison and David M. Kreps, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 20 (1979), 381-408. doi: 10.1016/0022-0531(79)90043-7. [10] David Hirshleifer, Avanidhar Subrahmanyam and Sheridan Titman, Feedback and the success of irrational investors, Journal of Financial Economics, 81 (2006), 311-338. doi: 10.1016/j.jfineco.2005.05.006. [11] Ioannis Karatzas and Steven E. Shreve, "Methods of Mathematical Finance," Springer-Verlag, New York, 1998. [12] Leonid Kogan, Stephen A. Ross, Jiang Wang and Mark M. Westerfield, The price impact and survival of irrational traders, Journal of Finance, 61 (2006), 195-229. doi: 10.1111/j.1540-6261.2006.00834.x. [13] John Lintner, The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47 (1965), 13-37. doi: 10.2307/1924119. [14] Robert C. Merton, An intertemporal capital asset pricing model, Econometrica, 41 (1973), 867-887. doi: 10.2307/1913811. [15] Stephen A. Ross, The arbitrage theory of capital asset pricing, Journal of Economic Theory, 13 (1976), 341-360. doi: 10.1016/0022-0531(76)90046-6. [16] Alvaro Sandroni, Do markets favor agents able to make accurate predictions?, Econometrica, 68 (2000), 1303-1341. doi: 10.1111/1468-0262.00163. [17] William Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. doi: 10.2307/2977928.

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##### References:
 [1] Armen A. Alchian, Uncertainty, evolution, and economic theory, Journal of Political Economy, 58 (1950), 211-221. doi: 10.1086/256940. [2] Fisher Black, Noise, Journal of Finance, 41 (1986), 529-543. doi: 10.2307/2328481. [3] Fisher Black and Myron Scholes, The valuation of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [4] Lawrence Blume and David Easley, If you are so smart, why aren't you rich? Belief selection in complete and incomplete markets, Econometrica, 74 (2006), 929-966. doi: 10.1111/j.1468-0262.2006.00691.x. [5] J. Bradford De Long, Andrei Shleifer, Lawrence H. Summers and Robert J. Waldman, The survival of noise traders in financial markets, Journal of Business, 64 (1991), 1-19. [6] Yuri Fedyk and Johan Walden, High-speed natural selection in financial markets with large state spaces, Working paper, Hass School of Business, UC Berkeley, 2007. [7] Milton Friedman, "Essays in Positive Economics," University of Chicago Press, Chicago, 1953. [8] Lars P. Hansen and Scott Richard, The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models, Econometrica, 55 (1987), 587-613. doi: 10.2307/1913601. [9] John M. Harrison and David M. Kreps, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 20 (1979), 381-408. doi: 10.1016/0022-0531(79)90043-7. [10] David Hirshleifer, Avanidhar Subrahmanyam and Sheridan Titman, Feedback and the success of irrational investors, Journal of Financial Economics, 81 (2006), 311-338. doi: 10.1016/j.jfineco.2005.05.006. [11] Ioannis Karatzas and Steven E. Shreve, "Methods of Mathematical Finance," Springer-Verlag, New York, 1998. [12] Leonid Kogan, Stephen A. Ross, Jiang Wang and Mark M. Westerfield, The price impact and survival of irrational traders, Journal of Finance, 61 (2006), 195-229. doi: 10.1111/j.1540-6261.2006.00834.x. [13] John Lintner, The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47 (1965), 13-37. doi: 10.2307/1924119. [14] Robert C. Merton, An intertemporal capital asset pricing model, Econometrica, 41 (1973), 867-887. doi: 10.2307/1913811. [15] Stephen A. Ross, The arbitrage theory of capital asset pricing, Journal of Economic Theory, 13 (1976), 341-360. doi: 10.1016/0022-0531(76)90046-6. [16] Alvaro Sandroni, Do markets favor agents able to make accurate predictions?, Econometrica, 68 (2000), 1303-1341. doi: 10.1111/1468-0262.00163. [17] William Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. doi: 10.2307/2977928.
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