A Small Open Economy DSGE model for an Oil Producer Economy: The Case of Venezuela∗ Jes´ us Morales†and Francisco S´aez‡. PRELIMINARY VERSION September 21, 2007

1

Introduction

Oil prices have risen dramatically over the last few years. While by the end of 2001 the price of WTI crude oil was about US$19 per barrel, in July 2006 it reached US$73, and in January 2007 it was US$48. Changes in oil prices have a direct and relevant impact on government budget for most oil producer countries. As a consequence, changes in oil prices can affect dramatically government expenditure and investment decisions. Although such shocks can generate large fluctuations in oil producer economies, little attention have been devoted to the analysis of the propagation of such shocks and to the optimal policy response of monetary policy for these economies. Most research has focused on the analysis of an oil-price shock from the point of view of an oil importer country. See, for example, Leduc and Sill((2004), (2006)) and Medina and Soto (2005). In this paper we aim to make a contribution on filling this gap by proposing a model that considers explicit a price-oil shock from the point of view of an oil producer economy. To this end, we extend the open economy DSGE model of Gal´ı and Monacelli (2005) by incorporating an oil shock and imperfect capital mobility on it.1 The model exhibit ∗

The views expressed are those of the individual authors, and do not necessarily reflect the position of

the Banco Central de Venezuela. We are responsible for any remaining errors. † Oficina de Investigaciones Econ´omicas, Banco Central de Venezuela, email: [email protected] ‡ Oficina de Investigaciones Econ´omicas, Banco Central de Venezuela, email: [email protected] 1 Dynamic stochastic general equilibrium models with nominal stickiness have become popular tools

1

nominal rigidities with Calvo-type staggered price-setting and two different transmission mechanisms of an oil shock: through direct transfers to the households and through changes in government investment in capital accumulation. A simple way to model an oil producer economy is to treat the oil resources as a pure rent associated to transfers from abroad. This strategy, however, could be problematic if the model assume perfect capital mobility or consider only one aggregate good. In this case, an increase in oil revenues reduces the non-oil output since the income effect induces people to enjoy more leisure. This result is not consistent with the empirical evidence in most oil producer economies which shows that oil booms are usually followed by an expansion in the non-oil sector. The assumption of perfect capital mobility, when the economy is driven only by terms of trade shocks, left unchange investment, and this is clearly against the cyclical patterns of the economy when the oil shock occurs. Modelling an oil producer economy, therefore, requires special features to account for the procyclical behavior of investment during a positive oil shock. These features could be related with (a) preferences, (b) capital mobility, (c) intensity in the use of capital, (d) domestic bias, (e) wealth effect, (f)exchange rate policy, and (g) expenditure switching. S´aez and Puch (2004) highlight the importance of preference. Based on a modification of the framework proposed by Bruno and Portier (1995), these author show that the GHH utility function together with some degree of imperfect capital mobility are important elements to represent the dynamic of output in a small open economy. These elements contribute to account for the degree of correlation of consumption and investment observed in the data. This is because with GHH preference the income effect does not affect leisure decisions. Besides, the output gap can go up, given that the imperfect capital mobility contributes to increase the domestic stock of capital. The law of capital accumulation implies that output reacts to the oil shock with a lag. In order to deal with this problem, in this paper we consider the possibility of variable capital utilization. In this paper, we also consider public capital as a factor of production which can amplify the impact of the oil shock on total factor productivity. This effect is reinforced by the demand side thought the inclusion of some domestic bias in public investment. Finally, the fiscal dominance in the economy also leads us to adopt an explicit structure for monetary policy analysis in recent years. See, for instance, Altig, Christiano, Eichenbaum and Linde (2005), Benigno and Benigno (2003), Christiano, Eichenbaum and Evans (2005), Clarida, Gali and Gertler (1999), Goodfriend and King (1998), Rotemberg and Woodford (1998), Schmitt-Grohe and Uribe (2001).

2

for modelling the link between the fiscal and the monetary sector. In the particular case of Venezuela, the main instrument for monetary policy has been the use of international reserves. This is because policymakers have frequently used changes in the exchange rate to deal with the expansion of high power money caused by the fiscal expenditure. This kind of policy rule imply a rise in the exchange rate market intervention when the monetary authority perceives an increasing inflation rate below the target. This behavior can have an effect over the economy dynamics very different from the one resulting from a simple Taylor rule. Moreover, with price stickiness an inflation target based on nominal anchor could induce fluctuations associated with the expenditure swishing. This strategy for modelling the monetary sector allows us to consider a comprehensive range of possibilities for the exchange rate flexibility and can be used to connect the model with the ’fear to floating’ literature. This structure also helps to connect the model with the analysis of economies with dual exchange rate, which still remain an unexplore area in the general equilibrium modelling literature. Some details of the model are still to be written and we are working on the calibration of the model.

3

2

The model

We build on the work of Gal´ı and Monacelli (2005) and extend their small open economy model by incorporating an oil shock and imperfect capital mobility on it. As in their model, households maximize a utility function consisting of consumption and leisure. However, in our model, households receive utility from the holding of cash balances. They consume and invest in baskets consisting of domestically produced and imported goods. In what follows, we sketch the model set up for households and firms. We also describe the behavior of the central bank and the fiscal authority.

