ESTUDIOS SOBRE LA ECONOMIA ESPAÑOLA

Eduardo L. Giménez José María Martín-Moreno

EEE 43

July 2001

FEDEA Fundación de Estudios de Economía Aplicada

http://www.fedea.es/hojas/publicado.html

Eduardo L. Giménez and José María Martín-Moreno† Universidade de Vigo

July 26, 2001

This paper examines the consequences of introducing a cash-in-advance constraint into a small open economy business cycle model for the Spanish case. A business cycle model is built extending Correia, Neves and Rebelo (1995) small open economy framework and Cooley and Hansen (1995) monetary economy. Money is introduced through a cash-in-advance constraint. The stochastic simulation of the model and its comparison to Spanish data show that the model is able to mimic i) the Dolado et al. puzzle, that is, the high volatility of private consumption for this economy; ii) the Dunlop-Tarshis observation, i.e., the negative correlation between real wages and hours worked; and iii) some cyclical features of the nominal dimension.

Keywords: Business Cycle, Cash-in-Advance Constraint, Small Open Economy. JEL E32, E37, E47, E52, F41

∗We specially thanks to Fuco Lores, Omar Licandro, Carlos de Miguel, Frank Portier, Luis Puch and Javier Vallés for their helpful and suggestions, and also to Carlos Borondo and Emmanuel Leao as well. We acknowledge the useful and insight comments by Victor Ríos-Rull, Michele Boldrin, Tim Kehoe, Jean-Olivier Hairault and the participants of the IV Workshop on Dynamic Macroeconomics, Soutomaior 1999, and Young Economists Conference 2000, Oxford. We acknowledge the scholarship CICYT (SEC99-1094).

†Address: Departamento de Fundamentos da Análise Económica e Historia Económica. Universidade de Vigo. E36200 Vigo (Spain). Fax: 34+986.812401 e-mail: <egimenez@uvigo.es> <jmartin@uvigo.es>

1 Introduction

The high volatility of private consumption relative to the volatility of output, the negative correlation between hours worked and real wages, and the correlation among money and prices with output are three prominent features of the Spanish business cycle.

First, the high relative volatility of private consumption in the Spanish economy, sometimes called the Dolado et al puzzle,¹ seems to be inconsistent with the economic theory of consumption.² This fact may indicate a large elasticity of intertemporal substitution together with a strong wealth efect, as indicated by Dolado et al. (1993), although some authors have suggested that the distinction between durable and non-durable consumption could help to explain this puzzle (e.g., Puch and Licandro, 1997, or Martín-Moreno, 1998). The latter means that the behavior of durable goods and investment over the cycle is very close and, therefore, the higher volatility of durable goods distorts the volatility of aggregate consumption. Spanish non-durable consumption data, however, still exhibit a high volatility.³ Models dealing with this issue could not satisfactorily explain this fact. Further modeling seems necessary to take into account some particular features of the Spanish economy. For example, the relevance of the efects of financial constraints in this economy, pointed out by Dolado et al (1993), gives rise to a model with the specification of a liquidity constraint, as proposed by Puch and Licandro (1997).

Second, the negative or weak correlation between hours worked and real wages, the socalled Dunlop-Tarshis observation, suggests co-movements on the supply and demand of labor in the economy. The Real Business Cycles theory, in the Kydland and Prescott (1982) tradition, sets technology shocks as the source of fluctuations of the economy. Since these shocks only shift the demand of labor, models of this type finds a high positive correlation between hours worked and real wages.⁴ Christiano and Eichenbaum (1992) introduced a government shock that afects individual preferences into an indivisible labor Real Business Cycle model for US economy. This modeling allows shifts of labor supply due to government shocks and shifts of labor demand due to technological shocks. However, it is a challenge to find alternative modelings where shocks on preferences are not present.⁵

Third, the cyclical features of some nominal variables have been described by Dolado, Sebastián and Vallés (1993) for the Spanish economy. However, there is no application to the introduction of money set-up in the real business cycle framework for this economy to reproduce these characteristics.⁶

¹Several authors have reported this taking the Quarterly National Accounts data in per capita terms: 1.13 (1970:I 1991:IV, Dolado et al, 1993); 1.07 (1970:I-1994:IV, Puch and Licandro, 1997); 1.14 (1976:III-1995:IV, Martín-Moreno, 1998).

²Theories like life-cycle (Modigliani, 1966) or permanent income (Friedman, 1956) suggest that households smooth consumption over the cycle. Hence volatility of consumption relative to volatility of output should be low.

³In the case of a closed economy 0.91 (1970:I-1994:IV), and 1.07 for the case of an open economy (1976:III-1995:IV).

⁴RBC models predict the correlation between real wages and hours to be in excess of 0.9, whereas the correlation in the data is negative or, essentially, zero.

⁵Observe that Puch and Licandro (1997), applying Christiano and Eichenbaum’s modeling for the Spanish economy, are not able to reproduce the existing correlation between wages and hours. Their best result is 0.83, while it is -0.28 in the data.

⁶Here it is worth of commenting the work by Andrés, López-Salido and Vallés (1999) in which through

This paper presents a monetary model which simultaneously reproduces these three previously mentioned features of the Spanish business cycle more accurately.

