Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Temporary Equilibrium

Postulate an elementary period or instant, which may be arbitrarily short. There is a set of parameters given or determined at its outset. They change only from one instant to the next. Within an instant some markets are cleared. In this temporary equilibrium the economy moves from instant to instant in accordance with the laws governing the behaviour of the parameters.

Hicks (1939, p. 122) stated that there will nearly always be some goods whose production can be changed within the instant. Applying this principle to macroeconomics Hicks (1937) treated labour as a perfectly variable factor for the individual entrepreneur, so that, in his interpretation, the Keynesian IS–LM equilibrium, or its full-employment counterpart, is the economy’s temporary equilibrium, with employment, output and interest rates determined within the instant, given the parametric stock of capital etc. This is still the standard temporary-equilibrium concept in macroeconomics. A point not lying on the IS curve is usually regarded as indicating a net excess demand for goods.

But there are two serious difficulties. First there is the well-known crux concerning Walras’ Law when there is involuntary unemployment in the IS–LM equilibrium. How can there be an excess supply of labour when there is no excess demand for anything else? The ingenious distinction made by Clower (1965, ch. 5) between ‘notional’ and ‘effective’ excess demands solves the problem formally, but prompts the question why it is required in macroeconomics when the rest of economics manages without it.

Secondly there is a strong case for assuming that labour, like capital, is a quasi-fixed factor for the individual entrepreneur, given or determined parametrically at the outset of each instant. For there are costs of hiring and firing people, and even of varying significantly hours worked, at short notice. But this suggests that macroeconomics should be based not on Hicks’s principle but on the Marshallian concept of a temporary equilibrium relative to a given state of expectations, in which market prices equate demands in each instant to outputs predetermined at its outset.

Actually a macroeconomic temporary equilibrium of this kind was devised long ago. One of its inventors was Keynes himself. Keynes’s economics was Marshallian in this respect from the Treatise on Money (1930, chs. 9–11) to the General Theory (1936) and beyond. The contrary belief regarding the General Theory expressed, for example, by Hicks (1965, pp. 64–6) will be shown to be incompatible with the evidence.

Keynes’s object in the Treatise on Money (1930, Preface, p. v) was to find a method of analysing dynamic processes towards and around a longer-run equilibrium. With the same end in view we shall present a model of temporary equilibrium under assumptions of constant returns to scale and labour-augmenting technical change, in order that a longer-run equilibrium may be one of steady growth. As in the Marshallian theory of relative prices, the dynamics will depend on revisions of short-term expected (or ‘normal’) prices when the prices of the temporary equilibrium turn out to be different from them. ‘Hicksian’ dynamics is somewhat pressed to find convincing substitutes for this lag, on which the Marshallian distinction between market and short-term normal prices is based. We shall show how it can be used in constructing a set of dynamic equations that accomplish Keynes’s objective in the Treatise on Money and enable us to put into a unified framework a great variety of macrodynamic theories.

But it may be useful to begin by expressing our general approach to aggregative analysis. The subject-matter of macro-economics is, we believe, the behaviour of index numbers, of final output, employment, the stock of capital, interest rates, the general price-level, etc. It is foolish to assume that their components are homogeneous, since index numbers are required just because they are not. We also dissent from the idea that there exists a fundamental non-aggregative system with which they should be consistent. The decision to be made is how far to disaggregate, not how to justify departures from this imaginary construct. Our purpose here will be served at the highest level of aggregation.

The index numbers are taken to reflect the average (or representative) behaviour and experience of economic agents. The deviations from the average are not predicted by the model, and so could not be inferred from it even if everyone knew it in detail.

Supply

We assume a closed economy, so that total money income equals the value of final output. Real final output Y = Kf(x), where K is the inherited stock of capital, x = N/K, and N is the demand for labour in efficiency units. For simplicity perfect competition is assumed. At the outset of an instant firms choose x by maximizing the profits expected to accrue in it. Thus optimum x depends on short-term expectations. If p is an index of prices expected for the instant and w an index of money wages per efficiency unit of labour, x maximizes pf(x) – wx. Necessary and sufficient conditions for an interior maximum are f′(x) = w/p and f′(x) < 0 Given p, w, and K, then, both output, Y, and the sum of expected money incomes, pY, are parameters for the instant.

Prices and Windfall Profits

Actual incomes may differ from pY. Let Q be the net sum of unexpected incomes deflated by pK. Thus money incomes deflated by K are p[f(x) + Q]. If π is the price level of final output, by definition πf(x) = p ([f(x) + Q], so that Q will turn out to be ⋛0 according as the market determines π to be ⋛p within the instant. Since output is completely inelastic within the instant, pQ = (π − p) f(x) = [πf(x) – wx] – [pf(x) − wx] is the net sum of unexpected or windfall profits deflated by K. (In the Treatise on Money Keynes apparently defined windfall profits as the excess of entrepreneurs’ actual over long-term normal renumeration (1930, pp. 124–5). The definition here follows from our having adopted his assumption in the General Theory (1936, ch. 5) that current employment of labour depends on short-term expectations, so that windfalls become the excess of actual over short-term expected profits.)