2.1

Households

A typical small open economy is inhabited by a representative household who seeks to maximize: U = E0

(∞ X t=0

 σ )  mt a m , β t ln (Ct − ψNtθ Xt ) + σ zt

(1)

where 1−γc Xt = Ctγc Xt−1 ,

(2)

and E0 denotes the expectation conditional on the information available at time zero. zt is a permanent technology shock and we assume that 0 < β < 1, θ > 0, ψ > 0, and σ > 0. The presence of the variable Xt implies that preferences are time non-separable in consumption and hours worked. We use the preferences of Jaimovich and Rebelo (2006) (the reasons will be explain in later versions)2 . Nt denotes hours of labour, and Ct is a composite consumption index defined by: η  η−1  η−1 η−1 1 1 η η η , Ct = (1 − αc ) η CH,t + αc CF,t

(3)

where CH,t is an index of consumption of domestic goods given by the CES function: ε Z 1  ε−1 ε−1 ε CH,t = CH,t (j)dj , (4) 0

2

These preferences nest as special cases the two classes of utility functions most widely used in the

business cycle literature. When γc = 1 we obtain preferences in the class discussed in King, Plosser and Rebelo (1988). When γc = 0 we obtain the preferences proposed by Greenwood, Hercowitz and Huffman (1988).

4

where j ∈ [0, 1] denotes the good variety. CF,t is an index of imported goods given by: γ Z 1 γ−1  γ−1 γ CF,t = Ci,t , (5) 0

where Ci,t is an index of the quantity of goods imported from country i and consumed by domestic households. It is given by an analogous CES function: ε  ε−1 Z 1 ε−1 . Ci,t = Ci,tε (j)dj

(6)

0

The parameter ǫ > 1 denotes the elasticity of substitution between varieties. Parameter αc ∈ [0, 1] is inversely related to the degree of home bias in preferences, and is thus a natural index of openness. Parameter η > 0 measures the substitutability between domestic and foreign goods, from the viewpoint of the domestic consumer, while γ measures the substitutability between goods produced in different foreign countries. Domestic households can accumulate capital and rent its services to domestic firms. They have access to three different types of assets: money mt , one-period noncontingent ∗ foreign bonds (denominated in foreign currency) Bp,t , and one-period domestic bonds bt

which pays out a risk free nominal return of ibt bt in the next period. The maximization of (1) is subject to two restrictions. The first one is a sequence of budget constraints and the second is the law of motion of capital. The sequence of budget constrains is Z 1  Z 1Z 1 ∗ εt Bp,t  ∗ (1 + τc ) PH,t (j)CH,t (j)dj + Pi,t (j)Ci,t (j)djdi + bt + ε B 0 0 0 (1 + i∗t )Θ Ptc Ytt t

+Ptc mt

+ (1 +

τk )Pti It

= (1 +

ibt−1 )bt−1

+ Wt Nt +

∗ εt Bp,t−1

+

c Pt−1 mt−1

+ Tt +

Rtk ut K t

+ Πt , (7)

for t = 0, 1, 2, . . . , where Pi,t (j) is the price of variety j imported from country i. Wt is the nominal wage, Tt are per-capita lump-sum net transfers from the government, Pti It is nominal investment in capital, Πt are profits, and εt is the nominal exchange rate. All the previous variables are expressed in units of domestic currency. The term Θ(.) determines the premium domestic households have to pay each time they borrow from abroad. It is a function of the aggregate net foreign asset position to GDP ratio,

εt Bt∗ . Ptc Yt

The fact that the premium depends on the aggregate net asset position

(and not the individual position) implies that households take it as an exogenous variable when optimizing. The presence of this risk premium guaranties a well defined steady state in the model, as shown by Schmitt-Grohe and Uribe (2001). 5

This implies that domestic households are charged a premium over the (exogenous) foreign interest rate i∗t if the domestic economy as a whole is a net borrower (Bt∗ < 0), and receive a lower remuneration on their savings if the domestic economy is a net lender (Bt∗ > 0). The optimal allocation of any given expenditure within each category of goods yields the demand functions: CH,t (j) = 

Ci,t (j) =



Pi,t (j) Pi,t

PH,t (j) PH,t

−ε

−ε

Ci,t ,

CH,t ,

(8)

∀i, j ∈ [0, 1],

(9)

where PH,t is the domestic price index defined by: PH,t =

Z

1

PH,t

1−ε

(j)dj

0

1  1−ε

,

(10)

and Pi,t is a price index for goods imported from country i defined by: Pi,t =

Z

1

Pi,t

1−ε

(j)dj

0

1  1−ε

.

(11)

In addition, the optimal allocation of expenditures on imported goods by country of origin implies: Ci,t =



Pi,t PF,t

−γ

CF,t ,

∀i ∈ [0, 1],

(12)

where PF,t is the price index for imported goods, defined by: PF,t ≡

Z

1

1−γ di Pi,t

0

1  1−γ

.

(13)

Finally, the optimal allocation of expenditures between domestic and imported goods are given by: CH,t = (1 − αc )



PH,t Ptc

−η

Ct ,

(14)

and CF,t = αc



PF,t Ptc

6

−η

Ct ,

(15)

where 1−η 1−η Ptc = (1 − αc )PH,t + αc PF,t

1
1−η

.

(16)

Using equations (8)-(16), we can write total consumption expenditures by domestic households as: Z 1 Z PH,t (j)CH,t (j)dj + 0

0

1

Z

1

Pi,t (j)Ci,t (j)djdi = PH,t CH,t + PF,t CF,t = Ptc Ct .

(17)

0

Thus, the period budget constraint (7) can be rewritten as: (1 + τc )Ptc Ct + bt +

∗ εt Bp,t  ∗  + Ptc mt + (1 + τk )Pti It ε B ∗ (1 + it )Θ Ptc Ytt t

= (1 +

ibt−1 )bt−1

+ Wt Nt +

∗ εt Bp,t−1

+

c Pt−1 mt−1

+ Tt + Rtk ut K t + Πt .

