Stability of the symmetric steady state is investigated as in Section 3. Holding L, constant, the proportional change in q with respect to 0K can be written as:
Again, there is one stabilizing force (the local competition effect shown in the first term in large parantheses) and two destabilizing forces (the last two terms). The first of the destabilizing terms corresponds to the demand link that stems from the expenditure shifting impact of production shifting. The last term reflects the cost-link stemming from the way in which production shifting (i.e. d0K>O) lowers F and raises F* via the variety linked cost effect. That is, an increase in the share of firms producing in the north lowers the northern 1-sector’s marginal cost by lowering the cost of intermediate inputs. This in turn increases the northern accumulation rate, and raises 0K. Notice that as ф approaches unity, the stabilizing force approaches zero faster than the destabilizing forces, so for some ф sufficiently close to unity, the symmetric equilibrium is unstable. Notice also that even when trade costs are prohibitive (ф=0), the symmetric equilibrium may be unstable when 1<а/(о-1). Intuitively, this implies that the cost-linkages in the I-sector must be very strong. In w’hat follows, we assume that this condition does not hold, so there is always a segment of for which the symmetric equilibrium is stable. Development of Technology
And find the range where this q* is less than unity. Since involves a non-integer power, we cannot analytically find the critical value of ф where this q*=l. Numerically, however, фи>| is simple to find. Importantly, фс’\ is always less than фса11, although the difference disappears as a gets large. This finding implies that for some range of trade costs, both the symmetric and core-periphery outcomes are stable. The stability properties of this modified model therefore resemble more closely the ones of the standard economic geography models. That is. there is no interior, non-symmetric steady state (as w’as the case in the basic model).
Setting to zero and solving for ф, we find that stability of the symmetric equilibrium is assured for ф below a critical value фса”.
Following the same procedure for checking the stability of the core-periphery equilibrium, we evaluate q* at 0K=1 to get.