سرمایه انسانی مخاطره آمیز و مالیات بر درآمد معوق سرمایه

We study the structure of optimal wedges and capital taxes in a dynamic Mirrlees economy with endogenous distribution of skills. Human capital is a private, stochastic state variable that drives the skill process of each individual. Building on the findings of the labor literature, we construct a tractable life-cycle model of human capital evolution with risky investment and stochastic depreciation. In this setting, we demonstrate the optimality of (a) a human capital premium, i.e., an excess return on human capital relative to physical capital, (b) a large intertemporal wedge early in the life-cycle, and (c) a non-zero intratemporal wedge even at the top of the skill distribution at all dates except the last date in the life-cycle. The main implication for the structure of optimal linear capital taxes is the necessity of deferred taxation of physical capital. The average marginal tax rate on physical capital held in every period is zero in present value. However, expected capital tax payments do not equal zero in every period. Necessarily, agents face negative expected capital tax payments early in the life-cycle and positive expected capital tax payments late in the life-cycle.

We study a dynamic Mirrlees economy in which agents’ lifetime skill profiles depend on a stochastic, endogenous, privately observed state variable which represents individual human capital. In this economy, we characterize the optimal allocation of human capital, labor, physical capital and consumption, and construct a tax system that implements this allocation in equilib- rium. Our model of human capital evolution is consistent with three main facts from the large em- pirical literature on human capital. First, numerous studies summarized in Carneiro and Heckman [6] document that most of human capital investment occurs early in the life-cycle. We incorporate this fact in our model by assuming that human capital investment can be undertaken only at the first date in the agent’s life-cycle. The lifetime skill profile of each agent in the economy is an increasing function of the initial human capital investment. 1 Second, it has been well documented in empirical studies that the returns on human capital investments are risky. 2 We capture this fact in our model by assuming that initial human cap- ital investment is subject to an idiosyncratic productivity shock, and the level of accumulated human capital is subject to idiosyncratic depreciation shocks throughout the agent’s life-cycle. For tractability, we assume an absorbing structure of the idiosyncratic shocks to human capital. 3 Once human capital of an agent is hit by an adverse shock, the skill profile of this agent diverts to a deterministic, exogenous, low skill profile. This assumption makes our economy a general- ization of the absorbing shock economy studied in Diamond and Mirrlees [11] and Golosov and Tsyvinski [17]. Third, the literature on human capital has long recognized the difficulty in distinguishing hu- man capital investment from ordinary consumption expenditure. 4 As well, this difficulty has been recognized in the ongoing policy debate on how to design the tax system in order to foster human capital accumulation. At the core of this measurement problem lies the fact that, in reality, there is a human capital investment component in ordinary consumption and a significant amount of consumption value in human capital investment activities such as education and training. Agents use a large variety of goods, services, and non-market activities as vehicles for their human capi- tal investment as well as for their consumption. It is difficult to measure the relative “loadings” of human capital investment and pure consumption embedded in a particular good or service. In or- der to capture this measurement problem in a model with a single consumption good, we assume that consumption and human capital investment are indistinguishable to an outside observer. 6 We derive two kinds of results. First, we obtain a set of results about the structure of optimal wedges in our environment. 7 Then, we derive results concerning the structure of optimal capital taxes. We characterize three types of wedges: the intertemporal and intratemporal wedges, which have their counterparts in the existing literature on dynamic Mirrlees economies, plus an addi- tional wedge that represents the optimality of a human capital premium. The intertemporal wedges are similar in our endogenous-skill economy to the intertempo- ral wedges characterizing optimal allocations in dynamic Mirrlees economies with exogenous skills (cf. Diamond and Mirrlees [11], Golosov, Kocherlakota and Tsyvinski [16]). At the opti- mal allocation, agents are savings-constrained and intertemporal wedges are positive. The usual martingale property of marginal utility, however, does not hold in our model in every period (cf. Rogerson [32]). In particular, early in the life-cycle when private human capital investment is made, the discounted inverse marginal utility of consumption is a strict supermartingale. This ef- fect reinforces the intertemporal wedge in our environment relative to the environments in which skills are exogenous. The structure of optimal intratemporal wedges obtained in our environment is different from those obtained in most Mirrlees economies. The usual no-distortion-at-the-top property does not hold at the optimal allocation of our economy. In particular, we find that, at all non-terminal dates in the life-cycle and for all agents at the top of the cross-sectional distribution of individual productivity, the marginal utility of an additional unit of consumption is strictly lower than the marginal disutility of effort necessary to produce this unit. Given any fixed life-cycle path of