2.2 The Environment
2.2.1 The baseline model
The monetary union consists of two countries; home and foreign. Each country has three different types of agents; households, intermediate goods rms and nal goods rms. Each type of agent in the monetary union is spread over the unit interval, with[0; n)residing in
the home country and(n;1]residing in the foreign country. That is, the size of the home
country isn, while that of the foreign country is(1 n).
Households across the monetary union have the same preferences and consume goods produced in both countries according to the Cobb-Douglas function:
Ct
(CH;t)n(CF;t)1 n
nn(1 n)1 n
Furthermore, nancial markets are assumed to be complete, implying that all house- holds across the union consume the same amount. Output in the two countries can still vary however, due to uctuations in the terms of trade, as suggested by the market clearing conditions:
YH;td =Tt1 nCt and YF;td =T n
t Ct (2.1)
Where the terms of trade has been de ned as the ratio of foreign nal goods prices over domestic, i.e. Tt PF;t=PH;t.
Intermediate rms use domestically supplied labour to produce inputs under imper- fect competition. They also face a xed probability (1 ) of changing there prices in
each period as in the Calvo (1983) model. Hence the price rigidities in our framework arise from the sector of intermediate goods.
Final goods rms are perfectly competitive and use baskets of intermediate goods produced in both countries according to the production function:
Yi;ts =Ait (1 i)1v(Xi H;t) 1 1v + ( i)1v(Xi F;t) 1 1v v v 1 (2.2) Where Xi
j;t is the basket of inputs supplied by intermediate rms in country j and
used by nal goods rms in countryi, fori; j =H; F. There is home bias in production so
that domestic rms generally use domestically produced inputs more ef ciently than they use the input basket produced abroad. Speci cally, we have:
H (1 n)! and 1 F n!
where! takes a value between zero and one and measures the degree of home bias.
The degree of openness in trade of intermediate goods has an inverse relationship with the level of home bias in production. When ! = 1, there is no home bias in production and
complete openness in trade and when! = 0, there is complete home bias and no trade in
intermediate goods.
We de ne union average variables as a linear combination of home and foreign values where each country is given a weight equal to its size:
XW nXH + (1 n)XF
Relative variables are de ned as the difference between foreign and domestic:
XR XF XH.
The reduced form of the system is described by the following conditions:3
yWt =Et(y W t+1) 1 rt Et( tW+1) rrt (2:6) H t = Yy W t (1 n) Ty R t + Et( Ht+1) +u H t (2.3) F t = Yy W t +n Ty R t + Et( Ft+1) +u F t (2.4) Tt= y R t (2.5) Tt=Tt 1+ Rt (2:9)
Where we have de ned Y ( + ) and T (1 + (1 !))(1 !) and
where is the household discount factor, is the coef cient of relative risk aversion and 3 Note that we have rearranged the supply schedules in(1:41)by using the fact that for the generic variable
X we have: aHXH +aFXF = aWXW +aRXR, where the coef cients solve aW = aH +aF and
measures workers' disutility from working. The sensitivity of in ation to changes in the marginal cost is determined by [(1 )(1 )]= and the natural rate of interest
measures expected changes to the natural level of union output, i.e.rrt Et( y W t+1).
In order to obtain analytical results for determinacy and learning stability in following sections, we need to partition our system into two independent subsystems. To do this we rst rewrite the two supply schedules(2:3)and(2:4)by using the de nitions for union and
relative variables and by substituting(2:5). The endogenous variables nyWt ; W
t ; Rt ; Tt
o
are then determined by:
yWt =Et(y W t+1) 1 rt Et( Wt+1) rrt (2.6) W t = Yy W t + Et( Wt+1) +u W t (2.7) R t = TTt+ Et( R t+1) +u R t (2.8) Tt=Tt 1+ Rt (2.9)
Note that there are two independent subsystems; (2:6) and (2:7) determine union
average variablesyWt and W
t , while(2:8)and(2:9)determine the relative variables Rt and
Tt. Because of this, monetary policy has no impact on relative variables. This distinguishes
our model from a closed economy, and the purpose of this chapter is to examine whether this alters the results from the closed economy literature on learning.
To further simplify our analysis, in what follows we combine the two equations de- scribing the relative system,(2:8)and(2:9), into the following condition for the terms of
trade:
Tt= Tt 1+ Et Tt+1 + uRt (2.10)
where 1
1+ + T. We now present the Taylor rules that are to be combined with (2:6),(2:7)and(2:10)when we look at determinacy and E-stability in following sections.