International labour mobility, business cycle and
inflation dynamics and monetary policy∗
INCOMPLETE - PLEASE DO NOT QUOTE
Morten Spange and Tony Yates†
Bank of England
June 10, 2008
Abstract
We analyse how labour mobility across countries affects inflation dynamics and international business cycle comovements, and how it affects
the design of optimal monetary policy. The analytical framework is a two
country dynamic stochastic general equilibrium model with sticky prices.
In our model, labour mobility tends to amplify the effect of productivity
shocks on output, with the effect on inflation being ambiguous. Despite
this, labour mobility does not affect the structural relationship between
output and inflation via the slope of the new Keynesian Phillips curve.
Our model also suggests that labour mobility reduces the cross-country
correlation of output, suggesting that this is not the likely cause of increases in such comovement observed in the data. We find that labour
mobility implies that in response to a productivity shock it will be optimal
to allow for a slightly larger pickup in consumer price inflation in order
to stabilise producer price inflation.
1
Introduction
A key feature of the world economy over the recent decade has been a tendency
towards increasing international integration. This process has had many manifestations, in goods and factor markets. Our paper studies the implications for
business cycles and monetary policy design of one of them: the increase in the
mobility of labour across national boundaries. This mobility is manifest in a
∗ The
views expressed in this document are those of the authors and not necessarily those of
the Bank of England. Comments and suggestions by Mick Grady, Stephen Millard, Katharine
Neiss, James Proudman and in particular Roman Sustek are gratefully acknowledged. We
are also grateful for help from Matthias Paustian with implementing the Dynare codes for
optimal policy.
† Mailing address: Bank of England, Threadneedlestreet, London EC2R 8AH, United Kingdom. e-mail: morten.spange@bankofengland.co.uk and tony.yates@bankofengland.co.uk
1
rise in the stock of immigrants in the US labour force from about 10 per cent
in 1990, to 15 per cent in 2005; and in Germany from about 5 per cent in 1980
to around 8 per cent in 2005.1 Barwell (2007) describes how estimated gross
migration flows for the United Kingdom totalled 400k in 1979, but had risen to
970k by 2005. This increased mobility no doubt has diverse and subtle causes,
such as, amongst other things: the spread of a common language (English?); a
fall in transport costs; a relaxation of the rules governing migration from one
nation to another, particularly in formerly communist countries, or governing
the terms under which residents from one country can take employment in another, particularly relevant following the accession of eastern European nations
into the European Union; a reduction in the costs of remitting capital from one
country to another.
We develop a two-country DSGE model with labour mobility. We study how
migration affects busincess cycle dynamics, the response of inflation to shocks,
and therefore monetary policy design. Our first concern, how increased labour
mobility affects the inflation process and monetary policy design, is inherited
from an already considerable body of work on the broader question of how
globalization has affected the inflation process. Mumtaz & Surico (2007) and
Ciccarelli & Mojon (2005) detect that inflation across countries has become
increasingly driven by what they call a ’global factor’. Borio & Filardo (2007)
claim to have found evidence that measures of ’global slack’ help explain inflation
in individual countries, and that the strength of this impact has increased,
though a subsequent study by Ihrig et al. (2007) argue that the evidence is not
robust.
The possibility that there is an increased correlation between inflation in a
single country and global forces has caused others to think through what theory
should tell us would be the effect of increased international openness on the
inflation process. Razin & Binyamini (2007) aim at providing a unified analysis
of the effects of globalisation on the Phillips curve and monetary policy, in a
New-Keynesian framework. They find that labour, goods, and capital mobility
flatten the tradeoff between inflation and activity. Bean (2006), Bernanke (2007)
and Woodford (2007), deal with an alarmist view of globalisation, which is that
it may cause central banks to lose control over their own inflation rates. In their
own ways, they emphasise a view that will obtain in the model we develop, which
is that though globalisation (for us read international labour mobility) affects
how shocks are transmitted into inflation, the central bank can and must still
choose the rate of inflation in the long run.
The flip side of inflation being more tightly related to global factors is that it
is less strongly related to domestic factors. Benati (2007) documents an increase
in the reduced form correlation of inflation with unemployment in the UK. He
appeals to the story told by L. Ball & Romer (1989) that lower inflation would
have increased the degree of price stickiness. A related literature has looked
at whether estimates of the relation between real quantities like the output
gap or marginal costs and inflation has weakened when seen through the lens
1 IMF
World Economic Outlook, April 2007, Chapter 5, Figure 5.2.
2
of structrual aggregate supply equations. Sbordone (2007) explains that the
increase in competition likely to have resulted from the increased openness in
product markets in the United States would not have led to a sizable reduction
in the relationship between marginal costs and inflation.
Our second concern is the increased international comovements of real variables over the business cycle that has been documented by, amongst others,
Stock & Watson (2005). There is evidence that this is due to increases in the
proportion of output that is traded, since as Baxter & Kouparitsas (2005) have
documented, countries that trade more with each other have outputs that comove more strongly. And that greater financial integration has increased the
degree of comovement between consumption and output in different countries:
see, for example Imbs (2006), Fund (2007).2 Our model allows us to explore
whether increased international labour mobility may have been a factor in bringing about this increased comovement in business cycles. As we will explain later,
we shall see that it cannot.
The model used in this paper is a two country dynamic stochastic general
equilibrium (DSGE) model with labour mobility. The share of households residing in each country is determined by optimising behaviour. Migration in real life
typically has implications for the pattern of spending: workers that leave country A to work in country B usually buy housing and other services in country
B that they formerly bought in country A. We capture this by enforcing that
there is a ‘home bias’ in consumption that, in crude terms, leads to spending on
goods produced in country A to be higher when more households live in that
country.