labor effort, an additional unit of human capital investment generates a gain in the expected future output. This gain provides a measure of return on human capital investment that is directly comparable with the rate of return on physical capital invest- ment. We find that, at the optimum, the return on human capital investment exceeds the return on physical capital investment, which demonstrates the optimality of a human capital premium. This inequality of the rates of return on the two types of capital constitutes an additional wedge, whichweterm asset return wedge . To our knowledge, this wedge is new to the literature on dynamic Mirrlees economies. The second group of results of the paper concern optimal taxes, i.e., taxes that implement the optimal allocation. We study a standard market equilibrium model in which agents freely trade capital and labor subject to capital and labor income taxes. We follow Kocherlakota [26] in our focus on tax systems that are history-dependent, non-linear in labor income, and linear in capi- tal. 8 Our main result concerning optimal taxes is the necessity of deferred taxation of capital. We demonstrate that if capital can only be taxed contemporaneously and capital taxes are differen- tiable, then implementation of the optimal allocation in a market equilibrium is impossible. Then, we show that there does exist an optimal tax system in which taxes on capital held early in the life cycle are deferred until later in the life-cycle, when all individual uncertainty has been resolved. The amount of deferred tax obligation is linear in capital saved during the initial period, when human capital investment is made. A key finding here is that the marginal tax rate that determines the deferred capital tax obligation has to be history-dependent. In particular, high deferred taxes are levied on agents with low labor income profiles, and those with high labor income profiles pay low deferred capital taxes. Intuition about these results comes from the incentive problem that determines the opti- mal allocation and, consequently, shapes optimal tax structures in our environment. In Mirrlees economies with exogenous skills (Kocherlakota [26], Albanesi and Sleet [3]), taxes, in addition to raising revenue, must provide incentives to prevent high-skilled agents from pretending to be low-skilled, i.e., from shirking. Savings and shirking are complementary: increasing one’s sav- ings makes shirking more attractive. Capital taxes, thus, are high for agents whose labor income is low, as low labor income is consistent with shirking. In our model, agents can become highly productive ex post only if they make sufficient human capital investment ex ante. Taxes, there- fore, must provide enough incentives not only to discourage shirking throughout the life-cycle, but also to encourage sufficient investment in human capital at the beginning of the life-cycle. Agents’ human capital investment and labor effort choices are private and complementary: if an agent plans to shirk, underinvesting in human capital increases the value of shirking. This value is further increased by over-saving. Similar to the exogenous skill case, high capital taxes on agents with low labor income (the suspected shirkers) eliminate this complementarity. However, due to the dynamic nature of agents’ deviation strategies (in which agents under-invest in human capital and over-save early, and shirk later in the life-cycle), partial labor income histories in general do not carry all the information needed for the tax system to efficiently deter these joint deviations. For this reason, taxes must be deferred until all relevant information becomes available. In our model, similar to Kocherlakota [26], it is optimal for the government revenue from tax- ation of capital to be zero. In our model, however, this does not imply that expected capital taxes are zero in every period. Since deferred taxes are necessary in our model, agents face negative expected capital taxes early in the life-cycle and positive expected capital taxes later in the life- cycle. The ex ante expected present value of lifetime capital taxes paid by every agent is zero, which implies that the present value of the government revenue from taxation of capital is zero. This result is intuitive. There is no reason for the government to raise revenue via distortionary capital taxation when lump-sum and non-distortionary (fully non-linear) labor income taxes are available. Under the optimal system, capital taxes are used purely to provide correct incentives to the agents. We also show that the volatility of marginal tax rates needed for the implementation of a given optimal allocation in our endogenous-skill environment is larger than that needed in a similar en- vironment in which skills are exogenous. This result follows from the difference in the structure of the incentive problem in the exogenous- and endogenous-skill models. Allowing for endoge- nous human capital accumulation through unobservable investment adds in our environment an extra dimension to the space of strategies that agents can use to deviate from the socially optimal pattern of behavior. With such an enhanced set of deviation opportunities, the incentive prob- lem of our environment is more severe, relative to environments in which skills are exogenous. This translates, at the optimum, into a larger intertemporal wedge between the shadow interest rate of consumption and the rate of return on physical capital investment early in the life-cycle when human capital investment is made. In order to support this wedge in equilibrium, capital taxes have to introduce more risk into the return on physical capital investment, which means that the present value of ex post marginal capital tax rates has to be more volatile when skills are endogenous.

