Consider two countries: Home and Foreign that produce two goods, cars and wheat. The production technologies are such that:
$q_{c} = K_{c}^{0.5} L_{c}^{0.5}$ and $q_{w} = 0.5 K_{w}^{0.5}L_{w}^{0.5}$ for Home. And for Foreign:
$q_{c}^{*} = 0.5 K_{c}^{0.5*} L_{c}^{0.5*}$ and $q_{w} = K_{w}^{0.5*}L_{w}^{0.5*}$
$K_{c}$ denotes the amount of capital used in the production of cars.
The asterix denotes the Foreign country. The endowments are:
$K_{c} + K_{w} = K_{c}^{*} + K_{w}^* = 1$ and $L_{c} + L_{w} = L_{c}^{*} + L_{w}^* = 1$
Preferences are homothetic and identical between the countries and given by $\frac{D_{c}}{D_{w}} = \frac{p_{w}}{p_{c}}$.
So both countries have the same endowments but their production technologies differ.
The first question is to find the autarky quantities and relative prices. This I have managed to do by setting up the profit maximisation problem in each sector, then finding the wage-rental ratio. And seeing as the Cobb-Douglas exponents are the same, I know that equal amounts of labor and capital will be used in the production of each good. I won't include the algebra but here are my wages and rental rates in each sector. For cars:
$w = 0.5 p_{c} (\frac{K_{c}}{L_{c}})^{0.5}$ $\quad$ (1)
$r = 0.5 p_{c} (\frac{L_{c}}{K_{c}})^{0.5}$ $\quad$ (2)
$w^* = 0.25 p_{c}^{*} (\frac{K_{c}^*}{L_{c}^*})^{0.5}$ $\quad$(3)
$r^* = 0.25 p_{c} (\frac{L_{c}^*}{K_{c}^*})^{0.5}$ $\quad$ (4)
And for the wheat sector:
$w = 0.25 p_{w} (\frac{K_{w}}{L_{w}})^{0.5}$ $\quad$ (5)
$r = 0.25 p_{w} (\frac{L_{w}}{K_{w}})^{0.5}$ $\quad$ (6)
$w^* = 0.5 p_{w}^{*} (\frac{K_{w}^*}{L_{w}^*})^{0.5}$ $\quad$(7)
$r^* = 0.5 p_{w} (\frac{L_{w}^*}{K_{w}^*})^{0.5}$ $\quad$ (8)
For the autarky case, dividing (1) by (3) and setting $K_{c} = L_{c}$ shows that for Home, the relative price, $\frac{p_{c}}{p_{w}} = 0.5$. Similarly for Foreign, $\frac{p_{c}^*}{p_{w}^*} = 2$. And using the preference function I can find the quantities of each good produced.
It is finding the free-trade relative price and quantities that is causing me some trouble. I know that in free-trade factor and output prices equalise and that world demand equals world production. I also know that Home has a comparative advantage in cars and Foreign in wheat (given the autarky relative prices). But I have tried for hours now to manipulate (1)-(8) but without much progress. Any suggestions as to how I can proceed?