DP2016-14

End ogenous Procyclicality of Labor
Produ ctivity, Employm ent, Rea l Wages
and Effort in Conditionally
Heteroskeda stic Su nspots
Unemployment Business Cycles with
Negishi-Solow Efficiency Wages
Jean-Michel GRANDMONTMarch 26, 2016

Endogenous Procyclicality of Labor
Productivity, Employment, Real Wages andEffort in Conditionally Heteroskedastic
Sunspots Unemployment Business Cycles with
Negishi-Solow Efficiency Wages∗

Jean-Michel Grandmont (1)March 2016∗

Early versions of parts of this work have been presented in various venues (Graig
Hiemstra Memorial Lecture, Annual SNDE Symposium, ENSAE, Paris ; Portuguese Economics Journal Annual Meeting, Plenary Lecture, Evora, Portugal; Augustin Cournot
Doctoral Days Lecture, Strasbourg ; Micro/Macroeconomics of the Labor Market International Meeting, Keynote Lecture, Cergy-Pontoise University, France). And in various
international conferences and seminars : at the universities of Tokyo, of Kyoto and of
Kobe in Japan; at the university of Leeds, UK; at the Center for Nonlinear Dynamics in
Economics and Finance (CeNDEF) in Amsterdam; at GREQAM in Marseille and at theCenter for Advanced Studies in Moscow. I am grateful for the comments of the participants of these conferences and to the members of CREST Macroeconomics seminar, andparticularly to Frédérique Bec, Xavier d’Haultfoeuille, Guy Laroque, Etienne Lehmann,
Jean-Michel Zakoian for helpful suggestions. I am also very grateful to Fanda TRAORE
for her efficient typing. As usual, I am sole responsible for any remaining shortcomings ofthe paper.
1
ICEF, Department of Economics, University Ca’ Foscari di Venezia at San Giobbe,
Italy, CREST (EXCESS, UMR CNRS 9194), Paris and RIEB fellow, University of Kobe,
Japan.

1

Abstract
This work introduces a new mechanism that is able to generate procyclicalcomovements of aggregate labor productivity, employment and real wages,
through endogenous variations of workers’ effort, in a simple model involving
structural unemployment, efficiency wages, financial market imperfections
and expectations driven conditionally heteroskedastic sunspots business cycles, near a locally indeterminate steady state. Owing to imperfect effort
monitoring, workers’ effort level equates their disutility of effort to their expected utility gain of not shirking, in terms of their earned real income,and of the resulting anticipated random consumption. A positive current(consumption) sunspot shock generates a countercyclical uncertainty shock,i.e. a drecrease of the anticipated sunspot volatility, and makes risk averse
workers more willing to provide ”precautionary effort” by increasing their
expected utility gain of not shirking. If workers’ relative prudence is small
and decreasing fast near the steady state, profit maximizing firms’ choice ofefficiency wage contracts generates significant endogenous procyclical variations of effort and employment, in particular when the capital-efficient labor
elasticity of substitution is smaller than 1.
Keywords : efficiency wages, unemployment, expectation driven business cycles, conditionally heteroskedastic sunspots, countercyclical uncertainty shocks, prudence, procyclical labor effort and productivity.
JEL classification : E00, E24, E32, J41.

