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 depends upon the rate at which new varieties are created. Since the steady state is marked by time-invariant L/s, implies that the growth rate of K” is time-invariant in steady state. In particular, the growth rate will either be the common g=g* (in the interior case), or north’s g (in the coreperiphery case). In either case, the steady-state values of investing in new units of К are:
Given this the regional q’s depend only on parameters and state variables, i.e.:
Optimizing consumers set expenditure at the permanent income hypothesis level in steady state. That is, they consume labor income plus p times their steady-state wealth, FK= 0K/A, and, F*K*= (1-0K)/A* in the north and in the south respectively”. Thus:
This relation between 0F and 0K can be thought as the optimal savings/expenditure function since it is derived from intertemporal utility maximisation.