11.4.1 Dynamic adjustment to a shock in the 2-bloc model In this section, we will provide the mathematics behind the dynamic adjustment to shocks in the 2-bloc model. We will use the example of the case where there has been a positive permanentdemand shock in bloc A (but not bloc B) in period 0. The intuition and explanation forthis example is contained in Section 11.2.5 of the main body of this chapter. We assume that bloc A and bloc B were in equilibrium in t = 1.So V11=AA?+q=ye [Jilzrg1zfl zr, =7ri1 = HT. and yf, =AB F a = ye. such that A" =AB. Then in t = 0, autonomous demand in bloc A increases permanently to AA]. All three actors first work out the changes to equilibrium values. Bloc A and bloc B were identical before the shock, so thatq = Oand F = AA ye = A8 ye. After the permanent demand shock in bloc A, the new medium-run equilibrium becomes: ye =AA' F" +I w=MVW _,/ AA' + A = 2 ye AA' _ AB and a! = . What happens in period 0? y,=ye+AA'AB 773:T+(y3_)e) =nT+(AA'AB). The central bank in bloc A, the central bank in bloc B and the foreign exchange market can now forecast that next period's Phillips curve in A will be nf=+(Vfye) Its also common knowledge that the central bank in bloc A has a monetary rule (MR) that defines the trade-off between output and inflation reductions each periodhence also next period, period 1.To simplify the notation, we assume at = ,6 = l: )4 Ye = (f NT) (monetary rule; bloc A) Putting period 1s Philllips curve and monetary rule equations together, all three rational actors can work out the combination of inflation and output in period 1 in bloc A that the central bank of bloc A will want to see. Thus:

=M A T n4=i>_:_) _ (HQHT) __ AA, AB yE_Ty2_ Now all three actors know the output level, yf, which the central bank in blocAwants to achieve next period. To do so the only instrument bloc A's central bank has at its disposal is r3. This has to be set to solve: =M-@+% The problem is that qo depends on r6 via the UIP condition: rgrg=Q1ECI0 dd= The CBs in both A and B must respect this set of equations. Summing both sides we get 00 A B _ / ~ E _ / A B -*'%: (rt rt) q qo smce JILL qt _ q . Moreover rt .r, > r . Hence f: (r? r5) = i (or r) (r? a = a go. Now assume that rA and q both converge to their new equilibrium levels F*',qiata proportional rate of A. The derivation of A follows the same method as in the small open economy (see the Appendix of Chapter 9), so we no longer assume ,6 = 1. L=(zr"21T)2+i+3(y*ye)2 => (MnT)+(y"ye)=o => (y? ye) = n}? H) 1 =Mwhr7:#l From this we derive A: 1 1 (y: _Ye) = E(JT-? nT) and (V14 -ye) = _m( _ JTT) MortHt #3 (ITSHT)_1+B' _ _ I A_u a_.r _, . Hence ((rf - V) (QB - r" )) = "L; Et = q qo- Slnce y? = ye that implies

iii-ye= = = (r,; F")(1+ 21:) = 4,1; _r")(:+:). This is the RX curve showing the relation along the equilibrium adjustment paththrough the relevant points of IS curves with different values of q. Notably the RX curve is shallower than the representative IS curve, implying that a given change in r has a greater impact on yin the open economy than in the closed. This is because the change in r both operates directly on y with coefficient 1 (or more generally a), and operates indirectly on y via its effect on changing q with coefficient 21;. Thus a much smaller change in r is needed in the open economy to have the same effect on y as in the closed economy. For example, if k = 0.5. then r needs to change by only 3/5 of the amount as in the closed economy. This is larger than in the small open economy case because of the bloc effect. When q appreciates initially, rB has to rise to keep ya in equilibrium. This requires a bigger change in rA than would be the case in the small open economy: in effect the rise in rA has triggered a rise in the world rate of interest, which would have been fixed in the small open economy. The slope of the RX line here is i: and in the small open economy case . The empirical implication here is that we might expect to see common patterns to interest rate changes across the world if there is a shock in any one big bloc.

Use the mathematics from Section 11.4.1 of the Appendix to derive the RX curve after a negative demand shock in bloc B.

textbook: Marcoeconomic: institutions, instability and the financial system