## Optimal Public Expenditure with Inefficient Unemployment

Pascal Michaillat, Emmanuel Saez

March 2017

March 2017

This paper 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 gap between the current and efficient rates of unemployment). Such optimal "stimulus spending" is described by a formula expressed with estimable sufficient statistics: the unemployment gap, the unemployment multiplier (the effect of public expenditures on unemployment), and the elasticity of substitution between public and private consumption. Using the formula, we obtain four results. (1) When the unemployment multiplier is positive and unemployment is inefficiently high, 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 increasing in the elasticity of substitution between public and private consumption. In particular, it is zero when extra public goods are useless, and it completely fills the unemployment gap when extra public goods are as valuable as extra private goods. (4) The formula for optimal stimulus spending remains the same whether taxes are distortionary or not.