Balassa$-$Samuelsson with tariffs $(35$ points$)$

Consider $an$ economy with two countries, China and the United States. China produces

nontraded goods and shoes. The United States produces nontraded goods and cars. Both shoes

and cars can $be$ traded. All types $of$ goods are produced $by$ perfectly competitive firms using

labor according $to$ production functions $of$ the form $Q_{i,t}^j=A_{i,t}^j{L}_{i,t}^j,$ where $A_{i,t}^{j}is$ country $i\text{'}s$

productivity $in$ sector $jat$ time $t.$

Let $P_{i,t}^j$ denote the price $of$ good $jin$ country $iin$ time $t.$ Let $N$ stand for nontraded goods, $S$ for

shoes, and $C$ for cars. The consumer price indices $in$ the United States $(denotedbyUS)$ and

China $(denotedbyCH)$ are given $by$

$P_{US,t}=(P_{US,t}^N{)}^{\frac{1}{4}}(P_{US,t}^C{)}^{\frac{1}{4}}(P_{US,t}^S{)}^{\frac{1}{2}}$

$P_{CH,t}=(P_{CH,t}^N{)}^{\frac{1}{4}}(P_{CH,t}^C{)}^{\frac{1}{2}}(P_{CCH,t}^S{)}^{\frac{1}{4}}$