2.3 The Results
3.1.4 Organization
In the following section, we set up the environment for our analysis. This includes a pre- sentation of the reduced form of the Benigno (2004) model, the interest rates rules to be considered, a brief summary of the methodology used to obtain the determinacy and E- stability results and a calibration of the model's underlying parameters. In section 3.3
we then present the results for the contemporaneous interest rate rules, while section 3.4
3.2 The Environment
We rst look at the underlying structure of the Benigno (2004) model and then present it in its reduced form. The monetary policy rules are then presented, which together with the core equations of the model give our monetary union system. Finally, we put forward a calibration for the models underlying parameters, which will be used in following sections when presenting our results.
3.2.1 The baseline model
Here, we outline the main features of the Benigno (2004) model.1 Consider a monetary union consisting of two countries; (H)ome and (F)oreign. The union contains a contin-
uum of households spread over the unit interval, where those residing in home country are spread over[0; n)while residents of the foreign country occupy(n;1]. That is, the size of
countryHisnand that of countryF is(1 n). Each household is both a consumer and a
monopolistically competitive producer. For simplicity, the population sizes are assumed to equate to the economic sizes of the two regions.
The consumer problem
Each household consumes the bundles produced in both countries and the elasticity of substitution between these two bundles is assumed to be one. In addition, the sov- ereign governments of the two regions each consume bundles produced in their respective
1 In this section, we summarize the main features of the framework. This serves our purpose and readers
countries. They also provide nominal lump sum transfers to households while imposing a proportional tax on nominal income.
Asset markets are complete domestically while positions can be taken in an in- ternationally traded bond. However, given the assumption of consumption preferences and assuming no initial holdings of the international bond, it becomes redundant. A corollary of this is that there is perfect risk sharing of consumption between regions, i.e.
CtH = CtF = Ct, for all t.2 Eventhough consumers in both countries are guaranteed the
same level of consumption, output can still vary due to uctuations in the terms of trade and different levels of government spending. The market clearing conditions are:3
YtH =Tt1 nCt+GHt and YtF =Tt nCt+GFt (3.1)
Where the terms of trade is de ned as the price of the foreign consumption bundle over the price of the home country consumption bundle, i.e. Tt
PF;t
PH;t.
Firms and price setting
Households produce a differentiated product that makes part of the domestic con- sumption bundle, where the elasticity of substitution across goods produced within a coun- try is greater than one, as in Dixit and Stiglitz (1977).
2 See the companionAppendix Aof Benigno (2004).
3 This expression differs from the market clearning conditions of the previous two chapters since there is
now government consumption. In this chapter we assume that government expenditure is not actively used to stabilize economic uctuations but is instead set at a steady state value. In the following chapter we consider the case in which scal policy is used in tandem with monetary policy to stabilize shocks.
Following Calvo (1983), each producer of countryifaces a xed probability(1 i)
of changing their price in each period, for i = H; F. The average length of a contract
1
1 i is therefore the same for all agents in countryibut not necessarily so for agents in
different countries. We then keep the average price contract length xed for the union as a whole (we de ne the union average length of a contract as 1
1 H
n 1
1 F
1 n
), while letting the relative contract length between the two countries vary, we can isolate the effect that heterogeneous price rigidities has on our results. In this chapter , we will without loss of generality assume that the home country has a higher degree of price rigidities than the foreign country.
Log-linearisation around steady state
We now look at the log-linearized form of the model under the assumption of sticky prices. For the generic variableX, we have de ned the log deviation from its steady state
value as x = x +x, where x is the exible price value and x the deviation from this
due to prices being sticky. We further de ne union average variables as; XW nXH +
(1 n)XF and relative variables as:XR XF XH.
The system is reduced to the following four equations:
yWt =Et(y W t+1) 1 (rt Et( Wt+1)) rrt (3.2) H t = (1 n)k H T (Tt) +kHCy W t + Et( Ht+1) (3.3)
F t = nk F T Tt +kCFy W t + Et( Ft+1) (3.4) Tt =Tt 1 + Ft H t Et Tt (3.5)
WhereyWt is the union output gap, W
t ; Ht and Ft are the union, home and foreign
country's respective in ation levels, rrt is the natural rate of interest familiar from the
closed economy literature and can be interpreted as a government expenditure shock over the natural level of output: rrt + Et( y
W
t+1 gWt ). The deviation of the terms of
trade from its natural level is denoted byTt.
The inverse of the relative risk aversion is given by; UCCC
UC , while the inverse
of the elasticity of producing goods is de ned as: VyyC
Vy . The supply schedule coef -
cients are then given by:
ki
C [(1 i )(1 i)= i] [( + )=(1 + )]
and
ki
T kCi [(1 + )=( + )] fori=H; F
where is the elasticity of substitution across goods produced within a region. Equa- tion(3:2)is an IS type curve for the union as a whole and is similar to those in the closed
economy literature. Equations(3:3) and(3:4)are the supply schedules of the two union
members and the last equation(3:5), is the condition for the terms of trade.
Because the terms of trade enter the supply schedules and hence affects in ation, there are three different distortions to the economy;(i)the deadweight loss from the inef-
cient price and production levels caused by monopolistic competition,(ii)the rigidity in prices caused by staggered price setting and(iii)the distortions to the terms of trade caus- ing disequilibrium of relative prices. Benigno (2004) assumes a subsidy in accordance with the framework to neutralize the rst of these three distortions.4 Monetary policy can then achieve an ef cient outcome only in the case where nominal rigidities are equal across the two regions. This is because only homogenous price rigidities implies that deviations of the terms of trade from equilibrium are neutral for the region as a whole. When this is the case we have thatkH
T = kFT = kT and thatkCH = kCF = kC, thus using the de nition of a
union variablexW nxH + (1 n)xF we can combine equations(3:3)and(3:4)to get a
union and a relative supply schedule:
W t =kCy W t + Et( Wt+1) (3.6) R t = kT Tt + Et( Rt+1) (3.7)
In this case the system can be partitioned, where(3:2)and(3:6)determine the union
variablesnyWt ; W t
o
and(3:5)and (3:7)determinen R t; Tt
o
. We show below that as in the previous chapter, the results from the closed economy literature on learning hold for this model when price rigidities are equal in the two countries.
In the more general case, when the average length of price contracts differ in the two regions, it is useful to write equation(3:2)as:
yWt =Et(y W t+1)
1 (r
t nEt( Ht+1) (1 n)Et( Ft+1)) rrt (3:20)
This yields a system with four equations determining the four state variables;
n
yWt ; H
t ; Ft; Tt
o
. While the rst three of these are free, the terms of trade is prede- termined.