We first study how international labour mobility affects the way the economy
responds to a productivity shock. Since labour is attracted to the country
with the higher productivity, we find that international migration amplifies the
response of output to the shock. The amplification effect of labour mobility also
leads to a larger fall in the terms of trade and the real exchange rate.
Next, we study how increased labour mobility affects the likely shape of the
New Keynesian Philips Curve. We find that this relationship would not appear
to change.
We move on to compute the optimal ‘Ramsey’ policy along the lines of
Schmitt-Grohe & Uribe (2007) and Coenen et al. (2008) and study how this
changes with increased labour mobility. We find that under a productivity shock
labour mobility implies that it will be optimal to allow for a slightly larger pickup
in consumer price inflation in order to stabilise producer price inflation. Under a
cost push shock labour mobility increases the volatility of inflation and output.
Our analysis to this point connects to two antecdents in the literature.
Woodford (2007) illustrates how a global and perfectly competitive market for
labour affects inflation dynamics, but yet leaves the domestic central bank still
able to control its own inflation rate in the long run. We also have a competitive
labour market, but the effective labour supply curve faced by our home country
2 We are grateful to Chris Peacock and Victoria Sapporta for drawing this literature to our
attention, and the implications that our model might have for it.
3
will be upward sloping, since extra migrants drive up the price of home-produced
necessities like land and housing (which we capture through the device of home
bias). Moreover, in our case movements will be restricted by the costs of migration. Bentolila et al. (2007) ask how immigrant workers may have affected the
Phillips Curve in Spain. They model the immigrant labour force as a distinct
labour market in which immigrants have lower bargaining power than natives.
In our model we abstract from any such distinction.
Finally, we illustrate the effect of increased labour mobility on the comovement of real variables at business cycle frequencies across countries. We find
that increased labour mobility descreases international business cycle comovements. This is to be expected, since as we observed above, increased labour
mobility causes flows of labour away from the country hit by low productivity
shocks to the other country that amplifies the effect of productivity on output.
Since we assume shocks to productivity are uncorrelated across countries, it
follows that these migrant flows reduce the comovement of outputs. We must
therefore conclude that the increased comovement observed in the data has some
other source.
The paper is structured as follows. In Section 2 we present the model and
characterise the behavior of households and firms. Section 3 discusses the calibration of the model, and in Section 4 we present the results on inflation dynamics. Section 5 studies the implications of labour mobility for optimal monetary
policy, and Section 6 concludes.
2
The model
Our analysis is conducted in a two-country dynamic stochastic general equilibirum model with sticky prices. The countries are denoted A and B. We assume
that the world is inhabited by a continuum of agents. The agents are infinitely
lived and form rational expectations. They maximise utiltiy with respect to
consumption, labour supply and country of residence, subject to a budget constraint.
In each country there is a continuum of monopolistically competitive firms
producing a single differentiated good. Output is produced subject to a production function with labour as the only input. The firms are owned by the
households who consequently receive all profits. Ex ante the countries are symmetric, but stochastic shocks will lead to temporary cross country productivity
differentials. This generates an incentive for international migration since households will prefer to live in the country with higher productivity.
To close the model we need to specify how monetary policy is conducted.
In Section 2 and 4 we assume that the central banks set a short term interest
rate according to a Taylor rule. In Section 5 we proceed to analyse optimal
monetary policy.
4
2.1
Consumers
In this section we outline the optimization problem facing the representative
household. Each household indexed with an i maximizes an infinite horizon
utility function. In each period the household chooses consumption and hours
worked. In addition, the household faces a choice between residing in country A
or country B. The optimization problem with respect to location is complicated
by the fact that the choice of country is a binary variable. Inspired by Devillanova (2001) we follow Hansen (1985) and Rogerson (1988) and convexify the
set of actions through the introduction of lotteries over the choice of country.
So in each period the consumer chooses the probability of staying in each of the
two countries. With preferences computed according to the expected utility of
outcomes we are back to solving a convex representative agent’s problem. Let
U denote utility. With Ct denote consumption and Ht denote hours worked.
With β being the discount factor, the discounted sum of a household’s stream
of future utility can then be written
∙
½
¸
∞
X
¢1−ξ2
1 ¡ A ¢1−ξ1
κ ¡
t
A
U0 (i) =
β λt (i)
+
C (i)
1 − Ht (i)
1 − ξ1 t
1 − ξ2
t=0
∙
¸
¢1−ξ2
1 ¡ B ¢1−ξ1
κ ¡
B
+ (1 − λt (i))
+
(1)
C (i)
1 − Ht (i)
1 − ξ1 t
1 − ξ2
−Mt (i)}
M (i) is a cost which arises if the consumer changes the probabilities in the
lottery from one period to the next. M therefore tends to reduce the migration
flows, and we interpret it as cost associated with migration. This cost is intended
to capture both the financial costs of travel and relocation of property as well
as the social costs associated with loss of contact with the local comunity. For
simplicity we assume that
M (i) =
2
[λt (i) − λt−1 (i)]
2
The households earn wage income from supplying labour services to the
representative firms. In addition, profits from the firms are distributed back to
the households. The households can invest in two types of one-period nominal
bonds, denominated in the currency of country A and B, respectively. We use
superscripts A and B to denote variables for consumers in those two countries.
