Demand, Supply and the Price Level in Macroeconomics

The Review of Economic Studies, Ltd.

Demand, Supply and the Price Level in MacroeconomicsAuthor(s): Hugh RoseSource: The Review of Economic Studies, Vol. 20, No. 1 (1952 - 1953), pp. 1-23Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2296158 .

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Demand, Supply and the Price Level in

Macroeconomics1 This article is divided into two parts. In Part I the well-known " neo-Marshallian"

theory of the firm is used to examine some of the problems that arise in the theory of investment, with special reference to the character of the investment relation contained in macroeconomic models. We shall discuss (i) the connexion between the rate of investment and the rate of interest, (2) the derivation of the acceleration principle and the Keynesian inducement to invest from the theory of the firm, and (3) the relation between the supply of output and the accumulation of capital. In Part II we try to show what modifications of current macroeconomic systems are implied by the results of our inquiry.

PART I i. The Theory of the Firm and Macroeconomics The analysis of planning over time has been conducted from two different points

of view, both developed from Marshall's Principles. There is on the one hand the neo- Marshallian theory itself, and on the other the so-called " intertemporal " theory of the firm, as in Dr. Hicks' Value and Capital.

The neo-Marshallian theory deals with the time factor by supposing the firm to have two plans, one for the short run, in which decisions are taken about current output and the employment of labour, materials, etc., and the entrepreneur is finding the best way of using a given capital equipment ; and one for the long run, in which decisions about altering the capital equipment itself are formed. The theory also has to specify the way in which these two plans are related to each other-how the short- term plan is adjusted towards the ultimate aims of the firm as set forth in the long-term plan, and the extent and manner in which these aims are based on experience in connexion with the fulfilment of short-term requirements.

In contrast the intertemporal theory conceives the firm as making only one plan, comprising decisions at once about current activity and about capital extension or retrenchment. The place of the problem of showing how two plans fit together is taken by that of specifying the relations between the plan adopted at one date and the modifications made necessary by the new circumstances in which the entrepreneur may find himself at subsequent dates.

Our decision to use the first of these approaches derives from a feeling of dis- satisfaction with the intertemporal theory as a basis for macroeconomic model building. We shall try to explain this, illustrating our remarks by reference to the Keynesian inducement to invest ; the closing paragraphs of this section will then offer some observations on the relevance of the neo-Marshallian theory to the demonstration of the acceleration principle.

As it is usually set forth, the intertemporal theory has the disadvantage, from the point of view of macroeconomics, of working without explicit reference to the concepts of capital and the rate of investment, We infer that investment is occurring when the firm increases its application of present inputs to the production of future output,

1 I wish to thank the following for encouragement, helpful discussion and advice, and for the generous sacrifice of their time which this has obviously involved: Professor J. R. Hicks, Professor D. G. Champer- nowne, Mrs. U. K. Hicks, Messrs. R. W. Clower and P. K. Newman, all of whom are either members of, or closely associated with, Nuffield College, Oxford. I also derived great benefit from correspondence with Mr. N. Kaldor.

1*



2 THE REVIEW OF ECONOMIC STUDIES

but, because the quantity of capital is not one of the variables in the problem, the actual rate of investment remains rather vague. This is no doubt not irremediable, but it is a difficulty.

More important is the intractability of the theory for yielding an investment relation that holds over time. All it directly tells us is the firm's investment plan at the initial date, i.e. a hypothetical series of quantities of investment corresponding to various assumptions about current and expected circumstances. We can get further than this only by taking into account the changes of plan that occur over time. Thus a macroeconomic investment equation derived from the intertemporal theory must incorporate special assumptions about the development of expectations, just because any macroeconomic system must hold over time if it is to be of any use. It should certainly be possible to carry this through, but it would be more difficult than with the neo-Marshallian theory, because of the complexity of the apparatus required to connect one planning date's decisions with those of subsequent dates.

It may be objected that, in the case of the Keynesian equation, we have a clear duty to use the intertemporal theory, on the grounds that that was what he himself had in mind. Dr. Hicks shows in Value and Capital (p. I98) that the marginal efficiency of capital concept may be regarded as a special case of substitution over time in the more general intertemporal theory. However, we cannot ignore certain affinities between Keynes' thought and the neo-Marshallian approach, in particular the distinction between long- and short-run expectations (General Theory, p. I48), and the impression left by those passages in which the long-run equilibrium of his system is analysed in terms of an approach to stationary conditions (e.g., p. 2I7 et seq.).

Finally there is the question of the relative capacity of the two approaches to incorporate the kinds of assumptions about expectations that are made in macroeconomic models. The intertemporal theory has a prima facie advantage here, in that it can take account of any pattern of expectations, while the other has only properly been developed for the case of " static " long-run expectations. It should, however, be capable of extension in terms of more complex hypotheses-for example, that the firm expects a certain trend of growth of demand in the long run. Moreover, the assumption of static long-run expectations is neither so unreal nor so irrelevant to macroeconomic models as it may at first appear.

Consider first the question of its realism. An entrepreneur's ideas of the future are, of course, uncertain. However, in order to make a definite decision about capital he has to form in his mind a definite level, or trend, of demand which he judges to be his " best estimate " in the circumstances. This estimate, which is as it were " crystal- lised " from the data he regards as relevant, we shall call the long-run level (or growth) of demand. The term " static long-run expectations " then distinguishes the case where what the firm has in mind is a level, as opposed to a trend of growth or decline, of demand in the future. The estimate is, of course, subject to revision, particularly in the light of changing current demand conditions, that is, it may be a function of these conditions, but this is not necessarily so. Our theory must, in fact, specify what relation holds between current conditions and the long-run estimate. The existence of a level of long-run demand in this sense gives meaning to the concept of desired economic capacity-the long-run level of output at which the firm is aiming-a concept that does not seem to be without empirical significance.

The long-run estimate may in some circumstances be an average level of expected demand: if firms disregard current changes in demand (perhaps below a certain magnitude) in making capital decisions, a natural interpretation would be that they thought of these changes as deviations about an average that was still regarded as relevant. On the other hand it may frequently be a maximum concept, rather than



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS 3

an average one-for example, the maximum level of demand that the entrepreneur judges he will be required. to cope with in the foreseeable future-c.f. the analysis of planned reserve capacity in Section 3 below. In this case the fact that current demand is less than long-run demand does not preclude an upward revision of the latter in the event of a rise in the former. Finally, where entrepreneurs revise their estimates whenever current conditions change-the usual supposition of induced investment theory-it would be clearly ridiculous to think of these estimates as expected averages, since the notion of an expected average carries with it the idea that deviations from it are expected to occur.

As to the relevance of the assumption of static long-run expectations, we intend subsequently to demonstrate that it is possible, using this assumption, to derive the Keynesian inducement to invest allowing both for the influence of expectations on the marginal efficiency of capital and for the ultimate tendency towards stationary conditions.

