Heckscher-Ohlin with different technologies 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?
As the companies doing the production will minimize costs in equilibrium
$$
\begin{align*}
|MRTS_c(K_c,L_c)| &amp; = r/w \\
\\
|MRTS_w(K_w,L_w)| &amp; = r/w
\end{align*}
$$
will hold. (Or you can also use the general form of your equations (1),(2),(5) and (6) to get this.) Given your particular production functions the marginal rates of technical substitution are
$$
\begin{align*}
|MRTS_c(K_c,L_c)| &amp; = \frac{L_c}{K_c} \\
\\
|MRTS_w(K_w,L_w)| &amp; = \frac{L_w}{K_w}.
\end{align*}
$$
Taken together, this means
$$
\frac{L_c}{K_c} = r/w = \frac{L_w}{K_w}.
$$
It follows that
$$
L_c + L_w = (K_c + K_w) \cdot r/w.
$$
The total amount of Labor and Capital are given in your exercise, and these will determine the factor price ratios as well as the $L_c/K_c = L_w/K_w$ ratio. Once you plug those in, your equation system will become linear.

Notice that this is true both in autarky and in free trade equilibrium, we have not used demand functions so far at all. Setting the appropriate demand equal to the appropriate production will give you the remaining equations you need to determine the equilibrium price ratio.