INTERNATIONAL RESOURCE ALLOCATION: Trade with Incomplete Information 3

Our model admits of two types of equilibria, integrated and non-integrated, where we define an integrated equilibrium (IE) as an equilibrium in which w = w* and a non-integrated equilibrium (NIE) as an equilibrium in which w * w* (hence w < w*). We will show that there exists an upper bound L/L * on the ratio of factor endowments for which an IE can obtain. But first we establish:

Remark L An integrated equilibrium with active domestic markets, if one exists, is equivalent to the equilibrium that would obtain in the integrated world economy. Proof: Let w be the wage that clears the world labor market in a given IE. If domestic markets are active, no producer can earn a lower profit than fclnfw), the domestic fall-back option, but since i|; = 1, neither can any producer earn a greater profit. World labor demand must then equal 2(-tc *(w)). It follows that in any IE w = W, the equilibrium wage in the IWE, and each producer earns \zt\n(W).
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For an integrated equilibrium to exist, both national labor markets must clear at W. With L> L*, this can only be accomplished by shifting enough labor demand from the foreign to the home country through successful international matching. The largest disparity in labor demands occurs when all international matches not strictly dominated by the fall-back domestic option are concluded, and all international labor demand falls on home labor. These conditions are naturally satisfied when the price of home labor is lower, and \|x is smaller than 1, and we have imposed them above. Thus the upper bound on the ratio of labor supplies compatible with an IE, L/L*, can be immediately obtained by substituting w = w* = W and = 1 in (11) and (11*) and dividing, yielding L/L* = 3. We can now state:
It builds intuition to give an alternative derivation for the value of L/L *. In an IE the producers of each country contribute -n'(W) to world labor demand. The probability that any producer makes a good match in the international market equals one-half. With equal wages, only good matches are acceptable and it follows that at most half of -tz'(W) can be transferred from the labor-scarce to the labor-abundant country, yielding labor demand equal to (3/2)(-n'(WJ) at home and (l/2)(-n'(WJ) abroad, or a ratio of labor demands equal to 3.