ABSTRACT
The Harrodian model assumes that investment demand as a ratio of capital stock is given in the short run, which we write as
I/K = g, (5)
where g is the short-run investment parameter which, since we abstract from depreciation, also denotes the rate of growth of capital. In the long run, following Flaschel et al. (1997), the model assumes that g adjusts according to the equation
dg/dt = ξ [ue – ud], (6)
where u = Y/K is a measure of capacity utilization, ue its expected value, ud its value desired or planned by firms, taken to be exogenously given, and ξ > 0 a speed of adjustment parameter which represents the rate at which firms adjust their investment plans in response to deviations of expected capacity utilization from their desired level. Expected capacity utilization, again following Flaschel et al. (1997) is assumed to change adaptively according to the dynamic equation
due/dt = β[u – ue], (7)
where β > 0 is an adjustment parameter representing the rate at which firms adjust their expected utilization rate due to deviations from the actual utilization rate. In the short run we assume that K, g and ue are given, and that the goods market clears through variations in output and hence, the capacity utilization rate. The short-run equilibrium level of capacity utilization, using equations (1) through (5), is given by
u = g____scσ , (8)
where (σ = z/(1 + z), the share of profits in income. An increase in g increases the short-run equilibrium level of capacity utilization through the standard multiplier mechanism, the multiplier being 1/s where the economy’s overall saving propensity, s = scσ. In the long run g and ue change according to equations (6) and (7), and the dynamics can be shown in the phase diagram in Figure 14.1. The sue line shows combinations of g and ue at which due/dt = 0 and its equation is obtained by substituting equation (8) into equation (7) and setting its right-hand side equal to zero. The vertical line at ud shows combinations of g and ue at which dg/dt = 0, as can be seen from equation (6). The vertical and horizontal arrows show the directions of change for g and ue, as can be verified from equations (6) through (8). The long-run equilibrium where both g and ue are stationary, that is where the two lines g = sue and ue = ud intersect at E, is found to be saddle-point
unstable. If g happens to be slightly above (below) the separatrix SS', g will eventually increase (decrease), moving further away from the long-run equilibrium value, sud. The saddle-point result formalizes Harrod’s notion of knife-edge instability. The long-run equilibrium rate of capital accumulation in this model is given by sud which can be written as Harrod’s s/v, since sud is the firms’ desired outputcapital ratio, the reciprocal of the desired capital-output ratio, v. The model shows that if g is greater (less) than the warranted rate of growth, gw /s/v, g will eventually move further away from it. The actual rate of investment, g, may seem to move towards gw for a while (if it starts with g > gw and ue < ud or with g < gw and ue > ud) but (upon crossing the ud line) it will eventually move further away. When g moves further away from gw it is easy to see that the actual rate of growth of output eventually will move away from the warranted rate of growth of output. The rate of growth of output in the model is given by g + û where the overhat denotes the rate of growth of a variable, which implies, from equation (8), that it is equal to g + ĝ. Since at long-run equilibrium ĝ = 0, the rate of growth of output is equal to g, so that s/v is also the warranted rate of growth of output. Since g moves further away from gw when it is above (below) it when ue > ud (ue < ud), it follows from equation (6) that the movement of g away from gw implies a movement of the actual rate of growth of output away from the warranted rate of growth of output. The specific assumption about expectations formation may be modified without changing the instability result. Suppose, for instance, that β in equation (7) approaches infinity, so that ue = u always holds, that is, there is perfect foresight on the part of firms. In this case equation (6) gets replaced by
dg/dt = ξ [u – ud]. (9)
Substituting equation (8) into it we find that long-run equilibrium for this model again occurs at the warranted rate of growth, but that an increase (decrease) in g above it implies that g rises (falls) monotonically, implying knifeedge instability. This Harrodian model has the obvious problem that it portrays the capitalist economy as being too unstable, poised on a sharp knife edge, from which the slightest departure would result in explosive growth or unstoppable stagnation. Harrod himself did not believe that the knife edge was so sharp. Although his presentation in Harrod (1939) and Harrod (1948) suggests a high degree of instability, as Neville (2003, 104) points out, Harrod objected strongly to the knifeedge terminology and said that “I hope that we shall hear no more of the ‘Harrod knife-edge’ ” (Harrod 1970, 741). Harrod (1973, 32-3) stated that actual economies did not exhibit such extreme instability. He wrote:
I have argued that an equilibrium growth path is normally unstable. A body is said to be in unstable equilibrium if, when pushed away from its position, it does not tend to return to it but to move further from it. If it is on a knifeedge a very tiny push would serve to push it away; but it would also be in unstable equilibrium if it were at the top of a shallow dome. Then a much larger push would be needed to set it moving. All depends on: (i) the gradient of declivity around it, about which, I think, I have not pronounced, and (ii) friction. In the economic case the amount of friction depends on built-in procedures, degree of conservatism, sensitivity to changes, changes in expectations, the kind of phenomena that affect expectations, etc. It needs empirical study, rather than theory, to evaluate the amount of friction.