The representative household operates subject to the following budget constraint
¢ A
R 1 A ¡ A¢ A
¡
Πt j dj
1 + iA
λt WtA HtA (i)
t−1 Bt−1 (i)
0
+
+
PtA
PtA
PtA
R 1 B ¡ B¢ B
¡
¢ B
εt 1 + iB
εt (1 − λt ) WtB HtB (i) εt 0 Πt j dj
t−1 Bt−1 (i)
+
+
+
(2)
PtA
PtA
PtA
B A (i) εt PtB
εt BtB (i)
(1 − λt ) CtB (i) +
= λt CtA (i) + t A +
A
Pt
Pt
PtA
5
where Btj is bonds issued in country j, paying a nominal rate of interest ijt , Πjt
is profits of the representative firm in country j and εt is the nominal exchange
rate (the price in country A’s currency of one unit of country B’s currency). As
we are considering a closed system the bonds are in zero net supply supply, i.e.
Z 1
Z 1
A
Bt (i) di =
BtB (i) di = 0
0
0
The setup with the lotteries implies that migration can be studied as a continuous problem, where the choice of location is the probability of staying in country
A, λt . Aggregation across individuals implies that population of country A and
B will be λt and (1 − λt ), respectively. All households in a country will act
identically, so in the following we will drop the index i.
The consumption bundles are defined over domestically produced and imported goods as a Dixit-Stiglitz aggregator, ie
χ
µ
¶ χ−1
¢ χ−1
χ−1
1 ¡
1
A
A
A
χ
χ
χ
χ
θ (CA,t )
+ (1 − θ) CB,t
(3)
Ct ≡
CtB
χ
µ
¶ χ−1
¢ χ−1
1 ¡
χ−1
1
B
B
χ
θ χ (CB,t
) χ + (1 − θ) χ CA,t
≡
(4)
where θ is the weight placed on domestically produced goods and χ is the
elasticity of substitution between domestically produced and imported goods.
When θ > 12 this reflects a home bias, which means that the consumer has a
relative preference towards consuming goods produced in the country where he
resides. This captures factors such as the necessity of buying nontradables like
land/housing and some services like haircuts locally while living in a country,
without modelling an explicit non-tradables sector.
The price index defined as the minimum cost of obtaining one unit of the
consumption bundle is given by
h ¡
i 1
¡
¢
¢
A 1−χ
B 1−χ 1−χ
PtA ≡ θ PA,t
+ (1 − θ) εt PB,t
(5)
1
"
#
µ
¶1−χ 1−χ
¡ B ¢1−χ
1 A
B
Pt
≡
θ PB,t
+ (1 − θ)
P
(6)
εt A,t
The sectoral consumption indices are generated by integrating over individaul
goods (brands)
Cji
≡
∙Z
1
Cji
(h)
η−1
η
0
η
¸ η−1
dh
, i = {A, B} , j = {A, B}
(7)
1
¸ 1−η
dh
, i = {A, B} , j = {A, B}
(8)
The consumer minimises the cost of obtaining one unit of the index. Assuming
that the law of one price holds, this leads to the following price indices
Pji
≡
∙Z
0
1
Pji
(h)
1−η
6
2.1.1
Consumer optimization
Maximising (1) subject to (2) produces the following conditions for CtA , CtB ,
HtA , HtB and λt
β
³ ´−ξ1 W j
t
Ctj
Ptj
³
´−ξ1 ³
´
j
Ct+1
1 + ijt
j
Pt+1
³
´−ξ2
= κ 1 − Htj
=
³ ´−ξ1
Ctj
Ptj
; j = {A, B}
; j = {A, B}
1 h¡ A ¢1−ξ1 ¡ B ¢1−ξ1 i
− Ct
Ct
1 − ξ1
¢1−ξ2 ¡
¢1−ξ2 i
κ h¡
− 1 − HtB
1 − HtA
+
1 − ξ2
+ [β (λt+1 − λt ) − (λt − λt−1 )]
¸
∙ A A
εt PtB B
Wt Ht (i) εt WtB HtB (i)
A
−
−
C
(i)
+
C
(i)
+Φt
t
PtA
PtA
PtA t
= 0
(9)
(10)
(11)
Now turn to the consumer’s intratemporal optimization problem. The DixitStiglitz indices for consumption (3) and (4) imply that demands for goods from
country A and B are given by
Ã
!−χ
A
¡
¢ A
PA,t
A
1 − ΓA,t Yt
= λt θ
CtA
(12)
PtA
Ã
!−χ
A
PA,t
CtB
+ (1 − λt ) (1 − θ)
εt PtB
¡
¢ B
1 − ΓB
B,t Yt
= λt (1 − θ)
Ã
+ (1 − λt ) θ
B
εt PB,t
!−χ
PtA
Ã
!−χ
B
PB,t
PtB
CtA
(13)
CtB
where PtA and PtB are defined in (5) and (6). Notice that the expressions on
the left hand side are the amount of output left for consumption once the price
adjustment costs Γt have been incurred, see discussion below.
2.2
Firms
Assume that the goods are produced by a continuum of firms in each country.
Each firm is the monopolistic producer of a single differentiated good. This
7
assumption justifies why output is demand determined once the price has been
set. For simplicity, we abstract from endogenous capital formation: firms use
labour as their only input, and the production function is assumed to exhibit
decreasing marginal returns to labour.3 We can therefore think of our model
as one with fixed capital. Labour input is the product of population size in
a particular country and the number of hours worked by a household in that
country.
The production functions are as follows
¢σ
¢σ
¡
¡
A
B
YtA (j) = ΨA
; YtB (j) = ΨB
(14)
t λt Ht (j)
t (1 − λt ) Ht (j)
Ψt is total factor productivity which is assumed to follow a stochastic AR(1)
processes (in logs), i.e.
