Unemployment Risks and Optimal Retirement in an Incomplete Market

We develop a new approach for solving the optimal retirement problem for an individual with an unhedgeable income risk. The income risk stems from a forced unemployment event, which occurs as an exponentially-distributed random shock. The optimal retirement problem is to determine an individual’s optimal consumption and investment behaviors and optimal retirement time simultaneously. We introduce a new convex-duality approach for reformulating the original retirement problem and provide an iterative numerical method to solve it. Reasonably calibrated parameters say that our model can give an explanation for lower consumption and risky investment behaviors of individuals, and for relatively higher stock holdings of the poor. We also analyze the sensitivity of an individual’s optimal behavior in changing her wealth level, investment opportunity, and the magnitude of preference for post-retirement leisure. Finally, we ﬁnd that our model explains a counter-cyclical pattern of the number of unemployed job leavers. 2013 and been with the for 3 for 2


Introduction
Starting from pioneering works by Merton (1969Merton ( , 1971, intertemporal models of optimal consumption and portfolio choice have evolved into some interesting generalizations. Among them, Bodie et al. (1992), Heaton and Lucas (1997), Koo (1998), Viceira (2001), Farhi and Panageas (2007) 2 Article submitted to Operations Research; manuscript no. OPRE-2013-08-441.R2 and others explored life-cycle models in which non-tradable labor income is incorporated into the problem of optimal consumption and portfolio choice. Bodie et al. (1992) examine the impact of labor flexibility on an individual's optimal behavior including consumption and risky investment under the assumption that the labor supply is certain over the life cycle. Heaton and Lucas (1997) and Koo (1998) investigate how uncertain income stream can affect an individual's optimal consumption and investment behavior. Further, Viceira (2001) illustrates an individual's optimal behaviors when she faces an exogenous shock of retirement, assuming that exogenous retirement shocks and mortality risks arrive in a random way, but with constant probabilities. Farhi and Panageas (2007) investigate an individual's optimal consumption, investment and retirement behaviors simultaneously under the assumption that the individual's income rate is certain while working. All these papers permit only one of the following two assumptions, not both: risky labor income or endogenous (or voluntary) retirement opportunity.
Considering both income risks and endogenous retirement opportunity in the classical life-cycle models is a complicated job. Only a few researchers such as Liu and Neis (2003), Bodie et al. (2004), Dybvig and Liu (2010), and Jang et al. (2013) have successfully resolved the optimal voluntary retirement problems with income risks, but they consider a complete market in which income risks can be hedged away by purchasing and selling some financial instruments traded in that market. This paper deals with both income risks and endogenous retirement in an incomplete market.
We assume individuals cannot eliminate income risks because they stem from exogenously forced unemployment events, and investigate the individual's optimal consumption, portfolio choice, and retirement behaviors in this serious situation.
Investigating the impact of income risks from unemployment events on an individual's optimal behaviors has been an important task for economists. Caroll (1992) shows that an unemployment event could have a major impact on an individual's current consumption and saving behaviors. Cocco et al. (2005) show that even a 0.05% probability of being unemployed in any given year has a large effect on an individual's portfolio choice, particularly early in life. Further, Wachter and Yogo (2010) stress the fact that unemployment risks could be significant for the poor in the sense that the optimal amount of risky investment increases at a sufficiently low wealth level.
Also, Lynch and Tan (2011) show that the amount invested in the risky assets could be relatively lower in a persistent unemployment state and assert that the unemployment state pays only 10% of permanent labor income. 1 This paper might give a significant impetus to clarify the effect of unhedgeable unemployment risks on an individual's optimal behaviors in an incomplete market.
From a technical standpoint, solving financial problems constructed in an incomplete market, such as pricing derivatives and choosing an investor's optimal consumption and investment strategies, is extremely complex due to the non-uniqueness of the equivalent martingale measure, and Alain Bensoussan, Bong-Gyu Jang, and Seyoung Park: Unemployment Risks and Optimal Retirement in an Incomplete Market Article submitted to Operations Research;manuscript no. OPRE-2013-08-441.R2 3 obtaining a closed-form solution of the problems is hardly possible. Just a little literature, such as Kim and Omberg (1996), Chacko and Viceira (2005), Sangvinatsos and Wachter (2005), and Liu (2007), have provided closed-form solutions of financial problems defined in an incomplete market.
It is well-known that there are two approaches for solving financial problems derived from an incomplete market: the dynamic programming approach (DPA) and the martingale approach (MA). Koo (1998) and Henderson (2005) exploit the DPA to solve optimal consumption and investment problems incorporating labor income risks. They derive a non-linear partial differential equation and solve it. He and Pearson (1991), Svensson and Werner (1993), Teplá (2000), and Keppo et al. (2007) apply the MA to solve financial problems in the presence of income risks in an incomplete market.
Most existing literature concerning income risks, no matter they are hedgeable or unhedgeable, model the income stream as a standard geometric Brownian motion, which permits use of the MA.
For instance, Karatzas et al. (1991) and He and Pearson (1991) suggest choosing the minimum local equivalent martingale measure with respect to unhedgable risks represented by a geometric Brownian motion (GBM). Further, Duffie et al. (1997) introduce a viscosity solution technique for solving portfolio choice problems in incomplete markets where a stochastic income evolved by a GBM cannot be hedged by utilizing a traded risky asset. However, we assume the income risks originally stem from forced unemployment events and such exogenous unemployment events occur following an exponential distribution with positive intensity. This kind of model is new in the optimal retirement literature, and a new approach is developed by using the DPA for solving the optimal retirement problem with the income risks. Specifically, we derive a differential equation and develop an iterative numerical method to solve it. As far as we know, this is the first paper to develop a method for solving the optimal retirement problem with a down-jump event of income which is modeled not using any Brownian motion.
With reasonably calibrated parameters we obtain some interesting features concerning an individual's optimal behaviors and income risks in the incomplete market. In our model, we find • income risks stemming from forced unemployment events might significantly lower an individual's consumption, investment in a risky asset and voluntary retirement wealth level, • income risks stemming from forced unemployment events might be an explanation for the findings of Cocco et al. (2005), Benzoni et al. (2007), and Lynch and Tan (2011), in that stock holdings in cash-on-hand can increase at a sufficiently low wealth level, • certainty equivalent wealth gain, the maximum wealth that an individual is willing to give up in exchange for the market without unemployment risks, decreases as wealth and/or investment opportunity grow(s), and Alain Bensoussan, Bong-Gyu Jang, and Seyoung Park: Unemployment Risks and Optimal Retirement in an Incomplete Market 4 Article submitted to Operations Research;manuscript no. OPRE-2013-08-441.R2 • certainty equivalent wealth gain has a bigger value for an individual with a higher preference of leisure after retirement.
The first finding about voluntary retirement wealth level could be an explanation of a countercyclical pattern of the number of unemployed job leavers who have voluntarily left their current jobs; the proportion of job leavers increases during economic recessions and decreases during economic expansions. 2 Our result says that soaring income risks due to forced unemployment events during economic recessions induce myopic investors 3 who have slightly smaller wealth than the voluntary retirement wealth level planned during the past economic expansion to enter early retirement.
Our paper is organized as follows. In section 2 we establish a financial market with forced unemployment events and formulate our problem in the market. We also introduce a new dual approach and an iterative numerical method to solve the problem. In section 3 we display some analytical results including optimal consumption and investment strategies of an individual, and in section 4 we show numerical implications of our model in a normal market. In section 5 we analyze the relationship between the number of job leavers and business cycles and in Section 6 we conclude the paper.

