Lecture notes on accumulation theories The mather (or father) of all growth models: Harrod’s

model and the instability problem

• Dual nature of investment: it generates capacity, it generates aggregate demand

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The model:

• S1 = sY1 (1) (multiplier)• I1 = vn (Y1E - Y0) (2) (accelerator)• S1 = I1 (3) (equilibrium in the commodity market:

there is dynamic equilibrium if I1 generates a growth of AD and capacity such as to generate an identical S1)

• vn = K/Y = K/Y is the desired (or required) capital coefficient, s marginal propensity to save

• "The axiomatic basis of the theory which I propose to develop consists of three propositions, namely: (a) that the level of a community income is the most important determinant of its supply of saving; (b) that the rate of increase of its income is an important determinant of its demand for saving; and (c) that demand is equal to supply. It thus consists in a marriage of the 'acceleration principle' and the 'multiplier' theory" (Harrod, 1939, p.43).

The warranted rate

• By simple substitution we get:

• vn (Y1E - Y0) = sY1

• or gw = (Y1E - Y0) /Y1 = s/vn

• gw is the rate of growth at which capacity savings (savings

forthcoming from existing capacity utilised at its normal or ‘expected

level) is equal to investment.

• Note that gw = gE, that is the economy will grow in equilibrium (I =

capacity saving) if the entrepreneurs expect a rate of growth (gE)

equal to gw.

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The warranted rate and the instability question

• gw = s/vn, is the so-called warranted growth rate . "The warranted rate of growth is taken to be that rate of growth which, if it occurs, will leave all parties satisfied that they have produced neither more nor less than the right amount" (1939, p.45).

• gE = s/vn , that is if the entrepreneurs expect a growth rate gE = gw = s/vn, then the economy grows with a goods market equilibrium (gA = gE)

• If the entrepreneurs knew gw, they would decide an investment level that would precisely generate gw. But they do not know it.

• Instability:• If gE > gW gA > gE and next time they will expect an even larger GE

• If gE < gW gA < gE and next time they will expect an even lower GE

• (a simple proof due to A.Sen in Jones)

A simple example

• Suppose v = 2, s = 0.2 so that gw = 0.1 (or 10%).• Suppose current output is 90, if ge = gw = 0.1, firms would invest 20

(= 2 x 10) and output at t+1 will indeed be 100 (we take the growth rate as a proportion of final output).

• If ge > gw, and they expect, say, 101, firms will invest 22 and output will be 110, so investors will feel that they anticipated too little (ga > ge > gw 0.2 > 0.11 > 0.1).

• So they will revise ge upward and ga will diverge even more from gw.

• Similarly if they anticipate 99, they will invest 18 and output at t+1 will be only 90

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Problems with Harrod

• Harrod’s model raised two questions:• gw is not a full employment path: this is a problem only for

neoclassical economists• gw is unstable, that is the economy does not gravitate around it (not

surprisingly, unless you believe in Say’s Law, there is no reason why in Harrod S = I).

• Instability seen through the lenses of the degree of capacity utilisation

• Normal degree of capacity utilisation : the average degree of capacity utilisation desired by the entrepreneurs

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Degree of capacity utilisation and the capital coefficient

un= Yn/Yf where Yf is the maximum physical output from a given capacity K. In general Yn < Yf and un < umax = 1. The main reason (Steindl) is that firms want to keep a margin of non utilised capacity to meet sudden peaks of demand; if peaks persist, they will increase capacity.

When gA > gw, it means that s/va> s/vn,, that is va = K/Ya < vn = K/Yn Ya > Yn

or, in terms of degree of capacity utilisation u, ua = Ya/Yf > un = Yn/Yf Ya > Yn

Read in the opposite direction: whenever gA > gw, Ya > Yn ua > un, the actual degree of capacity utilisation ua is higher than normal (so entrepreneurs invest even more to restore un , but in this way they aggravate the disequilibrium).

The opposite would of course happen when gA < gw (ua < un and investment would keep falling to absorb the less-than-normal u).

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Anticipation

• To anticipate what will follow when we shall consider the heterodox growth models, one way to accommodate Harrod’s instability is to say: if gA > gw and both va < vn and ua > un , then we take va and ua as the “new normal” capital coefficient and degree of capacity utilisation. But this sounds a post hoc ergo propter hoc kind of reasoning however adopted by the influential ‘neo-Kaleckian’ growth models.

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Two roads to get out from Harrod’s troubles

• Harrod influential in the 1950s (the market economy is unstable, then the State must plan the economy: France’s planning, in Italy Piano Vanoni, India’s planning). This is normative, but Harrod is unsatisfactory as a positive explanation, markets are unstable, but not explosive.