³
´
³
´
log Ψjt − Ψ = ϕ log Ψjt−1 − Ψ + υ jt ; j = {A, B}
(15)
where υ t are i.i.d. shocks.
Firms set prices in the currency of the producer country, and changing prices
from one period to the next is assumed to be costly as suggested by Rotemberg
(1982). We assume that this cost which we denote Γt is given by
φA
A
ΓA
A,t (j) =
2
Ã
A
(j)
PA,t
A
PA,t−1
(j)
!2
−1
(16)
with a similar adjustment cost function for country B.
The firms maximize the present value of an infinite stream of future profits.
Per period profits can be expressed as
A
A
A A
A
A A
ΠA
t (j) = PA,t (j) Yt (j) − λt Wt Ht (j) − PA,t Yt ΓA,t (j)
ΠB
t
(j) =
B
PB,t
(j) YtB
(j) − (1 −
λt ) WtB HtB
(j) −
B
PB,t
YtB ΓB
B,t
(17)
(j) (18)
To discount profits we use the stochastic pricing kernals
ρit,t+j ≡ β
Φit+j Pti
; i = {A, B}
i
Φit Pt+j
where the superscript i corresponds to the currency in which the profit stream
is denominated.
A
A
Let ΘA
1,t (j), Θ2,t (j), and Θ3,t (j), denote the lagrange multipliers on (14),
(12) and (16). The first order conditions for the country A firm with respect to
3 It is typical in this class of models to abstract from capital formation, see, for example,
Clarida et al. (2002).
8
A
PA,t
(j), YtA (j), HtA (j) and ΓA
A,t (j) are then given as follows
0 =
YtA (j)
−
ΘA
2,t
(j) PtA
+ (1 − λt ) (1 − θ) η
Ã
⎡
⎣λt θη
Ã
Ã
A
(j)
PA,t
A
PA,t
A
(j)
PA,t
A
PA,t
!−η−1
!−η−1
1
A
PA,t
!
A
(j)
PA,t
Ã
1
A
PA,t
A
PA,t
εt PtB
Ã
A
PA,t
PtA
!−χ
!−χ
⎤
CtB ⎦
CtA
(19)
1
−1
A
A
PA,t−1
(j)
PA,t−1
(j)
!
Ã
A
A
−PA,t+1
PA,t+1
(j)
(j)
A
A
A
+ρA
−
1
i2
h
t,t+1 Θ3,t+1 (j) Pt+1 φA
A (j)
PA,t
A (j)
PA,t
+ΘA
3,t
(j) PtA φA
A
A
A
A
A
PA,t
(j) − ΘA
1,t (j) Pt − Θ2,t (j) Pt = 0
¡ A ¢σ−1 σ
A A
λt = 0
−λt WtA + ΘA
1,t (j) Pt Ψt σ Ht (j)
A
−PA,t
2.3
(j) YtA
−
ΘA
3,t
(j) PtA
=0
(20)
(21)
(22)
Monetary policy
To close the model we need to specify how monetary policy is conducted. We assume that there is a central bank in each country setting interest rates according
to a Taylor rule, given by:
Ã
!
Ytj − Yetj
j
j
it = i + ς P pbt + ς Y
; i = {A, B}
(23)
Yetj
hj
Y j −Y
A ‘tilde’ denotes a flex price variable. So t Yh j t is the flex price output gap.
t
When computing the flex price equilibrium we need to take a stand on how
to treat endogenous state variables. In our case we have one endogenous state
variable, namely λt . We assume that in the flex price equilibrium population
shares are as in the corresponding sticky price equilibrium. Later we shall study
how the model behaves when we replace the Taylor rule with an assumption that
the central banks conduct optimal monetary policy.
2.4
Market clearing
The economy’s aggregate resource conditions can be written as
¡
¡
¢ A
¢ B
A
B
A
B
1 − ΓA
1 − ΓB
A,t Yt = λt CA,t +(1 − λt ) CA,t ;
B,t Yt = λt CB,t +(1 − λt ) CB,t
(24)
9
3
Calibration
We shall study a calibrated version of the model. Where possible, we take
values that are standard in the literature. We interpret one period to represent
a quarter and set the discount factor β = 0.99 so as to yield a steady-state annual
real interest rate of 4 %. The literature opperates with a range of values for
the parameters ξ 1 and ξ 2 , which determine the consumers’ relative risk aversion
with respect to consumption and leisure. Following the literature we set ξ 1 = 2.5
and ξ 2 = 5. We calibrate κ to ensure that in steady state the consumers spend
approximately one third of their time endowment working. This leads us to set
κ = 0.4.
For the elasticity of substitution between goods produced in different countries χ, Obstfeld & Rogoff (2000) report empirical estimates range between 3
and 6, but in recent literature the elasticity is usually assumed to be lower.
We choose χ = 2. We assume an elasticity of substitution between varieties of
the same type of goods of η = 5. This gives rise to a steady state markup over
marginal cost of 25%. To calibrate the relative weight on domestically produced
goods we set θ = 0.7. This gives rise to a steady state import share of 30%,
which roughly corresponds to that of the United Kingdom.
