SverigeS rikSbank
working paper SerieS
206
optimal Monetary policy
under Downward nominal
wage rigidity
Mikael Carlsson and Andreas Westermark
april 2007
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Optimal Monetary Policy under Downward Nominal Wage Rigidity∗
Mikael Carlsson†and Andreas Westermark‡
Sveriges Riksbank Working Paper Series
No. 206
Revised November 2007
Abstract
We develop a New Keynesian model with staggered price and wage setting where downward
nominal wage rigidity (DNWR) arises endogenously through the wage bargaining institutions. It is
shown that the optimal (discretionary) monetary policy response to changing economic conditions
then becomes asymmetric. Interestingly, in our baseline model we find that the welfare loss is
actually slightly smaller in an economy with DNWR. This is due to that DNWR is not an additional
constraint on the monetary policy problem. Instead, it is a constraint that changes the choice set
and opens up for potential welfare gains due to lower wage variability. Another finding is that the
Taylor rule provides a fairly good approximation of optimal policy under DNWR. In contrast, this
result does not hold in the unconstrained case. In fact, under the Taylor rule, agents would clearly
prefer an economy with DNWR before an unconstrained economy ex ante.
Keywords: Monetary Policy; Wage Bargaining; Downward Nominal Wage Rigidity.
JEL classification: E52, E58, J41.
∗
We are grateful to Nobuhiro Kiyotaki, Mathias Trabandt and seminar participants at Sveriges Riksbank, Norges
Bank, Uppsala University, the 2006 Meeting of the European Economic Association, Vienna, and the 2007 North American Winter Meeting of The Econometric Society, Chicago, for useful comments. We would also like to thank Erik
von Schedvin for excellent research assistance. We gratefully acknowledge financial support from Jan Wallander’s and
and Tom Hedelius’ Research Foundation and Westermark also from the Swedish Council for Working Life and Social
Research. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted
as reflecting the views of the Executive Board of Sveriges Riksbank.
†
Research Department, Sveriges Riksbank, SE-103 37, Stockholm, Sweden. e-mail: mikael.carlsson@riksbank.se.
‡
Department of Economics, Uppsala University, P.O. Box 513, SE-751 20 Uppsala, Sweden. e-mail:
andreas.westermark@nek.uu.se.
1
Introduction
A robust empirical finding is that money wages do not fall during an economic downturn, at least
not to any significant degree. A large number of studies report substantial downward nominal wage
rigidity in the U.S. as well as in Europe and Japan.1 Overall, the evidence points towards a sharp
asymmetry in the distribution of nominal wage changes around zero. That is, money wages rise but
they seldom fall. This may not have any noticeable real effects in periods with sufficiently high inflation
rates to allow for a reduction of real wages in response to adverse shocks without reducing nominal
wages. However, inflation rates have come down in many countries in recent decade(s) and periods
of very low inflation rates are no longer out of the picture. Recent examples are Japan, Sweden and
Switzerland which have all experienced prolonged episodes with average CPI-inflation rates below one
percent (see below). Still, downward nominal wage rigidity may not be a concern for real outcomes, if
it is not a feature of low inflation environments, as conjectured by e.g. Gordon (1996). However, the
empirical evidence shows that the downward rigidity of nominal wages persists even in low inflation
environments (see Agell and Lundborg, 2003, Fehr and Goette, 2005, and Kuroda and Yamamoto
2003a, 2003b). This, in turn, opens up for potentially important real effects of downward nominal
wage rigidity in the current era of low inflation rates.
The purpose of this paper is to study the implications for monetary policy in situations where
declining nominal wages are not a viable margin for adjustment to adverse economic conditions. To
this end, we develop a New Keynesian DSGE model that can endogenously account for downward
nominal wage rigidity. More specifically, this is achieved by introducing wage bargaining between
firms and unions as is done in Carlsson and Westermark (2006a), but modified in line with Holden
(1994). Then, downward nominal wage rigidity arises as a rational outcome.
In the model, price and wage setting are staggered. The main difference with our approach, relative
to standard New Keynesian DSGE models including an explicit labor market (see Erceg, Henderson
and Levin, 2000) is that we model wages as being determined in bargaining between firms and unions
(households).2 We follow Carlsson and Westermark (2006a), and assume that the household is attached
to a firm.
Wage bargaining is opened with a fixed probability each period, akin to Calvo (1983). Moreover,
1
The empirical evidence ranges from studies using data from personnel files presented in Altonji and Devereux (2000),
Baker, Gibbs, and Holmstrom (1994), Fehr and Goette (2005), and Wilson (1999), survey/register data in Altonji and
Devereux (2000), Akerlof, Dickens, and Perry (1996), Dickens, Goette, Groshen, Holden, Messina, Schweitzer, Turunen,
and Ward (2006), Fehr and Goette (2005), Holden and Wulfsberg (2007), Kuroda and Yamamoto (2003a, 2003b) to
interviews or surveys with wage setters like Agell and Lundborg (2003), and Bewley (1999), just to mention a few.
2
For this purpose, we must modify the simplifying assumption of Erceg, Henderson, and Levin (2000), that all
households work at all firms. Otherwise, each individual household works an infinitesimal amount at each firm, implying
that the effect of the individual household’s wage on firm surplus is zero. Thus, in the standard setup, there is no surplus
to be negotiated over, hence rendering bargaining irrelevant.
2
bargaining is non-cooperative as in the Rubinstein-Ståhl model, with the addition that if there is
disagreement but no party is willing to call a conflict, work takes place according to the old contract.
As argued by Holden (1994), this is in line with the labor market institutions in the U.S. and most
western European countries. Moreover, as in Holden (1994), there are costs associated with conflicts
in addition to costs stemming from impatience, such as disruptions in business relationships, startup
costs and deteriorating management-employee relationships. These costs sometimes render threats
of conflict non-credible, leading to agreement on the same wage as in the old contract. Since it is
reasonable to assume that these costs are much larger for firms than for workers, workers can credibly
threaten firms with conflict, whereas firms cannot. Since workers only use the threat to bid up wages,
downward nominal wage rigidity will result.
Given our setup, a non-linear restriction on wage inflation due to downward nominal wage rigidity
arises endogenously. Then, given the constraints from private sector behavior, the central bank solves
for optimal (discretionary) monetary policy.3
The optimal response to changing economic conditions is asymmetric, and not only in the wage
inflation dimension. Interestingly, the welfare loss is actually slightly smaller in an economy with
downward nominal wage rigidities in our baseline case. The reason is that downward nominal rigidity
is not an additional constraint on the problem. Instead, it is a constraint that changes the choice
set and opens up for potential welfare gains. Another finding is that the Taylor rule estimated by
Rudebusch (2002), provides a fairly good approximation of optimal discretionary policy in terms of
welfare under downward nominal wage rigidity. Experimenting with using the original Taylor (1993)
parameters for the Taylor rule indicates that the exact specification of the Taylor rule actually plays a
minor role for this property. In contrast, neither of these results seem to hold in the unconstrained case.
A corollary is that, under the Taylor rule, agents would clearly prefer an economy with downward
nominal wage rigidities to an unconstrained economy ex ante. That is, downward nominal wage
rigidity actually helps stabilizing the economy in the wage inflation dimension, whereas it does not
induce much more variation in inflation and the output gap.
In sections 1 and 2, we outline the model and discuss the equilibrium, respectively. In section 3,
we characterize the policy problem facing the central bank. Section 4 discusses optimal policy paths
for endogenous variables as well as the welfare implications of downward nominal wage rigidity under
optimal policy. Moreover, we also discuss the outcome of using a simple instrument (Taylor) rule
instead of the optimal policy. Finally, section 5 concludes.
3
We focus on the discretionary policy case, since this is closest to the actual practice of central banks.
3
1
The Economic Environment
The model outlined below is in many respects similar to that in Erceg, Henderson, and Levin (2000).
Goods are produced by monopolistically competitive producers using capital and labor. Producers set
prices in staggered contracts as in Calvo (1983). There are also some important differences, however.
In contrast to Erceg, Henderson, and Levin (2000), we follow Carlsson and Westermark (2006a),
and assume that a household is attached to each firm.4,5 Thus, firms do not perceive workers as
atomistic. In each period, bargaining over wages takes place with a fixed probability. Accordingly,
wages are staggered as in Calvo (1983), but, in contrast to Erceg, Henderson, and Levin (2000),
they are determined in bargaining between the household/union and the firm. Households derive
utility from consumption, real balances and leisure, earning income by working at firms and from
capital holdings. Below, we present the model in more detail and derive key relationships (for a full
derivation, see Appendix C and the Technical Appendix to Carlsson and Westermark, 2006a).
1.1
Firms and Price Setting
Since households will be identical, except for leisure choices, it simplifies the analysis to abstract away
from the households’ optimal choices for individual goods. Thus, we follow Erceg, Henderson, and
Levin (2000) and assume a competitive sector selling a composite final good, which is combined from
intermediate goods to the same proportions as those that households would choose. The composite
good is
Yt =
∙Z
1
Yt (f )
σ−1
σ
0
σ
¸ σ−1
,
(1)
where σ > 1 and Yt (f ) is the intermediate good produced by firm f . The price Pt of one unit of the
composite good is set equal to the marginal cost
1
⎡1
⎤ 1−σ
Z
.
Pt = ⎣ Pt (f )1−σ df ⎦
(2)
0
By standard arguments, the demand function for the intermediate good f , is
Yt (f ) =
µ
Pt (f )
Pt
4
¶−σ
Yt .
(3)
Several households could be attached to a firm, if these negotiate together.
There is no reallocation of workers among firms. This is obviously a simplifying assumption, but it enables us to
describe the model in terms of very simple relationships.
5
4
The production of firm f in period t, Yt (f ), is given by the following constant returns technology
Yt (f ) = At Kt (f )γ Lt (f )1−γ ,
(4)
where At is the technology level common to all firms and Kt (f ) and Lt (f ) denote the firms’ capital
and labor input in period t, respectively. Since firms have the right to manage, Kt (f ) and Lt (f ) are
optimally chosen, taking the rental cost of capital and the bargained wage Wt (f ) as given. Moreover,
as in Erceg, Henderson, and Levin (2000), the aggregate capital stock is fixed at K̄. Standard costminimization arguments then imply that the marginal cost in production is given by
M Ct (f ) =
Wt (f )
,
M P Lt (f )
(5)
where M P Lt (f ) is the firm’s marginal product of labor.6
1.1.1
Prices
The firm is allowed to change prices in a given period with probability 1 − α and renegotiate wages
with probability 1 − αw . In addition, any firm that is allowed to change wages is also allowed to
change prices, but not vice versa. Thus, the probability of a firm’s price remaining unchanged is αw α.
The latter assumption greatly simplifies our problem; in particular, it eliminates any intertemporal
interdependence between current and future price decisions via its effect on wage contracts for a
given firm. Besides convenience, this assumption is in line with the micro-evidence on price-setting
behavior presented in Altissimo, Ehrmann, and Smets (2006), where price and wage changes are to a
large extent synchronized in time (see especially their figure 4.4). Here, we assume that wage changes
induce price changes, since assuming the reverse would imply that the duration of wage contracts could
never be longer than the duration of prices, which seems implausible in face of the empirical evidence,
see section 3.1. Furthermore, since intertemporal interdependencies are eliminated, this allows us to
describe the goods market equilibrium by a similar type of forward looking new Keynesian Phillips
curve as in Erceg, Henderson, and Levin (2000) (see equation (21)).
The producers choose prices to maximize
max Et
pt (f )
s. t. Yt+k (f ) =
µ
∞
X
(αw α)k Ψt,t+k [(1 + τ ) Pt (f ) Yt+k (f ) − T C (Wt+k (f ) , Yt+k (f ))] (6)
k=0
¶−σ
Pt (f )
Pt+k
Yt+k ,
6
In contrast to Erceg, Henderson, and Levin (2000), the marginal cost is generally not equal among firms, since firms
face different wages out of steady state.
5
where T C (Wt+k (f ) , yt+k (f )) denotes the cost function, Ψt+k is the households’ valuation of nominal
profits in period t + k when in period t and τ is a tax/subsidy on output. The term inside the square
brackets is just firm profits in period t + k, given that prices were last reset in period t. The first-order
condition is
Et
∞
X
k
(αw α) Ψt,t+k
k=0
∙
¸
σ−1
(1 + τ ) Pt (f ) − M Ct+k (f ) Yt+k (f ) = 0.
σ
The subsidy τ is determined so as to set
σ−1
σ
(7)
(1 + τ ) = 1; that is, we assume that fiscal policy is used
to alleviate distortions due to monopoly price setting.7
1.2
Households
The economy is populated by a continuum of households, also indexed on the unit interval, which each
supplies labor to a single firm. This setup can alternatively be interpreted as a unionized economy
with firm-specific unions. In such a framework, each household can be considered as the representative
union member.
The expected life time utility of the household working at firm f in period t is given by
Et
½∞
P
β
s=t
s−t
∙
µ
¶
¸¾
Ms (f )
u (Cs (f )) + l
− v (Ls (f )) ,
Ps
(8)
where period s utility is additively separable in three arguments, final goods consumption Cs (f ), real
money balances
Ms (f )
Ps ,
where Ms (f ) denotes money holdings, and the disutility of working Ls (f ).8
Finally, β ∈ (0, 1) is the household’s discount factor.
The budget constraint of the household is
δ t+1,t Bt (f ) Mt (f )
Mt−1 (f ) + Bt−1 (f )
Wt (f ) Lt (f ) Γt
Tt
+
+ Ct (f ) =
+ (1 + τ w )
+
+ .
Pt
Pt
Pt
Pt
Pt Pt
(9)
The term δ t+1,t represents the price vector of assets that pays one unit of currency in a particular state
of nature in the subsequent period, while the corresponding elements in Bt (f ) represent the quantity
of such claims bought by the household. Moreover, Bt−1 (f ) is the realization of such claims bought in
the previous period. Also, Wt (f ) denotes the household’s nominal wage and τ w is the tax/subsidy on
labor income. Each household owns an equal share of all firms and the aggregate capital stock. Then,
7
Thus, we abstract from any Barro-Gordon type of credibility problems (see Barro and Gordon, 1983a, and Barro
and Gordon, 1983b).
8
In the Technical Appendix, we also introduce a consumption shock and a labor-supply shock as in Erceg, Henderson,
and Levin (2000). However, introducing these shocks does not yield any additional insights here. In fact, it can easily
be shown that under optimal policy, all disturbances in the model (introduced as in Erceg, Henderson and Levin, 2000)
can be reduced to a single disturbance term (being a linear combination of all these shocks).
6
Γt is the household’s aliquot share of profits and rental income. Finally, Tt denotes nominal lump-sum
transfers from the government. As in Erceg, Henderson, and Levin (2000), we assume that there exist
complete contingent claims markets (except for leisure) and equal initial wealth across households.
Then, households are homogeneous with respect to consumption and money holdings, i.e., we have
Ct (f ) = Ct , and Mt (f ) = Mt for all t.
1.3
Wage Setting
When a firm/household pair is drawn to renegotiate the wage, bargaining takes place in a setup similar
to the model by Holden (1994) and is here introduced in a New Keynesian framework following Carlsson
and Westermark (2006a). There are two key features of the bargaining model in Holden (1994). First,
there are costs of invoking a conflict, which are different from the standard costs in bargaining due
to impatience. Instead, they are caused by e.g., disrupting business relationships, startup costs and
deteriorating management-employee relationships (see Holden, 1994). Second, there is an old contract
in place at the firm and if no conflict is called and no new contract is signed, the workers work
according to the old contract. As pointed out by Holden (1994), this is a common feature of many
western European countries as well as of the U.S.
