مصرف در طول عمر و سرمایه گذاری: بازنشستگی و استقراض محدود

Retirement flexibility and inability to borrow against future labor income can significantly affect optimal consumption and investment. With voluntary retirement, there exists an optimal wealth-to-wage ratio threshold for retirement and human capital correlates negatively with the stock market even when wages have zero or slightly positive market risk exposure. Consequently, investors optimally invest more in the stock market than without retirement flexibility. Both consumption and portfolio choice jump at the endogenous retirement date. The inability to borrow limits hedging and reduces the value of labor income, the wealth-to-wage ratio threshold for retirement, and the stock investment.

How does retirement affect consumption and investment? Our paper offers two new results: • Consumption jumps at retirement because preferences are different when not working (and preferences are not additively separable over consumption and leisure). • Investment jumps at retirement because retirement is irreversible and human capital’s beta does not go to zero as the agent approaches the retirement boundary.The first result resolves an empirical puzzle in the literature, and the second result is an empirical prediction that is consistent with anecdotal evidence.1 We have other results that refine or extend ideas in the literature: • When wages are not related to market returns, agents’ tendency to work longer in expensive states in which the market is down gives labor income a negative beta that makes portfolio choice even more aggressive when young than predicted by the idea that the total portfolio equals bond-like human capital plus financial capital chosen to manage overall risk exposure. • The above result can be reversed when wages move with the market. • When the risk in human capital is due partly to wage uncertainty, hedging of human capital is less effective when it is not possible to borrow against future labor income or when the correlation between the wage rate and the market is low, and is dampened even when the no-borrowing constraint is not currently binding. These results are derived analytically in a consistent framework that yields rich empirical predictions and still hold even when labor income is unspanned by the financial market. We hope that these analysis and extensions will lend themselves to the study of policy questions in insurance, pensions, and retirement. We solve three models to isolate the effects on the optimal consumption and investment strategy of retirement flexibility with and without borrowing against future labor income. We derive almost explicit solutions (at least parametrically up to at most two constants) in all three models. Except for retirement flexibility and borrowing constraints these models share common features: irreversible retirement,2 a constant mortality rate, different marginal utility per unit of consumption before and after retirement, possibly stochastic labor income, bequest, and actuarially fair life insurance.3 The first model serves as a benchmark, it has an exogenous mandatory retirement date and allows limited borrowing.4 The second model considers voluntary retirement and also allows limited borrowing. This model seems intractable in the primal, so we solve it in the dual (i.e., as a function of the marginal utility of wealth) and obtain an explicit parametric solution up to a constant that is easy to determine numerically. We show that there exists a critical wealth-towage ratio above which it is optimal to retire. In addition, if labor income does not have a highly positive market exposure, human capital (the present value of future labor income) has a negative beta with any efficient portfolio (“the market”). This is because if the wage is nearly constant, itis optimal to work longer in expensive states, i.e., when the market is down, and work less when the market is up. In the absence of human capital risk, it is optimal to hold a constant fraction of total wealth (financial wealth plus human capital) in the market. With human capital risk, it is optimal to have an additional hedging component in the portfolio. Since human capital has a negative beta when retirement is flexible, the investor invests more aggressively in the market in this case to hedge against the human capital risk. If, however, the wage varies significantly with the market, the result can be reversed because human capital can now have a positive beta, which implies a less aggressive portfolio strategy since some required beta comes from labor income. The third model also considers voluntary retirement, but in contrast to the second case, prohibits the agent from borrowing against future labor income. This restriction limits hedging labor income risk in the stock market because the optimal hedge would cause financial wealth to become negative in many states. Limited hedging makes working longer less attractive and therefore optimal retirement comes sooner. As a result, the beta of human capital falls in magnitude. When the wage is nearly constant and thus human capital has a negative beta, limited hedging implies that the optimal portfolio is less aggressive than that in the second case, not only at the borrowing boundary but also away from the boundary. If the wage varies significantly with the market so that human capital has a positive beta, inability to hedge fully makes the optimal portfolio invest more in the market. When labor income is unspanned by the financial market, the effectiveness of hedging with stock investment is also reduced. Thus the unspanning of labor income produces a similar impact on the optimal stock investment to that of the no-borrowing constraint. It has been widely documented that consumption jumps at retirement (e.g., Banks et al. [1], Bernheim et al. [4]). Our model offers an explanation of this puzzle. The jump in consumption comes from the discrete change in labor supplied at retirement and a lack of additive separability of preferences over labor and consumption. For these reasons, the marginal utility of consumption at a given consumption level changes at retirement but the cost of consumption (implicit in the state-price density) is continuous, implying that optimal consumption jumps at retirement. Our model also predicts that portfolio weights jump at retirement, which is consistent with some anecdotal evidence. The jump in investment choice follows from the irreversibility of retirement and the curious fact that the beta of human capital does not approach zero as the retirement boundary nears (but is of course zero after retirement). The beta of the total wealth jumps at retirement and therefore the investment also jumps at retirement. Most of the existing literature does not produce discrete jumps of consumption or investment at retirement. For example, Lachance [14] considers a similar problem with endogenous