1

Introduction

The basic motivation for the introduction of technology shocks in real business cycles (RBC) macroeconomic models was essentially to be able to generate more or less pronounced procyclical comovements of labor productivity,
output, employment and real wages. With a standard neoclassical constantreturns to scale aggregate production function yt = AF (kt−1 , lt ), where yt
is output, kt−1 the initial capital stock and lt labor supply, aggregate laborproductivity as well as real wages are bound to be strongly countercyclicalsince AF (kt−1 , lt )/lt and AFl0 (kt−1 , lt ) must go down when lt goes up. By
contrast, these variables are empirically more or less procyclical, with employment (hours), output, being significantly more variable than observed1labor productivity, while evidence on real wages seems to be more ambiguous, as they may be strongly or only mildly procyclical, and in some cases
actually acyclical or even a little countercyclical (see e.g. Basu et al. (2001),
Bils et al. (2001), Messina et al. (2009), Solon et al. (1994), Uhlig et al.
(1996), Uhlig (2004), Verdugo (2016)). The way out in RBC models has been
to postulate exogenous (unexplained) stochastic shocks to either the global
productivity parameter At or to analogous technological parameters characterizing the input (capital and/or labor) productivities in the aggregate
production function F .
The alternative strategy chosen here is to take into account the fact thatthe labor input that enters the productive process is in the actual world a mixture of ”hours” worked and of ”effort”. Specially, ”efficient labor” is viewed as
lt = nt xt , where nt is the number of labor units employed (”hours”) and xt is”effort” that in endogenously supplied by workers. The issue is then whether
stochastic variations of the workers’ effort xt can generate empirically plausible comovements of labor productivity αt = yt /nt , employment nt (”hours”),
real wages, output. We also focus on the apparently relevant configuration
where such fluctuations are driven by endogenous self-fulfilling expectations(sunspot) shocks near a locally indeterminate steady state, rather than byexogenous shocks to fundamentals. Finally, we incorporate structural persistent unvoluntary unemployment in the picture by building on the insightful
efficiency wage arguments introduced long ago by Negishi (1979) and Solow(1979) (see also Yellen (1984)). The basic observation here is that in orderto show convincingly that Keynesian unemployment can persist, it is notenough to claim that employed workers would resist wage cuts, e.g. throughunions negotiations. One must explain why firms themselves are unwillingto cut down wages or hire unemployed workers at a lower wage. The core
efficiency wage argument is, in a nutshell, that workers’ effort supply (herext ) is bound to depend on real or nominal wages, so that firms’ profit maximization should lead them to choose persistently high real efficiency wage
contracts, explaining indeed structurally persistent unvoluntary unemployment. Firms empirical surveys evidence seems to confirm the strong presenceof such a mechanism in their wage setting and management behavior (Bewley(1999), Campbell et al. (1997), Stiglitz (1986)). There has been accordingly
a significant literature on the incidence of efficiency wages on the dynamics
of business cycles, but it has focused essentially on a RBC methodology, i.e.
on fluctuations generated by shocks to fundamentals (Collard et al. (2000),2Danthine et al. (1990, 2006), Uhlig (2004), Uhlig et al. (1996)). On the other
hand, there has been quite a few recent studies of the dynamic prperties of
expectations sunspots driven business fluctuations in the presence of unemployment, but they have essentially considered the case where wages are set
through negotiations with labor unions (Dos Santos Ferreira (2015), Dufourt
et al. (2008, 2009, 2013)). The present work seems to be the first to analyze the cyclical dynamical macroeconomic properties of expectations driven
sunspot business fluctuations, in the presence of persistent unemployment
based on explicit efficiency wage contracts arguments.
Specifically, we consider variable effort and efficiency wages as previously
introduced in Grandmont (2008) in a standard finance constrained economy,
as in Woodford (1986) and Grandmont et al. (1998), with a competitivemarket for output, two assets (capital and money) and heterogeneous households. At a deterministic steady state and nearby, ”capitalists” hold capital
as a positive interest bearing asset, and no money, which does not. ”Workers”, who are more impatient and face a finance (cash in advance) constraint,
hold money and no capital. Efficiency wages are introduced in a way that is
directly inspired from the ”shirking” formulation of Coimbra (1999), Alexopoulos (2004), Nakajima (2006) and Aloi et al. (2006), following the early
contribution of Shapiro et al. (1984). See also Danthine et al. (1990), Jullien
et al. (1998). Efficiency wage contracts specify the wage and the required effort level. There is imperfect monitoring : an employed worker who ”shirks”
faces a positive probability of getting caught, in which case he or she is fired
and gets unemployment insurance. Profit maximizing firms choose accordingly efficiency wage contracts involving reals wages high enough so as toinduce employed workers to provide the requested effort level, i.e. not to”shirk”. Hence the persistent occurrence of real wage downward ”rigidities”
and a resulting structural unemployment. To simplify matters, we abstract
away from any additional ”frictional” labor market imperfections such as
matching or search.
It was shown in Grandmont (2008) that the deterministic dynamics ofsuch an economy is bound to lead to a locally indeterminate steady state under a wide range of empirically relevant configurations. What we do here is toanalyze the dynamical cyclical properties of stochastic sunspot equilibria nearsuch a steady state. We first make precise the conditions needed to mae thedynamics of the economy, in particular of the firms’ profit maximizing choiceof efficiency wage contracts, well defined in such a stochastic context (section
32.1). We show next how to construct, and prove the existence of, stochastic
sunspot equilibria near a locally indeterminate steady state by adapting themethods and the results of Grandmont et al. (1998) (section 2.2). We thenshow that endogenous procyclical variations of workers’ effort (here xt ) areindeed able to generate a range of more or less significant procyclical comovements of aggregate labor productivity (here yt /nt ), employment nt (”hours”),
or real wages, that is wide enough to fit empirical observations, in particularin the empirically plausible case where the capital-efficient labor elasticity of
substitution is smaller than 1 (see Chirinko (2008)) (section 3.1.). We show
finally (section 3.2) that conditionally heteroskedastic sunspot shocks to theworkers’ consumption process do generate procyclical effort movements under natural conditions on their attitudes toward risk. The key observation
there is that the worker’ ”non-shirking schedule” determines at any time theireffort level by equating their disutility of effort to their expected utility gainof ”not-shirking”, expressed in terms of the wage or unemployment insurance
received, and thus of their ensuing anticipated random consumption. Effort
is thus similar to ”precautionary saving” in the sense that it enables workers to cope with the uncertainty of their future consumption, in the spiritof the early contributions of Leland (1968), Kimball (1990). If a positive
current (workers’ consumption) sunspot shock generates a countercyclicaluncertainty shock, i.e. a decrease of the anticipated sunspot volatility, thenit increases risk averse workers’ expected utility gain of ”not-shirking” for agiven effort level, and thus makes them more willing to provide ”precautionary effort” (as in Rothshild et al. (1970)). What we show here is that when
consumption sunspot shocks are conditionally heteroskedastic as above, andif the workers’ relative prudence is smaller than 2 and decreasing fast near
the steady state, the firms’ profit maximizing choice of efficiency wage contracts under the workers’ non-shirking schedule is indeed able to generate
the significant procyclical variations of effort we were looking for.
The ”real” mechanism put forward here is based on an analogy of workers’ effort with precautionary savings in the face of an uncertain future that
seems highly plausible. Beyond the voluntarily oversimplified structure of themodel that makes it unfit to full fledged empirical applications (e.g. nominal
variables behave badly, i.e. countercyclically, as in any cash in advance modelwith a constant money stock), the real mechanism linking conditionally heteroskedatic sunspot consumption shocks to procyclical ”precautionary effort”
seems to be empirically quite plausible. Indeed, while the stochastic nature4

of the consumption process has been recognized for long (Hall (1978)), conditional heteroskedasticity of uncertainty or volatility shocks seems to getincreasing theoretical and empirical attention. It has been long viewed asstrongly empirically relevant in the field of finance and the analysis of asset prices and stock returns (Francq et al. (2010), Nelson (1991)). Recent
findings that appear to uncover links between the stochastic properties ofconsumption, countercyclical macroeconomic uncertainty shocks and conditionally heteroskedastic financial shocks (Boguth et al. (2013), Bloom et al.
(2014), Tedongap (2016), Wang (2006)), do suggest that investigating further theoretically and empirically the real mechanism presented in this work
should be a fruitful avenue for future research.

2

Stochastic expectations driven unemployment business cycles

We define and characterize in this section stationary stochastic sunspot equilibria near a deterministic steady sate of the simple economy introduced inGrandmont (2008), that involves efficiency wages in a dynamical macroeconomic model with heterogeneous agents and financial constraints as in
Woodford (1986) and Grandmont et al. (1998).

2.1

Agents behavior

There are three types of agents : capitalists, workers and firms.
Capital holders
There is a continuum of identical small agents, of total size normalized to1, whose incomes are essentially generated by the returns of holding physicalcapital. To simplify matters, one considers the limit case where these agents
do not work, and are thus called ”capitalists”. They are further assumed
to maximize expected discounted intertemporal utility of consumption ct+j,cj
of the Cobb-Douglas type Et [Σ∞j=1 (βc ) log ct+j,c ], where the discount factor
satisfies 0 < βc < 1, given current and (possibly random) money prices ofconsumption (pt+j ), and real gross rates of returns on capital (Rt+j ), j ≥ 0.
There are two available assets : money, that does not bear interest, andcapital. We assume rational expectations and focus on equilibria near a5deterministic steady state satisfying Rt+1 = R̄ = 1/βc > 1 = pt /pt+1 for allt, so that Rt+j+1 > pt+j /pt+j+1 will hold along deterministic or stochastic
equilibria nearby for all t, j. Money is thus dominated as an asset along such
equilibria and capitalists choose accordingly to hold capital only, leading tobudget constraints of the form ctc + ktc = Rt kt−1,c in every period t and to
optimal consumption and capital investment plans given by
ctc = (1 − βc )Rt kt−1,c ,ktc = βc Rt kt−1,c .