The union and the firm only have incentives to call for a conflict when the negotiated contract
gives a higher payoff than the old contract. As soon as a conflict is called, payoffs are determined in a
standard Rubinstein-Ståhl bargaining game and the conflict costs are paid out of the parties’ respective
pockets. However, the costs of conflict imply that it is sometimes not credible to threaten with a
conflict in equilibrium. Specifically, if the difference between the old contract and the RubinsteinStåhl solution is small relative to the conflict cost, a party cannot credibly threaten with a conflict
and force the new contract into place. Then, no new agreement is struck and work continues according
to the old contract, resulting in nominal rigidity. If the difference is sufficiently large, however, then
conflict is a credible threat. Note, though, that there will be no conflicts in equilibrium, since it is
optimal to immediately agree on the Rubinstein-Ståhl solution, rather than waiting and enduring a
conflict.9
To derive only downward nominal rigidity, asymmetries in conflict costs are required. Specifically,
if the costs are large for the firm and negligible for the union, the firm can never credibly threaten
with a conflict (at least not close to the steady state), whereas the union can always do so when the
Rubinstein-Ståhl solution is larger than the old contract. In reality, conflict costs for the workers are
probably not zero, but small. Then, wages would be adjusted only if the Rubinstein-Ståhl solution
exceeded some threshold value ω̄ > Wt−1 (instead of ω̄ = Wt−1 ). For simplicity, we restrict the
9
That agreement is immediate follows from e.g. Rubinstein (1982).
7
attention to the case when conflict costs are zero for workers.10
Note that downward nominal rigidity implies that there is a potential relationship between wage
negotiations today and in the future. This interdependence comes from two sources. First, the
wage contract is a state variable in future negotiations and second, the wage set today affects prices
set in the future which, in turn, may affect future wage negotiations. The first interdependence is
eliminated by using the steady state distribution in the log linearization of the model (see Appendix
C for details and the caveat in section 2 for a further discussion). The second interdependence is
eliminated by the assumption that prices can be changed whenever wages are allowed to change.
Then, given these two steps, each wage negotiation can be analyzed separately, as in the standard
Calvo setup, thereby leading to a very simple and tractable framework. Note also that there will be
no intertemporal interdependence in price setting decisions for a given firm either. To see this, note
that since prices can be adjusted in any direction, the current price is not a state variable in future
price setting. Any interdependence in price setting over time must thus come via wage negotiations,
but such interdependence is ruled out by the assumption that prices change whenever wages change.
Unions
The union at a firm represents all workers at the firm and maximizes the welfare of all members.
Defining per-period utility (in the cash-less limiting case), for a given contract wage, as
Υt,t+k (f ) = u (Ct+k ) − v (Lt,t+k (f )) ,
(10)
where Lt,t+k (f ) denotes labor demand in period t+k when prices were last reset in period t. Moreover,
let
ζ t+k (dt+k (f )) = αw + (1 − αw ) Ft+k (dt+k (f )) ,
(11)
denote the probability that firm f ’s wages are unchanged in period t + k. The term Ft+k (dt+k (f ))
is then firm f ’s probability that the wage is not adjusted conditional on renegotiation taking place,
which is a function of
dt+k (f ) =
o (f )
Wt+k
,
Wt (f )
(12)
o (f ) denotes the unconstrained optimal wage in period
where Wt (f ) is the current contract and Wt+k
t + k for firm f , defined as the wage upon which parties would agree in period t + k if all conflict
u
denote union utility when the wage is
costs were temporarily removed in period t + k. Then, let Ut+k
10
A full explanation of downward nominal wage rigidity is likely to include several mechanisms that may be complementary. Studies like Bewley (1999), and others point towards psychological mechanisms involving fairness considerations
and managers’ concern over workplace morale. Moreover, the workers’ yardstick for fairness seems to be what happens
to nominal rather than real wages. However, here we focus on the fully rational explanation proposed by Holden (1994).
8
renegotiated in period t + k. Union utility in period t, Utu , is then a probability weighted discounted
sum of future per-period payoffs, i.e.
Utu = Et
∞
X
k=0
(αw αβ)k Υt,t+k (f ) + Et
∞ k−1
X
Y
ζ t+i (dt+i (f ))
(13)
k=1 i=0
⎤
⎡
∞
¢ kX
¡
¢
¡
u ⎦
.
(αw αβ)j Υt+k,t+k+j (f ) + 1 − ζ t+k (dt+k (f )) β k Ut+k
× ⎣ ζ t+k (dt+k (f )) − αw α β
j=0
To see the intuition behind the summations in (13), note that the first summation in (13) corresponds
to the case when prices are never changed in the future, whereas the second summation corresponds to
outcomes that include future price changes. To understand the second summation in (13), first note
that the terms inside the squared bracket are multiplied by the probability of the wage not having
Qk−1
ζ t+i (dt+i (f ))). Then, within a period, t + k, prices can
been changed up to period k − 1 (i.e. i=0
change in two ways. First, the price can change without the wage changing, which happens with
probability (ζ t+k (dt+k (f )) − αw α).11 Then, this probability is the weight for the utility associated
with a reset price in period t + k.12 The second way in which prices are changed in period t + k is if
the wage changes, which happens with probability (1 − ζ t+k (dt+k (f ))). Then, this probability is the
u is
weight for the utility associated with resetting the wage (and price) in period t + k. Note that Ut+k
in itself independent of the (unconstrained) wage bargained over today. Finally, for confirmation, we
note that the sum of probabilities inside the squared bracket at period t + k equals the probability of
prices being changed within period t + k (i.e., (1 − αw α)).
Firms
Let real per-period profits in period t + k, when the price was last rewritten in period t, be denoted as
µ
¶
Pto (f )
Wt (f )
φt,t+k (Wt (f )) = (1 + τ )
Yt+k (f ) − tc
, Yt+k (f ) ,
Pt+k
Pt+k
11
(14)
To understand this probability, note that we have the outcome that the price but not the wage changes in two cases:
First, if the firm is drawn for a price change but not for a wage change (which happens with probability (1 − α)αw ) and
second, if the firm is drawn for wage bargaining but downward nominal wage rigidity prevents a wage change (which
happens with probability (1 − αw )Ft+k (dt+k (f ))).
12
Although the utility from a reset price in period t + k is formulated as if the price would never again change in the
future, it is straightforward to show that the summations here keep track of outcomes where the price is changed more
than once.
9
where tc denotes real total cost. Firm payoff Utf is then
Utf
= Et
∞
X
(αw α)k ψ t,t+k φt,t+k (Wt (f ))
k=0
Ãk−1
∞
Y
X
+Et
k=1
+Et
∞
X
k=1
i=0
Ãk−1
Y
(15)
!
∞
¡
¢X
ζ t+i (dt+i (f )) ζ t+k (dt+k (f )) − αw α
(αw α)j ψ t+k,t+k+j φt+k,t+k+j (Wt (f ))
!
ζ t+i (dt+i (f ))
i=0
j=0
¡
¢
f
,
1 − ζ t+k (dt+k (f )) ψ t,t+k Ut+k
where the term ψt,t+k denotes how the households (which own an aliquot share of each firm) value
real profits in period t + k when in period t. The intuition behind the sums in (15) is analogous to
that of the sums in (13) discussed above.
Bargaining
Since the Rubinstein-Ståhl solution can be found by solving the Nash Bargaining problem, we can
solve for the unconstrained wage from
³ ´1−ϕ
,
max (Utu − Uo )ϕ Utf
Wt (f )
(16)
where ϕ is the household’s relative bargaining power and Uo its threat point. The threat point is
the payoff when there is disagreement (i.e., strike or lockout). The payoff of the firm when there is a
disagreement is assumed to be zero. Households are assumed to receive a share of steady-state (after
tax) income and not spend any time working. This interpretation of threat points is in line with a
standard Rubinstein-Ståhl bargaining model with discounting and no risk of breakdown as presented
in Binmore, Rubinstein, and Wolinsky (1986) (see also Mortensen, 2005, for an application of this
bargaining setup). A constant Uo leads to a very convenient and simple analysis; more complicated
models of threat points, e.g. based on workers having the opportunity to search for another job, could
also be introduced in this model. However, as argued by Hall and Milgrom (2005), the threat points
should not be sensitive to factors like unemployment or the average wage in the economy, since delay
is the relevant threat as opposed to permanently terminating the relationship between the firm and
the workers. For example, United Auto Workers permanently walking away from GM is never on the
table during wage negotiations, as pointed out by Hall and Milgrom (2005). The first-order condition
to problem (16) is
ϕUtf
∂Utf
∂Utu
+ (1 − ϕ) (Utu − Uo )
= 0.
∂W (f )
∂W (f )
(17)
Then, if Wto (f ) is the solution to the above problem, which is equal across all firms that are allowed
10
to renegotiate, the resulting wage for firm f with the old contract Wt−1 (f ) is
max{Wto (f ) , Wt−1 (f )}.
(18)
Thus, in the case that the unconstrained optimal wage is lower than the present wage contract, the
old wage contract prevails due to the conflict cost structure outlined above.
As in price setting, we eliminate the distortions that result from bargaining. Since there are two
instruments that can be used to achieve this, i.e., τ w and Uo , one of them is redundant. Here, we use
the method in Carlsson and Westermark (2006a), relying on adjusting τ and Uo to achieve efficiency.
1.3.1
Wage Evolution
Taking into account that firms cannot substitute across workers, the average wage is determined by
Z
Z
1
Wt−1 (f ) df + (1 − αw )
Wt−1 (f ) df
Wt−1 (f )>Wto (f )
Z
Wto (f ) df,
+ (1 − αw )
Wt = αw
0
(19)
Wt−1 (f )≤Wto (f )
where the second term of (19) is due to downward nominal wage rigidity.
1.4
Steady State
As discussed above, downward nominal wage rigidity is not likely to have any noticeable real effects
in periods with high inflation rates. However, inflation rates have come down in most countries in
recent decades and prolonged periods of very low inflation rates are no longer uncommon. In figure
1, we plot the CPI-inflation rate (fourth quarter-to-quarter) for Japan, Sweden and Switzerland and
put a shade on low inflation periods, identified as quarters where the five-point moving average of CPI
inflation is below 1 percent. As can be seen in figure 1, lengthy periods where the maneuvering space
for adjusting real wages without reducing nominal wages is seriously limited is very much a real world
possibility.
To set ideas and capture the main mechanisms at work, we focus on a zero steady state inflation
regime in this paper. It is possible to allow for a (small) positive steady state inflation rate. However,
in order to retain tractability, we then need to index wages and prices that cannot be changed.13 But
indexation implies that welfare is independent of the steady state inflation rate. To see this, note that
indexation implies that the downward nominal rigidity will be centered around the positive steady
13
Indexation is needed since it is otherwise impossible to eliminate expectations of variables for more than one period
ahead in the first-order conditions for wage and price setting. Thus, in the absence of indexation, it is necessary to keep
track of infinite sums.
11
Japan
Switzerland
Sweden
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
-1
-1
-1
-2
-2
93 94 95 96 97 98 99 00 01 02 03 04 05
-2
93 94 95 96 97 98 99 00 01 02 03 04 05
93 94 95 96 97 98 99 00 01 02 03 04 05
Figure 1: CPI-inflation rate in percentage units (fourth quarter-to-quarter). Shaded Regions indicate
periods with average inflation (five point moving average) below 1 percent.
state inflation rate instead of zero. Or, in other words, wages cannot grow slower than the steady state
inflation rate. This then gives rise to a an identical problem where downward wage rigidity binds just
as often as in the zero steady state case; hence the focus on a zero steady state regime here.
In the zero-inflation non-stochastic steady state, At is equal to its steady-state value, Ā. Moreover,
all firms produce the same (constant) amount of output, i.e. Y (f ) = Y , using the same (constant)
quantity of labor and all households supply the same amount of labor, i.e. L(f ) = L. Moreover, we
will have that C = Y and that B = 0. M and P are constant.
To find the steady state of the model, we use the production function (4) together with the
efficiency condition M P L = M RS (which holds due to having eliminated distortions as in Carlsson
and Westermark (2006a)) to solve for L and, in turn, Y and C.
2
Equilibrium
First, let the superscript
∗
denote variables in the flexible price and wage equilibrium, to which we
refer below to as the natural equilibrium, and a hat above a small letter variable denotes log-deviations
from the steady-state level of the variable. Linearizing around the steady state then gives the following
system of equations, where the parameters are given in Appendix B,
x̂t
π̂ t
π̂ ωt
ŵt
µ
´¶
1 ³b
∗
it − π̂ t+1 − rbt
= Et x̂t+1 −
,
ρC
¢
¡
γ
Πx̂t ,
= βEt π̂ t+1 + (1 − γ) π̂ ωt − βEt π̂ ωt+1 + Π (ŵt − ŵt∗ ) +
1−γ
½
¾
1 + αw
= max 0,
βEt π̂ ωt+1 − Ωw (ŵt − ŵt∗ ) − Ωx x̂t ,
2αw
= ŵt−1 + π̂ ωt − π̂ t .
12
(20)
(21)
(22)
(23)
For clarity, all parameters are defined to be positive.
Equation (20) is a standard goods-demand (Euler) equation which relates the output gap x̂t , i.e.
the log-deviation between output and the natural output level, to the expected future output gap and
the expected real interest rate gap (bit − π̂ t+1 − rbt∗ ), where bit denotes the log-deviation of the nominal
interest rate from steady state and rbt∗ is the log-deviation of the natural real interest rate from its
steady state.14 This relation is derived taking standard steps and using the households’ first-order
condition with respect to consumption, i.e., the consumption Euler equation.
The price-setting (Phillips) curve, equation (21), is derived using the firms’ first-order condition
(7), (see Carlsson and Westermark, 2006b, for details) and is similar in shape to the price-setting curve
derived by Erceg, Henderson, and Levin (2000), with the exception that current and expected future
wage inflation also enter the expression. Thus, price setting is affected by the real wage gap, i.e., the
log deviation between the real wage and the natural real wage (ŵt − ŵt∗ ), the output gap x̂t , future
inflation Et π̂ t+1 and current and future wage inflation π̂ ωt , Et π̂ ωt+1 . As can be seen from Carlsson and
Westermark (2006b), the relevant real marginal cost measure driving inflation depends on the real wage
gap in firms that actually change prices (and, naturally, capital prices and productivity). However,
since we are interested in a price-setting relationship expressed in terms of the economywide real wage
gap, we need to adjust for the fact that interdependence in price and wage setting implies that the
economywide real wage gap and the real wage gap in firms that actually change prices are different in
¡
¢
our model.15 This motivates the “correction term” (1 − γ) π̂ ωt − βEt π̂ ωt+1 . Thus, in expression (21),
the real wage change in firms that change prices has been decomposed into the aggregate real wage
change ŵt and wage inflation terms π̂ ωt , Et π̂ ωt+1 .
Equation (22) describes the wage setting behavior (see Appendix C and Carlsson and Westermark,
2006b, for details). From section (1.3) above, we know that wages are set according to (18). This
implies that wage inflation is non-negative and set according to the last term in the max operator
of (22) when positive. Hence, the max operator captures the restriction from wage setting in (22).