In the production function the parameter σ determines the labour share of
total revenue in steady state. To get a wage share of two thirds we set σ = 2/3.4
Following the literature we assume technology shocks to be fairly persistent and
B
set ϕ = 0.95. To calibrate the price rigidities we set φA
A = φB = 100. This is
lower than the value used by Harrison et al. (2005) for the stickiness of prices
of consumption and capital goods, but higher than the value used for export
goods. Faia & Monacelli (2008) have a stickiness parameter of 75. In the policy
rule we set ς P = 1.5 and ς Y = 0.5, the original benchmark values proposed by
Taylor (1993). The litterature provides no estimate of , which captures the
cost of migration. We have experimented with different values and decided to
set = 20. This implies that a 2.5 % productivity shock leads to a peak effect
on population of around 1.5%.
4
4.1
Results
Migration and inflation dynamics
In this section we consider the response of the economy to a productivity shock
in country A.5 The results are presented in the form of impulse responses,
which are collected in Appendix 2. The shock is assumed to be temporary but
persistent. We shall compare the response of the endogenous variables to the
4 Although literally we have no capital in the model, and it might seem curious, therefore,
to allocate a labour share of less than 1, in fact we wish our model to be thought of as one with
fixed capital, hence our assumption of diminishing returns, and therefore the labour share less
than 1.
5 The results are obtained with Dynare.
10
shock when migration is prohibited (λt = 12 ∀t) to the case in which labour is
internationally mobile (λt determined by (11)).
The positive shock to productivity in country A means that firm firms in
this country face a temporarily low marginal cost of production. This leads to
a fall in producer prices (b
pA
A,t < 0) and a fall in the terms of trade (τ t < 0). A
fall in producer prices and a fall in the terms of trade have opposing effects on
the consumer price level, but the net effect is that with the exception of the first
quarter following the shock, inflation is below its steady state. Price rigidity
implies that firms do not cut prices as quickly as they would otherwise have
done. This means that output picks up by less than in a flex price world, which
leads to the opening up of a negative output gap. Since the shock is assumed
temporary, all variables eventually converge back to their initial level.
Higher productivity in country A implies that residing there has become relatively more attractive. In the scenario where labour is internationally mobile,
this means that more agents migrate to country A (λt > 0). Notice that the
hump shaped responce of λt reflects that the cost of migration gives an incentive
to smooth the relocation process. This mimics the empirical observation that
migration flows tend to take place over time. The flow of workers into country
A amplifies the pickup in output. It also leads to a pickup in demand because of
the assumption of a home bias in consumption, but the supply effect dominates
the demand effect. Therefore, the terms of trade fall by more when labour is
internationally mobile as larger relative price adjustments are necessary to clear
the market.
Labour mobility also has implications for the extent to which wages and
profits respond to the shock. We find that migration leads to a sharper rise in
profits and a smaller increase in wages. This has nothing to do with bargaining
power since in both cases the labour market is perfectly competitive. Instead
it reflects that inward migration limits the pickup in the marginal product of
labour, which is what determines the wage. And since inward migration boosts
the supply capacity of firms, profits are higher when labour is mobile.
Table 1 summarises the effect of labour mobility on the unconditional variances of inflation, the output gap and the nominal interest rate for the two
different assumptions about labour mobility.
Table 1
¡ ¢
V ar ¡π A
t¢
V ar ¡ytA¢
V ar iA
t
Productivity shock
V ar|
Ratio V ar| l m o b i l e
l n o t m o b ile
0.96
2.36
0.97
The results confirm the intuition from the charts that labour greatly amplifies
the volatility of output.
11
4.2
The slope of the Phillips curve
A key question for monetary policymakers is whether international labour mobility alters the way inflation is affected by demand and supply conditions in
the economy. It has often been suggested that globalisation and the associated
pickup in international mobility of goods and labour will have implications for
the trade-off between inflation and the output gap. In particular, it has been
suggested that inflation will tend to be unaffected by the output gap since instead of leading to higher wages, labour market tightness will lead to an inflow
of migrants without a pickup in wage inflation.
To study the implications for the Phillips curve it is convenient to define a
measure of real marginal cost, RM CtA .
¡ A ¢1−σ −1 ¡ A ¢−1 ¡ A ¢−1
σ
Ht
PA,t
Ψt
RM CtA ≡ WtA λ1−σ
t
(25)
For a generic variable xt , let x
bt defnote the log deviation from steady state of
that variable. Based on (19) - (22) we can derive the log-linearized Phillips
curve (see appendix for details)
π
bA
A,t =
η−1
A
rmc
d t + βEt π
bA
A,t+1
φA
A
(26)
This is the familiar New Keynesian Phillips curve, which shows how current
inflation depends on real marginal costs and expected future domestic inflation.
As seen from (26), this is independent of openness and labour mobility. But
to get a clearer picture of how international labour mobility affects the Phillips
curve we need to establish the link bewteen real marginal cost and a measure
of the output gap.
To address this question we simulate the log-linearized model for 10,000
periods. It has been shown in related open economy models that the structural
Phillips curve typically involves a term in the terms of trade. We define the
terms of trade τ t as
A
PA,t
pA
A,t
τt ≡
=
B
εt PB,t
et pB
B,t
where
et ≡
εt PtB
PtA
is the real exchange rate. For a generic variable xt , let x
et defnote the log deviation from the corresponding ’natural’ value of that variable. In the workhorse
new Keynesian model natural levels are defined as the values that the variables
would take if prices and wages had always been completely flexible. A complicating factor of our model is that polulation λt is an endogenous state variable.
If we continue to calulate natural levels under the assumption that prices have
always been flexible, the natural levels will be conditioned on a value of λt which
is different from actual population. Instead we define natural rates under the
12
assumption that prices are flexible but population is equal to the value under
sticky prices.6
We then estimate (27), which has the same functional form as the structural
Phillips curve derived by Clarida et al. (2002).7
πA
etA + γ 2 · e
τ t + γ 3 · Et πA
A,t = γ 1 · y
A,t+1 + u3,t
(27)
The results for the scenario with and without labour mobility are reported in
Table 3.
