aggregate supply and demand: an explanation of chapter iii of the general theory

Aggregate Supply and Demand: An Explanation of Chapter III of the General TheoryAuthor(s): Paul WellsSource: The Canadian Journal of Economics and Political Science / Revue canadienned'Economique et de Science politique, Vol. 28, No. 4 (Nov., 1962), pp. 585-590Published by: Wiley on behalf of Canadian Economics AssociationStable URL: http://www.jstor.org/stable/139298 .

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NOTES AND MEMORANDA

AGGREGATE SUPPLY AND DEMAND: AN EXPLANATION OF CHAPTER III OF THE GENERAL THEORY*

PAUL WELLS

University of Illinois

IN Chapter iII of The General Theory of Employment, Interest and Money Keynes provides the reader with two conceptually different and unrelated summaries of his short-run theory of employment. He expresses these two brief accounts in terms of three macroeconomic relations: an aggregate demand function that is the sum of consumer and investment spending; an aggregate supply function that states the proceeds entrepreneurs must obtain from the sale of output if a given level of employment is to be maintained; and what we may call an "expected proceeds" function that relates the receipts entrepreneurs expect to receive from the sale of output to the volume of employment necessary to produce the output. Because Keynes does not show explicitly the connection between his two summaries, and because he employs two relations that today seem still to puzzle many economists, students of the General Theory are likely to find this important chapter difficult, if not im- possible, to understand. This is highly unfortunate, for we aim to show that this chapter constitutes a very eloquent and valuable epitome of Keynes's short-run theory of employment.

I. KEYNES's FIRST SUMMARY

The first of Keynes's brief accounts runs as follows:

It follows that in a given situation of technique, resources and factor cost per unit of employment, the amount of employment, both in each individual firm and industry and in the aggregate, depends on the amount of proceeds which the entrepreneurs expect to receive from the corresponding output. For entrepreneurs will endeavour to fix the amount of employment at the level which they expect to maximise the excess of the proceeds over the factor cost.

Let Z be the aggregate supply price of the output from employing N men, the relationship between Z and N being written Z = +(N), which can be called the Aggregate Supply Function. Similarly, let D be the proceeds which entrepreneurs expect to receive from the employment of N men, the relationship between D and N being written D = f(N), which can be called the Aggregate Demand Function.

Now if for a given value of N the expected proceeds are greater than the aggregate supply price, i.e. if D is greater than Z, there will be an incentive to entrepreneurs to increase employment beyond N and, if necessary, to raise costs by competing with one another for the factors of production, up to the value of N for which Z has become equal to D. Thus the volume of employment is given by the point of intersection between the aggregate demand function and the aggregate supply function; for it is at this point that the entrepreneurs' expectation of profits will be maximised.1

*The author is indebted to Professors Philip Cartwright and B. F. Haley and to the editorial readers of this JOURNAL for many helpful comments.

'J. M. Keynes, The General Theory of Employment, Interest and Money (London, 1949), 24-5.

585

Vol. XXVIII, no. 4, Nov., 1962



586 Canadian Journal of Economics and Political Science

To understand this passage we need to attach definite meanings to Keynes's aggregate supply function, Z = -(N), and to D = f(N), which Keynes calls, unluckily, an "aggregate demand" function, but which we shall call an "expected proceeds" function so as not to confuse this relation with the sum of consumer and investment spending.

1. The aggregate supply function is a relation between employment, N, and a sum of money receipts, Z, that states the aggregate revenues entrepreneurs must receive from the sale of output if they are to maintain a given level of employment. This function does not state the actual receipts entrepreneurs do receive from the sale of output that results from a given level of employment; it simply expresses, for each volume of employment, the minimal receipts which must be forthcoming to support various levels of employment.

To establish the exact relation between employment and the necessary receipts entrepreneurs must obtain we need to assume that: (a) the problem of aggregation has been solved, (b) perfect competition exists, (c) a given money wage rate w obtains, (d) varying amounts of X, final output measured in real terms, are produced with a fixed amount of capital equipment together with varying amounts of labour. Thus the short run aggregate production function may be written as X = X(N). In addition we shall make the custom- ary assumptions2 concerning the shape of this function, namely, that X' > 0, X" < 0, and X"' = 0.

Under conditions of perfect competition firms adjust their rates of output until marginal cost, w/X', equals the per-unit price of output, P. Thus we have the following microeconomic equilibrium condition holding for all firms in the economy.

(l.l) ~~~P X7r

Multiplying both sides of this equation by X, total physical output of the final good produced by all firms in the economy, gives us the corresponding macroeconomic equilibrium condition.