(18)

The households can increase their capital services Kt by investing It in additional ¯ t , taking one period to come into action, or by directly increasing the physical capital K ¯ t , where ut is the rate of utilization rate of the physical capital stock at hand Kt = ut K capital utilization. Both operations undertake a cost. The capital accumulation equation is given by K t+1 = (1 − δ(ut ))K t + S



It It−1



It , with K 0 > 0,

(19)

where S(It /It−1 ) represents adjustment costs that are incurred when the level of investment changes over time. We assume that S(γy ) = 1, S ′ (γy ) = 0, so there are no adjustment costs in the steady state, and that S ′′ (γy ) < 0. This adjustment cost formulation is proposed in Christiano et al. (2005). The function δ(ut ) represents the rate of capital depreciation. We assume that depreciation is convex in the rate of utilization: δ ′ (ut ) > 0, δ ′′ (ut ) ≥ 0. We assume that the basket invested by households is given by It =



1 η

η−1 η

1 η

η−1 η

(1 − αi ) IH,t + αi IF,t

η  η−1

,

(20)

where IH,t and IF,t are indexes of investment goods given by CES functions as in equations (4) and (5). The demand for the two types of investment goods from the households are IH,t = (1 − αi )



PH,t Pti

−η

It and IF,t = αi

7



PF,t Pti

−η

It ,

where the deflator of investment expenditure is given by: 1−η 1−η Pti = (1 − αi )PH,t + αi PF,t

Denote by Rt = H =E0

∞ X t=0

+Ptc mt

1 . 1+ibt

1
1−η

.

(21)

The Lagrangean of the household problem can be formulated as:

  σ ∗  εt Bp,t am m t θ  ∗ β ln (Ct − ψNt Xt ) + + λ1t (1 + τc )Ptc Ct + bt + ε B σ zt (1 + i∗t )Θ Ptc Ytt t

+ (1 +

t

τk )Pti It

− (1 +

ibt−1 )bt−1

− Wt Nt −     It 2 It , + λt K t+1 − (1 − δ(ut ))K t − S It−1

∗ εt Bp,t−1

−

c Pt−1 mt−1

− Tt −

Rtk ut K t

+ Πt

(22)

where the control variables of this problem are: Ct , Nt , Bp,t , bt , ut , mt , and It .

2.2

System of normalized equations

Because there exist a technology shock that grows at rate γy and a unit-root in the price level, a number of variables are non-stationary as they contain a real and nominal stochastic trend. We stationarize these variables by dividing all quantities with the trend level of technology zt = γyt and multiplying the Lagrangian multiplier with it. We let small letters indicate that a variable have been stationarized, and for the multiplier we introduce the notation λ1z,t = λ1t zt Ptc . The new variables are defined as: Ct = ct zt

Xt = xt zt

mt = mz,t zt

Yt = yt zt

It = it zt

Rtk = rtk Ptc

Wt = wt zt Ptc

∗ Bp,t = b∗p,t zt

8



The set of normalized first order condition are:  θ −1  Nt+1 ct xt+1 1 − γc ψNtθ c−1 t xt ct : − βψγc (1 − γc )Et + (1 + τc )λ1z,t = 0, at at+1 where at − ct + ψNtθ xt = 0, 1−γc  xt−1 γc = 0. and xt − ct γy (ψθxt Ntθ−1 ) − λ1z,t wt = 0. Nt : − at  ∗ b ∗ εt+1 πt+1 } = 0. Bp,t : εbt − (1 + it )Θ t Rt Et {b yt   λ1 −1 z,t+1 1 bt : λz,t Rt − βEt γy = 0. πt+1      rtk it it it i,c ′ It : (1 + τk )γt − ′ γy γy + S γy . . . S δ (ut ) it−1 it−1 it−1 "  (  2 #) k rt+1 it+1 it+1 ′ = 0. γy γy S +Et πt+1 ′ δ (ut+1 ) it it mt :

am (mz,t )σ−1 + λ1z,t (1 − Rt ) = 0,

between

and

Pty ,

and Rt = 1/(1 +

(24) (25) (26) (27) (28)

(29) (30)

c where πt+1 = Pt+1 /Ptc is the inflation rate of the economy, γtx,y =

Ptx

(23)

Ptx Pty

is the relative price

ibt ).

The normalized law of motion of capital is:   it k t+1 γy − (1 − δ(ut ))k t − S (31) γy it = 0. it−1

The steady state and the log-linearization of equations (23)-(31) are in the appendix.

2.3 2.3.1

Firms Technology

A typical firm in the home economy produces a differentiated good with a technology characterized by α2

Yt (i) = εat Kts (i)α1 (γty Nt (i))1−α1 K g,t − γyt Φ

(32)

where zt = γyt is a permanent technology shock, εat is a covariance stationary technology shock, Nt (i) and Kts (i) are labour and capital services hired by the ith firm, K g,t = Kg,t /K t is an externality that captures the roll of government capital in affecting private output,

9

and α1, α2 ∈ (0, 1). This form of the production function implies that the public services are complementary with the private inputs in the sense that an increase in Kg,t rises the marginal product of labour and private capital. The inclusion of the fixed cost γyt Φ ensure that profits are zero in steady state. The fixed cost is assumed to grow at the same rate as output do in steady state so that profits would not become positive because of the presence of monopoly power. For the stationary shock in (32) we assume the following process: ln(εat ) = (1 − ρz ) ln(εa ) + ρz ln(εat−1 ) + ηta ,

with ηta ∼ N (0, σa2 ).

(33)

The cost minimization problem facing the firm that produces de differentiated good i in period t, assuming that PH,t (i) is given, is: M = min s

Kt ,Nt (i)



Wt Nt (i) + Rtk Kts (i) + λt PH,t (i)[Yi,td − Yt (i)] .