A monetary business cycle model for the Spanish economy is built extending Correia, Neves and Rebelo (1995) small open economy framework, and the monetary economy by Cooley and Hansen (1995) and Hairault and Portier (1995). There are four agents in the economy: households, domestic firms, the government and the foreign sector. Only one good is produced in the economy, both by domestic firms and the foreign sector. Capital and labor are the required inputs for domestic production, and the labor supply is endogenous. The government finances its public deficit with lump-sum transfers and monetary emission. There exist three financial securities in the economy: domestic money, foreign money, and an international real bond. Domestic money is demanded by households, government and the foreign sector in order to buy the domestic produced good through a cash-in-advance constraint.⁷ The international real bonds, traded in an international market at an exogenous interest rate, are issued to get foreign currency and, later, to buy the good produced abroad. In order to avoid the existence of extra domestic money or make any presumption on the trade balance account, we formulate three key assumptions in this monetary small open economy: a) only households demand foreign money and, then, buy the good produced by the foreign sector; b) participation in the international bond market is restricted to domestic households and the foreign sector; and c) bonds are redeemed in the holders monetary unit, i.e., in domestic money or in foreign money. Three shocks hit the economy every period: a technological shock, a government expenditure shock and a monetary shock. Within each period there is a markets timing, where markets subsequently open and close. First the input markets open, labor and capital are hired, and then close. Next, financial markets open, and money and real bonds are bought with the nominal and real assets carried over from previous periods plus government transfers. Both monetary and public expenditure shocks are introduced into the economy through these transfers. Then, financial markets close and the goods market opens. Households, government and the foreign sector buy goods with their monetary holdings. Finally, after the goods market closes, households receive labor compensation and capital returns, and investment projects are carried out. The remaining wealth is transferred to the next period in monetary holdings.

The seminal papers by Kydland and Prescott (1982) and Long and Plosser (1983) present the technological shock playing a crucial role leading the business cycle. Two main lines of improvement of the standard real business cycle model can be distinguished. The first en riches the seminal model without denying the essential role of the technological disturbance. The second considers additional disturbances to account for the role of these shocks in the business cycle. In this paper we follow this second line and we add two demand side shocks to the standard neoclassical stochastic growth model: a public expenditure shock and a monetary shock. The former shock was introduced by Christiano and Eichenbaum (1992) to explain the Durlop-Tarshis observation, by moving the supply of labor. However, the applications to the Spanish economy (Puch and Licandro, 1997) were not successful since the government expenditure shock was not able to ofset the efect of the technology shock

a model in a small open economy with nominal and real rigidities they study the mechanism of monetary transmission on the liquidity efect and the exchange rate behavior for the largest European economies.

⁷The theoretical foundations of the basic cash-in-advance model of money are carefully laid out in Lucas and Stokey (1983, 1987), Svensson (1985) and Sargent (1987, Ch.5).

in the labor market.

The monetary shock was introduced following Cooley and Hansen (1995) and Hairault and Portier (1995). Both extend Hansen (1985) to be a monetary economy, where money is demanded by its transaction purpose through a liquidity constraint requirement. This second demand shock and this liquidity restriction were expected to have important efects on the supply of labor and the volatility of real variables. Our work, however, difers from these monetary papers due the introduction of a stochastic process for government expenditure and the consideration of a small open economy set-up. In this respect, we believe that Spain can be considered a small open economy because, firstly, its volume of exports and imports in terms of GDP is significant; and secondly, the world interest rate is not set in the economy.

The main results are the following. First, this analysis shows that the proposed model with a liquidity constraint is able to reproduce the high volatility of private consumption present in the Spanish economy. In particular, the modeling presented here improves on the previous work applied to this economy by Puch and Licandro (1997) and Martín-Moreno (1998) in a real model framework, with respect to the volatility of this variable.

In second place, the efect of money appears to be important. When money is introduced through a liquidity requirement, we show below that incorporating the monetary growth shock, which originates a shift in labor supply, contributes to explain the correlation between hours worked and real wages present in the data. This result is novel in the business cycle literature and it is explained by the magnitude of the volatility of the monetary aggregates in the Spanish economy.

Finally, the lack of correlation between monetary base and prices with the GDP is also well reproduced by the model.

The remainder of this paper is organized as follows. Section 2 displays the cyclical properties of the data that the model will attempt to replicate. Section 3 describes the model. Section 4 discusses the choice of parameter values. We report the main results in section 5 under the assumption that there are stochastic shocks on technology, public expenditure and monetary growth. Finally in the last section we present the conclusions.

2 Cyclical properties of the data

Table 1 summarizes the statistical properties of Spanish business cycles using quarterly data for the period 1976:III-1998:IV.⁸ The first column of the table gives the standard deviation (“volatility”) of several macroeconomic variables in per capita terms measured as deviations from the trend. The second column gives the correlation between each variable and output. Before the statistics were calculated, all the data were logged and detrended using the Hodrick-Prescott filter.

With regard to these variables, the key properties are the following:

⁸We choose this sample for two reasons. First, as indicated by Puch and Licandro (1997), from 1976 on the Spanish data “approximates the long run properties of the Spanish economy” (p.365). Second, the time series available for monetary variables are restricted to this period, since the Banco de España (the Spanish Central Bank) only supplies the monetary aggregates until the fourth quarter of 1998, when the peseta was incorporated into the Monetary Union. As a consequence of this event, the composition of monetary aggregates changed.

1. private consumption is more volatile than output;

2. the negative correlation between hours and real wages; and,

3. the acyclicity of money and prices.

[Insert Table 1]

3 The model

The model is a monetary extention of Correia, Neves and Rebelo (1995) small open economy with perfect international mobility of capital framework. We present a small open economy version of Cooley and Hansen (1995) and Hairault and Portier (1995, Model I) monetary model, where money is valued in equilibrium due to an exogenous requirement to buy cash goods with money, i.e., a cash-in-advance restriction.

The economy consists of four agents: a representative infinitely lived agent, a representative firm, the government and the foreign sector. For the sake of simplicity we will assume that all variables are stationary and the population is constant along time, so all variables are represented in per capita terms.