The Financial Market
Following the conventional models, we assume an individual can trade two assets in the financial market: a bond (or a risk-free asset) and a stock (or a risky asset). The bond price B t evolves by the relationship where the positive constant value r is considered to be a risk-free interest rate. On the other hand, the stock price S t follows where µ (µ > r) is the expected rate of the stock return, σ > 0 is the stock volatility, and W t is a standard Brownian motion defined on a suitable probability space.
We assume that the individual is in the workforce at the beginning and wants to retire voluntarily someday in the future. The individual receives income at the rate of I 1 from labor. She is exposed to forced (or involuntary) unemployment risks such that she loses her job by compulsion whenever an exogenous unemployment shock arrives before the voluntary retirement date. Accordingly, she obtains income at the rate of I 2 (I 2 < I 1 ) after the forced unemployment event. The forced unemployment event occurs following an exponential distribution with intensity δ, namely; for some time t ≥ 0 probability of {τ U ≤ t} = 1 − e −δt , Alain Bensoussan, Bong-Gyu Jang, and Seyoung Park: Unemployment Risks and Optimal Retirement in an Incomplete Market Article submitted to Operations Research;manuscript no. OPRE-2013-08-441.R2 5 where τ U is the forced unemployment time. 4 We have two risk sources: the market risk (or the Brownian motion W t ) and the unemployment risk (or the Poisson arrival time τ U ). The market risk is hedgeable and can be partially diversified away by controlling dollar investment amount in the stock, but the unemployment risk cannot be hedgeable. Elmendorf and Kimball (2000) and Gormley et al. (2010) emphasize the important role of insurance against large and negative wealth shocks such as unemployment risks on an individual's optimal investment strategies. However, private insurance markets for hedging labor income risks are not sufficiently competitive compared to other insurance markets (Cocco et al. 2005). From the realistic point of view, we are assuming that there is no financial vehicle to eliminate or diminish the forced unemployment risks, thus, the financial market is considered to be incomplete. For the technical simplicity, we assume that the Brownian motion and the Poisson arrival event are independent. 5

