Mathematical Social Sciences 62 (2011) 126–129

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Mathematical Social Sciences

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On the reservation wage under CARA and limited borrowingChristian Bauer ∗

Department of Economics, Ludwig-Maximilians-University Munich, Akademiestr. 1 / II, 80799 Munich, Germany

a r t i c l e i n f o

Article history:Received 17 March 2010Received in revised form29 September 2010Accepted 8 October 2010Available online 14 July 2011

a b s t r a c t

An individual’s optimal behavior in a continuous-time sequential job search model with savings andCARA preferences is characterized analytically. I isolate the effects of limited borrowing and nonnegativeconsumption as well as risk aversion on the reservation wage by using a system of ordinary differentialequations.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Recent empirical research has substantiated the existence ofstrict borrowing limits for a significant fraction of unemployedworkers (Browning and Crossley, 2001; Sullivan, 2002, 2008; Ren-don, 2006). Borrowing limits are important in explaining observedpatterns of job finding and search behavior (Algan et al., 2003;Bloemen and Stancanelli, 2001, 2005; Chetty, 2008; Stancanelli,1999). Chetty (2008), for example, shows that 60% of the increasein unemployment durations caused by unemployment insurancebenefits is due to the liquidity effects of the benefits rather thanthe distortion of the relative price of consumption and leisure. In asimilar vein, Rendon (2006), estimating a search model with lon-gitudinal survey data, argues that larger initial wealth and free ac-cess to liquidity increase the expected duration of unemploymentas well as future wage incomes.

This paper studies the implications of a borrowing limit onthe optimal behavior of a risk-averse worker in a continuous-timesequential job search model with savings. The goal is to provide arigorous analytical treatment of the impact of an unemployedindividual’s financial wealth on consumption and reservationwage, deciphering the effects of lower bounds on wealth andconsumption as well as risk aversion. A detailed understandingof these effects is important to assess real-world unemploymentpolicies which increasingly take an individual’s wealth intoaccount. This requires dynamic models where unemployedindividuals are not restricted to consume their benefits in eachinstant.

Allowing workers to borrow and save in an environmentwith search frictions in general poses an obstacle to theanalytical characterization of optimal behavior. An elegant wayout, employed by recent theoretical research, is to assume

∗ Tel.: +49 89 2180 6752.E-mail address: christian.bauer@lrz.uni-muenchen.de.

0165-4896/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.mathsocsci.2010.10.005

constant absolute risk aversion (CARA) preferences, unlimitedborrowing, and no nonnegativity constraint on consumption(see, e.g., Acemoglu and Shimer, 1999; Shimer and Werning,2007, 2008). Under this specific set of assumptions, a closed-form solution for consumption and the reservation wage exists.This solution includes the following properties: wealth entersconsumption linearly, wealth does not affect the reservationwage,and wealth and consumption fall below any real number withpositive probability.

Motivated by the empirical relevance of borrowing limits, andtheir implications for public policies, I characterize an unemployedindividual’s optimal behavior imposing CARA preferences and anexogenous lower bound on wealth. The immediate implicationis that consumption is no longer linear in wealth. Intuitively,the borrowing limit rules out debt levels of unemployed workerswhich are implied by a linear solution, while leaving consumptionof wealthy individuals relatively unaffected. This nonlinearityturns the reservation wage into a function of wealth.

The paper characterizes an individual’s optimal behaviorwithout imposing a linear solution. By doing this, it provides aformal proof of the view that limited borrowing by itself mitigatesthe impact of unemployment benefits on current consumption andthus leads to a declining sequence of reservation wages over theunemployment spell. The intuition is that, since limited borrowingincreases the marginal propensity to consume out of wealth aswealth declines, the reservation wage optimally declines withfalling wealth so as to offset the increase in marginal utility.

The remainder of the paper proceeds as follows. Section 2describes the model and its solution, first using a deterministicwage offer (Section 2.1), then including a wage offer distribution(Section 2.2). Section 3 concludes. All proofs are relegated to theAppendix.