The households

The economy is populated by a representative household, which obtains utility from consumption and leisure. It lives for infinite periods and takes prices as given at the markets in which they participate. The representative household maximizes its expected utility defined over stochastic sequences of consumption and labor

\[\mathcal {U} = E _ {0} \left\{\sum_ {t = 0} ^ {\infty} \beta^ {t} U (C _ {t}, N _ {t}) \right\}\]

where is the subjective rate of intertemporal discount. We assume that at every period the separable utility is identical and equal to:

\[U (C _ {t}, N _ {t}) = \frac {1}{1 - \sigma} \left[ (C _ {t} - \psi X _ {t} N _ {t} ^ {\nu}) ^ {1 - \sigma} - 1 \right]\]

where is the parameter of relative risk aversion, is the intertemporal elasticity substitution of labor supply and is positive. In order household preferences to be consistent with steady state growth the labor must increase at the level of technical progress . This technical progress is supposed to be associated with the eficiency of labor factor that grows at a constant rate (labor-augmenting technical progress Harrod-neutral):⁹

\[X _ {t + 1} = \gamma_ {x} X _ {t}\]

where the constant denotes the steady-state growth of the economy. The growthside of the model is, therefore, exogenous. To ensure that utility is finite, we assume that

X0 = 1 at t = 0

⁹Since this is a first order diference equation, we take as given X₀ = 1 at t = 0.

Households are endowed at each period with one unit of time that is allocated between work and leisure. They supply work to firms, and they also accumulate physical capital which is rented to firms. They are the owners of the firms and receive all the profits.

There are three securities in the economy: an international real bond foreign money and domestic money M. The representative agent has accessed to a perfectly competitive international capital market, where it can buy and sell international bonds at an (exogenous) international real interest rate . Then, at any level for the current account of the economy, an unbalanced in the trade balance can be ofset with purchases or sales of international bonds:

\[B _ {t + 1} = T B _ {t} + (1 + r ^ {*}) B _ {t}\tag{1}\]

where is the real trade balance.

The domestic firms

There exists only one international tradable good produced by the domestic firms and the foreign sector. The foreign sector supplies inelastically imported goods. A constant returnsto-scale technology produces the domestic output denoted by , combining the two production factors, the labor input and the aggregate capital stock . Given the assumption of constant returns, we can assume, without loss of generality, that there is only one competitive firm, that maximizes its profits subject to its own technology. In addition, the firm will make zero profits in equilibrium. Production can be described at the aggregate level by the production function:

\[Y _ {t} = F (Z _ {t}, K _ {t}, N _ {t}) = Z _ {t} K _ {t} ^ {1 - \alpha} (X _ {t} N _ {t}) ^ {\alpha}\]

where . The production level is afected by the level of the technical progress and by the productivity shock represented by . The technology shock consists of a single persistent component that evolves according to the law of motion that, in logaritms, follows an autoregresive of first order, :

\[l n Z _ {t} = \rho l n Z _ {t - 1} + \varepsilon_ {Z t}, \qquad 0 < \rho < 1\]

The random variable is normally distributed with zero mean and standard deviation. Production can be devoted, at zero cost, to the non-cash-good (the gross investment, , and to the cash-good (private consumption, , and public consumption, . The diference between production and domestic cash-good absorption is the trade balance That is, . The resources not consumed at each period are devoted to increase the stock of private physical capital of the following period. The investment at period t produces productive capital at period , and there is a cost of adjustment that depends on the net investment. The capital accumulation is given by the equation: , where is the constant rate of depreciation, so is the fixed capital consumption; and Φ is the adjustment cost function. This adjustment cost function is assumed to be homogeneous of degree zero This assumption allow us to transform the law of accumulation of capital into a per-capita individual restriction. In particular we take from Bruno and Portier (1995) the following quadratic functional form:

\[\Phi (K _ {t}, K _ {t + 1}) = \frac {\phi}{2} \left(\frac {K _ {t + 1} - K _ {t}}{K _ {t}}\right) ^ {2}\]

The government

The government uses the lump-sum real transfer revenue and money emission to finance its expenditure. The money injection can be used to directly finance public deficit or to reduce existing government debt. The government budget constraint holds at every period t:

\[G _ {t} + T _ {t} = \frac {M _ {t + 1}}{P _ {t}} - \frac {M _ {t}}{P _ {t}}\tag{2}\]

This study will consider two sources of fluctuation from the public sector: the per capita stock of money and the real government consumption, , which are assumed to be realizations of two exogenous stochastic processes correlated.

Analogous to Cooley and Hansen (1995) and Hairault and Portier (1995), the per capita money supply is assumed to grow at the rate in period t. That is , where is revealed at the beginning of period t. In addition, the real government consumption is a third source of fluctuation. The government expenditure is considered exogenous by the private sector and it has a stochastic component. The public consumption path and the monetary growth are assumed to be known by all agents in the economy. The monetary and public expenditure shocks are modeled according with the following first order autorregressive vector:¹⁰

\[\left[ \begin{array}{l} l n g _ {t} \\ l n \mu_ {t} \end{array} \right] = \left[ \begin{array}{l} g \\ \mu \end{array} \right] + \left[ \begin{array}{c c} \eta & v ^ {\mu} \\ v ^ {g} & \zeta \end{array} \right] \left[ \begin{array}{l} l n g _ {t - 1} \\ l n \mu_ {t - 1} \end{array} \right] + \left[ \begin{array}{l} \varepsilon_ {g t} \\ \varepsilon_ {\mu t} \end{array} \right]; \qquad \left[ \begin{array}{l} \varepsilon_ {g t} \\ \varepsilon_ {\mu t} \end{array} \right] \sim N \left(\left[ \begin{array}{l} 0 \\ 0 \end{array} \right], \left[ \begin{array}{c c} \sigma_ {g} ^ {2} & \varphi \\ \varphi & \sigma_ {\mu} ^ {2} \end{array} \right]\right)\tag{3}\]

where is the median of the stationary component of public expenditure and the average growth rate of money is equal to . The coeficient correlations are and . The parameter reflect the spillovers efect between these two sources of fluctuations. The contemporary correlation between innovations is . With this specification is guaranteed to be positive in every period, thus the cash-in-advance constraint is always binding. 11

