Exercise $2$

Within the framework $of$ the HOS model, $we$ have two countries A and $B,$ two goods $1$ and $2,$ produced

using two factors: capital $K$ and unskilled labor $L.y_{i}is$ the production $of$ good $i,K_{i}is$ the quantity

$of$ capital used $by$ sector i and $L_{i}is$ the quantity $of$ labor used $by$ sector $i.$ Production functions can

$be$ noted $as$ follows:

$y_1=K_1^C{L}_1^{(1-C)},$ and $,y_2=K_2^{(1-C)}{L}_2^C$

Capital endowments $Ket$ labor ones $Lof$ A and $B$ are given $by$: $K_A,L_A,K_B,L_B.$ Moreover, $we$

know that: $\frac{K_A}{L_A}>\frac{K_B}{L_B}.$ Good $1is$ the numeraire, therefore $pis$ the price $of$ good $2in$ terms

$of$ good $1,Yis$ the national income $in$ terms $of$ good $1,wis$ unit labor income $in$ terms $of$ good $1$

and $ris$ the unit capital income $in$ terms $of$ good $1.k_{i}is$ the capital intensity $of$ sector $i$:$k_i=\frac{K_i}{L_i}.$

Suppose that $d{1}_A=b_A{Y}_A$ and $d{1}_B=b_B{Y}_B,$ where $b_A$ and $b_B$ are preference parameters for the good

$1in$ country A and country $B$ respectively.

Express the relations that give the optimal allocation $of$ resources. Show how you get them.

Express also $k_1$ and $k_{2}as$ a function $of\frac{w}{r}.$

Give the relation linking $p$ and $\frac{w}{r}.$

Define the factor proportions theory. Using different values $ofC,$ give the different possibilities

$of$ specialization for countries A and $B$ when:

$a{b}_A=0\mathrm{.}5et{b}_B=0\mathrm{.}5$

$b{b}_A=0\mathrm{.}5et{b}_{B}is$ very high.

$In$ each case, explain whether the factor proportions theory $is$ verified $or$ not.