SERIEs (2010) 1:175–243DOI 10.1007/s13209-009-0011-x

ORIGINAL ARTICLE

MEDEA: a DSGE model for the Spanish economy

Pablo Burriel · Jesús Fernández-Villaverde ·Juan F. Rubio-Ramírez

Received: 6 May 2009 / Accepted: 22 October 2009 / Published online: 23 February 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract In this paper, we provide a brief introduction to a new macroeconometricmodel of the Spanish economy named MEDEA (Modelo de Equilibrio Dinámico de laEconomía EspañolA). MEDEA is a dynamic stochastic general equilibrium (DSGE)model that aims to describe the main features of the Spanish economy for policyanalysis, counterfactual exercises, and forecasting. MEDEA is built in the traditionof New Keynesian models with real and nominal rigidities, but it also incorporatesaspects such as a small open economy framework, an outside monetary authoritysuch as the ECB, and population growth, factors that are important in accounting foraggregate fluctuations in Spain. The model is estimated with Bayesian techniques anddata from the last two decades. Beyond describing the properties of the model, weperform different exercises to illustrate the potential of MEDEA, including historicaldecompositions, long-run and short-run simulations, and counterfactual experiments.

Keywords DSGE models · Likelihood estimation · Bayesian methods

JEL Classification C11 · C13 · E30

We thank David Taguas and Rafael Domenech for their support and encouragement during thedevelopment of MEDEA, Fillipo Ferroni for a very interesting discussion, and participants at numerousseminars for feedback. Beyond the usual disclaimer, we must note that any views expressed herein arethose of the authors and not necessarily those of the Banco de España, the Federal Reserve Bank ofAtlanta, or the Federal Reserve System. Finally, we also thank the NSF for financial support.

P. Burriel (B)Banco de España, Madrid, Spaine-mail: pburriel@bde.es

J. Fernández-VillaverdeUniversity of Pennsylvania, FEDEA, NBER, and CEPR, Philadelphia, PA, USA

J. F. Rubio-RamírezDuke University, Federal Reserve Bank of Atlanta, and FEDEA, Durham, NC, USA

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1 Introduction

In this paper, we provide a brief introduction to a dynamic equilibrium model of theSpanish economy named MEDEA (Modelo de Equilibrio Dinámico de la EconomíaEspañolA). This model was developed, solved, and estimated with Spanish data whilethe authors collaborated with the Economic Office of the President of Spain (OficinaEconómica del Presidente del Gobierno) from 2007 to 2008.

MEDEA is a dynamic stochastic general equilibrium (DSGE) model that aims todescribe the main features of the Spanish economy for policy analysis, counterfactualexercises, and forecasting. MEDEA is built in the tradition of New Keynesian modelswith real and nominal rigidities (see Christiano et al. 2005; Smets and Wouters 2003,and the book-length descriptions in Woodford 2003 and Galí 2008). New Keynesianmodels have proven to be a flexible framework that can incorporate many differenteconomic mechanisms of interest, to be rich enough for meaningful policy analysis,and to have a good forecasting track record. At the kernel of MEDEA, we have a neo-classical growth model with optimizing households and firms and long-run growthinduced by technological change and population growth. On top of this core, MEDEAhas rigidities of prices and wages, habit persistence in consumption, a set of adjust-ment costs (to investment, to exports, and to imports), a fiscal and monetary authoritythat determines a short-run nominal interest rate and taxes, and shocks to populationgrowth, technology, preferences, and policy that induce the stochastic dynamics of theeconomy.

In addition, MEDEA is designed to be a model for a small open economy thatbelongs to a currency area, in this case, the euro. The open economy aspects are cap-tured by the presence of exporting and importing firms with incomplete pass-throughand by the ability of the agents to save or borrow on foreign financial assets. Thecurrency area is modelled through a monetary authority that sets short-term nominalinterest rates by following a Taylor rule based on the economic performance of thewhole euro area.

MEDEA shares many features with other DSGE models developed at policy-mak-ing institutions for use as an input for their activities. Some examples are the FederalReserve Board (Erceg et al. 2006), the European Central Bank (Christoffel et al. 2008),the Bank of Canada (Murchison and Rennison 2006), the Bank of England (Harrisonet al. 2005), the Bank of Finland (Kilponen and Ripatti 2006; Kortelainen 2002), theBank of Sweden (Adolfson et al. 2005) the Bank of Spain (Andrés et al. 2006), and theSpanish Ministry of Economics and Finance (Ministerio de Economía y Hacienda)(Boscá et al. 2007). As such, MEDEA is a model that it is comparable to its peers andcan borrow from many years of experience.

At the same time, MEDEA has many new elements, some that are, in our opinion,interesting advances for DSGE modelling in general, and some that are important toadapt the model to Spain. We would like to highlight four of these:

1. MEDEA has stochastic growth coming from three sources: neutral technologi-cal progress, investment-specific technological progress, and population growth.These will allow us to capture two relevant characteristics of Spain in thelast decade: the low productivity growth and the large rise in immigration.

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By modelling population growth as a random walk in logs with a drift, we canexplore both the effects of changing the drift and the consequences of randomshocks (such as the unexpected arrival of more immigrants).

2. In comparison with other recent estimated DSGE models, we do not model theforeign world as an equilibrium outcome beyond the behavior of the EuropeanCentral Bank (ECB) through its Taylor rule. Spain is too small to have a signifi-cant impact on the world economy. We prefer to use the extra complexity to enrichother aspects of the model.

3. Fiscal policy. Most of the estimated DSGE models have been developed in cen-tral banks. Since monetary policy corresponds to their role, central banks havepaid particular attention to issues related to such policy, but the treatment of fiscalpolicy has been very parsimonious. We pay some detailed attention to the fiscalsector of the economy (although more work is needed), with three tax rates: oncapital income, labor income, and consumption. Given the current widespread useof fiscal instruments to fight the 2008–2009 recession, this aspect is particularlyinteresting.

4. At a more technical level, we design the solution of the model in such a way thatwe will be able to undertake higher-order approximations in the middle run ina relatively simple way. There is a growing body of literature that emphasizesthat there is much to be gained from a non-linear estimation of the model, bothin terms of accuracy and in terms of identification (see Fernández-Villaverde andRubio-Ramírez 2005, 2007; Fernández-Villaverde et al. 2006; An and Schorfheide2006, among several others). In the current version of the model, for computa-tional reasons, we solve the model by log-linearizing the equilibrium conditionsaround a transformed stationary steady state. However, our derivations are donewith the perspective of performing these higher-order approximations in the future(for example, contrary to common practice, we never substitute variables away tofind a Phillips curve, a strategy that works only up to a first-order approximation).

MEDEA is estimated by Bayesian methods. We follow the Bayesian paradigmbecause it is a powerful, coherent, and flexible perspective for the estimation ofdynamic models in economics (see An and Schorfheide 2006; Fernández-Villaverde2009, for surveys of the literature). First, Bayesian analysis is built on a clear set ofaxioms and it has a direct link with decision theory. The link is particularly relevantfor MEDEA since the model has been designed for applied policy analysis. Manyof the relevant policy decisions require an explicit consideration of uncertainty andasymmetric loss assessments. Consequently, the Bayesian approach provides a con-venient playground for risk management. Second, the Bayesian approach deals in atransparent way with misspecification and identification problems, which are perva-sive in the estimation of DSGE models (Canova and Sala 2006; Iskrev 2008). Third,the Bayesian estimators have desirable small sample and asymptotic properties, evenif they are evaluated by classical criteria (Fernández-Villaverde and Rubio-Ramírez2004). Fourth, priors allow us to introduce presample information and to reduce thedimensionality problem associated with the number of parameters. As Sims (2007)has emphasized, with any model rich enough to fit the data well, the use of priorsis essential to do any reasonable inference. Priors will be especially attractive for

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MEDEA because the deep changes in the structure of the Spanish economy over thelast several decades stop us from using data before the early 1980s, leaving us with arelatively short sample. Fifth, a likelihood-based method, such as our Bayesian esti-mation, allows us to recover the whole set of parameter values required for policy andwelfare analysis. Finally, Bayesian methods have important computational advantagesover maximum likelihood in large models like MEDEA. Simulating the posterior dis-tribution of the parameters is a much easier task than maximizing a highly dimensionallikelihood.

MEDEA can be employed for three main alternative purposes. First, we can use itto understand the dynamics of aggregate fluctuations. To illustrate this feature, in thispaper, we will show a decomposition of the last two business cycles among differentsources of variation. Second, we can use MEDEA for policy analysis, including coun-terfactuals and alternative policy experiments. We will perform several counterfactualsto illustrate the properties of the model. For instance, we will look at the effects ofchanges in the consumption tax rate and in the wage mark-up or the consequences ofalternative scenarios for population growth. Finally, we can use MEDEA for forecast-ing purposes. Even if DSGE models are not specifically designed with this goal inmind, their forecasting performance has been very satisfactory (see Edge et al. 2009;Christoffel et al. 2007, for the forecasting experience of the DSGE model used by theFederal Reserve System and the ECB, respectively). Because of space considerations,we will leave a thorough analysis of the forecasting properties of MEDEA for the nearfuture.

The structure of this paper is as follows. First, we outline the main structure ofthe model. Second, we describe MEDEA’s theoretical framework in detail. Third, wedefine the equilibrium of the economy, we transform this equilibrium into a stationaryone by appropriately changing the variables, and we solve the model by log-linearizingthe equilibrium conditions of the transformed model. Fourth, we build the likelihoodof the model. Finally, we estimate the model with data from 1986:1 to 2007:2 and wereport the results of a number of exercises undertaken with the model.

2 Outline of the model

MEDEA is a medium-scale model with 82 equations and 10 stochastic shocks. The82 equations include 16 equations that relate variables in the model to observables(although we will not use all of the observables in all of our exercises), the laws ofmotion for 37 state variables, and the equations determining 19 endogenous variablesthat are not states. Given this relatively large number of variables, it is worthwhile tooutline the basic structure of the model before we go into further details:

1. A continuum of households consume, save in domestic and foreign assets, holdmoney, supply labor, and set their own wages subject to a demand curve andCalvo’s pricing with partial indexation.

2. The labor of households is aggregated by a perfectly competitive labor packerwho sells the aggregated labor to the domestic intermediate good producers.

3. The final domestic good is manufactured by a final domestic good producer,which uses as inputs intermediate domestic goods.

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4. The consumption good is packed by a consumption good producer using the finaldomestic good and the final imported good.

5. Similarly, the investment good is packed by an investment good producer usingthe final domestic good and the final imported good.

6. Domestic intermediate goods producers rent capital and labor to manufacturetheir good and are subject to Calvo’s pricing with partial indexation.

7. The final imported good is packed by a final imported good producer usingintermediate goods produced by monopolistic competitors from a generic importgood, with incomplete pass-through specified as Calvo’s pricing with partialindexation.

8. The export goods are produced by monopolistic competitors who buy the finaldomestic good and differentiate it by brand naming. The exporters exhibit local-currency pricing that we specify as Calvo’s pricing with partial indexation.

9. There is a monetary authority, the ECB, that implements monetary policy. TheECB’s monetary policy fixes the one-period nominal interest rate of the euro areathrough open market operations, with the euro area inflation as target. The weightof the Spanish economy in this policy target is approximately 10%.

10. There is a government that implements fiscal policy to finance an exogenouslygiven stream of government consumption with taxes on capital and labor incomeand on consumption.

11. Finally, long-run growth of per capita income is induced by the presence of twounit roots, one in the level of neutral technology and one in the investment-specifictechnology. Moreover, there is stochastic population growth.

3 The model

We start now by describing the model. We will discuss each of type of agents (house-holds, distribution sector, intermediate good producers, foreign sector, the monetaryauthority, and the government) and how their actions aggregate.

3.1 Households

There is a continuum of households indexed by j ∈ [0, 1]. Each household is com-posed of Lt identical workers. The preferences of households are representable by thefollowing lifetime utility function, which is separable into per capita consumption,c jt , per capita real money balances, m jt/pt (where pt is the price of the domesticfinal good), and per capita hours worked, ls

j t (in terms of the proportion of the periodspent at work):

E0

∞∑

t=0

β t Lt dt

⎧⎪⎨

⎪⎩log(c jt − hc jt−1

)+ υ logm jt

pt− ϕtψ

(ls

j t

)1+ϑ

1 + ϑ

⎫⎪⎬

⎪⎭

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where E0 is the conditional expectation operator evaluated at time 0, β is the discountfactor, h is the parameter that controls habit persistence, and ϑ is the inverse of Frischlabor supply elasticity. The variable dt is an intertemporal preference shock, while ϕt

is a labor supply shock with laws of motion:

log dt = ρd log dt−1 + σdεd,t where εd,t ∼ N (0, 1),

log ϕt = ρϕ log ϕt−1 + σϕεϕ,t where εϕ,t ∼ N (0, 1).

Note that the preference shifters are common for all households. The preference shockdt changes the intertemporal first-order conditions, while the preference shock ϕt

moves the first-order conditions affecting labor supply and wage determination. Weinclude the shock dt to capture the changes in valuations between the present andthe future that the analysis of intertemporal wedges suggests as key for understand-ing aggregate fluctuations (Primiceri et al. 2006). We add the shock ϕt to model thechanges in labor supply that Hall (1997) and Chari et al. (2007) have pointed out asresponsible for a large proportion of the changes in employment over the businesscycle. We have selected a utility function where consumption appears in logs. Thus,the marginal relation of substitution between consumption and leisure is linear inconsumption and we have a balanced growth path with constant hours (King et al.1988).

The household’s size, Lt , follows a random walk with drift in logs:

Lt = Lt−1 exp(L + zL ,t

)where zL ,t = σLεL ,t and εL ,t ∼ N (0, 1).

Thus, the growth of population is given by

γ Lt = Lt

Lt−1= exp

(L + zL ,t

)

This process induces the first unit root in the model. However, this unit root will onlyaffect the absolute levels of the variables and not the per capita terms.

Households hold an amount a jt+1 of Arrow securities,1 an amount b jt of domesticgovernment bonds that pay a nominal gross interest rate of Rt , and an amount in domes-tic currency ext bW

jt of foreign government bonds from the rest of the world, which

pay a nominal gross interest rate of RWt � (·). The exchange rate, ext , is expressed in

terms of the domestic currency per unit of foreign currency. Following Schmitt-Grohéand Uribe (2003), the function � (·) represents the premium associated with buyingforeign bonds and it captures the costs (or benefits) for households of undertakingpositions in the international asset market. We assume that � (·) depends on the percapita holdings of foreign bonds in the entire economy with respect to nominal output

1 Households can trade on the whole set of possible Arrow securities within the country, indexed both bythe household j (since the household faces idiosyncratic wage-adjustment risk that we will describe below)and by time (to capture Spanish aggregate risk). The amount a jt+1 indicates the amount of those securitiesthat pay one unit of the domestic final good in event ω j,t+1,t purchased by household j at time t at a (real)price q jt+1,t .

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of the final domestic good:

bWt =

∫ 10 bW

jt d j

ptydt

Thus, as borrowers, households are charged a premium on the foreign interest rate

(that is, if bWt < 0, then �

(ext bW

t , ξbW

t

)< 1) and get a remuneration when they

act as lenders. Moreover, � (0) = 1, � (·)′ > 0, and � (·)′′ < 0. Domestic house-holds take bW

t as given when deciding their optimal holding of foreign bonds. Finally,revenues from the premium are rebated in a lump-sum to the foreign agents.2 Theholding cost of foreign debt is introduced to pin down a well-defined steady state forconsumption and assets in the context of international incomplete markets (otherwise,transient dynamics will have permanent effects) and it is motivated empirically bythe observation that Spain cannot fully insure against its idiosyncratic shocks in theinternational capital markets.3

Since we do not model the rest of the world, the evolution of the RWt is given by

an exogenous process:

RWt =

(RW)(1−ρRW

) (RW

t−1

)ρRWexp(σRW εRW ,t

)

In addition, there is a time-varying shock to the premium ξbW

t , with the followingprocess:

log ξbW

t = ρbW log ξbW

t−1 + σbW εbW ,t where εbW ,t ∼ N (0, 1),

With all this structure, the j-th household’s per capita budget constraint is givenby:

(1 + τc)pc

t

ptc jt + pi

t

pti j t + m jt

pt+ b jt

pt+ ext bW

jt

pt+∫

q jt+1,t a j t+1dω j,t+1,t

= (1−τw)w j t lsj t +(

rtu j t (1 − τk) + µ−1t δτk − µ−1

t �[u j t])

k jt−1 + 1

γ Lt

m jt−1

pt

+Rt−11

γ Lt

b jt−1

pt+ RW

t−1�(

ext bWt−1, ξ

bW

t−1

) 1

γ Lt

ext bWjt−1

pt+ 1

γ Lt

a jt + Tt + �t

where pt is the price of the domestic final good, pct is the price level of the consumption

final good, pit is the price level of the investment final good, w j t is the real wage in

2 However, since we do not model the equilibrium behavior of the rest of the world, the way in which thisrebate is distributed is irrelevant for our purposes.3 Some of the alternatives to this holding cost outlined by Schmitt-Grohé and Uribe (2003) are less attrac-tive for us. Complete international markets are empirically implausible and Uzawa preferences may inducecomplicated dynamic responses. The final possibility proposed by Schmitt-Grohé and Uribe, a quadraticadjustment cost on the level of the debt, would deliver results that are quantitatively nearly identical to ours.

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terms of the domestic final good, rt the real rental price of capital, also in terms of thedomestic final good, u j t > 0 the intensity of use of capital, µ−1

t �[u j t]

is the physi-cal cost of use of capital in resource terms, µt is an investment-specific technologicalshock to be described momentarily, Tt is a lump-sum transfer, �t are the profits ofthe firms in the economy, and τc, τw, and τk are the tax rates on consumption, wages,and capital income. Note that the tax on capital income is defined on the net returnof capital after depreciation δ and hence we include a tax credit µ−1

t δτk, expressedin resource terms. Also, note that we divide the per capita holdings of money andbonds carried into the period by the current population growth to express all quantitiesin current population per capita terms. Finally, we assume that � [1] = 0, �′ and�′′ > 0.

Investment i j t induces a law of motion for (per capita) capital held by the j-thhousehold:

γ Lt+1k jt = (1 − δ) k jt−1 + µt

(1 − S

[γ L

ti j t

i j t−1

])i j t

where S [·] is an adjustment cost function on the level of investment such that S [i ] =0, S′ [i ] = 0, and S′′ [·] > 0, and where i is the growth rate of investment along thebalance growth path. Note our capital timing: we index capital at the time its level isdecided. Also, the amount of per capita capital in the next period is a random variablebecause the population next period is also random (obviously, this randomness doesnot affect total capital at period t + 1, which is determined at time t).