Table 3
γ
b1
γ
b2
γ
b3
No migration
0.00275
0.072
0.99
Migration
0.00275
0.072
0.99
Firstly, our result confirm the usual finding that the Phillips curve tends to be
very flat in this class of models for reasonable degrees of price rigidity. But
we also find that for the correct specification in domestic inflation, the Phillips
curve is unaffected by the assumption about labour moblility. This contradicts
the perceived wisdom that increased labour mobility would flatten the trade-off
between inflation and the output gap. Notice that this conclusion is conditional
on the way we treat population when computing the output gap.
4.3
International output co-movements
Empirical evidence suggests that cross-country output correlations across the
industrialised economies picked up sharply at the time of the large oil prices
shocks of the 1970’s and have remained high since, see e.g. Kose et al. (2003)
and Fund (2007). In theory strong output co-movements may reflect that the
countries are hit by a common shock. This is likely to be behind the high output
correlations observed in the 1970’s. Alternatively, it may reflect the way that
shocks are transmitted across countries via trade linkages. It has been suggested
that the continuation of the strong co-movements of output may be linked to
increased international integration as this allows for stronger spill-over effects
of country specific shocks. To assess whether labour market integration has the
potential to contribute to the strong co-movements of output, Table 2 compares
the cross-country correlation of output under the different assumptions about
labour mobility. The results are based on simulations in which both countries
are subject to uncorrelated productivity or monetary policy shocks.
6 See Woodford (2003) chapter 5 for a discussion of how to treat endogenous state variables
when computing ’natural’ rates.
7 The precision with which these estimated are obtained suggests that (27) is indeed the correct structural Phillips curve. In the next version of the paper we intend to include derivations
to formally show this.
13
Table 2: output correlations
productivity shock
monetary policy shock
No migration
-0.36
-0.676
Migration
-0.78
-0.89
Firstly we notice that output is always negatively correlated across countries
for both shocks. Although this is a familiar feature of this type of models,
it is strongly at odds with the evidence. Moreover, the correlation becomes
even more negative when labour is allowed to migrate. This is because labour
mobility amplifies the effect of shocks on output. So the model considered here
would suggest that the labour mobility is not behind the high correlation of
output across industrialised countries.
5
Optimal monetary policy
In this section we study the implications of labour mobility for optimal monetary policy. The concept of optimal policy we adopt here is the so-called Ramsey
plan.8 The central banks therefore set interest rates in order to maximise (1),
taking as constraints the first order conditions resulting from private sector optimisation as well as the economy’s resource constraints. We study the response
of the economy under optimal policy to a productivity shock and a mark-up
shock.
5.1
Productivity shock
Starting with the productivity shock, we see that the central banks act to almost
fully stabilise domestic inflation. Thereby the firms avoid having to pay the
Rotemberg cost of changing their price, and it confirms the finding in open
economy models, that in response to a productivity shock, the central bank
should stabilise the price of domestic output, see e.g Gali & Monacelli (2005).
This illustrates that a productivity shock does not create a trade-off between
stabilising inflation and the output gap.9
With domestic prices inflation close to zero, consumer price inflation is determined by the relative price of imports. Consequently, as the terms of trade
deteriorates more when labour is internationally mobile (as explained in Section
4), CPI inflation is allowed to rise more in this scenario.following the shock.
8 We implement the Ramsey policy using a set of Dynare routines developed by A. Levin for
Levin et al. (2005). Other studies that implement the Ramsey approach in an open economy
setting include Coenen et al. (2008) and Faia & Monacelli (2008).
9 Note that under very strict assumptions it is optimal to perfectly stabilise inflation in
response to a productivity shock, see e.g. Gali & Monacelli (2005). Since these assumptions
are not satisfied in out model, domestic inflation is not fully stabilised.
14
5.2
Mark-up shock
The (negative) mark-up shock is modelled as an increase in the degree of competitiveness among intermediate goods producers η. The shock is assumed to
follow an AR(1) process
η t = ϕη ηt−1 + εη,t
with ϕη = 0.95. Unlike the productivity shock, the mark-up shock introduces a
trade-off between stabilising (domestic) inflation and the output gap. Whereas
the shock has no implications for the efficient levels of output and inflation, the
shock nevertheless implies that output tends to rise and inflation tends to fall.
Since output can only be brought down at the expense of even lower inflation,
the shock cannot be offset. So the central bank accepts a pickup in output and
a fall in inflation.
The fall in margins leads to a pickup in wages and an inflow of migrants.
Qualitatively this has the same amplification effect as under the productivity
shock. So aggregate variables such as output and the terms of trade respond by
more. Producer price inflation also falls by more when labour is mobile.
6
Conclusions
The paper provides a framework for studying the macroeconomic impact of
labour mobility and its implications for monetary policy. This enables us to
provide a rigorous analysis of a topical issue that central banks are worrying
about. We find that labour mobility tends to amplify the impact on output
of exogenous shocks on the economy. We also show that within our framework
labour mobility reduces the cross country correlation of output, both in response
to productivity shocks and monetary policy shocks. Finally, we have found that
labour mobility does not change the way that firms set prices as a function of
future expected marginal costs. So in that sense the slope of the Phillips curve
is unaffected.
Appendix 1: Derivations
The derivation of the Phillips curve is based on the firms first order conditions (19) - (22). First we aggregate across firms to get rid of the index j.