3.1 The markets timing

There exist five markets in this economy: the domestic and the foreign security markets (the international real bond, domestic and foreign money), the cash-goods market, the non-cashgood market, and the factor markets (labor and capital). The cash-in-advance framework forces the use of both domestic and foreign money to buy cash-goods. The goods produced by domestic firms and foreign sector can only be bought with domestic money and foreign money, respectively. The model presented here relies on three assumptions: only households demand foreign money and, hence, buy the good produced by foreign sector; domestic households and the foreign sector are the participants in the international traded bond market; and bonds are redeemed at the holders monetary unit, i.e., in domestic money and in foreign money.

1 + r∗. 1 + r*.

µ¯_γx γx

¹⁰There is lots of evidence pointing toward the endogeneity of monetary and fiscal policies.

¹¹This also holds in the steady state, since real return of money is lower than the real return of bonds

Time is divided in a sequence of discrete periods. A new period appears when the agents information set incorporates the simultaneous stochastic shocks (technological, public expenditure and monetary); i.e., there is symmetric information since the shocks are known by all agents. Within a period, markets open and then close with the following timing.

At the beginning of any period t the realization of the shocks is known. The factor markets open, and firms hire labor and capital to produce the perishable good and then these markets closed.

Next, the domestic and foreign security markets open (i.e., domestic and foreign money, and the international real bond markets). At this subperiod securities carried over from the previous period are redeemed. Money has no nominal returns and bonds have an exogenous real interest rate . The domestic agent’s net holdings of bonds at period t, is the diference between the bonds issued by the foreign sector at t to obtain domestic money, and the bonds issued by the domestic agent at t to demand foreign money , i.e., The portfolio position, whether long or short, will depend on the surplus or deficit of the trade balance.

Real bonds, domestic and foreign money are demanded in each security market. On the one hand real bonds allow transferring wealth to the future. On the other, monetary holdings will be required to buy goods. Households’ nominal resources are given by their money holdings from previous period , and the transfers given by government at the beginning of period,¹² . Some of these holdings will be devoted to buy goods produced by domestic firms and the remaining, will be dedicated to buy the international bond issued by the foreign agent –net of returns from previous holdings–, i.e., . This domestic money will allow the foreign agent to buy the good produced at the domestic firms, i.e., exports. Next, households issue the international security, net of returns, to buy foreign money: . This foreign money will buy the good produced by the foreign sector for household consumption. Given that the Purchasing Power Parity holds for every period , the restriction in this subperiod domestic monetary terms are given by

\[M _ {t} ^ {c} + P _ {t} \bar {B} _ {t + 1} = M _ {t} + P _ {t} T _ {t} + P _ {t} (1 + r ^ {*}) \bar {B} _ {t}\tag{4}\]

\[e M _ {t} ^ {f} + P _ {t} (1 + r ^ {*}) \bar {B} _ {t} ^ {f} = P _ {t} \bar {B} _ {t + 1} ^ {f}\tag{5}\]

Observe that, given our three assumptions, no extra domestic money nor any presumption on trade balance are needed.¹³

¹²The monetary and the public expenditure shocks are introduced into the economy through these real transfers.

¹³For example Palivos and Yip (1997) assume implicitly, in our same monetary small open economy set-up that trade balance is always positive. This allow them to consider government bonds, instead of international bonds, because of its always positivity. They do not model this capital account in the international payment balance.

Next securities markets close and domestic and foreign cash-goods markets open. Households, the government and the foreign sector exchange domestic money for the good produced at domestic firms. Analogously, households buy with foreign money the good supplied by the foreign sector. That is

\[P _ {t} C _ {t} = M _ {t} ^ {c} + e _ {t} M _ {t} ^ {f}\tag{6}\]

\[P _ {t} X _ {t} = M _ {t} ^ {b}\tag{7}\]

\[P _ {t} G _ {t} = M _ {t + 1} - M _ {t} - P _ {t} T _ {t}\tag{8}\]

Constraint (6) is due to the assumption that only households buy the imported good; that is Then the goods markets close. A reinterpretation in net terms can be available. The notational change permits us to present the foreign balance as in (1), i.e., . Then, summing up (4) and (5), and after the substitution of (6)–(8)

\[P _ {t} \left(C _ {t} + G _ {t} + T B _ {t}\right) \leq M _ {t + 1}\tag{9}\]

This is the cash-in-advance constraint. The inequality comes from the fact that households can demand money for other uses rather than transaction. In this model money only afects the equilibrium if this restriction is active. This constraint, however, will always be binding. Otherwise money should have a higher real return than bonds, i.e., , and therefore bonds will never be demanded.