For positive wage inflation rates, wage inflation increases with higher expected wage inflation. The
coefficient in front of Et π̂ ωt+1 , i.e.
negotiations
1+αw
2 β
(where
1+αw
2
1+αw
2αw β,
is the probability adjusted discount rate from the wage
is the (unconditional) steady-state probability that wages remain
unchanged in the next period) multiplied by
1
αw ,
which governs how relative wages today (conditional
on π̂ ωt > 0) feed into wage-inflationary pressure. Moreover, as in Erceg, Henderson, and Levin (2000),
wage inflation is influenced by the real wage gap and the output gap. Since the parameters associated
14
The nominal interest rate It is defined as the rate of return on an asset that pays one unit of currency under every
state of nature at time t + 1.
15
Specifically, since all firms that are allowed to change wages are also allowed to change prices, the share of wagechanging firms among the firms that change prices differs from the economywide average.
13
with these variables are determined by the bargaining problem, the size (and even the sign) of them
depend on e.g. the relative bargaining strength. See Carlsson and Westermark (2006a) for a detailed
discussion on wage setting in the unconstrained case.16
A caveat is in place here since the linearized wage-setting curve (22) is derived using the steady
state wage distribution. In general, since the last period’s wage is a state variable in today’s wage
setting problem, the aggregate wage outcome today will depend on the history of wage changes in the
economy, described by the wage distribution. However, starting from an initial distribution where all
firm/union pairs have the same wage, this will not be a problem when downward nominal wage rigidity
binds, since no one will reduce the wage anyway, although for periods beyond the first when wage
inflation is positive, the wage distribution potentially affects the aggregate wage inflation outcome.
We take this approach since it allows us to retain analytical tractability of the problem. Moreover, as
discussed above, this simplification should not lead us too far astray.
Finally, the evolution for the real wage (23) follows from the definition of the aggregate real wage
and states that today’s real wage is equal to yesterday’s real wage plus the difference between the
rates of wage and price change (π̂ ωt − π̂ t ).
As a comparison to the results from the economy with downward nominal wage rigidity, it is useful
to look at an economy where wages can adjust symmetrically. As shown in Carlsson and Westermark
(2006a), the unconstrained economy is described by (20), (21), (23) and replacing (22) with
∗
uc
π̂ ωt = βEt π̂ ωt+1 − Ωuc
w (ŵt − ŵt ) − Ωx x̂t
(24)
where, once more, the parameter definitions are given in Appendix B.
3
The Monetary Policy Problem
The central bank is assumed to maximize social welfare. Here, we focus on the discretionary policy
case. Although studying optimal policy is in essence a normative enterprise, given that no central
bank formally commits to a policy rule it is natural to focus on the discretionary case. Following
the main part of the monetary policy literature, we focus on the limiting cashless economy (see e.g.
Woodford (2003) for a discussion) with the social welfare function
Et
∞
X
t=0
µ
Z
β u (Ct ) −
t
0
16
1
¶
v (Lt (f )) df .
(25)
See also Carlsson and Westermark (2006a), for a detailed comparison between the unconstrained version of (22), i.e.
equation (24) below, and the wage setting curve resulting from the Erceg, Henderson, and Levin (2000) model.
14
Following Rotemberg and Woodford (1997), Erceg, Henderson, and Levin (2000), and others, we take
a second-order approximation to (25) around the steady state. This yields a standard expression for
the welfare gap (see Appendix C.5 for a detailed derivation, also c.f. Erceg, Henderson, and Levin
(2000)), i.e., the discounted sum of log-deviations of welfare from the natural (flexible price and wage
welfare level)
Et
∞
X
t=0
´
³
β t θx (x̂t )2 + θπ (π̂ t )2 + θπω (π̂ ωt )2 ,
(26)
where we have omitted higher order terms and terms independent of policy. As usual, θx < 0, θπ < 0
and θπω < 0 (see Appendix B for definitions). The first term captures the welfare loss (relative to
the flexible price and wage equilibrium) from output gap fluctuations stemming from the fact that
d will differ from mrs
mpl
d whenever x̂t 6= 0. However, even if x̂t = 0, there will be welfare losses due to
nominal rigidities. The reason is that nominal rigidities imply a non-degenerate distribution of prices
and wages. A non-degenerate distribution of prices and wages implies a non-degenerate distribution
of output across firms and working hours across households. This leads to welfare losses due to a
decreasing marginal product of labor and an increasing marginal disutility of labor.
Note that welfare only depends on variables x̂t , π̂ t and π̂ ωt which, in turn, can solely be determined
from equations (21) to (23). To find the optimal rule under discretion, the central bank then solves
the following problem
V (ŵt−1 , ŵt∗ ) =
max
ω
{x̂t ,π̂t ,π̂ t ,ŵt }
¢
¡
∗
,
θx (x̂t )2 + θπ (π̂ t )2 + θπω (π̂ ωt )2 + βEt V ŵt , ŵt+1
(27)
subject to equations (21) to (23), disregarding that expectations can be influenced by policy.
The wage inflation restriction (22) can be replaced by
1 + αw
βEt π̂ ωt+1 − Ωx x̂t − Ωw (ŵt − ŵt∗ ) − π̂ ωt ,
2αw
≥ 0.
π̂ ωt ≥
(28)
π̂ ωt
(29)
Note that the problem with the original max constraint (23) and the problem with inequality
constraints (28) and (29) need not be equivalent. It is obviously true that a solution (x̂t , π̂ t , π̂ ωt , ŵt ) to
the problem with the original max constraint also satisfies the two inequality constraints. However, it
is possible that there is a solution (x̂t , π̂ t , π̂ ωt , ŵt ) to the problem with inequality constraints, so that
none of the inequality constraints is binding, thus leading to a violation of the original max constraint.
However, this is ruled out by the following Lemma.
Lemma 1 At least one of the inequality constraints (28) and (29) must be binding.
15
Proof : See Appendix D. ¥
Thus, this possibility is ruled out by the above Lemma, thereby implying that the problems are
equivalent. The intuition for the result is the following. Since the constraints (28) and (29) both put
lower bounds on π̂ ωt and, as can be seen from expression (26), welfare is decreasing in π̂ ωt , the central
bank sets π̂ ωt as low as possible, implying that one of the inequality constraints (28) and (29) must
bind.
From the above, it follows that the central banks’ problem (27) gives rise to two systems depending
on whether the inequality constraint binds. These systems, in turn, consist of the case specific firstorder conditions for optimal policy and restrictions from private sector behavior (see Appendix D for
details).
3.1
Numerical Solution and Calibration
To solve the model, we find the paths for x̂t , π̂ t , π̂ ωt and ŵt that maximize welfare, as suggested by
Woodford (2003).17,18 As in Erceg, Henderson, and Levin (2000), we look at the effects of a technology
shock, which is assumed to follow an AR(1). It is straightforward to show that there is a positive
linear relationship between ŵt∗ and Ât .19 Then, if technology follows an AR(1) process, ŵt∗ also follows
an AR(1) process. We can thus model ŵt∗ as
∗
ŵt∗ = η ŵt−1
+ εt ,
(30)
where εt is an (scaled) i.i.d. (technology) shock with standard deviation σ .
For our numerical exercises, we follow Erceg, Henderson, and Levin (2000), and assume that
u (Ct ) =
¢1−χC
¡
1
,
Ct − Q̄
1 − χC
(31)
and that
v (Lt ) = −
¢1−χn
1 ¡
.
1 − Lt − Z̄
1 − χL
(32)
Here, we introduce Q̄ and Z̄ in order to facilitate the comparison with Erceg, Henderson, and Levin
17
We solve the problem in a different way than Erceg, Henderson, and Levin (2000), where an interest rate rule is
postulated and the parameters are chosen to maximize welfare.
18
To solve for the optimal instrument rule, the paths can be used together with the Euler equation and suitable criteria
for the shape of the rule; see Woodford (2003), for a discussion.
19
It is possible to allow for other shocks. In the Technical Appendix of Carlsson and Westermark (2006a), we also
introduce a consumption shock and a labor-supply shock as in Erceg, Henderson, and Levin (2000). However, introducing
these shocks does not yield any additional insights here. In fact, it can easily be shown that under optimal policy, all
disturbances in the model (introduced as in Erceg, Henderson and Levin, 2000) can be reduced to a single disturbance
term (being a linear combination of all these shocks).
16
(2000), by mimicking the preferences and the steady state of their model.20 The calibration of the
deep parameters, presented in Table 1, also follows Erceg, Henderson, and Levin (2000), when possible
(thus, e.g., we do not follow Erceg, Henderson and Levin, 2000, when calibrating α and αw , since they
have a different interpretation in our model).
Deep Parameters
dp
dw
β
γ
χC
χn
σ
η
σ
ϕ
Table 1: Baseline Calibration of the Model
Baseline values Derived Parameters Baseline values
Constrained (Unconstrained)
2
Π
0.505
uc
6
Ωx (Ωx )
0.005 (0.002)
)
0.110 (0.044)
0.99
Ωw (Ωuc
w
0.30
θx
−0.962
1.5
θπ
−1.043
−2.676 (−7.458)
1.5
θπω (θuc
πω )
uc
0.750 (0.600)
4
α (α )
)
0.667
(0.833)
0.95
αw (αuc
w
0.0067
0.5
Moreover, to find the steady state of the model, we also follow Erceg, Henderson, and Levin (2000) and
set: Q = 0.3163, Z = 0.03, K = 30Q and A = 4.0266. Then, using the scheme outlined in section (1.4)
we obtain L = 0.27. Thus, L and Z stand for about one quarter of the households’ time endowment.
Further, Y = C = 3.1627, giving rise to a steady state capital-output ratio of about three. Moreover,
to achieve symmetric Nash bargaining (equally shared surplus), we set the bargaining power of the
union ϕ to 0.5.
Here, we treat price and wage contract durations as deep parameters. The probabilities of price
and wage adjustment are then derived from price and wage contract durations. This is due to the
fact that when comparing economies with and without downward nominal wage rigidity, we can either
keep price and wage resetting probabilities fixed or price and wage contract durations fixed. We find it
natural to compare economies with the same contract durations. Letting dp and dw denote the duration
of price and wage contracts, respectively, we have dp = duc
p = 1/(1 − αw α) and dw = 1/(1 − (1 + αw )/2)
and duc
w = 1/(1 − αw ) with and without downward nominal wage rigidity. Starting with wage contract
duration, Taylor (1999), summarizes the evidence and argues that overall, the evidence points toward
a wage contract duration of about one year. However, Cecchetti (1987), found that average duration
increases in periods with low inflation, which is what we want to capture here. In fact, during the
1950s and 1960s when inflation was low in the U.S., the wage contract duration was about two years
for the large union sector. In the baseline calibration, we set the duration to six quarters, which
20
In the Technical Appendix of Carlsson and Westermark (2006a), where we allow for consumption and labor supply
shocks, Q̄ corresponds to the steady state value of a consumption shock and Z̄ to the steady state value of a labor-supply
shock.
17
is higher than suggested by Taylor (1999), but still being conservative relative to the low inflation
estimate from Cecchetti (1987).
For price contract duration, the micro evidence presented by Bils and Klenow (2004), suggests a
price duration of about five months, whereas the micro evidence presented by Nakamura and Steinsson
(2007), and the survey evidence in Blinder, Canetti, Lebow, and Rudd (1998), suggest about eight
months. In our baseline calibration, we set the price duration to two quarters which is in the range
given by the above studies.
Then, using the steady state solution together with the definitions for the derived parameters
(presented in appendix A) yields the values presented in Table 1.
It is interesting to see the high coefficient for wage inflation variance in the loss function (θπω ).
Starting with the constrained case, we see that the coefficient on wage inflation variance (θπω ) is about
three times larger than the coefficient on the variance in the output gap (θx ) and the coefficient on
the variance in inflation (θπ ). Thus, variation in wage inflation is associated with considerable welfare
losses. Moreover, when relaxing the downward nominal wage rigidity constraint, the coefficient on
wage inflation almost triples in size relative to the constrained economy. The reason is that since
wages never fall in the constrained economy there will be a cap on the relative wage misalignment
that a given wage inflation gives rise to.
To obtain a ball-park estimate of σ , we make use of a the Taylor (1993) rule estimated by
Rudebusch (2002) (see expression (33) below) as an approximation of actual monetary policy and
impose it on the unconstrained version of the model. Then, we set σ to match the standard deviation
of quarterly inflation in the model with the actual standard deviation of the U.S. quarterly CPI inflation
(1987:Q4-1999:Q4).21 This results in a standard deviation of the innovation to the ŵt∗ process of 0.0067
(= σ ).
Numerically, we solve the model by iterating on the policy functions and updating the value
function given the new policy functions. In the procedure, we also take into account how expectations
in the constraints are affected by this (see (121) in Appendix D).
A standard reference for algorithms with occasionally binding constraints is Christiano and Fisher
(2000). Unfortunately, we cannot use this algorithm since our model is slightly different. Specifically,
our problem includes expectations of the control variables in the constraints. The way in which we
take care of this problem is to use that the control variables are functions of the state variables (i.e.,
the policy functions) and rewrite the constraint set in terms of state variables only.22 This is related
21
We focus on inflation for the calibration, since this is the only variable we can directly observe without resorting to
some filtering technique.
22
Note that policy functions are potentially nonlinear, since there is a non-linear constraint to the problem (i.e.
constraint (22)).
18
to the method used in e.g., Soderlind (1999). However, instead of using the policy functions from
the previous iteration as is done in Soderlind (1999), we use the current policy functions, as in the
algorithm used in e.g., Krusell, Quadrini, and Rios-Rull (1996). Note that the algorithm in Krusell,
Quadrini, and Rios-Rull (1996) can be considered as analyzing a one-period deviation from a proposed
policy. Iteration finishes when there are no gains from deviating from the proposed policy. The full
algorithm is outlined in appendix A.
4
Optimal Policy V.S. Simple Rules
First, we solve the model for the calibration described above, both in the case with and without
downward nominal wage rigidity under optimal discretionary monetary policy. In figure 2, we plot
the impulse responses to a one standard deviation negative shock to the natural real wage ŵt∗ .
Output gap
Inflation
0.005
Optimal Discretionary Policy Constrained
Optimal Discretionary Policy Unconstrained
0.3
0
0.25
−0.005
0.2
−0.01
0.15
−0.015
0.1
−0.02
0.05
−0.025
−0.03
0
0
5
10
Time
15
−0.05
20
0
5
Wage Inflation
10
Time
15
20
15
20
Real Wage
0.08
−0.2
−0.25
0.06
−0.3
0.04
−0.35
0.02
−0.4
−0.45
0
−0.5
−0.02
−0.04
−0.55
0
5
10
Time
15
20
0
5
10
Time
Figure 2: Impulse responses to a one standard deviation negative shock in the natural real wage. Scale
corresponds to percentage units.