We include an investment-specific technological shock in our law of motion for cap-ital because we were convinced by Greenwood et al. (1997, 2000) that this mechanismis of key importance to quantitatively account for growth and aggregate fluctuations.The investment-specific technological shock follows an autoregressive process of theform:

µt = µt−1 exp(µ + zµ,t

)where zµ,t = σµεµ,t and εµ,t ∼ N (0, 1)

This process induces a second unit root in the model.

3.1.1 Household’s problem

Given our description of the household’s environment, the Lagrangian function asso-ciated with it is:

E0

∞∑

t=0

βt γ Lt

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

dt

⎧⎨

⎩log(c jt − hc jt−1

)+ υ logm jtpt

− ϕt ψ

(lsj t

)1+ϑ

1+ϑ

⎫⎬

⎭

−λ j t

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1 + τc)pct

ptc j t + pi

tpt

i j t + m jtpt

+ b jtpt

+ ext bWjt

pt+ ∫ q jt+1,t a j t+1dω j,t+1,t

− (1 − τw)w j t lsj t −

(rtu j t

(1 − τk

)+ µ−1t δτk − µ−1

t �[u j t])

k jt−1 − 1γ L

t

m jt−1pt

−Tt − �t

−Rt−11

γ Lt

b j t−1pt

− RWt−1�

(ext bW

t−1, ξbWt−1

)1

γ Lt

ext bWjt−1

pt− 1

γ Lt

a jt

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

−Q jt

{γ L

t+1k jt − (1 − δ) k jt−1 − µt

(1 − S

[γ L

ti j t

i j t−1

])i j t

}

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

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SERIEs (2010) 1:175–243 183

where γ Lt = ∏t

i=1 γ Li and they maximize over c jt , b jt , bW

jt , u j t , k jt , i j t , m jt , w j t ,

and lsj t ; while λ j t and Q jt are the Lagrangian multipliers associated with the budget

constraint and the evolution of installed capital, respectively.The first-order conditions of this problem (except for labor and wages) are:

dt(c jt − hc jt−1

)−1 − hEtβγ Lt+1dt+1

(c jt+1 − hc jt

)−1 = λ j t (1 + τc)pc

t

pt

λ j t = Et

{βλ j t+1

Rt

�t+1

}

λ j t = Et

⎧⎨

⎩βλ j t+1

RWt �

(ext bW

t , ξbW

t

)

�t+1

ext+1

ext

⎫⎬

⎭

rt = µ−1t �′ [u j t

]

(1 − τk)

q jtEtγL

t+1 = βEtγL

t+1

{λ j t+1

λ j t

((1 − δ) q jt+1 +

(rt+1u j t+1 (1 − τk)

+ µ−1t+1δτk − µ−1

t+1�[u j t+1

]))}

pit

pt= q jtµt

(1 − S

[γ L

ti j t

i j t−1

]− S′

[γ L

ti j t

i j t−1

]γ L

ti j t

i j t−1

)

+ Etβγ Lt+1q jt+1

λ j t+1

λ j tµt+1S′

[γ L

t+1i j t+1

i j t

]γ L

t+1

(i j t+1

i j t

)2

m jt

pt= dtυ

[βEtλ j t+1

Rt − 1

�t+1

]−1

.

where we have defined the (marginal) Tobin’s Q as q jt = Q jtλ j t

(the ratio of the twoLagrangian multipliers, or more loosely the value of installed capital in terms of itsreplacement cost), and substituted the Euler equation into the money balances equa-tion.4 For our analysis, the first order condition with respect to money holdings willnot be strictly needed: as we will explain later, the monetary authority will just issueenough money balances such that the optimality condition is satisfied given the allo-cation, prices, and the nominal interest rate.

The first order condition with respect to investment has a simple interpretation. IfS [·] = 0 (the case without adjustment costs), we get:

q jt = pit

pt

1

µt

4 We do not take first-order conditions with respect to Arrow securities since, in our environment withcomplete markets and separable utility in labor, their equilibrium price will be such that their demandensures that the household’s consumption does not depend on idiosyncratic shocks (see Erceg et al. 2000).

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184 SERIEs (2010) 1:175–243

that is, the marginal Tobin’s Q is equal to the replacement cost of capital (the relativeprice of capital) in terms of the domestic final good. Furthermore, if µt = 1 andpi

t = pt , as in the standard neoclassical growth model, q jt = 1.

3.1.2 Labor demand and wage decisions

The first-order conditions with respect to labor and wages are more involved. Thelabor used by intermediate good producers described below is supplied by a represen-tative competitive firm that hires the labor supplied by each household j , Ltls

j t . Thelabor supplier aggregates the differentiated labor types with the following productionfunction:

Ldt = Lt

⎛

⎝1∫

0

(ls

j t

) η−1η

d j

⎞

⎠

ηη−1

= Ltldt (1)

where 0 ≤ η < ∞ is the elasticity of substitution among different types of labor, ldt

is the per capita labor demand, and Ldt is the total labor demand.

The labor “packer” maximizes profits subject to the production function (1), takingas given all differentiated labor wages w j t and the aggregate wage wt :

maxl j t

wt Lt ldt −

1∫

0

w j t Lt lsj t d j

The first-order conditions of the labor “packer” are:

wtη

η − 1

⎛

⎝1∫

0

(ls

j t

) η−1η

d j

⎞

⎠

ηη−1 −1

η − 1

η

(ls

j t

) η−1η

−1 − w j t = 0 ∀ j

Dividing the first-order conditions for two types of labor i and j and integrating overall labor types, we get:

1∫

0

w j t lsj t d j = wi t

(lsi t

) 1η

1∫

0

(ls

j t

) η−1η

d j = wi t(lsi t

) 1η

(ls

j t

) η−1η

Using the zero profits condition implied by perfect competition:

wt ldt =

1∫

0

w j t lsj t d j

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SERIEs (2010) 1:175–243 185

and solving we obtain the per capita input demand function and aggregate wage:

lsj t =

(w j t

wt

)−η

ldt ∀ j

(2)

wt =⎛

⎝1∫

0

w1−ηj t d j

⎞

⎠

11−η

Idiosyncratic risk comes about because households set their wages following aCalvo’s setting. In each period, a fraction 1 − θw of households can change theirwages. All other households can only partially index their wages to past inflation ofthe final domestic good. Indexation is controlled by the parameter χw ∈ [0, 1]. Thisimplies that if the household cannot change its wage for τ periods, its normalized

wage is∏τ

s=1�

χwt+s−1�t+s

w j t . When new workers in the household begin to work, they areassigned a wage equal to the wage of the other workers in the household.

Thus, the relevant part of the Lagrangian for the household is:

maxw j t

Et

∞∑

τ=0

θτwβτ γ L

τ

⎧⎪⎨

⎪⎩−dt+τ ϕt+τψ

(ls

j t+τ

)1+ϑ

1+ϑ+λ j t+τ

τ∏

s=1

�χw

t+s−1

�t+s(1−τw) w j t l

sj t+τ

⎫⎪⎬

⎪⎭

s.t. lsj t+τ =

(τ∏

s=1

�χw

t+s−1

�t+s

w j t

wt+τ

)−η

ldt+τ

All households that can optimize their wages in this period set the same wage (w∗t =

w j t ∀ j that optimizes) because complete markets allow them to hedge the risk of thetiming of wage change. Hence, we can drop the j th from the choice of wages and λ j t .Similarly, the ratio of Lagrangians, λt+τ /λt , will be constant across households and,consequently, the marginal valuation of future income is also constant. The first-ordercondition of this problem is:

η − 1

η(1 − τw)w∗

t Et

∞∑

τ=0

(βθw)τ γ Lτ λt+τ

(τ∏

s=1

�χw

t+s−1

�t+s

)1−η (w∗

t

wt+τ

)−η

ldt+τ

= Et

∞∑

τ=0

(βθw)τ γ Lτ

⎛

⎝dt+τ ϕt+τψ

(τ∏

s=1

�χw

t+s−1

�t+s

w∗t

wt+τ

)−η(1+ϑ) (ldt+τ

)1+ϑ

⎞

⎠

Note that for those sums to be well defined (and, more generally, for the maximizationproblem to have a solution), we need to assume that (βθw)τ γ L

τ λt+τ goes to zero faster

than(∏τ

s=1 �χwt+s/�t+s−1

)1−ηgoes to infinity in expectation.

123

186 SERIEs (2010) 1:175–243

To express this equation recursively, we re-label each part of this equality as f 1t

and f 2t :

f 1t = η − 1

η(1 − τw)w∗

t Et

∞∑

τ=0

(βθw)τ γ Lτ λt+τ

(τ∏

s=1

�χw

t+s−1

�t+s

)1−η (wt+τ

w∗t

)η

ldt+τ

f 2t = Et

∞∑

τ=0

(βθw)τ γ L

τ dt+τ ϕt+τψ

(τ∏

s=1

�χw

t+s−1

�t+s

)−η(1+ϑ) (wt+τ

w∗t

)η(1+ϑ)(ldt+τ

)1+ϑ

and add the equality ft = f 1t = f 2

t .Then, in recursive form:

ft = η − 1

η(1 − τw)

(w∗

t

)1−ηλtw

ηt ld

t + βθwEtγL

t+1

(�

χwt

�t+1

)1−η (w∗

t+1

w∗t

)η−1

ft+1

and

ft =dtϕtψ

(w∗

t

wt

)−η(1+ϑ) (ldt

)1+ϑ + βθwEtγL

t+1

(�

χwt

�t+1

)−η(1+ϑ) (w∗

t+1

w∗t

)η(1+ϑ)

ft+1

Later, by solving for the dynamics of ft , we will be able to compute w∗t .

Finally, in a symmetric equilibrium and in every period, a fraction 1− θw of house-holds set w∗

t as their wage, while the remaining fraction θw partially index their priceto past inflation. Consequently, the real wage index evolves:

w1−ηt = θw

(�

χw

t−1

�t

)1−η

w1−ηt−1 + (1 − θw) w

∗1−ηt .

that is, as a geometric average of past real wage and the new optimal wage. Thisstructure is a direct consequence of the memoryless characteristic of Calvo pricing.

3.2 The distribution sector

The distribution sector is composed of two segments. At the end, there is a consump-tion good producer and an investment good producer, while at the source, a finaldomestic good producer aggregates all domestic intermediate goods to produce thefinal domestic good.

3.2.1 Final consumption and investment good producers

At the top of the distribution chain, there is a perfectly competitive consumption goodproducer and investment good producer who pack domestic consumption and invest-ment

(cd

t , idt

)with imported consumption and investment baskets

(cM

t , i Mt

)to generate

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SERIEs (2010) 1:175–243 187

final consumption and investment (ct , it ) using a technology described by:

ct =[(nc) 1

εc

(cd

t

) εc−1εc + (1 − nc) 1

εc

(cM

t

(1 − �c

t

)) εc−1εc

] εcεc−1

it =[(

ni) 1

εi(

idt

) εi −1εi +

(1 − ni

) 1εi(

i Mt

(1 − �i

t

)) εi −1εi

] εiεi −1

where there is a home bias in the aggregation, measured by nc and ni , which deter-mines the steady state degree of openness, and where εc (εi ) represents the elasticityof substitution between imported and domestic consumption (investment) goods. Inaddition, we assume that it is costly to change the share of imports of consumptionand investment in final production. This is modelled by adding a cost term (�c

t and �it )

to changing the import to consumption (investment) ratio in the production function:

�st = �s

2

(s M

t

st

/s M

t−1

st−1− 1

)2

for s = c, i

The producer of the final consumption good maximizes profits subject to the pro-duction function, taking as given the price of the final domestic good, of the importedconsumption goods (in domestic currency) pt , pM

t , and of the final consumption bas-ket pc

t . Due to adjustment costs, the problem becomes dynamic and the aggregatordiscounts future income with the pricing kernel βγ L

t+1λt+1λt

(below, when talking aboutdiscounting by intermediate good producers, we explain this point in more detail). Inthe case of consumption (and similarly for investment):

maxcd

t ,cMt

Et

∞∑

τ=0

βτ γ Lτ

λt+τ

λt

[pc

t ct − pt cdt − pM

t cMt

]

s.t. ct =[(nc) 1

εc

(cd

t

) εc−1εc + (1 − nc) 1

εc

(cM

t

(1 − �c

t

)) εc−1εc

] εcεc−1

Solving (and after some tedious algebra), we get the demand for the domestic andimported consumption good and the price of the final consumption good:

cdt = nc

(pt

pct

)−εc

ct

cMt = Et�

ct+1

(1 − nc)

(pM

t

pct

)−εc

ct

pct =

[nc (pt )

1−εc + Et�ct+1

(1 − nc) (pM

t

)1−εc] 1

1−εc

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188 SERIEs (2010) 1:175–243

where

Et�ct+1 =

[1−β (1 − nc)

1εc Et

γ Lt+1λt+1

λt

(pc

t

pMt

)�c

t+1

(ct+1

cMt+1

(1−�c

t+1

)) 1

εc�c ′

t+1

(�cM

t+1

)2

�ct+1

]−εc

(1 − �c

t) [

1−�ct −�c ′

t

(�cM

t�ct

)]−εc

3.2.2 Final domestic good producer

At the start of the distribution chain, we have the final domestic good producer thatproduces the final domestic good (yt ) by aggregating intermediate domestic goods(yi t ) with the following production function:

yt =⎛

⎝1∫

0

(yi t )ε−1ε di

⎞

⎠

εε−1

. (3)

where ε is the elasticity of substitution across domestic intermediate goods and all thevariables are expressed in per capita terms.

The final domestic good producer is perfectly competitive and maximizes prof-its subject to the production function (3), taking as given all intermediate domesticgoods prices pit and the final domestic good price pt . Following the same steps as forwages, we find the input demand functions and the aggregate price associated withthis problem:

yi t =(

pit

pt

)−ε

ydt ∀i,

pt =⎛

⎝1∫

0

p1−εi t di

⎞

⎠

11−ε

.

where ydt is the aggregate demand by the final good producer.

3.3 Intermediate good producers

At the bottom of the domestic production process, there is a continuum of intermedi-ate goods producers. Each intermediate good producer i has access to a technologyrepresented by a production function (expressed in per capita terms):

yi t = At kαi t−1

(ldi t

)1−α − φzt

where kit−1 is the capital rented by the firm, ldi t is the amount of the “packed” labor

input rented by the firm, and where At , the neutral technology level, follows the

123

SERIEs (2010) 1:175–243 189

process:

At = At−1 exp(A + z A,t

)where z A,t = σAεA,t and εA,t ∼ N (0, 1)

which induces a third unit root in the model, the second from technology. The parame-

ter φ, which corresponds to the fixed cost of production, and zt = A1

1−αt µ

α1−αt guarantee

that economic profits are roughly equal to zero in the steady state. We rule out theentry and exit of intermediate good producers. Long-run growth of domestic output

in per capita terms is determined by zt = A1

1−αt µ

α1−αt , which evolves as:

zt = zt−1 exp(z + zz,t

)where zz,t = z A,t + αzµ,t

1 − αand z = A + αµ

1 − α.

Intermediate goods producers solve a two-stage problem. In the first stage, takingthe input prices wt and rt as given, firms rent ld

i t and kit−1 in perfectly competitivefactor markets to minimize real cost:

minldi t ,kit−1

wt ldi t + rt kit−1

s.t. yi t ={

At kαi t−1

(ldi t

)1−α − φzt if At kαi t−1

(ldi t

)1−α ≥ φzt

0 otherwise

Assuming an interior solution, the intermediate good producers equate marginal pro-ductivity to input prices to get an optimal capital–labor ratio

kit−1

ldi t

= α

1 − α

wt

rt

and a real marginal cost:

mct =(

1

1 − α

)1−α ( 1

α

)αw1−α

t rαt

At

Note that both the optimal capital–labor ratio and the marginal cost do not depend oni : all firms receive the same technology shocks and all firms rent inputs at the sameprice.

In the second stage, intermediate good producers choose the price that maximizesdiscounted real profits. We assume they are under the same pricing scheme as house-holds. In each period, a fraction 1 − θp of firms can change their prices. All otherfirms can only index their prices to past inflation of the final domestic good price(�t = pt

pt−1

). Indexation is controlled by the parameter χ ∈ [0, 1], where χ = 0 is

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190 SERIEs (2010) 1:175–243

no indexation and χ = 1 is total indexation. The problem of the firms is then:

maxpit

Et

∞∑

τ=0

(βθp

)τγ Lτ

λt+τ

λt

{(τ∏

s=1

�χt+s−1

pit

pt+τ

− mct+τ

)yi t+τ

}

s.t. yi t+τ =(

τ∏

s=1

�χt+s−1

pit

pt+τ

)−ε

yt+τ

where the valuation of future profits is done with the common ratio of Lagrangian mul-tipliers λt+τ /λt (treated as exogenous by the firm). Since we have complete marketsin securities, this ratio is indeed the correct valuation on future profits.5 The solutionp∗

i t implies the first-order condition:

Et

∞∑

τ=0

(βθp

)τγ Lτ λt+τ

⎧⎨

⎩

⎛

⎝(1 − ε)

(τ∏

s=1

�χt+s−1

�t+s

)1−εp∗

t

pt

+ ε

(τ∏

s=1

�χt+s−1

�t+s

)−ε

mct+τ

)yt+τ

⎫⎪⎬

⎪⎭= 0 (4)

where we have dropped irrelevant constants and used the fact that we are in a sym-metric equilibrium. This expression nests the usual result in the fully flexible pricescase (θp = 0):

p∗t = ε

ε − 1pt mct+τ

that is, the price is equal to a mark-up over the nominal marginal cost.To express Eq. (4) recursively, we define g1

t and g2t :

g1t = λt mctyt + βθpEtγ

Lt+1

(�

χt

�t+1

)−ε

g1t+1

g2t = λt�

∗t yt + βθpEtγ

Lt+1

(�

χt

�t+1

)1−ε(

�∗t

�∗t+1

)g2

t+1

where �∗t = p∗

tpt

, and we get that (4) is equivalent to εg1t = (ε − 1)g2

t .Given Calvo’s pricing, the price index evolves:

(pt )1−ε = θp

(�

χt−1

)1−ε(pt−1)

1−ε + (1 − θp) (

p∗t

)1−ε

5 As in the household problem, we need(βθp

)τγ Lτ λt+τ /λt to go to zero sufficiently fast in relation to

the rate of inflation for the optimization problem to be well defined.

123

SERIEs (2010) 1:175–243 191

or, dividing by (pt−1)1−ε,

1 = θp

(�

χt−1

�t

)1−ε

+ (1 − θp) (

�∗t

)1−ε

3.4 Foreign sector

To better describe the foreign sector, we start by explaining the composition of theimport and export markets separately and then proceed to show how both types offirms set prices.