The acceleration theory has sometimes been put forward as though it were based on a purely technical relationship between investment and changes in output. However, most theorists recognise that changes in demand are the ultimate determinant ; output changes are used as being a more easily measurable index of demand changes. Even so, the theory is usually innocent of an explicit attempt to deal with expectations. The most complex form of the hypothesis, as presented in Dr. Hicks' Trade Cycle, seems to be based explicitly on the assumption of static long-run expectations: for~ he writes (p. 43), " There are good reasons for supposing that any rise in the level of output which businessmen regard as 'normal' will tend to induce an investment in new capital to support that increased output." And in any case Dr. Hicks' theory surely implicitly requires that assumption. For in it the rate of investment is a compound of the effects of past changes in output, and this is only likely to be the case if firms' investment plans suffer continual modification; it is difficult to see any reason why an anticipated series of changes in output should give rise to this " overlapping" effect. The overlapping is due to the changes in output being contrary to expectations. It might be produced in either of two ways:

i. If each change in output reflects a change in the firm's idea of the level of long- run demand, i.e. there are static expectations which are modified over time, the rate of investment at any time reflects the partial fulfilment of a series of plans, each one of which except the last is currently regarded as insufficient to accomplish the entrepreneur's long-run plan for output capacity.

2. If each change in output reflects a change in the firm's idea of the growth of long-run demand, the rate of investment at any time reflects the partial fulfilment of a series of plans, each one of which except the last is currently regarded as insufficient to accomplish what the entrepreneur regards as the appropriate growth of output capacity.

Of these two possibilities, the second can be ruled out as inappropriate to Dr. Hicks' analysis. For changes in expectations of the rate of growth of demand would themselves have consequences in entrepreneurial reactions that do not appear to be included in his schema, within which divergences from the equilibrium path are governed by the internal mechanics of the model, not by the inconsistency of entrepreneurial expectations with the facts. This indeed seems to be the main point at issue between Hicks and Harrod in their trade-cycle theories': the latter would make the divergence of the system from its equilibrium path due not to the values of the accelerator and multiplier coefficients, but to the inconsistency of entrepreneurs' expectations of the rate of growth with the actual rate of growth. Since this factor is absent from the

1 Harrod: " Notes on Trade Cycle Theory." Economic Journal, June, I95I.




Hicks theory we conclude that static long-run expectations are implied, and that our ensuing analysis is therefore relevant to his investment fun'ction.

2. The Rate of Investment and the Rate of Interest We now proceed to the task of deriving the induced investment equation from the

neo-Marshallian theory. We shall obtain it in its most general form, as it appears in Dr. Hicks' Trade Cycle, and also in the simpler version used by Professor Samuelson.' The object will be to observe the assumptions on which it is based, and to test the general validity of the relation in the light of the theory of the firm. On the way we shall have something to say on the subject of the rate of interest.

The demand for capital is a part of the long-term plan. Using an idea borrowed from Mr. Kaldor,2 we may divide the resources available to the firm into capital goods and current inputs. Capital goods yield services into the indefinite future after instal- lation, if continuously maintained and replaced: applications of current input of labour, materials, etc., have to be repeated every unit period. The state of the arts is unchanging, and there is a perfect market for loans.

The firm's capital plan is based on the maximisation of expected long-run discounted net revenue, i.e. the entrepreneur decides on the best level of long-run output to prepare for, and therefore the best amount of capital to employ, by the condition that the marginal efficiency of capital equal the expected long-run rate of interest. The capital plan will depend on r, the expected long-run interest rate, and D, the expected long-run state of demand. D is a cardinal index of the level of demand, such that an increase in D implies an increase in the marginal revenue obtained from a given output. Under perfect competition it will be the long-run expected price. If we bave a straight-line demand curve, there are two constants in the equation, the slope of the line and the point at which the line intercepts the price axis ; in that case D will be the latter, i.e. the price that would reduce expected sales to zero. Thus the capital plan is described by the equation:

K =K (D,r) ....... (i)

in which K varies directly with D and inversely with r.3 K is the value of capital goods, the valuation being either at initial cost or replacement cost, since the prices of such goods are supposed not to vary, and the two methods of valuation will therefore give identical results. The equation may be illustrated as in the Diagrams4 i and 2 on page 5. Let q stand for the slope of the curves in Diagram i, e for the slope of those in 2. Then q is almost always positive, and certainly so if there are constant returns to scale,3 while e is always negative. q is important for the acceleration principle, as it is a com- ponent of the acceleration coefficient, while e is important for the problem of the connexion between the rate of investment and the rate of interest ; and since they are both, in general, functions of r and D, being the partial derivatives of equation (i), it will be worth our while to say something more about the extent and manner of their variation.

1 Samuelson: " Interactions between the Multiplier Analysis and the Principle of Acceleration.' R.E. Stat., May, I939.

2 Kaldor: " Capital Intensity and the Trade Cycle." Economica, February, I939. 3 Strictly speaking q can be negative (K can vary inversely with D), unless there are constant returns

to scale. This would only be if K and output were " regressive " (see Hicks, Value and Capital, pp. 95-97). However, it is in practice inconceivable that capital as a whole should be a regressive factor.

4 In these diagrams r, < r2 < rY and D, < D2 < D.




i. The acceleration principle requires that q be constant with respect to D-that the curves in Diagram i be straight lines. It can be shown that this will only be so if there are constant returns to scale and the long-run demand curve is a straight line, but that it will hold approximately if there is perfect competition in the product market, long-run marginal costs are linear, and q itself is large.

K K

0 p r~~~~~~~:p i

zDi.qram 1. D ~Diagram 2.

2. With constant returns and a linear demand curve q will vary inversely with r, so that the curve system shown in Diagram i will have the form

K

Diagra-m3

In the general case, however, nothing can be said about the variation of q with r. 3. The effect of r on e is uncertain even if constant returns and a linear demand

curve are assumed. It seems likely that it will be small, i.e. that the curves in Diagram 2 will be approximately linear.