Starting with the unconstrained case, the negative shock drives down the natural real wage implying
that the actual real wage is higher than the natural real wage, thus initially causing a positive real-wage
gap. The real wage can be adjusted by changing inflation and wage inflation. Holding the inflation
rate above the wage inflation rate decreases the real wage. However, since it is costly to stabilize the
real wage gap, in terms of the implied variation in inflation, wage inflation and the output gap, it is
optimal not to fully compensate for the shock. For the same reason, optimality requires that inflation
and wage inflation should be kept (approximately) at opposite sides of zero (the path of x̂t must also
19
be considered). Thus, the optimal initial inflation response is positive, whereas the wage inflation
response is negative. But the difference between them is not sufficiently large to immediately fully
close the real-wage gap.
Given the AR(1) structure of the shock, the natural real wage increases towards the steady value
of zero after the initial negative shock. So at some point, the central bank needs to start increasing the
real wage in order to continue to stabilize the economy. This is also what we see after approximately
four quarters. For this purpose, the relationship between inflation and wage inflation needs to be
reversed, which also happens at this point in time.
In the constrained case, wage inflation cannot be used to initially lower the real wage. Instead,
optimal policy prescribes a stronger initial inflation reaction relative to the unconstrained case, and
also allow for a larger output gap in order to stabilize the real-wage gap. However, the initial optimal
inflation response is not sufficient to stabilize the real wage gap to the same extent as in the unconstrained case. This is reflected by the initial real wage response in the constrained case lying above
the real wage response in the unconstrained case.
In figure 3, we plot the resulting impulse responses to a one standard deviation positive shock to
ŵt∗ .
For the unconstrained case, the impulse responses are a mirror image through the horizontal
Output gap
Inflation
0.03
0.05
0
0.025
−0.05
0.02
−0.1
0.015
−0.15
0.01
−0.2
0.005
−0.25
0
−0.005
Optimal Discretionary Policy Constrained
Optimal Discretionary Policy Unconstrained
−0.3
0
5
10
Time
15
20
0
5
Wage Inflation
10
Time
15
20
15
20
Real Wage
0.08
0.06
0.5
0.04
0.02
0.4
0
0.3
−0.02
−0.04
0
5
10
Time
15
0.2
20
0
5
10
Time
Figure 3: Impulse responses to a one standard deviation positive shock in the natural real wage. The
scale corresponds to percentage units.
axis of the negative case. We now see that the initial optimal inflation and output gap responses are
smaller than in the unconstrained case. However, the initial wage inflation response is larger than
20
in the unconstrained case. Note that downward nominal wage rigidity does not only affect private
sector behavior but also the parameters in the loss function in making the parameter for wage inflation
smaller. Below we will decompose the effects on welfare from the change in the loss function from the
effects stemming from changes in private sector behavior. However, any asymmetry in the impulse
response paths across positive and negative shocks must stem from private sector behavior and not
from the (symmetric) loss function.
In figure 4, we plot the optimal interest rate path (in terms of deviations from steady state) for
the unconstrained and constrained case, respectively. As can be seen in figure 4, the optimal interest
Interest Rate Response (Negative Shock)
Interest Rate Response (Positive Shock)
0
0.2
−0.05
0.15
−0.1
0.1
−0.15
0.05
−0.2
Optimal Discretionary Policy Constrained
Optimal Discretionary Policy Unconstrained
0
0
5
10
Time
15
20
0
5
10
Time
15
20
Figure 4: Optimal interest-rate responses to a one standard deviation positive and negative shock in
the natural real wage. The scale corresponds to percentage units.
rate response is asymmetric in the constrained case. Especially, the interest rate response is larger in
the constrained case for negative shocks and smaller for positive shocks relative to the unconstrained
case.
Next, we turn to welfare analysis. Note first that it need not be the case that the model with
the downward nominal wage rigidity constraint necessarily leads to lower welfare relative to the unconstrained case. The reason is that this is not just an additional constraint on the problem, i.e., a
constraint that makes the feasible set smaller. Instead, it is a constraint that changes the choice set.23
Also, another factor is that the parameter on wage inflation variance in the welfare function changes
23
To see this, consider the relationship between, say π̂ω and x̂ in equation (22), treating other variables as constants.
Then, in the unconstrained case, the relationship is linear with slope Ωx . In contrast, in the constrained case, the
relationship is piecewise linear with zero slope below some critical value of x̂ and with slope Ωx above this value.
21
when imposing downward nominal wage rigidity. Thus, overall, the welfare effect from downward
nominal wage rigidity is ambiguous a priori.
To compute welfare, we construct sequences of shocks for 1000 periods and use these to find paths
for the variables x̂t , π̂ t , π̂ ωt and ŵt . Then, welfare is computed from these paths using the welfare
criterion (26), ignoring the periods t > 1000. This is repeated 1000 times to generate an approximation
of the expectation. Finally, to express the welfare loss as a fraction of steady state consumption, we
¢ ¢
¡ ¡
scale the welfare difference (26) by 1/ uC C̄, Q̄ C̄ .
In the unconstrained case, we find a welfare difference relative to the natural (flexible price and
wage) welfare level of 0.186 percentage units of steady state consumption. Interestingly, the result
for the constrained case is almost identical with a difference of 0.181. Thus, downward nominal wage
rigidity does not necessarily lead to large welfare losses, as is often considered. In fact, in our model we
find a (very) modest welfare gain. Now, since introducing downward nominal wage rigidity does not
only affect the behavior of the private sector, but also the parameter in the loss function on variation
in wage inflation (c.f. Table 1 above), it is interesting to try to isolate the effects. To this end, we both
solve for the unconstrained optimal policy, as well as calculate the welfare loss using the parameters
for the loss function from the constrained case. The resulting welfare difference is 0.178, which is
very close to the original results for the unconstrained case (0.186). Thus, the similarity between
the welfare outcomes in the unconstrained and constrained cases is not driven by the increase in the
loss-function parameter on wage inflation variance.
The intuition for the small welfare effects is that downward nominal wage rigidity is not all that
harmful since it may help keep down wage dispersion (see below). This, in turn, can be exploited by
the central bank when designing optimal monetary policy.
A Simple Rule
Next, we turn to analyzing the effects on welfare from relying on a simple instrument rule instead of
the optimal policy rule. To this end, we impose a Taylor (1993) rule, i.e.
bit = 1.24π̂ t + 0.33x̂t ,
(33)
where the parameters for (33) are calibrated to match the estimates in Rudebusch (2002). Note
that this rule does not take any asymmetry into account when setting the interest rate, although the
economy will react asymmetrically to positive and negative shocks, due to private sector behavior.24
24
To implement the Taylor rule, we replace the central bank’s first-order condition in systems (135) and (139) in
Appendix C with the sticky price Euler equation (20), where we have used the corresponding flexible-price Euler equation
to eliminate the real natural interest rate and the Taylor rule to eliminate the nominal interest rate. For the system
22
In figure 5, we plot the resulting impulse responses to a one standard deviation negative shock to
ŵt∗ when the nominal interest rate is governed by a Taylor rule as well as under optimal policy.
Output gap
Inflation
0.15
Optimal Discretionary Policy Constrained
Taylor Rule Constrained
0.3
0.1
0.25
0.05
0.2
0
0.15
0.1
−0.05
0.05
−0.1
0
0
5
10
Time
15
−0.05
20
0
5
Wage Inflation
10
Time
15
20
15
20
Real Wage
0.1
−0.2
−0.25
0.08
−0.3
0.06
−0.35
−0.4
0.04
−0.45
0.02
−0.5
−0.55
0
0
5
10
Time
15
20
0
5
10
Time
Figure 5: Impulse responses to a one standard deviation negative shock to ŵt∗ when the nominal
interest rate is governed by a Taylor rule as well as under the optimal policy. Scale corresponds to
percentage units.
Note that the impulse responses are not smooth as under optimal policy. The reason for this is that
the Taylor rule is not optimally chosen and has the same functional form for both the case when
downward nominal wage rigidity binds and when it does not. Interestingly, the Taylor rule responses
for inflation are in fact fairly similar to the optimal responses. However, for wage inflation, the Taylor
rule undershoots slightly when the constraint stops binding, all in all leading to substantial excess
volatility in the output gap. In figure 6, we plot the resulting impulse responses to a one standard
deviation positive shock to ŵt∗ . Once more, the Taylor rule responses for inflation are similar to the
optimal responses. However, the initial wage inflation response now overshoots. Once more, the
volatility in the output gap is substantially larger than in the optimal responses.
25
Next we turn to welfare. Note that it is not a trivial result that optimal discretionary policy
outperforms the Taylor rule, since the Taylor rule is, in fact, a commitment rule and hence, could
perform better than the optimal discretionary rule. However, we do find that optimal discretionary
under the Taylor rule, there is no need to iterate on the value function. Instead, we can directly solve the system for the
policy functions (i.e. we only do step 1 in the numerical algorithm outlined in Appendix A). Then, we can simulate the
model
and evaluate welfare as done above.
25
For brevity, we do not plot interest rate responses, since they are mainly a reflection of differences in the output
gap responses.
23
Output gap
Inflation
0.15
0.05
0
0.1
−0.05
0.05
−0.1
0
−0.15
−0.2
−0.05
−0.25
−0.1
Optimal Discretionary Policy Constrained
Taylor Rule Constrained
−0.3
0
5
10
Time
15
20
0
5
Wage Inflation
10
Time
15
20
15
20
Real Wage
0.1
0.08
0.5
0.06
0.4
0.04
0.02
0.3
0
0
5
10
Time
15
0.2
20
0
5
10
Time
Figure 6: Impulse responses to a one standard deviation positive shock to ŵt∗ when the nominal
interest rate is governed by a Taylor rule as well as under the optimal policy. Scale corresponds to
percentage units.
policy performs better than the policy prescribed by the Taylor rule. In terms of steady state consumption, the additional loss is about 0.07 percentage units of steady state consumption. But, all in
all, the Taylor rule seems to be a fairly good approximation of optimal discretionary monetary policy
in the presence of downward nominal wage rigidity.
Finally, we look at the impulse responses for the unconstrained economy versus an economy with
downward nominal wage rigidity under the Taylor rule. In figure 7, we plot the resulting impulse
responses to a one standard deviation negative shock to ŵt∗ . The key result here is that downward
nominal wage rigidity actually helps stabilize the economy in the wage inflation dimension, whereas it
does not induce much more variation in inflation and the output gap. A fairly similar result appears
from figure 8, where we plot the resulting impulse responses to a one standard deviation negative
shock to ŵt∗ . Most notably, the wage inflation variability is not substantially larger in the downward
rigid case. This is also the dimensions about which the households care most (c.f. table 1).
When looking at welfare differences, we find that welfare loss is quite a bit lower in the economy
with downward nominal wage rigidity relative to the unconstrained economy when monetary policy
is governed by the Taylor rule. In terms of steady state consumption, the welfare gain is about 0.14
percentage units of steady state consumption. Thus, an agent would in this case prefer an economy
with downward nominal wage rigidity over an economy with downward nominal wage flexibility ex ante.
24
Output gap
Inflation
0.15
Taylor Rule Unconstrained
Taylor Rule Constrained
0.3
0.1
0.25
0.05
0.2
0
0.15
0.1
−0.05
0.05
−0.1
0
0
5
10
Time
15
−0.05
20
0
5
Wage Inflation
10
Time
15
20
15
20
Real Wage
0.1
−0.2
−0.25
−0.3
0.05
−0.35
−0.4
−0.45
0
−0.5
−0.55
−0.05
0
5
10
Time
15
20
0
5
10
Time
Figure 7: Impulse responses to a one standard deviation negative shock to ŵt∗ when the nominal
interest rate is governed by a Taylor rule. Scale corresponds to percentage units.
Output gap
Inflation
0.15
0.05
0
0.1
−0.05
0.05
−0.1
0
−0.15
−0.2
−0.05
−0.25
−0.1
Taylor Rule Unconstrained
Taylor Rule Constrained
−0.3
0
5
10
Time
15
20
0
5
Wage Inflation
10
Time
15
20
15
20
Real Wage
0.1
0.5
0.05
0.4
0
0.3
−0.05
0
5
10
Time
15
0.2
20
0
5
10
Time
Figure 8: Impulse responses to a one standard deviation positive shock to ŵt∗ when the nominal
interest rate is governed by a Taylor rule. Scale corresponds to percentag units.
25
Evaluating the unconstrained economy with the loss function of the constrained economy suggests a
welfare gain from downward nominal wage rigidity of about 0.08 percent of steady state consumption.
Thus, once more, the above result is not just driven by the increase in the loss-function parameter on
wage inflation, when relaxing the constraint while keeping the duration of price and wage contracts
constant. Instead, private sector behavior plays an important part in explaining the results.
Wage-Inflation Distributions and Welfare
To see the consequences for welfare, we illustrate the wage-inflation distributions in four histograms;
see figure 9. First, if we vary policy, i.e., compare the Taylor rule and optimal policy, the distribution
Optimal Policy Unconstrained
−0.4
−0.2
0
0.2
Emprical Taylor Rule
0.4
−0.4
Optimal Policy Constrained
−0.4
−0.2
0
0.2
−0.2
0
0.2
0.4
Empirical Taylor Rule Constrained
0.4
−0.4
−0.2
0
0.2
0.4
Figure 9: Histograms of simulated wage-inflation paths.
is more dispersed under the Taylor rule. This seems to hold irrespectively whether there are downward
nominal wage rigidities or not. If we fix the policy regime, on the other hand, the distribution seems
to have lower variability with downward nominal wage rigidities than without.
4.1
Robustness
A key parameter for welfare evaluation is the standard deviation of the innovation to ŵt∗ , σ . We now
consider increasing σ by 50 percent. The results of this experiment are presented in Table 2. We see
that this does not change the qualitative conclusions from our baseline simulation, in terms of welfare
rankings. But increasing the shock size does increase the welfare loss in all four cases.
26
Unc.
Con.
Unc.
Con.
Opt.
Opt.
Taylor
Taylor
Table 2: Robustness - Shock Size
Welfare Differences Relative to Flex Price
Baseline (σ = 0.0067) Big Shocks (σ = 0.0101)
−0.1856
−0.4176
−0.1809
−0.4125
−0.3872
−0.8723
−0.2508
−0.6118
The values are expressed in terms of percentage units of steady state consumption.
In the baseline calibration, the wage-contract duration is six quarters. Since this may seem to
be on the high side (even though it is in line with the evidence of wage-contract durations in lowinflation environments), we also see to what extent our results are robust to varying the wage-contract
duration. In table 3, we present the results from decreasing the wage duration by a quarter. As can
be seen in the table, decreasing the duration of the wage contracts leads to slightly smaller welfare
Table 3: Robustness - Contract Durations
Welfare Differences Relative to Flex Price
Contract Durations dp = 2, dw = 6
dp = 2, dw = 5
Unc. Opt.
−0.1856
−0.1759
Con. Opt.
−0.1809
−0.1761
Unc. Taylor
−0.3872
−0.3767
Con. Taylor
−0.2508
−0.2436
The values are expressed in terms of percentage units of steady state consumption.
losses in all cases. Also, note that the welfare ranking between the optimal policy in the constrained
and unconstrained case shifts to a tiny advantage for the unconstrained case. However, the key point
that downward nominal wage rigidity does not imply large welfare losses is robust.
