Consider first interior steady states where both nations are investing (innovating), so q=l and q*=l. Using and in, q=q*=l gives a second relation between 0K and 0r which we can think of as the optimal investment relation. Together with the optimal saving relation of, it produces three solutions: The first is the symmetric case. The second and third roots – which correspond to interior, non-symmetric steady states – are economically relevant only for a narrow range of ф. In particular, the second and third solutions converge to 1/2 as ф approachs a particular value which we call фс;и (for reasons that become clear below). For levels of ф below фа”, the second and third solutions are imaginary and so are irrelevant. For levels of ф above another critical value (defined explicitly below), the second solution is negative and the third solution exceeds unity, so both are economically irrelevant.

Given, the remaining aspects of the interior steady state are easily calculated.
In particular, solving q=l for g and then using : L,* is found by a similar procedure. Note that for the symmetric case (0K=l/2): Using the second and third roots from in yields analytic solutions for L, in the interior non-symmetric cases, but the expressions are too unwieldy to be revealing.

Expression yields values of 0K that are economically irrelevant when ф exceeds a critical level. For such ф’5,0к has two types of solutions: the core-periphery outcome (0K=O or 1), or the symmetric outcome (note that 0K= 1 /2 solves q=q*=l for all ф). The critical value, call it фср (a mnemonic for core-periphery), is established by noting that at the core-periphery outcome 0K=1, q=l and q*<l. That is, continuous innovation is profitable in the north since V=F, but V*<F* so no southern M-firm would choose to setup.* Using,,, and 0K, q* with 0K=1 simplifies to: (2£+р)ф

The ф that solves q*=l defines the endpoint of the core-periphery set, namely: фср = 2Z +p – s](2L+9)2-Ak2L{L+9) (2_14) 2X(L +p)

Note that although there are two roots, only one is economically relevant.
Using 0K=1, the remaining aspects of the core-periphery steady state are simple to calculate. In particular, since 0K=U q=l and q*<l, we have: Figure 3 summarizes the various steady states and their dependence on trade costs. North’s share of world K, 0K, is on the vertical axis and all conceivable levels of trade free-ness are shown with the interval on the horizontal axis. As noted above, the symmetric case exists for all ф, but the core-periphery outcome (either 0K=1 or 0) is an equilibrium only for ф>фср. The final type of steady state, the interior, non-symmetric case, is shown as the bowed line.  