Finally, after the cash-goods market closes, households receive income from the domestic firms, and the non-cash-goods market opens. Households collect their income from input factors: employee compensation , and the gross operating surplus , where and are, respectively, the nominal wage and the nominal interest rate of capital. With this income -in nominal terms- and money not spent on consumption in the cash-goods market at period t, i.e. , household purchase the investment non-cash-good . The remaining money is held over to the next period

\[P _ {t} I _ {t} + M _ {t + 1} = W _ {t} N _ {t} + R _ {t} K _ {t} + \left[ M _ {t + 1} - P _ {t} \left(C _ {t} + G _ {t} + T B _ {t}\right) \right]\]

Finally, substitution of (1) and (2) onto this equation yields the representative household’s budget constraint (in real terms):

\[B _ {t + 1} + \frac {M _ {t + 1}}{P _ {t}} + C _ {t} + I _ {t} = w _ {t} N _ {t} + r _ {t} K _ {t} + \frac {M _ {t}}{P _ {t}} + B _ {t} (1 + r ^ {*}) + T _ {t}\tag{10}\]

where and are now real variables. Analogously, the substitution of (1) and (2) onto (9) gets the cash-in-advance constraint

\[C _ {t} \leq \frac {M _ {t}}{P _ {t}} + B _ {t} (1 + r ^ {*}) + T _ {t} - B _ {t + 1}\tag{11}\]

3.2 Agents’ problem

A) Representative Household.

The problem has three state variables and ; and five decision variables 7

and

max

s.t.

\[\begin{array}{r l} \max & \{C _ {t}, N _ {t}, B _ {t + 1}, M _ {t + 1}, K _ {t + 1}, I _ {t} \} _ {\tau = 0} ^ {\infty} E _ {0} \sum_ {t = 0} ^ {\infty} \beta^ {t} U (C _ {t}, N _ {t}) \\ \text {s.t.} & C _ {t} + I _ {t} + B _ {t + 1} + \frac {M _ {t + 1}}{P _ {t}} \leq w _ {t} N _ {t} + r _ {t} K _ {t} + \frac {M _ {t}}{P _ {t}} + B _ {t} (1 + r ^ {*}) + T _ {t} \\ & C _ {t} \leq \frac {M _ {t}}{P _ {t}} + B _ {t} (1 + r ^ {*}) + T _ {t} - B _ {t + 1} \\ & I _ {t} = K _ {t + 1} - (1 - \delta) K _ {t} + \Phi (K _ {t}, K _ {t + 1}) \\ & \text {given} K _ {0}, B _ {0}, M _ {0} \end{array}\tag{12}\]

Recall that the population does not grow, hence the aggregate investment function could be presented in per capita terms.

B) Representative Firm.

The firm seeks to maximize profits, which are equal to:

\[\begin{array}{r l} \max _ {K _ {t}, N _ {t}} & Y _ {t} - w _ {t} N _ {t} - r _ {t} K _ {t} \\ \mathrm{s.t.} & Y _ {t} = F (Z _ {t}, K _ {t}, N _ {t}) = Z _ {t} K _ {t} ^ {1 - \alpha} N _ {t} ^ {\alpha} \end{array}\tag{13}\]

where .

C) Government.

\[G _ {t} + T _ {t} = \frac {M _ {t + 1}}{P _ {t}} - \frac {M _ {t}}{P _ {t}}\tag{14}\]

From the agents’ problem we obtain the following optimality conditions,

\[\begin{array}{r c l} \left(C _ {t} - \psi X _ {t} N _ {t} ^ {\nu}\right) ^ {- \sigma} & = & \lambda_ {t} + \mu_ {t} \\ \lambda_ {t} + \mu_ {t} & = & \beta (1 + r ^ {*}) E _ {t} [ \lambda_ {t + 1} + \mu_ {t + 1} ] \\ \lambda_ {t} \left(1 + \gamma_ {x} ^ {t} \frac {\phi}{K _ {t}} \left(\frac {K _ {t + 1}}{K _ {t}} - \gamma_ {x}\right)\right) & = & \beta E _ {t} \left[ \lambda_ {t + 1} \left((r _ {t + 1} + 1 - \delta) + \gamma_ {x} ^ {t + 1} \frac {\phi}{K _ {t + 1}} \left(\frac {K _ {t + 2} - \gamma_ {x} K _ {t + 1}}{K _ {t}}\right) \frac {K _ {t + 2}}{K _ {t + 1}}\right) \right] \\ \lambda_ {t} & = & \beta E _ {t} \left[ (\lambda_ {t + 1} + \mu_ {t + 1}) \frac {P _ {t}}{P _ {t + 1}} \right] \end{array} \tag {15}\]

\[\psi \nu N _ {t} ^ {\nu - 1} X _ {t} (C _ {t} - \psi X _ {t} N _ {t} ^ {\nu}) ^ {- \sigma} = \lambda_ {t} w _ {t}\]

jointly with the resource restriction (10) and the cash-in-advance constraint (11).

The first order conditions for the firms are:

\[\begin{array}{r c l} \frac {\partial F (Z _ {t} , K _ {t} , N _ {t})}{\partial K _ {t}} & = & r _ {t} \\ \frac {\partial F (Z _ {t} , K _ {t} , N _ {t})}{\partial N _ {t}} & = & w _ {t} \end{array}\]

Finally the transversality conditions are:

\[\begin{array}{r c l} \lim _ {t \to \infty} E _ {0} (\beta^ {t} \lambda_ {t} K _ {t + 1}) & = & 0 \\ \lim _ {t \to \infty} E _ {0} (\beta^ {t} \lambda_ {t} B _ {t + 1}) & = & 0 \\ \lim _ {t \to \infty} E _ {0} (\beta^ {t} \lambda_ {t} M _ {t + 1}) & = & 0 \end{array}\]

3.3 Definition of the equilibrium

A Competitive equilibrium is given by a set of decision rules , where is the state variables vector of representative household; the firm’s capital and labor demand, where is the state variables vector of the firm; such that, given and the exogenous path processes , it holds:

[1] Given maximizes the intertemporal problem of the representative agent (12).