Importing firms

The import sector is composed of two segments. At the end, a distributor (oraggregator) mixes differentiated imported goods

(yM

it

)to produce the final imported

basket(yM

t

), while at the source, a continuum of importing firms buy the foreign

homogeneous final good in the international markets at price pWt and turn it into a

differentiated import good through a differentiating technology or brand naming.The distributor (or aggregator) produces the final imported good

(yM

t

)from differ-

entiated imported goods(yM

it

)with the following production function:

yMt =

⎛

⎝1∫

0

(yM

it

) εM −1εM di

⎞

⎠

εMεM −1

.

where εM is the elasticity of substitution across foreign final goods and all the variablesare expressed in per capita terms. Following the same steps as for domestic prices, wefind the price of the imported final basket and the import demand functions:

yMit =

(pM

it

pMt

)−εM

yMt ∀i,

pMt =

⎛

⎝1∫

0

(pM

it

)1−εMdi

⎞

⎠

11−εM

The total per capita amount of imported goods is thus given by:

Mt =1∫

0

yMit di

Exporting firms

The export sector consists of a continuum of firms that buy the final domestic goodand differentiate it by brand naming. Then, they sell the continuum of differentiatedgoods to importers from the rest of the world. Each exporting firm faces the following

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192 SERIEs (2010) 1:175–243

demand for its products:

yxi t =

(px

it

pxt

)−εx

yxt ∀i,

where both prices are expressed in the foreign currency of the export market. Theexport deflator is

pxt =

⎛

⎝1∫

0

(px

it

)1−εx di

⎞

⎠

11−εx

Therefore, the total amount of exported good is given by:

xt =1∫

0

yxi t di

Finally, appealing to symmetry, we assume that the world demand of our exports is:

yxt =

(px

t

pWt

)−εW

yWt .

and we have that:

yxi t =

(px

it

pxt

)−εx(

pxt

pWt

)−εW

yWt ∀i.

The evolution of world demand is exogenously given by:

yWt =

(yW)(1−ρyW

) (yW

t−1

)ρyWe

(σyW εyW ,t

)

and world inflation by:

�Wt =

(�W

)(1−ρ�W) (

�Wt−1

)ρ�W

e

(σ

�W ε�W ,t

)

Price-setting in the foreign sector

To allow for incomplete exchange rate pass-through to import and export prices, weassume that importing and exporting firms in the foreign sector face price sticki-ness à la Calvo. Since the problem faced by both types of firms is similar, we willdescribe them together. In particular, in each period, a fraction 1 − θM (1 − θX )

123

SERIEs (2010) 1:175–243 193

of importing (exporting) firms can change their prices. All other importing (export-ing) firms can only index their prices to past inflation of the final imported (for-

eign) good

(�M

t = pMt

pMt−1

(�W

t = pWt

pWt−1

)). Indexation is controlled by the parameter

χM , χX ∈ [0, 1], where χM , χX = 0 is no indexation and χM , χX = 1 is totalindexation.

Since importing (exporting) firms buy the homogeneous foreign (domestic) goodat price pW

t (pt ) in the world (domestic) market, their real marginal cost, in domestic

(foreign) currency terms, is equal to mcMt = pW

t ext

pMt

(mcx

t = ptext px

t

).

The problem of importing (exporting) firm i is then:

maxp f m

it

Et

∞∑

τ=0

(βθ f m

)τγ Lτ

λt+τ

λt

{(τ∏

s=1

(�

f mt+s−1

)χ f m p f mit

p f mt+τ

− mc f mt+τ

)y

f mit+τ

}

s.t. y f mit+τ =

(τ∏

s=1

(�

f mt+s−1

)χ f m p f mit

p f mt+τ

)−ε f m

yf m

t+τ for f m = M, x

Proceeding as in the case of domestic prices we get:

gM1t =

⎡

⎢⎣λt

ext pWt

pMt

yMt +

βθMEtγL

t+1

((�M

t)χM

�Mt+1

)−εM

gM1t+1

⎤

⎥⎦

gM2t =

⎡

⎣λt�M∗t yM

t + βθMEtγL

t+1

((�M

t

)χM

�Mt+1

)1−εM(

�M∗t

�M∗t+1

)gM2

t+1

⎤

⎦

gx1t =

⎡

⎣λt

ptext px

tyx

t +βθxEtγ

Lt+1

((�W

t)χx

�xt+1

)−εx

gx1t+1

⎤

⎦

gx2t = λt�

x∗t yx

t + βθxEtγL

t+1

((�W

t

)χx

�xt+1

)1−εx(

�x∗t

�x∗t+1

)gx2

t+1

εMgM1t = (εM − 1)gM2

t ; εxgX1t = (εx − 1)gX2

t

where �M∗t = pM∗

t

pMt

and �x∗t = px∗

tpx

t.

Given Calvo’s pricing, the import and export price indices evolve as:

1 = θM

((�M

t−1

)χM

�Mt

)1−εM

+ (1 − θM )(�M∗

t

)1−εM

1 = θx

((�W

t−1

)χx

�xt

)1−εx

+ (1 − θx )(�x∗

t

)1−εx

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194 SERIEs (2010) 1:175–243

Evolution of net foreign assets

To close the foreign sector, we have to determine the evolution of net foreign assets.The balance of payments, including the premium on foreign holdings, evolves asfollows:

1∫

0

ext bWjt d j = RW

t−1�(

ext bWt−1, ξ

bW

t−1

)ext

1

γ Lt

1∫

0

bWjt−1d j + ext px

t yxt − ext pW

t Mt

where we have used the fact that:

1∫

0

pxity

xi t di =

1∫

0

pxit

(px

it

pxt

)−εx

yxt di = (px

t

)εx yxt

1∫

0

(px

it

)1−εx di = pxt y

xt

3.5 The monetary authority

Monetary policy is controlled by the ECB, which sets the nominal interest rates forthe euro area (Rt ) according to the Taylor rule:

Rt

R=(

Rt−1

R

)γR

⎛

⎜⎝(

�E At

�E A

)γ�

⎛

⎜⎝γ L E A

tyE A

t

yE At−1

exp(L E A + yE A

)

⎞

⎟⎠

γy⎞

⎟⎠

1−γR

exp(ξm

t

)

through open market operations, where �E A represents the euro area target level ofinflation, R euro area steady state nominal gross return of capital, and yd the euroarea steady state gross growth rate of yE A

t . The term ξmt is a random shock to monetary

policy that follows ξmt = σmεmt where εmt ∼ N (0, 1). The presence of the previous

period interest rate, Rt−1, is justified because we want to match the smooth profile ofthe interest rate over time observed in the data. Note that R is beyond the control ofthe monetary authority, since it is equal to the steady state real gross return of capitalplus the target level of inflation.

The Spanish economy contributes to the euro area inflation and output according toits relative size. Ideally, we would like to account for how shocks to the Spanish econ-omy affect euro area variables and through those, to Rt . Unfortunately, in practice, itis difficult to solve a DSGE model taking Spain’s behavior as implied by the modeland the rest of the euro data as given. This is because, given the small weight of Spainin the euro area aggregate (10%), the indeterminacy region of such a model is so largethat the model becomes nearly useless. One way to solve this problem is to build amodel of a two-country small open monetary area, like BEMOD (Andrés et al. 2006).However, we avoid this route because we fear it would make us lose focus.

Instead, we adopt two alternative approaches depending on the objective of theexercise. In the first approach, we assume the domestic economy has an independentlocal monetary policy that sets the nominal interest rate. This is equivalent to setting

123

SERIEs (2010) 1:175–243 195

a weight of 1 for Spain’s aggregates in the Taylor rule above. This would correspondto the time before the euro area was set up and we use it when estimating the modelover the whole sample period (1986–2007). We also employ it when doing a his-torical decomposition of shocks and in some counterfactual exercises. In the secondapproach, we assume Spain is too small to have a significant influence on the ECB’sTaylor rule. Thus, changes in Spanish conditions do not affect Rt , which is determinedby the Taylor rule above, evaluated at the observed (or forecasted) values of euro areavariables. This is the approach we use when estimating over the most recent period(1997–2007) or when we do policy analysis related to the current situation.

3.6 The government problem

The per capita government budget constraint is:

bt = gt zt

ydt

+ Tt

ydt

+mt−1pt−1

γ Lt yd

t �t+ Rt−1bt−1

γ Lt �ty

dt

−(

rtu j t − µ−1t δ)

τkkt−1

ydt

− τwwtldt

ydt

− τcpc

t

pt

ct

ydt

−mtpt

ydt

where we have redefined the level of outstanding debt as a proportion of nominal

output as bt =∫ 1

0 b jt d j

ptydt

.6 The level of real government consumption appears multi-

plied by zt to keep it a stationary share of output, which is exogenous and determinedaccording to a stochastic process:

log gt = (1 − ρg)

log g + ρg log gt−1 + σgεg,t where εg,t ∼ N (0, 1),

Fiscal policy must be designed to prevent the level of debt from exploding. Since alltax rates are assumed to be constant, we assume that lump-sum taxes as a proportionof output in per capita terms ( Tt

yt) respond sufficiently to prevent deviations of the level

of debt as a proportion of output (bt ) from target

(b =

(b

pyd

)):

Tt

ydt

= T0 − T1

(bt − b

)

3.7 Aggregation

To close the model, we need aggregation conditions for each of the markets consid-ered: goods, labor, import, and export markets. In the case of the goods market, we

6 One can show that the government budget constraint is correct by inserting it into the households’ budgetconstraint (evaluated at the aggregate level), which implies that all of the tax terms except gt cancel out.

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196 SERIEs (2010) 1:175–243

start from the expression for per capita aggregate demand of the domestic final good:

ydt = cd

t + idt + gt zt + µ−1

t � [ut ] kt−1 + xt ,

Then, we use the equation determining the demand for each intermediate producer’s

good

(yi t =

(pitpt

)−ε

ydt

)and the production function

(yi t = At kα

i t−1

(ldi t

)1−α)

, and

integrating over all firms, we get the aggregate condition of the goods market:

nc(

pt

pct

)−εc

ct + ni(

pt

pit

)−εi

it + gt zt + µ−1t � [ut ] kt−1 + xt

= At (ut kt−1)α(ldt

)1−α − φzt

vpt

where vpt = ∫ 1

0

(pitpt

)−ε

di measures the impact of the price distribution on output.

To get this result, we have used the fact that all of the intermediate good producershave the same optimal capital-labor ratio and that by market clearing:

ldt =

1∫

0

ldi t di

ut kt−1 =1∫

0

kit−1di

To close the labor market, we start with the demand for household’s labor services(ls

j t =(

w j twt

)−η

ldt

)and integrate over all households to get the condition equating

(per capita) aggregate labor demand (ldt ) and supply (lt )

ldt = 1

vwt

lt .

where vwt = ∫ 1

0

(w j twt

)−η

d j measures the impact of the wage distribution on employ-ment.

In the case of the imported goods market, using the definition of aggregate imports(Mt = ∫ 1

0 yMit di) and the fact that its production is distributed to households as

imported consumption and investment (yMt = cM

t + i Mt ), we get:

Mt = vMt

[Et�

ct+1

(1 − nc)

(pM

t

pct

)−εc

ct + Et�it+1

(1 − ni

)( pMt

pit

)−εi

it

]

123

SERIEs (2010) 1:175–243 197

where vMt = ∫ 1

0

(pM

it

pMt

)−εM

di measures the impact of price dispersion in the output

of this sector and

Et�st+1 =

[1−β (1−ns)

1εs Et

γ Lt+1λt+1

λt

(ps

t

pMt

)�s

t+1

(st+1

sMt+1

(1−�s

t+1

)) 1

εs�s ′

t+1

(�sM

t+1

)2

�st+1

]−εs

(1−�s

t) [

1−�st −�s ′

t

(�sM

t�st

)]−εs

for s = c, i .In the exported goods market, using the definition of exports (xt = ∫ 1

0 yxi t di) and

the demand functions for their output

yxi t =

(px

it

pxt

)−εx(

pxt

pWt

)−εW

yWt

we have that aggregate exports are equal to

xt = vxt y

xt = vx

t

(px

t

pWt

)−εW

yWt

where vxt = ∫ 1

0

(px

itpx

t

)−εxdi .

Finally, by the properties of price indices under Calvo’s pricing mechanism, pricedispersions evolve according to:

vpt = θp

(�

χt−1

�t

)−ε

vpt−1 + (1 − θp

) (�

∗t

)−ε

vwt = θw

(wt−1

wt

�χw

t−1

�t

)−η

vwt−1 + (1 − θw)

(�∗w

t

)−η

vMt = θM

((�M

t−1

)χM

�Mt

)−εM

vMt−1 + (1 − θM )

(�M∗

t

)−εM

vxt = θx

((�W

t−1

)χx

�xt

)−εx

vxt−1 + (1 − θx )

(�x∗

t

)−εx.

4 Equilibrium and model solution

The definition of equilibrium in this economy is standard and we omit it in the interestof space. Since there is growth in the model induced by technological change, most of

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198 SERIEs (2010) 1:175–243

the variables are growing on average along the equilibrium path.7 Before we can solvethe model, we need to rescale all the relevant variables to obtain a system of stationaryvariables. Hence, we define ct = ct

zt, λt = λt zt , rt = rtµt , qt = qtµt , xt = xt

zt,

wt = wtzt

, w∗t = w∗

tzt

, kt = ktzt µt

, mt = mtzt

, ydt = yd

tzt

, and the growth rates zt = ztzt−1

,

At = AtAt−1

, µt = µtµt−1

, L t = LtLt−1

. In addition, we express all the relative prices in

terms of the vector(

pct

pt,

pit

pt,

pMtpt

,ext px

tpt

and ext pWt

pt

). To solve the model, we find the

steady state and log-linearize the equilibrium conditions around it. For completeness,the full set of non-linear and log-linearized equilibrium conditions is included in theappendix.

4.1 The steady state

Now, we find the deterministic steady state of the model. We know several of its prop-

erties. First, the law of one price must hold,(

ex pW

p

)= 1. Second, the exchange rate

is assumed to be constant (�ex = 1), which means that the domestic nominal interest

rate is equal to the world nominal interest rate(

R = RW = �zβ

), and the net foreign

asset position is assumed to be equal to zero (expressed in domestic currency), so

that nominal exports equal nominal imports(

x(

ex px

p

)= vx

(ex pW

p

)M)

. Third, let

z = exp (z), µ = exp(µ

), A = exp (A) , and γ L = exp (L) . Also, given the

definition of c, xt , wt , w∗t , and yd

t , we have that c = i = w = w∗ = yd = z ,and u = d = ϕ = 1 and g = g.

In addition, we need to choose functional forms for all of the adjustment costfunctions in the model: � [·], S [·], �s [·] and �

(b∗). For � [u] , we pick: � [ut ] =

�1 (ut − 1) + �22 (ut − 1)2. We select �1 as a function of the other parameters of the

model to normalize u = 1 in steady state. Therefore, � [1] = 0 and �′ [1] = �1.The investment adjustment cost function is expressed in terms of quadratic deviationswith respect to the average growth of investment:

S

[γ L

tit

it−1zt

]= S

[it

it−1

]= κ

2

(γ L

tit

it−1zt − i

)2

.

Then, along the balanced growth path, S[γ L z

] = S [i ] = S′ [i ] = 0. Finally, theimports adjustment cost function along the balanced growth path is

�st = �s

2

((s M

t

st

)/(s M

t−1

st−1

)− 1

)2

,

7 Population growth does not appear explicitly in the equilibrium since the variables are already expressedin per capita terms.

123

SERIEs (2010) 1:175–243 199

thus �s = �s ′ = 0. With respect to the adjustment cost of the premium for holdingforeign assets, we assume:

�(

ext b∗t , ξb∗

t

)= e

(−�b∗

(ext b∗t −exb∗)+ξb∗

t

)

.

Since in the steady state the domestic and world interest rates are the same, R =R∗�

(exb∗, 0

), we have �

(exb∗, 0

) = e0 = 1 and �′ (exb∗) = −�b∗e0 = −�b∗

.

Therefore, using these results and the equilibrium conditions we can simplify thesteady state to the following set of equations determining ld , while all the rest of thevariables are recursive to these:

k =(

α

1 − α

w

rzµ

)ld = �ld .

yd =Az kα

(ld)1−α − φ

v p

i =(

γ L − (1 − δ)

µz

)k

c = 1

nc

(pc

p

)−εc

⎡

⎣yd − ni(

pi

p

)εi

i − vx

(ex pW

p

)

(ex px

p

) M − g

⎤

⎦

M = vM

⎡

⎢⎣(1 − nc)

⎡

⎣

(pM

p

)

(pc

p

)

⎤

⎦−εc

c +(

1 − ni)⎡

⎣

(pM

p

)

(pi

p

)

⎤

⎦−εi

i

⎤

⎥⎦

λ =(

z − hβγ L

z − h

)1

c (1 + τc)pc

p

1 − βθw z η(1+ϑ)�η(1−χw)(1+ϑ)γ L

1 − βθw z η−1�−(1−χw)(1−η)γ L=

ψ(

w∗w

)−ηϑ (ld)ϑ

η−1η

(1 − τw) w∗λ

Or alternatively, after some algebra, we have the following equation on ld :

1 − βθw z η(1+ϑ)�η(1−χw)(1+ϑ)γ L

1 − βθw z η−1�−(1−χw)(1−η)γ L=

ψ(

w∗w

)−ηϑ (ld)ϑ

η−1η

(1 − τw) w∗

(z − h

z − hβγ L

)

∗{

(1 + τc)

c

(pc

p

)1−εc[[

Az �α (v p)−1 − i

(pi

p

)εi(

γ L−(1−δ)µz

)�]

ld

−φ (v p)−1 − g

]}

This is a non-linear equation that we solve for ld with a root finder. Note that w∗, w,(pc

p

),(

pi

p

), and v p are functions of parameters of the model, � and �M are param-

eters to be estimated, and �W and yW are exogenously given.

123

200 SERIEs (2010) 1:175–243

1990 1995 2000 2005

0

0.01

0.02

0.03

picobs

1990 1995 2000 2005−0.02

0

0.02

gyobs

1990 1995 2000 2005

−0.010

0.010.02

gcobs

1990 1995 2000 2005−0.04−0.02

00.020.040.06

gwhobs

1990 1995 2000 2005−0.02

0

0.02

gldhobs

1990 1995 2000 20050

0.02

0.04

Rnobs

1990 1995 2000 2005−2

0

2x 10

−3 gLobs

1990 1995 2000 2005

−0.02

−0.01

0

0.01gyWobs

1990 1995 2000 2005

−0.1

0

0.1

piWobs

Fig. 1 Data series used in the estimation

5 Estimating the model

As motivated in the introduction, we will confront our model with the data usingBayesian methods. Formally, we stack all the parameters in the model in the vector� ∈ � and we elicit a prior distribution for them, p(�). The model implies a like-lihood p(Y T |�) given some observed data, Y T = {y1, . . . , yT }. Then, we have theposterior distribution of �:

p(�|Y T

)∝ p

(Y T |�

)p(�),

where “∝” indicates proportionality. The posterior summarizes the uncertainty regard-ing the parameter values and it can be used for point estimation once we have specifieda loss function. For example, under a quadratic loss, our point estimates will be themean of the posterior. Since the posterior is also difficult to characterize, we generatedraws from it using a Metropolis-Hastings algorithm. We use the resulting empiricaldistribution to obtain point estimates, standard deviations, etc.