(34)

where Rtk is the nominal rental rate per unit of capital services, Wt is the nominal wage, and Yi,td is good’s i demand for the given price PH,t (i). The first order conditions of (34) with respect to Kts and Nt are ∂M α2 = Rtk − λt PH,t (i)εat α1 Kts (i)α1 −1 (γyt Nt (i))1−α1 K g,t = 0, s ∂Kt (i) ∂M α2 = Wtk − λt PH,t (i)εat (1 − α1 )Kts (i)α1 (γyt )1−α1 Nt (i)−α1 K g,t = 0. ∂Nt (i)

(35) (36)

Combining the first order conditions and using the fact that the capital-labor ratio is equal across firms, we obtain the following condition: Kts = where Kts =

R1 0

Kts (i)di and Nt =

R1 0

α1 Wt Nt , 1 − α1 Rtk

(37)

Nt (i)di.

The Lagrangian multiplier in equation (34), λt PH,t (i), can be interpreted as the nominal cost of producing one additional unit of the domestic good. This nominal marginal cost is the same across firms and equals to:   α1  1−α1
−α2 a −1 1 1 M Ct = (εt ) , Wt1−α1 (Rtk )α1 (γyt )α1 −1 K g,t α1 1 − α1

and the real marginal cost in normalized form equals:   α1  1−α1
−α2 a −1 M Ct 1 1 mct = (εt ) . = γtc,H wt1−α1 (rtk )α1 K g,t PH,t α1 1 − α1 10

(38)

(39)

2.3.2

Price-setting

Following Calvo (1983) we assume that each individual firm resets its price with probability 1−ξp each period, independently of the time elapsed since its last price adjustment. Hence, a measure 1 − ξp of randomly selected firms set new prices each period, while a fraction ξp is not allow to reoptimize its prices, and its price in period t + 1 is indexed to last period domestic inflation, πH,t =

PH,t , PH,t−1

and the steady state inflation rate. Let Ptnew denote the

price set by a firm i adjusting its price in period t. With probability ξps , for s = 0, 1, 2, . . . , a firm is not allow to change its price during s periods ahead, the price in period t + s will  Q s  κp l−κp new be Pt+s (i) = Pt l=1 πH,t+l−1 πH,ss , where κp is an indexation parameter. Since all firms resetting prices in any given period will choose the same price, we henceforth drop the i subscript. When setting a new price in period t a firm seeks to maximize the current value of its dividend stream, conditional on that price being effective: ) # ( " s  ∞  Y X l−κ κ p πH,ssp − M Ct+s yt+s πH,t+l−1 (ξp )s Et (β s vt+s ) Ptnew max new Pt

s=0

(40)

l=1

subject to 

yt+s = 

Ptnew

Qs

l=1



κ

l−κ

p πH,ssp πH,t+l−1

PH,t+s

 −ε 

d , yt+s

where the firm is using the stochastic discount factor β s vt+s to make profits conditional upon utility. β is the discount factor, vt+s the marginal utility of the households’ nominal income in period t + s, which is exogenous to the firms, and M Ct+s is the firm’s nominal marginal cost. The first order condition of the firms’ optimization problem can be written as ∞ X

ξps Et

s=0

or

∞ X

(

(ξp β) Et

where µ∗ =

s Y

yt+s l−κ κp πH,ssp yt+s + εM Ct+s new πH,t+l−1 (β s vt+s ) (1 − ε) Pt l=1

s

s=0

"



#)

= 0,

κ    PH,t+s−1 p (l−κp )s ∗ new πH,ss − µ M Ct+s = 0. vt+s yt+s Pt PH,t−1

ε . ε−1

11

Defining mct+s ≡

M Ct+s , PH,t+s

Pbtnew ≡

Ptnew , t πH,ss

condition in term of stationary variables " ( ∞ X t+s vt+s )yt+s Pbtnew (ξp β)s Et (πH,ss s=0

and PbH,t+s ≡ PbH,t+s−1 PbH,t−1

!κp

PH,t+s t+s , πH,ss

we can write the first order

− µ∗ PbH,t+s mct+s

#)

= 0.

Log-linearizing the previous condition around the steady state we obtain   ∞ new X e e e e s − κp PbH,t−1 = (1 − ξp β) Pbt f t+s − κp PbH,t+s−1 + PbH,t+s , (ξp β) Et mc s=0

where mc f t+s is the log deviation of the real marginal cost from its steady state. This

expression can be written into a more compact form as     new   new eb eb eb eb e e bH,t + (1 − ξp β)mc f t . (41) bH,t = ξp β Et P t+1 − P H , t − ξp βκp π P t − P H,t−1 − π

From the aggregate price index (10) follows that     e new eb eb eb eb P H,t = ξp κp P H,t−1 − P H,t−2 + P H,t−1 + (1 − ξp ) Pbt .

(42)

Combining equations (41) and (42) yields the following aggregate Phillips curve e π bH,t =

β κp e (1 − ξp ) (1 − ξp β) e Et {π bH,t+1 } + π bH,t−1 + mc f t. 1 − βκp 1 − κp β ξp (1 − βκp )

(43)

Note that for the case of no indexation, κp = 0, with zero inflation in the steady state, πH,ss = 1, this relation reduces to a Phillips curve as in Gal´ı and Monacelli (2005), π eH,t = βEt {e πH,t+1 } +

1−ξp (1 ξp

− ξp β)mc f t.

12

2.4

Government behavior

The government budget constraint is given by c

i

εt ∆Rest + Pt g Ctg + Pt g Itg + ibt−1 bt−1 + Tt = τc Ptc Ct + τk Pti It + εt Ytp + (bt − bt−1 ) + (Mt − Mt−1 ),

(44)

where Ctg and Itg are government consumption and investment, Ytp is oil revenues in foreign currency, ∆Rest is changes in international reserves, bt is issuance of new bonds by the government, and ∆Mt = Mt − Mt−1 is the change in money supply. Government capital evolves according to g Kt+1 = Itg + (1 − δg )Ktg , δg ∈ (0, 1).