(1)Workers
There is another continuum, also normalized to 1, of small identicalagents, called ”workers”. Each individual worker supplies one unit of labortime, but effort supply xt ≥ 0 varies continuously and endogenously, while aworker’s consumption is noted ctw ≥ 0. Instantaneous utility of consumption
is noted U (ctw ) and disutility of effort V (xt ). Workers are assumed to maximize expected discounted utility over an infinite horizon, with a discountfactor 0 < βw < 1.
To simplify matters, we assume that there is also a continuum of isolated
identical local labor markets with a single firm dealing in each such local
market and that workers are uniformly spread over them. This is in order
to abstract away from the complexities involved in possible wage competition among firms or in possible workers migration (”search”) accross local
labor markets. Firms’ maximizing profit behavior (to be described precisely
shortly) will result in an efficiency wage contract (wt , xt ) specifying the nominal wage wt > 0 paid to an employed worker, under the requirement thatthe effort level xt > 0 is to be provided. Firms’ behavior will generate also ademand nt > 0 of labor units. We focus here on unemployment, so at date t,
the proportion 0 < nt < 1 of individual workers is offered the efficiency wage
contract (wt , xt ). As we shall see, such a contract will be designed to induce
the workers to make indeed the requested effort level (”incentive compatibility”) : no worker will turn down the job offer, so that in any period there are
0 < nt < 1 workers employed and 0 < 1 − nt < 1 unvoluntarily unemployed.
There is unemployment insurance, so employed workers are paid the nominal
wage wt > 0, while unemployed workers get the income 0 < νwt < wt , therate of compensation 0 < ν < 1 being constant and known by all agents.
To simplify matters, it is assumed that unemployment insurance is fundedby taxing the incomes of employed and unemployed workers at the uniform
6

rate 0 < 1 − dt < 1. Then by definition, what is taxed away from employedworkers is equal to the disposable income of unemployed workers, that is(1 − dt )nt wt = (1 − nt )dt νwt , which determines the rate dt from the wage
rate, the insurance rate and employment (wt , ν, nt ). An important simplifying feature of this approach is that the total amount of after tax income
gotten at t by all, employed and unemployed, workers, must be equal to theinitial pre-taxes wage bill nt wt > 0.
We focus on intertemporal stochastic equilibria near a deterministic steadystate, along which stationary consumptions of employed and unemployedworkers are noted c̄w > 0 and νc̄w > 0. We assume throughoutU 0 (νc̄w ) > U 0 (cw ) > βw R̄U 0 (νc̄w ) > βw R̄U 0 (c̄w ) ,(2)

where the steady state real gross rate of return of capital satisfies R̄ = Rt =1/βc for all t. These inequalities will hold whenever a) workers discount the
future much more than capitalists, 0 < βw < βc < 1, b) the unemployment
insurance rate ν is relatively close to 1, and c) workers marginal utility for
consumption U 0 (c) does not decrease too fast. Condition (2) implies thatfor every possibly random consumption path near that steady state one has
U 0 (ctw ) > βw Et [Rt+1 U 0 (ct+1,w )] : a worker’s marginal utility cost of investingat t in one unit of capital exceeds its marginal expected utility gain, and this,no matter which current or expected employment state for this worker. One
gets accordingly the standard conclusion that the more impatient agents,
i.e. workers here, save by holding cash and no capital, while the situation isreversed for capitalists.
It is further assumed, here as in Woodford (1986) or Grandmont et al.
(1998), that workers are subject to a finance constraint stating that they will
get their disposable money wage or insurance income in cash at the end of
the period but cannot borrow against it. Such a cash-in-advance constraint
is binding at a deterministic steady state and thus nearby. Workers’ behavior
will display accordingly a simple two-periods overlapping generations structure : ”young” workers do keep in period t their current disposable incomein the form of cash balances, in order to spend these to consume at timet + 1, when ”old”. We assume a constant total money stock M > 0, so thatin equilibrium.
MM= nt ωt and= nt ct+1,w + (1 − nt )νct+1,w ,
pt
pt+1
7

(3)

where ωt = wt /pt is the real wage.
Firms
There is a continuum (again, normalized to 1) of identical small firms.
Each firm combines at t the services of initial capital kt−1 ≥ 0 and ”efficient”
labor lt = nt xt ≥ 0 to produce output according to the constant returns toscale production function yt = AF (kt−1 , lt ) ≥ 0. For l > 0, the reduced production function y/l = Af (a) = AF (a, 1) is assumed to satisfy the standard
condition:(2.a) The reduced production function y/l = Af (a) = AF (a, 1) is asmooth function of the capital-efficient labor ratio a = k/l ≥ 0, with f (0) = 0
and f 0 (a) > 0, f 00 (a) < 0 for a > 0. The marginal productivity of capitalAρ(a) = Af 0 (a) decreases from +∞ to 0, whereas the marginal productivityof efficient labor Aω(a) = A(f (a) − af 0 (a)) increases from 0 to +∞, when agoes up from 0 to +∞.
Firms seek to maximize their real profitωtlt − ρt kt−1 .
xt
Given the real rental rate of capital services ρt > 0 and the real wage perunit of effort ωt /xt > 0, a firm’s optimum kt−1 > 0, lt > 0 must satisfy the
first order conditions.
AF (kt−1 , lt ) −ρt = Aρ(at ) > 0 and ωt /xt = Aω(at ) > 0.

(4)

Capital stock is assumed to depreciate at the constant rate 0 < δ < 1. The
real gross rate of return on capital Rt is thus equal to R(at ) = Aρ(at ) + 1 − δ.
Finally, profit maximization makes firms seek to design an efficiency wage
contract (wt , xt ) that in effect minimizes the real cost of efficient labor ωt /xt ,under the constraint that it should maintain the workers’ incentive to provide
actually the effort level xt .
Efficiency wages contracts
The set up considered here, as in Grandmont (2008), is of the ”noshirking” variety, in the spirit of the early contribution of Shapiro and Stiglitz(1984). There is imperfect effort monitoring : a worker hired at t under the8contract (wt , xt ) who shirks, i.e. provides no effort, faces the probability
0 < θ < 1 to be caught. A non-shirking employed worker gets the after tax
wage income dt ωt pt , which he keeps in cash in order to spend it on consumption next period ct+1,w = dt ωt pt /pt+1 . An employed worker who is caught
shirking is fired and receives the after tax unemployment insurance νdt ωt pt ,leading to a consumption one period later equal to cut+1,w = νct+1,w . Effort
involves a disutility V (x), while consumption generates utility U (c). Workers will have no incentive to shirk if the expected utility gain of not shirking
outweights the disutility of effort :V (xt ) − V (0) ≤ θEt [U (ct+1,w ) − U (νct+1,w )] where ct+1,w = dt ωt pt /pt+1 . (5)In period t, each individual small firm takes as given current information
about dt , pt and the (rational) anticipation of the future price pt+1 , hencein particular of the possibly random sunspot market perturbation εt+1 =[(1/pt+1 )−Et (1/pt+1 )]/Et (1/pt+1 ), assumed to be small with Et (εt+1 ) = 0. A
firm’s offer of a wage wt (or equivalently of a real wage ωt = wt /pt ) affects thenits workers’ consumption essentially through the expected mean consumptionc̄t+1,w = Et (ct+1,w ) = dt ωt pt Et (1/pt+1 ), actual consumption being ct+1,w =c̄t+1,w (1 + εt+1 ). As the non-shirking condition (5) is bound to be binding at
the optimum choice, this leads toLemma 1. Given the distribution ε̃t+1 of random (sunspots) market perturbations εt+1 = [(1/pt+1 ) − Et (1/pt+1 )]/Et (1/pt+1 )], with arbitrarily small
support −a ≤ εt+1 ≤ b and Et (εt+1 ) = 0, each firm chooses an efficiency
wage contract at t so as to maximize effort xt ≥ 0 per unit of expected mean
consumption c̄t+1,w = Et (ct+1,w ), i.e. xt /c̄t+1,w , under the non-shirking schedule (N SS)
V (xt ) − V (0) = Φ(c̄t+1,w ; ε̃t+1 )(6)