Excess Demand for Final Output

We assume pK-deflated planned investment and saving to be functions I(Q, r, x) and S(Q, r, x) Planned saving is expected income minus planned consumption. r is an index of the general level of real interest rates. The pK-deflated excess demand for final output is

$$ {X}_g = I\left(Q,r;x\right)-S\left(Q,r;x\right)-Q $$

The subscript g is for ‘goods’. The semicolon preceding x indicates that it is a parameter for the instant. I Q may be negative, since unexpectedly high prices may induce disinvestment in inventories. The sign of S Q is ambiguous: a negative income effect may be outweighed by a positive substitution effect of unusually high or low π in relation to p. So is the sign of S r . But we assume that I Q – S Q – 1 and I λ – S r are both negative. I x is non-negative, but is positive if long-term expectations of profit move in the same direction as short-term expectations of it. Finally S x , which has the sign of the marginal propensity to save, is assumed to be positive.

π will rise or fall (given p) according as X g is positive or negative. So, therefore, will Q.

Excess Flow Demand for Money

There is a central-banking system. We abstract from the note issue. Commercial banks’ reserves at the Central Bank, deflated by pK, are R. The public’s pK-deflated demand for commercial banks’ deposits is a function, L(Q, r; x, λ) where λ is the parametric expected rate of inflation of p. L r is negative and so is L λ . L Q may be zero. In any case its sign is ambiguous. There may be a positive income effect. But since a portion of loans is normally kept on deposit, when a rise in Q reduces the demand for inventories (and correspondingly the demand for bank loans) the borrowers’ demand for deposits may also be reduced. Finally L x may be negative. For the rise in expected profits with x may increase confidence, reducing the demand for liquidity. (Transactions demand is already largely accounted for by expressing the demand as a ratio to pK.)

The public as a whole can make deposits whatever it wishes them to be by altering its borrowings from the banks. There can be no inevitable net creation of ‘derivative’ deposits by the banks themselves as they attempt to remove a net surplus of reserves, when the public commands the volume of bank loans at the banks’ current loan rates. For a discussion of the genesis of deposits see Rose (1985, section 4).

It is convenient, but not essential, to assume that deposits are momently equal to the stock demand for them. The banks, however, have, at the outset of an instant, reserves that are not, in general, what they need. Let c be their desired ratio of reserves to deposits, assumed constant for simplicity. When cL differs from R they try to reduce the gap during the instant by active net hoarding. Its extent is assumed to be ß(cL – R), where ß is a positive adjustment coefficient.

But there may also be passive net hoarding. The theory of the precautionary demand for money suggests that, since the terms for unexpected transactions between money and securities at short notice are apt to be worse than those for expected transactions between them, the optimum strategy should involve a temporarily passive response to unexpected net receipts, i.e., passive net hoarding of them. Now unexpected net receipts arise when Q is non-zero. We therefore assume that passive net hoarding is αQ(0 ≤ α ≤ 1), with α constant.

The pK-deflated excess flow demand for money (reserves and deposits) is therefore

$$ {X}_{\mathrm{m}} = \beta \left[ cL\left(Q,r;\;x,\lambda \right)-R\right]+\alpha Q-\dot{R} $$

the subscript m is for ‘money’.

Walras’ Law

Since final output is a parameter, the temporary equilibrium is an equilibrium of exchange. The sum of the values of the excess demands for goods, securities, and money must be zero. The excess demands for factors are irrelevant during the instant, owing to the assumption that factor employments are fixed at its outset. The problem encountered in the Hicksian theory simply does not arise here.

Excess Flow Demand for Loanable Funds

The excess supply of securities is the excess demand for lonable funds, whose pK-deflated value is X f . Therefore by Walras’ Law

$$ {X}_{\mathrm{f}}=I\left(Q,r;x\right)-S\left(Q,r;x\right)-Q+\beta \left[ cL\left(Q,r;x,\lambda \right)-R\right]+\alpha Q-\dot{R} $$

The subscript f is for ‘funds’. r will rise or fall according as X f is positive or negative.

The Temporary Equilibrium with Parametric R

If the Central Bank sets R for the instant, \( \dot{R}=0 \). The adjustment of r and π (or equivalently Q) puts X f and X g to zero, establishing a unique equilibrium if, in addition to the inequalities I Q  − S Q −1 < 0, I r  − S r  < 0, and L r  < 0, the condition (I Q  − S Q −1) L r  − (I r  − S r ) L Q  > 0 is satisfied.

The equations are

$$ \begin{array}{l}I\left({Q}^{*},{r}^{*};x\right)-S\left({Q}^{*},{r}^{*};x\right)\hfill \\ {}={Q}^{*}\alpha \left[I\left({Q}^{*},{r}^{*};x\right)-S\left({Q}^{*},{r}^{*};x\right)\right]\hfill \\ {}=\beta \left[R- cL\left({Q}^{*},{r}^{*};x,\lambda \right)\right]\hfill \\ {}\kern8.5em {\pi}^{*}=p+p\left({I}^{*}-{S}^{*}\right)\hfill \end{array} $$

(The asterisks indicate equilibrium values.) The first is Keynes’s Fundamental Equation (viii) (1930, p. 138). The third is the form assumed by his Fundamental Equation (iv) (1930, p. 137) when windfalls are defined as in section “Prices and Windfall Profits” above. The second is more general than its counterpart in Keynes. For he assumed that there is no passive net hoarding, i.e., that α = 0. The consequence is his ‘liquidity preference’ theory of interest, L (Q *, r * ; x, λ) = R/c.

He held to this aspect of his temporary equilibrium not only in the Treatise on Money and immediately after it (Keynes 1973a, pp. 224–5) but also in and after the General Theory. The net demand for funds represented by I *-S * is matched by net loans from windfalls exactly equal to it. Thus in a letter to Hawtrey written soon after the publication of the General Theory he insisted that an increase in investment would not directly raise r * because it would raise the demand for securities by precisely the same amount (Keynes 1973b, p. 12).