The Retirement Problem
The retirement problem explored in this paper can be thought of as a variation of the problem investigated by Farhi and Panageas (2007), but it allows an individual to be exposed to forced unemployment risks.
The individual has the following time-additive utility function of Cobb-Douglas type: where c(t) is per-period consumption and l(t) is leisure at time t, and a is a weight for consumption satisfying 0 < a < 1. We consider a binomial choice of leisure, in which the individual either works full time or she is retired permanently. 6 More specifically, the individual enjoys leisure l(t) = l 1 while she is working and l(t) = l 2 (l 1 < l 2 ) when she retires. We assume that the wage rate w is constant and, then, the individual gets an income of I 1 = w(l 2 − l 1 ) > 0 per unit time during working status. We also assume she gets I 2 > 0 (I 1 > I 2 ) per unit time after retirement. 7 The assumption of a positive income after retirement reflects the fact that most countries provide unemployment allowances and other public welfare services for retired people.
The wealth process X(t) of the individual is given by where π is the dollar amount invested in the stock and θ represents the Sharpe ratio (µ − r)/σ. The individual accumulates wealth at the rates of rX − c + I 1 (rX − c + I 2 ) before (after, respectively) voluntary or involuntary retirement. Note that the individual is exposed to forced unemployment risks and, thus, her labor income rate will decrease from I 1 to I 2 at the unemployment event. She is also exposed to the market risk stemming from stock investment and simultaneously compensated by the market premium, π(µ − r).
We assume that the individual can borrow money with her human capital. Following Friedman (1957) and Hall (1978) we define human capital h as the present value of future labor income discounted by the risk-free interest rate r: 8 Then we impose a natural wealth constraint as the following: 9 This implies that the individual can consume and invest in the stock as long as her wealth level is above −I 1 /r. If the wealth level approaches −I 1 /r, she cannot consume and invest in the stock any more, i.e., consumption c and risky investment π should be zero. We exclude such trivial case and just consider the cases in the presence of the wealth constraint (1). We call the consumption and investment strategies satisfying the wealth constraint admissible strategies.
We normalize leisure prior to retirement as l 1 = 1, then the utility function during working should be U 1 (c) ≡ U (1, c) = ln c.
We also let which represents the preference for leisure after retirement. The retirement problem is to find the maximum of the individual's expected utility for consumption. The individual would like to maximize the utility by controlling her consumption c, risky portfolio π, and voluntary retirement time τ , i.e., the individual willingly obtains the following value function: where X(0) = x > −I 1 /r is initial wealth of the individual and β > 0 is the individual's subjective discount rate. By utilizing the conditional expectation of τ U , we can rewrite the individual's the value function Φ(x) given by (2) as the following: 10 where In fact, U 2 (z) is the value function of the classical Merton's problem with infinite investment horizon under the condition that the individual has a logarithmic utility and an income stream I 2 forever.
In our model, it is possible for an individual to involuntarily retire with negative wealth at the forced unemployment date. It makes our problem difficult to be well-defined because U 2 is not defined in the region of (−I 1 /r, −I 2 /r]. We extend our problem into the problem defined in the region of (−I 1 /r, ∞), which contains some negative values of wealth, by assuming that the postretirement value function U 2 (z) is continuous at z = 0 and the first derivative U 2 (z) equals to zero for z < 0. 11 Specifically, we assume that

