Utility optimization implies that a constant fraction a of northern consumption expenditure E falls on M-varieties with the rest spent on T. Northern optimization also yields unitary elastic demand for T and the CES demand functions for M varieties:
where s, is variety j’s share of expenditure on all M-varieties in the north, E is northern expenditure and the p’s are consumer prices. The optimal northern consumption path satisfies the Euler equation E/E=r-p (r is the north’s rate of return on investment) and a transversality condition. Southern optimization conditions are isomorphic.
On the supply side, free trade in T equalizes nominal wage rates as long as both regions produce some T (always true as long as a is not too large). Taking home labour as numeraire and defining px as T’s price, T=pT*=w=w*=l,2 As for the M-sector, we choose units such that aM=l-l/o. As usual M-sector optimal pricing is given then by p=T and p*=x where p and p* are typical local and export market prices, respectively. Southern M-firms have analogous pricing rules.
With monopolistic competition, equilibrium operating profit is the value of sales divided by a. Rearranging, using the optimal pricing rules:
where E” is world expenditure, 0,: is north’s share of Ew, and ф=т!'”. Here ф is a mnemonic for the ‘free-ness’ (phi-ness) of trade since trade gets freer as ф rises from ф=0 (prohibitive trade costs) to ф=1 (costless trade). Also, В is a mnemonic for the ‘bias’ in northern M-sector sales since В measures the extent to which the value of sales of a northern variety (namely, pc+p*c*) exceeds average sales per variety worldwide (namely, aEw/Kw). The expression for 7i* is analogous. Note that the definition of В permits a decomposition of 7Г changes into global developments (measured by К w) and local developments (measured by 0K and 0,.).
Finally, differentiating its definition, the law of motion for 0K is:
Note that the search for steady states is simplified by inspection of. By definition 0K=O in steady state, so the model has only two types of long-run equilibria: those in which g=g* (nations accumulate capital at equal rates), and those in which 0K equals either unity or zero. We refer to these tw’o types as, respectively, the interior and core-periphery outcomes. (To avoid repetition, we consider only the core-in-north case, i.e. 0K=1.)