Engaging with Discreteness: Gambling, Under-Participation, and Lotteries in Economies with Indivisible Goods

We study an endowment economy with heterogeneous agents and two complementary consumption goods, one of which is indivisible. Although agents have standard concave preferences, the indivisibility gives rise to Friedman-Savage convexity in indirect utility. Agents care about relative, not just absolute, wealth, and about the magnitude of wealth differences as well as rankings. While agents dislike small risks, they like large ones. Poorer agents exhibit a preference for lottery-like assets with negative expected returns and also invest a smaller share of wealth than richer agents in assets with positive expected returns. If the difference in wealth is large, poor agents play a lottery among themselves with a single winner. In consequence, richer agents have a higher expected return on wealth and inequality is expected to increase. While initial inequality affects investment choices, it has only a minor effect on ultimate inequality. Therefore, standard prescriptions for reducing inequality may have little effect.