In the literature of staggered wages (Taylor, 1979, 1980; Blanchard, 1986; Ball and Cecchetti, 1991) the discount factor is neglected in the workers’ loss function.  Yet discounting is to be viewed as an extra piece of micro-foundation with implications for discretionary monetary policy.  We revisit the issue and show that discounting in the model of staggered wages actually lowers the time consistent steady inflation

Bonini, Patricia

Da Silva, Sergio

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Staggered wages, inflation, and
discounting
Patricia Bonini and Sergio Da Silva
Federal University of Santa Catarina
2007
Online at http://mpra.ub.uni-muenchen.de/1979/
MPRA Paper No. 1979, posted 3. March 2007
Staggered wages, inflation, and discounting 
 
 
Patricia Boninia, Sergio Da Silvaa* 
 
aDepartment of Economics, Federal University of Santa Catarina, 88049970 Florianopolis SC, Brazil 
 
 
 
Abstract 
 
In the literature of staggered wages (Taylor, 1979, 1980; Blanchard, 1986; Ball and 
Cecchetti, 1991) the discount factor is neglected in the workers’ loss function.  Yet 
discounting is to be viewed as an extra piece of micro-foundation with implications for 
discretionary monetary policy.  We revisit the issue and show that discounting in the model 
of staggered wages actually lowers the time consistent steady inflation. 
 
Keywords: Staggered wage model; Time consistent steady inflation; Discounting 
 
JEL classification: E12; E31; E52 
 
 
 
1. Introduction 
 
In the literature of staggered wages (Taylor, 1979, 1980; Blanchard 1986; Ball and 
Cecchetti, 1991) the discount factor is neglected in the workers’ loss function.  Yet 
discounting is to be viewed as an extra piece of micro-foundation with implications for 
discretionary monetary policy.  Both wage contracts in an imperfectly competitive labor 
market and distorting taxes on labor income generate an equilibrium wage above the 
socially efficient level.  This inefficiency gives the monetary authority an incentive to 
create surprise inflation.  Along the lines of Ball and Cecchetti (1991), we derive the 
discretionary time consistent inflation rate by considering the influence of discounting.  We 
find that a positive money growth rate in the presence of discounting pushes the average 
real wage below its optimal level, shrinks the gap between the market level of output and 
that socially efficient, and thus diminishes the incentives for the policymaker to create 
surprise inflation.  Thus considering discounting in the model of staggered wages actually 
lowers the time consistent steady inflation. 
 Section 2 revisits the benchmark model.  Section 3 extends the model to consider 
discounting.  Section 4 derives the equilibrium time consistent inflation rate, and then 
calibrates the extended model.  Section 5 concludes. 
 
                                                 
* Corresponding author. 
   Tel./Fax: +55 48 3238 1219. 
   E-mail address: professorsergiodasilva@gmail.com 
2. Model 
 
We take the nominal contract setup in which firms are on their labor demand curves 
(Gray, 1976; Fischer, 1977; Taylor, 1979).  The perfectly symmetric firms are both price 
( p ) and wage ( w ) takers when maximizing profits, so that aggregate output supply is 
 
( )
1t t t
y w pαα= −− ,                                                                                                   (1)  
 
where all variables are in logs, and parameter (0,1)α ∈  is the degree of returns to scale. 
Aggregate demand is given by the quantity equation with constant velocity of 
money, i.e. t t tm p y− = .  Then the aggregate price level that balances the output market is 
 
(1 )t t tp w mα α= + − ,                                                                                                 (2) 
 
where the money supply grows steadily at rate µ , i.e. tm tµ= . 
The labor market is imperfectly competitive, and there is no perfect labor mobility.  
The economy has two sectors, each of them with a pool of workers that are immobile 
thanks to their specific skills.  With perfect symmetry across the sectors, individual wages 
match the average wage.  Labor supply depends positively on the real wage, and labor 
contracts target *w , which is the wage clearing the labor market in the absence of nominal 
rigidities.  By leaving out constants one gets *t tw p=  in equilibrium.  The contract wage tx  
is renewed at time period t.  Half of the labor force minimizes the losses coming from the 
deviations from equilibrium, i.e. 
 