But if α is positive I *-S * is not fully matched by net loans from windfalls. Active net dishoarding must fill the gap, viz. αI *-S *, and r * must stand above or below the level corresponding to L * = R/c according as I *-S * is positive or negative. This is essentially the ‘loanable funds’ theory of interest, for which see, e.g., Robertson (1940, pp. 1–20).

The Question of Say’s Law

If rational conduct does imply that α is positive, there is a decisive answer to the question whether aggregate demand must be a determinant of the economy’s behaviour, or equivalently whether the ‘classical’ theory of interest (Keynes 1936, ch. 14) must be wrong. (For a fuller account of this subject see Rose (1985, pp. 1–17).) If for each pair of values of x and λ we can find a stock of reserves with which the temporary-equilibrium equations become

$$ \begin{array}{c}I\left(0,{r}^{*};x\right)=S\left(0,{r}^{*};x\right)\\ {} cL\left(0,{r}^{*};x,\lambda \right)={R}^{*}\\ {}\kern5em {\pi}^{*}=p\end{array} $$

with r * > 0, the answer is no. The ‘classical’ theory of interest becomes valid, and, since Q* = 0, aggregate money demand, p[Y + K (I * − S *)], and money income, π * Y, are equal to and determined by the given sum of expected incomes, pY. If such an R * could always be found, inflation, fluctuations, unemployment there might be, but none of them due to movements of aggregate demand for output. Moreover the appropriate level of reserves can be found and sustained ‘without the necessity for any special intervention or grandmotherly care on the part of the monetary authorities’ (Keynes 1936, p. 177). In effect Say’s Law of Markets can be imposed whenever we wish; for the market mechanism itself will guarantee that supply, pY, creates its own demand. To impose it the Central Bank should stand passively ready to deal in securities with the member banks, at their current prices, in exchange for reserves. Both convenience and economic incentive will induce the banks to accomplish their active net hoarding via the Central Bank, the incentive being the tendency of security prices to move against them if they go to the market instead. Thus they will adjust their reserves to the “demand” for them in accordance with the equation \( \dot{R}=\beta \left( cL-R\right) \). But then, since α is positive, the second equation in section “The Temporary Equilibrium with Parametric R” above implies I * = S *. The market cannot support a non-zero I *-S * when the banks provide it with no active net dishoarding.

But if α were zero the second equation in section “The Temporary Equilibrium with Parametric R” would not imply Q * = 0 when R * = cl *. Instead there would be many possible equilibria. Which of them would eventuate would depend on which value of R * were fortuitously reached in the adjustment to cL *. The Central Bank’s policy could not succeed in imposing Say’s Law. It would simply render indeterminate the equilibrium at which I *- S * was matched by net loans from windfalls. No wonder Keynes was so insistent on his ‘liquidity preference’ theory of interest!

In a system with no Central Bank, all money consisting of the notes and deposits of non-colluding commercial banks holding each others’ deposits as reserves, Say’s Law would always rule if α were positive. For if R = c = 0 then Q * = 0.

Process Analysis

Comparative Statics of Temporary Equilibrium

Let m be the ‘potential’ supply of deposits, R/c. We shall refer to it as the supply of money deflated by pK.

The temporary equilibrium implies functions

Q *(x, m, λ) and r *(x, m, λ). The signs of their partial derivatives are of the first importance in process analysis. What can be learnt about them from the formulae obtained by differentiating the equations of section “The Temporary Equilibrium with Parametric R” and applying Cramer’s Rule, together with the inequalities assumed there?

Definite signs are attached to r * m , r * λ , Q * m , and Q * λ . The first two are negative, of course, and in consequence the last two are positive.

Sign \( {Q}_x^{*}=\mathrm{sign}\left[\left({I}_r-{S}_r\right){L}_x-\left({I}_x-{S}_x\right){L}_r\right] \). It may easily be positive; for the marginal inducement to invest, I x , may exceed the marginal propensity to save, S x , and L x may be negative (see section “Excess Flow Demand for Money”).

Sign \( {r}_x^{*}=\mathrm{sign}\left[\left(\alpha +\beta c{L}_Q\right)\left({I}_x-{S}_x\right)-\left({I}_Q-{S}_Q-1\right)\beta c{L}_x\right] \). If L Q is zero and if, as one might expect, ß is large, sign \( {r}_x^{*}=\mathrm{sign}\;{L}_x \).

Since the banks’ desired cash ratio will make no further explicit appearance, the letter c will be given a new definition in section “The Equations of Motion” below.

Capital Accumulation

Since the goods markets are cleared, actual and planned investment are equal, Therefore \( \dot{K}/\ K={I}^{*}\left(x,m,\lambda \right) \). An essential requirement i snome theories of growth and all ‘overinvestment’ theories of the business cycle (Haberler 1937, ch. 3) is that I * x should be non-negative. Now \( {I}_x^{*}={I}_x+{I}_r{r}_x^{*}+{I}_Q{Q}_x^{*} \), so that all is well if I r and L x are negative and |I Q | is small. In a Say’s-Law regime \( {I}_x^{*}=\left({I}_x{S}_r-{I}_r{S}_x\right)/\left({S}_r-{I}_r\right) \), which is almost surely positive. The other two partials are positive if I r is negative and |I Q | small.