A new convex-duality approach
We utilize the conventional dynamic programming approach to resolve our retirement problem in an incomplete financial market. For a fixed stopping time τ , we define Then the value function Φ(x) given by (3) is rewritten as To solve the optimal stopping problem (4) we use the variational inequality approach. Variational Inequalities for the problem of this kind have been introduced by Bensoussan and Lions (1982) and Øksendal (2007). We will refer to Øksendal (2007) in the paper. 12 In fact, we can derive the following inequality: for any x > −I 1 /r. region and the strict inequality holds in the stopping region. Moreover, if the strict inequality in the second inequality in (5) holds, i.e., if the value function prior to voluntary retirement is strictly larger than the value function after retirement, then the individual is in the continuation region and delay voluntary retirement. If the value function prior to retirement approaches the value function after retirement (i.e., the equality holds in the second inequality in (5)), then the individual is in the stopping region and, hence, optimally enters voluntary retirement. Note that for each value of x > −I 1 /r one of the first two equalities must hold, and, thus, we need the third equality in (5).
The continuation region and stopping region are determined by the so-called critical wealth level, over which it is optimal for an individual to enter voluntary retirement. Thus, we conjecture that the retirement problem can be solved by finding an optimal stopping boundary, or equivalently, a free boundaryx, which can be characterized by value matching and smooth pasting conditions.
The problem can be formulated as follows: wherex is the critical wealth level. If we find φ(x) satisfying C 1 and piecewise C 2 , verifying the inequalities in the variational inequality (5) everywhere, then φ(x) in (6) is indeed a solution of the variational inequality (5). 13 Further, it is straightforward to verify that the solution φ(x) of (5) is equivalent to the solution Φ(x) of our optimal stopping problem (4) (see Theorem 10.4.1 in Øksendal, 2007). 14 Now, we introduce a dual variable λ; we define the marginal value of the value function φ(x) as the variable λ. Then the critical wealth levelx has an inverse relationship between the variable λ by the last equation in (6). Specifically, If we differentiate the first equation in (6) with respect to x and take the left derivative coefficient We introduce a modification of the conventional convex-duality approach by Karatzas and Shreve (1998) to solve our incomplete market problem. At first, we define a function G(·) which is the so-called convex-dual function 15 by Hence, we get the following relationships: For the notational convenience, we let G λ(x) = G and λ(x) = λ. Then by using the relationships given by (9), the equation (7) becomes If we rearrange the equation (10) and rewrite the last equation in (6) by using the definition (8) of the convex-dual function G, we obtain the following equations for λ >λ: where We add one more constraint, which implies the individual's marginal utility λ goes to infinity as initial wealth x goes down to − I 1 r , which is the lower bound for initial wealth x.
Technically, for the case of δ = 0, without any forced unemployment risks, the problem formulated by equations in (11) and (12) has an analytic solution, whereas the problem for the case of δ > 0 is unlikely to have an explicit solution. To solve the problem for δ > 0, we first verify the existence of a solution satisfying equations in (11) and (12)  2.3.2. The iterative method First, we define α δ > 0 and α * δ < 0 as the two roots of We conjecture the general solution of (11) as Article submitted to Operations Research; manuscript no. OPRE-2013-08-441.R2 subject to Putting the relationship in (13) into (11), we get for λ >λ and for Here, The derivations of the relationships in (14) and (16) are in an online Appendix. Further, in Appendix 7.2 we prove the uniqueness of G(·) and the free boundaryλ, and the strictly decreasing property of G(·).
Putting this G(λ) into (16), we getλ. Suppose δ = 0, but has a sufficiently small value. 17 We exploit G(λ) andλ for the case where δ = 0 as the initial values of our iteration method.

Lower and Upper Bounds for Critical Wealth Level
Even though it is hard to have a closed form of the threshold levelx in the optimal stopping problem (6), we can derive analytical lower and upper bounds.
To simplify notation, set L δ to be the left hand side of (16) and ψ δ (λ) to be its right hand side, that is, we let Moreover, we define two functions which can be lower and upper bounds of ψ δ (λ) respectively.
The upper and lower bounds for the critical wealth levelx, at which the individual enters voluntary retirement, also can be written as if we use the result in the proposition and the definition of function G(·) in (8). Notice that φ δ (λ 1 δ ) becomes φ δ (λ 0 δ ), or equivalently, the upper and lower boundaries in the proposition are identical, where intensity δ is zero. For this case, the boundaries become the corresponding critical wealth level described in Farhi and Panageas (2007), in which individuals are not exposed to any forced unemployment risk.

