ONLINE APPENDIX Constrained Efficiency in a Human Capital Model

This document contains supplemental materials for “Constrained Efficiency in a Human Capital Model ”. We discuss the implementation of the constrained efficient allocation, formally analyze the effects of consumption poor household’s relative income sources on pecuniary externalities, provide a further sensitivity analysis, and discuss the effects of the time investment model and the effects of different market structure. We also provide the proof of propositions in the main text and formally derive the planner’s first-order conditions in an infinite horizon model. ∗Correspondence: Department of Economics, University of Rochester, 280 Hutchison Road, Rochester, NY 14627. Email: [email protected] A Implementation of the constrained efficient allocation In this section, we discuss how we can implement the constrained efficient allocation. The constrained efficient allocation can be attained if the government directly modifies each consumer’s savings and human capital investments. We can also implement the constrained efficient allocation as a competitive equilibrium using a tax(subsidy)-transfer(lump-sum tax) system that is history dependent and induces no reallocation of income across individuals or realization of idiosyncratic shocks. In the two-period model with additive idiosyncratic shock, history-dependent taxes and transfers are represented by the initial heterogeneity-dependent taxes and transfers. The tax(subsidy)transfer(lump-sum tax) system is then completely characterized by the linear proportional tax rates of capital income and labor income (τk(k, h), τl(k, h)) and transfers (T (k, h, e)), which are functions of initial wealth and human capital (k, h) and idiosyncratic shock realization e. Proposition 5. There exists a triple of history-dependent capital income tax rates, labor income tax rates, and income transfers (τk(k, h), τl(k, h), T (k, h, e)) that implements the constrained planner’s allocation. Proof By setting capital income tax rates and labor income tax rates in the following way, the Euler equation of the competitive equilibrium and the constrained planner’s problem become equivalent: τk(k, h) = − ∆k (Fk − δ)Es[u′(c1)|k, h] (1) τl(k, h) = − ∆h fHE[u′(c1)|k, h] (2) Then, the following transfer function, which imposes no income transfer across agents, implements the constrained efficient allocation. T (k, h, e) = τkrk(k, h) + τlw[h(k, h) + e] „ Or alternatively, we can implement the same allocation using linear tax rates1 of capital income and linear subsidy rates of human capital (τk(k, h), s(k, h)), lumpsum taxes in period 0 T0(k, h), and lumpsum transfers in period 1 T1(k, h), such that τk(k, h) is set as in (1) and s(k, h) is set to s(k, h) = βgxh∆h u′(c0(c0(k, h))) . 1That is, with subsidy, the cost of investing xh in human capital is (1 − s(k, h))xh(k, h)