The Dynamics of Short-Term Expectations

Three forces act on p from one instant to the next, namely expected inflation, the excess of windfall profits over windfall losses, and what we may call cost push. Their action is expressed by

$$ \begin{array}{cc}\hfill \dot{p}/p=\lambda +H\left({Q}^{*}\right)+\sigma \left(\dot{w}/w-\lambda \right),\hfill & \hfill 0\le \sigma \le \hfill \end{array}1,. $$

with σ constant. H is an increasing function and H(0) = 0, because windfalls cause trial-and-error revision of short-term expectations. When Q * = 0 and \( \dot{w}/w-\lambda \) the inflation of expected prices equals the expected inflation of them, λ. The cost push term, \( \sigma \left(\dot{w}/w-\lambda \right) \), allows for the possibility that when the index of efficiency wages rises or falls, firms expect prices to rise or fall in other affected industries, diverting demand to or from their own industry.

The Dynamics of Efficiency Wages

Similarly three forces act on w, namely expected inflation, the excess demand for labour, and the indexation of wages to expected prices. Their action is expressed by

$$ \begin{array}{cc}\hfill \dot{w}/w=\lambda +F\left(x/\upsilon \right)+\sigma \left(\dot{p}/p-\lambda \right),\hfill & \hfill 0\le \tau \le \hfill \end{array}1, $$

with τ constant. Let N s be the supply of labour in efficiency units and υ be N s/K. Then x/υ = N s/K. When x = υ unemployment equals unfilled vacancies. The corresponding unemployment rate is the ‘natural rate, kept in being by the break-up of old jobs and imperfect information about the new jobs that replace them. (Firms with vacancies use their workers more intensively while seeking to fill them, so that the vacancies do not preclude the production of Y Kf(x).) The unemployment rate is a decreasing function of x/υ. Unemployment is involuntary when x/v is <1. F is a non-decreasing function with F(1) = 0.

The Equations of Motion

Logic requires στ<1; for \( \dot{w}/w \)  /  \( \dot{p}/p \) cannot be both exclusively determined by x/υ and exclusively determined by Q *. Therefore the development of the economy is governed by the following equations:

$$ \begin{array}{c}\dot{x}/x=aH\left[{Q}^{*}\left(x,m,\lambda \right)\right]-bF\left(x/v\right)\\ {}\dot{p}/p-\lambda =cH\left[{Q}^{*}\left(x,m,\lambda \right)\right] - gF\left(x/v\right)\\ {}\dot{w}/w-\lambda =hH\left[{Q}^{*}\left(x,m,\lambda \right)\right]-cF\left(x/v\right)\\ {}\dot{v}/v=n-{I}^{*}\left(x,m,\lambda \right).\end{array} $$

The first is from the derivative of log f′(x) = log w/p with respect to time. The second and third combine the equations of sections “The Dynamics of Short-Term Expectations” and “The Dynamics of Efficiency Wages.” The fourth is from \( \dot{v}/\ v = {\dot{N}}^{\mathrm{s}}/{N}^{\mathrm{s}}-\dot{K}/K \), with n defined as N s /N s, the growth of the supply of labour in efficiency units. The coefficients are as follows:

$$ \begin{array}{c}\begin{array}{cc}\hfill a=\phi \left(1-\tau \right)/\left(1-\sigma \tau \right)\ge 0\hfill & \hfill \mathrm{with}\;\phi =-{f}^{\prime }(x)/x{f}^{{\prime\prime} }(x)>0;\hfill \end{array}\\ {}\begin{array}{cc}\hfill b=\phi \left(1-\sigma \right)/\left(1-\sigma \tau \right)\ge\ 0;\hfill & \hfill c=1/\left(1-\sigma \tau \right)>0;\hfill \end{array}\\ {}\begin{array}{cc}\hfill g=\sigma /\left(1-\sigma \tau \right)\ge\ 0;\hfill & \hfill h=\tau /\left(1-\sigma \tau \right)\ge 0.\hfill \end{array}\end{array} $$

In conjunction with particular assumptions about the behaviour of m, λ, and n, these equations enable us to capture the essential characteristics of many macrodynamic theories and to display their interrelationships.

Processes with a Constant Labour–Capital Ratio

If v = N s/K is a constant, \( \overline{\upsilon} \), the last equation in section “The Equations of Motion” disappears. Two interpretations are possible: either the change in υ over the relevant period is negligible, or labour-augmenting technical progress equals the growth of capital per worker. Processes with constant υ can therefore be regarded as occurring in relation either to a short-period equilibrium without technical change or to a long-period equilibrium with endogenous growth. The formal structure is the same in both cases.

Keynes’s General Theory

Expectations and Short-Period Equilibrium

In the General Theory the temporary equilibrium converges to a Marshallian short-period equilibrium with no technical change. Keynes imagines two ways by which it may be reached. In the General Theory for the most part he assumes as a short cut that short-term expectations are always fulfilled (Keynes 1973a, pp. 602–3). At the outset of an instant, entrepreneurs, correctly anticipating the aggregate demand-price, choose the employment, x, that will maximize their actual profit, π* f(x) − wx, since p = π. This is the case of the ‘instantaneous multiplier’; Y * is determined at the outset of each instant so as to make Q * = 0, i.e., I * = S *, within it. However he does not insist on this. If short-term expectations are not always fulfilled, p is adjusted by trial and error from one instant to the next. This process, along with the assumption that during it the economy is in the temporary equilibrium, is actually contemplated at one point in the General Theory itself (Keynes 1936, pp. 123–4), and indeed later he wished that he had made more of it there (Keynes 1973b, pp. 180–1). We may also wish he had; for by not doing so he originated the myth that he was himself rejecting the Treatise on Money’s Marshallian conception of temporary equilibrium in favour of Hick’s conception of it.