[2] Given and the technology , maximizes (13), i.e., their profits subject to their own technology.

[3] Markets Clear:

Goods Markets

\[Y _ {t} = C _ {t} + I _ {t} + G _ {t} + T B _ {t}\]

Trade Balance

\[T B _ {t} = B _ {t + 1} - B _ {t} (1 + r ^ {*})\]

Labor Market

\[N _ {t} ^ {d} = N _ {t} ^ {s}\]

Monetary Market

\[M _ {t + 1} = M _ {t} + P _ {t} (T _ {t} + G _ {t})\]

3.4 The steady state

In order to study the properties of the model we must first describe the steady state, i.e., without the presence of stochastic shocks. This steady state will characterize the long run properties of the economy as well as provide us with the parametric value of the variables without growth. We will find the solution of the competitive equilibrium around these stationary variables once we have introduced the stochastic shocks.

It is worth writing the decentralized problem and the steady state as a function of the stationary variables in the convergence direction in a non-stochastic set-up, i.e., , where will represent any variable without growth. Then, a steady-state equilibrium is an equi librium in which the absence of uncertainty is assumed , and where grow at the same rate and the other variables are constant. From the optimality conditions of the described problem the steady state of the economy can be found

from the following equations:

\[1 = \beta \gamma_ {x} ^ {- \sigma} (1 + r ^ {*})\tag{16}\]

\[N = \left[ \frac {\alpha Z N ^ {\alpha - 1} k ^ {1 - \alpha}}{\psi \nu} \frac {\gamma_ {x}}{\bar {\mu}} \frac {1}{1 + r ^ {*}} \right] ^ {\frac {1}{\nu - 1}}\tag{17}\]

\[1 + \phi \left(\frac {\gamma_ {x} - 1}{k}\right) = \beta \gamma_ {x} ^ {- \sigma} \left[ (r + 1 - \delta) + \phi \left(\frac {\gamma_ {x} - 1}{k}\right) (1 + \gamma_ {x}) \right]\tag{18}\]

\[c = \frac {m}{p} - g - t b\tag{19}\]

\[c + (\gamma_ {x} - 1 + \delta) k + (\gamma_ {x} - 1) ^ {2} \frac {\phi}{2} + g + t b = Z N ^ {\alpha} k ^ {1 - \alpha}\tag{20}\]

\[t b = - (1 + r ^ {*} - \gamma_ {x}) b\tag{21}\]

The first equation is a condition to be held by the parameters of the model. It relates the growth rate of the economy with the subjective rate of the intertemporal discount of consumers. This condition can be justified from the general equilibrium perspective: if the real world were several little and identical economies as the one described here, the interest rate at equilibrium in world markets would be endogenous and could be found at (16). The number of working hours to be supplied by the representative agent at every period can be found in equation (17). Equation (18) relates the capital-labor ratio with the international interest rate. The cash-in-advance constraint at the steady state is given by equation (19). The resource restrictions and the accumulation of foreign securities, equations (20) and (21) respectively, set the private consumption and the trade balance.

4 Calibration

The model described above, under the assumption of rational expectations, presents condi tions of stochastic optimality and non-linearities that make it impossible to have an analytical solution of the variables. Therefore we must find a numerical solution to characterize a stochastic process of such variables from the realization of the structural shocks. Later, this will allow us to analyze the economic relations among the variables of the model in a stochastic dynamic competitive equilibrium.

Following Kydland and Prescott’s approach, we calibrate the model based on microeconomic evidence and also on long-run considerations, i.e., we require that values for the parameters be chosen in such a way that the model steady state values are close to average for the Spanish economy over the data period being studied. This principle is based on the assumption that the Spanish economy is moving around its balanced growth path during the period, which implies that stationary variables move around the observed averages.¹⁴

Table 2 reports the parameter values from the quarterly data of reference for the Spanish economy at the considered period.

¹⁴This is one of the reasons why the period 1976:III-1998:IV is chosen. Puch and Licandro (1997) pointed out that Spanish data are able to reproduce the long-run properties of the Spanish economy better from 1976 on.

[Insert Table 2]

Following Christiano and Eichenbaum (1992), we fix the individual’s productive time endowment to 1369 hours per quarter. To calibrate the labor share we follow the European Economy (1994). The quarterly world interest rate is taken from the value suggested by Kydland and Prescott (1982). The parameterization of preferences is borrowed from Greenwood et al. (1988), , and from Mendoza . All other parameters, except were chosen in order that the balanced growth path conditions of the model should hold for the average data for the period 1976:III-1998:IV; that is, the quarterly gross growth rate the depreciation rate , the subjective rate of discount ψ and the international foreign bonds Parameter is chosen so that the variability of investment relative to output would be suitably reproduced by the model.

To complete the calibration, the parameters corresponding to technology, government expenditure and monetary growth stochastic processes have to be set. First, the public expenditure is the Public Consumption series at the National Accounts for the period considered. Second, the monetary growth rate are obtained making use the Monetary Base. We choose this monetary aggregate because we consider it is consistent with the present model where money is demanded only for transaction purposes. This variable, however, has been widely used throughout the period of study to carry out diferent monetary policies. Therefore two outliers can be reported: one in IV:83, and the other in VI:89. This finding coincides with the turning period of tight monetary policy carried out by Spanish policymakers.