2
2
1
( )tt t t
t
L x pρ
=
= −∑                                                                                                    (3) 
 
subject to 1t tx x += .  This constraint means that the contracts fix a wage for two periods.  
The other half of the labor force minimizes the losses L  at the previous time period 1t − , 
and fixes the wage at 1t tx x− = .  Parameter (0,1]ρ ∈  is the discount factor.  In the 
benchmark staggered wage model it is neglected to its borderline case 1ρ = . 
A distorting tax is imposed on labor income (Ball and Cecchetti, 1991).  The 
equilibrium pre-tax *w  is then set above the efficient level by an amount wˆ .  The latter is a 
function of the tax rate and the parameter of preference over consumption and leisure 
entering the labor supply, and of the parameter of technology (entering the labor demand).  
The efficient real wage is that attained in the equilibrium with no taxes, i.e. ˆEt tw p w− = − .  
It is negative because * 0w p− =  after dropping constants.  As *w  is set above the efficient 
level, aggregate employment and output are set below their efficient level. 
The monetary authority is concerned with output inefficiency (Barro and Gordon, 
1983).  While wage setters target *w p= , the monetary authority cares about the loss 
provoked by the departure of the contract wage from Ew .  This loss is 2ˆ( )tt t tx p wρ= − +?  
for the sector renewing the contract in t.  The social loss averaged across firms in the first 
and second period is then 
 
  2 2 21
1 1ˆ ˆ( ) ( )
2 2t t t t t
x p w x p wρ ρ −= − + + − +? .                                                            (4) 
 
The monetary authority chooses the money each period taking the expectations of current 
and future money as given, so that equilibrium is discretionary.  Since the time paths of 
wages and price are both linear functions of µ , they are also functions of the sequence of 
surprises, i.e. { }t tmδ µ= − .  Both the workers and the authority have rational expectations. 
 
3. The model with discounting 
 
Relaxing the assumption that 1ρ =  in the workers’ loss function (3) leads to a new 
rule of contract revision: 
 
1
1
1 1
e
t t tx p p
ρ
ρ ρ += ++ + .                                                                                           (5) 
 
Considering (2) and (5), a solution for tx  and 1tx −  exists for homogeneous staggering.  Its 
stable characteristic root ( 1λ < ) is 
 
2 2 2(2 )(1 ) (2 ) (1 ) 4
2
α ρ α ρ α ρλ αρ
− + − − + −= .                                                        (6) 
 
It is implied that 0λ ρ∂ ∂ >  for 1ρ < .  Thus taking discounting into account raises λ . 
As the supply of money grows steadily and there is no surprise, the time paths of the 
contract wages are the first terms in the right hand side of (7) and (8): 
 
1
0
( ) (1 )jt t j
j
x t zµ λ δ∞ + −
=
= + + −∑                                                                                  (7) 
 
and 
 
1
1
( 1) (1 )jt t j
j
x t zµ λ δ∞− −
=
= + − + −∑ ,                                                                            (8) 
 
where 
 
2
(1 )[1 ( 1)] (1 )
(1 )(1 )(1 )
z ρ λ λ λρ λ ρρ λ λρ
+ − + − += + − + .                                                                      (9) 
 
The summation terms in (7) and (8) give the effect of a monetary surprise δ  on the wages.  
And term z  summarizes the effect of discounting on the time paths of the contract wages.  
Neglecting the discount factor ( 1ρ = ) yields 1 2z = , but 1 2z <  if 1ρ < .  With 
discounting, the non surprise time paths of aggregate real wage and output are 
 
1( 1)
2t t
w p zα µ − = − −                                                                                         (10) 
 
and 
 
1
2N
y zα µ = −   .                                                                                                   (11) 
 
The average real wage in (10) falls short of its optimal level * 0t tw p− =  over the contract 
period because µ  is positive.  And the equilibrium steady state output Ny  is pushed 
upward; as a result, the gap between equilibrium output and its efficient level is shrunk.  
This diminishes the incentive for the monetary authority to create surprise inflation. 
 