retirement. However, in contrast to this paper, she assumes that the utility function is additively separable in labor and consumption, which implies that consumption is continuous across retirement. Financial advisors often advise investors to invest more in the stock market when young and to shift gradually into the riskless asset as they age. Two main justifications are provided in the literature. Bodie et al. [5] (BMS) show that if investors can frequently change working-hours, then labor income will be negatively perfectly correlated with the stock market and therefore the young should invest more in the stock market, because they can work longer hours if market goes down. However, working-hours are typically inflexible and consistent with this, an extensive empirical literature shows that labor income has a very low correlation with the stock market (e.g., Heaton and Lucas [9]). Therefore this working-hour flexibility is unlikely the main justification for the traditional advice. In contrast, Jagannathan and Kocherlakota [11] (JK) argue that total capital is human capital (which is bond-like) plus financial capital (whose market risk can bechosen). To keep the overall mix constant, financial capital has to have a high beta on market risk when young (when total wealth consists mostly of human capital) but a more modest beta on market risk when old (when total wealth consists mostly of financial capital). Our analysis contributes to this literature by providing two more important factors that affect the validity of the traditional advice. Our model implies that portfolio choice is not just a function of age and the relative size of financial wealth and earnings capacity may be more important. If we neglect these factors, we show that retirement flexibility has a positive effect on the validity of the traditional advice and the no-borrowing constraint has a negative one. Our analysis complements BMS by showing that even though wage rate itself might be uncorrelated with the stock market, human capital can be significantly negatively correlated with the market given voluntary retirement. This retirement flexibility, like working-hour flexibility, can make it optimal for the young to invest more in the stock. Compared to JK, our analysis suggests that the traditional advice may be valid for an even larger class of labor income distributions, because retirement flexibility induces an incremental negative correlation between human capital and the stock market and this negative correlation can offset some positive correlation between wage rate and the stock market. However, this observation must be placed in the context that in addition to age the optimal portfolio strategy also depends on other factors such as financial wealth and earnings capacity. In contrast to our model, neither BMS nor JK considers the retirement decision, which is arguably one of the most important life-cycle decisions. In addition, our model accounts for the inflexibility of labor supply choice before retirement, the imperfect correlation between labor income and stock return, and the ability to choose when to retire. This paper also contains technical innovations that lead to explicit parametric solutions (up to two constants) and analytical comparative statics. In particular, we combine the dual approach of Pliska [20], He and Pagès [8], and Karatzas and Wang [12] with an analysis of the boundary to obtain a problem we can solve in a parametric form even if no known solution exists in the primal problem. Having an explicit solution allows us to derive analytically the impact of parameter changes and, more importantly, allows us to prove a verification theorem (given in the companion paper Dybvig and Liu [6]), showing that the first-order (Bellman equation) solution is a true solution to the choice problem. Proving a verification theorem in our model is more subtle than it might seem, because of (1) the nonconvexity introduced by the retirement decision, (2) the market incompleteness (from the agent’s view) caused by the no-borrowing constraint, and (3) the technical problems caused by utility unbounded above or below. In particular, traditional verification theorems based on dynamic programming (Fleming–Richel) or a separation theorem (Slater condition) do not seem to apply to our model, and instead our proof uses a hybrid of the two techniques, patched together using optional sampling.5 The literature on life-cycle consumption and investment is extensive. Jun Liu and Neis [17] and Basak [2] consider the optimal consumption and investment problem with endogenous working hours. Similar to Lachance [14], they do not consider any borrowing constraint against future labor income. Sundaresan and Zapatero [21] investigate the effects of pension plans on retirement policies with an emphasis on the valuation of pension obligations. Khitatrakun [13] shows that individuals not affected by institutional constraints respond to a positive wealth shock by retiring or expecting to retire earlier than previously expected. Gustman and Steinmeier [7] also find a positive correlation between wealth and retirement. These findings are consistent with the em-pirical implications of this paper, in particular the implication that a worker will retire when the wealth-to-income ratio is high enough. The rest of the paper is organized as follows. Section 2 describes the formal choice problems used in most of the paper. Section 3 presents analytical results and comparative statics. Section 4 discusses the case with unspanned labor income. Section 5 provides graphical illustration and more discussion of the main results. We offer some discussions on possible further extensions in Section 6 and Section 7 closes the paper.

We have constructed tractable models to examine how retirement flexibility and a borrowing constraint affect life-cycle consumption and investment. Retirement flexibility causes human capital to correlate negatively with the stock market if labor income does not have highly positive market exposure. Both consumption and portfolio choice jump at the voluntary retirement date because of the difference in preferences and the option to work longer before retirement. The inability to borrow against future income limits the value of retirement flexibility and, concomitantly, the stock investment. Our models suggest that those whose labor income has low market exposure should subscribe to the traditional life-cycle investment rule. However, for those, such as entrepreneurs, whose labor income has highly positive market exposure, the opposite of the traditional rule applies. We abstract from many age-dependent and institutional factors, such as health and social securities, and focus on the pure effect of wealth and earnings. Empirically testable implications are largely consistent with existing empirical studies.We hope these models will prove useful for analyzing retirement, pension, and insurance