K means desired capital, to be distinguished from the actual capital at any moment. The two will only coincide in stationary conditions. K will vary over time whenever the firm revises its notion of r or D. Any such revision gives rise to a marginal investment plan, dK/dt K'(t), the extra capital planned at t as compared with the plan in the preceding period. The general expression for the marginal investment plan at t is found by differentiating equation (I) with respect to t. Remembering that q and e are the partial derivatives of K in equation (i), we thus obtain:

K'(t) q D'(t) + er'(t) ............... (2)

Suppose for simplicity that any marginal investment plan is fulfilled at once, without any time spread. In that case the actual capital will always equal last period's planned capital, and the rate of investment will coincide with the current marginal plan, since the rate of investment, 1(t), is the rate of increase of actual capital. We should then have

I(t) = q D'(t) + er'(t) ............... (3)

This is the fundamental relation on which the acceleration principle depends. We have still to introduce more complex assumptions about the spread of the marginal investment plans' execution over time, and also to connect investment with changes in output. But this simple statement brings out one interesting point: the rate of investment depends not only on the rate of change of demand but also on the rate of change of the long-term expected interest rate. This may seem surprising, since it is usual to think of investment as depending on the level of this rate. One's natural reaction is to suspect that we have somewhere slipped in a special assumption that produces this odd result. But is it really so odd ? Perhaps the reason why we usually think of investment as being related to the level of r is that the intertemporal theory of the firm encourages us to think of the investment equation as a " hypothetical " relationship, referring to the different rates of investment contemplated by the entrepreneur at any given moment for different values of r. In this sense it is certainly true that the lower is the anticipated rate of interest as compared with what it has been in the past, the larger will the investment plan be: in the same sense there would be a hypothetical relation between the rate of investment at any given moment and the level of demand anticipated. Even in this sense, however, the absolute level of the rate of interest (or the conditions of demand) has no significant bearing upon the rate of investment ; a rate of interest of 2 per cent would not necessarily be more stimulating to investment than a rate of 5 per cent, if at the time of planning the 2 per cent rate had ruled sufficiently long for the firm to have adjusted its capital to that rate. It is the capital plan, not the investment plan, for which the absolute levels of r and D are significant. This may point to an explanation of the apparent indifference of entrepreneurs to the level of the rate of interest in forming their investment plans, without the necessity to bring in risk factors, etc.

When we abandon the "hypothetical" view, and consider the rate of investment over time, it is changes in r that are relevant, just as it is changes in D : the acceleration principle applies in the case of both variables. We can, it is true, make special assumptions that will give us a relation between I, r and D that holds over time, and we shall have to do so for the Keynesian investment function. But in general if the accelerator relation holds between I and D it holds equally between I and r. Provided entrepreneurial expectations of the long rate do not undergo large or frequent changes, it is permissible to follow the acceleration theorists in neglecting the second term of equation (3). This -would also be possible if the coefficient e were small (though it should be observed that, strictly speaking, e could not be zero unless q were also zero).




Our previous argument requires modification in one respect, however. It is possible that the levels of r and D should influence the rate of investment, in so far as they influence the magnitude of the coefficients q and e. The important case here is that of r?s influence on q ; for the acceleration theory requires us to make assumptions that render q constant with respect to D, and little can be said of r's influence on e. (The effect of D on e is mathematically equivalent to that of r on q, if the usual assumptions are made about the continuity of our original function, K.)

We have previously observed that the same assumptions as serve to make q constant with respect to D imply an inverse relation between r and q. In that case it would be true to say that a rate of interest of 2 per cent that had ruled for some time would cause a given change in demand to be more stimulating of investment than would a rate of 5 per cent. But it should be remembered that this only necessarily applies with linear demand and constant returns to scale.

3. The Acceleration Principle As a rule, of course, we may expect a significant interval to elapse between the

starting and completion of a marginal investment plan. Typically, therefore, the rate of investment at any moment will have to be described by a more complicated relation than equation (3) above. Suppose it takes an interval of time of length T to bring a marginal investment plan from its inception to its completion, some proportion of it being fulfilled at each point of time within this interval. If we confine our attention to upward revisions of the capital plan, that is, to positive marginal investment plans, the simplest, and possibly most realistic, assumption to make is that any given marginal plan, once embarked upon, must be carried through to completion without revision. Once an order for equipment has been placed, a firm that finds it wants a larger quantity of such goods during the execution of the order will have to place another order; it will not in general be able to adjust the order that is in process of fulfilment. In that case the rate of investment at t will be a compound of those parts of the marginal investment plans currently in process of fulfilment that fall due to be executed at t. Our equation becomes

rT I (t) j fi(z) K' (t-z) dz ........................... (4)

We may shorten out notation conveniently by writing this

I =f j(z) K'dz ........................... (4a)

In (4), K' (t-z) is the marginal investment plan begun at t-z and j (z) is the proportion of it that is currently executed (at t). Substituting for the K' from equation (2), we obtain:

I =fq.j (z).D'dz + e.j(z).r'dz ........................... (5)

Let us henceforward suppose that changes in r are sufficiently small for us to be able to neglect the second term on the right of (5) and the influence of changes in r on q. In that case, if we assume constant returns to scale and a linear demand law, so that q is constant, we may write equation (5):

I q fj (z) .D' dz ............. (6)




Similarly, for the simple case where investment plans have no time spread, equation (3) becomes:

I =qD' ............. t (7)

If, on the other hand, we assume perfect competition and allow that q is sufficiently large for us to be able to treat it as constant, we obtain the two corresponding relations:

1 = q j(z) P'dz .(8)

and I =q P' .(9)

where P is the long-run expected price of the commodity. The acceleration principle, however, is commonly put forward as a relation not

between investment and changes in demand or price but between investment and changes in output. Does our theory of the firm give support to this view ?

i. Clearly there is such a relation between investment and changes in output capacity, that is, the long-run level of output planned. For the marginal investment plan can then be written:

K' = v x( (Io)

where v is constant under constant returns, and probably approximately so in any case ; and x is output capacity. The substitution of this expression in (4) gives us a relation between investment and changes in capacity. It would, of course, only be relevant in an analysis of long-run development, in which it would be valid to think of firms as always producing to capacity.

2. If we seek a relation between investment and changes in current output, we have to introduce, first, a relation between long- and short-run demand or price and, second, the firm's short-run output plan: Long- and Short-Run Expectations: Induced investment is said to occur when the formation of the long-run estimates which give rise to investment activity is influenced by current events. Let d be an index of short-run demand analogous to D, the index of long-run demand. Then a sufficient condition for changes in d to influence the capital plan is that D be some function of d. In what follows we shall concentrate on two simple cases:

Case I. Firms expect current demand conditions to continue indefinitely. Here D = d, and indeed the short- and long-run demand curves coincide.

Case 2. Firms habitually regard short-run demand as below long-run demand. For simplicity, suppose that both long- and short-run demand curves are straight lines with the same slope, and that the entrepreneur conceives the long-run curve to lie above the short-run curve by a constant absolute amount. In this case D = d plus a constant.

Under perfect competition Case I implies the coincidence of long- and short-run price, while Case 2 implies that long-run price is short-run price plus a constant.

In both of these cases a change in short-run demand implies an equal change in long-run demand (D' = d'). Whichever case is assumed to hold, therefore, we may regard equations (6) to (9) as connecting the rate of investment to past and present changes in short-run demand conditions (price). The Short-Run Output Plan: The entrepreneur bases his short-run output plan on the maximisation of the net revenue obtainable from operating the capital actually possessed at any moment. Short-run output is thus, in general, an increasing function of both d (or price) and the firm's actual capital, Q.