Money Wages and Employment

Keynes claims as a fundamental objection to the ‘classical’ theory the postulate that the real wages, w/p, on which employment, x, depends are directly affected by labour’s bargaining about money wages (Keynes 1936, p. 13). Keynesian unemployment is involuntary in a special way: it cannot be directly eliminated by flexibility of money wages. This dogma is first enunciated in the Treatise on Money (Keynes 1930, p. 167), where changes in w have no direct tendency to bring about non-zero profits, Q *, because, so long as they are not allowed to affect interest rates, they cause a proportionate change in the price level, π* = p + p (I * − S *) f(x). But that is so only if they induce a proportionate change in expected prices, p, leaving w/p, and so x, unaffected. In fact he is assuming full cost push, σ = 1, so that, in section “The Equations of Motion” \( \dot{x}/x=aH\left({Q}^{*}\right) \). Changes in employment are due solely to the effect of Q * on short-term expectations of prices in terms of wage units, p/w, not at all to changes in the wage unit, w, itself, except in so far as they may affect the parameters determining Q *.

Not a strong foundation for a general theory! Nevertheless there is a good reason for retaining this possibility in our process analysis. In the Hicksian temporary equilibrium the real wage is likewise determined independently of the money wage so long as m is given. When post-Keynesians who adopt the Hicksian viewpoint allow for some degree of money-wage flexibility, the qualitative behaviour of their models will be just as if there were full cost push.

The Trial-and-Error Process

If, for simplicity, one treats as a parameter the supply of money ‘in terms of wage units’, so that m = kf′(x) with k a positive constant, the process, with parametric λ, is \( \dot{x} \)/x = aH [Q *(x, kf′(x); λ)], \( \dot{p} \)/p − λ = c [H (Q *) + F (x/\( \overline{\upsilon} \))], \( \dot{w} \)/w − λ = cF (x/\( \overline{\upsilon} \)); for g = c when σ = 1, and h = 0 because no wage-indexation is assumed. In the equilibrium [which is stable if m \( {Q}_x^{*}+{Q}_m^{*}k{f}^{{\prime\prime} }(x) \) is negative] Q * = 0, x *< \( \overline{\upsilon} \), and (\( \dot{p} \)/p)* − λ = (\( \dot{w} \)/w)* − λ = F (x */\( \overline{\upsilon} \)). Thus Keynes really needs to assume wage inflexibility below full employment, F (x/\( \overline{\upsilon} \)) = 0 for x < \( \overline{\upsilon} \), in addition to σ = 1. Otherwise the equilibrium would be upset by a systematic error about expected inflation. The underemployment equilibrium is then I (0, r *, x *) = S (0, r *, x *), L (0, r *, x *; λ) = m * = kf′(x *), (\( \dot{p} \)/p)* = (\( \dot{w} \)/w)* = λ, with x * < \( \overline{\upsilon} \).

Say’s Law

If m * is such that Q * = 0 for all x and λ, the process is \( \dot{x} \)/x = −bF (x/ \( \overline{\upsilon} \)), \( \dot{p} \)/p − λ = gF (x/ \( \overline{\upsilon} \)), \( \dot{w} \)/w − λ = cF (x/ \( \overline{\upsilon} \)). When σ is less than unity and F is strictly increasing there is a convergence to equilibrium at the natural unemployment rate, with inflation of p and w at the rate λ, which is not determined by the system. The equilibrium equations are I (0, r *, x *) = S (0, r *, x *), x * = \( \overline{\upsilon} \), L (0, r *, x *; λ) = m *, (\( \dot{p} \)/p)* = (\( \dot{w} \)/w)* = λ.

Keynes (1936, p. 26) maintained that Say’s Law would imply indeterminacy of x. Indeed it would under his assumption σ = 1, for then b = 0. However his allegation, that in these circumstances competition between entrepreneurs would lead to full employment, is a nonsequitur, as Hawtrey pointed out to him (Keynes 1973b, pp. 31–2).

Full Wage-Indexation

If τ = 1 then a = 0. The process is \( \dot{x} \)/x = −bF (x/    \( \overline{\upsilon} \)), \( \dot{p} \)/p − λ = cH[Q * (x,m,λ)] + gF (x/ \( \overline{\upsilon} \)), \( \dot{w} \)/w − λ = c[H (Q *) + F (x/\( \overline{\upsilon} \)). As under Say’s Law, there is convergence to the natural unemployment rate. But, whereas Say’s Law leaves inflation indeterminate, full wage-indexation offers a painless means of manipulating it by changing the supply of money.

Underemployment Equilibrium in a Growing Economy

The Keynesian equilibrium of section “The Trial-and-Error Process” can be interpreted as one of endogenous growth with involuntary unemployment. This extension is due to Domar (1946, pp. 137–47). Actually he used the ‘extreme Keynesian’ assumptions that I is determined by entrepreneurs’ animal spirits, with I x  = I r  = 0, and that S = sf(x) with s a positive fraction, so that money has no effect on them. In his equilibrium (which is obviously stable, given I) I = sf (x *), and the ratio of actual output, Y * = Kf (x *), to normal capacity output, P = Kf (\( \overline{\upsilon} \)), is less than unity unless I is large enough to imply x * = \( \overline{\upsilon} \).

Business Cycles with a Constant Labour-Capital Ratio

A Purely Monetary Theory of Cycles

The appellation is taken from Haberler (1937, ch. 2), where he expounds Hawtrey’s theory, contrasting it with overinvestment theories, in which changes in υ are an essential feature. The following version generalizes a model constructed by Phillips (1961, pp. 360–70) but conveying ideas much like those expressed by Hawtrey. For his first statement of them see Hawtrey (1928, ch. 5).