5.1 Data

We use time series for 9 variables to estimate MEDEA (see Fig. 1): real GDP growth(gyobs), real private consumption growth (gcobs), total employment in hours growth(gldhobs), real compensation per hour growth (total compensation/total hours/GDPdeflator) (gwhobs), consumption deflator inflation (picobs), total population over 16years of age growth (gLobs), euro area nominal interest rate (Rnobs), inflation ofSpanish competitors prices (piWobs), and Spain’s foreign demand growth (gyWobs).All the time series are taken from national accounts published by INE, except for theforeign-sector variables and the nominal interest rate, which come from the database

123

SERIEs (2010) 1:175–243 201

developed for the REMS model (BDREMS).8 We have excluded real investment (or theinvestment deflator) from the baseline estimation since it has grown in the last decadeat an unprecedented pace, mainly due to the construction sector, which we believewould be difficult to explain using a model without housing and financial frictions(see Andrés and Arce 2008, for a theoretical model tackling this issue). Neverthe-less, we check the robustness of this baseline estimation by adding real investment,investment deflator inflation, or public consumption.

The choice of the sample period over which to estimate the parameters of themodel is controversial. There have been significant changes in the Spanish econ-omy since the mid-nineties, mainly related to the set-up of the euro area but alsoto the increase in labor force participation and the large immigration flows. Somepapers in the literature have thus decided to use only the period since the euro areawas conceived, that is, from 1997 onward. In this way, these papers avoid havingto deal with structural breaks in the sample and with the change in the implemen-tation of monetary policy. However, since it is likely that the impact of the creationof the monetary union lasted for several years after 1997, it is not certain that thestructural break problem will disappear. Moreover, this will not avoid the other struc-tural changes such as immigration. The main drawback of this approach is, how-ever, that the sample becomes fairly short, probably requiring tighter priors in theestimation.

Instead, in this paper we have decided to proceed in three stages. First, we use datafor the full sample 1986–2007, as if Spain had an independent monetary policy duringthis period. This allows us to set fairly loose priors and let the data speak up as muchas possible. We had to drop the data before 1986 because the changes in the structureof the Spanish economy in the early eighties were too substantial. Second, we checkthe stability of the point estimates by estimating the model separately for two subs-amples but maintaining the assumption of an independent monetary authority: onefor the period before the euro area was set up (1986–1996) and the other from 1997onward. Third, the model is re-estimated over the most recent subsample assumingSpain has no independent monetary policy; that is, the interest rate is exogenous andthe exchange rate is constant.

The model incorporates economic growth. Therefore, to take the model to the data,it is not necessary to transform our observables. Instead, we add transition equations toour state space representation relating model and empirical variables. These transitionequations account for the following differences. First, in the log-linearized version ofthe model, all variables are expressed as log deviations with respect to their steady statevalue. Second, all variables in the model are expressed in per capita terms dividing bythe population (Lt ). Third, some real variables in the model are made stationary bydividing by zt . Therefore, in the case of real per capita variables, like real GDP per

capita, the growth rate of the empirical variable(yd,O

t

)is equal to the growth rate of

8 The BDREMS database is described in Boscá et al. (2008) and is available for open download at: http://www.sgpg.pap.meh.es/SGPG/Cln_Principal/Presupuestos/Documentacion/BasedatosmodeloREMS.htm.

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202 SERIEs (2010) 1:175–243

the model variable (in per capita terms)(yd

t

)plus population growth (γ L

t ):

log yd,Ot = � log yt + γ L

t

But the variable included in the stationary log-linearized version of the model is yt ,which has been made stationary by dividing by technology and expressed as a devi-ation with respect to the steady state (yt = log yt

zt− log y

z ). The same point applies

to the growth rate of technology ( zt = log ztzt−1

− log z ). Considering all of this, thetransition equation for real per capita variables, such as output, is

log yd,Ot = � log yt + γ L

t = �yt +zt + γ Lt + log z + γ L

An exception is employment in hours, which is stationary in per capita terms in themodel (ld

t = log ldt − log l), so we only have to add population growth:

log ld,Ot = � log ld

t + γ Lt = �ld

t + γ Lt + γ L

In the case of nominal variables, such as inflation and interest rates, model and empir-ical variables are the same, so we express them as deviation with respect to the steadystate:

log �c,Ot = �c

t + log �c

log ROt = Rt + log R

5.2 Calibration and prior distributions

The model has 59 parameters, 12 of which are calibrated and the remaining 47 esti-mated. The calibration is shown in Table 1. Several theoretical and empirical reasonsexplain why one may not want to estimate all the parameters of the model. First, someparameters are difficult to identify with the model structure, such as the discount factorβ. This parameter is set to be consistent with an annualized nominal interest rate of2.5% and an inflation objective of 2%, so that the steady state annual nominal interest

rate(

R = RW = �zβ

)is 4.5%. Second, other parameters such as the depreciation rate

δ or the labor share α, are better estimated using micro data, while others would requireadding more data series to the estimation, such as the three tax rates (τc, τw, τk). Third,there are parameters that are irrelevant for the model solution, such as the coefficientof money demand in the utility function, υ. Finally, the parameters of the Taylor ruleare set equal to the standard estimation results for the euro area (Christoffel et al.2008). The two fiscal parameters have not yet been included in the estimation and thusare set to their empirical values.

The first vertical panel of Table 2 summarizes our assumptions regarding prior dis-tributions for the estimated parameters. Our approach has been to set priors as looseas possible. Therefore, for most parameters, we have chosen as our priors uniform

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SERIEs (2010) 1:175–243 203

Table 1 Calibrated parameters

Value Reason Value Reason

β 0.99 Difficult to identify γy 0.125 Taylor rule U.E.

υ 0.1 Irrelevant τc 0.113 Data on taxes

δ 0.0175 Micro data τw 0.341 Data on taxes

α 0.3621 Micro data τk 0.219 Data on taxes

γR 0.8 Taylor rule U.E. g 0.17 –

γ� 1.7 Taylor rule U.E. b 0.40 –

distributions with a range covering all the theoretically feasible values. In particu-lar, we have set a range of (0, 1) , for the labor supply coefficient, price and wageCalvo and indexation parameters, adjustment cost parameters, autoregressive coef-ficients (except the labor supply shock for which we have chosen (0, 0.9)), and thestandard deviations of shocks. In the case of the elasticity of substitution parameters,we have set a range of (6, 10). In the case of the habits coefficient and the home biascoefficients, we have imposed stronger priors by assuming beta distributions, becausethe data moved them toward unrealistic parameter values. Finally, we also set betadistributions for the parameters determining growth in the model, to help identifythem.9

5.3 Estimation results

The right-hand panel of Table 2 presents the estimation results for the full sample(1986Q1–2007Q4). Table 3 presents the results for the two subsamples considered(1986Q1–1996Q4 and 1997Q1–2007Q4) as well as the model without independentmonetary policy. The columns of both tables report the mean, mode, median, standarddeviation, and the 5th and 95th percentiles of the posterior distribution of the param-eters. All of them are computed using a Metropolis-Hastings algorithm in Dynare,based on a Markov chain with 5 million draws, with the first 2.5 million draws beingdiscarded as burn-in draws, and the appropriate acceptance ratio (Roberts et al. 1997).

We start by studying the goodness of the estimation. We have implemented stan-dard convergence diagnostic tests, which show that the draws of the posterior samplingconverged for all of the estimated parameters (see Mengersen et al. 1999). Details areavailable upon request. Moreover, the smoothed estimates of the innovation compo-nent of structural shocks (see Fig. 2) are clearly stationary. Note that the variance ofthe shocks seems to have fallen in the second part of the sample, with the exception ofthe population growth shock, which has increased, in line with the rise in populationgrowth in Spain over the last decade.

Another way to check the quality of the estimation is by comparing the prior andposterior distributions of each parameter, as shown by Figs. 3, 4, 5, 6, 7. In general,

9 Unfortunately, for some parameters (in particular, those related to the open sector of the economy),uniform priors seem to bring insufficient information for some empirical exercises. We need a more thor-ough assessment of the robustness of the empirical estimates with respect to priors.

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204 SERIEs (2010) 1:175–243

Table 2 Prior and posterior distribution of structural parameters [full sample (86q1–07q4)]

Parameter Prior distribution Posterior distribution

Type Mean s.d. Mean Mode s.d. Median 5% 95%

Preferences

Habits h Beta 0.70 0.10 0.847 0.795 0.010 0.847 0.831 0.864

Labour supply coef. ψ Uniform 0.50 0.29 6.772 6.744 0.059 6.792 6.673 6.847

Frisch elasticity ϑ Uniform 1.55 0.84 1.835 1.970 0.110 1.840 1.648 2.003

Adjustment costs

Investment κ Uniform 25.0 14 28.954 28.995 0.042 28.960 28.887 29.016

Fixed cost of production � Uniform 0.50 0.29 0.127 0.051 0.058 0.127 0.034 0.223

Capital utilization �2 Uniform 1.00 0.58 0.248 0.461 0.017 0.246 0.219 0.273

Risk premium �bW Uniform 0.50 0.29 0.832 0.859 0.064 0.827 0.742 0.959

Import consumption �c Uniform 0.50 0.29 0.449 0.259 0.153 0.440 0.211 0.692

Import investment �i Uniform 0.50 0.29 0.618 0.647 0.046 0.629 0.538 0.691

Elasticities of substitution

Domestic goods ε Uniform 8.00 1.15 8.577 8.480 0.132 8.575 8.396 8.800

Import goods εM Uniform 8.00 1.15 8.787 8.727 0.064 8.778 8.690 8.892

Export goods εX Uniform 8.00 1.15 9.491 9.437 0.101 9.498 9.321 9.622

World goods εW Uniform 8.00 1.15 6.791 7.300 0.064 6.785 6.689 6.900

Consumption goods εc Uniform 8.00 1.15 7.512 7.671 0.044 7.517 7.441 7.585

Investment goods εi Uniform 8.00 1.15 7.851 8.056 0.207 7.867 7.595 8.108

Labour types η Uniform 8.00 1.15 7.758 7.706 0.057 7.754 7.670 7.865

Price and wage and price setting

Calvo dom. prices θp Uniform 0.50 0.29 0.898 0.904 0.001 0.898 0.897 0.900

Calvo exp. Prices θX Uniform 0.50 0.29 0.330 0.327 0.032 0.335 0.272 0.378

Calvo imp. θM Uniform 0.50 0.29 0.064 0.050 0.053 0.046 0.000 0.147

Calvo wages θw Uniform 0.50 0.29 0.457 0.235 0.025 0.457 0.417 0.501

Indexation dom. prices χp Uniform 0.50 0.29 0.004 0.003 0.004 0.003 0.000 0.008

Indexation imp. Prices χM Uniform 0.50 0.29 0.064 0.287 0.047 0.055 0.000 0.148

Indexation exp. Prices χX Uniform 0.50 0.29 0.027 0.013 0.022 0.020 0.000 0.062

Indexation wages χw Uniform 0.50 0.29 0.961 0.967 0.031 0.969 0.919 1.000

Fiscal policy

Fiscal rule coeff. T1 Uniform 0.05 0.03 0.051 0.051 0.028 0.047 0.015 0.100

Home bias

In consumption nc Beta 0.70 0.10 0.962 0.813 0.0127 0.962 0.943 0.982

In investment ni Beta 0.50 0.20 0.072 0.100 0.0129 0.071 0.053 0.094

Growth rates

Inv. especific tech. µ Beta 0.004 0.001 0.004 0.003 0.001 0.004 0.002 0.006

General technology A Beta 0.004 0.001 0.004 0.002 0.001 0.004 0.002 0.006

Population γ L Beta 0.004 0.001 0.003 0.003 0.001 0.003 0.002 0.004

Autorregressive coefficients of shocks

Intertemp. preferences ρd Uniform 0.50 0.29 0.978 0.900 0.006 0.978 0.969 0.989

Hours preferences ρψ Uniform 0.45 0.26 0.895 0.800 0.005 0.897 0.889 0.900

Public consumption ρg Uniform 0.50 0.29 0.979 0.978 0.011 0.982 0.967 0.990

Foreign prices ρπw Uniform 0.50 0.29 0.361 0.366 0.040 0.363 0.285 0.422

Foreign demand ρyW Uniform 0.50 0.29 0.033 0.459 0.029 0.025 0.000 0.070

World interest rate ρRW Uniform 0.50 0.29 0.876 0.963 0.042 0.869 0.802 0.957

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SERIEs (2010) 1:175–243 205

Table 2 continued

Parameter Prior distribution Posterior distribution

Type Mean s.d. Mean Mode s.d. Median 5% 95%

Standard deviations of shocks

Inv. especific tech. σµ Uniform 0.50 0.29 0.403 0.300 0.037 0.400 0.347 0.463

General technology σA Uniform 0.50 0.29 0.009 0.012 0.001 0.009 0.008 0.010

Population σL Uniform 0.50 0.29 0.001 0.001 0.0001 0.001 0.001 0.001

Intertemp. preferences σd Uniform 0.50 0.29 0.174 0.933 0.046 0.161 0.109 0.250

Hours preferences σψ Uniform 0.50 0.29 0.266 0.137 0.028 0.262 0.223 0.313

Public consumption σg Uniform 0.50 0.29 0.062 0.076 0.009 0.062 0.047 0.076

Interest rate σR Uniform 0.50 0.29 0.003 0.003 0.0002 0.003 0.003 0.004

Foreign prices σπw Uniform 0.50 0.29 0.044 0.044 0.004 0.044 0.038 0.049

Foreign demand σyW Uniform 0.50 0.29 0.145 0.150 0.013 0.144 0.124 0.165

World interest rate σRW Uniform 0.50 0.29 0.005 0.001 0.003 0.005 0.000 0.009

Table 3 Prior and posterior distribution of structural parameters (subsamples)

Parameter 1986Q1–1996Q4 1997Q1–2007Q4 1997Q1–2007Q4 (ex MP)

Mean 5% 95% Mean 5% 95% Mean 5% 95%

Preferences

Habits h 0.861 0.836 0.886 0.827 0.796 0.863 0.886 0.851 0.911

Labour supply coef. ψ 7.005 6.653 7.336 6.982 6.815 7.168 6.642 6.541 6.736

Frisch elasticity ϑ 1.909 1.675 2.093 1.775 1.564 1.940 2.274 2.072 2.437

Adjustment costs

Investment κ 29.722 29.477 29.997 29.052 28.946 29.138 28.964 28.873 29.067

Fixed cost of production � 0.760 0.626 0.894 0.209 0.097 0.338 0.024 0.000 0.058

Capital utilization �2 0.244 0.208 0.278 0.394 0.335 0.454 0.687 0.637 0.749

Risk premium �bW 0.827 0.689 0.987 0.865 0.692 1.000 0.000 0.000 0.001

Import consumption �c 0.036 0.000 0.081 0.501 0.172 0.860 0.436 0.249 0.533

Import investment �i 0.503 0.293 0.635 0.758 0.644 0.874 0.889 0.828 0.972

Elasticities of substitution

Domestic goods ε 7.494 6.904 7.901 8.618 8.353 8.809 8.947 8.771 9.088

Import goods εM 8.690 8.446 8.946 8.741 8.618 8.857 8.356 8.291 8.426

Export goods εX 9.553 9.374 9.727 9.591 9.440 9.774 9.177 9.126 9.216

World goods εW 6.676 6.354 6.950 7.067 6.886 7.183 6.649 6.597 6.691

Consumption goods εc 8.736 8.425 8.945 7.874 7.765 8.046 7.958 7.871 8.038

Investment goods εi 7.266 7.130 7.435 8.371 8.199 8.499 8.282 8.210 8.327

labour types η 7.784 7.565 8.101 7.860 7.768 7.960 8.032 7.982 8.097

Price and wage and price setting

Calvo dom. prices θp 0.901 0.898 0.905 0.895 0.893 0.897 0.529 0.490 0.573

Calvo exp. Prices θX 0.241 0.130 0.365 0.148 0.053 0.259 0.592 0.486 0.673

Calvo imp. Prices θM 0.077 0.000 0.150 0.297 0.125 0.455 0.285 0.254 0.339

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206 SERIEs (2010) 1:175–243

Table 3 continued

Parameter 1986Q1–1996Q4 1997Q1–2007Q4 1997Q1–2007Q4 (ex MP)

Mean 5% 95% Mean 5% 95% Mean 5% 95%

Calvo wages θw 0.418 0.328 0.538 0.527 0.486 0.573 0.030 0.000 0.075

Indexation dom. prices χp 0.006 0.000 0.014 0.007 0.000 0.016 0.031 0.000 0.081

Indexation imp. Prices χM 0.344 0.232 0.433 0.201 0.089 0.336 0.028 0.000 0.052

Indexation exp. Prices χX 0.096 0.000 0.196 0.058 0.000 0.126 0.180 0.095 0.250

Indexation wages χw 0.767 0.592 0.953 0.891 0.741 1.000 0.954 0.921 1.000

Fiscal policy

Fiscal rule coeff. T1 0.043 0.000 0.084 0.052 0.009 0.100 0.051 0.017 0.091

Home bias

In consumption nc 0.940 0.905 0.972 0.884 0.842 0.919 0.860 0.809 0.903

In investment ni 0.109 0.073 0.145 0.148 0.112 0.197 0.088 0.032 0.140

Growth rates

Inv. especific tech. µ 0.004 0.002 0.006 0.004 0.002 0.006 0.004 0.002 0.006

General technology A 0.004 0.002 0.006 0.004 0.002 0.006 0.004 0.002 0.006

Population γ L 0.003 0.002 0.004 0.003 0.002 0.004 0.003 0.002 0.004

Autorregressive coefficients of shocks

Intertemp. preferences ρd 0.954 0.934 0.973 0.973 0.958 0.990 0.883 0.816 0.990

Hours preferences ρψ 0.875 0.847 0.900 0.888 0.873 0.900 0.861 0.831 0.891

Public consumption ρg 0.623 0.523 0.707 0.899 0.817 0.990 0.974 0.961 0.990

Foreign prices ρπw 0.278 0.175 0.398 0.170 0.064 0.306 0.030 0.000 0.051

Foreign demand ρyW 0.129 0.015 0.239 0.059 0.000 0.125 0.025 0.000 0.059

World interest rate ρRW 0.900 0.823 0.990 0.739 0.548 0.990 0.954 0.925 0.990

Standard deviations of shocks

Inv. especific tech. σµ 0.662 0.575 0.764 0.185 0.147 0.222 0.179 0.100 0.268

General technology σA 0.011 0.009 0.013 0.003 0.003 0.004 0.012 0.009 0.014

Population σL 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.001 0.002

Intertemp. preferences σd 0.109 0.079 0.138 0.093 0.052 0.137 0.101 0.054 0.222

Hours preferences σψ 0.326 0.208 0.505 0.131 0.097 0.166 0.072 0.054 0.088

Public consumption σg 0.057 0.006 0.092 0.037 0.029 0.044 0.598 0.558 0.636

Interest rate σR 0.004 0.003 0.005 0.002 0.002 0.002 0.143 0.074 0.240

Foreign prices σπw 0.049 0.040 0.057 0.037 0.031 0.044 0.037 0.031 0.044

Foreign demand σyW 0.221 0.176 0.259 0.081 0.065 0.097 0.160 0.116 0.196

World interest rate σRW 0.010 0.000 0.016 0.001 0.000 0.002 0.001 0.001 0.001

the results show that the data are very informative about the posterior distribution ofthe parameters. An exception is the elasticity of substitution of investment goods (εi ),with a twin-peaked posterior distribution, although both peaks imply fairly similarestimates. More relevant is the fact that the data contain little information on the pos-terior distribution of the steady state growth rate of technology and population (theposterior lies on top of the beta prior), the coefficient of the fiscal rule (T1), and theadjustment cost parameter of imported consumption (�c). In the case of the growthrates, we have set fairly tight priors centered on the sample means of observed growth

123

SERIEs (2010) 1:175–243 207

1990 1995 2000 2005

−0.02

0

0.02

0.04

epsA

1990 1995 2000 2005

−1

0

1epsd

1990 1995 2000 2005

−0.2

0

0.2

epsg

1990 1995 2000 2005−2

0

2x 10

−3 epsL

1990 1995 2000 2005−5

05

1015

x 10−3 epsm

1990 1995 2000 2005

−0.5

0

0.5

epsmu

1990 1995 2000 2005

−0.5

0

0.5

epsphi

1990 1995 2000 2005−0.1

0

0.1

epspiW

1990 1995 2000 2005−0.01

0

0.01epsRWn

1990 1995 2000 2005−0.4

−0.2

0

0.2

epsyW

Fig. 2 Smoothed estimates of innovations

rates of population and output per capita. Given this result, one should be cautiouswhen making inferences about the relative importance in the observed data of the driftsin technological growth (neutral and investment specific) included in the model.