2. Model and solution

Consider a continuous-time partial equilibrium model ofsequential job search with savings. An unemployed worker

C. Bauer / Mathematical Social Sciences 62 (2011) 126–129 127

receives job offers with random wage draws w at Poisson rate α.1There is no job separation and no on-the-job search. The individualmaximizes her expected lifetime utility at time t ,

U(t) = Et

∫∞

te−ρ(τ−t)u (c(τ )) dτ , (1)

where u(c) = −e−γ c is the instantaneous utility from consump-tion c and ρ is the discount rate. The individual has access to ariskless asset at rate r . Throughout the paper, the borrowing andlending interest rates are assumed to be equal. Moreover, the uni-form interest rate is equal to the rate of time preference (r = ρ).2

The unemployed individual receives constant benefits b. Usingconsumption as the numéraire, her asset holdings evolve accordingto

au = ra + b − cu. (2)

When employed, the worker lives in a stationary world. Hence,ce = ra + w, gae = 0, and the value of being employed isV e(w, a) =

u(ce(w,a))r (see Appendix A.1). When unemployed, the

worker chooses consumption and a reservation wage to maximize(1) subject to (2) and a no-Ponzi game condition that limits thegrowth rate of debt to r (at this stage a, c ∈ R).

2.1. Deterministic wage offer

Suppose that there is only a single deterministic wage offer w.Before turning to the general solution, I followShimer andWerning(2008) and derive a closed-form solution by going through averification theorem (cf. Merton, 1969, 1971, among others). Ifan optimal Markov control exists, the Hamilton–Jacobi–Bellman(HJB) equation is rV u(a) = maxcu [u(cu) + (ra + b − cu)V u′(a)],where V u(a) is the value of being unemployed when holding aunits of wealth. The first order condition (f.o.c.) for consumption,u′(cu) = V u′(a), defines the policy function cu = cu(a). We easilyverify that the linear guesses (g for guess and the δi’s are constants)cug = δ0a+δ1 andV u

g (a) = δ2u(cug ) satisfy both the f.o.c. and theHJBequation if δ0 =

1δ2

= r and δ1 solves (b − δ1)γ rα

= e−γ (w−δ1) − 1,which determines a unique b < δ⋆

1 < w.3 We will see below thatδ⋆1 , i.e. the fraction of labor income used to finance consumption,is equal to the reservation wage in the case where wages aredrawn from a distribution and there is no lower bound on wealthand consumption. This linear solution (which e.g. corresponds toShimer and Werning (2008) without a wage distribution) impliesau = ra + b − (ra + δ⋆

1) = b − δ⋆1 < 0. Wealth falls during

the unemployment spell and consumption declines along with it.Consumption becomes negative at a = −

δ⋆1r . If we impose ad

hoc that consumption must be nonnegative, debt still continues togrow if it reaches δ⋆

1r (if cu = 0, au = ra+ b < 0 for a < −

br ). Debt

exceeds any real number with positive probability.We now turn to the general solution. That is, we describe the

optimal behavior of an unemployed worker by a two-dimensionalsystem of differential equations. The first equation is given by thelaw ofmotion forwealth in (2). The second equation is given by the

1 Under CARA, recall of offers is not optimal and can be ignored.2 These assumptions are restrictive but necessary to maintain analytical

tractability. If, in a general equilibrium framework, the interest rate is endogenized,it will generally fall short of the rate of time preference since the borrowingconstraint implies an additional motive to save. I thank an anonymous referee forpointing out this limitation (cf. footnote 5).3 The left hand side falls in δ1 , intersecting the horizontal axis at δ1 = b from

above. The right hand side is monotonically increasing in δ1 , equals e−γw at δ1 = 0,and intersects the horizontal axis at δ1 = w from below.

