Long-run growth in this model is driven by the ceaseless accumulation of knowledge capital resulting in an ever greater range of M-varieties. Given preferences, this ceaseless expansion of variety raises real consumption continually. While technology and output in the traditional sector is stagnant, the expansion of M-varieties forces up the price of T relative to that of the composite good CM. The value of the two sectoral outputs thus grows in tandem.
Stage-One’s Growth and Investment Rates
By definition, the initial interior solution entails symmetry, i.e.,0E=0K=l/2 and, as long as ф <фса1, this outcome is stable. The steady-state rate of К accumulation during this phase is found using the expression for L, from in, to get:
This common rate of К-accumulation is unaffected by the level of trade costs, ф.
Although we cannot analytically characterize the transitional dynamics of a system with three non-linear differential equations, we can say that a continuing rise in trade freeness would raise 0K until the core-periphery outcome is the only stable long-run equilibrium. Of course, southern knowledge never disappears entirely, so the core-periphery outcome is only reached asymptotically (the number of southern varieties remains fixed, but the value of these drops forever towards zero due to the ceaseless introduction of new northern varieties).
Once the core-periphery outcome is reached – or more precisely, once we can approximate 0K as unity – the world economy enters a third distinct phase. For trade costs lower than this point, the world economy behaves as it did in the first phase. That is to say, making trade less costly has the usual static effects, but no location effects.
As history would have it, the cost of doing business internationally has declined sharply since the 18th century. While this trend seems obvious and irreversible with hindsight, it was not obviously predictable in advance, nor was it monotonic. During many periods, and sometimes for decades at a time, the trend was reversed. From 1929 to 1945, for example, international trade became increasingly difficult. Restoration of peace and the founding of GATT allowed the trend to resume, but this was not a foregone conclusion in, say, 1938.
Location and Trade Costs: A Punctuated Equilibrium
Following Krugman and Venables, this section considers the implications of lowering the cost of trade (as captured by the parameter t). To keep the analysis as sharp as possible, we take prohibitive trade costs as our initial condition.
When trade costs are high the symmetric equilibrium is stable and gradually reducing trade costs ^ф>0) produces standard, static effects – more trade, lower prices, and higher welfare (more on this below). There is, however, no impact on industrial location, so during an initial phase, the global distribution of industry appears unaffected by ф.
The stability test for the core-periphery equilibrium case is slightly different since the core-periphery outcome entails 0K=1> q=landq*<l. The procedure, therefore, is to find the range of ф where q*<l, when 0K=1 • Since this is exactly the procedure used in Section 2 to determine the range of the core-periphery, we see that the core-periphery steady state is stable w’herever it exists. To examine the stability of the interior-non-symmetric steady state, we adopt the same procedure.
Namely, we study dq/50K evaluated at 0K given by. Given the complex nature of dq/50K and, we cannot sign the derivative analytically. However, for reasonable values of the parameters the derivative is negative. This finding is robust to sensitivity analysis on parameter values.
Finally, we tu...
The first and third term are positive, so they represent the destabilizing forces, namely the demand-linked and growth-linked circular causalities (respectively). The negative second term reflects the stabilizing local-competition effect. Clearly, reducing trade costs ^ф>0) erodes the stabilizing force more quickly than it erodes the destabilizing demand-linkage. Moreover, trade free-ness ф does not affect the strength of growth-linkage (third term).
To isolate the two distinct cycles of circular causality, suppose, for the sake of argument, that the demand-linkage is cut, so d0E/d0K=O. In this case, 3q/d0K is positive and the system is unstable when А.<2ф/(1+ф2). This shows that growth-linked circular causality can by itself produce total agglomeration when trade costs are low enough. (Recall that 0<ф< 1 is a measure of the free-ness of trade, so ф=1 indicates costless trade). To see the dependence of growth-linked circular causality on localized knowledge spillovers, note that with A=1 and d0E /d0K=O, the symmetric equilibrium is always stable. At the other extreme, w’hen spillovers are purely local (A=0), the symmetric equilibrium is never stable even without the demand linkage.
Finally the criti...
The appendix shows that in the neighbourhood of the symmetric equilibrium, the linearized system has two positive and one negative real roots when ф is less than a critical value. For this range of ф’з the system is saddle path stable, since only 0K is a nonjumper.
For ф beyond the critical value, the linearized system as three positive eigenvalues, so the symmetric equilibrium is unstable. As it turns out, however, an informal approach to stability provides the same answer with greater intuition.
Specifically, to study the symmetric equilibrium’s stability, we exogenously increase 0K by a small amount and check the impact of this perturbation on the regional q’s, allow ing expenditure shares to adjust according to. In particular, using,,, and, the steady-state q can be expressed as a function 0K and L,. Holding L, constant for the moment, the partial derivative of interest is <3q/d0K from ^=[0к,Ьк,0Е[0к];ф].
The symmetric equilib...
Although an equal division of M-varieties is always an equilibrium, it need not be stable, as the economic geography literature has emphasised. Indeed, in our model two cycles of’circular causality’ tend to de-stabilize the symmetric equilibrium.
The first is the well-known demand-linked cycle in which production shifting leads to expenditure shifting and vice versa. The particular variant present in our model is based on the mechanism introduced by Baldwin. To see the logic of this linkage, consider a perturbation that exogenously shifts one M-sector firm from the south to the north. Firms are associated with a unit of capital and capital-earnings are spent locally, so ‘production shifting’ leads to ‘expenditure shifting’*. Other things equal, this expenditure shifting raises northern operating profits and lowers southern operating profits due to a market-size effect. This tends to raise q and lower q* thereby speeding north’s accumulation and retarding south’s. The initial exogenous shift thus leads to another round of production shifting and the cycle repeats. As w’e shall see, if trade costs are sufficiently low, demand-linked circular causality alone can de-stabilize the symmetric equilibrium.
The second link is the growth-linked circular c...
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.
The simplest way of analysing this model is to take L as numeraire (as assumed above) and L,. L,*, and 0K as state variables. The L,’s indicate labour devoted to creating new K, so they are the regional levels of real investment. While there may be many ways of determining investment in a general equilibrium model, Tobin’s q-approach (Tobin, 1969) is a powerful, intuitive, and well-known method for doing just that. The essence of Tobin’s approach is to assert that the equilibrium level of investment is characterised by the equality of the stock market value of a unit of capital – which we denote with the symbol V – and the replacement cost of capital. PK. Tobin takes the ratio of these, so what trade economists would naturally call the M-sector free-entry condition (namely V=PK) becomes Tobin’s famous condition qsV/PK=l.
The denominator of Tobin’s q is the price of new capital. Due to I-sector competition, northern and southern prices of К are F and F* (respectively). Calculating the numerator of Tobin’s q (the present value of introducing a new variety) requires a discount rate. In steady state, E=0 in both nations, so the Euler equations imply that r=r*=p, (‘bars’ indicate steady-state values). Moreover from, the present value of a new variety also depen...
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 compe...