Moving to the point estimates, a number of findings are worth noting. First, theestimates of the utility parameters are quite standard. The data strongly support a highestimate of habit persistence, which is not surprising given the persistence of observedconsumption, while fixed costs of production are very close to zero. The Frisch elas-ticity posterior mean of 1.83 is in line with most estimations for other countries andrather plausible once we think about both the intensive and the extensive margin oflabor supply.

Second, the estimated elasticities of substitution between different types of inter-mediate goods produced are relatively similar, implying a mark-up between 13.5% inthe case of domestic goods and 12% in the case of export goods, while the wage mark-up is somewhat higher, at around 15%. This is not surprising given the rigidities ofthe Spanish labor market, where wages are mainly set by insiders with long-termcontracts and thus high bargaining power. The estimates are also similar for thedemand elasticity of substitution between imported and domestically produced goods.In contrast, the adjustment cost parameter associated with changing the import con-tent varies substantially across both types of goods. We estimate that the adjustmentcost is much higher for investment. Moreover, the data are very informative aboutthis. This is evidence of the technological constraints that the Spanish economy stillfaces in areas such as advanced capital goods or IT, which require a large importcontent.

123

208 SERIEs (2010) 1:175–243

0 0.2 0.4 0.6 0.80

200

400

600SE_epsA

0 0.2 0.4 0.6 0.80

5

10

SE_epsd

0 0.2 0.4 0.6 0.80

20

40

SE_epsg

0 0.2 0.4 0.6 0.80

2000

4000

SE_epsL

0 0.2 0.4 0.6 0.80

500

1000

1500

SE_epsm

0 0.2 0.4 0.6 0.80

5

10

SE_epsmu

0 0.2 0.4 0.6 0.80

5

10

15

SE_epsphi

0 0.2 0.4 0.6 0.80

50

100

SE_epspiW

0 0.2 0.4 0.6 0.80

50

100

SE_epsRWn

Fig. 3 Prior and posterior distributions of the structural parameters

0 0.2 0.4 0.6 0.80

10

20

30

SE_epsyW

0.4 0.6 0.80

10

20

30

40

h

0 2 4 6 80

5

10

psi

0 20 400

5

10

kappa

0 0.2 0.4 0.6 0.80

2

4

6

8

phi

0 0.5 1 1.50

10

20

30phi2

0.5 1 1.5 2 2.50

1

2

3

4

vatheta

6 7 8 90

2

4

6

epsilon

6 8 10 12 14 16 180

2

4

6

8

epsilonM

Fig. 4 Prior and posterior distributions of the structural parameters (continued)

123

SERIEs (2010) 1:175–243 209

6 7 8 90

1

2

3

4

epsilonx

6 7 8 90

5

10

epsilonW

6 7 8 90

5

10

epsilonc

6 7 8 90

2

4

epsiloni

6 7 8 90

2

4

6

8

eta

0 0.2 0.4 0.6 0.80

100

200

300

400

thetap

0 0.2 0.4 0.6 0.80

5

10

15

thetax

0 0.2 0.4 0.6 0.80

5

10

15thetaM

0 0.2 0.4 0.6 0.80

5

10

15

thetaw

Fig. 5 Prior and posterior distributions of the structural parameters (continued)

0 0.2 0.4 0.6 0.80

100

200

chi

0 0.2 0.4 0.6 0.80

5

10

chiM

0 0.2 0.4 0.6 0.80

10

20

chix

0 0.5 10

5

10

15

20

chiw

0 0.05 0.10

5

10

15

T1

0.4 0.6 0.8 10

10

20

30

nc

0 0.2 0.4 0.6 0.80

10

20

30

ni

0 0.5 10

2

4

6

8

gammabW

0 0.2 0.4 0.6 0.80

1

2

3

gammac

Fig. 6 Prior and posterior distributions of the structural parameters (continued)

123

210 SERIEs (2010) 1:175–243

0 0.2 0.4 0.6 0.80

5

10

15gammai

0 5 10

x 10−3

0

100

200

300

400

lambdamu

0 5 10

x 10−3

0

100

200

300

400

lambdaA

0 2 4 6

x 10−3

0

200

400

600

800

lambdaL

0 0.5 10

20

40

60

rhod

0 0.2 0.4 0.6 0.80

50

100

150

rhophi

0 0.5 10

20

40

60

rhog

0 0.2 0.4 0.6 0.80

5

10

rhopiW

0 0.2 0.4 0.6 0.80

5

10

15

20

rhoyW

0 0.5 10

5

10

15

rhoRWn

Fig. 7 Prior and posterior distributions of the structural parameters (continued)

Third, on the nominal side, we find important differences across sectors of theeconomy. The estimate of the Calvo parameter is very high for intermediate domesticgoods, although not different from values generally obtained for the euro area (Smetsand Wouters 2003), but quite low for the import and export intermediate goods. Thesame is true for wages. In contrast, indexation is very close to one for wages,whichseems the direct consequence of the (ex-ante) indexation mechanism inherent inSpanish wage agreements. However, indexation is non-existent in prices of domesti-cally produced goods, while indexation is a bit higher in the import and export sectors.These differences across sectors of production and the labor market are strongly sup-ported in the data.

Fourth, the estimation confirms the evidence in input-output tables that in Spainthere is a much stronger home bias in consumption than investment (remember ourearlier comment about the technological constraints faced by our economy). However,the point estimates are too large given the micro evidence. Nevertheless, the data arevery informative about this result, since imposing tighter priors does not change the

123

SERIEs (2010) 1:175–243 211

results, even when data on real investment or on the investment deflator are used (seeTable 5).

Finally, the point estimates for the autoregressive parameters of shock processesshow that domestic shocks are very persistent, especially those related to demand,public consumption, and preferences. This may suggest that the model has some dif-ficulty endogenously generating the level of persistence present in the data and, thus,it opts for these exogenous shocks to be highly persistent. Alternatively, one couldargue that this is the consequence of a structural break in the data, but this hypothesisis rejected when the estimation is performed recursively over the final sample (seeTable 4). In comparison, the foreign demand and inflation shocks have much lowerpersistence.

5.4 Subsample analysis and robustness

Comparing the results across the two subsamples (see Table 3), we observe that thepoint estimates are rather similar for most parameters. This suggests that our fearsabout pervasive structural breaks over the most recent years may have been exagger-ated. The recursive estimation in Table 4 confirms this impression. The exception isthe standard deviation of shocks, which have all fallen markedly, except for the pop-ulation shock, which has increased. This was already noticeable in the graph of theinnovations for the whole sample period. This seems to be another manifestation ofthe “great moderation” that the western economies experienced from 1984 to 2007(McConnell and Pérez-Quirós 2000; Stock and Watson 2003).

In addition, the estimate of the adjustment cost of the import content of consumptionand investment goods is larger for the most recent sample. Finally, the estimated elas-ticities of substitution suggest a more competitive economy since 1997, with slightlylower steady-state mark-ups, especially for domestic goods (13 vs. 15%), and moreflexible prices, while wages have become stickier and more persistent.

When the model is estimated assuming an exogenous monetary policy (see thirdpanel of Table 3), the estimates of open economy and monetary policy parameterschange markedly. In particular, the mark-up on imports, exports, and world goodsrises and the premium on foreign interest rate falls, while domestic prices becomemore flexible and competitive. Moreover, since the interest rate does not react toSpanish economic conditions, the standard deviation of the Taylor rule shock risesgreatly. This suggests that more informative priors for these parameters may help themodel to deliver a more consistent performance.

Finally, several robustness checks have been performed. First, a recursive estima-tion (adding two years every time) over the most recent period (see Table 4) confirmsthat there are very few signs of structural instability in our sample, since most parame-ters change little and gradually, with the exception of the fixed cost of production andthe indexation of import prices. Second, adding real investment to the estimation hasa significant impact on the point estimates (see the second panel of Table 5), reducingmost steady-state mark-ups (except for domestic prices, which increase) and increasingprice indexation parameters. This is not surprising given that investment has grownvery quickly during the last economic cycle, especially due to housing investment.

123

212 SERIEs (2010) 1:175–243

Table 4 Prior and posterior distribution of structural parameters (recursive estimation)

Parameter Prior distribution Mean of posterior distribution

Type Mean s.d. 86–96 86–98 86–00 86–02 86–04 86–06 86–07

PreferencesHabits h Beta 0.70 0.10 0.861 0.848 0.850 0.854 0.845 0.847 0.847

Labour supply coef. ψ Uniform 0.50 0.29 7.005 6.684 6.671 6.603 6.827 6.859 6.772

Frisch elasticity ϑ Uniform 1.55 0.84 1.909 2.040 1.927 2.017 1.826 1.857 1.835

Adjustment costs

Investment K Uniform 25.0 14 29.722 28.949 28.984 28.950 28.857 28.911 28.954

Fixed cost of production � Uniform 0.50 0.29 0.760 0.039 0.039 0.048 0.120 0.096 0.127

Capital utilization �2 Uniform 1.00 0.58 0.244 0.238 0.243 0.243 0.244 0.290 0.248

Risk premium �bW Uniform 0.50 0.29 0.827 0.871 0.962 0.894 0.974 0.825 0.832

Import consumption �c Uniform 0.50 0.29 0.036 0.415 0.330 0.517 0.133 0.259 0.449

Import investment �i Uniform 0.50 0.29 0.503 0.618 0.794 0.867 0.654 0.621 0.618

Elasticities of substitution

Domestic goods ε Uniform 8.00 1.15 7.494 8.302 8.527 8.627 8.435 8.505 8.577

Import goods εM Uniform 8.00 1.15 8.690 8.696 8.674 8.751 8.518 8.592 8.787

Export goods εX Uniform 8.00 1.15 9.553 9.339 9.257 9.293 9.386 9.394 9.491

World goods εW Uniform 8.00 1.15 6.676 7.107 7.390 7.318 7.026 7.094 6.791

Consumption goods εc Uniform 8.00 1.15 8.736 7.668 7.818 7.660 7.722 7.782 7.512

Investment goods εi Uniform 8.00 1.15 7.266 7.799 8.001 7.847 8.215 8.254 7.851

Labour types η Uniform 8.00 1.15 7.784 7.800 7.820 7.767 7.688 7.563 7.758

Price and wage and price setting

Calvo dom. prices θp Uniform 0.50 0.29 0.901 0.900 0.900 0.899 0.900 0.900 0.898

Calvo exp. Prices θX Uniform 0.50 0.29 0.241 0.305 0.330 0.289 0.367 0.320 0.330

Calvo imp. Prices θM Uniform 0.50 0.29 0.077 0.040 0.186 0.135 0.156 0.129 0.064

Calvo wages θw Uniform 0.50 0.29 0.418 0.404 0.365 0.415 0.431 0.410 0.457

Indexation dom. prices Xp Uniform 0.50 0.29 0.006 0.004 0.005 0.005 0.003 0.003 0.004

Indexation imp. Prices XM Uniform 0.50 0.29 0.344 0.278 0.063 0.167 0.348 0.187 0.064

Indexation exp. Prices XX Uniform 0.50 0.29 0.096 0.036 0.039 0.020 0.035 0.041 0.027

Indexation wages Xw Uniform 0.50 0.29 0.767 0.927 0.905 0.972 0.932 0.965 0.961

Fiscal policy

Fiscal rule coeff. T1 Uniform 0.05 0.03 0.043 0.048 0.060 0.050 0.046 0.056 0.051

Home bias

In consumption nc Beta 0.70 0.10 0.940 0.975 0.975 0.965 0.971 0.970 0.962

In investment ni Beta 0.50 0.20 0.109 0.065 0.054 0.069 0.066 0.053 0.072

Growth rates

Inv. especific tech. µ Beta 0.004 0.001 0.004 0.004 0.004 0.004 0.004 0.004 0.004

General technology A Beta 0.004 0.001 0.004 0.004 0.004 0.004 0.004 0.004 0.004

Population yL Beta 0.004 0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003

Autorregressive coefficients of shocks

Intertemp. preferences ρd Uniform 0.50 0.29 0.954 0.989 0.989 0.987 0.989 0.986 0.978

Hours preferences ρψ Uniform 0.45 0.26 0.875 0.896 0.895 0.893 0.896 0.897 0.895

123

SERIEs (2010) 1:175–243 213

Table 4 continued

Parameter Prior distribution Mean of posterior distribution

Type Mean s.d. 86–96 86–98 86–00 86–02 86–04 86–06 86–07

Public consumption ρg Uniform 0.50 0.29 0.623 0.920 0.973 0.964 0.938 0.976 0.979

Foreign prices ρTTw Uniform 0.50 0.29 0.278 0.341 0.380 0.324 0.422 0.366 0.361

Foreign demand ρyW Uniform 0.50 0.29 0.129 0.111 0.035 0.038 0.094 0.043 0.033

World interest rate ρRW Uniform 0.50 0.29 0.900 0.869 0.963 0.954 0.864 0.925 0.876

Standard deviations of shocks

Inv. especific tech. σµ Uniform 0.50 0.29 0.662 0.415 0.428 0.446 0.398 0.376 0.403

General technology σ A Uniform 0.50 0.29 0.011 0.009 0.009 0.010 0.009 0.009 0.009

Population σ L Uniform 0.50 0.29 0.001 0.001 0.001 0.001 0.001 0.001 0.001

Intertemp. preferences σd Uniform 0.50 0.29 0.109 0.433 0.503 0.376 0.355 0.343 0.174

Hours preferences σψ Uniform 0.50 0.29 0.326 0.246 0.200 0.285 0.239 0.206 0.266

Public consumption σg Uniform 0.50 0.29 0.057 0.064 0.067 0.069 0.061 0.063 0.062

Interest rate σR Uniform 0.50 0.29 0.004 0.003 0.003 0.003 0.003 0.003 0.003

Foreign prices σTTw Uniform 0.50 0.29 0.049 0.046 0.047 0.046 0.048 0.044 0.044

Foreign demand σyW Uniform 0.50 0.29 0.221 0.164 0.160 0.164 0.168 0.144 0.145

World interest rate σRW Uniform 0.50 0.29 0.010 0.006 0.006 0.007 0.006 0.005 0.005

However, our model is not able (and it was not designed) to account for the boomin housing investment. Third, adding the investment deflator or public consumptionaffects the estimation results only marginally (see the last two panels of Table 5).

6 Applications

In this section, we consider a number of properties and applications of our model toillustrate the contributions that MEDEA can make to policy analysis. First, we brieflydescribe the basic properties of the model. In the interest of space, we offer onlysome concise information. Second, we show how the model can help in understandingthe evolution of the Spanish economy over the last several decades by interpretinghistorical developments through the lens of equilibrium theory. Many of the answersthat MEDEA will give us are not surprising and either have been suggested beforeby the literature or fit into our economic intuition (although it is comforting to havea quantitative corroboration), but others will be relatively new. Third, we evaluatethe impact and dynamics after a change in some relevant steady state parameters.Finally, we illustrate how MEDEA can be used to conduct alternative scenarios forobserved variables. This exercise is particularly important for the assessment of policyinterventions by the government and the monetary authority.

6.1 Model properties

Table 6 reports the steady state ratios implied by our point estimates. The ratios arecomparable to the ones observed in the data, and in the case of ratios for which the

123

214 SERIEs (2010) 1:175–243

Tabl

e5

Prio

ran

dpo

ster

ior

dist

ribu

tion

ofst

ruct

ural

para

met

ers

(add

ing

obse

rvab

les)

Para

met

erB

asel

ine

Add

real

inve

stm

ent

Add

inve

stm

entd

eflat

orA

ddpu

blic

cons

umpt

ion

Mea

n5%

95%

Mea

n5%

95%

Mea

n5%

95%

Mea

n5%

95%

Pref

eren

ces

Hab

itsh

0.84

70.

831

0.86

40.

926

0.91

20.

940

0.85

70.

826

0.89

10.

820

0.80

60.

831

Lab

our

supp

lyco

ef.

ψ6.

772

6.67

36.

847

6.74

76.

561

6.87

66.

753

6.57

76.

856

6.72

46.

695

6.75

1

Fris

chel

astic

ityϑ

1.83

51.

648

2.00

32.

571

2.29

42.

755

1.79

91.

663

1.95

81.

928

1.90

51.

950

Adj

ustm

entc

osts

Inve

stm

ent

K28

.954

28.8

8729

.016

28.7

8028

.565

28.9

8928

.817

28.7

6228

.893

28.9

9228

.977

29.0

06

Fixe

dco

stof

prod

uctio

n�

0.12

70.

034

0.22

30.

220

0.02

60.

346

0.08

80.

009

0.16

10.

016

0.00

00.

036

Cap

italu

tiliz

atio

n�

20.

248

0.21

90.

273

0.34

40.

285

0.40

60.

233

0.20

70.

260

0.47

10.

460

0.48

5

Ris

kpr

emiu

m�

bW0.