From (22) we get
A
A
A
ΘA
3,t Pt = −PA,t Yt
From (21) we get
¡ ¢−1 −1 ¡ A ¢1−σ
1−σ
A
ΘA
WtA ΨA
σ
Ht
1,t Pt = λt
t
(28)
and (20) can be rewritten as
A
A
A
A
ΘA
1,t Pt = PA,t − Θ2,t Pt
15
(29)
Now insert (29) into (28) and re-arrange to get
¡ ¢−1 −1 ¡ A ¢1−σ
1−σ
A
A
ΘA
WtA ΨA
σ
Ht
2,t Pt = PA,t − λt
t
(30)
A
A
A
Now insert the expressions for ΘA
1,t Pt and Θ2,t Pt from (28) and (30) into (19)
and use that
Ã
Ã
!−χ
!−χ
A
A
PA,t
PA,t
A
A
Yt = λt θ
Ct + (1 − λt ) (1 − θ)
CtB
PtA
εt PtB
to get
³
¡ A ¢−1 −1 ¡ A ¢1−σ ´ 1
A
A
− λ1−σ
W
σ
0 = YtA − PA,t
τ A YtA
Ψt
Ht
t
t
PA,t
Ã
!
A
PA,t
1
A
−PA,t
YtA φA
−1
A
A
A
PA,t−1
PA,t−1
!
Ã
A
A
PA,t+1
PA,t+1
ρA
t+1 A
A
A
+β A PA,t+1 Yt+1 φA
−
1
³
´2
A
ρt
PA,t
A
PA,t
(31)
Now derive an expression for real marginal cost. We have that
Y
λH
= Ψ (λH)σ =⇒
1
= Y σΨ
−1
σ
Total costs of production therefore equals
1
costs = W A Y σ Ψ
−1
σ
implying that marginal costs become
¡ ¢ −1 1 ¡ A ¢ 1−σ
∂ cos ts
σ
= W A ΨA σ
Y
∂Y
σ
W A λH A
=
σY A
Defining real marginal costs relative to producer prices, we therefore have
MC
RM C ≡
=
¡
¢1−σ −1 ¡ A ¢−1 ¡ A ¢−1
W A λH A
PA
Ψ
= W A λ1−σ H A
σ
A
A
σPA Y
(32)
where we have used the production function (14) to substitute in for Y . Substituting (25) into (31) we get
0 = YtA − (1 − RM Ct ) τ YtA
Ã
!
A
A
PA,t
PA,t
A A
−Yt φA
−
1
A
A
PA,t−1
PA,t−1
!Ã
Ã
!2
A
A
PA,t+1
PA,t+1
ρA
t+1 A
A
+β A Yt+1 φA
−1
A
A
ρt
PA,t
PA,t
16
(33)
Defining producer price inflation
we can write
pbA
A,t ≡
A
PA,t
A
PA,t−1
−1=
¢
pA
A,t ¡
1 + pbA
t −1
A
pA,t−1
0 = 1 − (1 − RM Ct ) τ
¡
¢
A
A
−φA
A π A,t 1 + π A,t
+β
(34)
A
¢¡
¢2
ρA
t+1 Yt+1 A ¡ A
φA π A,t+1 1 + π A
A,t+1
A
A
ρt Yt
which we further rewrite as
RM Ct
h
¡
¢
A
A
= τ −1 τ − 1 + φA
A π A,t 1 + π A,t
A
¡ A
¢¡
¢2
ρA Yt+1
φA
−β t+1
1 + πA
A,t+1
A π A,t+1
A
A
ρt Yt
¸
Now differentiate the lhs and the rhs wrt the endogenous variables (rmct =
log (RM Ct ))
∂lhst
= RM Ct
∂rmct
£¡
¢
¤
∂rhst
A
= τ −1 φA
1 + πA
A
A,t + π A,t
A
∂π A,t
h¡
A
¢2
¡
¢i
ρA Yt+1
∂rhst
A
= −τ −1 β t+1
φA
+ 2π A
1 + πA
A,t+1
A,t+1 1 + π A,t+1
A
A
A
A
∂π A,t+1
ρt Yt
We do not need to differentiate with respect to (functions of) Y or ρ since these
derivatives will be zero in ss because πA
A = 0. Note that
ρA
t+1
ρA
t
¯
A ¯
ρt+1 ¯
¯
ρA
t
SS
=
¡ A ¢−ξ1
PtA ΦA
PtA Ct+1
t+1
= A ¡ ¢−ξ =⇒
A
1
Pt+1
ΦA
Pt+1 C A
t
t
= 1
Evaluated in steady state the derivatives are
¯
∂lhst ¯¯
τ −1
=
∂rmct ¯SS
τ
¯
¯
∂rhst ¯
= τ −1 φA
¯
A
¯
∂π A
A,t SS
¯
∂rhst ¯¯
= −τ −1 βφA
¯
A
¯
∂π A
A,t+1
SS
17
This implies that the log-linearized Phillips curve can be written
τ −1
rmc
dt
τ
π
bA
A,t
−1
= τ −1 φA
bA
βφA
bA
A,t − τ
A,t+1 ⇐⇒
Aπ
A Et π
=
τ −1
rmc
d t + βEt π
bA
A,t+1
φA
A
18
Appendix 2: Charts - Taylor rule - prod shk
Productivity
Output
2.5
3.00
2.50
2
No
migration
1.5
2.00
1.50
Migration
1
1.00
0.5
0.50
0
1
11
21
31
41
51
61
71
81
91
0.00
1
11
21
Output gap
31
41
51
61
71
81
91
Consumer price inflation
0.00
1
11
21
31
41
51
61
71
81
0.05
91
-0.10
No
migration
-0.30
0
71 81
No 91
migration -0.05
Migration
-0.40
Migration
-0.20
1
11
21 31 41 51 61
-0.1
-0.50
-0.15
-0.60
Producer price inflation
Terms of trade
0
1
11
21
31
41
51
61
71
81
91
-0.05
0
1
11
21
31
41
51
61
71
81
91
-0.2
-0.4
-0.1
No
migration
Migration
-0.15