We have assumed, in order to render the coefficient q constant, that long-run marginal costs are either constant or linear. This means that the short-run marginal cost curve corresponding to any given capital equipment, or " plant ", will have the same (constant) slope as the long-run curve up to the point of full capacity, though it will lie below it because interest and depreciation do not enter into short-run marginal costs. From the point of plant capacity the short-run curve will rise more steeply. Let us introduce the following additional assumptions: (i) The short-run marginal revenue curve is a straight line; (2) The short-run marginal cost curve corresponding to each plant is linear beyond the point of full capacity; (3) The steeper portions of the marginal cost curves corresponding to all sizes of plant are parallel. With linear long-run marginal costs it will follow that equal successive increases in capital involve equal successive displacements of these steeper portions to the right. This is illustrated in Diagram 4, where LMC is the long-run marginal cost curve, SMC1, SMC2, SMC3 are three plant curves corresponding to three sizes of capital, Q1, Q2, Q3, such that Q3- Q= Q2- Q1. The upper parts of the plant curves are equidistant parallel straight lines.

Costs

sIcI SMC2 src3

LIC

0 Output 0 Diztag?am 4. O'tw

In this case the output plan has the simple form

Y =kd + hQ .......... ... . (II)

or, under perfect competition

Y = kp + hQ ..(2)

where Y is output, k and h are positive constants, and p is short-run price. k is the reciprocal of the excess of the slope of the short-run marginal cost curve over that of the short-run marginal revenue curve, that is, if c is the slope of the cost curve, m the slope of the revenue curve (m < o),

k = I(C- m) ...............(.. . . (I3) The time differentials of (nI) and (I2) are .

Y' =kd + hI (.4) and

Y' =kp + hI ......... (5)



o0 THE REVIEW OF ECONOMIC STUDIES

Consider the magnitude of the coefficient h. A change in demand expectations induces a change in desired capacity and therefore a certain marginal investment plan. This plan is, by equations (io) and (2)

K' v x' = q D'. Let us first limit our attention to the case where the long and short run demand curves coincide, Case i above. The increase in actual output that will result from this change in demand taken in isolation, when the investment plan has been fulfilled, will then be equal to the increase in capacity, and is given by:

Y' k D' + h K' x'. By combining these equations we obtain the expression

q -kv -hqv =o ................................... (i6) Now it can be shown that

q = v/(C -m) .................................. (I7) where C is the slope of the long-run marginal cost curve and m is the downward slope of the marginal revenue curve. Substituting in (i6) the values of k and q given by (I3) and (I7) respectively, we have:

C - ...................... (i8) c - m

Since stability requires that both C - m and c - m be positive, h must be less than I/v. We may take it that v is always positive, for it is inconceivable that capital as a whole should be a regressive factor. It follows that h is positive so long as the short-run marginal cost curve rises more steeply than the long-run marginal cost curve. It would only be zero where the two curves had the same slope, which would mean that it was just as easy to increase output by a given amount by working existing equipment more intensively as by extension of plant. But when long-run demand coincides with short- run demand a firm will only extend plant if its present plant capacity is regarded as too small, and this implies that demand is such as to cause the firm to be working its present plant beyond the point of full capacity, on the steeply rising part of the short-run cost curve ; so that in Case I h must be positive if investment is taking place.

Now it is only in the case where h = o that we can find a relation between the rate of investment and changes in output that takes the form postulated by Dr. Hicks in his Trade Cycle (p. I82). If h = o we can substitute Y' = k D' into equations (6) and (7) to obtain:

I j J f (z) Y' dz

= v j (z) Y'dz (by (i6)) ........................... (i9)

and for the simpler case: I - v Y' ..................... (20)

If, on the other hand, h is positive, it is only in the simple case that we can get the accelerator equation in its customary form. For the substitution of Y' = k D' + h I in (6) and (7) gives us:

I= i (z) (Y -hI) dz ....................... (2I)

and

I k q hY v Y' (by (i6)) ........................,., (22)



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS II

The interpretation of (2I) is that it is not in general possible to write the investment equation in the form of equation (I9), because not all changes in output generate investment ; some part of any change in output is likely to be the result of previous investment. In the simple case of instantaneously fulfilled marginal investment plans the difficulty can be circumvented by reducing the value of the acceleration coefficient below what it would be if investment-induced output were ignored. But this is not possible in the general case. Even the simple case breaks down if we allow a lag between changes in demand and the investment induced by them. For if

It = q (Dt-_ Dt-2), and Yt- - Yt-2 = k(Dt-1 - Dt-2) + h It-1, we have

It *-~ (Yt-1 yt-2 h It_l).

Within a macroeconomic system, where Yt = Ct + It (C being consumption), it will always be open to us to solve for I in terms of past values of Y if consumption is a function of income. But the resulting investment equation will be more complex than that postulated by acceleration theorists. In any case there is little point in solving for I in terms of income in this way ; it is natural to set out our models in terms of relations that are independent of each other.

But that constitutes also an objection to writing investment as a function of changes in current output even in the simple case where it is possible to do so. It involves the combination of two relations, both of which have a place in macroeconomic models-the investment equation and the aggregate supply function. It is confusing, therefore, not to state them both explicitly.

The difficulty here of course arises over the measurement of demand and changes in demand for the economy as a whole. The chief attraction of the capital-output relation probably lies in that it side-steps the problem. But why should we have to side-step it ? Let us at least' partially face it by making the assumption of perfect competition and so writing investment as a function of changes in the general level of prices, or rather the ratio of prices to money wages. In addition we must include the aggregate supply equation in our models. We shall then be able to allow not only for the output-inducing effects of investment via capacity extension, but also for the course of the general price level of output, in models based on the acceleration principle. In Part II we shall endeavour to develop this idea a little further.

There is, however, one possibility of justifying an output accelerator, of the form of equation (I9) above, not so far considered. It will be observed that we have not yet made use of the expectation pattern which we labelled " Case 2." It may well be that firms habitually plan to have a reserve of capacity on hand, to operate a plant that is, strictly speaking, too large for the output which it is intended to produce, so that in the event of a sudden increase in the volume of orders the extra output can be produced smoothly, without encountering sharply rising costs. In diagrammatic terms, firms plan to operate on the part of their short-run cost curves that is parallel with their long-run curve, not at the point of full capacity. If so, we may say that they regard short-run demand as below long-run demand, and the latter is perhaps best interpreted as the maximum level of demand that they deem they will be called on to satisfy. In the simplest terms, therefore, planning for excess capacity can be analysed by making the assumption of Case 2. Now so long as the increases in current demand are not so rapid nor of such a magnitude as to absorb the excess capacity which the firm plans to maintain, the coefficient h will be zero, and equation (I9) will apply. But clearly it cannot be supposed that this will be the case over the whole course of the upswing of a trade cycle.