(1.2) PX = w X x Since the product PX is simply the sum total of all expenditures on final output, and is, in equilibrium, equal to Z, the sum of money receipts necessary to support the amount of employment necessary to produce an output X, we may rewrite equation (1.2) as

(1.3) Z= X x This equation states the necessary sum of money, Z, entrepreneurs must receive if they are to maintain a rate of output X. The relation we seek to define, however, is between the variables Z and N, not between Z and X. To find this relation we replace X in equation (1.3) by XN, where X denotes the

21t should be noted that the above assumptions are mine and not Keynes's.



Notes and Memoranda 587

average productivity of labour. Thus the product XN is simply the total output produced by a given amount of labour N. This substitution yields the exact form of Keynes's elusive aggregate supply function.

(1.4) z wX N

This equation states the minimal proceeds, Z, entrepreneurs must earn from the sale of output if the established level of employment, N, is to be maintained. If the actual receipts accruing to entrepreneurs exceeds (falls short of) Z, then employment will rise (fall) until equation (1.4) is satisfied.

Because X/X', the elasticity of output with respect to employment, decreases as employment increases,3 this relation is drawn, in Figure 1, convex to the abscissa. This means that in order for higher levels of employment to be established proportionately larger increments in expenditure and receipts are necessary. An additional property of this function is that it is not defined for points beyond Nf (Figure 1), the size of the labour force.

2. The expected proceeds function states the proceeds, D, entrepreneurs expect to realize from the sale of output resulting from the employment of a certain amount of labour. These proceeds are simply the product of the price expected by entrepreneurs, P, and the physical output they produce, X. If we assume that the expected price is a diminishing function of output, then both _P and X, and their product D, are functions of the level of employment. This function is concave to the abscissa (Figure 1) because of diminishing marginal returns in production and because of the assumed inverse relation between the expected price and employment. Thus, in the minds of entrepreneurs, given increments in employment produce constantly decreasing additions to expected proceeds.

From the definitions we have given these two relations, and from the implicit assumption of Keynes that expected proceeds will equal realized revenues, it follows that expected profits will be maximized if sufficient labour is employed to equate expected proceeds, D, to the aggregate supply of output, Z. For if D = Z, then PX = (w/X')X, and the expected price will equal the marginal cost of production. Thus the intersection of these two curves at point a (Figure 1) illustrates the expected equilibrium level of employment, Ne, according to the first of Keynes's two summaries. As can be seen from the diagram this equilibrium is stable barring changes in any of the parameters of the system. At a higher (lower) level of employment, marginal cost would exceed (fall short of) the expected price and entrepreneurs would make the appropriate adjustment in their rates of output.

Strangely enough, in this account of his theory of employment Keynes makes no direct use of one of his major contributions to economic theory, The Principle of Effective Demand, for in this summary the unknowns are

3The assumptions made above that X' > 0, X" < 0, and X"' = 0 guarantee that X/X' decreases as N increases. The proof is as follows: Z = (wX/X')N = (w/X')X. Thus

2X(X")2 X"1 dZ/dN = w[j -XX"/(X')2]. Hence d2Z/dN2 = w - - X , which inspection shows

L (XI), XIj to be positive.




$

Z 1 (N) DI

ID=f(N) 0z Ze=De-- - - -

Nf Z +(N)

O N, Nf

FIGURE 1 FIGURE, 2

simply employment, output, and expected proceeds. The relations necessary to determine their equilibrium values are the aggregate supply function, the production function, and the price expectations of entrepreneurs.

I I. THE SECOND SUMMARY

Keynes expresses his second summary in the forms of five separate points. They are:

(1) In a given situation of technique, resources and costs, income (both money-income and real income) depends on the volume of employment N.

(2) The relationship between the community's income and what it can be expected to spend on consumption, designated by Di, will depend on the psychological characteristic of the community, which we shall call its propensity to consume. That is to say, consumption will depend on the level of aggregate income and, therefore, on the level of employment N, except when there is some change in the propensity to consume.

(3) The amount of labour N which the entrepreneurs decide to employ depends on the sum (D) of two quantities, namely Di, the amount which the community is expected to spend on consumption, and D2, the amount which it is expected to devote to new investment. D is what we have called above the e;ffective demand.

(4) Since Di + D2 =D = +(N), where 0 is the aggregate supply function, and since, as we have seen in (2) above, Di is a function of N, which we may write %(N), depending on the propensity to consume, it follows that +(N) -%(N) = D2.

(5) Hence the volume of employment in equilibrium depends on (i) the aggregate supply function, 0, (ii) the propensity to consume, X, and (iii) the volume of investment, D2. This is the essence of the General Theory of Employment.4

4General Theory, 28-9.