No
migration
-0.2
Migration
-0.6
-0.8
-1
-1.2
-0.25
-0.3
19
-1.4
-1.6
Population
Real exchange rate
No
migration
Migration
0.7
1.6
0.6
1.4
1.2
0.5
Migration 1
0.8
0.4
0.3
0.6
0.2
0.4
0.1
0.2
0
0
1
11
21
31
41
51
61
71
81
1
91
11
21
31
41
Wages
51
61
71
81
91
Profits
No
migration
0.7
1.4
0.6
1.2
Migration
1
0.8
0.6
No
migration
0.5
Migration
0.3
0.4
0.2
0.2
0.1
0
1
11
21
31
41
51
61
71
81
91
-0.2
0
1
11
21
Hours worked
31
41
51
61
71
81
11
21
31
41
51
61
71
91
71
0.9
0.8
No
0.7
migration 0.6
0.5
Migration
0.4
0.3
0.2
0.1
0
81 91
Consumption
0
1
0.4
81
91
-0.1
-0.2
No
migration
-0.3
-0.4
-0.5
Migration
-0.6
-0.7
-0.8
-0.9
20
1
11
21
31
41
51
61
Appendix 3: Charts - Ramsey plan - prod shk
Productivity
Output
3.00
2.5
2.50
2
No
migration
1.5
2.00
1.50
Migration
1
1.00
0.5
0.50
0.00
0
1
11
21
31
41
51
61
71
81
1
91
11
21
31
41
51
61
71
81
91
Consumer price inflation
Nominal interest rate
0.5
0.00
1
11
21
31
41
51
61
71
81
91
-0.02
0.4
-0.04
0.3
-0.06
No
migration
-0.08
Migration
-0.10
No
migration
-0.12
Migration
-0.14
1
11
21
31 41
51
61
71
81 91
-0.16
Producer price inflation
0
11
21 31
41
51 61
71
0
-0.1
0
1
11
21
31
41
51
61
71
81
91
-0.2
-0.4
81 91
-0.002
No
migration
Migration
0.1
Terms of trade
0.002
1
0.2
-0.004
No
migration
-0.006
Migration
-0.008
-0.01
21
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
Population
Real exchange rate
0.7
0.6
No
migration
0.5
0.4
Migration
0.3
0.2
0.1
0
1
11
21
31
41
51
61
71
81
1
91
11
21
31
41
Wages
51
61
71
Profits
No
migration
0.35
1.8
1.6
1.4
1.2
1
Migration
0.3
No
migration 0.25
0.2
Migration
0.15
0.8
0.6
0.4
0.2
0
1
11
21
31
41
51
1.8
1.6
1.4
1.2
Migration
1
0.8
0.6
0.4
0.2
0
81 91
61
71
81
0.1
0.05
0
1
91
11
21
Hours worked
31
41
51
61
71
81
91
Consumption
0
1
11
21
31
41
51
61
71
81
91
1
-0.05
0.8
No
migration
0.6
Migration
0.4
-0.1
No
migration
Migration
-0.15
-0.2
-0.25
0.2
-0.3
-0.35
22
0
1
11
21
31
41
51
61
71
81
91
Appendix 4: Charts - Ramsey plan - cp shk
Competitiveness
Output
3.50
25
3.00
20
15
10
No
migration
2.50
Migration
1.50
2.00
1.00
5
0.50
0.00
0
1
11
21
31
41
51
61
71
81
1
91
11
21
Nominal interest rate
Migration
11
21
31
41
51
61
71
81
41
51
61
71
81
91
Consumer price inflation
No
migration
1
31
91
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
0.05
0
1
11
21 31 41 51 61
71 81 91
-0.1
No
migration
Migration
0
21
31
41
51
61
71
81
-0.25
91
0
1
11
21
31
41
51
61
71
81
91
-0.2
-0.05
-0.4
-0.1
-0.15
No
migration
-0.2
No
migration
Migration
-0.2
Terms of trade
0.05
11
-0.15
-0.3
Producer price inflation
1
-0.05
-0.25
-0.6
-0.8
Migration
-1
-0.3
-0.35
-1.2
-0.4
-1.4
23
Population
Real exchange rate
71
4.5
4
3.5
3
Migration
2.5
2
1.5
1
0.5
0
81 91
71
81
0.6
No
migration
0.5
Migration
0.3
0.4
0.2
0.1
0
1
11
21
31
41
51
61
71
81
1
91
11
21
31
41
Wages
51
61
Profits
No
migration
4
Migration
3
3.5
0
1
11
21
31
41
51
61
91
-0.1
No
-0.2
migration
-0.3
Migration
-0.4
2.5
2
1.5
1
-0.5
0.5
-0.6
0
1
11
21
31
41
51
61
71
81
91
-0.7
Hours worked
Consumption
1.4
0.4
1.2
0.35
No
0.3
migration
0.25
1
No
migration
0.8
Migration
0.4
Migration 0.2
0.6
0.15
0.1
0.2
0.05
0
1
11
21
31
41
51
61
71
81
91
0
1
24
11
21
31
41
51
61
71
81
91
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