Exercise $2.(From$ Schmitt$-$Groh$e,$ Uribe and Woodford $2022)$ Consider two countries, say the $U.S.$ and Japan. Both countries produce tradables and nontradables. Suppose that $at$ some point $in$ time the production technology $in$ the $U.S.is$ described $by$ QTUS $=$ aUS $T$ LNUS; with aUS $T=0\mathrm{.}4$ and QNUS $=$aUS $N$ LNUS; with aUS $N=0\mathrm{.}1,T$ andaUS where QTUS and QNUS denote, respectively, output $of$ tradables and nontradables $in$ the $U.S.,$ aUS $N$ denote, respectively, labor productivity $in$ the tradable and the nontrad able sector, and LTUS and LNUS denote, respectively, the amount $of$ labor employed $in$ the tradable and nontradable $sectorsin$ the $U.S.$ The total supply $of$ labor $in$ the $U.S.is$ equal $to1,so$ that $1=$ LTUS and LNUS. $At$ the same point $in$ time, production possibilities $in$ Japan are given $byQJT=0\mathrm{.}2LJT1$ and $QJN=0\mathrm{.}2LJN,$ where the superscript $J$ denotes Japan. The total supply $of$ labor $in$ Japan $is$ also equal $to1.$ Assume that $in$ each country wages $in$ the traded sector equal wages $in$ the nontraded sector. Suppose that the price index $in$ the $U.S.,$ which $we$ denote $by$ PUS, $is$ given $by$ PUS $=$ where PUS $T$ and PUS PUS $T$ PUS $N,N$ denote, respectively, the dollar prices $of$ tradables and nontradables $in$ the $U.S.$ Similarly, the price index $in$ Japan $is$ given $byPJ=$ where Japanese prices are expressed $in$ yen. $PJTPJN,1.$ Calculate the dollar$-$yen real exchange rate, defined $ase=EP\frac{J}{P}US,$ where $E$ denotes the dollar$-$yen exchange rate $(dollar$ price $of1$ yen$).$ The answer $to$ this question $is$ a number, but show your work. $2.$ Suppose that the $U.S.$ labor productivity $in$ the traded sector, aUS $T,$ grows $ata3$ per cent rate per year, whereas labor productivity $in$ the nontraded sector, aUS $N,$ grows $at1$ percent per year. Assume that labor productivities $in$ Japan are constant over time. Calculate the growth rate $of$ the real exchange rate. Provide $an$ intuitive explanation $of$ your result.