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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?

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    $\begingroup$ And why would you set $K_{c} = L_{c}$ in the autarky case? $\endgroup$ – Giskard May 9 '18 at 12:34
  • $\begingroup$ I should have been clearer; I didn't actually just set them equal to each other. I found that this is the case in autarky. Essentially, since factors are mobile across sectors we know that wages equalise (same for rental rates). Then by setting the wage-rental ratios equal to each other in each sector, I found that $K_{c} = K_{w}$ thus, $K_{c} = 1/2$. And the same for labour, $L_{c} = L_{w}$ so $L_{c} = 1/2$. But couldn't I also argue that because the exponents are the same for both factor, that they are used in the same proportion? $\endgroup$ – BenBernke May 9 '18 at 12:55
  • $\begingroup$ If $K_c = K_w = L_c = L_w = 1/2$ then factor prices are not equal according to your own equations, e.g. (1) and (5). $\endgroup$ – Giskard May 9 '18 at 13:07
  • $\begingroup$ I must be missing something. From (1), $w = 0.5 p_{c}$ and from (5), $w = 0.25 p_{w}$. By setting the wages equal to each other I find the relative price, $p_{c}/p_{w} = 0.5$. $\endgroup$ – BenBernke May 9 '18 at 13:25
  • $\begingroup$ You are correct, I made a mistake in my calculations, sorry. $\endgroup$ – Giskard May 9 '18 at 13:47
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As the companies doing the production will minimize costs in equilibrium $$ \begin{align*} |MRTS_c(K_c,L_c)| & = r/w \\ \\ |MRTS_w(K_w,L_w)| & = r/w \end{align*} $$ 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 $$ \begin{align*} |MRTS_c(K_c,L_c)| & = \frac{L_c}{K_c} \\ \\ |MRTS_w(K_w,L_w)| & = \frac{L_w}{K_w}. \end{align*} $$ 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.

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  • $\begingroup$ Many thanks for the reply. I have two questions: 1) I understand the $\frac{L_{c}}{K_{c}} = r/w$ part. But I don't quite see how it follows that $\frac{L_{c} + L_{w}}{K_{c} + K_{w}} = r/w$. 2) Using your method I was able to derive the autarky values. But for the international equilibrium, will be demand function now be $\frac{D_{c} + D_{c}^*}{D_{w} + D_{w}^*} = p_w/p_c$? $\endgroup$ – BenBernke May 9 '18 at 14:36
  • $\begingroup$ Yes, this is just basic math. You can show for any positive numbers $a,b,c,d,\lambda$ that if $a/b = \lambda$ and $c/d = \lambda$ then $(a+c)/(b+d) = \lambda$. Try $(a,b,c,d) = (1,3,2,6)$ and you'll see. I think this answers both your questions 1) and 2). $\endgroup$ – Giskard May 9 '18 at 15:00
  • $\begingroup$ Ah, of course. Using the fact that $\frac{L_{c}}{K_{c}} = 1 = \frac{L_{w}}{K_{w}}$ ($r/w = 1$ from the last equation). I find that 0.5 of each factor is used (as in the autarky). Then $q_{c} = 0.5, q_{c}^* = 0.25$ and $q_{w} = 0.25, q_{w}^* = 0.5$. And since world demand has to equal world production, I use the new demand function and find that the relative price is one. But a relative price of one seems counter-intuitive because a higher relative price should result in Home increasing its production of cars for export right? Now I have found that Home produces the same amount in both cases... $\endgroup$ – BenBernke May 9 '18 at 15:27
  • $\begingroup$ In the free trade equilibrium $L_c = K_c = 1/2$ does in fact result in different factor prices, so there you will have to look for different factor levels. $\endgroup$ – Giskard May 9 '18 at 15:36
  • $\begingroup$ Ok now I'm really confused... I just saw my tutor and he says that the autarky prices are correct. But according to him, in the trade equilibrium, Home specialises completely in cars, so $q_{w} = 0$. I am inclined to think he has miscalculated because $L_{w} = K_{w} \neq 0$ $\endgroup$ – BenBernke May 9 '18 at 16:45

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