[Insert Figure 1]

To take into account the two outliers we filtered the series using one dummy variable for each exogenous outlier. Then, making use (3), we estimate the VAR with the corresponding parameters. The estimates obtained are:

\[\left[ \begin{array}{c c} \eta & v ^ {\mu} \\ v ^ {g} & \zeta \end{array} \right] = \left[ \begin{array}{l l} 0. 9 8 5 5 & 0. 0 0 6 2 \\ (0. 0 1 2) & (0. 0 1 2) \\ 0. 0 4 9 6 & 0. 3 2 9 1 \\ (0. 0 3 4) & (0. 0 5 7) \end{array} \right]; \left[ \begin{array}{c c} \sigma_ {g} ^ {2} & \varphi \\ \varphi & \sigma_ {\mu} ^ {2} \end{array} \right] = \left[ \begin{array}{c c} 0. 0 0 5 8 ^ {2} & 0. 1 2 3 7 \\ 0. 1 2 3 7 & 0. 0 3 0 1 ^ {2} \end{array} \right]\]

where the numbers in parenthesis are standard errors. Observe that the spillovers between both shocks are not significantly diferent from zero.

Finally, to calibrate the corresponding technological stochastic process parameters we compute the Solow residual for this economy. It is worth noting that the Solow residual found from our reference data shows too high of a standard deviation and serial correlation in order to reproduce the volatility of the GDP for the Spanish economy. Hence we calibrate and in such a way that the model reproduces the volatility of our production measurement in the presence of the three shocks.¹⁶

¹⁵This parameter of relative risk aversion is usually taken in the literature. See, for example, Mendoza (1991). On the other hand, Prescott (1986) points that this parameter might not be much higher than 1.

¹⁶Kollintzas and Vassilatos (1996), Correia et al. (1995) and Mendoza (1991) reproduce this volatility in the presence of the technology shock only. On the other hand, McCallum (1989, p.28-29) pointed out that when the adjustment costs and the fluctuations in the terms of trade are taken into account, the Solow residual is not a suitable proxy for the productivity shocks.

5 Main findings

In this section we discuss our empirical results for the model under consideration. Table 3 displays a summary of statistics for the simulated economy with only technology shocks operating, with public expenditure and technology shocks and, finally, the results with monetary, technology and public expenditure shocks. These are the averages of statistics computed from 100 simulations of 90 periods, taken logarithms, and filter each simulated time series using H-P filter.

[Insert Table 3]

Firstly, we report that the behavior of the variables with only a productivity shock is very similar to the behavior of the variables shown with both productivity and public expenditure shocks. Although with a diferent model than ours, this result is similar to Puch and Licandro (1997) for the Spanish calibration, whilst Christiano and Eichenbaum (1992) improve in reducing the correlation between hours and real wages.

In evaluating the model with these diferent shocks, we focus our attention on the three features present in the Spanish economy data. First, the high volatility of consumption relative to output. As has been stated in Section 2, private consumption is highly volatile in the Spanish economy, 1.15. With the technological shock and the public expenditure shock, the volatility reproduced in the model for this variable is 0.91. Although the model does not reproduce completely the volatility of this variable, it is worth noting that the artificial economy reproduces its relative volatility higher than what was found in standard real business cycles, like Puch and Licandro (1997) and Martín-Moreno (1998) for the Spanish economy, or for other models, such as the one proposed by Bruno and Portier (1995) for the French economy. However, if we consider the implications of adding a monetary growth shock to this model, the first important finding is that this relative volatility of private consumption is accurately reproduced. This improvement was suggested by Puch and Licandro (1997) and Dolado et al. (1993), when they pointed out that a better empirical performance along this dimension may need either an alternative specification of the households’ preferences or some specification of liquidity constraint. Both characteristics have been introduced in the present paper.

With respect to the role the labor market plays in the Spanish economy, the model is able to reproduce the volatility of hours relative to the volatility of productivity of labor in the presence of technology and public expenditure shocks. However, the model fails dramatically at approximating the observed correlation between hours and productivity, like in the relative volatility of hours. The incorporation of a shock on monetary growth into the analysis produces an outstanding improvement in the performance of the model along this dimension. On the one hand, the volatility of hours increases about 50%; and although it is not able to reproduce the volatility of this variable, the improvement is important. On the other hand, the introduction of monetary shocks helps to explain the correlation between hours and real wages present in the Spanish data. Observe that in the previous case, the volatility of government spending can not compensate for the efect of technological shocks. That is why the model predicts a strong positive correlation between hours and real wages (0.99). In the presence of the three shocks, the demand moves rightwards due to the technological shock, and the supply moves leftwards due to monetary and public expenditure shocks. The correlation found between hours and real wages, as in the Spanish data, suggests that the efect on the labor supply is fairly strong.

Finally, the model understates the volatility of money and overstates the volatility of prices. However, the acyclicity of both variables is accurately reproduced in this monetary model.

6 Conclusions

The aim of this paper is a theoretical and empirical study of the consequences of introducing a cash-in-advance constraint into a small open economy business cycle model for the Spanish economy. A business cycle model is built extending Correia, Neves and Rebelo (1995) small open economy framework and Cooley and Hansen (1995) monetary economy. Money is introduced through a cash-in-advance constraint. The relevant questions to be answered for the Spanish economy were the following: can a monetary model in a small open economy framework explain the higher volatility of private consumption?; is this model able to reproduce the negative correlation between hours and real wages?; and, how well does this monetary model reproduce the nominal dimension? To answer these questions the model is parameterized, simulated and calibrated with a set of measurements of the aggregate variables of the Spanish economy.