Φ(c̄t+1,w ; ε̃t+1 ) = Et [ϕ(c̄t+1,w (1 + εt+1 ))] with ϕ(c) = θ[U (c) − U (νc)].

(7)

where

The mechanisms underlying the workers’ (non-)shirking behavior (6),(7)are strongly related to the curvatures of the functions involved. That is a)their aversion toward effort (curvature of V (x)) and b) their attitude toward9risk (curvature of U (c)). The latter goes essentially through the degree ofconcavity of ϕ(c) = θ[U (c) − U (νc)], which stands for the expected utility
gain of making the right effort while taking into account the risk of beingcaught, with probability θ, if shirking, this risk being present even in theabsence of sunspots shocks. In this respect, we shall assume throughout, as
in Grandmont (2008) for the deterministic case εt+1 ≡ 0(2.b) Disutility of effort V (x) is smooth for 0 < x < x̄, where x̄ isthe maximum (possibly infinite) effort level, increasing and strictly convex,
V 0 (x) > 0, V 00 (x) > 0, with V 0 (x) going to +∞ when x tends to x̄. There isa fixed cost of effort, limx→0 V (x) = x0 > V (0) = 0.
(2.c) Utility of consumption U (c) is snooth for c ≥ 0, increasing andstrictly concave, U 0 (c) > 0, U 00 (c) < 0, with marginal utility U 0 (c) decreasingfrom +∞ to 0 when consumption c goes up from 0 to +∞. Utility for consumption displays not too large relative risk aversion, RU (c) = −cU 00 (x)/U 0 (c) <1, or equivalently cU 0 (c) is increasing.
(2.d ) The fixed cost of providing effort is positive but not too large, whereas
the disutility of the maximum effort level x̄ is high0 < x0 < lim ϕ(c) = θ[U (c) − U (νc)] < lim V (x)
c→+∞

x→x̄

Assumption (2.c) implies that ϕ(c) is increasing, and thus that the righthand side of the NSS (6) is also increasing in c̄t+1,w . Adding assumptions(2.b) and (2.d ) ensures next that this NSS (6) determines the effort level xt
as a function of expected mean consumption c̄t+1,w , given the distributionε̃t+1 of the random sunspots disturbance, xt = H(c̄t+1,w ; ε̃t+1 ), where : 1)
H(0; ε̃t+1 ) = 0, 2) xt is well defined whenever c̄t+1,w > c0t+1 where c0t+1 >0 is the minimum expected mean consumption required to cover the fixed
disutility cost of providing effort, i.e. x0 = Φ(c0t+1 ; ε̃t+1 ), in which casext = H(c̄t+1,w ; ε̃t+1 ) = V −1 [Φ(c̄t+1,w ; ε̃t+1 )] > 0 for c̄t+1,w > c0t+1 > 0,(8)

and 3) inf c>c0t+1 H(c̄t+1,w ; ε̃t+1 ) = 0.
Finally, workers’ incentives to provide an effort level xt > 0 in the NSS(6),(7) do bear similarities with the motives to invest in ”precautionary10

savings” when facing risky future income. So we might expect that somemeasure of workers’ attitudes toward risk such as ”prudence” might play arole in the present framework as well. This is indeed the case in the following assumption (2.e) , which ensures that ϕ(c) is strictly concave, i.e.
c2 ϕ00 (c) = θ[c2 U 00 (c) − ν 2 c2 U 00 (νc)] < 0. This implies that Φ(c̄t+1,w ; ε̃t+1 ) has
the same property with respect to c̄t+1,w ≥ 0, as well as H(c̄t+1,w ; ε̃t+1 ) =V −1 [Φ(c̄t+1,w ; ε̃t+1 ] for c̄t+1,w > c0t+1 .
(2.e) cU 0 (c) is strictly concave (i.e. −c2 U 00 (c) is increasing) or equivalently relative prudence is not too large, i.e. PU = −cU 000 (c)/U 00 (c) < 2.
Figure 1
Under these assumptions, the non-shirking schedule (8) can be simply
represented graphically as in Figure 1. It is then not difficult to verify that
under these conditions, profit maximization, as specified in Lemma 1, generates consistently a unique efficiency wage contract at time t (the proof of
the analogous Proposition 1 in Grandmont (2008), Appendix II) for the deterministic case εt+1 ≡ 0, applies to the case at hand with sunspots shocks
with only marginal adaptations).
Proposition 1. Under assumptions (2.b), (2.c), (2.d), (2.e), given
the distribution of anticipated small sunspots random perturbations εt+1 =[(1/pt+1 ) − Et (1/pt+1 )]/Et (1/pt+1 ), there is a unique profit maximizing efficiency wage contract that maximizes effort per unit of expected mean consumption xt /c̄t+1,w > 0 under the non-shirking conditions (6),(7). It is characterized by the first order condition
xt V 0 (xt ) = c̄t+1,w Φ0c (c̄t+1,w ; ε̃t+1 ).

(9)

We shall assume throughout (2.b) to (2.e). In the case of a constantelasticity (CRRA) specification U (c) = c1−η /(1 − η), where η = RU (c) =PU (c) − 1, assumptions (2.c) and (2.e) are satisfied whenever 0 < η < 1.