Finally, we have also experimented to see how sensitive the results are to the exact values of the
parameters in the Taylor rule. In table 4, we present the result from using the original values from
Taylor (1993) (i.e., β π̂ = 1.5 and β x̂ = 0.5). When using the Taylor (1993) parameter values for
Table 4: Robustness - Taylor Rule Parameters
Welfare Differences Relative to Flex Price
Taylor Rule β π̂ = 1.24, β x̂ = 0.33 β π̂ = 1.5, β x̂ = 0.5
Unc. Taylor
−0.3872
−0.3224
Con. Taylor
−0.2508
−0.2604
The values are expressed in terms of percentage units of steady state consumption.
the Taylor rule, the welfare loss decreases without downward nominal wage rigidity, while the welfare
loss almost remains unchanged when downward nominal wage rigidities are present. Thus, the results
indicate that downward nominal wage rigidity makes the welfare outcome less sensitive to the exact
calibration of the Taylor rule.
27
Overall, the key points from the previous section seem to be robust.
5
Concluding remarks
In this paper, we study the implications for optimal monetary policy when declining nominal wages
do not constitute a viable margin for adjustment to adverse economic conditions. To this end, a
New Keynesian model is developed that can endogenously account for downward nominal wage rigidity. This is achieved by introducing wage bargaining between firms and unions in the model as in
Holden (1994). Under asymmetric conflict costs, downward nominal wage rigidity arises as a rational
endogenous outcome.
Focusing on optimal discretionary monetary policy, we show that when money wages cannot fall,
the optimal policy response to changing economic conditions becomes asymmetric. More specifically,
inflation and the output gap respond more when the downward nominal wage rigidity constraint binds.
Interestingly, in our baseline case the welfare loss is actually slightly smaller in an economy with
downward nominal wage rigidities. The reason is that downward nominal rigidity is not an additional
constraint on the problem. Instead, it is a constraint that changes the choice set and opens up for
potential welfare gains. Another effect of downward nominal wage rigidity is that the loss function
parameter for wage inflation variation is changed, although this latter effect seems to play a small
role in explaining the welfare effects of downward nominal wage rigidity (at least under optimal
policy). We also find that the Taylor rule estimated by Rudebusch (2002), provides a fairly good
approximation of optimal discretionary policy in terms of welfare under downward nominal wage
rigidity. Experimenting with using the original Taylor (1993), parameters for the Taylor rule indicates
that the exact specification of the Taylor rule actually plays a minor role for this property. In contrast,
the Taylor rule does not provide such a good approximation of optimal policy in the unconstrained case.
A corollary then is that, under the Taylor rule, agents would clearly prefer an economy with downward
nominal wage rigidities rather than an unconstrained economy ex ante. That is, since downward
nominal wage rigidity actually helps stabilizing the economy in the wage inflation dimension and
hence, reduces wage variability, but does not induce much more variation in inflation and the output
gap.
28
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31
Appendix
A
Numerical Algorithm
The main outline of the algorithm follows algorithm 12.2 in Judd (1998). We first define N ⊂ R2
nodes over the state space.
Step 0. Guess policy functions Ui0 , i.e., parameter values in second-order complete polynomials
π̂ 0 (ŵt−1 , ŵt∗ ) ,
π̂ ω0 (ŵt−1 , ŵt∗ ) ,
(34)
x̂0 (ŵt−1 , ŵt∗ ) ,
and value function
V 0 (ŵt−1 , ŵt∗ ) .
(35)
Then proceed to step 2.
Step 1. Consider systems (135) and (139) derived by using the first three first-order conditions in
(130) to eliminate the Lagrange multipliers in the last first-order condition, together with the three
constraints (21) to (23) (see the discussion in section 4 and Appendix C) We find the new guess
for the policy functions Uil+1 by solving for these from systems (135) and (139), respectively, with a
collocation method. While solving for policy functions, we take into account that the policy functions
affect the expectations.
Step 2. Compute current period utility Pil+1 for i = 1, . . . , N , given policy function guesses Uil+1 .26
Step 3. Update the value function V l+1 (ŵt−1 , ŵt∗ ) using
´
³
l+1 −1
V l+1 = I − βQU
P l+1 ,
where QU
l+1
(36)
is the transition matrix defined by the new guess for the policy functions for inflation
and wage inflation and the flow equation for real wages.
°
°
Step 4. If °V l+1 − V l ° < ε stop. Otherwise, go to step 1.
B
Parameter Definitions
The parameters in the equilibrium relations are defined as
∗
In terms of Judd (1998) p. 416, compute π yi , Uil+1 where Uil+1 consists of π̂t , π̂ω
t and x̂t and yi = (ŵt−1 , ŵt )i ,
l+1
l+1
which gives Pi = π yi , Ui
.
26
32
x
Ωx = Π1 Φ
Φd ,
uc Φx
Ωuc
x = Π1 Φd ,
Ωw = Π1 ΦΦwd ,
uc Φw
Ωuc
w = Π1 Φd ,
(37)
where
¡
Π1 = 1 −
Π=
1−αw α
αw α
1+αw
2 β
¢ 1−αw
αw
,
1−αw
Πuc
1 = (1 − αw β) αw ,
(38)
(1 − αw αβ) ,
and Φx , Φw and Φd are defined as (see the Appendix C below for details)
1
v̄L L̄
(ϕ (2 + (1 + ρL ) εL ) − (1 + εL )) v̄L L̄ − ϕ
(1 − γ)2
Φd = εL
(39)
σ−1
1−γ
µ
¶
¶
µ
εL
1 − σγ
1 v̄L L̄
ρC − ρL
− (1 − ϕ) σγ v̄L L̄ + ϕ
(1 − σγ − ρC (1 − γ))
Φx =
−ϕ
σ−1
1−γ
σ−11−γ
¶
µ
εL
1 v̄L L̄
(1 − σρL ) + (1 − ϕ) (1 − γ) σ v̄L L̄ + ϕ
(1 + εL )
Φw = − −ϕ
σ−1
σ−11−γ
where
εL = − (γ + σ (1 − γ)) .
(40)
Let
ρC = −
C̄
L̄
ūCC C̄
v̄LL L
.
= χC
= −χn
and ρL = −
ūC
v̄L
1 − L̄ − Z̄
C̄ − Q̄
(41)
The loss function parameters in (26) are then defined as
θx
θπ
θ πω
µ
¶
1
γ
C̄
ūC −ρC + ρL
−
=
,
2
1−γ 1−γ
v̄L L̄ σ (γ (1 − σ) + 1 + γ) 1
,
= −
2
1−γ
Π
µ
¶
v̄L L̄ −γεL + (1 − γ) σ (1 + γ (1 − σ))
σ (γ (1 − σ) + 1 + γ)
.
= −
− (1 − γ)
2
Π1
Π
(42)
and for the unconstrained case we need to replace θπω with θuc
π ω defined as
θuc
πω
v̄L L̄
=−
2
µ
−γεL + (1 − γ) σ (1 + γ (1 − σ))
σ (γ (1 − σ) + 1 + γ)
− (1 − γ)
Πuc
Π
1
33
¶
.
(43)
C
Some Derivations
Most of the derivations in the paper are similar to those in Carlsson and Westermark (2006b). Here,
we present the derivations that must be modified due to downward nominal wage rigidity. Thus, the
changes concern the sections where wage setting is a part. Hence, wage setting itself must be modified,
as well as part of the steady state analysis and the welfare computations.
C.1
Wages
As in Carlsson and Westermark (2006b), we use gradient notation (∇) to indicate derivatives. We can
write the derivative of Uut as
∇W Uut = Et
×
∞
X
k=0
∞
X
j=0
(αw αβ)k ∇W Υt,t+k + Et
∞ k−1
Y
X
k=1 i=0
(αw αβ)j ∇W Υt+k,t+k+j + Et
¡
¢
ζ t+i (dt+i ) ζ t+k (dt+k ) − αw α β k
∞ X
k−1
X
k=1 i=1
⎛
⎜
⎝
k−1
Y
m=0
m6=i
⎞
⎟
ζ t+m (dt+m )⎠ (1 − αw ) ∇W ζ t+i (dt+i )
⎞
⎛
∞
¢ kX
¡
¢
¡
(αw αβ)j Υt+k,t+k+j + 1 − ζ t+k (dt+k ) β k Uut+k ⎠
× ⎝ ζ t+k (dt+k ) − αw α β
(44)
j=0
+Et
∞
X
k=1
Ãk−1
Y
i=0
⎛
⎞
∞
X
ζ t+i (dt+i ) ∇W ζ t+k (dt+k ) β k ⎝
(αw αβ)j Υt+k,t+k+j − Uut+k ⎠
!
j=0
and the derivative of Uft as
∇W Uft
= Et
∞
X
k=0
×
∞
X
j=0
(αw α)k ψ t,t+k ∇W φt,t+k + Et
∞ k−1
Y
X
k=1 i=0
(αw α)j ψ t,t+k+j ∇W φt+k,t+k+j
¡
¢
ζ t+i (dt+i ) ζ t+k (dt+k ) − αw α
⎞
⎛
∞ X
k−1
k−1
X
Y
⎟
⎜
+ Et
ζ t+m (dt+m )⎠ ∇W ζ t+i (dt+i )
⎝
k=1 i=1
m=0
m6=i
(45)
⎞
⎛
∞
¢X
¡
¢
¡
(αw α)j ψ t,t+k+j φt+k,t+k+j (W (f )) + 1 − ζ t+k (dt+k ) ψ t,t+k Uf,t+k ⎠
× ⎝ ζ t+k (dt+k ) − αw α
j=0
+Et
Ãk−1
∞
Y
X
k=1
i=0
⎞
⎛
∞
X
ζ t+i (dt+i ) ∇W ζ t+k (dt+k ) ⎝
(αw α)j ψ t,t+k+j φt+k,t+k+j − ψ t,t+k Uf,t+k ⎠ .
!
j=0
34
Note that, evaluating at the steady state distribution, we have
∇W Uft
= Et
∞
X
(αw α)k ψ t,t+k ∇W φt,t+k
k=0
∞ k−1
Y
X
+Et
k=1 i=0
(46)
∞
¡
¢X
ζ t+i (dt+i ) ζ t+k (dt+k ) − αw α
(αw α)j ψ t,t+k+j ∇W φt+k,t+k+j
j=0
with ∇W φt+k,t+k+j < 0 and hence, ∇W Uft < 0.
Then, using the first-order condition (17) in steady state, we have
∇W Uut = −
and hence
1 − ϕ Uu,t − Uo
∇W Uft > 0
ϕ
Uf,t
1−ϕ
∂W (f )
ϕ +1
=
(Uu,t − Uo )∇W Uft > 0.
∂ϕ
SOC
{z }| {z }
|
| {z }
(+)
(−)
(47)
(48)
(−)
The ideal wage for the workers is when ϕ = 1 and the ideal wage for the firm is when ϕ = 0. Thus, the
desired wage of the workers is larger than the ideal wage for the firm. Thus, for small shocks around
the the steady state, the union would never want to reduce the wage.
C.2
Steady state
We now turn to the (non-stochastic) steady state of the model.27 Note that the steady state of the
real variables is the same in the flexible price model and the sticky price model. In the steady state,
R, C, Y (f ) and B are constant. Moreover, B = 0. M and P also grow at the rate π̄, i.e., we have
Pt+1
Pt
= π̄ and I¯ = R̄π̄.
Now, let us analyze the Nash Bargaining solution in steady state. Since Ft (dt ) = 0 for all dt ≥ 1
and Ft (dt ) = 1 for all dt < 1, the first-order condition (17) is well defined for any W (f ) 6= Pt w̄, given
that all other variables are at their steady state values28 , and we have
¡
¢
Ξ (W (f )) = ϕŪf (W (f )) ∇W Ūu (W (f )) + (1 − ϕ) Ūu (W (f )) − Ūo ∇W Ūf (W (f )) ,
(49)
where Ūu (W (f )) etc. indicates that all variables except W (f ) are at steady state levels, noting that
the steady state value of ψ t,t+k is ψ̄ k = β k . When W (f ) > Pt w̄, we have dt+k < 1 and Ft+1 (dt+k ) = 1
27
28
That is, a situation where the disturbance At is equal to its mean value at all dates.
If all other variables are at their steady state values dt = 1 ⇐⇒ W (f ) = Pt w̄.
35
and hence, ζ t+i (dt+k ) = 1
Ūu (W (f )) =
Ūf (W (f )) =
1
Ῡ (W (f )) ,
1−β
1
φ̄ (W (f )) .
1−β
(50)
Letting
Ūo =
1
Ῡo
1−β
(51)
and
¡
¢
ΞD (W (f )) = ϕφ̄ (W (f )) ∇W Ῡ (W (f )) + (1 − ϕ) Ῡ (W (f )) − Ῡo ∇W φ̄ (W (f )) .
(52)
When W (f ) > Pt w̄, we have
Ξ (W (f )) =
¢
1
ϕ
1−ϕ ¡
∇W Ῡ (W (f )) +
φ̄ (W (f ))
Ῡ (W (f )) − Ῡo ∇W φ̄ (W (f ))
1−β
1−β
1−β
and hence
Ξ (W (f )) =
µ
1
1−β
¶2
ΞD (W (f )) .
(53)
(54)
When W (f ) < Pt w̄, we have
Ξ (W (f )) =
¶2
1
ΞD (W (f )) +
1 − αw β
¡
¢
¤
β (1 − αw ) £
+
2 ϕŪf (Pt w̄) ∇W Ῡ (W (f )) + (1 − ϕ) Ūu (Pt w̄) − Ūo ∇W φ̄ (W (f )) .
(1 − αw β)
µ
As Pt w̄ → W (f ), the term on the second row converges to ϑΞD (Pt w̄) where ϑ > 0, since Ūf (Pt w̄)
and Ūu (Pt w̄) converges to Ūf (W (f )) and Ūu (W (f )), respectively. Hence, we must have
ΞD (Pt w̄) = 0
in steady state. Since per-period utility and profits are concave and continuously differentiable in
wages, there is a unique wage such that this holds.
36
As in Carlsson and Westermark (2006a), we have
Ῡ = ū − v̄,
¡
¢
¡
¢
Ῡo = u C̄ − bw̄L̄, Q̄ − v 0, Z̄ ,
φ̄ = τ ȳ,
∂ Ῡ
∂W (f )
∂ φ̄
∂W (f )
= ūC (1 + τ w ) (1 − (γ + σ (1 − γ)))
= − (1 − γ)
(55)
w̄L̄
L̄
+ v̄L (γ + σ (1 − γ))
,
W (f )
W (f )
tc
,
W (f )
and ΞD (W (f )) = 0 can be written as
¡
¢
ϕτ ȳ [ūC (1 + τ w ) (1 − σ) w̄ (1 − γ) + v̄L (γ + σ (1 − γ))] L̄ − (1 − ϕ) ū − v̄ − Ῡo (1 − γ) ȳ = 0. (56)
Potentially, the parties can settle on wages where ΞD (W (f )) ≤ 0. However, in the case when
prices are flexible, there is a unique wage agreement on the wage where ΞD (W (f )) = 0. We focus on
the same point here.
C.2.1
Efficiency
Since there are two distortions and three instruments, one instrument is redundant. Here we use the
same method as in Carlsson and Westermark (2006a), relying on adjusting τ and Ῡo , i.e., the per
period version of Ūo , to achieve efficiency.
C.3
Optimal Wages and the Wage Setting “Phillips” Curve
Here, it is important to distinguish the period when the wage contract was last rewritten, the period
when the price was last changed and the current period. Therefore, we use the following notation
xtt+k,t+k+j to denote the value of variable x in period t + k + j, when the wage contract was last
renegotiated in t and the price was last changed in t + k.