832

0.74

20.

959

0.76

30.

572

1.00

00.

249

0.01

00.

489

0.87

50.

867

0.88

4

Impo

rtco

nsum

ptio

n�

c0.

449

0.21

10.

692

0.48

20.

321

0.63

80.

012

0.00

00.

026

0.25

70.

245

0.26

4

Impo

rtin

vest

men

t�

i0.

618

0.53

80.

691

0.51

80.

342

0.72

60.

782

0.62

51.

000

0.69

50.

676

0.71

5

Ela

stic

ities

ofsu

bstit

utio

n

Dom

estic

good

sε

8.57

78.

396

8.80

07.

771

7.66

87.

888

8.68

68.

538

8.80

28.

488

8.47

98.

497

Impo

rtgo

ods

εM

8.78

78.

690

8.89

29.

207

9.13

49.

315

8.54

78.

463

8.63

98.

774

8.76

38.

784

Exp

ortg

oods

εX

9.49

19.

321

9.62

29.

716

9.40

410

.000

9.46

69.

291

9.63

39.

413

9.39

59.

437

Wor

ldgo

ods

εW

6.79

16.

689

6.90

07.

572

7.46

97.

669

6.17

46.

000

6.42

27.

264

7.25

37.

278

Con

sum

ptio

ngo

ods

εc

7.51

27.

441

7.58

57.

776

7.65

87.

891

7.76

07.

709

7.81

87.

679

7.66

87.

691

Inve

stm

entg

oods

εi

7.85

17.

595

8.10

88.

740

8.50

49.

065

7.79

57.

526

7.97

58.

057

8.04

78.

065

Lab

our

type

sη

7.75

87.

670

7.86

57.

517

7.41

57.

615

7.67

37.

419

7.88

37.

692

7.68

17.

703

Pric

ean

dw

age

and

pric

ese

tting

Cal

vodo

m.p

rice

sθ

p0.

898

0.89

70.

900

0.87

80.

869

0.88

70.

899

0.89

70.

900

0.90

20.

901

0.90

3

Cal

voex

p.Pr

ices

θX

0.33

00.

272

0.37

80.

081

0.00

00.

147

0.55

20.

477

0.63

80.

342

0.32

80.

355

Cal

voim

p.Pr

ices

θM

0.06

40.

000

0.14

70.

543

0.20

30.

806

0.04

20.

000

0.08

30.

083

0.04

80.

102

123

SERIEs (2010) 1:175–243 215

Tabl

e5

cont

inue

d

Para

met

erB

asel

ine

Add

real

inve

stm

ent

Add

inve

stm

entd

eflat

orA

ddpu

blic

cons

umpt

ion

Mea

n5%

95%

Mea

n5%

95%

Mea

n5%

95%

Mea

n5%

95%

Cal

vow

ages

θw

0.45

70.

417

0.50

10.

355

0.27

10.

445

0.44

10.

375

0.51

60.

304

0.29

60.

313

Inde

xatio

ndo

m.p

rice

sχ

p0.

004

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00.

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000

0.00

4

Inde

xatio

nim

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ices

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40.

000

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80.

584

0.44

50.

683

0.13

50.

000

0.21

30.

325

0.31

40.

336

Inde

xatio

nex

p.Pr

ices

χX

0.02

70.

000

0.06

20.

295

0.18

90.

373

0.06

10.

000

0.15

70.

006

0.00

00.

012

Inde

xatio

nw

ages

χw

0.96

10.

919

1.00

00.

973

0.94

11.

000

0.94

40.

884

1.00

00.

988

0.98

00.

998

Fisc

alpo

licy

Fisc

alru

leco

eff.

T1

0.05

10.

015

0.10

00.

053

0.00

70.

095

0.05

70.

014

0.10

00.

009

0.00

00.

019

Hom

ebi

as

Inco

nsum

ptio

nnc

0.96

20.

943

0.98

20.

928

0.87

70.

975

0.87

50.

827

0.91

50.

847

0.83

20.

859

Inin

vest

men

tni

0.07

20.

053

0.09

40.

036

0.01

10.

057

0.01

70.

006

0.02

90.

010

0.00

40.

020

Gro

wth

rate

s

Inv.

espe

cific

tech

.

µ0.

004

0.00

20.

006

0.00

40.

002

0.00

60.

004

0.00

20.

006

0.00

30.

001

0.00

4

Gen

eral

tech

nolo

gy

A0.

004

0.00

20.

006

0.00

40.

002

0.00

60.

004

0.00

20.

006

0.00

20.

001

0.00

4

Popu

latio

nyl

0.00

30.

002

0.00

40.

003

0.00

20.

004

0.00

30.

002

0.00

40.

003

0.00

20.

004

Aut

orre

gres

sive

coef

ficie

nts

ofsh

ocks

Inte

rtem

p.pr

efer

ence

sρ

d0.

978

0.96

90.

989

0.98

40.

977

0.99

00.

972

0.95

30.

989

0.99

00.

989

0.99

0

Hou

rspr

efer

ence

sρψ

0.89

50.

889

0.90

00.

810

0.72

20.

900

0.89

40.

885

0.90

00.

899

0.89

80.

900

Publ

icco

nsum

ptio

nρ

g0.

979

0.96

70.

990

0.97

90.

970

0.98

70.

736

0.69

20.

783

0.84

90.

813

0.87

3

Fore

ign

pric

esρ

TTw

0.36

10.

285

0.42

20.

072

0.00

00.

141

0.29

00.

146

0.45

30.

349

0.33

90.

361

Fore

ign

dem

and

ρyW

0.03

30.

000

0.07

00.

288

0.16

90.

388

0.04

70.

000

0.08

00.

440

0.42

80.

450

Wor

ldin

tere

stra

teρ

RW

0.87

60.

802

0.95

70.

918

0.86

90.

967

0.98

80.

986

0.99

00.

987

0.98

40.

990

123

216 SERIEs (2010) 1:175–243

Tabl

e5

cont

inue

d

Para

met

erB

asel

ine

Add

real

inve

stm

ent

Add

inve

stm

entd

eflat

orA

ddpu

blic

cons

umpt

ion

Mea

n5%

95%

Mea

n5%

95%

Mea

n5%

95%

Mea

n5%

95%

Stan

dard

devi

atio

nsof

shoc

ks

Inv.

espe

cific

tech

.σµ

0.40

30.

347

0.46

30.

322

0.27

80.

361

0.30

20.

266

0.34

90.

304

0.29

70.

312

Gen

eral

tech

nolo

gyσ

A0.

009

0.00

80.

010

0.00

80.

007

0.01

00.

008

0.00

70.

009

0.01

10.

009

0.01

2

Popu

latio

nσ

L0.

001

0.00

10.

001

0.00

10.

001

0.00

10.

001

0.00

10.

001

0.00

10.

001

0.00

1

Inte

rtem

p.pr

efer

ence

sσ

d0.

174

0.10

90.

250

0.23

00.

151

0.30

40.

161

0.08

40.

246

0.89

90.

862

0.92

2

Hou

rspr

efer

ence

sσψ

0.26

60.

223

0.31

30.

251

0.16

40.

361

0.24

80.

174

0.34

00.

159

0.15

20.

169

Publ

icco

nsum

ptio

nσ

g0.

062

0.04

70.

076

0.81

70.

737

0.89

20.

454

0.38

20.

514

0.18

00.

165

0.19

2

Inte

rest

rate

σR

0.00

30.

003

0.00

40.

003

0.00

30.

004

0.00

30.

003

0.00

30.

003

0.00

30.

004

Fore

ign

pric

esσ

TTw

0.04

40.

038

0.04

90.

042

0.03

60.

047

0.04

30.

038

0.04

90.

043

0.03

70.

049

Fore

ign

dem

and

σyW

0.14

50.

124

0.16

50.

148

0.12

90.

168

0.12

50.

107

0.14

30.

159

0.14

60.

170

Wor

ldin

tere

stra

teσ

RW

0.00

50.

000

0.00

90.

100

0.07

50.

130

0.05

30.

007

0.10

10.

016

0.01

40.

018

123

SERIEs (2010) 1:175–243 217

Table 6 Steady state ratiosModel Data Model Data

k/y 10.1 – T/y 0.24 –

c/y 0.62 0.59 R 1.01 –

i/y 0.29 0.25 r 0.03 –

m/y 0.33 0.30 q 1.12 –

x/y 0.29 0.24

5 10 15 20

1

2

3x 10

−3 gyobs

5 10 15 20

5

10

15

x 10−4 gcobs

5 10 15 202468

101214

x 10−4 giobs

5 10 15 20−10

−5

0

x 10−3 gldhobs

5 10 15 20−1

0

1x 10

−3 gwhobs

5 10 15 20

1

2

3

x 10−3 gMobs

5 10 15 20

0

2

4x 10

−3 gxobs

5 10 15 20−4

−2

0

x 10−4 picobs

5 10 15 20−3

−2

−1

0x 10

−3 piMobs

5 10 15 20−2

−1

0

1

x 10−4 Rnobs

5 10 15 20−3

−2

−1

x 10−3 rer

IRF shock epsA

Fig. 8 Impulse response function to a neutral technology shock

data are less precise (such as the capital/output ratio), we have values that are com-parable to the numbers usually employed in the literature. In addition, in the class ofDSGE models to which MEDEA belongs, small changes in steady state ratios haveonly second-order effects on the dynamics of aggregate variables.

Figures 8, 9, 10, 11, 12, 13 show the impulse response functions (IRFs) to someof the stochastic shocks of the model, as well as the 5 and 95% confidence bandsimplied by the posterior distribution of parameters. Since there is growth in the modeldue to technological progress and the increase in population, the real variables in oursolution are expressed in per capita efficiency units. In some cases, mainly supplyside shocks, the behavior of variables defined in this way may seem confusing. Thus,we show instead in the simulations the growth rates of real variables expressed in the

123

218 SERIEs (2010) 1:175–243

5 10 15 20

0

0.02

0.04

gyobs

5 10 15 20−15−10

−50

x 10−3 gcobs

5 10 15 20

0.0050.01

0.0150.02

0.025

giobs

5 10 15 20−0.02

00.020.040.060.08

gldhobs

5 10 15 20

−5

0

5x 10

−3 gwhobs

5 10 15 20

0.010.020.030.040.05

gMobs

5 10 15 20

−0.020

0.020.040.060.08

gxobs

5 10 15 20

6

8

x 10−3 picobs

5 10 15 20−0.03−0.02−0.01

00.01

piMobs

5 10 15 20

5

10

x 10−3 Rnobs

5 10 15 20

−0.04

−0.02

rer

IRF shock epsmu

Fig. 9 Impulse response function to an investment-specific technology shock

same units as in the data. That is, for example gyobs is equal to total real GDP growth.In all cases, we show a one-standard-deviation shock to the corresponding innovation.In all Figs. 8, 9, 10, 11, 12, 13, the order of variables (from left to right and from topto bottom) is output growth, consumption growth, investment growth, hours growth,wage per hour growth, imports growth, exports growth, consumption deflator, importsdeflator, nominal interest rate, and real interest rate.

Figure 8 reports the IRFs for a neutral technology shock. In a rather standard way,output, consumption, and investment respond positively to the shock. Hours fall atimpact (with sticky prices and wages, the demand for total labor services is rigid inthe short run and since, thanks to the technological shocks, we need less labor toproduce the same output, firms hire fewer workers), but they recover in the secondquarter and become positive. Prices and the nominal interest rate go down becauseof higher productivity (marginal cost falls and the monetary authority is less worriedabout inflation).

Figure 9 plots the IRFs for an investment-specific technological shock. Here, invest-ment goes up rapidly, but since the economy is not more productive in the short run,it has to do so at the expense of lower consumption and longer hours. Note that theimpact on hours is the opposite of that in the previous case: now it goes up and thenfalls. Our model, thus, reproduces the insights of Fisher (2006), who emphasizes thatthe response of hours to technological shocks depends on the specifics of the techno-

123

SERIEs (2010) 1:175–243 219

5 10 15 2002468

x 10−4 gyobs

5 10 15 200

5

10

x 10−4 gcobs

5 10 15 202

4

6

8

x 10−5 giobs

5 10 15 200

2

4

6

x 10−4 gldhobs

5 10 15 20

−2

−1

0

x 10−4 gwhobs

5 10 15 20

1

2

3x 10

−4 gMobs

5 10 15 20

0

1

2

x 10−4 gxobs

5 10 15 20

−3

−2

−1

x 10−5 picobs

5 10 15 20

−2

−1

0x 10

−4 piMobs

5 10 15 20−2

−1

0

1x 10

−5 Rnobs

5 10 15 20

−2−1.5

−1−0.5

x 10−4 rer

IRF shock epsL

Fig. 10 Impulse response function to a population growth shock

logical process assumed by the model. Imports rise because we want to invest moreand exports increase at impact (to later fall) because the investment-specific techno-logical shock makes the national investment product relatively cheap in the worldmarket. Consumption prices increase because fewer resources are concentrated in itsproduction.

The IRFs to a population growth shock are drawn in Fig. 10. Output, investment,and imports grow (we have more workers and we need more capital for them). Interest-ingly, the consumption deflator goes down because the arrival of new workers lowerswages at impact. Figure 11 displays the IRFs of a labor supply shock. Figure 11 isnearly the opposite view of Fig. 10: a labor supply shock reduces hours worked forall levels of wages, and therefore, it works in nearly the same way as an increase inpopulation. Figs. 10 and 11 suggest that part of the reason why Spain has had suchhigh levels of investment and imports over the last decade is that there have been largeimmigration flows.

Figures 12 and 13 show the responses of the economy to two important policyshocks: a shock to monetary policy and a shock to government consumption. Twoaspects are relevant. One, both shocks have an expansionary effect (as conventionallydone, Fig. 12 is expressed in terms of a rise in the interest rate, so to think about a fallin the nominal rate, the reader only needs to flip the lines). Second, the expansionaryeffect is, however, rather small. For instance, the multiplier of a shock to government

123

220 SERIEs (2010) 1:175–243

5 10 15 20

−4

−2

0

x 10−3 gyobs

5 10 15 20−6

−4

−2

0

x 10−3 gcobs

5 10 15 20

−4

−2

0

x 10−3 giobs

5 10 15 20

−4

−2

0

x 10−3 gldhobs

5 10 15 20

0

0.02

0.04

0.06

gwhobs

5 10 15 20

−4

−2

0

x 10−3 gMobs

5 10 15 20

−8−6−4−2

0x 10

−3 gxobs

5 10 15 20

12345

x 10−3 picobs

5 10 15 20

12345

x 10−3 piMobs

5 10 15 20

2

4x 10

−3 Rnobs

5 10 15 20

−4

−2

0

x 10−3 rer

IRF shock epsphi

Fig. 11 Impulse response function to a labour supply shock

consumption is slightly less than 0.8 and it rapidly falls to zero. Moreover, shocksto government consumption are associated with falls in consumption and investment(given the low impact multiplier, this is nearly an accounting truism) and increases inprices. Thus, MEDEA does not support the view that increases in government con-sumption are effective tools for stabilizing output. On the positive side, Fig. 12 tellsus that monetary shocks seem effective in controlling inflation in a relatively fastway.

6.2 Historical decompositions

MEDEA can be used to investigate what the driving forces have been behindSpanish economic growth during the last three decades by decomposing the observedGDP growth into the contributions of the structural shocks. The summary resultsare reported in Fig. 14 and in Table 7. To facilitate the presentation, we group theshocks into five categories: technology shocks, labor shocks, demand shocks, fiscaland monetary policy shocks, and foreign shocks. Then, Figs. 15 and 16 and Table 8decompose the contribution of labor supply shocks into labor participation (prefer-ence between work and leisure) and population growth (creation of new households in

123

SERIEs (2010) 1:175–243 221

5 10 15 20

−3−2−1

0

x 10−3 gyobs

5 10 15 20

−8−6−4−2

02

x 10−4 gcobs

5 10 15 20−15

−10

−5

0

x 10−4 giobs

5 10 15 20

−4

−2

0

x 10−3 gldhobs

5 10 15 20

−4

−2

0

x 10−3 gwhobs

5 10 15 20−15

−10

−5

0

x 10−4 gMobs

5 10 15 20−10

−5

0

x 10−3 gxobs

5 10 15 20−6

−4

−2

x 10−4 picobs

5 10 15 20−3−2−1

01

x 10−3 piMobs

5 10 15 200

1

2

3

x 10−3 Rnobs

5 10 15 20−3

−2

−1

x 10−3 rer

IRF shock epsm

Fig. 12 Impulse response function to a monetary policy shock

the economy), and the contribution of technology into neutral and investment-specificcomponents.

Looking at the period as a whole, the main contributors to growth have been thelabor supply, mainly population, and demand shocks. Each of them accounts foraround 40% (1.3 percentage points, p.p. hereafter) of real GDP growth. Productiv-ity is the third factor in importance explaining over 15% (0.5 p.p.). The remainingshocks explain little over the long run, something that we could have expected froma neoclassical growth model (such as the one at the core of our paper). This mainpicture presents a scenario of a Spanish economy that has enjoyed many years of goodshocks (immigration, incorporation of women and younger workers into the labormarket, low real interest rates, positive world demand, and moderate energy prices),but that has not broken free from the historical constraints of low productivity and poorinnovation.

Nevertheless, the contributions have been different over time. We will divide theanalysis into three relevant periods: boom in the late 1980s, the crisis of 1993–95, andthe expansion since then until 2007. The boom of the late eighties was characterizedby large productivity growth but also by a rise in labor supply, mainly population asthe large cohorts of the 1960s joined the labor market and women started to searchfor jobs in the market, but also by higher participation, and positive demand shocks

123

222 SERIEs (2010) 1:175–243

5 10 15 20

02468

x 10−3 gyobs

5 10 15 20

−15

−10

−5

0x 10

−4 gcobs

5 10 15 20

−4

−2

0x 10

−4 giobs

5 10 15 20

0

5

10

x 10−3 gldhobs

5 10 15 20

0123

x 10−3 gwhobs

5 10 15 20

−4

−2

0x 10

−4 gMobs

5 10 15 20

−6

−4

−2

0x 10

−4 gxobs

5 10 15 20

2

4

6x 10

−4 picobs

5 10 15 20

2

4

x 10−4 piMobs

5 10 15 20

2

4

x 10−4 Rnobs

5 10 15 20

−6

−4

−2

x 10−4 rer

IRF shock epsg

Fig. 13 Impulse response function to a government consumption shock

-4 %

-2 %

0%

2%

4%

6%

8%

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

productivity labourfiscal & monetary policy foreignpreferences GDP

Sources of GDP growth in Spain

(in percentage)

Fig. 14 Sources of GDP growth in Spain

(probably related to the reduction in uncertainty after the large crisis of the transition todemocracy and the fall in oil prices). Each of these elements explains about one-thirdof the increase in real GDP, while fiscal policy accounts for only around 5%. Monetary

123

SERIEs (2010) 1:175–243 223

Table 7 Sources of GDP growth in Spain

Period GDP growth Average contribution to GDP growth(%)

Productivity Labour Preferences Policies Foreign(%) (%) (%) (%) (%)

1986–91 3.82 1.11 1.41 1.49 0.03 −0.22

1992–94 0.75 1.53 −0.59 0.10 −0.28 −0.01

1995–07 3.57 −0.03 1.83 1.50 0.08 0.20

1986–07 3.26 0.49 1.38 1.30 0.02 0.06

policy and foreign shocks contributed negatively, limiting GDP growth by around 0.15and 0.2 p.p. on average, respectively. Those two are likely explained by the tough standthat the Bank of Spain took against inflation with a policy of competitive disinflationthat brought high real interest rates and high value of the peseta.