Optimal Consumption and Investment Strategies
We find the optimal consumption and investment strategies by exploiting the terms ofλ, B(·) and G(·), which can be obtained by the iterative numerical method in the previous section.
Theorem 1. The optimal consumption c and risky portfolio π are given as where λ * (x) is a decreasing function with respect to wealth x and the solution of Proof. The relationship (8) and the first-order conditions with respect to c(t) and π(t), which were used in deriving the variational inequality (5), yield the optimal consumption c(t) and risky portfolio proportion π(t). Q.E.D.
The optimal consumption, c M , of the classical Merton's (1969Merton's ( , 1971 problem in the presence of the income I 1 is represented as so the marginal propensity to consume (MPC) out of wealth, ∂c t ∂x , is constant. However, in our model without any forced unemployment risk, namely δ = 0, the first and second terms of the right hand side of (21), which spring up due to the voluntary retirement event, yield the optimal This implies the MPC out of wealth should be positive and the optimal consumption is a concave function with respect to wealth. Since B(λ) > 0 for δ = 0, the individual with voluntary retirement option consumes relatively less than the one in the classical Merton problem because she is likely to accumulate wealth by cutting down her consumption to enter retirement early. Moreover, since λ * is a decreasing function with respect to wealth level the effect of the voluntary retirement option on her consumption behavior becomes significant as her wealth increases. This might be an explanation of the finding that individuals could dramatically decrease their consumption near the critical wealth level (Farhi andPanageas 2007, Dybvig and. The third term of the right hand side of (21) consists of two parts which are closely associated with the impact of income risks stemming from forced unemployment events. Notice that the first (integral) part gives a negative effect to the individual's consumption and the second (integral) part affects it conversely. As an individual's wealth approaches the critical wealth levelx, λ * (x) gets closer toλ, and subsequently, in the limit case the first part disappears and the second part has a fixed value. Hence, if an individual's wealth reaches near the critical wealth level, the third term of the right hand side in (21) could make a positive impact on the individual's consumption and offset the negative impact of the second term of the right hand side in (21), which stands for the voluntary retirement option value. So it might be possible for people facing forced unemployment risks to consume more than individuals not exposed to those, near retirement time.
14 Article submitted to Operations Research; manuscript no. OPRE-2013-08-441.R2 However, using the first and second terms of the right hand side of (22) we can get the optimal risky portfolio, π V R , for the case where there exists no forced unemployment risk, i.e., δ = 0: Notice that the second term of (23) is associated with voluntary retirement and is a positive and increasing function with respect to initial wealth x. An individual permitted voluntary retirement is willing to take more risk than one considered in the classical Merton's set-up. Also it seems that the risky investment, π V R , increases according to the growth of an individual's wealth level. Intuitively, people who have slightly smaller wealth than the critical wealth level give their attention to retire voluntarily as soon as possible, so they tend to invest more in risky assets even though they may end up losing relatively large amounts of money. Similar observations were reported by Farhi and Panageas (2007) and Dybvig and Liu (2010).
The third and fourth terms of the right hand side of (22) are closely associated with involuntary unemployment. Notice that the third term is negative, so it decreases an individual's stockholding, while the forth term consisting of two integral parts urges stock investment.
We can get an upper bound of π where individual's wealth goes up to a sufficiently close level to the critical wealth level and a lower bound where individual's wealth goes down to zero: Proof. We first utilize inequality G(µ) − I 1 r + + I 2 r ≤ max 1 βλ , I 2 r , and take the limit of λ * ↓λ to derive the last term of the right hand side of the first inequality. On the other hand, the last term of the right hand side of the second inequality is derived if we use G(µ) ≥ 0 for all µ and take the limit of x ↓ 0. Q.E.D.

Baseline Parameters
The baseline parameters are fixed to r = 3.71%, which is the annual rate of return from rolling over 1-month T-bills during the time period of 1926-2009 18 and we assume β has the same value as r.
We utilize µ = 11.23% and σ = 19.54%, which are the return and standard deviation of the world's large stocks during the time period of 1926-2009. 19 We set K = 3 following the assumption used by Dybvig and Liu (2010), I 1 = 1, and I 2 = 0.10 following the results of Lynch and Tan (2011), who conclude the unemployment state pays only 10% of permanent labor income. Table 1 shows critical wealth levelx's for various parameter values, such as forced unemployment intensity δ, expected rate µ of stock return, stock volatility σ, and leisure K. It is obvious that an individual would be better off entering voluntary retirement wherever her wealth level is not less thanx, so that she can enjoy more leisure after retirement. Table 1 shows that such critical wealth levelx decreases as forced unemployment intensity δ increases. Intuitively, individuals with a higher δ enter the voluntary retirement stage earlier even though the wealth level at retirement is not relatively high, because they willingly submit to such utility losses in exchange for avoiding utility losses stemming from forced unemployment risks.

Critical Wealth Level
On the other hand, the critical wealth level increases as µ increases or σ or K decreases, ceteris paribus. After retirement, an individual in our model faces a tradeoff between utility gains owing to the increase of leisure and utility losses from the significant reduction of income. In a financial market with a higher expected rate of stock return, an individual is willing to postpone her voluntary retirement because a better investment opportunity makes her worry less about forced unemployment risks. Similarly, a lower stock volatility also provides her with a better investment opportunity, so she worries less about forced unemployment risks. Evidently, a lower quantity of leisure after retirement hinders an individual from entering voluntary early retirement.  Figure 1 shows the amounts of optimal consumption and investment in the risky stock as a function of initial wealth for several values of forced unemployment intensity δ. An individual's consumption c grows as initial wealth level x becomes higher, but it falls as her forced unemployment possibility δ grows. A higher demand on precautionary savings against a higher forced unemployment risk could be an explanation of the latter observation. (See Carroll, 1992;Malley and Moutos, 1996;Gruber, 1997). 20 In terms of optimal risky investment, Figure 1 shows an interesting feature. Admittedly, if wealth is not small enough, the optimal risky investment increases as initial wealth increases. However, for poor people this might not be true; up to some wealth level, some of them with a higher forced unemployment possibility lessen their investment in the risky stock even though they have more wealth. This observation implies that forced unemployment risks could be an important explanation for the findings of Benzoni et al. (2007) and Lynch and Tan (2011), in that they find the stock holdings can increase at a sufficiently low wealth level. Moreover, the result seems to be consistent with that of Cocco et al. (2005), who address whether the amount of optimal risky investment can rise due to unemployment risks when wealth level is low enough. Figure 2 says that optimal consumption to wealth ratio and optimal risky investment to wealth ratio decrease at decreasing rates as retirement time approaches, or equivalently, as x goes tox. 21

