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A.F. Newman / Journal of Economic Theory 137 (2007) 11 – 26 13
2. Model There is a single good which is produced from labor according to the stochastic production function f (L, ), where L is the amount of labor hired (or worker effort) and is a random variable indexing the state of the world, representing the risks that inhere in the production and sale of goods. Raising raises both the total and marginal products of labor, that is, f (L, ) and f1(L, ) are increasing in (for instance, is a multiplicative noise). These risks are independent and identically distributed across firms. The production function satisfies the standard properties: f (0, ) = 0, f11 < 0 < f1, limL→0 f1(L, ) = ∞, limL→∞ f1(L, ) = 0. Each firm also requires the effort e of an entrepreneur. We could suppose that the entrepreneur expends this effort in coordinating production, marketing output, finding suppliers, etc. It enters the production technology only through the distribution of . There is a continuum of agents, indexed by the unit interval. In order for the self-selection underlying the Knightian story to have any relevance, agents must differ in some way. One way to do this is to focus on preferences, allowing, for instance, that all the variation be indexed by some utility parameter; this is the tack taken by Kihlstrom and Laffont [12]. Here we shall follow a special case of this approach, which is to assume that agents have identical preferences and vary instead in some other characteristic, namely initial wealth. Since standard assumptions, such as decreasing absolute risk aversion, then place restrictions on the relationship between risk attitudes and marginal incentives, this approach has the advantage of generating additional testable implications almost for free. Thus, let all agents have identical preferences represented by the von Neumann–Morgenstern utility u(y)−e, where y0 is realized income and e is the effort expended. The income utility u(·) has the usual properties: u > 0 > u with decreasing absolute risk aversion (i.e. uu > (u)2). In order to ensure that only risk sharing is at issue here, it is convenient to assume that u(·) is unbounded below. 2 Agents vary in the amount of initial assets or wealth a, the distribution of which is exogenous. An agent may choose one of two occupations in which to expend his effort: either he can become a worker, earning a sure wage w (so that his income is a +w), or he can be an entrepreneur, hiring L units of labor and earning the residual profit from production, which we denote y(). Note that in general the amount of labor that an entrepreneur wishes to hire could depend not only on the wage, but also on his characteristics, which in this case are limited to his wealth a. Make the following assumptions on L, , and e: entrepreneurs may hire any nonnegative amount of labor, is drawn from a finite set, and e is a binary variable taking values in {0, 1} (the last assumption allows us to abstract from issues having to do with how effort levels might also change with parameters of the model; the independence of the effort cost function from occupation facilitates focus on risk-bearing as the determinant of occupational choice). Denote by q() the probability of when e = 1 and p() when e = 0; q(·) stochastically dominates p(·) and the two distributions have common support.We assume throughout that it is optimal for both workers and entrepreneurs to set their effort levels equal to 1. 2 This is a standard condition (see e.g. [8]) that ensures that no nonnegativity constraints on income bind at an optimum, which in turn purifies the model of any surplus extraction effect (in the usual formulation of the principal–agent model where the principal offers the contract, it guarantees that the agent’s participation constraint binds). It also effectively lets us ignore financing issues (i.e. we assume entrepreneurs always have enough income ex post to pay their workers; with utility unbounded below, this will never be a problem), so that we can focus purely on the role of risk aversion in the occupational choice. |
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