C.4
Case 1 Wages are adjusted
In this section, we derive the wage setting “Phillips” curve. Assume that the downward nominal wage
rigidity constraint does not bind. Log-linearizing the first-order condition (17) gives
0 =
t
\
ϕ∇W Ūu ∇
W Uu
+ (1 − ϕ)
+ (1 − ϕ)
Ã
1
Ūu − Ūo
∇W Ūf Ūu Ûut − ¡ ¢2 ∇W Ūf Ūf Ûft
Ūf
Ūf
Ūu − Ūo
t
\
∇W Ūf ∇
W Uf .
Ūf
37
!
(57)
The four terms in the above expressions are29
Ūu Ûut
= Et
∞
X
(αw αβ)k Υ̂tt,t+k
k=0
∞ µ
X
+Et
k=1
Ūf Ûft
= Et
∞
X
1 + αw
2
(58)
¶k−1 µ
¶ X
∞
1 − αw
k
β
αw (1 − α) +
(αw αβ)j Υ̂tt+k,t+k+j ,
2
j=0
t
(αw α)k ψ̄ k φ̄φ̂t,t+k (W (f ))
k=0
∞ µ
X
+Et
k=1
t
\
∇W Ūu ∇
W U u = Et
1 + αw
2
∞
X
(59)
¶k−1 µ
¶ ∞
1 − αw X
t
(αw α)j ψ̄ k+j φ̄φ̂t+k,t+k+j (W (f )) ,
αw (1 − α) +
2
j=0
t
\
(αw αβ)k ∇W Υ∇
W Υt,t+k
k=0
∞ µ
X
+Et
k=1
1 + αw
2
(60)
¶k−1 µ
¶ X
∞
t
1 − αw
\
βk
αw (1 − α) +
(αw αβ)j ∇W Υ∇
W Υt+k,t+k+j ,
2
j=0
and
∞
´
³
³
´
X
t
k
t
\
[
=
E
(α
α)
U
ψ̄
∇W Ūf −∇
∇
φ
−
∇
φ
t
w
W f
W t,t+k +
k W
k=0
∞ µ
X
+Et
k=1
×
∞
X
j=0
1 + αw
2
¶k−1 µ
¶
1 − αw
αw (1 − α) +
2
(61)
´
³
t
[
(αw α)j ψ̄ k+j ∇W φ −∇
W φt+k,t+k+j .
Leading the first-order condition for wages one period, multiplying with αw β and taking the
expectation at t gives
¶
µ
¶
µ
1
1 + αw
1 + αw
\
t+1
t
t+1
t
\
βEt ∇W Uu
βEt Ûu
+ (1 − ϕ)
∇W Ūf Ūu Ûu −
0 = ϕ∇W Ūu ∇W Uu −
2
2
Ūf
Ã
µ
¶!
1 + αw
Ūu − Ūo
t+1
t
βEt Ûf
+ (1 − ϕ) − ¡ ¢2 ∇W Ūf Ūf Ûf −
(62)
2
Ūf
µ
µ
¶¶
1 + αw
Ūu − Ūo
\
t+1
t
\
βEt −∇W Uf
∇W Ūf −∇W Uf −
.
+ (1 − ϕ)
2
Ūf
29
The last summation in the expressions contains all future price changes during the present wage contract.
38
We define n̂t (f ) as the log-linearized relative wage (nt (f ) =
W (f )
Wt ).
Let
µ
¶
¢
1 + αw ¡
t+1
ω
t
β Et n̂ (f ) + Et π̂ t+1
n̂ (f ) −
∆n̂ (f ) =
w
2
1 − 1+α
2 β
1
t
(63)
and, from Carlsson and Westermark (2006b)
n̂t =
Z
n̂t (f ) df =
αw
π̂ ω .
1 − αw t
(64)
Moreover, we need to distinguish the period where the wage contract was last rewritten for the
terms φ̂t+k,t+k+j and Υ̂t+k,t+k+j as well as for the corresponding derivatives. As for firm payoff
and union utility, we indicate the wage contract period with superscripts, i.e., we use the notation
t
φ̂t+k,t+k+j and Υ̂tt+k,t+k+j .
We eliminate the two distortions in the economy stemming from monopoly power in the intermediate goods market and from union bargaining power in the labor market as in Carlsson and Westermark
(2006a). Following the method in Carlsson and Westermark (2006b), we then have
Φd ∆n̂t + Φx x̂t + Φw (ŵt − ŵt∗ ) = 0,
(65)
where
Φd
Φx
Φw
µ
¶
1
(ϕ (2 + (1 + ρL ) εL ) − (1 + εL )) − ϕ (1 − γ) v̄L L̄,
=
εL
σ−1
µ
¶
¶
¶
µµ
1 − σγ
1 − σγ − ρC (1 − γ)
εL
ρC − ρL
− (1 − ϕ) σγ + ϕ
v̄L L̄,
=
−ϕ
σ−1
1−γ
(σ − 1) (1 − γ)
µ µ
¶
¶
εL
1 + εL
=
− −ϕ
v̄L L̄.
(1 − σρL ) + (1 − ϕ) (1 − γ) σ + ϕ
σ−1
(σ − 1) (1 − γ)
(66)
Using (66), when dividing expression (65) with Φd and dividing through with 1 + εL , we get
Φx
Φd
=
Φw
Φd
=
³
´
γ
− ρC + (1 − ϕ) σ 1−γ
´
³
,
εL
(1
+
ρ
ε
)
+
1
−
(1
−
ϕ)
ε
ϕ 1+ε
L
L
L
L
´
³
εL
1
ϕ 1+εL (1 − σρL ) + 1−γ
+ (1 − ϕ) σ
´
³
.
εL
(1
+
ρ
ε
)
+
1
−
(1
−
ϕ)
ε
ϕ 1+ε
L
L L
L
ϕ
1−σγ 1+ρL εL
1−γ 1+εL
(67)
Using (63) and (64)
Z
1
t
∆n̂ (f ) df =
1−
1+αw
2 β
µ
¶
αw
1 1 + αw
ω
ω
βEt π̂ t+1 ,
π̂ −
1 − αw t
1 − αw
2
39
(68)
and using (38), the labor market Phillips curve is
π̂ ωt
C.4.1
1 + αw
βEt π̂ ωt+1 − Π1
=
2αw
µ
¶
Φx
Φw
∗
x̂t +
(ŵt − ŵt ) .
Φd
Φd
(69)
Case 2. Wages are not adjusted
With negative shocks, we get
n̂t = π̂ ωt = 0.
C.5
(70)
Welfare
First, consider the relationship between relative prices and wages. This can be derived from the
price setting relationship; see Carlsson and Westermark (2006b). As in the Technical Appendix, we
consider the relative price q̂t (f ) and the relative wage in period t (where prices were last rewritten in
some period s < t). By similar arguments as in the Technical Appendix of Carlsson and Westermark
(2006a), we then have the following relationship between relative prices and wages
q̂t (f ) = (1 − γ) n̂t (f ) + K 0 .
(71)
Since prices and wages are not fully flexible, variances are persistent. We want to find the variance
today as a function of previous variances and inflation. For this purpose, let us express varf (log Pt (f ))
and varf (log Wt (f )) in terms of squared inflation and wage inflation. Let P̄t = Ef log Pt (f ). We have
¡
¢2 ¡
¢2
varf (log Pt (f )) = Ef log Pt (f ) − log π̄ − P̄t−1 − ∆P̄t
(72)
where
∆P̄t = P̄t − log π̄ − P̄t−1 .
(73)
Let us rewrite ∆P̄t in terms of inflation. Since log Pt = Ef log Pt (f ) = P̄t , we have
30
∆P̄t = log Pt − log π̄ − log Pt−1 = log π t − log π̄ = π̂ t .
30
This follows from using a first-order Taylor approximation of the price level Pt1−σ =
]
P̄ + (1 − σ) P̄ −σ P̄ P̂t − P̄ =
1
0
Since P̄ = P̄ (f ), we have
P̂t =
]
U1
0
(74)
Pt (f )1−σ df
df.
P̄ + (1 − σ) P̄ −σ P̄ P̂t (f ) − P̄
1
P̂t (f ) df
0
or, by the definition of P̂t and P̂t (f), we have (74) (they are constructed as log-deviations around the same mean).
40
Similarly, we have
31
∆W̄t = log Wt − log π̄ ω − log Wt−1 = π̂ w
t .
(75)
We can write varf (log Pt (f )) as
¡
¢2 ¡
¢2
Ef log Pt (f ) − log π̄ − P̄t−1 − ∆P̄t
¡
¡
¢2
¢2
= αw αEf log π̄Pt−1 (f ) − log π̄ − P̄t−1 + (1 − α) αw Ef log Pto (Wt (f )) − log π̄ − P̄t−1
Z
¡
¢2
+ (1 − αw )
Ef log Pto (Wt (f )) − log π̄ − P̄t−1 dF (dt,t )
(76)
Wt−1 (f )>Wto (f )
Z
¡
¢2
¢2
¡
Ef log Pto (Wto (f )) − log π̄ − P̄t−1 dF (dt,t ) − ∆P̄t ,
+ (1 − αw )
Wt−1 (f )≤Wto (f )
recalling that superscript o indicates an optimally chosen price and wage. When evaluating at the
steady state, using that when wages are changed, they are the same for all firms, i.e., Wto (f ) = Wto
for all f , and hence optimal prices are the same, we get
32
¢2
¡
¢2 ¡
Ef log Pt (f ) − log π̄ − P̄t−1 − ∆P̄t
¡
¢2
= αw αEf log π̄Pt−1 (f ) − log π̄ − P̄t−1
¢2
¡
+ (1 − α) αw Ef log Pto (Wt (f )) − log π̄ − P̄t−1
¡
¢2 ¡
¢2
+ (1 − αw ) log Pto (Wto ) − log π̄ − P̄t−1 − ∆P̄t
(77)
when log Wto − log π̄ w − W̄t−1 ≥ 0 and
¡
¢2 ¡
¢2
Ef log Pt (f ) − log π̄ − P̄t−1 − ∆P̄t
¡
¢2
= ααw Ef log π̄Pt−1 (f ) − log π̄ − P̄t−1
¡
¢2 ¡
¢2
+ ((1 − α) αw + (1 − αw )) Ef log Pto (Wt (f )) − log π̄ − P̄t−1 − ∆P̄t
(78)
otherwise, i.e., when log Wto − log π̄ w − W̄t−1 < 0.
¡
¢2
Let us express ∆P̄t in terms of Ef log Pto (Wt (f )) − log π̄ − P̄t−1 and ∆W̄t . First, suppose that
31
This follows from using a first-order Taylor approximation of the price level Wt =
Ŵt =
]
1
Ŵt (f ) df,
U1
0
Wt (f) df ;
0
or, by the definition of Ŵt and Ŵt (f) (they are constructed as log-deviations around the same mean);
log Wt = Ef log Wt (f ) = W̄t .
32
A caveat is in place since our approach disregards any effects stemming from that the wage distribution may be
non-degenerate. This dramatically reduces the complexity of the problem, c.f. the discussion of the wage distribution in
section 2.
41
log Wto − log π̄ w − W̄t−1 ≥ 0. We have
¡
¢
∆P̄t = P̄t − log π̄ − P̄t−1 = (1 − α) αw Ef log Pto (Wt (f )) − log π̄ − P̄t−1
¡
¢
+ (1 − αw ) Ef log Pto (Wto ) − log π̄ − P̄t−1 .
(79)
Note that, from the firms’ optimal capital and labor choice and the optimal pricing decision, we can
write the optimal price as Pto =
(W (f ))1−γ where
only depend on aggregate variables. We thus
can write
log Pto (Wto ) = log Pto (π̄ ω Wt−1 (f )) + (1 − γ) (log Wto − log π̄ ω Wt−1 (f ))
(80)
Using that we have Wt (f ) = π̄ ω Wt−1 (f ) for firms that do not change prices and since Ef (log π̄ ω Wt−1 (f )) =
log π̄ ω + W̄t−1 , we have
¡
¢
∆P̄t − (1 − γ) ∆W̄t = ((1 − α) αw + (1 − αw )) Ef log Pto (Wt (f )) − log π̄ − P̄t−1
(81)
and
¡
¢
∆W̄t = (1 − αw ) log Wto − log π̄ ω − W̄t−1
(82)
when log Wto − log π̄ w − W̄t−1 ≥ 0.
When log Wto − log π̄ w − W̄t−1 < 0, we have
¡
¢
∆P̄t = P̄t − log π̄ − P̄t−1 = ((1 − α) αw + (1 − αw )) Ef log Pto (Wt (f )) − log π̄ − P̄t−1 ,
(83)
and ∆W̄t = 0. Note that we always have
¡
¢2
Ef log Pto (Wt (f )) − log π̄ − P̄ p̄t−1
¡
¢2
= Ef log Pto (Wt (f )) − Ef log Pto (Wt (f )) + Ef log Pto (Wt (f )) − log π̄ − P̄t−1
¡
¢2
= varf log Pto (Wt (f )) + Ef log Pto (Wt (f )) − log π̄ − P̄t−1 .
(84)
Using (81) and (84) when log Wto − log π̄ w − W̄t−1 ≥ 0, we have
¡
¢2
¡
¢2
= varf log Pto (Wt (f )) + Ef log Pto (Wt (f )) − log π̄ − P̄t−1
Ef log Pto (Wt (f )) − log π̄ − P̄t−1
¡
¢2
∆P̄t − (1 − γ) ∆W̄t
o
.
(85)
= varf log Pt (Wt (f )) +
((1 − α) αw + (1 − αw ))2
42
and, using (83) and (84)
¡
¢2
Ef log Pto (f ) − log π̄ − P̄t−1 = varf log Pto (Wt (f )) +
¡
¢2
1
∆
P̄
t
((1 − α) αw + (1 − αw ))2
(86)
otherwise.
Furthermore, by a similar method as in the Technical Appendix of Carlsson and Westermark
(2006b), using the fact that log Pto (Wto ) is the same for all firms that change wages and the loglinearization of log Pto (Wto ) i.e., (80) we can write
¢2
log Pto (Wto ) − log π̄ − P̄t−1
(87)
¡
¢2
= Ef log Pto (π̄ ω Wt−1 (f )) − log π̄ − P̄t−1 + Ef ((1 − γ) (log Wto − log π̄ ω Wt−1 (f )))2
¡
¢
+2 (1 − γ) Ef log Pto (π̄ ω Wt−1 (f )) − log π̄ − P̄t−1 (log Wto − log π̄ ω Wt−1 (f ))
¡
where, using (84)
¡
¢2
Ef log Pto (π̄ ω Wt−1 (f )) − log π̄ − P̄t−1
¡
¢2
1
= varf log Pto (Wt (f )) +
2 ∆P̄t − (1 − γ) ∆W̄t ,
((1 − α) αw + (1 − αw ))
(88)
using (75);
Ef ((1 − γ) (log Wto − log π̄ ω Wt−1 (f )))2 = (1 − γ)2
µ
¶
¡
¢2
1
ω 2
∆
W̄
+
(π̄
)
var
log
W
(f
)
t
t−1
f
(1 − αw )2
(89)
and, using (75), (81) and (84) that
¡
¢
(90)
Ef log Pto (π̄ ω Wt−1 (f )) − log π̄ − P̄t−1 (log Wto − log π̄ ω Wt−1 (f ))
¢
¡
1
1
=
∆W̄t − (π̄ ω )2 (1 − γ) varf log Wt−1 (f ) .