Optimal Consumption and Risky Investment
The decreasing properties of optimal consumption and risky investment with respect to wealth ratio seem to result from the intense aspirations toward voluntary early retirement under our set-up.
Notice also that the decreasing rate becomes bigger as forced unemployment possibility δ increases.
This fact indicates that our model can reflect the higher intense aspirations toward voluntary early retirement of an individual facing a higher forced unemployment risk. The second result in Figure 2 is compatible with the observation of Polkovnichenko (2007): an individual with a higher unemployment risks has to invest more in stock by using savings to finance consumption, because she cannot expand income much by choosing labor supply in the unemployment state. Economists have a consensus that the flexibility in labor supply causes higher investment in stock (e.g., Bodie et al., 1992), and our model says that forced unemployment risks might lead to a much higher allocation to stock when the wealth of an individual is close to the critical wealth level.

Certainty Equivalent Wealth Gain
We define certainty equivalent wealth gain (henceforth, CEWG) to be the maximum wealth level that an individual in our model is willing to give up in exchange for the market without forced unemployment risks. We now investigate how much CEWG is in a normal economic situation and how sensitive CEWG is when µ, σ, and K change.
Definition 1. ∆(x) is the certainty equivalent wealth gain at initial wealth level x if it satisfies whereφ(x) is the value function φ(x) with δ = 0. δ means a higher critical wealth level, so the individual has more time to prepare for forced unemployment risks; thus CEWG should be smaller. The sensitivity of CEWG for various K, preference for leisure, is illustrated in Figure 4. It seems that CEWG is much bigger for a bigger preference for leisure if an individual's wealth is far less than her critical wealth level. This implies individuals with a higher preference of leisure have more stress if their wealth level is relatively low compared with their critical wealth level. This tendency is mitigated as their wealth gets closer to the critical wealth level.

Economic Downturns and Unemployment Risks
In 2000 our model decreases as forced unemployment risk increases, implying a higher forced unemployment risk could compel individuals to retire at a lower wealth level. Generally, forced unemployment risks are relatively high during economic recessions and relatively low during economic expansions, so we might conclude that soaring forced unemployment risks during economic recessions induce myopic individuals who have slightly smaller wealth than their critical wealth level planned during the past economic expansion to enter early retirement.
More specifically, our model can show the behaviors of myopic individuals under different market conditions. Table 4 shows critical wealth levels for various δ and I 1 under two different market conditions, say, 'economic expansions' and 'economic recessions'. Ang and Bekaert (2002) show that the expected stock return and stock volatility in the US market was µ = 0.1394 and σ = 0.1313 during the economic expansions and µ = 0.1394 and σ = 0.2600 during the economic recessions, and we utilize the same parameters for Table 4. 22 In the table, myopic individuals who draw up their retirement plan solely reflecting the current financial market conditions seem to retire at a much lower wealth level during the economic recessions than during the economic expansions, even though the force unemployment possibility and their income rate do not change. For instance, an individual with I 1 = 1 and δ = 0.02 makes a plan to retire at the wealth level of 206.3668 during the economic expansions, however, the critical wealth level gets much smaller to 74.2596 during the economic recessions.
Admittedly, a smaller income rate I 1 or a bigger forced unemployment possibility δ during economic recession periods is most likely to consolidate the reduction of critical wealth level. 23 Since an individual with an asset portfolio and labor income is exposed to a bigger income risk stemming from forced unemployment events and worse financial market conditions during economic recessions, she inevitably lowers her voluntary retirement wealth level in order to get a relatively higher utility gain (mostly obtained from more leisure time) after retirement. For example, according to Alain Bensoussan, Bong-Gyu Jang, and Table 4 Critical