11

2.2

Stationary stochastic sunspot unemployment equilibria

We characterize now how stationary stochastic sunspot unemployment equilibria are generated arbitrarily near a locally indeterminate deterministic
steady state of this economy. All fundamental real characteristics of thesystem are constant over time. Expectations driven fluctuations are generated by endogenous sunspots shocks in every period to the agents’ expectations about money prices of the good. These sunspots shocks are
summarized by a stationary stochastic process, with arbitrarily small support, {εt = [(1/pt ) − Et−1 (1/pt )]/Et−1 (1/pt )} for all t. From Proposition
1, this determines fully the dynamics of effort and expected mean consumption (xt , c̄t+1,w ), and thus of employed workers’ consumption throughct+1,w = c̄t+1,w (1 + εt+1 ). To complete the description of the dynamics, one
needs to look at the evolution of capital stocks k and of the capital-efficientlabor ratios a = k/l. Specifically,
Definition 1. Given a stationary stochastic process of sunspots shocks
εt = [(1/pt ) − Et−1 (1/pt )]/Et−1 (1/pt ) with arbitrarily small support, the dynamics of the economy is described bya) the stochastic process of effort levels xt > 0 and of employed workers expected mean consumption c̄t+1,w = Et (ct+1,w ), where ct+1,w = c̄t+1,w (1 + εt+1 ),generated by firms’ profit maximizing choice of efficiency wage contracts, as
in Proposition 1,b) the two-dimensional stochastic process of capital stocks kt and of capitalefficient labor ratios at = kt−1 /lt driven by
kt = βc R(at )kt−1 with R(at ) = Aρ(at ) + 1 − δ(10)kt Aω(at+1 )
kt−1
= g(
; xt , c̄t+1,w )(1 + εt+1 )
at+1
at

(11)

where

g(lt ; xt , c̄t+1,w ) =lt
lt
c̄t+1,w + (1 − )νc̄t+1,w .
xt
xt

(12)

Indeed, (10) stand for the capital accumulation equation of capital holders(1), while (11), (12) are mere translations of the money market equilibrium
12

relations (3). The left hand side of (11) being in fact M/pt+1 = nt+1 ωt+1 =lt+1 ωt+1 /xt+1 together with lt+1 = kt /at+1 (by definition), and real wage perunit of effort being equal to Aω(at+1 ) from the firms’ profit maximization(4). The right hand side of (11), (12) being simply the second part of (3),with nt = lt /xt and ct+1,w = c̄t+1,w (1 + εt+1 ).
The deterministic dynamics of the economy is obtained when there are nosunspots shocks to expectations, εt ≡ 0 for all t. In such a case, from Proposition 1, xt = x∗ and ct+1,w = c∗ are constant. The deterministic dynamics (10),(11), (12) reduces then to a two-dimensional deterministic difference equation (kt , at+1 ) = G(kt−1 , at ). Geometrically, stochastic sunspots equilibria
are the result of the composition of this deterministic dynamics in the plane(k, a), followed by a ”vertical” random perturbation of the variable at+1 , that
results from the addition of the sunspot shock εt+1 in (11), together with thesmall stochastic perturbation of (x∗ , c∗ ) to (xt , c̄t+1,w ) resulting from the impact of sunspots shocks on efficiency wage contracts as in Proposition 1. The
constructive analysis of stationary stochastic sunspots equilibria arbitrarilynear locally (in)determinate deterministic equilibria presented in Grandmont
et al. (1998, Section 3) applies therefore formally here. In particular,
Proposition 2. Consider a locally indeterminate deterministic steadystate with εt ≡ 0, xt = x∗ > 0, ct+1,w = c∗ > 0, kt = k̄ > 0, at = ā > 0 for allt, with unemployment n̄ < 1. Then for any arbitrarily small neighborhood ofthat steady state, one can construct infinitely many stationary sunspot equilibria as in Definition 1, that stay in that neighborhood for all t, by choosing
an arbitrary stationary stochastic process of sunspots shocks {εt }, and scaling
it down appropriately.
This follows by applying the arguments of Proposition 3.1 and Theorem
3.2 in Grandmont et al. (1998).

3

Procyclicality of labor productivity, real wage, employment and effort, and heteroskedastic sunspots

We derive in this section conditions under which labor productivity yt /nt , employment nt , real wages ωt are procyclical along stationary stochastic sunspot
13equilibria in this economy. If effort were constant, xt = x∗ for all t, as would
be the case in a standard neoclassical model, efficient labor lt = nt x∗ would
essentially coincide with employment. In such a case, labor productivity
yt /nt = Af (at )x∗ ), real wages ωt = Aω(at )x∗ are anticyclical. Clearly, torecover procyclicality of labor productivity yt /nt = Af (at )xt , of real wages
ωt = Aω(at )xt , without introducing exogenous shocks to the productivity
parameter A, one needs effort xt itself to be variable and endogenously procyclical (Section 3.1). It is also clear from Proposition 1 that one needs
non-vanishing stochastic sunspots shocks εt 6= 0 to be able to generate sucha picture : in the case of the deterministic dynamics of the economy (εt ≡ 0
for all t), effort and employed workers consumption are bound to stay constant, xt = x∗ , ctw = c∗ for all t. By contrast, in the case of non-vanishingsunspots (εt 6= 0), procyclicality of effort xt should obtain if the incentiveof employed workers to provide effort is stimulated when εt goes up today.
From (6),(7), in Lemma 1, one may expect that this will occur, under someconditions on workers’ attitudes toward risk and aversions toward effort, if
a higher current sunspot shock εt generates an anticipation of a less risky
future sunspot shock εt+1 , since then the expected utility of not shirking
Et [ϕ(ct+1,w )] gets larger as illustrated in Fig.1 (Section 3.2).

3.1

Comovements of productivity, employment, real
wages and variable effort

We assume throughout conditions (2.a), (2.b), (2.c), (2.d ), (2.e), and consider a stationary stochastic sunspot equilibrium near a locally indeterminate
deterministic steady state xt = x∗ > 0, ctw = c∗ > 0, kt = k̄ > 0, at = ā > 0
for all t. Such a sunspot equilibrium is constructed, as in Proposition 2 andDefinition 1, from a given stationary stochastic process of small sunspots
shocks {εt }. We look at period t and analyze the consequences of a small positive shock dεt > 0, given history up to time t − 1. This sunspot shock affects
directly workers current consumption ctw = c̄tw (1 + εt ), with c̄tw = Et−1 [ctw ]
dεt
dctw
=
.
(and νctw for those who where unemployed at t − 1), so thatctw
1 + εtFrom the money market equilibrium condition (11) adapted to period t,
kt−1 Aω(at )= nt ωt = (nt−1 + (1 − nt−1 )ν)c̄tw (1 + εt )(13)
at
it is clear that the impact of dεt > 0 will rest significantly on the local
14response of the marginal productivity of efficient labor Aω(a) at the steadystate and nearby. We recall the useful standard relationsεω (a) =s(a)aρ0 (a)1 − s(a)aω 0 (a)=, ερ (a) =
=−
ω(a)
σ(a)
ρ(a)
σ(a)