4. The Keynesian Inducement to Invest

The theory of the firm implies that rather special assumptions have to be made to give rise to a dependence of the rate of investment on the levels of the rate of interest and income. Our present purpose is to find a relation of this kind that not only holds over time but also fits the kind of situation Keynes seems to have had in mind. It may be asked: did Keynes intend his investment function to hold over time, to be more than a " hypothetical " relationship ? Did he not simply mean that at the begin- ning of the year, say, entrepreneurs make their investment plans on the basis of the rate of interest and level of demand expected over the year, and does not his investment function simply describe the alternative plans that would be made for various levels of r and Y ? Surely not: for his system purports to explain reactions over time. It tells us, for example, that, other things being equal, an increase in the quantity of money lowers the rate of interest, which stimulates investment, etc.-that is, it des- cribes a chain of events. For Keynes' " comparative statics" were no parlour game, but a basis for policy recommendations.

In The Keynesian Revolution, Professor Klein, considering the problem we have before us, bases his solution on the inclusion of the rate of investment, as well as the stock of capital, as a variable on which the long-run level of output depends from a technical point of view-he includes the rate of investment in the production function. (See his appendix, p. 196.) He distinguishes between the productivity of new capital goods and old ones, as a first step towards a more comprehensive distinction between the productivity of capital goods of all " age-groups." Without quarrelling with the rationale of this distinction, we must question its propriety in this context. For in the first place new capital goods do not necessarily constitute net investment-they may be replacing old ones that wear out. And, secondly, there is no evidence that the distinction was in Keynes' mind. Nevertheless, we agree that from a formal point of view the solution is on the right lines: the rate of investment must find its way into the production function if we are to get the relation we want between I, r and Y. How are we to get it there ?

We have previously remarked that for Keynes the long run is conceived as an approach to stationary conditions. On this basis his analysis would seem to be meant to apply to the situation arising as a result of a disturbance of a previous stationary state, and to hold over the period elapsing between the initial disturbance and the point of time at which, if no further disturbances ensued, stationary conditions would be restored. If the analysis opens with a disturbance of equilibrium in period t, it applies over the interval from t to t + T, where T is the time taken to execute the marginal investment plan formed as a result of the disturbance. In fact, of course, fresh disturbances will be likely to occur between t and t + T.

It appears that Keynes implicitly assumes that the execution of any marginal investment plan takes place at a constant time-rate. In that case equation (4) is special- ised by the fact that j (z) = i/T, and the rate of investment at any moment between the initial disturbance at t and t + T will be

K-Qe T . (23)

where K is currently desired capital, Qe is the initial capital, and is therefore a constant in the analysis. Thus we can substitute for K in the production function

K IT + Qe.

The maximisation of long-run discounted net revenue, now yields an i'nvestment plan,



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS T3

which will be the same as our previously discussed capital plan (equation (i)), except for the constant Qe/T and the fact that K is itself to be divided by T, i.e.

K(D, r) - Qe T (24...........)......................... (24

The rate of investment is a function of the levels of r and D ; under perfect competition it will be a function of the levels of r and p, the price of the product. Of course, even now the rate of investment is still implicitly a function of the difference between two demand (or price) levels and the difference between two levels of r. For let re and De be the levels of r and D that would make K equal to Qe, i.e. the levels of r and D that ruled in the equilibrium existing before the initial disturbance. We may write (24) in the form:

(D, r) - K (De, re) , which is sometimes soluble explicitly as

I-I(DI-De, r-re) .(25) such that I o when D = De and r = re.

Since an increase in desired capital is, on our present assumptions, also an increase in the rate of investment, the firm's maximisation of discounted net revenue implies that both the marginal efficiency of capital and the marginal efficiency of investment are made equal to r; for these two marginal efficiencies come to the same thing in this case. Keynes was therefore quite justified in making no distinction between these two concepts.

Keynes seems to have regarded the marginal efficiency of capital schedule as dependent on the current level of demand, and post-Keynesian writers have interpreted this to mean that investment should be made a function of the level of current income. Let D (or p) represent current demand (or price). The output equation, as given by equations (ii) and (I2) above, may be written in the more general form:

Y = Y (DiQ) ........ . . (26) where Q is the actual capital currently possessed, to be distinguished from Qe, the initial capital. So long as we may treat Q as a constant, this gives us a relation between Y and D, which enables us to write equation (25) in the form:

1I- F (Y - Ye,r-re) ................................... ... (27) This is the Keynesian investment function. It depends on the assumption that Y is a satisfactory index of the level of demand, which will only be approximately true over a period short enough for Q to be treated as constant. In the greater part of his analysis Keynrs does explicitly assume " constant capital equipment." Nevertheless, he does at times refer to a longer run in which capital accumulates. For this purpose equation (27) is inappropriate ; we should have to go back to equation (25.)

PART II We turn our attention now to some of the macroeconomic implications of the

argument of Part I. Throughout we adhere to the following assumptions : i. Competition is perfect. This means that we shall be dealing with the level of,

and changes in the level of, prices rather than " demand." This is not strictly necessary. It should be possible to define a demand index for the economy as a whole, which would be a weighted average of individual demand indices. But there are difficulties, and as we are not at present concerned with this aspect of macroeconomic theory, we shall stick to the simpler hypothesis.

2. A constant ratio of the price level of consumption goods to that of investment goods, so that we need only include one price level, that of output as a whole.




3. All investment is capital-widening. Since it is not assumed that the price level of investment goods is constant in relation to the wage level, this amounts to supposing that the substitution of capital for labour or labour for capital in response to changes in their relative value is sufficiently slight to be neglected.

4. Wages are " sticky " while there is unemployment of labour, and flexible upwards from the point of full employment. The price variable to be introduced is the ratio of the price level of output to the money wage level.

5. The rate of interest is omitted from the models. 6. All the relations are linear.

5. Capital Accumulation in the Keynesian System

A linear version of the ordinary Keynesian model may be written

Demand Supply C =cY+K ys kp + hQ I V(Y Ye) ND nY yD C + I Ns = constant

* yD K Vye I -C v

yD yS

..................................................... ........................(X8) in which C is real consumption and c the marginal propensity to consume ; I is real investment (net) and v is the " marginal propensity to invest" ; Y, yD, and ys are respectively income, aggregate demand for income (or output) and aggregate supply of it. Ye is the level of income at which net investment would be zero. Q is the actual quantity of capital, ND and Ns are the demand and supply of labour, n is a positive constant. k and h are positive constants as in equation (I2) of Part I. p is the ratio of the price level of output to the money wage level. Finally K is consumption that is independent of income.

The system can be regarded as determining Y, p and N, and their determination can be represented diagrammatically as follows

N

De jr-.n 5 . N Diagrtim 5. Ym w Darn6.