Our main findings are that the introduction of both a liquidity constraint and a monetary growth rate shock into the analysis contributes to the explanation of these stylized facts about the Spanish economy: i) the so-called Dolado et al. puzzle, that is, the high volatility of private consumption; ii) the Dunlop-Tarshis observation, i.e., the negative correlation between hours and productivity; and iii) some cyclical features of the nominal dimension.

References

1. 1. J. Andrés, J.D. López-Salido and J. Vallés (1999) “The liquidity efect in a Small Open Economy Model”, Working Paper. Estudios sobre la Economía Española, 21, FEDEA.
2. 2. Bruno and F. Portier (1995) “A small open economy RBC model: the French economy case”, in Pierre-Yves Henin (ed.) Advances in Business Cycle Research. With Application to the French and US Economies, Springer, Berlin.
3. 3. L.J. Christiano and M. Eichenbaum (1992) “Current Real Business Cycles Theories and Aggregate Labor Market Fluctuation”, American Economic Review, 82, pp.430- 450.
4. 4. T.F. Cooley and G. Hansen (1995) “Money and the Business Cycle” in Cooley (ed.), Frontiers in Business Cycle Research. Princeton University Press
5. 5. I. Correia, J.C. Neves and S. Rebelo (1995) “Business Cycle in a Small Open Economy”, European Economic Review, 39, pp.1089-1113.
6. 6. J. J. Dolado, M. Sebastián and J. Vallés (1993) “Cyclical Patterns of the Spanish Economy”, Investigaciones Económicas, vol.XVII (3) pp.445-473.
7. 7. European Economy (1994) n.58
8. 8. M. Friedman (1956) A Theory of the Consumption Function, National Bureau of Economic Research. New York.
9. 9. J. Greenwood, Z. Hercowitz and G. Hufman (1988) “Investment, Capacity Utilization and the Real Business Cycle”, American Economic Review, 78, pp.971-987.
10. 10. J.O. Hairault and F. Portier (1995) “Cash-In-Advance Constraint and the Business Cycle”, in Pierre-Yves Henin (ed.) Advances in Business Cycle Research. With Application to the French and US Economies, Springer, Berlin.
11. 11. G. Hansen (1985) “Indivisible Labor and the Business Cycle,” Journal of Monetary Economics, 16, pp.309-327.
12. 12. T. Kollintzas and V. Vassilatos (1996) “A Stochastic Dynamics General Equilibrium Model for Greece” CEPR Dicussion Paper, 1518.
13. 13. F. Kydland and E. Prescott (1982) “Time to Build and Aggregate Fluctuation” Econometrica, 50, pp.1345-1370.
14. 14. J.B. Long and C.I. Plosser (1983) “Real Business Cycle” Journal of Political Economy, 91, pp.1345-1370.
15. 15. R. Lucas and N. Stockey (1983) “Optimal Fiscal and Monetary Policy in an Economy without Capital.” Journal of Monetary Economics, 12, pp.55-93.
16. 16. R. Lucas and N. Stockey (1987) “Money and Interest in a Cash-in-Advance Economy.” Econometrica, 55, pp.491-514.
17. 17. J. M. Martín-Moreno (1998) “Ciclos Reales en Economías Abiertas. Una Aplicación al caso Español”, Moneda y Crédito, n.207, pp.87-113.
18. 18. B. T. McCallum (1989) “Real Business Cycles Models”, in R. J. Barro (ed.) Modern Business Cycle Theory, Harvard University Press, Cambridge. pp.16-50.
19. 19. E. G. Mendoza (1991) “Real Business Cycles in a Small Open Economy” American Economic Review, 81, pp.791-818.
20. 20. F. Modigliani (1966) “The Life Cycle Hypothesis of Saving, the Demand for Wealth and the Supply of Capital”, Social Research, vol.33 n.2.
21. 21. T. Palivos and C. K. Yip (1997) “The Gains from Trade for a Monetary Economy once again” Canadian Journal of Economics, XXX, n.1 pp.208-223.
22. 22. E. Prescott (1986) “Theory Ahead of Business Cycle Measurement” Quarterly Review of the Federal Reserve Bank of Minneapolis, 10 (4), pp.9-33.
23. 23. L. A. Puch and O. Licandro (1997) “Are there any Special Feature in the Spanish Business Cycle?”, Investigaciones Económicas, XXI(2), pp.361-394.
24. 24. T. Sargent (1987) Dynamic Macroeconomy Theory. Cambridge, Mass. Harvard University Press.
25. 25. L. E. O. Svensson (1985) “Money and Asset Prices in a Cash-in-Advance Economy.” Journal of Political Economy, 93, pp.919-944.

Appendix

Table 1 Properties of the detrended Spanish data 1976:III - 1998:IV.

GDP–real GDP, 1986 pts.; Consumption–private consumption, 1986 pts; Money–monetary base; Prices–Consumer Price Index, all items, base 1986; Hours–number of hours worked. ∗ We made use the series built by Puch and Licandro (1997). However, if alternative data base were considered the results would be similar. For example, the sample of hours worked reported by EPA (Active Population Survey) for 1976:III - 1998:II the correlation with labor productivity is -0.23.

Table 2.- Parameters of the Economy

Calibration criteria: [1] Set from the model at steady state, [2] External information, [3] Sample average, [4] calibration of the 2nd order moments, and [5] properties of the stochastic process.

Figure 1: Monetary Base Growth Rate

Table 3.- Simulation

Where we represent [1] Technological shock, [2] Government expenditure shock and [3] Monetary growth shock.

All the statistics for the models are averages, across 100 simulated data sets, each with 90 observations. For each variable x we show the relative volatility as the percentage of the standard deviation of the variable x with respect to volatility of output, and is the contemporaneous correlation of the GDP with the variable x.