Two Additions to Lucas's 'Inflation and Welfare'

Abstract

This work adds to Lucas (2000) by providing analytical solutions to two problems that are solved only numerically by the author. The …rst part uses a theorem in control theory (Arrow's su¢ ciency theorem) to provide su¢ ciency conditions to characterize the optimum in a shopping-time problem where the value function need not be concave. In the original paper the optimality of the …rst-order condition is characterized only by means of a numerical analysis. The second part of the paper provides a closed-form solution to the generalequilibrium expression of the welfare costs of in ‡ation when the money demand is double logarithmic. This closed-form solution allows for the precise calculation of the di¤erence between the general-equilibrium and Bailey's partial-equilibrium estimates of the welfare losses due to in ‡ation. Again, in Lucas's original paper, the solution to the general-equilibrium-case underlying nonlinear di¤erential equation is done only numerically, and the posterior assertion that the generalequilibrium welfare …gures cannot be distinguished from those derived using Bailey's formula rely only on numerical simulations as well.