Consider the following simpli$ed$ version $of$ the model $we$ studied $in$ class.

The world consists $of$ two countries, called home and foreign. All variables $(except$ interest

rates$)$ are $inlogs$ and measure deviations from zero$-$shock equilibrium levels. Foreign variables are

denoted with $an$ asterisk.

Output $in$ each country $(y$ for home, $y$ for foreign$)is$ a function $of$ employment $(n,n)$ and $a$

worldwide productivity shock $x$:

$y=(1)nx$; $(1)$

$y=(1)nx$; $(2)$

where and $xis$ identically and independently distributed with zero mean.

Labor demand $in$ each country $is$ determined $by$ optimality conditions for $rm$ behavior that

equate real wages $to$ the marginal product $of$ labor. $Inlogs,$ these conditions are:

$wp=nx$; $(3)$

$wp=nx(wherew$ and $w$ are the nominal wages, and $p$ and $p$ are product prices.

Consumer price levels $(q,q)$ are given $by$:

$q=ap+(1a)(e+p)=p+(1a)z$; $(5)$

$q=a(pe)+(1a)p=paz$; $(6)$

where $ais$ the share $of$ spending $on$ the home good $by$ consumers $in$ each country

the nominal exchange rate $(unitsof$ home currency per unit $of$ foreign$),$ and $zis$ the terms $of$ trade,

$denedbyze+pp(unitsof$ the home good per unit $of$ the foreign good$).$

Expenditure equilibrium conditions for the home and foreign goods are:

$y=(1a)z+"[ay+(1a)y][ar+(1a)r]$ ; $(7)$

$y=az+"[ay+(1a)y][ar+(1a)r]$ ; $(8)$

where and and $r$ and $r$ are the $ex$ ante real interest rates.

Denoting nominal interest rates with i and $i,r$ and $r$ are determined $by$:

$r=i(E(q+1)q)$ ; $(9)$

$r=iEq$

$+1$

$q$ ; $(10)$

where $E(q+1)(Eq$

$+1$

$)is$ the expected value $of$ the home $(foreign)$ CPI one period ahead based

$on$ the currently available information.

Optimal bond holding behavior $in$ the two countries implies uncovered interest rate parity $(UIP)$:

$ii=E(e+1)e$: $(11)$

Proceeding $asin$ the slides, use the $ex$ ante real interest rate equations $(9)$ and $(10),$ UIP

$(11),$ the CPI equations $(5)$ and $(6),$ and the $denitionof$ the terms $of$ trade $ze+pp$

$to$ show that $r=r,so$ that equations $(7)$ and $(8)$ can $be$ rewritten $as$:

$y=(1a)z+"[ay+(1a)y]r$; $(12)$

$y=az+"[ay+(1a)y]r$: $(13)4)$

Denoting money demand $(equalto$ money supply $in$ equilibrium$)$ with $mat$ home and $min$

the foreign country, money market equilibrium $in$ each country requires:

$m=p+y$; $(14)$

$m=p+y$: $(15)$

Note: $We$ are simplifying the model $we$ studied $in$ class $by$ removing the $eectof$ interest rates $on$

money demand. The money market equilibrium conditions above are thus analogous $to$ those $in$

Barry Eichengreen$s$ model $of$ policy interactions under the interwar Gold Standard.

Proceeding $asin$ the slides, show that prices and employment $in$ each country are such that:

$p=w+n+x$; $(16)$

$p=w+n+x$; $(17)$

and

$n=mw$; $(18)$

$n=mw$: $(19)$

Assume that $rms$ and workers $in$ each country agree $to$ wages set $at$ the end $of$ the previous

period $to$ minimize the expected squared deviation $of$ employment from the zero shock equilibrium

$in$ each country. $In$ other words, $wis$ chosen $to$ minimize $E1$

$n2=2$ and $wis$ chosen $to$ minimize

$E1$

$n2=2,$ where $E1$ denotes the expectation conditional $on$ information available $at$ the end

$of$ the previous period.

Assume that the exchange rate $isexible,$ and central banks use the respective money supplies

$as$ their policy instruments. Central banks choose money supplies $to$ minimize loss functions that

depend $on$ the squared deviations $of$ CPI $ination$ and employment from their zero shock levels. $In$

other words, policymakers have $no$ motive $to$ move their money supplies other than responding $to$

shocks:

$LCB=1$

$2$

$q2+(1)n2$ ; $(20)$

$LCB$

$=1$

$2$

$q2+(1)n2$ ; $(21)$

where Assume that central banks care more about $ination$ than employment, $i.e.$

$>1=2.$

Proceeding $asin$ the slides, show that the assumptions $we$ are making imply that wage setting

results $inw=w=0.$

Use a superscript $Dto$ denote the $dierence$ between home and foreign variables $(for$ instance,

$mDmm).$ Use the money market equations $(14)$ and $(15),$ the expenditure equations

$(12)$ and $(13),$ equations $(16)-(19),$ and the result about wage setting above $to$ show that the

exchange rate $is$ determined $by$:

$e=1(1)$

$mD$:

Why didn$twe$ have $to$ use the UIP equation $(11)as$ part $of$ exchange rate determination like

$in$ the slides? $(Hint$: Think about the money demand equations and compare the exchange

rate solution above $to$ the one $in$ the slides.$)$