We consider a small open economy producing two goods i $=1,2.$ These goods

are produced using only one factor of production, labor. We will denote by ai

the quantity of labor required for the production of one unit of good i$.$ We will

denote by L the labor endowment of the country. The representative consumer

in the country has preferences given by the utility function

U $($C$1,$ C$2)=\backslash$alpha Log C$1+(1\backslash$alpha $)$ Log C$2$

We let pi denote the price of good i and wi denote the wage rate in sector i$.$

$1.$ Derive the relationship determining the competitive supply of this economy, given by the vector Q $=($Q$1,$ Q$2)$ of produced quantities of both

goods, as a function of prices p$1$ and p$2.$ Represent graphically the relative supply schedule of good $1,$ Q$1$

Q$2$

$,$ as a function of the relative price p$1$

p$2$

$.$

Determine the labor market clearing rate w.