(14)

where s(a) = af 0 (a)/f (a) is the share of capital in total income and σ(a) isthe capital-efficient labor elasticity of substitution. We assume throughtout(3.a) σ(ā) > s(ā)
at the deterministic steady state and thus nearby. From Grandmont (2008,
Proposition 4) this is a necessary condition for local indeterminacy. It implies
in particular that ω(a)/a is decreasing, so that in view of (13), following thepositive sunspot shock dεt > 0, at = kt−1 /lt goes downdεtdat
σ(at )
dlt
=−
=
>0
lt
at
σ(at ) − s(at ) 1 + εt(15)whereas output yt = AF (kt−1 , lt ) and capital kt = βc R(at )kt−1 go up. On theother hand, as in any standard neoclassical model, efficient labor productivity
αte = yt /lt = Af (at ) should go down. Here one will get that plain laborproductivity αt = yt /nt = Af (at )xt becomes procyclical if effort xt itself issufficiently procyclical. Similarly, in order to get real wages ωt = Aω(at )xt
to become procyclical, one needs also effort to be procyclical enough. By
contrast, in order to get employment nt = lt /xt to be procyclical one needs
that effort xt be not too much procyclical. Specifically, one gets easily bystraight (log)differenciation :Proposition 3. Consider a stationary stochastic sunspot equilibriumnear a locally indeterminate deterministic steady state, driven by the process{εt } of stationary stochastic sunspots shocks. Following a small increasedεt > 0 of the sunspot shock in period t,
1) Labor productivity αt = yt /nt = Af (at )xt will be procyclical if effort xt
dαt> 0 if and only ifincreases sufficiently, i.e.
αt
dxt
dat
s(at )σ(at ) dεt> −s(at )=.
xt
at
σ(at ) − s(at ) 1 + εt

15

2) The real wage ωt = Aω(at )xt will be procyclical if effort xt goes updωt> 0 if and only if
sufficiently, i.e.
ωtdxts(at ) dats(at )
dεt
>−
=
.
xt
σ(at ) at
σ(at ) − s(at ) 1 + εt3) Employment nt = lt /xt will be procyclical if effort xt does not increasednt
too much, i.e.
> 0 if and only if
nt
dxt
dlt
σ(at )
dεt
<
=
.
xt
lt
σ(at ) − s(at ) 1 + εt
The above findings illustrate the potential relevance of introducing variable procyclical effort xt as a key factor toward explaining a few dynamic
properties of observed macroeconomic variables corresponding to αt , nt , wt .
Empirically, these are more or less procyclical, with employment nt being significantly more variable than observed labor productivity αt = yt /nt , whereas
the evidence on real wage ωt seems to be mixed, as it may be only mildlyprocyclical and in some cases actually acyclical or even mildly anticyclical(see e.g. Basu et al. (2001), Bils et al. (2001), Messina et al. (2009), Solonet al. (1994), Uhlig et al. (1996), Verdugo (2016)). The difficulties to obtain such properties within standard neoclassical models or new-Keynesianmodels with nominal rigidities have been a significant motivation to rely on
exogenous shocks to the global productivity parameter A in standard RBC
models. The following corollary demonstrates that one can indeed easily reproduce such empirical regularities through endogenous procyclical variationsof effort xt in an efficiency wages model as considered here, in the empirically
relevant case where the elasticity of capital-efficient labor substitution σ issomewhat less than 1 (see Chirinko (2008)).
Corollary 1. Consider the benchmark configuration where the real wage
s
dεt
dxt
'
,
ωt is approximately acyclical at the steady state, i.e.
xt
σ − s 1 + εtwhere s and σ stand for s(at ), σ(at ) evaluated at the steady state. Employment nt varies then procyclically like the current sunspot consumption shock
εt while the covariation of labor productivity αt = Af (at )xt is given by
dεt
dctw
dnt
σ − s dαt=='.
1 + εtctw
nts(1 − σ) αt16

Labor productivity αt is then procyclical when s < σ < 1, with employment
being significantly more variable than labor productivity when σ is relatively
close to 1. With the income capital share set at s = 0.3, one gets dnt /nt '2.5dαt /αt when σ = 0.6, or dnt /nt ' 8.3dαt /αt when σ = 0.8.
Mild procyclical (resp. countercyclical) variations of the real wage arethen obtained from small increases (resp. decreases) of dxt /xt from the above
benchmark acyclical configuration.

3.2

Conditionally heteroskedastic sunspots and procyclical effort

We study now the mechanisms than can generate a procyclical increase ofeffort dxt > 0 as a result of a positive sunspot consumption shock dεt =dctw > 0 at time t, as described in the previous section. From Proposition
1, along the stationary sunspot equilibrium under consideration, the current
effort level xt and employed workers’ expected mean consumption c̄t+1,w =Et [ct+1,w ] are determined by the anticipated distribution of the random futuresunspot shock εt+1 , through the firms’ choices of an efficiency wage contract.
Clearly, if the given sunspots shocks processes {εt } were i.i.d., the resulting
equilibrium sequence (xt , c̄t+1,w ) would be deterministic and constant overtime. To get procyclical effort variations, it is thus necessary that the current
shock εt affects the distribution of εt+1 . Specifically, in view of the NSSS (8)V (xt ) − V (0) = Et [ϕ(c̄t+1,w (1 + εt+1 ))],

(16)

where the worker’s expected utility of not shirking ϕ(c) = θ[U (c) − U (νc)] is
strictly concave under assumption (2.e) (relative prudence PU (c) =−cU 000 (c)/U 00 (c) is low), we need that a current positive sunspot shock dεt > 0
decreases the risk involved in future consumption ct+1,w = c̄t+1,w (1 + εt+1 ) so
as to increase the workers’ incentives to provide effort (i.e. to make the righthand side of the NSS (16) to go up, as illustrated in Fig.1). We shall assume
accordingly(3.b) The stationary sunspot shocks process {εt } is conditionally heteroskedastic : a positive increase of the current shock εt make the variance of the future shock vt+1 = Et [ε2t+1 ] to decrease, i.e. dεt > 0 implies
dvt+1 = d(Et [ε2t+1 ]) < 0.

17

Assumption (3.b) (heteroskedastic sunspots) and (2.e) (low relative prudence) do ensure that effort per unit of real wage, or of expected mean
consumption xt /c̄t+1,w , will be procyclical. Since current effort xt plays arole similar to ”precautionary saving”, that guarantees the expected mean
consumption c̄t+1,w and aims at coping with the risky future sunspot shock
εt+1 , one should expect that actual movements of xt and c̄t+1,w will depend
significantly on further properties of the workers’ attitudes toward risk oreffort, as they condition the curvatures of the NSS (16) in Fig. 1, and of thefirms’ FOC
xt V 0 (xt ) = Et [c̄t+1,w (1 + εt+1 )ϕ0 (c̄t+1,w (1 + εt+1 ))](17)

As we are going to see shortly, in the constant elasticity specification of
the workers’ utility function U (c), where relative prudence PU (c) is constant,effort xt is actually constant and determined independently of the sunspot
distribution εt+1 , while expected mean consumption c̄t+1,w is countercyclical(Lemma 2). We show by contrast that effort is procyclical (dxt > 0 as aresult of a positive shock dεt > 0) if relative prudence PU (c) is small anddecreasing fast on an interval containing the steady state consumptions c∗and νc∗ . We also show that expected mean consumption c̄t+1,w is procyclicalin such a prudence configuration if the workers’ relative degree of aversion toeffort is small, while it may become countercyclical if it is large (Proposition
4). The strength of these comovements is larger, the larger of the degreeof heteroskedasticity postulated in (3.b). We end up with an illustration byconsidering the HARA family for the workers’ utility function (Corollary 2).
Lemma 2. In the constant elasticity (CRRA) case for the workers’ utility, U (c) = c1−η /(1 − η) with 0 < η < 1, the optimum efficiency wage
conditions (16), (17) reduce to(1 − η)(V (xt ) − V (0)) = θ(1 − ν 1−η )Et [(c̄t+1,w (1 + εt+1 ))1−η ] = xt V 0 (xt ).
The resulting effort is constant and equal to the deterministic steady state
effort level xt = x∗ for all t. Expected mean consumption c̄t+1,w is thencountercyclical.
The proof follows immediately from expliciting the right hand sides of(16), (17), which become in fact identical in this particular specification.
18

To deal with the relevant general case where relative prudence PU (c) is decreasing near the deterministic steady state, we proceed by considering thesecond order Taylor expansions of the dynamics (16), (17) of the efficiency
wage contract (xt , c̄t+1,w ) near that deterministic steady state, with respect tothe small anticipated sunspot consumption shock εt+1 . This makes apparent
the impact of its variance vt+1 = Et [(ε2t+1 )]. To simplify the exposition, when
there is no ambiguity, we use the notations x = xt , c = c̄t+1,w , vε = vt+1 :1V (x) − V (0) ' ϕ(c) + c2 ϕ00 (c)vε ,
2

(18)

1
xV 0 (x) ' cϕ0 (c) + (c2 ϕ00 (c) + c3 ϕ000 (c))vε .
(19)2
Proposition 4. Consider a positive current consumption shock dεt > 0.
It generates, under assumption (3.b), a decrease of the anticipated variance,
dvt+1 < 0.
1) The resulting effort level is procyclical, i.e. dxt > 0, if the workers’
relative prudence PU (c) is low and decreasing fastεPU (c) = cPU0 (c)/PU (c) < PU (c) − 2 < 0(20)

on a whole interval (γ1 , γ2 ) including the deterministic steady state consumptions c∗ and νc∗ , and thus along the stationary stochastic sunspot equilibrium
nearby. The larger the inequalities in (20), and the larger in absolute value
the heteroscedasticity dvt+1 /dεt < 0 in (3.b), the larger the procyclical variation dxt /dεt > 0.
2) Expected mean consumption is also procyclical, i.e. dc̄t+1,w > 0, under the same inequalities (20) on relative prudence, if the relative degreeof aversion to effort is small at the deterministic steady state, RV (x∗ ) =x∗ V 00 (x∗ )/V 0 (x∗ ) < 1. It is countercyclical, dc̄t+1,w < 0, if aversion to effort is large, RV (x∗ ) > 1, and relative prudence does not decrease too fast,
PU (c) − 2 < εPU (c) ≤ 0, on an interval (γ1 , γ2 ) for consumption including c∗and νc∗ .
The detailed proof rests on a straightforward differenciation of (18), (19),and is given in the appendix. It is useful to illustrate intuitively here, without
getting into details, the role and meaning of the assumptions made on relative
19

prudence for the behavior of the workers’ expected utility gain of not shirking
ϕ(c) = θ[U (c) − U (νc)]. As already noted in assumption (2.e), a low relative
prudence PU (c) < 2 means that −c2 U 00 (c) is increasing. As this prevails overan interval (γ1 , γ2 ) that includes the steady state consumptions c∗ and νc∗ ,this implies that c2 ϕ00 (c) = θ[c2 U 00 (c) − ν 2 c2 U 00 (νc)] is negative at c = c∗ ,and that it gets larger in absolute value when PU (c) gets smaller on thatinterval (γ1 , γ2 ). The elasticity of relative prudence has similar consequenceson the sign and magnitude of c3 ϕ000 (c) = θ[c3 U 000 (c) − ν 3 c3 U 000 (νc)] at c∗ .
The elasticity of c3 U 000 (c) is equal to 2 − PU (c) + εPU . So a fast decreasing
relative prudence as in (20) on an interval (γ1 , γ2 ) ensures that c3 U 000 (c) is
decreasing fast on that interval. In that case, if (γ1 , γ2 ) includes c∗ and νc∗ ,c3 ϕ000 (c) is also negative at c∗ , and gets larger in absolute value when εPU (c) <PU (c) − 2 < 0 gets more and more negative over (γ1 , γ2 ). Analytically, thecoefficients of the anticipated variance vε of the future consumption sunspot
shock εt+1 in (18), (19) become more and more negative, which ensures thatdxt > 0 when dvt+1 < 0.
On the economic side , current effort may be viewed as some sort ofcostly ”precautionary saving” that seeks to guarantee the expected mean
consumption c̄t+1,w , while coping with the risk involved in the future sunspot
consumption shock εt+1 . A lower relative prudence PU (c) < 2 means that the
expected utility gain of not shirking ϕ(c) is more concave (displays more risk
aversion). This implies that a decrease in the risk of the anticipated sunspot
shock makes workers more willing to provide the requested ”precautionary
effort” : the expected utility gain of not shirking in the NSS (16) gets larger
for a given expected mean reward c̄t+1,w (the curve representing the NSS (16)goes up in Fig.1). On the other hand, making relative prudence to decrease
faster (its elasticity εPU (c) more negative) diminishes ϕ000 (c), i.e. makes thedecreasing ”marginal expected utility of not shirking” ϕ0 (c) > 0, in (17) or in(9), ”more concave”, i.e. to display ”less prudence”. So this should also makeworkers more willing at the margin to provide the requested ”precautionary
effort” following a decrease in the risk of the future sunspot shock εt+1 . For insuch a case, the marginal expected utility of not shirking in (17) reacts morepositively to a risk decrease, given expected mean consumption c̄t+1,w . What
Proposition 4 states is that when relative prudence is low and decreases fast
as in (20), over an interval of consumptions (γ1 , γ2 ) containing c∗ and νc∗ ,these two combined effects imply that dvt+1 < 0 leads to dxt > 0.
We end up with an application to the analytical specification where the20workers’ utility for consumption displays Hyperbolic Absolute Risk Aversion(HARA), i.e.
U (c) = (c − a)1−η /(1 − η) and AU (c) = −U 00 (c)/U 0 (c) = η/(c − a),on the consumption interval (γ1 , γ2 ) containing the steady state consumptionsc∗ and νc∗ as in Proposition 4. As Lemma 2 established that effort wasconstant, xt = x∗ , in the CRRA specification a = 0, we focus here on thecase where a > 0. It is then immediate to verify by direct inspection :Corollary 2. Let (γ1 , γ2 ) be an interval containing the workers’ steadystate consumptions, 0 < γ1 < νc∗ < c∗ < γ2 . Assume that the workers’
utility for consumption is of the HARA family, U (c) = (c − a)1−η /(1 − η) on
that interval, with 0 < η < 1 and γ1 > a > 0. Theni) assumption (2.c), i.e. RU (c) < 1, is met on (γ1 , γ2 ) if and only if(1 − η)γ1 > a.
ii) assumption (2.e), i.e. PU (c) < 2, is met on (γ1 , γ2 ) if and only if(1 − η)γ1 > 2a.
iii) condition (20), i.e. εPU (c) < PU (c) − 2, is met on (γ1 , γ2 ) if and only
if (1 − η)γ2 < 3a.