The crisis of the early nineties was characterized by a very strong negative laborsupply shock, mainly due to the large increase in unemployment, which the modelinterprets as a reduction in labor participation. This mechanism limited growth byalmost 1.9 p.p. over this period. Labor shocks did not become positive again until1998. At the same time, the Spanish economy suffered a fairly large negative demandshock, lasting from the end of 1991 until mid 1993, that reduced GDP growth byaround 1 p.p.

In contrast, the long period of continuous real GDP growth experienced since themid-nineties was mainly explained by favorable labor supply and demand shocks,probably a manifestation of immigration, changes in the age composition of the popu-lation, and the adoption of the euro and the associated historically low real interest rates.Technology shocks limited growth until 2001, a moment after which its contributionbecame positive, although rather small.10 In addition, monetary policy shocks and for-eign shocks have had a positive but much smaller contribution. Figure 15 suggests thatboth types of labor supply shocks have been very important, contributing on average1.9 p.p., which represents over 50% of GDP growth since 1995, reaching 3.5 p.p. inthe early 2000s, with population growth accounting for over 40% of growth and laborparticipation around 10%.

6.3 Permanent shocks

Another application of DSGE models is to trace the consequences of permanentchanges in some variables or parameters.11 For instance, we can evaluate the effects

10 There is the caveat, however. Since the new workers joining the labor in the last decade were likely tohave lower human capital than the existing workers, MEDEA might be picking up a composition effect thatbiases downward the contribution of technological shocks.11 The parameters of MEDEA are behavioral, in the sense that they have a clear interpretation rootedin economic theory but they are not necessarily structural in the sense of being invariant to the class ofinterventions we might be interested in. See Fernandez-Villaverde and Rubio-Ramirez (2008) for a moredetail discussion.

123

224 SERIEs (2010) 1:175–243

-4 %

-3 %

-2 %

-1 %

0%

1%

2%

3%

4%

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

labour supply population labourSources of GDP growth in Spain

(in percentage) LABOUR FACTOR

Fig. 15 Sources of GDP growth in Spain. Labour factor

-2%

-1%

0%

1%

2%

3%

1986 19 88 1990 1992 1994 1996 1998 20 00 2002 2004 2006

general technology investment technology

productivity

Sources of GDP growth in Spain (in percentage) PRODUCTIVITY

69

Fig. 16 Sources of GDP growth in Spain. Technology

Table 8 Sources of GDP growth in Spain (labour and productivity)

Period Average contribution to GDP growth

Labour Population Labour Productivity General Investment(%) (%) supply (%) (%) technology (%) technology (%)

1986–91 1.41 1.00 0.41 1.11 1.06 0.06

1992–94 0.59 1.27 −1.86 1.53 2.33 −0.80

1995–07 −1.83 1.41 0.42 −0.03 0.69 −0.72

1986–07 1.38 1.28 0.10 0.49 1.01 −0.52

of a reduction in distortionary taxation and the impact of an increase in competitionin the labor or goods market. These are but two out of many other exercises of thekind we can select. However, these two are particularly illustrative given our currentdownturn.

There are three types of distortionary taxes in MEDEA: a tax on capital income,on labor income, and on consumption. The first panel of Table 9 reports the long-run

123

SERIEs (2010) 1:175–243 225

Tabl

e9

Lon

gru

nef

fect

sof

ape

rman

entr

educ

tion

inta

xra

tes,

pric

esan

dw

age

mar

k-up

s

Var

iabl

eB

asel

ine

stea

dyst

ate

%C

hang

eof

stea

dyst

ate

valu

es

Lev

elch

ange

Cap

italt

axL

abou

rin

com

eta

xV

AT

Wag

em

ark-

upPr

ice

mar

kup

Pric

em

arku

pPr

ice

mar

kup

dom

estic

good

sim

port

edgo

ods

expo

rted

good

s

0.22

–1.0

00.

34–1

.00

0.11

–1.0

015

–0.5

013

–0.5

013

–0.5

012

–0.5

0

Out

put

1.40

70.

360.

460.

280.

130.

010.

180.

18

Con

sum

ptio

n0.

861

0.16

0.55

0.33

0.15

−0.3

40.

110.

11

Inve

stm

ent

0.40

80.

820.

400.

240.

110.

730.

430.

44

Exp

orts

0.34

80.

570.

350.

210.

100.

431.

271.

28

Impo

rts

0.39

00.

490.

300.

180.

080.

381.

091.

10

Em

ploy

men

t0.

410

0.06

0.47

0.28

0.13

0.38

0.03

0.03

Wag

e1.

996

0.29

−0.0

2−0

.01

−0.0

10.

410.

140.

14

Wag

e*ho

urs

0.81

80.

350.

450.

270.

130.

790.

170.

17

Ren

talp

rice

ofca

pita

l0.

032

−0.5

10.

040.

030.

010.

05−0

.25

−0.2

5

Mar

gina

lcos

t0.

863

0.00

0.00

0.00

0.00

0.28

0.00

0.00

Tobi

n’s

q1.

109

0.06

0.04

0.02

0.01

0.05

−0.2

1−0

.22

Tra

nsfe

rs0.

373

−0.2

2−1

.44

−2.1

60.

200.

490.

160.

16

Term

sof

trad

e0.

990

−0.0

8−0

.05

−0.0

3−0

.01

−0.0

60.

27−0

.17

123

226 SERIEs (2010) 1:175–243

0 5 10 15 200

2

4

6

8x 10

−3

yci

0 5 10 15 200

0.5

1

1.5x 10

−3

MX

0 5 10 15 20−10

−5

0

5x 10

−5

πc

πi

Rn

0 5 10 15 20−2

0

2

4

6x 10

−4

rerex

0 5 10 15 20−15

−10

−5

0

5x 10

−3

wld

5 10 15 20

−3

−2

−1

0x 10

−3

rmc

1% VAT reduction

Fig. 17 Transitional dynamics after a permanent reduction of 1% in the consumption tax rate

impact of unexpectedly reducing each of these taxes by 0.5 p.p. To save on space, andsince changes in the VAT have been proposed by many economists (and implementedin the UK) as a fiscal policy tool, we will concentrate on describing the effects of cut-ting the consumption tax. An unexpected reduction in the VAT by 1 p.p. has a long-runpositive effect on the Spanish economy, by increasing output per capita in efficiencyunits and hours worked. Higher labor input pushes up the marginal productivity ofcapital and increases investment. On the demand side, the increase in the paymentsto capital and the rise in real compensation per worker (the rise in hours compensatesfor the fall in real wages) increase households’ income and consumption. In order toequilibrate demand and supply, the terms of trade (px/pM ) fall to improve the exter-nal position. Figure 17 draws the transitional dynamics after the shock (to make thecomparison easier, in the charts, all variables are expressed as differences with respectto the initial steady state). Most variables move smoothly to the new steady state. Theexception is the real wage, which falls initially below its long-run level and then risesback toward the new equilibrium.

The degree of competition in the goods and labor markets is determined by themark-up over prices or wages in each case. The second panel of Table 9 reports thelong-run impact of unexpectedly reducing each of these mark-ups by 0.5 p.p. Increas-ing competition in the labor market reduces the wage per hour and increases thenumber of hours worked, expanding output per capita in efficiency units. Higher laborinput pushes up the marginal productivity of capital and increases investment. The

123

SERIEs (2010) 1:175–243 227

Tabl

e10

Lon

gru

nef

fect

sof

ape

rman

entc

hang

ein

seve

ralp

aram

eter

s

Var

iabl

eB

asel

ine

stea

dyst

ate

%C

hang

eof

stea

dyst

ate

valu

es

Lev

elch

ange

Hab

itsL

abou

rE

last

icity

Ela

stic

ityE

last

icity

Con

sum

ptio

nIn

vest

men

tN

eutr

alte

ch.

Inve

stm

ent

Popu

latio

nsu

pply

ofla

bour

ofsu

bst.

ofsu

bst.

hom

ebi

asho

me

bias

grow

thra

tesp

ecifi

cte

ch.

grow

thra

teco

effic

ient

subs

t.co

nsum

ptio

ngo

ods

inve

stm

entg

oods

grow

thra

te0.

80–0

.10

6.74

–10.

001.

97–1

.00

7.67

–3.0

08.

06–3

.20

0.81

–10.

000.

10–1

0.00

0.00

23–1

.00

0.00

26–1

.00

0.00

32–1

.00

Out

put

1.40

70.

11−1

.49

0.54

0.04

0.06

−0.6

7−0

.43

−0.5

1−0

.55

−0.6

9

Con

sum

ptio

n0.

861

0.12

−1.7

60.

640.

020.

02−0

.91

−0.7

7−0

.74

−0.7

7−0

.87

Inve

stm

ent

0.40

80.

09−1

.29

0.47

0.08

0.13

−1.4

5−0

.86

−1.2

7−1

.30

−1.5

7

Exp

orts

0.34

80.

08−1

.12

0.40

−0.3

3−0

.36

6.41

5.21

6.56

6.54

6.35

Impo

rts

0.39

00.

07−0

.97

0.35

−0.2

8−0

.31

5.51

4.48

5.64

5.62

5.45

Em

ploy

men

t0.

410

0.11

−1.5

10.

550.

010.

01−0

.21

−0.1

7−0

.21

−0.2

2−0

.22

Wag

e1.

996

−0.0

10.

08−0

.03

0.02

0.04

−0.4

4−0

.24

−0.2

7−0

.31

−0.4

4

Wag

e*ho

urs

0.81

80.

10−1

.44

0.52

0.03

0.05

−0.6

5−0

.41

−0.4

9−0

.52

−0.6

6

Ren

talp

rice

ofca

pita

l0.

032

0.01

−0.1

40.

05−0

.04

−0.0

80.

770.

420.

630.

610.

77

Mar

gina

lcos

t0.

863

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.05

0.02

0.00

Tobi

n’s

q1.

109

0.01

−0.1

20.

04−0

.04

−0.0

70.

670.

370.

690.

690.

67

Tra

nsfe

rs0.

373

0.06

−2.2

90.

830.

020.

05−0

.46

−0.2

4−0

.21

−0.2

9−0

.54

Term

sof

trad

e0.

990

−0.0

10.

16−0

.06

0.04

0.05

−0.8

5−0

.69

−0.8

7−0

.86

−0.8

4

123

228 SERIEs (2010) 1:175–243

0 5 10 15 200

0.5

1

1.5

2x 10

−3

yci

0 5 10 15 200

1

2

3

4x 10

−4

MX

0 5 10 15 20−3

−2

−1

0

1x 10

−5

πc

πi

Rn

0 5 10 15 20−5

0

5

10

15x 10

−5

rerex

0 5 10 15 20−3

−2

−1

0

1x 10

−3

wrmc

5 10 15 20

2

4

6x 10

−3

kld

0.5% wage markup reduction

Fig. 18 Transitional dynamics after a permanent reduction of 0.5 p.p. in the wage mark-up

demand side is very similar to the case of a reduction of consumption taxes. Duringthe transition to the new steady state (see Fig. 18), the real wage initially undershootsits long-run level and then recovers.

We complete this subsection with Table 10, which reports the long-run effects ofchanges in several parameters of the model.

6.4 Alternative scenarios

The historical decomposition of GDP growth showed that population growth createdby immigration has been one of the important determinants of economic growth dur-ing the recent expansion, explaining around 1.4 p.p. of GDP growth (see Table 11).However, this contribution has not been constant over time. Instead, Fig. 19 shows thatthere has been an important change in the long-run population growth rate during thesample, increasing from 0.28% on average over the period 1986–96 to 0.35% sincethen. Therefore, an interesting question is what would GDP growth have been hadthis rise in population growth not taken place (for example, if the conservative andsocialist governments had followed a more restrictive immigration policy).

123

SERIEs (2010) 1:175–243 229

Table 11 Alternative scenario for population growth

Period Baseline Change in alternative scenario

GDP Population Population GDP Population Populationgrowth (%) growth (%) contribution (%) growth (%) growth (%) contribution (%)

1986–96 2.75 0.284 1.11 −0.24 0.000 −0.26

1997–07 3.76 0.353 1.42 −0.79 −0.067 −0.67

1986–07 3.26 0.318 1.28 −0.48 −0.033 −0.47

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

population growth (alt esc) population growth

subsamples mean sample mean

Fig. 19 Alternative scenario for population growth

This important question can be easily answered with MEDEA.12 In particular, wewould like to see what would have happened if population growth followed the alter-native scenario depicted in Fig. 19 (discontinuous line). In that figure, we assume thatpopulation growth followed a path similar to the historical one, but shifted downward,so that the long-run mean is equal to the one in the first part of the sample. Figure 20and Table 11 show the impact of this alternative scenario. We find that had populationgrown at this alternative slower pace, GDP growth would have been 0.5 p.p. lower.That is, the population growth shock experienced in Spain in the current decade addedhalf a percentage point to annual output growth.

12 We want to be careful, though, since this exercise assumes that our parameters and the shocks recov-ered by the estimation are structural in the sense of being invariant to the change in population growth.This assumption may be problematic, although, unfortunately, difficult to test. We thank Fabio Canova foremphasizing this caveat to us.

123

230 SERIEs (2010) 1:175–243

�4 %

�2 %

0%

2%

4%

6%

8%

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

productivity labour

fiscal & monetary policy foreignpreferences GDP baselineGDP

Sources of GDP growth in Spain (in percentage)

Fig. 20 Alternative scenario for population growth. Sources of growth

7 Conclusions

In this paper, we have introduced MEDEA, a DSGE model of the Spanish economy,and estimated it with data from the last several decades. To illustrate the potentialitiesof MEDEA, we have applied the model to policy analysis and counterfactual evalu-ations. We think that our enterprise has been a success. We now have an operationalmodel of the Spanish economy, rich enough for detailed study and yet sufficientlyconcise to be solved and estimated with off-the-shelf software and a regular worksta-tion. The estimates are reasonable and they tell us important lessons about how oureconomy works. Some we suspected, such as the differences in behavior of investmentgood imports versus consumption imports, some we did not, such as the small punchof fiscal policy. The model’s performance as a forecasting tool (not a primary designconsideration, but a relevant aspect nevertheless) still needs more time before we canestablish it.

There are, however, many dimensions along which we would like to improve ourwork and make DSGE models an important element in the toolbox of policy makers.In particular, we will like to:

1. Incorporate a richer specification of fiscal policy, including tax and transfer shocks,a distinction between public consumption and public investment, and public capi-tal in the production function. The recent active use of fiscal policy as an instrumentto stabilize the economy suggests that we need a more detailed understanding ofthe propagation effects of fiscal policy in Spain.

2. Specify a social security system through the device of stochastic aging of house-holds. As the Spanish population ages over the next decades and the social securitysystem is strained to its limits, we need to know how the steady state and aggregatefluctuations will be affected by this aging and by possible re-adjustments in thesystem.

123

SERIEs (2010) 1:175–243 231

3. Model energy consumption more explicitly. Given the large exposure of the Span-ish economy to oil shocks, this seems to be an important mechanism for under-standing aggregate fluctuations.

4. Pay more attention to the behavior of the labor market. The Spanish economy’sbiggest open problem has been, for over three decades, its schizophrenic labormarket, a heritage of darker eras of our economic policy that no government hasdared to tackle. Beyond bitterly complaining about it, our task as macroecono-mists is to add to our models a more realistic description of our outmoded set oflabor market institutions.

5. Incorporate a financial sector. The recent financial crisis highlights how we wantto trace the effects of different financial shocks on the economy and how to designmacroeconomic policies that help to correct the problems caused by these financialshocks.

6. Estimate the model non-linearly and allow for stochastic volatility of the shocksand possible parameter drifting.

Hopefully, the support of research institutions and of the profession in general willallow us to see MEDEA or one of her descendants grow over the next years.

Appendix A: Equilibrium conditions

We present now the full set of equilibrium conditions.

