Equilibrium Risk Premia for Risk Seekers

1. Equilibrium Risk Premia for Risk Seekers Douglas W. Blackburn Andrey D. Ukhov Indiana University

2. Motivation and Background • Risk Seeking Behavior • Friedman and Savage (1948) • Kahneman and Tversky (1979) • Green and Rydqvist (1997,1999) • Jackwerth (2000) • Risk Premium Puzzle • Mehra and Prescott (1985)

3. Research Question • Can individuals exhibit risk seeking behavior while at the same time exist in an economy that demands a risk premium?

4. Results • Yes! • An economy of homogeneous risk seekers, under perfect competition, will exhibit risk neutral behavior. • If agents’ wealth is distributed over an interval, then the economy’s indifference curve is strictly convex and differentiable.

5. Results • For every risk averse economy there exists a supporting economy comprised entirely of risk seekers that replicates this economy.

6. Our Economy • Utility functions are convex and time separable. • Individuals are risk seekers. • Concave indifference curves. • Individuals choose between the X-good today and the Y-good tomorrow. • There is a fixed quantity of X and Y. • Perfect Competition – Aumann (1964)

7. Homogenous Agents • N agents have same convex utility function and same initial endowment. • Economy efficiently allocates Ymax of Y and no X to all investors. • Strategy – Trade the X good for the maximum amount of Y possible while maintaining each individual’s current utility.

8. Agent’s Indifference Curve y ymax For each agent i=1 to N x A x B x xmax

9. Efficient Allocation: X∈(0,Xmax] N-1 agents hold all X or all Y. One agent holds both X and Y.

10. Social Indifference Curve Two Agent Case Y Ymax Agent 1 Agent 1 holds all X Agent 2 holds all Y Agent 2 Xmax X

11. Social Indifference Curve Five Agent Case Y Ymax Xmax X

12. Perfect Competition • Allowing N ∞ while holding Ymax and Xmax constant: • Each agent’s initial endowment of Y becomes smaller. • The “humps” of each agent’s indifference curve become arbitrarily small. • The social indifference curve converges to a straight line – risk neutrality.

13. Heterogeneous Agents • All agents have the same utility function • Agents are divided into two wealth classes – the rich and the poor. • The rich are initially endowed with a larger quantity of the Y-good than the poor.

14. Efficient Allocation • Two Cases: • Indifference Curves Have Same Curvature. • Then rates of substitution are the same across both wealth classes. • Indifference Curves Curve At Different Rates. • Rates of substitution are not the same. • Allocate the X good to the wealth class with the greatest rate of substitution.

15. Efficient Allocation: X∈(0,Xmax] y Rich Poor x

16. Social Indifference Curve Y Poor: NP=2 Rich: NR=2 X

17. Social Indifference Curve Y Poor: NP ∞ Rich: NR ∞ X

18. Social Indifference Curve Y Ymax Xmax X

19. Risk Averse Economy • Suppose the economy is risk averse. • Social indifference curve is convex and differentiable. • By following our line of reasoning backwards, we can build an economy of risk seekers, with a particular wealth distribution, that replicates the risk averse economy.

20. Conclusions and Implications • An economy of risk seekers can, in the aggregate, demand a risk premium. • The distribution of wealth and the budget constraint may be of same importance as the individual’s utility function. • Caution must be taken when making implications about individuals using aggregate data.