∆P̄t − (1 − γ) ∆W̄t
(1 − α) αw + (1 − αw )
1 − αw
From (71) we have varf log Pto (Wt (f )) = (1 − γ)2 varf log Wt (f ). Then, using (88), (90) and (89) in
(87), (77) and (85), we have that
varf (log Pt (f )) = αw αvarf (log Pt−1 (f )) + (1 − α) αw (1 − γ)2 varf log Wt (f )
¶
µ
¢2
¢2
¡
1−α
αw
2¡
(1 − γ) ∆W̄t
+
α ∆P̄t +
(1 − α) αw + (1 − αw )
1 − αw
43
(91)
when log Wto (f ) − log π̄ w − W̄t−1 ≥ 0 and, from (78) and (86);
varf (log Pt (f )) = αw αvarf (log Pt−1 (f )) + ((1 − α) αw + 1 − αw ) (1 − γ)2 varf log Wt (f ) (92)
¢2
¡
αw α
+
∆P̄t ,
(1 − α) αw + (1 − αw )
otherwise.
For wages, using a similar method as in (72), we can write when ∆W̄t > 0
¡
¢2 ¡
¢2
varf (log Wt (f )) = Ef log Wt (f ) − log π̄ w − W̄t−1 − ∆W̄t
(93)
and
varf (log Wt (f )) = varf (log Wt−1 (f )) ,
(94)
otherwise. When ∆W̄t > 0, we have
¡
¢2 ¡
¢2
Ef log Wt (f ) − log π̄ w − W̄t−1 − ∆W̄t
¡
¢2
= αw Ef log π̄ w Wt−1 (f ) − log π̄ w − W̄t−1
¡
¢2
¢2 ¡
+ (1 − αw ) log Wto (f ) − log π̄ w − W̄t−1 − ∆W̄t .
(95)
Using (75), we have
varf (log Wt (f )) = αw varf (log Wt−1 (f )) +
¢2
αw ¡
∆W̄t .
1 − αw
(96)
Using ∆W̄t = π̂ w
t gives
varf (log Wt (f )) = αw varf (log Wt−1 (f )) +
´
³
αw
2
3
(π̂ w
)
+
o
kξk
1 − αw t
(97)
when log Wtopt (f ) − log π̄ w − W̄t−1 ≥ 0 and
´
³
varf (log Wt (f )) = varf (log Wt−1 (f )) + o kξk3 ,
otherwise.
44
(98)
C.5.1
Loss Function
We focus on the limiting cashless economy. The social welfare function is then
E0
∞
X
β t SWt
(99)
t=0
where
SWt = u (Ct , Qt ) −
Z
1
v (Lt (f ) , Zt ) df.
(100)
0
As in the Technical Appendix of Carlsson and Westermark (2006a), the total welfare difference is
E0
∞
X
t=0
β t (SWt − SWt∗ ) = −E0
+E0
∞
´
¡
¢
v̄L L̄ X t ³
β σvarf P̂t (f ) + γ 2 + 2σγ (1 − γ) varf Ŵt (f ) (101)
2 t=0
∞
´2
´
³
Λ∗ C̄ X t ³
β Ŷt − Ŷt∗ + tip + o kξk3 .
2 t=0
In this expression, the price and wage variances are different than in Carlsson and Westermark
(2006b), due to the downward nominal wage rigidities, since these are affected by wage setting. Given
the path π̂ w , let Tt (π̂ w , r) denote the number of times wage inflation is positive between t and s
w
Tt (π̂ w , s) = |{r : π̂ w
r > π̄ , t ≥ r ≥ s}| .
(102)
Repeatedly substituting (97) and (98) into themselves (forwardly) using (74), starting at 0 gives
´
³
Tt (π̂ w ,0)
varf (log W−1 (f )) +
varf (log Wt (f )) = αw
t
´
³
αw X Tt (π̂w ,s)−1 w 2
αw
(π̂ s ) + o kξk3 . (103)
1 − αw s=0
Multiplying by β t on both sides, using that varf (log W−1 (f )) is independent of policy and summing
from 0 to infinity gives
t
´
³
αw X Tt (π̂w ,s)−1 w 2
β
β varf (log Wt (f )) =
αw
(π̂ s ) + tip + o kξk3 .
1 − αw s=0
t=0
t=0
∞
X
t
∞
X
t
(104)
Taking the expectation at t = 0 and using the fact that the expression is evaluated at the steady state
and hence that the probability that the constraint binds is
E0
∞
X
t=0
β t varf (log Wt (f )) =
1
2
gives
∞
´
³
X
1
αw
3
t
w 2
.
E
β
(π̂
)
+
tip
+
o
kξk
0
t
w
1 − αw 1 − β 1+α
2
t=0
45
(105)
Now consider prices again and expressions (91) and (92). These can be rewritten as
varf (log Pt (f )) = αw αvarf (log Pt−1 (f )) + (1 − α) αw (1 − γ)2 varf log Wt−1 (f )
(106)
(1 − α) αw
αw α
2
(π̂ t )2 +
(1 − γ)2 (π̂ w
+
t )
(1 − α) αw + (1 − αw )
(1 − αw ) ((1 − α) αw + (1 − αw ))
when log Wto (f ) − log π̄ w − W̄t−1 ≥ 0 and, from (86) and (78)
varf (log Pt (f )) = αw αvarf (log Pt−1 (f )) + ((1 − α) αw + 1 − αw ) (1 − γ)2 varf log Wt−1 (f )
αw α
+
(π̂ t )2 ,
(107)
(1 − α) αw + (1 − αw )
otherwise.
Repeatedly substituting this expression into itself (forwardly), starting at 0, taking expectations
at time 0 and evaluating at the steady state gives, using that the previous variance is passed through
with
1
2
(1 − α) αw + 12 ((1 − α) αw + (1 − αw ))
E0 varf (log Pt (f )) = E0
t−1
X
(αw α)
s=0
t
X
+E0
s=0
t
X
+E0
t−s
¶
¶
µµ
1 − αw
2
(1 − γ) varf (log Ws (f ))
(1 − α) αw +
2
(αw α)t−s (1 − γ)2
(αw α)t−s
s=0
(1 − α) αw
2
(π̂ w
s)
(1 − αw ) ((1 − α) αw + (1 − αw ))
(108)
´
³
αw α
(π̂ s )2 + tip + o kξk3 .
(1 − α) αw + (1 − αw )
Multiplying by β t on both sides, using that varf (log W−1 (f )) is independent of policy and summing
from 0 to infinity gives
E0
∞
X
t=0
¢
∞
w
(1 − γ)2 X t
(1 − α) αw + 1−α
2
β varf (log Pt (f )) = β
β varf (log W wt (f ))
E0
1 − βαw α
¡
t
t=0
∞
X
(1 − γ) (1 − α) αw
2
E0
β t (π̂ w
t )
(1 − βαw α) (1 − αw ) ((1 − α) αw + (1 − αw ))
t=0
2
+
+
(109)
∞
´
³
X
αw α
E0
β t (π̂ t )2 + tip + o kξk3 .
(1 − βαw α) ((1 − α) αw + (1 − αw ))
t=0
Using (109) and (105) in (101) gives
E0
∞
X
t=0
β t (SWt − SWt∗ ) = E0
∞
X
t=0
´
³
β t Lt + tip + o kξk3
(110)
where
Lt = θx (x̂t )2 + θπ (π̂ t )2 + θπω (π̂ ωt )2 ,
46
(111)
and using similar methods as in Carlsson and Westermark (2006b), and expression (38) we get
θx
θπ
θ πω
¶
µ
γ
1
C̄
ūC −ρC + ρL
−
,
=
2
1−γ 1−γ
v̄L L̄ σ (γ (1 − σ) + 1 + γ) 1
,
= −
2
1−γ
Π
µ
¶
σ (γ (1 − σ) + 1 + γ)
v̄L L̄ −γεL + (1 − γ) σ (1 + γ (1 − σ))
.
− (1 − γ)
= −
2
Π1
Π
(112)
Note that θx < 0, θπ < 0 and θπω < 0.
D
Optimal Discretionary Policy
To find the optimal rule under discretion, the central bank solves the following problem
V (ŵt−1 , ŵt∗ ) =
max
ω
{x̂t ,π̂ t ,π̂t ,ŵt }
´
³
¡
¢
∗
θx (x̂t )2 + θπ (π̂ t )2 + θπω (π̂ ωt )2 + βEt V ŵt , ŵt+1
+ tip + o kξk3 (113)
subject to
x̂t
π̂ t
π̂ ωt
ŵt
µ
´¶
1 ³b
∗
it − π̂ t+1 − rbt
= Et x̂t+1 −
,
ρC
¡
¢
γ
Πx̂t ,
= βEt π̂ t+1 + (1 − γ) π̂ ωt − βEt π̂ ωt+1 + Π (ŵt − ŵt∗ ) +
1−γ
¾
½
1 + αw
= max 0,
βEt π̂ ωt+1 − Ωw (ŵt − ŵt∗ ) − Ωx x̂t ,
2αw
= ŵt−1 + π̂ ωt − π̂ t .
(114)
(115)
(116)
(117)
Note that the max restriction can be replaced by
1 + αw
βEt π̂ ωt+1 − Ωw (ŵt − ŵt∗ ) − Ωx x̂t ,
2αw
≥ 0.
π̂ ωt ≥
(118)
π̂ ωt
(119)
The reason for the restriction π̂ ωt ≥
ω
1+αw
2αw βEt π̂ t+1
− Ωx x̂t − Ωw (ŵt − ŵt∗ ) is the following. First, if
the desired wage change is positive, it is set according to the first-order condition, implying that the
condition holds with equality. Second, if the desired wage change is negative, downward rigidity kicks
in so that wages are higher than those prescribed by the first-order condition.
47
We solve by using the Lagrange method. The Lagrangian is
¡
¢
∗
L = θx (x̂t )2 + θπ (π̂ t )2 + θπω (π̂ ωt )2 + βEt V ŵt , ŵt+1
µ
¢
¡
−λπt βEt π̂ t+1 + (1 − γ) π̂ ωt − βEt π̂ ωt+1 + Π (ŵt − ŵt∗ ) +
ω
−λw
t (ŵt−1 + π̂ t − π̂ t − ŵt )
µ
¶
1 + αw
πω
ω
∗
ω
−μt
βEt π̂ t+1 − Ωx x̂t − Ωw (ŵt − ŵt ) − π̂ t
2αw
−μ0t (−π̂ ωt ) .
γ
Πx̂t − π̂ t
1−γ
¶
(120)
The first-order conditions are
γ
ω
Π + μπt Ωx ,
1−γ
π
0 = 2θπ π̂ t + λt + λw
t ,
0 = 2θx x̂t − λπt
ω
+ μπt + μ0t ,
0 = 2θπω π̂ ωt − λπt (1 − γ) − λw
¶
µt
¡
¢
∂ π̂ ωt+1
∂ π̂ t+1
∗
π
0 = βEt V1 ŵt , ŵt+1 − λt βEt
− (1 − γ) βEt
+Π
∂ ŵt
∂ ŵt
µ
¶
∂ π̂ ωt+1
1 + αw
πω
+λw
−
μ
βE
− Ωw .
t
t
t
2αw
∂ ŵt
(121)
Note that we restrict the attention to Markov-perfect equilibria, i.e., we do not consider any equilibria
with reputational effects that could arise from complex non-Markov behavior. However, we need to
take into account that the real wage is an endogenous state variable. Therefore, expected inflation
and expected wage inflation will depend on lagged real wages in equilibrium. An implication of this
is that when designing monetary policy, even in the absence of a commitment mechanism, the central
bank should take into account how changes in the real wage today affect private sector expectations.
We also have the complementary slackness conditions
ω
μπt
µ
¶
1 + αw
ω
∗
ω
βEt π̂ t+1 − Ωx x̂t − Ωw (ŵt − ŵt ) − π̂ t
= 0,
2αw
μ0t (−π̂ ωt ) = 0.
(122)
Note that the original constraint (116) and the inequality constraints (118) and (119) need not be
equivalent. It is obviously true that a π̂ ωt satisfying the original constraint satisfies the two inequality
constraints. However, it is possible that there is a solution to the problem with inequality constraints
so that none of the inequality constraints is binding, leading to a violation of the original constraints.
The following Lemma shows that at least one of the inequality constraints must be binding, ruling
out this possibility.
Lemma 1. At least one of the inequality constraints (118) and (119) must be binding.
48
Proof : We prove this by contradiction. Suppose that
1 + αw
βEt π̂ ωt+1 − Ωx x̂t − Ωw (ŵt − ŵt∗ ) ,
2αw
> 0.
π̂ ωt >
π̂ ωt
(123)
ω
Then μπt = μ0t = 0 and the first-order conditions simplifies to
γ
Π,
1−γ
0 = 2θπ π̂ t + λπt + λw
t ,
0 = 2θx x̂t − λπt
(124)
,
0 = 2θπω π̂ ωt − λπt (1 − γ) − λw
µt
¶
¡
¢
∂ π̂ ωt+1
∂ π̂ t+1
∗
π
0 = βEt V1 ŵt , ŵt+1 − λt βEt
− (1 − γ) βEt
+ Π + λw
t .
∂ ŵt
∂ ŵt
Using the first three first-order conditions, we get
2θx x̂t + (2θπ π̂ t + 2θπω π̂ ωt )
1
Π = 0.
1−γ
(125)
Suppose that we change π̂ ωt and keep (ŵt − ŵt∗ ) fixed. To ensure that the two equality constraints
hold, we require that, for the second constraint,
1=
dπ̂ t
dπ̂ ω
t
= 1 and, for the first constraint,
dx̂t
dπ̂ω
t
=
dx̂t
1
Π
.
1 − γ dπ̂ ωt
1−γ
Π
(126)
The effect on the objective of a change in π̂ ωt is then
¶
µ
¶
µ
dπ̂ t
1−γ
dx̂t
ω
ω
ω
+ 2θπ (π̂ t ) + 2θπω (π̂ t ) dπ̂ ωt = 0
2θx (x̂t ) ω + 2θπ (π̂ t ) ω + 2θπω (π̂ t ) dπ̂ t = 2θx (x̂t )
dπ̂ t
dπ̂ t
Π
(127)
using expression (125). From the last first-order condition we have
µ
¶
¡
¢
∂ π̂ ωt+1
∂ π̂ t+1
2θπ π̂ t + 2θπω π̂ ωt
∗
βEt
− (1 − γ) βEt
+Π
βEt V1 ŵt , ŵt+1 = −
γ
∂ ŵt
∂ ŵt
2θπ π̂ t + 2θπω π̂ ωt
.
−2θπ π̂ t −
γ
(128)
The effect on the right-hand side of the change in π̂ ωt , while keeping real wages constant, is
¶
¶
µ
µ
Π+1
− 2θπ dπ̂ ωt .
