This note provides an analytical confirmation and a refinement of [Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68 (62), 247–274 (March)] numerical findings regarding the characterization of optimality in the shopping-time model presented in that paper. The original numerical analysis concludes that a coefficient of risk aversion (σ) greater than 0.01 is sufficient for optimality. Here we use Arrow’s sufficiency theorem to confirm this result and, more importantly, to show without more calculations how changes in parameters can affect it.

Several works in the shopping-time and in the human-capital literature, due to the non-concavity of the underlying Hamiltonian, use first-order conditions in dynamic optimization to characterize necessity, but not sufficiency, in intertemporal problems. This note selects some works in these two areas and shows that optimality can be characterized, and some results quantitatively improved, by means of an application of Arrow’s [Arrow, K. J., 1968. Applications of control theory to economic growth. In: Dantzig, G.B., Veinott Jr., A.F. (Eds.), Mathematics of the Decisions Sciences. American Mathematical Society, Providence, RI] sufficiency theorem.

Multidimensional measures of the welfare costs of inflation have been employed in the literature without an explicit concern of how the demand for the respective monetary assets are generated and without an investigation of the respective integrability conditions. This note establishes conditions under which such welfare measures are well defined.

Two papers published in this journal (Jantti, 1994, and Mocan, 1999), among others, find empirical evidence that “increases in structural unemployment have a substantial aggravating impact on income inequality.” The main point of this work is to show that standard job-search models can help us understand this empirical regularity. As a byproduct of the analysis, the paper also provides a closed-form general expression that enables direct calculation of the Gini coefficient of wage-income inequality as a function of any arbitrary initial distribution of wage offers. Three numerical examples illustrate the results.

This paper investigates which properties money-demand functions must satisfy so that they are consistent with Lucas’s [Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68, 247–274] versions of the Sidrauski and the shopping-time models. We conclude that shopping-time-integrable money-demand functions are necessarily also Sidrauski-integrable, but that the converse is not necessarily true, unless a boundedness assumption on the nominal interest rate is made. Both the log–log with an interest-rate elasticity greater than or equal to one and the semi-log money demands may serve as counterexamples. All the models and results are also extended to the case in which there are several assets in the economy performing monetary functions.