# PRODUCTIVITY DIFFERENCES: The Basic Model 2

Therefore, increases in Nh (Nl) improve the productivity of skilled (unskilled) workers in all sectors. Nh and Nl are the only state variables of this economy.

R&D (in the North) leads to the discovery of new machine types (blueprints). We assume that technical change is directed, in the sense that the degree to which new technologies are skill-complementary is determined endogenously (see Acemoglu, 1998). Some firms improve technologies complementing unskilled workers, while others work to invent skill-complementary machines. The cost of discovering a new machine complementing workers of group z (z ~ L от H) is 1 /фг units of final output, so Nz = фг • Xz where Xz denotes total output devoted to improving the technology of group z. We assume that фг — ф(хг), фг
Analysis
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We first take the technology variables NL and TV# as given and characterize the equilibrium in the North, and we continue to suppress country indexes. We also assume that there is no commodity trade between the North and the South. We start with an intuitive lemma. As with other proofs, the proof of this lemma is in Appendix A.

Goods with higher indexes produced with unskilled labor have lower productivity, and command higher prices. The converse is true for skilled goods. Equation (8) is then obtained using the consumer indifference condition. It exploits the fact that goods markets have to clear in the North and the South separately.

To fully characterize the equilibrium for given NL and AT#, we must determine J. Good J can be produced by either skilled or unskilled workers, and must yield zero profit in either case, thus, when i = J, both (6) and (7) apply. This implies: