Constrained Inefficiency and Optimal Taxation with Uninsurable Risks

• Author: ATSUSHI KAJII,TOMOYUKI NAKAJIMA,PIERO GOTTARDI
• Date: 01 February 2016
• Published date: 01 February 2016
• DOI: http://doi.org/10.1111/jpet.12135

CONSTRAINED INEFFICIENCY AND OPTIMAL TAXATION

WITH UNINSURABLE RISKS

PIERO GOTTARDI

European University Institute and Universit`

a Ca’ Foscari

ATSUSHI KAJII

Kyoto University and Singapore Management University

TOMOYUKI NAKAJIMA

Kyoto University and CIGS

Abstract

When individuals’ labor and capital income are subject to uninsurable

idiosyncratic risks, should capital and labor be taxed, and if so how? In

a two-period general equilibrium model with production, we derive a

decomposition formula of the welfare effects of these taxes into insur-

ance and distribution effects. This allows us to determine how the sign

of the optimal taxes on capital and labor depend on the nature of the

shocks and the degree of heterogeneity among consumers’ income, as

well as on the way in which the tax revenue is used to provide lump-sum

transfers to consumers. When shocks affect primarily labor income and

heterogeneity is small, the optimal tax on capital is positive. However,

in other cases a negative tax on capital is welfare-improving.

1. Introduction

The main objective of this paper is to investigate the welfare effects of investment and

labor income taxes in a two-period production economy with uninsurable background

risk. More precisely, we examine whether the introduction of linear, distortionary

taxes or subsidies on labor income and/or on the returns from savings are welfare-

improving, and what is then the optimal sign of such taxes. This amounts to studying

the Ramsey problem in a general equilibrium setup. We depart however from most of the

literature on the subject1for the fact that we consider an environment with no public

1See Chari and Kehoe (1999) for a survey.

Piero Gottardi, Department of Economics, European University Institute, Villa San Paolo, Via della

Piazzuola 43, 50133 Florence, Italy (piero.gottardi@EUI.eu). Atsushi Kajii, Institute of Economic Re-

search, Kyoto University, Kyoto 606-8501, Japan (kajii@kier.kyoto-u.ac.jp). Tomoyuki Nakajima, Insti-

tute of Economic Research, Kyoto University, Kyoto 606-8501, Japan (tomoyuki.nakajima@gmail.com).

We thank Naoki Yoshihara for his valuable comments. Gottardi acknowledges financial support

from the EUI; Kajii from the Inamori Foundation and JSPS Grant-in-Aid for Scientific Research No.

(S)20223001; and Nakajima from the Murata Science Foundation and the JSPS Grant-in-Aid for Scien-

tific Research No. 23530218. Nakajima thanks the hospitality of the Federal Reserve Bank of Atlanta.

All errors and omissions are ours

Received October 5, 2014; Accepted October 6, 2014.

C2014 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (1), 2016, pp. 1–28.

1

2 Journal of Public Economic Theory

expenditure, where there is no need to raise tax revenue. Nonetheless, optimal taxes

are typically nonzero; even distortionary taxes can improve the allocation of risk in the

face of incomplete markets. Then the question is which production factor should be

taxed: we want to identify the economic properties which determine the signs of the

optimal taxes on production factors.

A possible answer may come from the following consequence of the agents’ precau-

tionary motive for saving: under uninsurable risk, this motive implies that savings and

hence capital accumulation will be higher compared to the situation where markets are

complete. This point was made in an influential paper by Aiyagari (1995, p. 1160) and

in fact various papers thereafter suggested that with incomplete markets, the precaution-

ary saving motive leads to overaccumulation of capital and hence that a positive tax on capital

is welfare-improving.2

However, the comparison between the level of capital accumulation with and with-

out complete markets has no clear welfare implication. If there were a policy tool that

could implement the complete market allocation, there would be little doubt for the

policy maker to adopt such a policy as far as attaining efficiency is concerned. However,

since a tax and subsidy scheme of the kind mentioned above is not sufficient for com-

pleting the markets, this comparison tells little about the welfare effects of taxation, not

to mention whether or not capital should be taxed.

To properly assess whether or not positive taxes on capital are welfare-improving

when markets are incomplete, one should rather compare the competitive equilibria

with and without taxes, keeping the other parts of the market structure, in particular

the set of available financial assets, fixed. In this paper we carry out this second-best

exercise. We find that subsidizing capital may indeed be welfare-improving in a situa-

tion where the equilibrium stock of capital is higher than when markets are complete.

This finding does not rely on the presence of upward-sloping demand curves, so that

subsidizing capital further increases its level but nevertheless raises consumers’ welfare.

More generally, we identify conditions, regarding the nature of the shocks and the de-

gree of heterogeneity among consumers, under which it is optimal to tax or subsidize

capital and labor, for different specifications of the tax policy. In the rest of this section,

we outline the structure of this paper and summarize our findings.

Section 2 describes the economy. Production occurs only in the second period,

using capital and labor, traded in competitive markets. There is a continuum of con-

sumers, who may differ in terms of their initial income as well as of their preferences.

Each consumer has to choose how much to work in the second period and his savings

in the first period, invested to obtain capital, the only saving instrument. Idiosyncratic

shocks affect the productivity of the work of each consumer and possibly the return on

his savings.

A first component of the welfare effects of the introduction of linear taxes on capital

and labor is the effect of the change of equilibrium factor prices induced by the change

in savings and labor supply. To isolate and analyze this effect, in Section 3 we consider

an ideal situation where consumers’ savings or labor supply can be directly modified.

This problem was previously investigated by D´

avila et al. (2012) in a similar setup, but

they assume an exogenous labor supply and labor productivity shocks only.

The main and novel finding we report here is the derivation of a decomposition for-

mula: the welfare effects per marginal changes in the stock of capital or labor usage

2See for instance, Ljungqvist and Sargent (2004, chapter 15, pp. 535–536), Mankiw, Weinzierl, and

Yagan (2009, pp. 167–168 ).