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS IS

In Diagram 5 the vertical line yD iS the aggregate demand " curve ", the upward- sloping line is the aggregate supply curve. Their intersection determines p. Diagram 6 tells us the level of employment corresponding to the level of income " determined" in Diagram 5.

The analogy with price-determination in a particular industry is clear. Indeed, the analogy only fails to be complete because the aggregate demand curve is completely inelastic with respect to p-which would certainly never be the case in a particular industry. The result is that Diagram 5 determines only the ratio of prices to wages, the level of income being predetermined by the saving-investment relation. But the assumption of complete inelasticity of aggregate demand with respect to p is certainly not necessary, nor indeed is it likely to hold in reality. Quite apart from the effect of changes in p on investment we should expect consumption-saving decisions to be somewhat affected by the level of real wages. In fact the original Keynesian formula- tion, in which the equalisation of saving and investment determines income in terms of wage units, amounts to the assumption that yD is a rectangular hyperbola, that aggregate demand has an elasticity of unity with respect to p.

Except in the case where entrepreneurs keep enough reserve capacity in hand to render h zero, the equation for ys is dependent on Q as well as p, so that our aggregate supply curve's position will alter if Q alters. Strictly speaking, Q cannot be constant so long as I is positive ; but Keynes supposes that it can be treated as constant in short-period analysis.

Suppose, however, that we wish to analyse the effects of capital accumulation on p, Y and N. The investigation would be worth while only if it could be held that T, the " investment period," is long enough for the economy to move towards stationary equilibrium within that period. Let us assume that this is so.

The curve Ys will shift to the right by an amount h per unit of net investment, and will go on moving to the right until net investment becomes zero. It is perhaps less immediately obvious that changes in Q will also affect yD. We have already mentioned the reason for this (Section 4 above): it is due to the fact that the investment equation itself is not independent of the aggregate supply function. For, under the assumption of linearity and neglecting the rate of interest, the original form of the investment equation is, from (25) above:

I - q(P-pe) ......... (29) where q is a constant. Our relation I = v (Y - Ye) is obtained from (29) by substi-

tuting from the aggregate supply function, p = k Qandpe = - hQ whence

I = q (y _ Ye) = v (Y - Ye), where v = q/k. Since Ye. depends on Q, the relation

I = v (Y - Ye) is also dependent on Q, and so, therefore, is aggregate demand as a whole.

The solution is clearly to discard the derivative investment function, replacing it by equation (29). We now have aggregate demand as a function of p

D K + q(p -e) I - C(30)

The aggregate demand curve is still vertical below Pe, but slopes upwards above it. (Strictly speaking it will be vertical at a price below Pe, at the price which induces the maximum possible rate of disinvestment.)



i6 THE REVIEW OF ECONOMIC STUDIES

As capital accumulates the supply curve moves to the right, and p falls towards Pe. This discourages net investment, and therefore reduces income and employment, until the economy lapses into a low-level stationary state with p = pe, and, net investment being zero, the supply curve's movement comes to an end. See Diagram 7.

0 Diaqrami 7:

In this model income and employment fall as the economy approaches the stationary state. This effect would be mitigated, and might even be reversed, if falling prices encouraged consumption. If the consumption function is:

C-=cY-bp + K ...................(3') where b is a positive constant, aggregate demand becomes:

yDKR+ (q-b) p-qpe.(2

Its slope, above pe, will have the sign of (q -b), as shown in Diagram 8.

P~~~~~~~~~y j X

Di aqraTn 8



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS

The movement of p and Y over time is easily obtained as follows : Since the price- level moves to keep aggregate demand and supply equal, we must have not only

_D N yS yD1 _D, = (q- b) yD = ys, but also yD, = ys'. Now y = ( . and Ys = kp + hI

'. -

Pe

-

(- Pe). ,, ......................................(33) where e is the base of the system of natural logarithms, and where

- hq (I - c) u--_ k (i -- ) < o ....................................(34) U =k (i-c) - (q-b).<(34)

since stability requires that k > q -- b

that is, that the slope of the supply curve be

greater than that of the demand curve. Prices decline at a constant geometric rate. From equation (32) :

= I c +-(p- pe) ....................................(35)

KI- bpe that is, income grows (q < b) or declines (q > b) to the level at the same

constant rate. If the economy is allowed to sink into stationary equilibrium through the effect

of sagging prices on investment, the new equilibrium will be characterised by a smaller capital and output-capacity than that originally planned; for the stationary conditions are due only in part to the fulfilment of capital plans : in part they are due to a downward revision of these plans as prices fall. If prices could be prevented from falling, during the fulfilment of the original investment plans at least, a subsequent stationary state would be one in which the originally desired increase in capital and output capacity had been achieved. How, then, could the fall in prices be prevented ?

One way would be for the Government to undertake or sponsor autonomous investment programmes additional to those already planned by the private sector, What pattern of autonomous investment would be required to keep prices from falling ? It can be shown that, since productive investment adds both to supply and to demand, autonomous investment would have to grow at a constant geometric rate. For with p' # o the rate of change of aggregate demand is:

yD, - (q--b)p'+ A'( y D - bA .......................................... (36)

where A is autonomous investment, and the rate of change of aggregate supply is ys' = kp' + hI

= kp' + hq (p - Pe) + hA

Putting yD = ys', we obtain :

{ k(i - c) - (q - b) }p' + hq (i - c) (p - pe) = A' - h (i - c) A

The solution of this for p' = o is : A = (Ao + q (po - pe) ) Eh(-c)t - q (po - pe) ...................... (37)

To keep prices from falling it would be necessary for autonomous investment to grow geometrically at the rate h (i - c), and for income and the employment of labour to grow, in consequence, at the same rate. This result is very similar to that of Domar 2 Vol 2o

I7



i8 THE REVIEW OF ECONOMIC STUDIES

and Tsiang,l who argue that in a capitalist economy full employment of capital can only be maintained if investment and income grow geometrically at the rate h (I - c), because of the " dual nature of the investment process." But it is not quite the same. Domar's thesis seems to be that since prices either do not fall, owing to monopoly, or should be prevented from falling and so discouraging investment, equilibrium between the growth of demand and supply requires that investment and income grow at the rate h (i - c) per unit of time. Moreover, although it is recognised that this result depends upon the assumption of constant prices, the implication is that the relaxation of this assumption would not materially affect the conclusion.2 Our own analysis indicates that the whole purpose of geometrically growing investment is to keep prices from falling ; if they were allowed to fall, this would not disrupt the equilibrium between demand and supply-they would fall in order to preserve that equilibrium-but it would discourage investment and the creation of capacity, and might discourage employment of labour, if there were not a sufficient offset through increased consumption. Further, the sag of prices might equally well be avoided by increasing non-productive expenditure, that is by means of projects that would create demand without adding to supply. It can be shown that in this case only arithmetically increasing autonomous expenditure and income are implied. For, from equation (32), the rate of change of aggregate demand when autonomous expenditure, K, is variable, is

yD, K'+(q -b)p I-C

K' -

K -c for o. I - C

And similarly, for p' - o, YsI hI = hq (po - Pe)

Whence, with yDt - ySTS K' = h(i - c) I = h(i - c) q (pe - Pe) = constant .... (38) In this case investment, so far from growing at a constant geometric rate, is maintained at a constant level. The annual increase in Government expenditure will have to be h (i - c) times the constant level of investment.