Deferred taxes are a common feature of capital income tax systems currently used in many countries. Our paper provides a theoretical rationale for the use of such solutions. Our results show that it is necessary to use deferred taxes in a setting in which information relevant for the assessment of tax is revealed gradually over time. In a dynamic Mirrlees economy, we show that when human capital accumulation is taken into account in a way consistent with three important empirical facts about the life-cycle prop- erties of individual-specific human capital, deferred taxes are a necessary part of any optimal differentiable tax system. In our model, in which human capital investment is private, risky, and non-separable from consumption, there is a three-way complementarity between shirking, under- investing in human capital, and over-saving. This complementarity requires that a portion of tax on capital income obtained by agents early in the life-cycle be deferred until late in the life- cycle, when more information about agents’ private human capital decisions is available through the observation of longer labor income histories. Long histories of high labor income are con- sistent with high effort and high human capital investment. The deferred tax assessed on agents with such observed histories is low. Histories of low labor income, in contrast, are consistent with over-consumption and under-investment in human capital early in the life-cycle and shirking at later dates in the life cycle. Therefore, the deferred tax assessed on agents with such observed histories is high. Our results do not depend on several assumptions that we make for the ease of exposition. First, we assume that all agents are ex ante identical in our model. Our results go through with minor changes when ex ante agent heterogeneity is incorporated into the model, as long as these differences in individual characteristics are publicly observable. Second, the model can be eas- ily modified to replace the period-by-period resource constraint with the present value resource constraint. Third, in the market/tax implementation mechanism we consider, capital is the only asset that agents trade. If financial claims markets with observable trades are added to the model,our results go through without change, with all wealth (physical capital and financial claims) receiving the same tax treatment. In this paper, we focus on tax systems in which agents’ tax obligations depend linearly on their capital holdings. Linear capital taxes have been of central interest in the optimal capital taxation literature, and our focus on such taxes makes our results comparable with the results obtained in this literature (most readily with Kocherlakota [26]). Also, with linear capital taxes we can demonstrate the necessity of deferred capital taxes using a simple argument based on examining the Euler equations of the agents’ utility maximization problem. The same argument can be used to obtain this result under any tax system differentiable in capital (recall footnote 22). Under a tax system non-differentiable in capital, our argument based on the Euler equation cannot be used. It is clear, however, that deferred capital taxes are needed in some non-differentiable tax systems and not needed in other. On the one hand, consider a tax system constructed from the optimal linear system we characterize by introducing a small kink in the amount of the tax levied on capital held on the equilibrium path in our implementation at t = 1. With this kink sufficiently small, the values of the joint deviation strategies involving over-saving can be made arbitrarily close to the values of these strategies under linear taxes. Thus, under such a non- differentiable system, deferred capital taxes continue to be necessary. On the other hand, no capital taxes (including deferred) are necessary if the government controls capital accumulation directly, which it can in all dynamic Mirrlees economies in which capital is observable. Our results suggest that deferred taxation may have a role in other contexts in which informa- tion relevant for the provision of incentives becomes available gradually over time.