Conclusion
We developed a new approach for solving the optimal retirement problem for an individual with an unhedgeable income risk. The income risk stems from a forced unemployment event, which occurs as an exponentially-distributed random shock. The optimal retirement problem is to determine an individual's optimal consumption and investment behaviors and optimal retirement time simultaneously. Our approach for solving the problem originated from the combination of the DPA and the convex-duality approach, but we introduced a slightly different convex-dual function of the individual's value function from the conventional ones, and we also provided an efficient iterative numerical method.
Reasonably calibrated parameters show that our model can give an explanation for lower consumption and risky investment behaviors of individuals, and for relatively higher stock holdings of the poor. Exploiting the concept of CEWG, we glanced at an individual's optimal behaviors in changing her wealth level, investment opportunity, and the magnitude of preference of postretirement leisure.
Finally, we find our model gives an explanation of a counter-cyclical pattern of the number of unemployed job leavers. It could provide an evidence that soaring forced unemployment risks during economic recessions induces people who have slightly smaller wealth than the critical wealth level planned during the past economic expansion to enter early retirement.
An interesting extension of our paper is to introduce a continuous-time Markov regime-switching model to discuss the effects of economic recessions and economic expansions on an individual's Article submitted to Operations Research;manuscript no. OPRE-2013-08-441.R2 optimal strategies. We leave solving the retirement problem in a regime switching framework to the reader as an extension for future research.

Appendix
This version of the appendix contains the statements of important theorems without the proofs.
An extended appendix concerning the details of the proofs is available online.
7.1. The Existence of a Solution to (11) and (12) We assume that the subjective discount rate β equals to the risk-free interest rate r. This assumption is only used when verifying the existence of a solution to (11) and (12). Actually, if one wants to relax it, it is necessary to take a restriction on the free boundaryλ to verify the existence. By taking the assumption of β = r, we can show the existence of a solution to (11) and (12) for anŷ λ > 0. Most importantly, without the verification for the existence of a solution we could find a numerical solution satisfying (11), (12) and (13) in a wide range of parameters.
We can provide a theorem concerning this issue.
Theorem 2. We assume that β = r. Then for anyλ > 0, there exists a solution to the problem formulated by equations in (11) and (12), which has the boundedness such that where n is a natural number, C * n and C * * n are constants.
The following theorem permits us to take a monotonically-decreasing G(λ) under suitable parameter conditions.
For the unique solutionλ of (16), the corresponding G(λ) is also the unique solution of (11).

