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

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)

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

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.

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

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)

Homogenous Agents N agents have same convex utility function and same initial endowment. Economy efficiently allocates Y max 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.

Agent’s Indifference Curve y x x A x B x max y max For each agent i=1 to N

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

Social Indifference Curve Y X Y max X max Two Agent Case Agent 1 holds all X Agent 2 holds all Y Agent 1 Agent 2

Social Indifference Curve Y X Y max X max Five Agent Case

Perfect Competition Allowing N  ∞ while holding Y max and X max 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.

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.

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.

Efficient Allocation: X ∈ (0,X max ] y x Poor Rich

Social Indifference Curve Y X Poor: N P =2 Rich: N R =2

Social Indifference Curve Y X Poor: N P  ∞ Rich: N R  ∞

Social Indifference Curve Y X Y max X max

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.

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.