Thus it is not true that geometric growth is implied by the very nature of capitalistic production, if equilibrium is to be maintained, even if by " equilibrium " we mean " constant prices." And even if all expenditures except private consumption are " productive," geometric growth is required not to maintain equilibrium between demand and supply, but to keep prices constant, if thalt is desired. Domar's analysis is misleading in so far as it ignores the part played by capital accumulation in exerting a downward (or at least relieving an upward) pressure on prices.

6. Productive Investment, Price Changes and the Accelerator. In Part I, Section 3, it was maintained that, while an output accelerator is open

to the objection that it ignores the productivity of investment, a demand or price accelerator is a more reasonable hypothesis from the point of view of the theory of the firm. In this section we shall take as our starting point the model of Hicks' Trade Cycle and examine the results of replacing his output accelerator by a price accelerator

1 Domar: "Capital Expansion, Rate of Growth and Employment." Econometrica, April, I946; and " Expansion and Employment." A. E. R., March, I947. Tsiang : " A Rehabilitation of the Time Dimension of Investment in Macrodynamic Analysis." Economica, August, I949; and " Accelerator, Theory of the Firm, and the Business Cycle," Q. J. E., August, I95I.

2" Expansion and Employment," p. 36.



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS i9

and introducing explicitly the aggregate supply function. So long as we are not speci- fically interested in the cycle-generating properties of lagged systems, we shall find it more convenient to work, as we have done hitherto, in terms of differential, rather than difference, equations.

The Hicks' system may be written r2=T

c fc(z)Y(t-z)dz+K= fc(z)Ydz+K ,Z=o

0 I1= vfJ(z) Y' (t-z)dz +A = v 1 (z) Y'dz+ A

Y C + I

.,. Y- c(z)Ydz-v j(z)Y'dz=A+K ...................... (39)

C is consumption, I investment, c (z) (z _ o . . . T) are the consumption coefficients. The coefficients v and j (z) (z = o . . . T) have been explained in Part I, Section 3; see in particular equation (I9) above. A and K are autonomous investment and consumption respectively.

The argument of Part I, Section 3, leads to the conclusion that the Hicks system is a special case of a system with a price accelerator, as follows:

C {c(z) Ydz + K (a)

I =q J(z)P'(t-z)dz+A-q j(z)p'dz+A (b)

Y=kp + hQ (c) *Y' kp' + hI (d)

Y = C + I (e) ........................ .(40)

(a) is the consumption equation of system 39. (b) gives investment as a linear function of past changes in p, the ratio of the price level of output to money wages: see equation (8) above. (c) and (d) are the aggregate supply function and its time derivative respectively: see equations (I2) and (I5) above.

After the elimination of prices from this system we obtain the differential equation in Y:

z=T y=T

Y - c (z) Ydz- q{j(z) Y'dz + k (z) Ydz- J{ f (z) c (y) Y (t - z - y) dzdy z=O y=o

A + K ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. . o... .. .. .. .. .. .. .. .. .. (4I)

Here the variable y is a time variable with the same range as z (i.e. from zero to T).

If we put h = o and write q/k = v in (4I), it reduces to the Hicks equation, (39) above. In other words, Dr. Hicks has implicitly assumed that investment does not, by increasing capacity, give rise to an enlarged output stream, but only affects output via the multiplier mechanism. To make sense of this specialisation we should have to suppose either that the productivity of investment is of the second order of smalls or that firms keep a capacity reserve. Perhaps the best justification of the Hicks system is a combination of these two suppositions: firms plan for reserve capacity, which is

2*




sufficient in the early stages of the upswing to render h zero ; when at length demand outstrips capacity the magnitude of h is sufficiently small to be neglected.'

By differentiating system 40 with respect to time we can eliminate output instead of prices, obtaining the differential equation :

kp' + hq {j(z)P'dz - k fc(z)P'dz - hq Jc(z)j(y)p"dydz - q fj(z) P"dz

A + K ... (42)

That is, we now have two " basic equations " to our model: one in output and one in price changes.

The question we have to ask now is: does the replacement of Dr. Hicks' system by our system of equations (the replacement of (39) by (40)) make any appreciable difference to the Hicks theory ? There are really two questions here: (i) is the magnitude of h sufficient to make the Hicksian equation (39) gravely inaccurate, and (2) what kind of difference in behaviour of the economic system is involved when we substitute equations (4I) and (42) for (39) ?

(i) We discussed the limits between which the coefficient h must lie in Section 3 (in particular equations (i6)-(i8)). There it was shown that, for the individual firm, h must be positive but less than the ratio of equilibrium output capacity to equilibrium capital. For the economy as a whole, however, a further limitation is implied: h must be less than unity ; for in a macroeconomic system output and capital are measured in terms of the same units, for example units of value, and it is inconceivable that an increment in capital should increase the value of output by as much as the value of the investment itself. Indeed at a guess we might suppose that h is of the order of magnitude of 5 per cent per annum. This means that the degree of inaccuracy involved in the implicit assumption of Dr. Hicks and others that h is zero cannot be very large.

(2) However, it is still worth while to investigate the kind of difference made by the recognition that h is positive, especially as, to the extent that investment is con- trolled by central policy, its productivity is to some degree susceptible of choice. We shall consider first the effect on the equilibrium path, and then the effect on the deviations from equilibrium.

(a) The Equilibrium Path. We shall examine this under the assumption of continuity ; the results will be materially the same in a lagged system. Suppose that autonomous investment increases geometrically at a given rate, g: A A&Egt, Ao being a given constant. Then, if the coefficients of the model were such as to render it stable, income and prices would also sooner or later settle down to geometric growth at the same rate. That is, we should have the equations for income and price-changes respectively.

Ye -Eeg ....... (43)

Pe =peg ....... (44)

1 This is, of course, not the same line of reasoning as that pursued by Mr. Kaldor in his recent review of Dr. Hicks' book (Economic Journal, December, I95I), to show that Dr. Hicks' autonomous investment is " investment which does not generate output capacity, like digging holes in the ground or building pyramids." Mr. Kaldor argues that the rate of growth of autonomous investment cannot properly be regarded as an independent variable, but must be related to the technical ratio between capital and output ; since Hicks does assume the rate of growth to be independent, what is growing autonomously in the Hicks model cannot be autonomous investment at all, but must be unproductive expenditure, not A, but K (pp. 84I-847). Our point is that all investment in the Hicks model is unproductive, because the output resulting from investment makes no appearance whatever.