Fig. 1. Consumption and wealth dynamics.

evolution of optimal consumption (see Appendix A.2):

cu = −α

γ[1 + u(ra + w − cu)]. (3)

The dynamics of the system are illustrated in Fig. 1. Wealthis constant on ζa(a) ≡ ra + b. Consumption is constant onζc(a) ≡ ra + w. Consumption and wealth diverge from ζc andζa, respectively. Hence, the optimal path lies between ζa(a) andζc(a) (consumption is a normal good).4 The closed-form solutionabove is the trajectory consistent with the terminal conditioncu

−

δ⋆1r

= 0. The dashed line depicts the solution including

a nonnegativity constraint on consumption. Suppose further thatwealth must not fall below a ≡ −

br . Such a borrowing limit

provides a globally saddle-path stable steady state at (c, a) =

(0, a). The policy function must then be strictly increasing andstrictly concave in wealth, see Fig. 1. Given initial asset holdingsa(0) ≥ a, consumption starts on this trajectory and declinestoward the steady state. If the unemployed does not leaveunemployment, her consumption level reaches zero in finite time,and then remains constant until she finds a job. Intuitively, limitedborrowing rules out debt levels which are implied by a linearsolution while leaving consumption of very wealthy individualsvirtually unaffected.

2.2. Random wage offers

With this background, consider wage offers drawn from adistribution F(w) with support [0, ∞) and F ′(w) > 0 for somew > 0. This gives rise to a reservation wage w that leaves theindividual indifferent between accepting the job and remainingunemployed: V e(a, w) ≡ V u(a). Implicit differentiation usingV e(a, w) =

u(ra+w)

r (w ≥ w) and V u′(a) = u′(cu(a)) shows

dwda

=

[u′(cu(a))

u′(ce(a, w))− 1

]r. (4)

The reservation wage is independent of wealth if cu(a) = ce(a, w)and increasing in wealth if cu(a) < ce(a, w). If the model is solvedwithout lower bounds on wealth and consumption, guessing andverifying a linear policy function in fact yields cu⋆(a) = ra +

w⋆= ce(w⋆, a) (Shimer andWerning, 2008, Proposition 1), so that

dwda = 0.5 As shown in Appendix A.1, the reservation wage is then

4 Above ζc(a), cu → ∞ while a → −∞. Below ζa(a), cu → −∞ while a → ∞.Neither case can be part of a feasible optimal program.5 If r = ρ, an analogous condition to (4) applies, but an employed individual

no longer holds a constant amount of assets (e.g., with a deterministic wage offer,optimal consumption evolves according to ce =

r−ρ

γand cu(a) =

r−ρ

γ−

αγ[1 +

u(ce(a; w) − cu(a))]). Then, a system of partial differential equations, together

128 C. Bauer / Mathematical Social Sciences 62 (2011) 126–129

determined by

w⋆= b +

α

γ r

∫∞

w⋆

[1 + u(w − w⋆)]dF(w) (> b). (5)

Implicit differentiation proves that w⋆ decreases uniformly in theArrow–Pratt measure of absolute risk aversion at all wealth levels,∂w⋆

∂γ< 0. Analogously to the previous section, w⋆ > b implies

−a > x for all x > 0 with positive probability. Lower boundson wealth and consumption require optimal consumption to benonlinear and turn the reservation wage into a function of wealth.

To prove this assertion, consider again the evolution of optimalconsumption (see Appendix A.2):

cu = −α

γ

∫∞

w(a)

[1 −

u′(ce(a, w))

u′(cu(a))

]dF(w). (6)

Using u′(c) = −γ u(c) and ce = ra + w in (6), consumption isconstant on

cu0 (a) = ra + (u)−1∫

∞

w(a)u(w)

dF(w)

1 − F(w(a))

, (7)

i.e. if the instantaneous income from benefits and dissaving (b −

a = cu − ra) equals the certainty equivalent of the lottery overfuture wages.

Proposition. Suppose a ≥ a ≡ −br and c ≥ 0. Then, consumption,

wealth, and the reservation wage are strictly declining over theunemployment spell until the stationary point (c, a) = (0, a) isreached. The reservation wage (which is increasing in wealth) exceedsthe unemployment benefits at all wealth levels.

The proof is in Appendix A.3. To interpret this finding, noticethat an increase in unemployment benefits raises cu⋆ one-by-one

∂cu⋆∂w⋆

∂w⋆

∂b = 1. With sufficient liquidity, unemployment benefits

provide a strong incentive to remain unemployed, waiting forbetter offers. Limited access to borrowing mitigates this ‘‘benefit-to-mouth’’ effect and increases the need to accept a job as wealthdeclines.