4. Calibration 
 
Now we derive the equilibrium time consistent inflation rate to show how 
discounting affects the time consistent steady inflation.  Under rational expectations, δ  
affects the monetary authority’s loss function (4) through (7) and (8).  The time consistent 
discretionary equilibrium requires zero surprise inflation each period, and this means 
imposing 00
( ) 0
t
dL t
d δδ
∞
== =∑  on the authority’s problem, which leads to 
 
wˆdµ =                                                                                                                    (12) 
 
where 
 
2( ) (1 )(1 )
(1 )[(2 (1 )] [ (1 ) 2 ]
d ρ λ α ρ λρ ν α λ ν α λ λ
+ − + += − − + + + −                                                          (13) 
 
and 
 
1
2
z zν α  = + −   .                                                                                                   (14) 
 
The time consistent steady inflation is then a function of ρ  (and of wˆ  and α ).  Ball and 
Cecchetti’s (1991) result is a particular case arising if 1ρ =  in (13), and 1 2zν = = . 
Table 1 shows our model calibrated with selected values for ρ  and α , including 
the favorite ones in the real business cycle literature, i.e. .25.96ρ =  and .75α = .  The 
values of d  are lower for 1ρ < , thereby showing that if discounting is considered in (3) 
and (4), the time consistent steady inflation µ  is reduced. 
 
5. Conclusion 
 
We extend the standard staggered wage model to consider discounting.  By deriving 
the novel equilibrium time consistent inflation rate, and then calibrating the resulting 
model, we find that discounting actually lowers the time consistent steady inflation. 
 
Table 1. The effect of discounting in the staggered wage model 
 
 α = .40 α = .60 α = .75 α = .80 
 
ρ = 1     
λ .127 .223 .333 .382 
z  ½ ½ ½ ½ 
d  1.549 1.265 1 .894 
 
.25.96ρ =      
λ .128 .226 .335 .384 
z  .496 .494 .490 .487 
d  1.541 1.258 .994 .888 
 
ρ = .9     
λ .134 .237 .351 .402 
z  .456 .434 .412 .368 
d  1.469 1.195 .938 .835 
 
ρ = .85     
λ .137 .243 .360 .412 
z  .432 .399 .365 .297 
d  1.429 1.027 .906 .698 
 
Notes 
ρ  = discount factor 
α  = degree of returns to scale 
λ  = characteristic root 
z  = (see equation (9)) 
d  = (see equation (13)) 
References 
 
Barro, R., Gordon, D., 1983. A positive theory of monetary policy in a natural rate model. 
Journal of Monetary Economics 91, 589−610. 
 
Ball, L., Cecchetti, S.G., 1991. Wage indexation and discretionary monetary policy. 
American Economic Review 81, 1310−1319. 
 
Blanchard, O., 1986. The wage-price spiral. Quarterly Journal of Economics 101, 543−563. 
 
Fischer, S., 1977. Long term contracts, rational expectations and the optimal money supply 
rule. Journal of Political Economy 85, 163−190. 
 
Gray, J., 1976. Wage indexation: A macroeconomic approach. Journal of Monetary 
Economics 3, 221−335. 
 
Taylor, J., 1979. Staggering price setting in a macro model. American Economic Review 
69, 108−113. 
 
Taylor, J., 1980. Aggregate dynamics and staggered contracts. Journal of Political 
Economy 88, 1−24. 