• Note a “traditional” feature of Harrod’s model: s positively influences gw and growth is endogenous (in the sense that it depends on preferences about present an future consumption). Perfect for a neoclassical economist, were not for the absence of full employment. Solow’s neoclassical growth model (1956) recovers full employment, but losses, somewhat paradoxically, the influence of s on g. Endogenous growth theory (that actually begun in the early 1960s) is just on having both.

• Two early ways out form Harrod’s problems: Solow’s neoclassical growth model and the Cambridge Equation (later followed by other heterodox schools)

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Two roads to get out from Harrod’s troubles II

• Main characteristic of the neoclassical approach is the dependence of investment from savings, more specifically from full-capacity savings, savings that spring from a productive capacity that both is fully (or normally) employed, and also fully employs all the labour supply.

• Main characteristic of (most of ) the alternative approaches is what Garegnani named (after Kaldor) the “Keynesian Hypothesis” (KH) that investment are independent from savings both in the short and in the long run.*

• This is the striking difference between neoclassical economists of all persuasions (including neo-Keynesians a là Blanchard, Stiglitz, Krugman, De Long etc): aggregate demand determines output both in the short and in the long period, or more precisely, it is capacity that adjusts to aggregate demand both in the short and in the long period. We shall begin from neoclassical growth story.

• * However, there are a number of “heterodox models” that are “keynesian” in the short-run and “Classical” (in the sense of Say’s Law) in the long-run (e.g. Shaikh, Foley and Michl; Dumenille and Levy; Kurz and Salvadori?). We shall not study them.

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After Harrod

Harrod

Solow Cambridge equation

EGTNeo-kaleckian

and/versus Sraffians

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Before we consider some heterodox models, let us remind some critical aspects of neoclassical growth theory

• Solow’s fundamental equation:

•

knsyk )(

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The Solowian warranted growth rate

• At k*, sy = ( + n)k, and k=0.

• Note that in equilibrium gw = sy/k= ( + n)k that is s/v* = ( + n), where v* is the long-run value of v.

• Neglecting , gw = n, a stable, full employment path

• In Solow the actual growth rate gravitates towards the warranted rate

• This depends of factors’ substutability

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Gravitation towards the equilibrium On the left of k*: sy > ( + n)k, i falls, k rises.On the right of k*: sy < ( + n)k, i rises, k falls.

• “In its basic form the neo-classical model depends on the assumption that it is always possible and consistent with equilibrium that investment should be undertaken of an amount equal to full-employment savings. The mechanism that ensures this is as a rule not specified. Most neo-classical writers have, however, had in mind some financial mechanism. In the ideal neo-classical world one may think of there being a certain level of the rate of interest (r) that will lead entrepreneurs, weighing interest cost against expected profits, to carry out investment equal to full-employment savings.” Hahn & Metthews EJ dec. 1964

• On the left of k*, since sy > ( + n)k, then sy/k > ( + n), and given that k/y = va, GA = s/va > ( + n), where va is the actual value of v.

• va has to rise to realise the neoclassical warranted equilibrium path s/ v* = ( + n).

• And indeed (from the left) the increase of k implies a fall of v = y/k=f(k)/k next slide

Solowian stability

• Why (on the left of the stationary state k*) when k rises,then v also rises (so that ga = s/v falls)?

• va = k/y: when k rises,also y rises,but given the falling marginal productivity of capital, it rises less than proportionally than k. So v

rises and va v* (A visual inspection is enough, for a given k, y is smaller).

• we see that the neocl. subst. mech.s entail the gravitation towards a full employment path: the fall in the interest rate induce the entrepreneurs to adopt more capital intensive techniques

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Solowian stability

• Note that this result depends upon what Hahn & Metthews pointed out, a decreasing demand function for investment (capital) negatively elastic to the interest rate. The capital theory controversy has shown that this function can have any shape. So Solow’s model is not robust, it’s deadly wrong.

• Seen from a different perspective: there is no rigorous foundation to the proposition that a falling interst rate induce firms to adopt more capital intensive techniques. So no increase of k when we are on the left of k*. Rigorously, k can rise, then fall, then rise again. No convergence, no stability, no full employment growth.

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Comparative statics: a change in s entails higher y and k but not a higher gw: a disappointing result (it came as a surprise)

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Just a word about Endogenous Growth Theory

• Many empirical studies show that there is a positive correlation between I/Y (the investment share) and g (the growth rate).

• Through neoclassical lenses (according to which I = S) this is a correlation between s = S/Y and g.

• But in Solow a rise of s only affect y not g!.• From the very early sixties this is the start of EGT: a desperate

attempt to establish a connection between s and G.• Not surprisingly, most of EGT is a return to Harrod (g is

“endogenous” in Harrod, endogenous in the specific marginalist sense that it depends on s, on preferences between present and future consumption)

• If curious, please see my papers on EGT (see my web page)