Verification for the Optimal Stopping Problem
The verification for our optimal stopping problem (4) is executed in the following two steps: we first verify that the solution, φ(x), to the variational inequality (5) is the solution to the optimal stopping problem (4). Next, we verify that the solution to the free boundary problem (6) satisfies the variational inequality (5). We reuse some notations and definitions in Øksendal (2007).
First, we fix a domain G in R k and let be an Itô diffusion in R k . Define Let f : R k → R and g : R k → R be continuous functions satisfying Let T denote the set of all stopping times τ ≤ τ G . Consider the following problem: Find Φ(y) and τ * ∈ T such that Note that since J 0 (y) = g(y) we have Φ(y) ≥ g(y) for all y ∈ G.
Article submitted to Operations Research; manuscript no. OPRE-2013-08-441.R2 We can now formulate the variational inequalities. As usual we let (σσ T ) ij (y) ∂ 2 ∂y i ∂y j be the partial differential operator. Now, we can state Theorem 10.4.1 in Øksendal (2007).
Theorem 6. (Variational inequalities for optimal stopping) a) Suppose we can find a function φ : G → R such that Suppose Y t spends 0 time on ∂D a.s., i.e.
(iii) E y τ G 0 χ ∂D (Y t )dt = 0 for all y ∈ G and suppose that (iv) ∂D is a Lipschitz surface, i.e. ∂D is locally the graph of a function h : Moreover, suppose the following: (v) φ ∈ C 2 (G\∂D) and the second order derivatives of φ are locally bounded near ∂D Then φ(y) ≥ Φ(y) for all y ∈ G. b) Suppose, in addition to the above, that s. the probability law of Y t for all y ∈ G and (ix) the family {φ(Y τ ); τ ≤ τ D , τ ∈ T } is uniformly integrable with respect to the probability law of Y t for all y ∈ G. Then is an optimal stopping time for this problem.
Next, we provide a theorem verifying that the solution φ(x) to the free boundary problem (6) satisfies the variational inequality (5).
7.4. Convergence of the Iterative Procedure 7.4.1. Proof of the convergence We show that the approximation function G(·) obtained from the iterative procedure converges to the to the implicit equation (14) by using the Banach fixed-point theorem.
Consider a set X = [λ, ∞) which is the domain of λ(·). Since the set R of real numbers is complete, the set B(X, R) of all bounded functions f : X → R is a complete metric space with the supremum norm Note that the set C b (X, R) consisting of all continuous bounded functions f : X → R is a closed subspace of B(X, R), so that, C b (X, R) is also a complete metric space. Hence, the continuous and decreasing function G(λ), which is a solution to the differential equation (11) satisfying should be in C b (X, R).    Table 5 shows the numerical solutions of Φ i (x) (i = 0, 1, ..., 5) for various initial wealth levels.
It shows that the numerical results apparently seem to be convergent and are bounded by U 2 (x) and Φ 0 (x). On the other hand, if we letλ i (i = 1, 2, ...) be the free boundary obtained from the i-th iteration, then we can observe in Table 6 that the numerical results for the free boundary seem to be convergent and stay between the lower bound of λ 0 δ and the upper bound of λ 1 δ (see the inequality (17) in Proposition 3.1) for various unemployment intensity δ's.  Table 5 Numerical solutions for various initial wealth levels: δ = 0.01, β = 0.0371, r = 0.0371, µ = 0.1123, σ = 0.1954, K = 3, I1 = 1, and I2 = 0.10 are used for parameter values.
1. Cocco et al. (2005) define unemployment state as the state of zero income, however, Gakidis (1998) shows that even unemployed individuals may have other sources of income (e.g., income coming from unemployment benefits and social welfare). 3. Obviously, it might be reasonable to consider business cycles (therefore, non-myopic investors) in our model if we want to get some market-consistent results. Usually, researchers take a Markov regime-switching model to investigate the effect of business cycles on investors' optimal behaviors, but solving our problem in a regime-switching framework is much complex because regime risks are added in our model. In this paper we focus on optimal behaviors of myopic investors, and leave solving our problem in a regime-switching framework to the reader as an extension for future research. As a matter of fact, many existing literature analyze myopic investors' optimal behaviors (see, e.g., Gompers 1994, Marston and Craven 1998, Edmans 2009) and find out some economically meaningful results from their behaviors.
4. The forced unemployment time considered in this paper is not an optimal stopping time given by the set of information of stock price movements but a random time, resulting in market incompleteness.
5. We can allow a correlation between the Brownian motion and the Poisson arrival event as follows: assume that where dδ t = δ t (µ δ t dt + σ δ t dW * t ).
Here, µ δ t and σ δ t are functions with respect to time variable t and W * t is a standard Brownian motion such that dW t · dW * t = ρdt, for ρ ∈ [−1, 1].
Article submitted to Operations Research; manuscript no. OPRE-2013-08-441.R2 In this case, however, the stochastic intensity δ can violate the condition of ∞ 0 δ t dt ≤ 1, because the intensity δ is a stochastic process with a random drift. We leave the problem in the presence of a correlation between the Brownian motion and the Poisson arrival event as an extension for future research.
6. Bodie et al. (1992) and Choi et al. (2008) allow a continum of choice between labor and leisure.
However, Farhi and Panageas (2007) and Dybvig and Liu (2010) support empirical evidence that labor supply is largely indivisible. Concerning leisure choice (or equivalently, labor supply), we take the assumption of Farhi and Panageas (2007) and Dybvig and Liu (2010).
7. If we take the definition of unemployment in Lynch and Tan (2011), I 2 is about 10% of I 1 .
8. It is possible to consider perceived unemployment risk in determining the discount rate. For instance, we can replace r by r + δ, and show the discount rate of this type in the individual's modified objective function (3).
9. See Dybvig and Liu (2010), and Park and Jang (2014) to find the effects of various wealth constraints, e.g., a non-negative wealth constraint and a negative wealth constraint respectively, on an individual's optimal behaviors. Investigating the effect of a non-negative wealth constraint with unemployment risk would be an important research subject, and it was examined by Jang and Park (2015).
10. For the details of the derivation, see an online Appendix.
11. It guarantees the boundedness of the first derivative of the value function everywhere except zero. Roughly speaking, the investor with such preference is risk-averse for positive values of wealth and indifferent for how much they borrow up to the wealth constraint of X(t) > − I 1 r . Piecewise connected utility functions of this kind are seen in lots of articles in economics. For example, Venter (1983) introduces a utility function which is constant for negative values of wealth, and also says that it is designed to reflect bankruptcy laws.
12. Bensoussan and Lions (1982) showed the connection between optimal stopping and variational inequality. For the details of the derivation of the variational inequality for our problem, see Appendix 7.3.
13. For the details, see Theorem 7 in Appendix 7.3.
14. In Appendix 7.3, we provide the verification theorem for optimal stopping.
15. The reason for why we call the function G the convex-dual function is that it is a dual function of the marginal utility of objective function φ and a convex function. Actually, the convexity of G is a conjecture and should be verified. As we compared to the traditional convex-duality approach The research of the 1st author is supported by the Research Grants Council of HKSAR (CityU 500111).