Staggered wages, inflation, and discounting

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveStaggered wages, inflation, anddiscountingPatricia Bonini and Sergio Da SilvaFederal University of Santa Catarina2007Online at http://mpra.ub.uni-muenchen.de/1979/MPRA Paper No. 1979, posted 3. March 2007Staggered wages, inflation, and discounting   Patricia Boninia, Sergio Da Silvaa*  aDepartment of Economics, Federal University of Santa Catarina, 88049970 Florianopolis SC, Brazil    Abstract  In the literature of staggered wages (Taylor, 1979, 1980; Blanchard, 1986; Ball and Cecchetti, 1991) the discount factor is neglected in the workers’ loss function.  Yet discounting is to be viewed as an extra piece of micro-foundation with implications for discretionary monetary policy.  We revisit the issue and show that discounting in the model of staggered wages actually lowers the time consistent steady inflation.  Keywords: Staggered wage model; Time consistent steady inflation; Discounting  JEL classification: E12; E31; E52    1. Introduction  In the literature of staggered wages (Taylor, 1979, 1980; Blanchard 1986; Ball and Cecchetti, 1991) the discount factor is neglected in the workers’ loss function.  Yet discounting is to be viewed as an extra piece of micro-foundation with implications for discretionary monetary policy.  Both wage contracts in an imperfectly competitive labor market and distorting taxes on labor income generate an equilibrium wage above the socially efficient level.  This inefficiency gives the monetary authority an incentive to create surprise inflation.  Along the lines of Ball and Cecchetti (1991), we derive the discretionary time consistent inflation rate by considering the influence of discounting.  We find that a positive money growth rate in the presence of discounting pushes the average real wage below its optimal level, shrinks the gap between the market level of output and that socially efficient, and thus diminishes the incentives for the policymaker to create surprise inflation.  Thus considering discounting in the model of staggered wages actually lowers the time consistent steady inflation.  Section 2 revisits the benchmark model.  Section 3 extends the model to consider discounting.  Section 4 derives the equilibrium time consistent inflation rate, and then calibrates the extended model.  Section 5 concludes.                                                   * Corresponding author.    Tel./Fax: +55 48 3238 1219.    E-mail address: professorsergiodasilva@gmail.com 2. Model  We take the nominal contract setup in which firms are on their labor demand curves (Gray, 1976; Fischer, 1977; Taylor, 1979).  The perfectly symmetric firms are both price ( p ) and wage ( w ) takers when maximizing profits, so that aggregate output supply is  ( )1t t ty w pαα= −− ,                                                                                                   (1)   where all variables are in logs, and parameter (0,1)α ∈  is the degree of returns to scale. Aggregate demand is given by the quantity equation with constant velocity of money, i.e. t t tm p y− = .  Then the aggregate price level that balances the output market is  (1 )t t tp w mα α= + − ,                                                                                                 (2)  where the money supply grows steadily at rate µ , i.e. tm tµ= . The labor market is imperfectly competitive, and there is no perfect labor mobility.  The economy has two sectors, each of them with a pool of workers that are immobile thanks to their specific skills.  With perfect symmetry across the sectors, individual wages match the average wage.  Labor supply depends positively on the real wage, and labor contracts target *w , which is the wage clearing the labor market in the absence of nominal rigidities.  By leaving out constants one gets *t tw p=  in equilibrium.  The contract wage tx  is renewed at time period t.  Half of the labor force minimizes the losses coming from the deviations from equilibrium, i.e.  221( )tt t ttL x pρ== −∑                                                                                                    (3)  subject to 1t tx x += .  This constraint means that the contracts fix a wage for two periods.  The other half of the labor force minimizes the losses L  at the previous time period 1t − , and fixes the wage at 1t tx x− = .  Parameter (0,1]ρ ∈  is the discount factor.  In the benchmark staggered wage model it is neglected to its borderline case 1ρ = . A distorting tax is imposed on labor income (Ball and Cecchetti, 1991).  The equilibrium pre-tax *w  is then set above the efficient level by an amount wˆ .  The latter is a function of the tax rate and the parameter of preference over consumption and leisure entering the labor supply, and of the parameter of technology (entering the labor demand).  The efficient real wage is that attained in the equilibrium with no taxes, i.e. ˆEt tw p w− = − .  It is negative because * 0w p− =  after dropping constants.  As *w  is set above the efficient level, aggregate employment and output are set below their efficient level. The monetary authority is concerned with output inefficiency (Barro and Gordon, 1983).  While wage setters target *w p= , the monetary authority cares about the loss provoked by the departure of the contract wage from Ew .  This loss is 2ˆ( )tt t tx p wρ= − +A  for the sector renewing the contract in t.  The social loss averaged across firms in the first and second period is then    2 2 211 1ˆ ˆ( ) ( )2 2t t t t tx p w x p wρ ρ −= − + + − +A .                                                            (4)  The monetary authority chooses the money each period taking the expectations of current and future money as given, so that equilibrium is discretionary.  Since the time paths of wages and price are both linear functions of µ , they are also functions of the sequence of surprises, i.e. { }t tmδ µ= − .  Both the workers and the authority have rational expectations.  3. The model with discounting  Relaxing the assumption that 1ρ =  in the workers’ loss function (3) leads to a new rule of contract revision:  111 1et t tx p pρρ ρ += ++ + .                                                                                           (5)  Considering (2) and (5), a solution for tx  and 1tx −  exists for homogeneous staggering.  Its stable characteristic root ( 1λ < ) is  2 2 2(2 )(1 ) (2 ) (1 ) 42α ρ α ρ α ρλ αρ− + − − + −= .                                                        (6)  It is implied that 0λ ρ∂ ∂ >  for 1ρ < .  Thus taking discounting into account raises λ . As the supply of money grows steadily and there is no surprise, the time paths of the contract wages are the first terms in the right hand side of (7) and (8):  10( ) (1 )jt t jjx t zµ λ δ∞ + −== + + −∑                                                                                  (7)  and  11( 1) (1 )jt t jjx t zµ λ δ∞− −== + − + −∑ ,                                                                            (8)  where  2(1 )[1 ( 1)] (1 )(1 )(1 )(1 )z ρ λ λ λρ λ ρρ λ λρ+ − + − += + − + .                                                                      (9)  The summation terms in (7) and (8) give the effect of a monetary surprise δ  on the wages.  And term z  summarizes the effect of discounting on the time paths of the contract wages.  Neglecting the discount factor ( 1ρ = ) yields 1 2z = , but 1 2z <  if 1ρ < .  With discounting, the non surprise time paths of aggregate real wage and output are  1( 1)2t tw p zα µ − = − −                                                                                         (10)  and  12Ny zα µ = −   .                                                                                                   (11)  The average real wage in (10) falls short of its optimal level * 0t tw p− =  over the contract period because µ  is positive.  And the equilibrium steady state output Ny  is pushed upward; as a result, the gap between equilibrium output and its efficient level is shrunk.  This diminishes the incentive for the monetary authority to create surprise inflation.  4. Calibration  Now we derive the equilibrium time consistent inflation rate to show how discounting affects the time consistent steady inflation.  Under rational expectations, δ  affects the monetary authority’s loss function (4) through (7) and (8).  The time consistent discretionary equilibrium requires zero surprise inflation each period, and this means imposing 00( ) 0tdL td δδ∞== =∑  on the authority’s problem, which leads to  wˆdµ =                                                                                                                    (12)  where  2( ) (1 )(1 )(1 )[(2 (1 )] [ (1 ) 2 ]d ρ λ α ρ λρ ν α λ ν α λ λ+ − + += − − + + + −                                                          (13)  and  12z zν α  = + −   .                                                                                                   (14)  The time consistent steady inflation is then a function of ρ  (and of wˆ  and α ).  Ball and Cecchetti’s (1991) result is a particular case arising if 1ρ =  in (13), and 1 2zν = = . Table 1 shows our model calibrated with selected values for ρ  and α , including the favorite ones in the real business cycle literature, i.e. .25.96ρ =  and .75α = .  The values of d  are lower for 1ρ < , thereby showing that if discounting is considered in (3) and (4), the time consistent steady inflation µ  is reduced.  5. Conclusion  We extend the standard staggered wage model to consider discounting.  By deriving the novel equilibrium time consistent inflation rate, and then calibrating the resulting model, we find that discounting actually lowers the time consistent steady inflation.  Table 1. The effect of discounting in the staggered wage model   α = .40 α = .60 α = .75 α = .80  ρ = 1     λ .127 .223 .333 .382 z  ½ ½ ½ ½ d  1.549 1.265 1 .894  .25.96ρ =      λ .128 .226 .335 .384 z  .496 .494 .490 .487 d  1.541 1.258 .994 .888  ρ = .9     λ .134 .237 .351 .402 z  .456 .434 .412 .368 d  1.469 1.195 .938 .835  ρ = .85     λ .137 .243 .360 .412 z  .432 .399 .365 .297 d  1.429 1.027 .906 .698  Notes ρ  = discount factor α  = degree of returns to scale λ  = characteristic root z  = (see equation (9)) d  = (see equation (13)) References  Barro, R., Gordon, D., 1983. A positive theory of monetary policy in a natural rate model. Journal of Monetary Economics 91, 589−610.  Ball, L., Cecchetti, S.G., 1991. Wage indexation and discretionary monetary policy. American Economic Review 81, 1310−1319.  Blanchard, O., 1986. The wage-price spiral. Quarterly Journal of Economics 101, 543−563.  Fischer, S., 1977. Long term contracts, rational expectations and the optimal money supply rule. Journal of Political Economy 85, 163−190.  Gray, J., 1976. Wage indexation: A macroeconomic approach. Journal of Monetary Economics 3, 221−335.  Taylor, J., 1979. Staggering price setting in a macro model. American Economic Review 69, 108−113.  Taylor, J., 1980. Aggregate dynamics and staggered contracts. Journal of Political Economy 88, 1−24. 

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Staggered wages, inflation, and discounting

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