(45)

We follow Kollintzas and Vassilatos (2000) assuming that fiscal policy follows feedback rules and react endogenously to changes in oil revenues. In particular, government consumption, government investment, and transfers in normalized form are determined by  p εc εt yt g p ct = yss , (46) cg sc p yss Pt  p εi εt yt g p it = yss , (47) p ig si yss Pt  p εT yt εt p s yss , (48) Tt = p c T Pt yss where Ctg = γyt cgt , Itg = γyt igt , Tt = Ptc γyt T t , sj , j = c, i, T , are the shares of each of the components of public expenditure to the steady state value of oil revenues, and εj , j = c, i, T , are the elasticities of public expenditure to oil revenues changes. We assume that oil revenues is given by Ytg = γyt ytp , where ytp follows an stochastic process defined by an autoregressive process of order one p p ) + ρp ln(yt−1 ) + εpt , with εpt ∼ N (0, σp2 ). ln(ytp ) = (1 − ρp ) ln(yss

The basket consumed and invested by the government are given by Ctg

η   η−1 1 η−1 η−1 1 g g η η η η + αcg (CF,t ) = (1 − αcg ) (CH,t )

13

(49)

and Itg

=



(1 − αig )

1 η

η−1 g (IH,t ) η

+

1 η

η−1 g ) η αig (IF,t

η  η−1

,

g g g g where CH,t , CF,t , IH,t , and IF,t are indexes of consumption and investment goods given by

CES functions as in equations (4) and (5). The demand for the two types of consumption goods from the government are −η −η   PF,t PH,t g g g Ct and CF,t = αcg Ctg , CH,t = (1 − αcg ) c c Pt g Pt g and for the investment goods are −η −η   PF,t PH,t g g g It and IF,t = αig Itg , IH,t = (1 − αig ) ig ig Pt Pt where the deflator of consumption expenditure is given by: c

1−η 1−η Pt g = (1 − αcg )PH,t + αcg PF,t

and the deflator of investment expenditure is given by: i

1−η 1−η + αig PF,t Pt g = (1 − αig )PH,t

2.5

1
1−η

1
1−η

,

(50)

.

(51)

The Central Bank

Because in the particular case of Venezuela the main instrument for monetary policy has been the use of international reserves, we assume that the central bank controls the supply of foreign currencies to the private sector through the use of the following rule  ηvd πt s vdp,ss zt . vdp,t = πss

(52)

The change in international reserves is then defined as ∆Rest = Rest+1 − Rest = −vdg,t − vdp,t ,

(53)

where vdg,t is the flow of the government balance of payments, minus government net exports, and is defined by vdg,t = −xng,t , where government net exports is given by g g ∗ xng,t = Ytp − PF,t (CF,t + IF,t ).

14

(54)

The variable vdp,t denotes the private sector balance of payments, which balanced through movements between the private sector and the central bank, that is, vdp,t = vdsp,t for all t; and is given by



vdp,t = 

Bt∗ (1 +

i∗t )Θ



εt Bt∗ Pt Yt



∗   − Bt−1 − xnp,t ,

(55)

where net exports by the private sector is !−γ PH,t PH,t ∗ ∗ xnp,t = YF,t − PF,t (CF,t + IF,t ). ∗ εt εt PF,t Note that equation (53) represents the evolution of net foreign assets. [Discuss the role of the elasticity parameter ηvd ] The change in the stock of domestic money ∆Mt equals the change in the Central Bank domestic credit ∆Dt plus the change in the domestic currency value of international reserve εt ∆Rest ,3 ∆Mt = ∆Dt + εt ∆Rest .

(56)

Finally, we assume that changes in domestic credit follow the rule Dt+1 = ρt Dt .

2.6

Aggregation

Before closing the model we derive some useful aggregate expressions. 2.6.1

Aggregate demand R1 Let Yt ≡ 0 Yt (j)dj represent an index for aggregate domestic output. Goods market

clearing in the home economy requires

Z

1
i i Yt (j) = CH,t (j) + IH,t (j) + + + CH,t (j) + IH,t (j) di 0 # !−γ −ε " 
PH,t PH,t (j) g g ∗ ∗ , (57) + IF,t CF,t CH,t + IH,t + CH,t + IH,t + = ∗ PH,t εt PF,t g CH,t (j)

3

g IH,t (j)

It is assume that the central bank does not monetize changes in the domestic currency value of

international reserves arising from changes in the official exchange rate.

15

i i for allj ∈ [0, 1] and for all t, where CH,t (j) + IH,t (j) denotes country i’s demand for good j

produced in the home economy. To obtain the second equality we use the demand equations (8),(9), and (12); and the analogous equations for investment, government expenditure, and foreign countries, together with our assumption of symmetric preferences across countries, which implies that i i (j) = (j) + IH,t CH,t



PH,t (j) PH,t

−ε

PH,t ∗ εt PF,t

!−γ

∗ , YF,t

∗ ∗ ∗ where YF,t ≡ CF,t + IF,t is the optimal allocation of any foreign country on goods produced

in the home economy. Using the equations for domestic demand and plugging equation (57) into the definition of aggregate domestic output we obtain !−γ