Notes and Memoranda 589

Figure 2 illustrates our interpretation of this summary, which the reader will recognize as the more familiar and which the writer believes to be the more valuable of the two accounts. The propensity to consume, D1, and the volume of investment expenditures, D2, together with the 45-degree line determine the equilibrium level of aggregate demand, Ye. The aggregate supply function, shown in the lower half of Figure 2, is defined as before. This relation shows that if Ne workers are employed, aggregate supply or output valued at marginal cost, Ze, will equal aggregate demand. And if aggregate supply equals aggregate demand, then entrepreneurs will have maximized their realized profits. This is so because if YF = Ze, then PeXe = (w/Xe')Xe, and the per-unit price of output established in the markets as a result of an aggregate demand of Ye and a physical output of X, will equal the marginal cost of production (W/Xe').

Although "the volume of employment in equilibrium depends on (a) the aggregate supply function, X, (b) the propensity to consume, x, and (c) the volume of investment, D2," there is a good deal more to Keynes's second summary than either his penultimate sentence in the above quotation or our graphic interpretation would indicate. To bring out the full generality of this account we shall render a brief mathematical statement of it.

Changing Keynes's notation slightly, so as to conform with modern usage, we find that the complete set of unknowns implicitly contained in the second summary is: (a) I, investment spending; (b) C, consumption; (c) Z, aggregate supply or national income; (d) Y, aggregate demand; (e), N, employment; (f) X, final output measured in physical units; and (g) P, the per-unit price of output. The equations that determine the equilibrium values of these unknowns are:

(2.1) I=I (2.2) C = C(Z)

(2.3) Z= Y (2.4) Y C+ I

(2.5) N Z

(2.6) X X(N)

(2.7) p Y= w ~x x

Equation (2.1) expresses the fact that investment expenditures are given. The second equation is the propensity to consume, and the third is the well. known 45-degree line which expresses the productive behaviour of entrepreneurs in a rather meaningless fashion. It merely states the simple fact that entrepreneurs adjust their production to match their sales. The fourth equation defines aggregate demand.

These first four equations of the above set are sufficient to determine the equilibrium values of consumption, investment, aggregate demand, and national income. They do not suffice, however, to determine the value of




employment, physical output, and the per-unit price of output. To solve for these additional important and interesting variables the final three equations are needed. Equation (2.5) is the inverse of the aggregate supply function and is called an "employment function" by Keynes.5 It states the amount of labour needed to produce any given national income. Alternatively, it shows the amount of employment various levels of aggregate demand will establish. Equation (2.6) is the production function and equation (2.7) the price-determining function. In short, these final three equations reveal how a given aggregate demand and national income are resolved into a price level, a physical output, and a level of employment.

From the above two summaries we may conclude that Chapter iii of the General Theory teaches us, contrary to many of the popularized accounts of this book, that expectations and supply, as well as demand, play important roles in Keynes's short run theory of employment. Though one may agree with Patinkin's oft-repeated charge that Keynesian economics "overlooks the supply side of the commodity market" 6 it is clear that the General Theory does not.7

5Ibid., pp. 280-91. 6D. Patinkin, Money, Interest, and Prices (Evanston, 1956), 222, 237, and 251; also "In-

voluntary Unemployment and the, Keynesian Supply Function," Economic Journal, LIX, 360-83.

7Additional support for this conclusion is to be found in chapters xx and XXI of the General Theory, and in H. Neisser, "Keynes's Aggregate Supply Function: Further Comments," Economic Journal, LXXI, 850-2.

INSTITUTE FOR NORTHERN STUDIES: SCHOLARSHIPS

THE INSTITUTE FOR NORTHERN STUDIES is offering for the 1963-64 academic session, six scholarships, and possibly more, for students of marked ability interested in research problems to do with Northern Canada. This work may be in any field of the biological, physical or social sciences or the humanities and will be carried out under the supervision of a department of the University of Saskatchewan.

Applicants who have no post-graduate experience are eligible to apply for scholarships of the value of $1,800.00 for twelve months, and those with at least one year of post-graduate research training are eligible to apply for scholarships of the value of $2,200.00 for twelve months. Essential equipment and travelling expenses incurred while carrying out a research project will be looked after by the Institute.

The Institute will also consider meeting essential equipment and travelling expenses incurred by a graduate student who holds a scholarship from some other source.

Further information and application forms may be obtained from the Secre- tary, Institute for Northern Studies, University of Saskatchewan, Saskatoon. The application forms and supporting documents must be received not later than February 1, 1963.



aggregate supply and demand: an explanation of chapter iii of the general theory

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