− (2θπ + 2θπω )
γ
(129)
Since the term within the parenthesis is positive and dπ̂ ωt can be changed in any direction, this
implies that the last first-order condition is violated. Since the change led to an unaffected value
49
of the problem, there is another change in the variables π̂ t , π̂ ωt , x̂t and ŵt which leads to a strict
improvement in the value of the problem, ruling out optimality of a solution where both inequality
constraints holds strictly as in (123). ¥
First, suppose π̂ ωt > 0. Then, we get μ0t = 0 from the complementary slackness condition. The
first-order conditions then simplify to
γ
ω
Π + μπt Ωx ,
1−γ
π
0 = 2θπ π̂ t + λt + λw
t ,
0 = 2θx x̂t − λπt
ω
π
0 = 2θπω π̂ ωt − λπt (1 − γ) − λw
t + μt ,
µ
¶
¡
¢
∂ π̂ ω
∂ π̂ t+1
∗
0 = βEt V1 ŵt , ŵt+1
− λπt βEt
− (1 − γ) βEt t+1 + Π
∂ ŵt
∂ ŵt
¶
µ
ω
∂ π̂
1 + αw
πω
βEt t+1 − Ωw .
+λw
t − μt
2αw
∂ ŵt
(130)
We also have the constraints
¢
¡
π̂ t = βEt π̂ t+1 + (1 − γ) π̂ ωt − βEt π̂ ωt+1 + Π (ŵt − ŵt∗ ) +
ŵt = ŵt−1 + π̂ ωt − π̂ t ,
1 + αw
βEt π̂ ωt+1 − Ωx x̂t − Ωw (ŵt − ŵt∗ ) .
π̂ ωt =
2αw
γ
Πx̂t ,
1−γ
(131)
Using the first three first-order conditions gives
2θx x̂t − λπt
γ
ω
Π + μπt Ωx = 0,
1−γ
2θπ π̂ t + λπt + λw
= 0,
t
π
2θπω π̂ ωt − λπt (1 − γ) − λw
t + μt
We get;
⎛
λπt
⎜
⎜ w
⎜ λt
⎝
ω
μπt
⎞
⎛
⎟
⎜
⎟
⎜
⎟ = −2 ⎜
⎠
⎝
γ
− 1−γ
Π
0
1
1
− (1 − γ) −1
Ωx
ω
= 0.
⎞−1 ⎛
⎟
⎟
0 ⎟
⎠
1
(132)
θx x̂t
⎜
⎜
· ⎜ θπ π̂ t
⎝
θπω π̂ ωt
⎞
⎟
⎟
⎟.
⎠
(133)
We can write the Lagrange multipliers as functions of x̂t , π̂ t and π̂ ωt and hence, we can eliminate the
Lagrange multipliers from the last first-order condition
¶
µ
¡
¢
∂ π̂ ωt+1
∂ π̂ t+1
∗
π
ω
− (1 − γ) βEt
+Π
0 = βEt V1 ŵt , ŵt+1 − λt (x̂t , π̂ t , π̂ t ) βEt
∂ ŵt
∂ ŵt
µ
¶
∂ π̂ ωt+1
1 + αw
ω
ω
πω
+λw
βE
− Ωw .
(x̂
,
π̂
,
π̂
)
(x̂
,
π̂
,
π̂
)
−
μ
t
t t t
t t t
t
t
2αw
∂ ŵt
50
(134)
We then get the following system of equations
0 =
π̂ t =
ŵt =
π̂ ωt =
¶
µ
∂ π̂ ωt+1
∂ π̂ t+1
− (1 − γ) βEt
+Π
βEt
∂ ŵt
∂ ŵt
µ
¶
∂ π̂ ωt+1
1 + αw
ω
ω
πω
+λw
(x̂
,
π̂
,
π̂
)
(x̂
,
π̂
,
π̂
)
−
μ
βE
− Ωw ,
t t t
t t t
t
t
t
2αw
∂ ŵt
¡
¢
γ
Πx̂t ,
βEt π̂ t+1 + (1 − γ) π̂ ωt − βEt π̂ ωt+1 + Π (ŵt − ŵt∗ ) +
1−γ
ŵt−1 + π̂ ωt − π̂ t ,
1 + αw
βEt π̂ ωt+1 − Ωx x̂t − Ωw (ŵt − ŵt∗ ) .
2αw
¡
∗
βEt V1 ŵt , ŵt+1
¢
− λπt
(x̂t , π̂ t , π̂ ωt )
(135)
ω
Second, suppose that π̂ ωt = 0. Then, we get μπt = 0 from the complementary slackness condition.
The first-order conditions then simplify to
γ
Π,
1−γ
0 = 2θπ π̂ t + λπt + λw
t ,
0 = 2θx x̂t − λπt
(136)
0
0 = −λπt (1 − γ) − λw
t + μt ,
µ
¶
¡
¢
∂ π̂ ωt+1
∂ π̂ t+1
∗
π
0 = βEt V1 ŵt , ŵt+1 − λt βEt
− (1 − γ) βEt
+ Π + λw
t
∂ ŵt
∂ ŵt
or
⎛
λπt
⎜
⎜ w
⎜ λt
⎝
μ0t
⎞
⎛
⎜
⎟
⎜
⎟
⎟ = −2 ⎜
⎝
⎠
γ
− 1−γ
Π
0
0
⎞−1 ⎛
⎟
⎟
1
1 0 ⎟
⎠
− (1 − γ) −1 1
θx x̂t
⎜
⎜
· ⎜ θπ π̂ t
⎝
0
⎞
⎟
⎟
⎟.
⎠
(137)
Thus, we can write the Lagrange multipliers as functions of x̂t and π̂ t and hence, once more, eliminate
them from the last first-order condition. The last first-order condition becomes
¶
µ
¢
¡
∂ π̂ ωt+1
∂ π̂ t+1
π
∗
− (1 − γ) βEt
+ Π + λw
βEt V1 ŵt , ŵt+1 − λt (x̂t , π̂ t ) βEt
t (x̂t , π̂ t ) = 0.
∂ ŵt
∂ ŵt
(138)
We then get the following system of equations
¶
µ
∂ π̂ ωt+1
∂ π̂ t+1
− (1 − γ) βEt
+ Π + λw
0 =
(x̂t , π̂ t ) βEt
t (x̂t , π̂ t ) ,
∂ ŵt
∂ ŵt
¢
¡
γ
π̂ t = βEt π̂ t+1 + (1 − γ) π̂ ωt − βEt π̂ ωt+1 + Π (ŵt − ŵt∗ ) +
Πx̂t ,
(139)
1−γ
ŵt = ŵt−1 + π̂ ωt − π̂ t ,
¡
∗
βEt V1 ŵt , ŵt+1
¢
− λπt
π̂ ωt = 0.
51
earlier working papers:
for a complete list of working papers published by Sveriges riksbank, see www.riksbank.se
Evaluating Implied RNDs by some New Confidence Interval Estimation Techniques
by Magnus Andersson and Magnus Lomakka .................................................................................... 2003:146
Taylor Rules and the Predictability of Interest Rates
by Paul Söderlind, Ulf Söderström and Anders Vredin ....................................................................... 2003:147
Inflation, Markups and Monetary Policy
by Magnus Jonsson and Stefan Palmqvist ......................................................................................... 2003:148
Financial Cycles and Bankruptcies in the Nordic Countries by Jan Hansen.......................................... 2003:149
Bayes Estimators of the Cointegration Space by Mattias Villani ........................................................ 2003:150
Business Survey Data: Do They Help in Forecasting the Macro Economy?
by Jesper Hansson, Per Jansson and Mårten Löf ............................................................................... 2003:151
The Equilibrium Rate of Unemployment and the Real Exchange Rate:
An Unobserved Components System Approach by Hans Lindblad and Peter Sellin ........................... 2003:152
Monetary Policy Shocks and Business Cycle Fluctuations in a
Small Open Economy: Sweden 1986-2002 by Jesper Lindé ............................................................... 2003:153
Bank Lending Policy, Credit Scoring and the Survival of Loans by Kasper Roszbach ........................... 2003:154
Internal Ratings Systems, Implied Credit Risk and the Consistency of Banks’ Risk
Classification Policies by Tor Jacobson, Jesper Lindé and Kasper Roszbach ........................................ 2003:155
Monetary Policy Analysis in a Small Open Economy using Bayesian Cointegrated
Structural VARs by Mattias Villani and Anders Warne ...................................................................... 2003:156
Indicator Accuracy and Monetary Policy: Is Ignorance Bliss? by Kristoffer P. Nimark .......................... 2003:157
Intersectoral Wage Linkages in Sweden by Kent Friberg .................................................................... 2003:158
Do Higher Wages Cause Inflation? by Magnus Jonsson and Stefan Palmqvist ................................. 2004:159
Why Are Long Rates Sensitive to Monetary Policy by Tore Ellingsen and Ulf Söderström .................. 2004:160
The Effects of Permanent Technology Shocks on Labor Productivity
and Hours in the RBC model by Jesper Lindé..................................................................................... 2004:161
Credit Risk versus Capital Requirements under Basel II: Are SME Loans and Retail
Credit Really Different? by Tor Jacobson, Jesper Lindé and Kasper Roszbach .................................... 2004:162
Exchange Rate Puzzles: A Tale of Switching Attractors
by Paul De Grauwe and Marianna Grimaldi ...................................................................................... 2004:163
Bubbles and Crashes in a Behavioural Finance Model
by Paul De Grauwe and Marianna Grimaldi ...................................................................................... 2004:164
Multiple-Bank Lending: Diversification and Free-Riding in Monitoring
by Elena Carletti, Vittoria Cerasi and Sonja Daltung .......................................................................... 2004:165
Populism by Lars Frisell ..................................................................................................................... 2004:166
Monetary Policy in an Estimated Open-Economy Model with Imperfect Pass-Through
by Jesper Lindé, Marianne Nessén and Ulf Söderström ..................................................................... 2004:167
Is Firm Interdependence within Industries Important for Portfolio Credit Risk?
by Kenneth Carling, Lars Rönnegård and Kasper Roszbach ................................................................ 2004:168
How Useful are Simple Rules for Monetary Policy? The Swedish Experience
by Claes Berg, Per Jansson and Anders Vredin .................................................................................. 2004:169
The Welfare Cost of Imperfect Competition and Distortionary Taxation
by Magnus Jonsson ........................................................................................................................... 2004:170
A Bayesian Approach to Modelling Graphical Vector Autoregressions
by Jukka Corander and Mattias Villani .............................................................................................. 2004:171
Do Prices Reflect Costs? A study of the price- and cost structure of retail payment
services in the Swedish banking sector 2002 by Gabriela Guibourg and Björn Segendorf .................. 2004:172
Excess Sensitivity and Volatility of Long Interest Rates: The Role of Limited
Information in Bond Markets by Meredith Beechey ........................................................................... 2004:173
State Dependent Pricing and Exchange Rate Pass-Through
by Martin Flodén and Fredrik Wilander............................................................................................. 2004:174
The Multivariate Split Normal Distribution and Asymmetric Principal
Components Analysis by Mattias Villani and Rolf Larsson ................................................................. 2004:175
Firm-Specific Capital, Nominal Rigidities and the Business Cycle
by David Altig, Lawrence Christiano, Martin Eichenbaum and Jesper Lindé ...................................... 2004:176
Estimation of an Adaptive Stock Market Model with Heterogeneous Agents by Henrik Amilon ........ 2005:177
Some Further Evidence on Interest-Rate Smoothing: The Role of Measurement
Errors in the Output Gap by Mikael Apel and Per Jansson................................................................. 2005:178
Bayesian Estimation of an Open Economy DSGE Model with Incomplete Pass-Through
by Malin Adolfson, Stefan Laséen, Jesper Lindé and Mattias Villani ................................................ 2005:179
Are Constant Interest Rate Forecasts Modest Interventions? Evidence from
an Estimated Open Economy DSGE Model of the Euro Area by Malin Adolfson,
Stefan Laséen, Jesper Lindé and Mattias Villani ............................................................................... 2005:180
Inference in Vector Autoregressive Models with an Informative
Prior on the Steady State by Mattias Villani ...................................................................................... 2005:181
Bank Mergers, Competition and Liquidity by Elena Carletti, Philipp Hartmann
and Giancarlo Spagnolo .................................................................................................................... 2005:182
Testing Near-Rationality using Detailed Survey Data
by Michael F. Bryan and Stefan Palmqvist ........................................................................................ 2005:183
Exploring Interactions between Real Activity and the Financial Stance
by Tor Jacobson, Jesper Lindé and Kasper Roszbach......................................................................... 2005:184
Two-Sided Network Effects, Bank Interchange Fees,
and the Allocation of Fixed Costs by Mats A. Bergman .................................................................... 2005:185
Trade Deicits in the baltic States: How long will the party last?
by Rudolfs Bems and Kristian Jönsson .............................................................................................. 2005:186
real exchange rate and Consumption fluctuations follwing trade liberalization
by Kristian Jönsson ........................................................................................................................... 2005:187
Modern forecasting Models in action: improving Macroeconomic analyses at Central banks
by Malin Adolfson, Michael K. Andersson, Jesper Lindé, Mattias Villani and Anders Vredin .......... 2005:188
bayesian inference of general linear restrictions on the Cointegration Space by Mattias Villani ..... 2005:189
forecasting performance of an open economy Dynamic Stochastic general equilibrium Model
by Malin Adolfson, Stefan Laséen, Jesper Lindé and Mattias Villani ................................................ 2005:190
forecast Combination and Model averaging using predictive Measures
by Jana Eklund and Sune Karlsson .................................................................................................... 2005:191
Swedish intervention and the krona float, 1993-2002
by Owen F. Humpage and Javiera Ragnartz .................................................................................... 2006:192
a Simultaneous Model of the Swedish krona, the US Dollar and the euro
by Hans Lindblad and Peter Sellin..................................................................................................... 2006:193
testing theories of Job Creation: Does Supply Create its own Demand?
by Mikael Carlsson, Stefan Eriksson and Nils Gottfries ..................................................................... 2006:194
Down or out: assessing the welfare Costs of Household investment Mistakes
by Laurent E. Calvet, John Y. Campbell and Paolo Sodini ................................................................ 2006:195
eficient bayesian inference for Multiple Change-point and Mixture innovation Models
by Paolo Giordani and Robert Kohn ................................................................................................. 2006:196
Derivation and estimation of a new keynesian phillips Curve in a Small open economy
by Karolina Holmberg .......................................................................................................................... 2006:197
technology Shocks and the labour-input response: evidence from firm-level Data
by Mikael Carlsson and Jon Smedsaas ................................................................................................ 2006:198
Monetary policy and Staggered wage bargaining when prices are Sticky
by Mikael Carlsson and Andreas Westermark ..................................................................................... 2006:199
the Swedish external position and the krona by Philip R. Lane ..........................................................2006:200
price Setting transactions and the role of Denominating Currency in fX Markets
by Richard Friberg and Fredrik Wilander ............................................................................................. 2007:201
the geography of asset holdings: evidence from Sweden
by Nicolas Coeurdacier and Philippe Martin ....................................................................................... 2007:202
evaluating an estimated new keynesian Small open economy Model
by Malin Adolfson, Stefan Laséen, Jesper Lindé and Mattias Villani .................................................. 2007:203
the Use of Cash and the Size of the Shadow economy in Sweden
by Gabriela Guibourg and Björn Segendorf ......................................................................................... 2007:204
bank supervision russian style: evidence of conlicts between micro- and macroprudential concerns by Sophie Claeys and Koen Schoors ................................................................... 2007:205
Website: www.riksbank.se
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Monetary Policy under Downward Nominal Wage Rigidity
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