PH,t ∗ YF,t ∗ εt PF,t  −η −η −η   PH,t PH,t PH,t = (1 − αc ) Ctg Ct + (1 − αi ) It + (1 − αcg ) c Ptc Pti Pt g !−γ  −η PH,t P H,t ∗ +(1 − αig ) Itg + YF,t . (58) ig ∗ εt PF,t Pt

g g Yt = CH,t + IH,t + CH,t + IH,t +

2.7

Market clearing conditions

2.7.1

Aggregate resource constraint

The domestic goods market clears when the aggregate demand from the households, the private sector, and the government given by (58) equals domestic production. From equations (32), (37), and (58) the equilibrium resources constraint from the production perspective must satisfies   1−α1 −η −η PH,t PH,t 1 − α1 Rtk t t 1−α1 α2 K g,t − γy Φ = (1 − αc ) Ct + (1 − αi ) It (γy ) α1 Wt Ptc Pti !−γ −η −η   PH,t PH,t PH,t g g ∗ YF,t . (59) It + +(1 − αcg ) Ct + (1 − αig ) c i ∗ εt PF,t Pt g Pt g

εat Kts



16

or, in normalized form:   1−α1 −η −η PH,t PH,t 1 − α1 rtk α2 k g,t − Φ = (1 − αc ) ct + (1 − αi ) it α1 wt Ptc Pti !−γ −η  −η  ∗ P PH,t P H,t H,t g g ∗ zt y , (60) i + +(1 − αcg ) ) c + (1 − α i c t t g F,t i ∗ εt PF,t zt Pt g Pt g

εat kts



where that Yt∗ has been scaled with zt∗ which is the reason why zt∗ /zt appears in the formula. zt∗ is supposed to follow a similar process as zt . We assume that z˜t∗ =

zt∗ zt

is a

stationary shock which measures the degree of asymmetry in the technological progress in the domestic economy versus the rest of the world. By assuming z0∗ = z0 = 1 this implies ∗ that the technology levels must be the same in steady state, z˜ss = 1. We assume that the

asymmetric technology shock follows the process (log-linearized) ∗ z˜ˆt+1 = ρz˜∗ zˆ˜t∗ + εz˜∗ ,t+1 .

2.7.2

Foreign economy

We follow Adolfson, Laseen, Linde and Villani (2007) in assuming that foreign output and ∗ interest rate are exogenously given. Let Xt∗ = [ yF,t i∗t ]′ , where i∗t is foreign interest rate ∗ and yF,t foreign output. The foreign economy is modelled as a VAR model, ∗ 2 F0 Xt∗ = F (L)Xt−1 + εX ∗ ,t , with εX ∗ ,t N (0, σX ∗ ).

2.8

Calibration

To be written

2.9

Simulations

To be written

2.10

Conclusions

To be written

17

References Adolfson, Malin, Stefan Laseen, Jesper Linde, and Mattias Villani (2007) ‘Bayesian estimation of an open economy dsge model with incomplete pass-through.’ Journal of International Economics 72(2), 481–511 Altig, David, Lawrence Christiano, Martin Eichenbaum, and Jesper Linde (2005) ‘Firmspecific capital, nominal rigidities and the business cycle.’ NBER Working Papers 11034, National Bureau of Economic Research, Inc, January Benigno, Gianluca, and Pierpaolo Benigno (2003) ‘Price stability in open economies.’ Review of Economic Studies 70(4), 743–764 Bruno, C., and F. Portier (1995) A small open economy RBC model: the french economy case en Pierre-Yves Henin, advances in business cycle research ed. (Springer) Calvo, Guillermo A. (1983) ‘Staggered prices in a utility-maximizing framework.’ Journal of Monetary Economics 12(3), 383–398 Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans (2005) ‘Nominal rigidities and the dynamic effects of a shock to monetary policy.’ Journal of Political Economy 113(1), 1–45 Clarida, Richard, Jordi Gali, and Mark Gertler (1999) ‘The science of monetary policy: A new keynesian perspective.’ Journal of Economic Literature 37(4), 1661–1707 Gal´ı, Jordi, and Tommaso Monacelli (2005) ‘Monetary policy and exchange rate volatility in a small open economy.’ Review of Economic Studies 72(3), 707–734 Goodfriend, Marvin, and Robert G. King (1998) ‘The new neoclassical synthesis and the role of monetary policy.’ Technical Report Greenwood, Jeremy, Zvi Hercowitz, and Gregory W Huffman (1988) ‘Investment, capacity utilization, and the real business cycle.’ American Economic Review 78(3), 402–17 Jaimovich, Nir, and Sergio Rebelo (2006) ‘Can news about the future drive the business cycle?’ NBER Working Papers 12537, National Bureau of Economic Research, Inc, September

18

King, Robert G., Charles I. Plosser, and Sergio T. Rebelo (1988) ‘Production, growth and business cycles : I. the basic neoclassical model.’ Journal of Monetary Economics 21(2-3), 195–232 Kollintzas, Tryphon, and Vanghelis Vassilatos (2000) ‘A small open economy model with transaction costs in foreign capital.’ European Economic Review 44(8), 1515–1541 Leduc, Sylvain, and Keith Sill (2004) ‘A quantitative analysis of oil-price shocks, systematic monetary policy, and economic downturns.’ Journal of Monetary Economics 51(4), 781–808 Leduc, Sylvain, and Keith Sill (2006) ‘Monetary policy, oil shocks, and tfp: accounting for the decline in u.s. volatility.’ Technical Report Medina, Juan Pablo, and Claudio Soto (2005) ‘Oil shocks and monetary policy in an estimated dsge model for a small open economy.’ Working Papers Central Bank of Chile 353, Central Bank of Chile, December Rotemberg, Julio J., and Michael Woodford (1998) ‘An optimization-based econometric framework for the evaluation of monetary policy: Expanded version.’ NBER Technical Working Papers 0233, National Bureau of Economic Research, Inc, May Schmitt-Grohe, Stephanie, and Martin Uribe (2001) ‘Stabilization policy and the costs of dollarization.’ Journal of Money, Credit and Banking 33(2), 482–509 S´aez, Francisco J., and Luis A. Puch (2004) ‘Shocks externos y fluctuaciones en una econom´ıa petrolera.’ Serie Documentos de Trabajo 59, Banco Central de Venezuela