There are four assumptions: (i) F is strictly increasing; (ii) the ratio of the nominal money-suppy to K grows at the constant rate μ; (iii) people expect inflation to be μ, i.e., λ = μ; (iv) the equilibrium is stable.

Since m is the supply of money deflated by pK, \( \dot{m} \)/m = μ −\( \dot{p} \)/p by (ii). But from the second equation of motion in section “The Equations of Motion” we have \( \dot{p} \)/p = λ + cH + gF, so that \( \dot{m} \)/m = μ − λ − cH − gF = −cH–gF by (iii). Hence the dynamic system in x and m is

$$ \begin{array}{l}\dot{x}/x=aH\left[{Q}^{*}\left(x,m;\mu \right)\right]-bF\left(x/\overline{\upsilon}\right)\dot{m}/m=\hfill \\ {}-cH\left[{Q}^{*}\left(x,m;\mu \right)\right] - gF\left(x/\overline{\upsilon}\right),\hfill \end{array} $$

with the equilibrium x * = \( \overline{v} \), I (0, r *, x *) = S (0, r *, x *), L (0, r *, x * ; μ) = m *. Notice that changes in μ have no real effect on it, merely altering m *.

There is local stability if \( \overline{v}\ \left[a{H}^{\prime }(0){Q}_x^{*} - b{F}_x\right] - {m}^{*}c{H}^{\prime }(0){Q}_m^{*} \) is negative. Thus even if the first term, representing the effect of x on \( \dot{x} \), is positive, the second term, representing the effect of p on the course of real balances, and therefore on the course of interest rates, can (and we are assuming will) outweigh it. For Hawtrey the first term is positive. A shock induces a cumulative expansion (or contraction), which is eventually reversed because a growing shortage (or abundance) of money increases (or reduces) interest rates.

The discriminant of the linearized system is

$$ D={\left\{\overline{\upsilon}\left[a{H}^{\prime }(0){Q}_x^{*} - b{F}_x\right] - {m}^{*}c{H}^{\prime }(0){Q}_m^{*}\ \right\}}^2-4\left( ag+bc\right)\overline{\upsilon}{m}^{*}\ {H}^{\prime }(0){F}_x{Q}_m^{*}\ . $$

It implies that there will be oscillations if, ceteris paribus, Q * m is large. For \( \partial D/\partial {Q}_m^{*} \) is negative when the stability condition is satisfied.

Examination of D reveals a very interesting point. With full cost push (b = 0) higher wage-flexibility (larger F x ) must, ceteris paribus, induce more rapid oscillations. Compare Keynes (1936, pp. 269–71). (Phillip’s model, in which a coefficient ß corresponds with our F x , has this Keynesian characteristic.) For it increases the frequency of the turning points induced by the monetary factor without damping the cumulative process. But when b is positive high enough wage-flexibility eliminates the cumulative process entirely. No oscillations can occur.

Staglation Cycles

There have been periods during which inflation and the unemployment rate have risen or fallen simultaneously. Three assumptions are sufficient to explain this phenomenon: (i) expectations of inflation are adaptive: \( \lambda =\gamma \left(\dot{p}/p-\lambda\ \right) \) with γ positive and constant; (ii) monetary policy is to decrease (or increase) m when λ rises (or falls): m = m (λ;θ) with m λ negative; θ is a shift parameter with m θ positive; (iii) the equilibrium is stable.

We have then

$$ \dot{x}/x=aH\left[Z\left(x,\lambda; \theta \right)\right] - bF\left(x/\overline{\upsilon}\right) $$
$$ \dot{\lambda}=\gamma \left\{cH\left[Z\left(x,\lambda; \theta \right)\right]+gF\left(x/\overline{\upsilon}\right)\right\}, $$

Where Z (x, λ;θ) is Q * [ x, m (λ;θ), λ] The equilibrium equations are x * =\( \overline{\upsilon} \), I (0, r *, x *) = S (0, r *, x *), and L (0, r *, x *, λ *) = m (λ * ;θ).Observe that changes in θ affect only λ *.

The equilibrium is locally stable if Z λ and \( \overline{\upsilon} \)[aH′(0) Z x  − bF x ] + γ cH′(0) Z λ are both negative. The first condition is satisfied if and only if m λ is more negative than L λ .The authorities must ensure that real interest rates move in the same direction as λ The second condition guarantees that the course of real interest rates eventually dissipates the cumulative expansions and contractions that may occur if \( {Z}_x={Q}_m^{*} \) is large.

If there are oscillations the turning points are due to the Central Bank’s policy. As in the previous model higher wage-flexibility increases their frequency if b is zero, but weakens the cumulative forces if b is positive.

A shock due to a change in θ must initially cause x and λ to move in the same direction. But, whereas λ tends to a new equilibrium, x must tend back to the original \( {x}^{*}=\overline{\upsilon} \). There must therefore be a period during which x and λ move in opposite directions, and since the inflation of both expected and actual prices tends to λ *, there must also be a period during which inflation and the unemployment rate move in the same direction.

Keynesian Overinvestment Cycles

Henceforward we assume that \( {\dot{N}}^s/N=n \) is a constant, thereby resuscitating the fourth equation in section “The Equations of Motion.”

Purely monetary theories fail to reproduce two observed features of business cycles: (1) The unemployment rate continues to fall (or rise) after entrepreneurs’ expected profit-rates have begun to fall (or rise). (2) The real efficiency wage is not a monotonically increasing function of the unemployment rate. But overinvestment theories with a variable unemployment rate do reproduce them.