DEMAND, SUPPLY AND PRICE LEVEL IN MACROECONOMICS 2I

where Ye is equilibrium income, Pe' is the equilibrium change in prices, and E and P are as yet undetermined constants. By substituting the above expression for Ye and that for autonomous investment in 4I, and the expression for Pe' and the autonomous investment equation in (42) (see Hicks' Trade Cycle, pp. I83-4), and ignoring K, we obtain the expressions:

Ao E

I -fc (z) t-gz dz- {g-h (I-fc (y) 6-gy dy)} (z) e-gz dz (45)

{g - h (i - c(y) E-gY dY)}Ao p~~~~~~~~~~~~~ (46) k (I- c (z) e-gsdz) -q{g_I (i- c (y) e-gydy) jfi (z) e-9z dz

The expression for E is identical with that given by Dr. Hicks (Trade Cycle, p. I84) if we put h = o. When h is positive, however, the value of E is smaller than that given by Dr. Hicks, which may then be regarded as an upper limit to E. E will clearly be

smaller the smaller is the excess of g over h (i - c (y) e-gY dy). Inspection of (46)

shows that E will be smaller, for any given values of Ao, q and k, the smaller is P, i.e. the smaller is the " extent " of the rise in prices along the equilibrium path. Since the amount of induced investment in the system depends upon the extent to which prices rise, we see that the reason why E will be smaller the smaller is the excess of g

over h (I - C (y) e-gy dy) is that, the smaller this excess is, the less induced invest-

ment will there be along the equilibrium path. If g were equal to h (I - C (y) e-gy dy)

prices would be constant along the equilibrium path and induced investment zero. It is interesting to compare this last result with our earlier analysis of capital

accumulation in the Keynesian model. There we found that prices would be constant if investment grew geometrically at the rate h (i - c). If in our present model investment and consumption were generated without time spread, i.e. if we had C - cY and I = qp' + A, the expressions for E and P would be:

Ao E= I -c-q- g - h (i - c)}

p {g - h (i - c) }Ao k (i - c) - {g-.h (I - C)}

whence prices would be constant and induced investment zero if g = h (i - c).

Thus the amount of induced investment along the equilibrium path is not determined solely by the marginal propensity to save, the accelerator coefficient and the level of autonomous investment and its rate of growth, but also by the magnitude of h in relation to the rate of growth of autonomous investment and the marginal propensity to save. Even when entrepreneurs are disposed to carry out large investment programmes in response to current increases in the conditions of demand for their products, they will not have the incentive to do so if the economy is on its equilibrium path and autonomous investment is sufficiently productive to prevent prices from rising in relation to costs.



THE REVIEW OF ECONOMIC STUDIES

The significance of this modification becomes apparent when we consider the possible efficacy of a policy designed to keep the economy moving along the full employment ceiling. Suppose that the level and rate of growth of autonomous investment are sufficient to set the system moving along its ceiling. In the Hicksian model this will involve induced investment growing at the same rate as autonomous investment; for P must be > o with h = o. Thus, if autonomous investment is only just sufficient to achieve the ceiling, a collapse may occur through the difficulty of securing smoothly increasing autonomous investment : a sudden " trough" in the autonomous investment " line " would be enough to cause a collapse of induced investment and a slump. For safety, therefore, autonomous investment would have to be sufficient to keep employment " over-full ". (Trade Cycle, p. I67). Once we recognise, however, that the volume of induced investment depends on the magnitude of h, another possibility is, in theory, available, that autonomous investment may be sufficiently productive for induced investment to be practically zero along the ceiling. If this were so, a sudden trough in autonomous investment would be unable to do much damage, and over- employment would become unnecessary. Our modification points the way, therefore, to a greater faith than Dr. Hicks himself has in a policy designed to eliminate cyclical movement rather than one which aims merely to quieten it. Its advantage over the over-employment solution is that induced investment is eliminated not by repressing an existing desire for it, but by preventing the emergence of the desire. But the prac- tical difficulties are obviously very great.

(b) Deviations from Equilibrium. An exhaustive analysis of this subject will not be attempted. We shall be content to examine a simple model with consumption depending on one previous income and induced investment depending on one previous change in output, that is, a model corresponding to Dr. Hicks' " elementary case" (Trade Cycle, p. 67 and pp. I85-6). Our model is :

Ct = (I- S) Yt- It= q (pt-,-pt-2) +At

Yt- Yt- = k (pt- pt-,) + hlt-1

Y t = Ct + It ................................................. (47) where s is the marginal propensity to save. The basic difference equation in output is:

Yt- {I-s+

q (I -h) }Yt- +{ I-h (i- s) } Yt-2 = At .... (48)

Write yt for the deviation of Yt from its equilibrium value Yt. Then, putting v = q/k, we have :

yt - {i-s + (I - h)}yt-1 + v{I h (I -s)}yt-2 = o ........(49)

with the auxiliary equation : 2 {I -S + V (I h)} + V{-h(I - ...............(50)

Equations (49) and (50) reduce to those given by Dr. Hicks (Trade Cycle, p. i85) when h = o.

The roots of the auxiliary equation will be complex if v {I - h (i - s) } lies between certain limits as follows:

I- x/ s (I + hv) < v{ i - h (I - s) } < I + V s (i + hv)) , and real

if it lies outside these limits. The case where it lies below the lower limit is that of

22




stable convergence of the deviations to zero: if it lies above the upper limit we have explosive, non-cyclical deviations. If it lies within the limits, the deviations will be

cyclical, the cycles being damped, steady or explosive according as V{ - (I-S) > I. L. J

A comparison with Dr. Hicks' results yields the following conclusions: (i) The Middle Point-the value of v which gives steady cycles-is no longer

simply at v = i, but above this: that is, the effect of a positive h is, by weakening the accelerator mechanism, to require a larger v, in conjunction with any given propensity to save, to upset a tendency to damping.

(2) The condition that v{ i - h (i - s) } lie below the lower limit is that v (i - h) + 2 Vs (I + hv) - s < i ; and the left-hand side of this inequality will be smaller the larger is h. Similarly, the condition that it lie above the upper limit is that v (i - h) - 2 Vs (I + hv) -s > I ; and the left-hand side of this expression too will be smaller the larger is h. Thus, given the coefficients v and s, the first condition will be more easily satisfied, the second less easily satisfied, the larger is h. In other words, the more productive investment is, the more " likely " become stable non-cyclical solutions as opposed to damped oscillations, and the less " likely " become complete explosions as opposed to explosive oscillations. As we should expect, investment productivity is a stabilising factor.

Oxford. HUGH ROSE.



Demand, Supply and the Price Level in Macroeconomics

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