3. Conclusion

The paper characterized optimal behavior of an unemployedindividual in a standard sequential job search model with savingsand CARA preferences, where the only history dependence comesfrom the individual’s financial wealth. In the benchmark casewhere a, c ∈ R, consumption is linear. This implies that changesin unemployment benefits and absolute risk aversion affectconsumption equally at all wealth levels and that the reservationwage is constant over the unemployment spell.

The general policy function for consumption is given by thesolution to a two-dimensional system of ordinary differentialequations in wealth and consumption. Along the specific solutionconsistent with limited borrowing, the lower bound on wealthmitigates the impact of unemployment benefits on currentconsumption and, with consumption being concave in wealth,leads to a declining sequence of reservation wages over theunemployment spell.

More research, both theoretical and empirical, into theinteracting effects of savings and risk aversion on search behavioris needed to put far-reaching unemployment policies, which takeindividuals’ wealth holdings into account, on solid ground.

with the indifference condition characterizing w, must be solved. The impact of theborrowing constraint described above can be expected to carry over qualitativelyto this case, since, with CARA utility and a, c ∈ R, r = ρ only shifts the levels ofconsumption. For this benchmark case, Shimer and Werning (2008) show that alevel-shift does not affect the reservation wage because it multiplies intertemporalutility by a constant.

Acknowledgments

Financial support from the Deutsche ForschungsgemeinschaftthroughGRK801 is gratefully acknowledged.Many thanks to KlausWälde for his encouragement and suggestions. I am grateful fordetailed comments by an anonymous referee and an associateeditor. This manuscript has also benefited from comments by LutzArnold, Alfred Maußner, Andrea Schrage, and Nicolas Sauter.

Appendix. Proofs

A.1. Optimal behavior

The value function of an employed worker is rV e(a) =

maxce{u(ce)+[ra+w− ce]V e′(a)}. The f.o.c. reads u′(ce) = V e′(a).The ‘‘educated guesses’’ ceg = γ0a + γ1 and V e

g (a) = γ2u(ceg) solvethe f.o.c. and the HJB equation if γ0γ2 = 1 and r

γ0= 1 − [(r −

γ0)a+w −γ1]γ2. As wealth drops out for γ0 = r, ce = ra+w andV e

=u(ce)r (anaturally exceeds the lowest admissiblewealth level).

If a, c ∈ R, the unemployed’s HJB equation rV u= maxcu{u(cu) +

(ra + b − cu)V u′(a) + α

∞

0 max[V e(a, w) − V u(a), 0]dF(w)} andf.o.c., V e

=u(ce)r , and V e(w, a) ≡ V u(a) analogously verify cu =

κ0a + κ1, V u(a) = κ2u(cu(a)) for κ0 =1κ2

= r and κ1 = w⋆.Substituting these expressions and the f.o.c. in the HJB equationdelivers (5).

A.2. Evolution of optimal consumption

Substituting u′(cu) = V u′(a) in the unemployed’s HJB equation,differentiating, and using the f.o.c., and −α[V e(a, w(a)) −

V u(a)]w′(a) = 0 gives

[ra + b − cu(a)]V u′′(a)

= −α

∫∞

w(a)[V e′(a, w) − V u′(a)]dF(w). (A.1)

Differentiating V u′(a) using the Change of Variable Formula (CVF,cf. Sennewald and Wälde, 2006; Sennewald, 2007a,b; Øksendal,2003) and canceling 1 − F(w(a)) yields

dV u′(a) = V u′′(a)dau

+

[∫∞

w(a)[V e′(a, w) − V u′(a)]dF(w)

]dqα

where dqα is the increment of the Poisson process. Substitutingwith the f.o.c.’s in both employment states, dau = (ra + b −

cu)dt , and using the resulting expression in (A.1) (dqα = 0 for anindividual who does not find a job), we get

du′(cu) = −α

∫∞

w(a)[u′(ce) − u′(cu)]dF(w)dt. (A.2)