Natural and Warranted Rates of Growth

The natural rate is n, the sum of the growth rates of the supply of workers and efficiency per worker. The term ‘warranted rate’ was introduced by Harrod (1939, pp. 14–33) to designate a rate of growth of output which, if it occurs, will leave all parties satisfied that they have produced the right amount (ibid., p. 16). Several formulae are given for it there, and also in Harrod (1948, Lecture 3) and Harrod (1952, Essay 14), depending on alternative assumptions about the determinants of planned investment and planned saving. But the alternatives have one thing in common, namely that these plans are not significantly influenced by monetary policy; either the real rate of interest cannot easily be changed, or the plans are inelastic with respect to it (Harrod 1952, pp. 95–100). Theories involving the warranted rate have an ‘extreme Keynesian’ bias.

In our equations of motion assume (i) b = 0; (ii) F is zero on a large interval around x = υ; (iii) Q * and I * depend only on x; (iv) τ = 0. Then \( \dot{x}/x=aH\left[{Q}^{*}\ (x)\right] \), \( \dot{p}/p-\lambda = cH\left[{Q}^{*}\ (x)\right] \). \( \dot{w}/w=\lambda \), and \( \dot{\upsilon}/\upsilon =n-{I}^{*}\ (x) \). The warranted rate is \( \dot{Y}/Y={I}^{*} \)with Q * = 0, for it is justified by the realization, on the average, of short-term expectations.

Now as it stands this system is quite useless. The warranted rate is divorced from the natural rate, so that there is almost surely no equilibrium. But the defect can be remedied if either I or S can be assumed to depend on υ.

Autonomous Consumption

A rationale for making S depend on υ was given by Matthews (1955, pp. 75–95), who suggested that planned consumption from a given income increases with the unemployment rate. Support for the unemployed is at the expense of planned saving. Such changes in consumption are ‘autonomous’ in that they are not in response to changes in income. Thus S * = S * (x,υ), with S * v negative. The system

$$ \dot{x}/x=aH\left[{Q}^{*}\ \left(x,\upsilon \right)\right] $$
$$ \dot{\upsilon}/\upsilon =n-{I}^{*}\ (x) $$

is assumed to have a unique equilibrium, n = I * (x *) = S * (x *,υ*), with underemployment, i.e., x * < v *.

Shock-Induced Oscillations

Assume that the equilibrium is stable. This is the case if \( {Q}_x^{*} = {I}_x^{*} - {S}_x^{*} \) is negative and I * x is positive at the equilibrium point. It can be shown that there will be oscillations if |Q * x | is sufficiently small. A shock induces overinvestment cycles, in that during the boom the growth of capital is excessive (I * > n).The upper turning point is reached when the consequential fall in υ pushes S * above I *. Similarly the lower turning point is reached when the rise in υ, due to an excess of n over I * during the slump, pushes S * below I *. For this kind of theory see Samuelson (1939, pp. 75–8). In his version n is zero and autonomous consumption spending is by the government.

Self-Exciting Oscillations

Three conditions are sufficient for these: (i) The equilibrium is unstable but I * x is positive; (ii) nevertheless Q * x is negative for high and low values of x, say because short-term expected profit seems a less trustworthy guide to investment planning when it has moved far from its equilibrium; (iii) H′(Q *) is so large that the changes in x when Q * is non-zero are much larger than the changes in υ when I * differs from n. By (i) the equilibrium is surrounded by centrifugal forces, and is almost surely not the initial state. By (ii) there are turning points for x, because when x and υ are moving in opposite directions they combine to reduce windfalls, |Q * x |. By (iii) there are turning points for υ, because of the rapidity with which net overinvestment, |I * − n|, is reduced when x and υ are moving in the same direction.

This essentially is Kaldor’s theory (Kaldor 1940, pp. 78–92). Only the first two conditions are given in his text, but the third is implicit there, and is explicitly stated in his appendix (Kaldor, p. 90).

Autonomous Investment

Some investment may grow at the natural rate, n. Then I * = J(x) + Ae nt/K, where KJ(x) is ‘induced’ investment, Ae nt is ‘autonomous’ investment, and A is a positive constant. Since \( {e}^{nt}={N}^S/{N}_0^S,{I}^{*}=J(x)+\left(A/{N}_0^S\right)\upsilon \), or, more generally, \( {I}^{*}={I}^{*}\left(x,\upsilon \right) \)with I * υ positive. The system

$$ \dot{x}/x=aH\left[{Q}^{*}\ \left(x,\upsilon \right)\right] $$
$$ \dot{v}/v=n-{I}^{*}\ \left(x,\upsilon \right) $$

is assumed to have a unique equilibrium, n = S * (x *) = I * (x *,υ*), with x * < υ*.

Shock-Induced Oscillations

Assume that the equilibrium is stable. This is so if \( {x}^{*}a{H}^{\prime }(0){Q}_x^{*} - {\upsilon}^{*}\ {I}_x^{*} \) is negative and I * x (x*, υ*) is positive. It can be shown that there must be oscillations if, ceteris paribus, Q * x is large. Overinvestment (underinvestment) leads to an upper (lower) turning point as changes in υ push I * below (above) S *. For this alternative to the autonomous-consumption story see Kalecki (1939, Essay 6). He assumes that n is zero.

