This appendix contains additional results on using scanner data to estimate inflation rates at the household level. There are three sections. Section 1 shows cross-sectional distributions of Fisher and Paasche inflation rates. Section 2 shows the evolution over time of measures of dispersion of Fisher and Paasche inflation rates. Section 3 shows cross-sectional distributions of two-year inflation rates measured with Fisher and Paasche indexes.

We use scanner data to estimate inflation rates at the household level. Households' inflation rates are remarkably heterogeneous, with an interquartile range of 6.2 to 9.0 percentage points on an annual basis. Most of the heterogeneity comes not from variation in broadly defined consumption bundles but from variation in prices paid for the same types of goods - a source of variation that previous research has not measured. The entire distribution of household inflation rates shifts in parallel with aggregate inflation. Deviations from aggregate inflation exhibit only slightly negative serial correlation within each household over time, implying that the difference between a household's price level and the aggregate price level is persistent. Together, the large cross-sectional dispersion and low serial correlation of household-level inflation rates mean that almost all of the variability in a household's inflation rate over time comes from variability in household-level prices relative to average prices for the same goods, not from variability in the aggregate inflation rate. We provide a characterization of the stochastic process for household inflation that can be used to calibrate models of household decisions.

We use an extended Barro-Becker model of endogenous fertility, in which parents are heterogeneous in their labor productivity, to study the efficient degree of consumption inequality in the long run. In our environment a utilitarian planner allows for consumption inequality even when labor productivity is public information. We show that adding private information does not alter this result. We also show that the informationally constrained optimal insurance contract has a resetting property—whenever a family line experiences the highest shock, the continuation utility of each child is reset to a (high) level that is independent of history. This implies that there is a non-trivial, stationary distribution over continuation utilities and there is no mass at misery. The novelty of our approach is that the no-immiseration result is achieved without requiring that the objectives of the planner and the private agents disagree. Because there is no discrepancy between planner and private agents' objectives, the policy implications for implementation of the efficient allocation differ from previous results in the literature. Two examples of these are: 1) estate taxes are positive and 2) there are positive taxes on family size.

This paper studies optimal taxation of earnings when the degree of tax progressivity is allowed to vary with age. The setting is an overlapping-generations model that incorporates irreversible skill investment, flexible labor supply, ex-ante heterogeneity in the disutility of work and the cost of skill acquisition, partially insurable wage risk, and a life cycle productivity profile. An analytically tractable version of the model without intertemporal trade is used to characterize and quantify the salient trade-offs in tax design. The key results are that progressivity should be U-shaped in age and that the average marginal tax rate should be increasing and concave in age. These findings are confirmed in a version of the model with borrowing and saving that we solve numerically.

What shapes the optimal degree of progressivity of the tax and transfer system? On the one hand, a progressive tax system can counteract inequality in initial conditions and substitute for imperfect private insurance against idiosyncratic earnings risk. On the other hand, progressivity reduces incentives to work and to invest in skills, distortions that are especially costly when the government must finance public goods. We develop a tractable equilibrium model that features all of these trade-offs. The analytical expressions we derive for social welfare deliver a transparent understanding of how preference, technology, and market structure parameters influence the optimal degree of progressivity. A calibration for the U.S. economy indicates that endogenous skill investment, flexible labor supply, and the desire to finance government purchases play quantitatively similar roles in limiting optimal progressivity. In a version of the model where poverty constrains skill investment, optimal progressivity is close to the U.S. value. An empirical analysis on cross-country data offers support to the theory.

There is now an extensive literature regarding the efficient design of incentive mechanisms in dynamic environments. In this literature, there are no exogenous links across time periods because either privately observed shocks are assumed time independent or past private actions have no influence on the realizations of current variables. The absence of exogenous links across time periods ensures that preferences over continuation contracts are common knowledge, making the definition of incentive compatible contracts at a point in time a simple matter. In this paper, we present general recursive methods to handle environments where privately observed variables are linked over time. We show that incentive compatible contracts are implemented recursively with a threat keeping constraint in addition to the usual temporary incentive compatibility conditions.