4Conclusion

As noted in the introduction, the basic initial motivation behind the introduction of exogenous technology shocks in Real Business Cycles (RBC) macroeconomic models was their potential to generate more or less pronounced comovements of labor productivity, output, employment, real wages. Beyond
the issue of evaluating whether this research avenue was empirically fully
successful (see e.g. Francis et al. (2005), Gali (1999)), this work considers
an alternative theoretical mechanism that would rely instead on endogenous
procyclical variations of workers’ effort. The novelty of the proposed mechanism is to mix two approaches that have been studied only separately inthe literature. Namely, 1) the occurrence of persistent structural unemployment through efficiency wage contracting arguments of the ”no-shirking”
variety, and 2) the generation of endogenous expectations driven businesscycles through stochastic sunspot (consumption) shocks near a locally indeterminate steady state. Workers’ non-shirking schedule reflects the equality
21

of their disutility of effort with their expected utility gain of not shirking,measured in terms of the resulting anticipated random consumption they canobtain from the real income they get. If there is conditional heteroskedasticity of sunspot consumption shocks, so that a current positive sunspot shock
generates a decrease of the anticipated sunspot consumption volatility, risk
averse workers are more willing to provide ”precautionary effort” for a given
real wage. We show further that if workers’ relative prudence is less than
2 and decreasing fast near the steady state, the efficiency wage contracts
chosen by firms do induce then workers to provide more effort, so that it candisplay significant procyclical movements that are potentially useful to match
empirical regularities without having to appeal to exogenous shocks to fundamentals. Even though the voluntarily (over)simplified structure of the model
does not make it quite fit to fullfledged empirical investigations, the proposed
real mechanism linking conditionally heteroskedastic sunspot shocks to procyclical effort movements appears to be quite plausible in terms of workers’
behavior. Such conditional heteroskedasticity of uncertainty shocks havebeen found to be quite relevant in financial analysis and the study of asset
returns (Nelson (1991), Francq et al. (2010). Moreover, potential links between such conditionally heteroskedastic shocks and related stochastic properties of consumption, or countercyclical macroeconomic uncertainty shocks,
have received recent increasing attention (Boguth et al. (2013), Bloom et al.
(2014), Tedongap (2016), Wang (2006)). The real mechanism presented here
appears accordingly to be a potentially fruitful avenue to investigate further.

22

Appendix

Proof of Proposition 4
Differenciation of (18), (19) with respect to x = xt , c = c̄t+1,w , vε =Et [(εt+1 )2 ] yields, after neglecting 3rd order terms such as vε dc,
V 0 dx1' ϕ0 dc + c2 ϕ00 dvε ,2

1(V 0 + xV 00 )dx ' (ϕ0 + cϕ00 )dc + (c2 ϕ00 + c3 ϕ000 )dvε ,2which can be solved into
(V 0 cϕ00 − xV 00 ϕ0 )dx '1 3 00 2[c (ϕ ) − c2 ϕ00 ϕ0 − c3 ϕ000 ϕ0 ]dvε ,2

xV 00 01 2 00
xV 00
1
00
ϕ
−
cϕ
)dc
'
[c
ϕ
(1
−
) + c3 ϕ000 ]dvε .
0
0
V
2
V
2
In the above, again up to 3rd order terms, all these expressions can be evaluated at the deterministic steady state values x = x∗ , c = c∗ . From assumption(2.b), the disutility of effort is increasing and convex, V 0 > 0, V 00 > 0, whilefrom assumption (2.e) , c2 ϕ00 (c) = θ[c2 U 00 (c) − ν 2 c2 U 00 (νc)] < 0 at c = c∗ if−c2 U 00 (c) is increasing, or PU (c) < 2, over an interval (γ1 , γ2 ) containing c∗and νc∗ . In order to get dx/dvε < 0, hence dxt /dεt > 0 under assumption(3.b), it is therefore sufficient that(c3 ϕ000 (c) = θ[c3 U 000 (c) − ν 3 c3 U 000 (νc)] < 0 at c = c∗ .
A sufficient condition is that c3 U 000 (c) be decreasing in the interval (γ1 , γ2 ).
The elasticity of c3 U 000 (c) = −c2 PU (c)U 00 (c) is 2 + εPU (c) − PU (c), so theinequalities (20) over (γ1 , γ2 ) are indeed sufficient for dxt /dεt > 0. The
derivative dxt /dvt+1 < 0 will be larger in absolute value whenever −c2 U 00 (c)and c3 U 000 (c) increase more strongly, hence whenever the inequalities in (20)are larger, over the range (γ1 , γ2 ).
Similarly, one will get dc/dvε < 0, hence dc̄t+1,w /dεt > 0 under assumption (3.b), if c3 ϕ000 (c) < 0 at the steady state c∗ , i.e. under the same inequalities (20) over (γ1 , γ2 ), provided that RV (x∗ ) < 1. By contrast, one will get
23

dc/dvε > 0 if RV (x∗ ) > 1 and if c3 ϕ000 (c) > 0 at c = c∗ . A sufficient condition
for that is that c3 U 000 (c) is increasing, i.e. PU (c) − 2 < εPU (c) ≤ 0 over theinterval (γ1 , γ2 ) containing c∗ and νc∗ .

24

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