Self-Exciting Oscillations

A persistent cycle follows from assumptions similar to those of Hicks (1950); cf. also Goodwin (1951, pp. 1–17): (i) The equilibrium is unstable. (ii) There is a full-employment ceiling, a rigid x barrier, C, such that x ≤ Cυ.It is a constraint on x that is binding so long as its free motion would violate it. (iii) There is a value of x, viz. ξ < x *, such that I * x (x, υ) is positive for all x > ξ but is zero for all x ≤ ξ.For induced gross investment in fixed capital cannot be negative, and further induced disinvestment in inventories would disrupt the productive process (cf. Hicks 1950, p. 104).

The cycle is attained in finite time from any non-equilibrium initial state. It has a floor implied by the fact that, if in its course the situation I *(x,υ) = n occurs when x ≤ ξ,υ, must remain constant until x has risen above ξ. The floor value of υ is the solution to I (ξ,υ) = n. The cycle must hit either the ceiling or the floor, but need not hit both.

Non-Keynesian Overinvestment Cycles

Henceforth we assume σ < 1 and some flexibility of money wages.

Oscillations with Imperfect Wage-Flexibility

F is strictly increasing, and there are positive constants q and l (q > 1 > l) such that F tends to + ∞ as x/υ tends to q, and to – ∞ as x/υ tends to l.

A ‘Non-Monetary’ Theory

Under a Say’s-Law regime

$$ \dot{x}/x=-bF\left(x\upsilon \right) $$
$$ \dot{\upsilon}/\upsilon =n-{I}^{*}\ (x), $$

Where \( {I}_x^{*}\kern0.5em =\left({I}_x{S}_x - {I}_r{S}_x\right)\left({S}_r - {I}_r\right) > 0 \) (see section “Capital Accumulation”). The equilibrium, n = I * (υ*), is globally stable, but there will be shock-induced oscillations if F′ is small and I * x is large in its neighbourhood. For the analysis and a comparison with Cassel’s theory see Rose (1969, section III).

There will also be such oscillations if the elasticity of substitution between labour and capital (and therefore b) is small. The model then reproduces approximately Goodwin’s growth cycle (Goodwin 1967, pp. 54–8). (If, as he assumes, the elasticity is zero, and in addition all profits are saved and all wages consumed, every solution will be periodic in w/p and υ).

If, however, wages were perfectly flexible the system would reduce to \( \dot{x}/x=n-{I}^{*}\ (x) \) which is Solow’s growth model (Solow 1956, pp. 58–94).

A Monetary Theory

Let monetary policy be to sustain a constant m. The system

$$ \dot{x}/x=aH\left[{Q}^{*}\ \left(x;m,\lambda\ \right)\right] - bF\left(x/\upsilon \right), $$
$$ \dot{\upsilon}/\upsilon =n-{I}^{*}\ \left(x;m,\lambda\ \right) $$
$$ \dot{p}/p-\lambda =cH+gF $$

has only a ‘quasi-equilibrium’ if λ is arbitrarily given: for \( \dot{x}=\dot{\upsilon}=0 \) does not imply \( \dot{p}/p=\lambda \). To avoid this systematic error about long-run inflation we assume that the public foresees the value λ must take if \( {\left(\dot{p}/p\right)}^{*} \) is to equal it. The equilibrium will then be n = I (0, r *, x *) = S (0, r *, x *), L (0, r *, x *, λ *) = m, v * = x *.

The interesting characteristic of this model is that, if the equilibrium is unstable and if I * x is everywhere positive, there must be self-exciting oscillations whose amplitude can be quite small. For the details see Rose (1967, pp. 153–73).

An Equilibrium Theory of Business Cycles

Once upon a time cycles were thought to arise from unsustainable alternations in the structure of the production, brought about by inappropriate and unanticipated changes in the supply of money. Wage inflexibility was not an essential ingredient. This position, held by Hayek (1935, Lecture III) and a cohort of ‘Austrian’ economists, is surveyed in Haberler (1937, pp. 31–67). Recently, Lucate duce, there has been a remarkable attempt to recapture it (Lucas 1975, pp. 1113–44).

The assumptions in our version of it are as follows: (i) there is continuous full employment; (ii) the growth rate of nominal money per unit of capital is a constant, μ. (iii) λ = μ (iv) there is no cost push (σ = 0).Therefore

$$ \dot{x}/x=n-{I}^{*}\ \left(x,m;\mu \right) $$
$$ \dot{m}/m=-H\left[{Q}^{*}\ \left(x,m;\mu \right)\right] $$

The equilibrium is almost certainly stable, but there can be oscillations if I * x and Q * m are small and Q * x is negative.

For simplicity we tell the story as if n = μ = λ = 0. Equilibrium is disturbed by an unanticipated increase in nominal money. Interest rates fall, creating an investment boom and net windfall profits (‘forced saving’). The investment boom increases capital, output, and capital intensity, K/Y = 1/f(x), and is only weakly checked by the larger capital (lower x). But net windfalls raise p (reduce m) and so interest rates rise, eventually leading to an upper turning point for K and K/Y. Net windfalls are still positive, but, once K begins to fall, both higher interest rates and lower K (higher x) convert them into net losses. Now both K is falling and there are net windfall losses. But these reduce p and so interest rates fall, leading to a lower turning point for K and K/Y. Finally lower interest rates and higher K create net windfall profits once again, and a new boom of investment and windfalls begins.

This version may not please Lucas and his school. Persistent, recurrent, and unexploited profit opportunities are anathema to them. But for the inhabitants of their archipelago there persist also recurrent, unexploited profits to be made by discovering what is happening on other islands. Indeed the situations are not dissimilar. In our case what needs to be discovered is not only whether Q is positive or negative but also the whereabouts of its components, which are not predicted by the model.

See Also