# The phase space of the world economy

In this section, I partition the space (NB,NA) in order to distinguish the portions of the space in which the dynamics of populations and capitals of the two countries can be characterized. That is, I will define the phase space of the dynamics of capitals and populations of the world economy. More precisely, in this section, I will define only the different portions of the phase space, but they will be determined in an appendix, Section 5.A1. Figure 5.1 represents the phase space. On the horizontal axis, there is labour in country В which, when multiplied by x, gives the amount of (corn-)capital in country B. On the vertical axis, there is labour in country A which, when multiplied by x, gives the amount of (corn-)capital in country A. Both countries produce corn since both countries have plots of land of the best quality: only a country that does not have plots of land of the best quality may be specialized in cloth production when the marginal productivity of labour in the corn sector is lower than ap2, even if the population employed in corn production is nought. By contrast, it may be the case that only one country produces cloth.

Even though both countries share the same agricultural technology, their production functions for corn turn out to be different because their high- to low-quality land ratio is different. It must be stressed that if only high-quality land is employed in each country, that is, if low-quality land is left idle in country T, then the production functions

forcorn are the same in both countries. This is so if 0 < NA <(m /

By contrast, if NA >(mIa)-m, then the low-quality land is cultivated in country A and the production functions for corn are inevitably different. Hence, it is of crucial importance to determine the portion of the space (NB,NA) in which the two countries share the same production functions. This will be done by determining the locus Figure 5.1 The phase space when m = 1/2, a = 2, /3 = 1/2

in which NA = NAy, = (m / a)i-«i. Below this locus, low-quality land is not cultivated in country A, whereas above this locus it is cultivated in country A. Along the locus x(l + rA) = mNAl~ and this value equals ap2 if cloth is produced in country A; otherwise, it may be larger than ap2- The locus is represented by a purple line in Figure 5.1 and 1 will refer to it as the purple line.

Some points of the space (Nb,Na) cannot be reached (on the assumption that the initial conditions never require negative investments). Since the growth rate of population of a country may be nought in some points of the space (NB,NA), the population of that country cannot grow enough to reach some other points. In particular, the world economy cannot reach all the points of the space (Nb,Na ) that are:

• • above and on the right of the locus in which the rate of profits is nought in both countries;
• • above the locus in which the rate of profits of country A is nought and the rate of profits of country В is positive;
• • on the right of the locus in which the rate of profits of country В is nought and the rate of profits in country A is positive.

These loci are all represented as a brown line in Figure 5.1 and I will refer to their set as the brown line. Along this line the rate of profits in at least one country equals zero and therefore ap2 = x, and this value is not lower than the marginal productivity of labour in corn production and is equal to it in any country in which cloth is produced.

The portion of space (NB,NA) bounded by the brown line is the relevant part of the space, and we ignore what happens beyond it. This portion can be divided into three parts. In the central part both countries produce cloth, and therefore, they share the same rate of profits and the same growth rate. Consequently, within this part NA and NB grow in proportion, and therefore, any trajectory of populations and capitals in this area lies along a straight line passing through the origin. Note that along these trajectories the common rate of profits is decreasing. This portion of the phase space will be referred to as the no specialization area.

In the part to the right of the no specialization area, cloth is produced only in country В, whereas country A is specialized in corn production. This portion of the phase space will be referred to as the A specialized area. In the corn-specialized country, the rate of profits and the rate of growth are higher than in country B. As a consequence, any trajectory of populations and capitals in this area is along a curve whose tangent in each point is steeper than the slope of the straight line passing from the point in which the tangent is calculated and the origin. Thus, both the first and the second derivatives are positive. Note that along these trajectories the rate of profits in country A is decreasing, whereas the rate of profits in country В may be increasing and is certainly increasing when this line is almost vertical. By the way, in the no-home bias case, that we have excluded, in which capitalists invest only in the country with the highest rate of profits, the mentioned curve would be a vertical straight line. In Figure 5.1, the A specialized area is separated from the no specialization area by a red line. Along the red line both countries share the same rate of profits, but only country В produces cloth.

The part to the left of the no specialization area will be referred to as the В specialized area since cloth is produced only in country A. Since the rate of profits and the rate of growth are higher in country B, any trajectory of populations and capitals in this area is along a curve whose tangent in each point is less deep than the slope of the straight line passing from the point in which the tangent is calculated and the origin. Thus, the first derivative is positive, whereas the second derivative is negative. Note that along these trajectories the rate of profits in country В is decreasing, whereas the rate of profits in country A

may be increasing and is certainly increasing when this line is almost horizontal. By the way, in the no-home bias case, in which capitalists invest only in the country with the highest rate of profits, the above- mentioned line would be horizontal. In Figure 5.1, the В specialized area is separated from the no specialization area by a blue line. Along the blue line both countries share the same rate of profits, but only country A produces cloth.