Let υ ≡ (u′(cu))−1 such that dυ = dcu. Applying the CVFto υ yields dυ(u′(c)) = υ ′(u′(c))du′(cz(a))dt + [υ(u′(cu)) −

υ(u′(ce))]dqα . Using dυ = dcu, υ ′= u′′(cu(a))−1, dqα = 0,

and du′(cu) from (A.2), we find cu = −α

u′′(cu)

∞

w(a)[u′(ce) −

u′(cu)]dF(w). Inserting u′′(cu) = −γ u′(cu) and u′(ce)u′(cu) = −u(ce −

cu) gives (3).

A.3. Proof of the proposition

To begin, note that cu(a) = ra + b − a(a) ≥ 0 implies cu(a)= 0 since a = −

br and accessing −a(a) > 0 is not feasible at

a = a. Now verify w(a) > b. Naturally, w(a) ≥ b (in particular,accepting w < b at a violates the borrowing limit). Suppose,to the contrary, that w(a) = b. The unemployed’s maximized

C. Bauer / Mathematical Social Sciences 62 (2011) 126–129 129

HJB equation, evaluated at a, is rV u(a) = u(cu(a)) + [ra + b −

cu(a)]V u′(a) + α

∞

w(a)[Ve(a, w) − V u(a)]dF . Substituting with the

candidate solution for w and V u(a) = V e(w(a), a) =u(ra+w(a))

r =

u(0)r implies a contradiction, since

∞

b [u(w − b) − u(0)]dF = 0.Intuitively, by accepting w = b at a = a, the worker is not betteroff, but looses the option of getting w > b.

Consider next the dynamic behavior. First, note that cu0 (a) >

ζa(a). This follows from u′(w(a) − b) =

∞

w(a) u′(w(a)−b)dF

1−F(w(a)) ≥∞

w(a) u′(w−b)dF

1−F(w(a)) = u′(cu0 (a)) (the last equality uses (7) and u′=

−γ u), which, using u′′ < 0 and w(a) > b, implies cu0 (a) ≥ w(a) −

b > 0 = ζa(a). Continuity implies that cu0 > ζa(a) for a slightlylarger than a. For these wealth levels it follows that cu, au < 0since both zero growth loci are unstable (whereby cu < 0 belowcu0 and au < 0 above ζa) and paths above cu0 and below ζa(a) canbe ruled out in analogy to footnote 4. Moreover, for larger wealthlevels a′, optimal consumption rules out cu(a′) = ra′

+ b sincethis would imply consumption to be constant at a′. But then (7)must hold for a′ and the candidate consumption level, i.e. u′(ra′

+

b) =

∞

w(a′) u′(ra′+w)dF

1−F(w(a′)) . This, however, implies

∞

w(a′) u′(b)dF =

∞

w(a′) u′(w)dF , a contradiction. Taken together, cu < cu0 since cu0

is unstable and au < 0 is implied by cu > ra + b at a > a.Hence, au, cu < 0 over the unemployment spell. Together withcu(a) = 0 (< cu⋆(a)), these consumption and wealth dynamicsimply that cu(a) is strictly increasing and strictly concave in (a, c)-space. In particular, the slope of the policy function consistent withthe borrowing limit strictly exceeds the slope of the linear solution,∂cu∂a > ∂cu⋆

∂a = r at all finite wealth levels (and approaches r fromabove as a → ∞, i.e. when the impact of current labor incomeand the borrowing constraint become negligible and cu and cu⋆converge). Since w′ is continuous and, at the lowest feasiblewealthlevel, w′(a) > 0 (from (4) since u′(0) > u′(ra + w(a))), excludingw′

= 0 at any a′ implies w′ > 0 at all finite wealth levels. Suppose,to the contrary, that w′(a′) = 0. Then, (4) implies cu(a′) = ra′

+ w,

i.e. ∂cu∂a (a′) = r , a contradiction. Thus, w′(a) > 0, so that the

reservation wage is strictly decreasing over the unemploymentspell since au < 0.

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