• The first-order conditions of the household:

dt

(ct − h

ct−1

zt

)−1

− hEtβγ Lt+1dt+1 (ct+1zt+1 − hct )

−1 = λt (1 + τc)

(pc

t

pt

)

(5)

λt = Et

{β

λt+1

zt+1

Rt

�t+1

}(6)

λt = Et

⎧⎨

⎩βλt+1

zt+1

RWt �

(ext bW

t , ξbW

t

)

�t+1

ext+1

ext

⎫⎬

⎭ (7)

rt = �′ [ut ]

(1 − τk)(8)

qtEtγL

t+1 = βEtγL

t+1

{λt+1

λt zt+1µt+1

((1 − δ) qt+1 + rt+1ut+1 (1 − τk)

+δτk − �[ut+1

])}

(9)

(pi

t

pt

)= qt

(1 − S

[γ L

tit

it−1zt

]− S′

[γ L

tit

it−1zt

]γ L

tit

it−1zt

)

+ Etβqt+1λt+1

λt zt+1S′[γ L

t+1it+1

itzt+1

](γ L

t+1it+1

itzt+1

)2

(10)

123

232 SERIEs (2010) 1:175–243

(mt

pt

)= dtυ

[βEt

λt+1

zt+1

Rt − 1

�t+1

]−1

(11)

ft = η − 1

η(1 − τw)

(w∗

t

)1−ηλt (wt )

η ldt

+βθwEtγL

t+1

(�

χwt

�t+1

)1−η (w∗

t+1

w∗t

zt+1

)η−1

ft+1 (12)

ft = dtϕtψ

(w∗

t

wt

)−η(1+ϑ) (ldt

)1+ϑ

+βθwEtγL

t+1

(�

χwt

�t+1

)−η(1+ϑ) (w∗

t+1

w∗t

zt+1

)η(1+ϑ)

ft+1 (13)

• The intermediate domestic firms that can change prices set them to satisfy:

g1t = λt mct y

dt + βθpEtγ

Lt+1

(�

χt

�t+1

)−ε

g1t+1 (14)

g2t = λt�

∗t y

dt + βθpEtγ

Lt+1

(�

χt

�t+1

)1−ε(

�∗t

�∗t+1

)g2

t+1 (15)

εg1t = (ε − 1)g2

t (16)

where they rent inputs to satisfy their static minimization problem:

ut

ldt

kt−1 = α

1 − α

wt

rtzt µt (17)

mct =(

1

1 − α

)1−α ( 1

α

)α

w1−αt rα

t (18)

• The importing and exporting domestic firms that can change prices set them tosatisfy:

gM1t = λt

⎡

⎢⎣

(ext pW

tpt

)

(pM

tpt

)

⎤

⎥⎦ yMt + βθMEtγ

Lt+1

((�M

t

)χM

�Mt+1

)−εM

gM1t+1 (19)

gx1t = λt

yxt(

ext pxt

pt

) + βθxEtγL

t+1

((�W

t

)χx

�xt+1

)−εx

gx1t+1 (20)

gM2t = λt�

M∗t yM

t + βθMEtγL

t+1

((�M

t

)χM

�Mt+1

)1−εM(

�M∗t

�M∗t+1

)gM2

t+1 (21)

gx2t = λt�

x∗t yx

t + βθxEtγL

t+1

((�W

t

)χx

�xt+1

)1−εx(

�x∗t

�x∗t+1

)gx2

t+1 (22)

123

SERIEs (2010) 1:175–243 233

εMgM1t = (εM − 1)gM2

t ; εxgx1t = (εx − 1)gx2

t (23)

• Wages and prices evolve as:

1 = θw

(�

χw

t−1

�t

)1−η (wt−1

wt zt

)1−η

+ (1 − θw)

(w∗

t

wt

)1−η

, (24)

1 = θp

(�

χt−1

�t

)1−ε

+ (1 − θp)�∗1−ε

t (25)

1 = θM

((�M

t−1

)χM

�Mt

)1−εM

+ (1 − θM )(�M∗

t

)1−εM(26)

1 = θx

((�W

t−1

)χx

�xt

)1−εx

+ (1 − θx )(�x∗

t

)1−εx(27)

• Monetary authority follows its Taylor rule:

Rt

R=(

Rt−1

R

)γR

⎛

⎜⎝(

�t

�

)γ�

⎛

⎜⎝γ L

tyd

t

ydt−1

zt

exp(L + yd

)

⎞

⎟⎠

γy⎞

⎟⎠

1−γR

exp(ξm

t

)(28)

While the government’s policy comprises transfers to households and the level ofdebt, which are determined by the government’s budget constraint:

bt = gt

ydt

+ Tt

ydt

+ mt−1

pt−1

1

γ Lt yd

t zt�t+ Rt−1bt−1

ydt−1

γ Lt �t y

dt zt

− (rtut − δ) τkkt−1

ydt zt µt

− τw

wt ldt

ydt

− τc

(pc

t

pt

)ct

ydt

− mt

pt

1

ydt

(29)

and the fiscal policy rule:

Tt

ydt

= T0 − T1

(bt − b

)(30)

• Net foreign assets evolve as:

ext bWt = RW

t−1�(�ext ext−1bW

t−1, ξbW

t−1

)�ext

ydt−1

γ Lt zt�t y

dt

ext−1bWt−1

+(

ext pWt

pt

)εW (ext pxt

pt

)1−εW(yW

t

ydt

)−(

ext pWt

pt

)(Mt

ydt

)(31)

123

234 SERIEs (2010) 1:175–243

• Aggregate imports and exports evolve as:

Mt =vMt

⎡

⎢⎣Et�ct+1

(1 − nc)

⎡

⎢⎣

(pM

tpt

)

(pc

tpt

)

⎤

⎥⎦

−εc

ct + Et�it+1

(1 − ni

)⎡

⎢⎣

(pM

tpt

)

(pi

tpt

)

⎤

⎥⎦

−εi

it

⎤

⎥⎦

(32)

xt = vxt

⎡

⎢⎣

(ext px

tpt

)

(ext pW

tpt

)

⎤

⎥⎦

−εW

yWt (33)

where

Et �st+1 =

⎡

⎣1 − β (1 − ns)1εs Et

γ Lt+1 λt+1

λt zt+1

⎡

⎣

(pst

pt

)

(pM

tpt

)

⎤

⎦�st+1

(st+1

sMt+1

(1−�s

t+1

)) 1

εs�s ′

t+1

(�sM

t+1

)2

�st+1

⎤

⎦−εs

(1 − �s

t) [

1 − �st − �s ′

t

(�sM

t�st

)]−εs

for s = c, i and the distribution of relative prices of imports and exports:

vMt = θM

((�M

t−1

)χM

�Mt

)−εM

vMt−1 + (1 − θM )

(�M∗

t

)−εM(34)

vxt = θx

((�W

t−1

)χx

�xt

)−εx

vxt−1 + (1 − θx )

(�x∗

t

)−εx. (35)

The production of importing and exporting firms is:

yMt = cM

t + i Mt (36)

yxt =

⎡

⎢⎣

(ext px

tpt

)

(ext pW

tpt

)

⎤

⎥⎦

−εW

yWt (37)

while the demands for consumption and investment imports relative to the corre-sponding domestic components are:

cMt

cdt

= Et�ct+1 (1 − nc)

nc

(pM

t

pt

)−εc

(38)

i Mt

i dt

= Et�it+1

(1 − ni

)

ni

(pM

t

pt

)−εi

(39)

123

SERIEs (2010) 1:175–243 235

• Markets clear:

nc(

pct

pt

)εc

ct +ni(

pit

pt

)εi

it +gt +� [ut ]kt−1

µt zt+ xt

=Atzt

(ut kt−1

)α (ldt

)1−α−φ

vpt

(40)

where

lt = vwt ld

t (41)

vpt = θp

(�

χt−1

�t

)−ε

vpt−1 + (1 − θp

)�∗−ε

t (42)

vwt = θw

(wt−1

wt zt

�χw

t−1

�t

)−η

vwt−1 + (1 − θw)

(w∗

t

wt

)−η

(43)

and

EtγL

t+1kt − (1 − δ)kt−1

zt µt−(

1 − S

[γ L

tit

it−1zt

])it = 0 (44)

Finally, aggregate consumption and investment evolve as:

ct =[(nc) 1

εc

(cd

t

) εc−1εc + (1 − nc) 1

εc

(cM

t

(1 − �c

t

)) εc−1εc

] εcεc−1

(45)

it =[(

ni) 1

εi(

i dt

) εi −1εi +

(1 − ni

) 1εi(

i Mt

(1 − �i

t

)) εi −1εi

] εiεi −1

(46)

• Relative consumption and investment prices evolve as:

pct

pt=[

nc + �cM

t

(1 − nc)

(pM

t

pt

)1−εc] 1

1−εc

(47)

pit

pt=[

ni + �i M

t

(1 − ni

)( pMt

pt

)1−εi] 1

1−εi

(48)

123

236 SERIEs (2010) 1:175–243

• The definitions of inflation rates are the following:

�ct =

(pc

tpt

)

(pc

t−1pt−1

)�t ; �it =

(pi

tpt

)

(pi

t−1pt−1

)�t

�Mt =

(pM

tpt

)

(pM

t−1pt−1

)�t �xt =

(ext px

tpt

)

(ext−1 px

t−1pt−1

) �t

�ext(49)

�Wt =

(ext pW

tpt

)

(ext−1 pW

t−1pt−1

) �t

�ext

• The growth rate of technology:

zt = A1

1−αt µ

α1−αt (50)

Appendix B: Log-linearized equilibrum conditions

The full set of log-linearized equilibrum conditions is:

(1 − hβγ L

z

)(λt + ( pc

t − pt))

= dt − hβγ L

zEt dt+1 − 1 + h2βγ L

z2(1 − h

z

) ct + h

z(

1 − hz

)ct−1

+ βhγ L

z(

1 − hz

)Etct+1 − h

z(

1 − hz

)zt (51)

λt = Et

{λt+1 + Rt − �t+1

}. (52)

Etλt+1 + R∗

t − Et�t+1 − �b∗ (ext + b∗

t

)− �b∗ξb∗

ξb∗t + Et�ex t+1 − λt = 0

(53)

r t = �2

�1ut . (54)

qt = Etλt+1 + β (1 − δ)

zµEtqt+1 + β�1

zµ(

pi

p

)Etr t+1. (55)

(pi

t − pt

)+ κ

(γ L z

)2 (�i t +zt + γ L

t

)= qt + βκ

(γ L)3

z2Et �i t+1 (56)

123

SERIEs (2010) 1:175–243 237

(mt − pt) = dt −

(R

R − 1

)Rt − Et

(λt+1 − �t+1

)(57)

ft =(

1 − βθwγ L zη−1�−(1−η)(1−χw)) (

(1 − η) w∗t + λt + ηwt + ld

t

)

+βθwγ L�−(1−η)(1−χw) zη−1Et

(ft+1 − (1 − η)

(�t+1 − χw�t + �w∗

t+1

))

(58)

ft =(

1 − βθwγ L zη(1+ϑ)�η(1+ϑ)(1−χw))

×(

dt + ϕt + η (1 + ϑ)(wt − w∗

t

)+ (1 + ϑ)ld

t

)

+βθwγ L zη(1+ϑ)�η(1+ϑ)(1−χw)Et(

ft+1 + η (1 + ϑ)

×(�t+1 − χw�t + �w∗

t+1

))(59)

g1t =

(1 − βθpγ

L�ε(1−χ)) (λt + mct + yd

t

)

+βθpγL�

ε(1−χ)t Et

(g1

t+1 + ε(�t+1 − χ�t ))

. (60)

g2t =

(1 − βθpγ

L�−(1−ε)(1−χ)) (λt + �∗

t + ydt

)

+βθpγL�−(1−ε)(1−χ)

Et

(g2

t+1 − (1 − ε)(�t+1 − χ�t

)− (�∗t+1 − �∗

t

)).

(61)

g1t = g2

t . (62)

ut +kt−1 − ldt = wt −r t +zt + µt . (63)

mct = (1 − α)wt + αr t . (64)

gM1t =

(1 − βθMγ L

(�M

)εM (1−χM ))

×(λt +

(ex t + pW

t − pt

)−(

pMt − pt

)+ yM

t

)

+βθMγ L(�M

)εM (1−χM )

tEt

(gM1

t+1 + εM

(�M

t+1 − χM�Mt

)). (65)

gM2t =

(1 − βθMγ L

(�M

)−(1−εM )(1−χM ))(λt + �M∗

t + yMt

)

+βθMγ L(�M

)−(1−εM )(1−χM )

Et

⎛

⎝gM2

t+1 − (1 − εM )(�M

t+1 − χM�Mt

)

− (�M∗t+1 − �M∗

t

)

⎞

⎠ .

(66)

123

238 SERIEs (2010) 1:175–243

gM1t = gM2

t . (67)

gx1t =

(1 − βθxγ

L(�W

)−εx χx (�x)εx

)(λt + yx

t − (ex t + pxt − pt

))

+βθxγL(�W

)−εx χx (�x)εx

Et

(gx1

t+1 + εx

(�x

t+1 − χx�Wt

)). (68)

gx2t =

(1 − βθxγ

L(�W

)(1−εx )χx (�x)−(1−εx )

)(λt + �x∗

t + yxt

)

+βθxγL(�W

)(1−εx )χx (�x)−(1−εx )

Et

(gx2

t+1−(1−εx )(�x

t+1−χx�Wt

)

− (�x∗t+1 − �x∗

t

)

).

(69)

gx1t = gx2

t . (70)

θw�−(1−χw)(1−η) z−(1−η)

(1 − θw) (�w∗)1−η

(�t − χw�t−1 + �w

t + zt) = w∗

t − wt . (71)

θp�−(1−ε)(1−χ)

(1 − θp

)(�∗)(1−ε)

(�t − χ�t−1

) = �∗t . (72)

θM(�M

)−(1−εM )(1−χM )

(1 − θM ) (�M∗)(1−εM )

(�M

t − χM�Mt−1

)= �M∗

t . (73)

θx(�W

)(1−εx )χx(�x )−(1−εx )

(1 − θx ) (�x∗)(1−εx )

(�x

t − χx�Wt−1

)= �x∗

t . (74)

Rt = γR Rt−1 + (1 − γR)(γ��t + γy(yd

t +zt + γ Lt ))

+ ξmt . (75)

bt + ydt =

(m

p

)1

byd�γ L z

((mt−1 − pt−1) − �t −zt − γ L

t

)

−(

m

p

)1

byd

(mt − pt)

+ 1

βγ L

(Rt−1 +bt−1 + yd

t−1 − �t −zt − γ Lt

)− τw

wld

byd

(wt + ld

t

)

−τk (r − δ)

zµ

(k

byd

) (kt−1 −zt − µt

)− τkr

zµ

(k

byd

) (r t + ut

)

− τc

(pc

p

)(c

byd

) ((pc

t − pt)+ct

)+ g

bydgt + T0

bT t (76)

123

SERIEs (2010) 1:175–243 239

T

yd

(T t − ydt

)= −

(T1b

T0

)bt (77)

(ext + b∗

t

) = 1

γ Lβ

(ext−1 + b∗

t−1

)

+(

ex pW

p

)(M

yd

)((1 − εW )

[ (ex t + px

t − pt)

− (ex t + pWt − pt

)]

− Mt + yWt

)

(78)

Mt = vMt + yM

t (79)

xt = vxt − εW

[(ex t + px

t − pt)−

(ex t + pW

t − pt

)]+ yW

t (80)

vMt = θM

(�M

)εM (1−χM ) (εM (�M

t − χM�Mt−1) + vM

t−1

)

−(

1 − θM

(�M

)εM (1−χM ))

εM�M∗t (81)

vxt = θx

(�W

)−εx χx (�x)εx

(εx (�

xt − χx�

Wt−1) + vx

t−1

)

−(

1 − θx

(�W

)−εx χx (�x)εx

)εx�

x∗t (82)

yMt =

(cM

cM + i M

)cM

t +(

i M

cM + i M

)i

Mt (83)

yxt = −εW

[(ex t + px

t − pt)−

(ex t + pW

t − pt

)]+ yW

t (84)

cMt −cd

t = −ac1Et

(cM

t − ct

)+ ac

2Et

(cM

t+1 − ct+1

)− εc

(pM

t − pt

)

(85)

iMt −id

t = −ai1Et

(i M

t − it

)+ ai

2Et

(i M

t+1 − it+1

)− εi

(pM

t − pt

)

(86)

(ydv p)(v

pt + yd

t

)= A

zkα(

ld)1−α

(At −zt + α

(ut +kt−1

)+ (1 − α)l

dt

).

(87)

y d y dt = nc

(pc

p

)εc

c(ct + εc

(pc

t − pt))+ ni

(pi

p

)εi

i(

i t + εi

(pi

t − pt

))

+ ggt + �1k

µzut + xxt , (88)

123

240 SERIEs (2010) 1:175–243

lt = vwt + ld

t , (89)

vpt = θp�

ε(1−χ)(ε(�t − χ�t−1) + v

pt−1

)

−(

1 − θp�ε(1−χ)

)ε�∗

t . (90)

vwt = θw�η(1−χw) zη

(η(�t − χw�t−1 + wt − wt−1 +zt

)+ vwt−1

)

−(

1 − θw�η(1−χw) zη)

η(w∗t − wt ). (91)

kt = (1 − δ)

zµγ Lkt−1 +

(1 − 1 − δ

zµγ L

)i t − (1 − δ)

zµγ L

(zt + µt

). (92)

ct = (nc) 1εc

(cd

c

) εc−1εccd

t +⎡

⎣1 − (nc) 1εc

(cd

c

) εc−1εc

⎤

⎦

1εc

cMt (93)

i t =(

ni) 1

εi

(i d

i

) εi −1εi i

dt +

⎡

⎣1 −(

ni) 1

εi

(i d

i

) εi −1εi

⎤

⎦

1εi

iMt (94)

(pc

t − pt) =

(1 − nc

1 − εc

)⎛

⎝

(pM

p

)

(pc

p

)

⎞

⎠1−εc

×[

(1 − εc)(

pMt − pt

)

−ac1Et

(�cM

t − �ct)+ ac

2Et(�cM

t+1 − �ct+1)]

(95)

(pi

t − pt

)=(

1 − ni

1 − εi

)⎛

⎝

(pM

p

)

(pi

p

)

⎞

⎠1−εi

×[

(1 − εi )(

pMt − pt

)

−ai1Et(�i M

t − �it)+ ai

2Et(�i M

t+1 − �it+1)]

(96)

�ct = ( pc

t − pt)− ( pc

t−1 − pt−1)+ �t (97)

�it =

(pi

t − pt

)−(

pit−1 − pt−1

)+ �t (98)

�Mt =

(pM

t − pt

)−(

pMt−1 − pt−1

)+ �t (99)

�xt = (ext + px

t − pt)− (ext−1 + px

t−1 − pt−1)+ �t − �ex t (100)

�Wt =

(pW

t + ext − pt

)−(

pWt−1 + ext−1 − pt−1

)+ �t − �ex t (101)

zt =At + αµt

1 − α(102)

123

SERIEs (2010) 1:175–243 241

�Wt = ξ�W

t (103)

R∗t = ξ R∗

t (104)

yWt − ξ

yW

t = 0 (105)

dt = ρd dt−1 + σdεd,t (106)

ϕt = ρϕϕt−1 + σϕεϕ,t (107)

gt = ρggt−1 + σgεg,t (108)

ξmt = σmεm,t (109)

γ Lt = σLεL ,t (110)

At = σAεA,t (111)

µt = σµεµ,t (112)

ξ�W

t = ρ�W ξ�W

t−1 + σ�W ε�W ,t (113)

ξyW

t = ρyW ξyW

t−1 + σyW εyW ,t (114)

ξ R∗t = ρR∗ ξ R∗

t−1 + σR∗εR∗,t (115)

ξb∗t = ρb∗ ξb∗

t−1 + σb∗εb∗,t (116)

�Ot − log � = �t (117)

wOt − wO

t−1 − log z = (wt − wt−1)+zt (118)

ROt − log R = Rt (119)

yd,Ot − yd,O

t−1 − log z − log γ L =(yd

t − ydt−1

)+zt + γ L

t (120)

cOt − cO

t−1 − log z − log γ L = (ct −ct−1)+zt + γ L

t (121)

i Ot − i O

t−1 − log z − log γ L =(

i t −i t−1

)+zt + γ L

t (122)

�c,Ot − log �c = �c

t (123)

�i,Ot − log �i = �i

t (124)

ld,Ot − ld,O

t−1 − log γ L =(

ldt − ld

t−1

)= γ L

t (125)

M Ot − M O

t−1 − log z − log γ L =(Mt − Mt−1

)+zt + γ L

t (126)

x Ot − x O

t−1 − log z − log γ L = (x t − xt−1)+zt + γ L

t (127)

�M,Ot − log �M = �M

t (128)

�x,Ot − log �x = �x

t (129)

�W,Ot − log �W = �W

t (130)

R∗,Ot − log R∗ = R∗

t (131)

yW,Ot − yW,O

t−1 − log z − log γ L =(yW

t − yWt−1

)+zt + γ L

t (132)

123

242 SERIEs (2010) 1:175–243

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