This paper proposes a search-and-matching model of unemployment in which jobs are rationed: the labor market does not clear in the absence of matching frictions. This job shortage arises in an economic equilibrium from the combination of some wage rigidity and diminishing marginal returns to labor. In recessions, job rationing is acute, driving the rise in unemployment, whereas matching frictions contribute little to unemployment. Intuitively in recessions, jobs are lacking, the labor market is slack, and recruiting is easy and inexpensive, so matching frictions do not matter much. In a calibrated model, cyclical fluctuations in the composition of unemployment are large.

This article proposes a theory of optimal public expenditure when unemployment is inefficient. The theory is based on a matching model. Optimal public expenditure deviates from the Samuelson rule to reduce the unemployment gap (the difference between current and efficient unemployment rates). Such optimal “stimulus spending” is described by a formula expressed with three sufficient statistics: the unemployment gap, the unemployment multiplier (the decrease in unemployment achieved by increasing public expenditure), and the elasticity of substitution between public and private consumption. When unemployment is inefficiently high and the multiplier is positive, the formula yields the following results. (1) Optimal stimulus spending is positive and increasing in the unemployment gap. (2) Optimal stimulus spending is zero for a zero multiplier, increasing in the multiplier for small multipliers, largest for a moderate multiplier, and decreasing in the multiplier beyond that. (3) Optimal stimulus spending is zero if extra public goods have no value, it becomes larger as the elasticity of substitution increases, and it completely fills the unemployment gap if extra public goods are as valuable as extra private goods.