19

APPENDIX A

Households equations

Para cualquier nivel de consumo Ct , cada familia compra una composici´ on de bienes dom´esticos e importados en el periodo t para minimizar el costo total de su canasta de consumo. Por lo tanto, cada familia resuelve el problema Z 1 Z min PH,t (j)CH,t (j)dj + {CH,t ,Ci,t }i,j∈[0,1]

0

0

1Z 1



Pi,t (j)Ci,t (j)djdi ,

0

(A-1)

Sujeto a la restricci´ on expresada por la ecuaci´ on (3). Este problema puede ser planteado de la siguiente forma Lagranjeana: Z 1  Z 1Z 1 F = min PH,t (j)CH,t (j)dj + Pi,t (j)Ci,t (j)djdi {CH,t ,Ci,t }i,j ∈[0,1] 0 0 0   η ) η−1

1

1

η−1

η +λt (1 − αc ) η CH,t + αcη CF,tη

η−1

.

(A-2)

Las condiciones de primer orden del problem est´ an dados por: ∂F ∂CH,t (j) ∂F ∂Ci,t (j) ∂F ∂Ci,t

− η+1

1

−1

= PH,t (j) + λt (1 − αc ) η Ct−1 CH,t η CH,tε (j) = 0, − η+1 η

1

= Pi,t (j) + λt αcη Ct−1 Ci,t

−1

Ci,t ε (j) = 0,

−1

1

−1 = 0. = Pi,t + λt Ct−1 αcη Ci,tγ CF,t

(A-3) (A-4) (A-5)

La funci´ on de consumo definida por la ecuaci´ on (8) se obtienen de: Por la ecuaci´ on (A-3) se tiene que: − η+1

1

−1

PH,t (j) = −λt (1 − αc ) η Ct−1 CH,t η CH,tε (j).

(A-6)

Por lo que: −1

CH,tη (j) PH,t (j) . = − η1 PH,t (i) CH,t (i)

(A-7)

Despejando CH,t y usando la definici´ on de las ecuaciones (4) y (10) se llega a la funci´ on de demanda de consumo expresada en la ecuaci´ on (8). Ahora bien, la ecuaci´ on (9) se obtiene de los siguientes procedimientos: Como: 1

− η+1

−1

Pi,t (j) = −λt αcη Ct−1 CF,t η Ci,t ε (j),

20

∀j ∈ [0, 1],

(A-8)

entonces, −1

Ci,tη (j) Pi,t (j) . = − η1 Pi,t (i) Ci,t (i)

(A-9)

Despejando Ci,t y utilizando la definici´ on de la ecuaci´ on (6) y (11) se obtiene la funci´ on de demanda de consumo de la ecuaci´ on (9). De la definici´ on denotada por la ecuaci´ on (12) proviene: Por ecuaci´ on (A-5): −1

1

−1 Ci,tγ . Pi,t (j) = −λt αcη Ct−1 CF,t

(A-10)

Por lo tanto: −1

Ci,tγ Pi,t = −1 Pi0 ,t Ci0 ,tγ

∀i, i0 ∈ [0, 1].

(A-11)

Utilizando la definici´ on de las ecuaciones (5) y (13) se obtiene la ecuaci´ on (12) Las ecuaciones (14) y (15) provienen de: min

{CH,t ,Ci,t }i,j∈[0,1]

[PH,t CH,t + Pi,t Ci,t ]

(A-12)

Sujeto a la ecuaci´ on (3). El Lagranjeano de este problema esta dado por: h η−1 i η η−1 1 1 η−1 η + αcη CF,tη G = PH,t CH,t + PF,t CF,t + λt (1 − αc ) η CH,t

(A-13)

Obteniendo as´ı, las siguientes condiciones de primer orden:

1 −1 ∂G = PH,t + λt (1 − αc ) η CH,tη Ct−1 = 0, ∂CH,t (j)

(A-14)

1 −1 ∂G = PF,t + λt αcη Ct−1 CF,tη = 0, ∂CF,t (j)

(A-15)

lo cual nos permitir´ a obtener las definiciones expuesta en la ecuaci´ on (14) y (15), ya que igualando las ecuaciones (A-14) y (A-15) se tiene que: CF,t =

i F,t −η

hP

PH,t

21

αc CH,t . 1 − αc

(A-16)

Despejando CF,t de la ecuaci´ on (3) y sustituyendo en (A-16) se obtiene la ecuaci´ on (14), realizando la misma operaci´ on con la ecuaci´ on (3) pero en este caso despejando CH,t y sustituyendo en (A-16) se obtiene la ecuaci´ on (15). El problema de los hogares puede expresarse de la forma Lagrangiana:    ∞ ∗  X εt Bp,t am mt σ  ∗ H =E0 + λ1t (1 + τc )Ptc Ct + bt + β t ln (Ct − ψNtθ Xt ) + ε B σ zt (1 + i∗ )Θ tc t t=0

t

Pt Yt

 ∗ c +Ptc mt + (1 + τk )Pti It − (1 + ibt−1 )bt−1 − Wt Nt − εt Bp,t−1 − Pt−1 mt−1 − Tt − Rtk ut K t + Πt     It 2 (A-17) + λt K t+1 − (1 − δ(ut ))K t − S It , It−1

∗ , b , u , m y I . donde las variables de control son: Ct , Nt , Bp,t t t t t

Las siguientes condiciones de primer orden de este problema estan dadas por: ∂H ∂Ct

=

(A-18)

1−γc At = Ct − ψNtθ Xt y Xt = Ctγc Xt−1

donde ∂H ∂Nt ∂H ∂mt ∂H ∂Bp,t

θ C −1 X βψγc (1 − γc )Nt+1 1 − γc ψNtθ Ct−1 Xt t+1 t + (1 + τc )λ1t Ptc − = 0, At At+1

(ψθXt Ntθ−1 ) − λ1t Wt = 0. At   1 mt σ−1 = am + λ1t Ptc − βλ1t+1 Ptc = 0. zt zt λ1t εt  ∗  − βλ1t+1 εt+1 = 0. = ε B ∗ (1 + it )Θ Ptc Ytt = −

(A-19) (A-20) (A-21)

t

∂H ∂bt ∂H ∂ut ∂H ∂It

= λ1t Rt − βλ1t+1 = 0.

(A-22)

′

(A-23) = −λ1t Rtk + λ2t δ (ut ) = 0.         2  It It It It+1 It+1 ′ ′ = (1 + τk )λ1t Pti − λ2t S +S + βλ2t+1 S It−1 It−1 It−1 It It2 = 0. (A-24)

donde Rt = 1/(1 + ibt ). Despejando βλ1t+1 de la ecuaci´ on (A-22) y sustituyendo en la ecuaci´ on (A-20) se obtiene:  σ−1 mt am + zt λ1t Ptc (1 − Rt ) = 0. (A-25) zt De igual manera, despejando βλ1t+1 de la ecuaci´ on (A-22) y sustituyendo en la ecuaci´ on (A-21) se obtiene que: εt − (1 +

i∗t )Θ



εt Bt∗ Ptc Yt

22



Rt εt+1 = 0.

(A-26)

Ahora bien, despejando λ2t de la ecuaci´ on (A-23) y sustituyendo en la ecuaci´ on (A-24) se tiene que: (1 +

τk )λ1t Pti

   2       k βλ1t+1 Rt+1 λ1t Rtk It It It It+1 It+1 ′ ′ = 0, − ′ + ′ +S S S δ (ut ) It−1 It−1 It−1 δ (ut+1 ) It It2

(A-27)

por ecuaci´ on (A-22)se tiene que: βλ1t+1 = λ1t Rt ,

(A-28)

entonces, la ecuaci´ on (A-28) se puede reescribir de la siguiente manera: "     2 #    k R k R I R I I I I ′ ′ t t+1 t t+1 t t t+1 S (1 + τk )Pti − ′ t + ′ = 0. +S S δ (ut ) It−1 It−1 It−1 It It δ (ut+1 )(1 + ibt ) (A-29) Por lo tanto, el sistema de ecuaciones para el problema de los hogares esta definido de la siguiente forma:  σ−1 mt + λ1t Ptc zt (1 − Rt ) = 0. (A-30) mt : am zt   εt Bt∗ ∗ Bt : εt − (1 + it )Θ Rt εt+1 = 0. (A-31) Ptc Yt "     2 #    k R k R I R I I I I ′ ′ t t+1 t t+1 t t t+1 S ut : (1 + τk )Pti − ′ t + ′ +S S δ (ut ) It−1 It−1 It−1 It It δ (ut+1 )(1 + ibt ) = 0. ct :

1−

(A-32)

γc ψNtθ Ct−1 Xt At

+ (1 + τc )λ1t Ptc −

βψγc (1 −

θ C −1 X γc )Nt+1 t+1 t

At+1

= 0,

(A-33)

At :

At − Ct + ψNtθ Xt = 0,

(A-34)

Xt :

1−γc Xt = Ctγc Xt−1 .

(A-35)

Nt : λt :

(ψθXt Ntθ−1 )

− λ1t Wt = 0. At λ1t Rt − βλ1t+1 = 0.

−

(A-36) (A-37)

De la ecuaci´ on (18) se obtiene: (1 + τc )Ptc Ct + bt +

∗ εt Bp,t  ∗  + Ptc mt + (1 + τk )Pti It ε B ∗ (1 + it )Θ Ptc Ytt t

∗ c −(1 + ibt−1 )bt−1 − Wt Nt − εt Bp,t−1 − Pt−1 mt−1 − Tt + Rtk ut K t + Πt = 0.

De la ecuaci´ on (19) se obtiene: K t+1 − (1 − δ(ut ))K t − S

23



It It−1



It = 0.

(A-38)

(A-39)

Sistema de Ecuaciones en Estado Estacionario 1 θ −1 css xss ] + (1 + τc )λ1z,ss = 0. [1 − (1 + β(1 − γc ))γc ψNss ass θ ass − css + ψNss xss = 0.

(A-41)

−γc γc −1 γy = 0, 1 − cγssc xss

(A-42)

θ−1 −ψθxss Nss

− λ1z,ss wss = 0.  ∗  b 1 − (1 + i∗ss )Θ ss Rss = 0. yss Rss − β/(γy πss ) = 0.  i k rss P − (1 + τk ) = 0. P c ss δ ′ (uss ) ass

(A-40)

(A-43) (A-44) (A-45) (A-46)

am (mz,ss )σ−1 + λ1z,ss (1 − Rss ) = 0.

(A-47)

[γy − 1 + δ(uss )]k ss − iss = 0.

(A-48)

donde las funciones que a continuaci´ on se presentan conservan la siguiente extructura:

δ(ut ) = δ0 uw t .

(A-49)

Θ (x)

=

(A-50)

S (x)

=


con φb , b > 0. exp −φb (x − b)    g1 g1 . exp (−g2 (x − γy )) − 1 + 1 + g3 g1 exp (x − γy ) + g2 g2

Cabe destacar que la funci´ on S(.) preserva las siguientes propiedades: S(γy ) = 1 S ′ (γy ) = 0 S ′′ (γy ) = g3 g1 (1 + g2 ).

24

(A-51)