a simple macroeconomic model with monopolistic firms

A SIMPLE MACROECONOMIC MODEL WITH MONOPOLISTIC FIRMS

NICHOLAS ROWE‘

This paper presents a simple macroeconomic model in which firm’ outputs are imperfect substitutes, and explores the macroeconomic implications of monopolistic competition. The model is classical in some respects, but Keynesian in others. Multiple or unstable equilibria are not unlikely. Permanent price controls will, in principle, be desirable, since they allow a permanent and eficient increase in aggregate output. Small costs of price adjustment may induce large deviations of output from the natural rate. Fiscal policy will generally afiect aggregate output, but the sign and magnitude of the government expenditure multiplier cannot be determined a priori.

1. INTRODUCTION

This paper presents a simple macroeconomic model of an economy with monopolistic firms.’ In contrast to the assumption of competitive markets normally made in macroeconomics, we assume that each firm’s output is an imperfect substitute for the output of every other firm.

The model has been designed to provide a close counterpart of the simple classical macroeconomic model, dropping only the assumption of competition, in order to examine the extent to which abandoning that particular assumption alone alters the standard predictions of the classical macroeconomic model. In particular, aggregate demand is given by the fixed velocity equation of exchange MV = PY. This feature is not shared, for instance, by the monopolistic macroeconomic models of Hart [ 19821, Snower [1983] and Weitzman [1982].2 Hart’s model is of a barter economy. Snower and Weitzman osten- sibly model monetary exchange economies, but at the same time fail to incorporate any notion of the demand for money and the effect of real balances

* Assistant Professor, Carleton University, Ottawa. I would like to thank Ron Bodkin, Steve Ferris, Peter Howitt, Tim Lane, Tom Rymes and Richard Sweeney for their comments on earlier drafts. Responsibility for errors and opinions is my own.

1. I make no distinction between the monopolistic competition of Chamberlin [I9331 and the imperfect competition of Robinson [1933].

2. The original version of this paper appeared as a Carleton University working paper in August 1984. While revising this paper in April 1986 I received a copy of a working paper by Startz (19861 which also cites a related dissertation by Kiyotaki [19851, both of which explore the macroeconomic implications of monopolistic competition. I have not had time to digest those works and compare their findings to my own. Weitzman [1985] constructs a macroeconomic model of monopolistic competition based on a particular functional form of consumers’ preferences, which he used to explore profit sharing. Sweeney [1974] is the first economist (to my knowledge) who explicitly adopts monopolistic competition as a basis for macroeceonomic analysis.

83 Economic Inquiry Vol. XXV, January 1987, 83-102

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on aggregate demand, assuming instead that the demand for goods is identically equal to wage income, all profits being “saved” (in what form?). It is this feature, and not the assumption of monopolistic power (or increasing returns), which accounts for their non-standard results-the wage-price spiral in Snower, and the indeterminacy of real income in Weitzman.3

Our aim is not to provide a rigorous and general model. Instead we seek to provide a basic conceptual framework to aid our intuition in thinking about the macroeconomic implications of monopolistic power. Given this motiva- tion, we start by merely positing the demand function for an individual firm. However, since one’s intuition is generally assisted by more than one way of looking at the same thing, we also present an alternative heuristic in the space of consumers’ indifference curves.

The potential payoffs to such a model, despite (or perhaps because of) its simplicity, are exemplified by using it to address four particular topics. First, we show that the existence of a unique, stable equilibrium cannot easily be assumed if the economy is monopolistic. It is quite possible that there be multiple equilibrium levels of prices, aggregate output and unemployment. Second, the model provides a simple “best case” argument in favor of permanent and economy-wide price controls. Third, the model provides a more hospitable environment for explaining macroeconomic fluctuations as the result of an assumed temporary rigidity of prices, requiring less unpalatable assumptions to yield more plausible predictions than its competitive counterpart. Fourth, we show that fiscal policy will generally affect real output, but that the sign and order of magnitude of the government expenditure multiplier cannot be determined a priori.

II. THE MODEL

This section presents a version of the model in which firms’ demand functions are simply posited. Section IV presents an alternative representation in terms of consumers’ preferences.

The economy consists of a fixed number of firms,4 each of which produces a good which is an imperfect substitute for the good produced by each other firm. This assumption implies that the product space is fixed, with each firm having a monopoly on the production of one product.

3. In both models all behavioral functions are homogenous of degree zero in nominal prices, so that the equilibrium price level is necessarily indeterminate. Since both also set profits equal to zero by free entry, desire expenditure is identically equal to income for all levels of real income, so that real income is indeterminate. Exactly the same results could be. derived from a textbook “Keynesian Cross” model, with no government, investment, or trade, by assuming a consumption function in which consumption is identically equal to income.

4. The assumption that the number of firms is fixed carries different implications at the macroeconomic level than it does at the level of a particular industry. Resources may enter or leave a particular industry, but not a closed economy. When we allow new firms to enter a particular industry, we allow them to bring with them resources that had previously been elsewhere em- ployed. We cannot do this in a macroeconomic analysis. Thus holding the number of firms fixed is not as awkward an assumption as it might first appear. From where would new firms get their resources, if existing remurces are already owned by existing firms?

ROWE: MONOPOLISTIC MACROECONOMICS 85

Firm i’s demand function is:

The quantity demanded of firm i’s good, yi, depends positively on real aggregate demand, Y, and negatively on its own price, P,, relative to the general price level. Although each firm produces a qualitatively different good, firms are “identical” in the sense that each faces the same demand function. This enables us to analyze the economy in terms of the representative firm.

We define the price level as the simple average of the prices charged by the n firms:

n

P = 2 Pi/n. i=l

With identical firms, if each firm charges the same price (Pc = P) then each

D(Y, 1) = Y/n. (3)

gets an equal share of aggregate demand:

Firms have zero costs, but have a fixed capacity level of output 8: yi I ij = Y/n. (4)

(There are no explicit factor markets, and it is perhaps easiest to think of each firm as a producer cooperative, with no alternative uses for its inputs.)

Each firm sets its price to maximize profits, taking aggregate demand and the general price level as given. (With a large number of firms, this is nearly equivalent to assuming that each firm takes other firms’ prices as given, for the general price level and aggregate demand are then insignificantly affected by any one firm’s price.)

Since we are not primarily concerned with aggregate demand, we will simply assume that nominal aggregate demand is determined by the nominal money supply, M , times a fixed (income) velocity of circulation, V :

PY = M V . (5) Any alternative theory, provided real aggregate demand depends negatively on the price level and approaches infinity as the price level approaches zero, will give qualitatively similar result^.^

5. For convenience I have posited an aggregate demand function separately from individual firms’ demand functions. This enables us to see more immediately that real output and relative prices are determined independently of the aggregate demand function. An alternative procedure would be to write firm’s demand functions to incorporate directly the real money supply and relative prices as arguments. Setting relative prices at their equilibrium levels of unity then determines a relationship between firms’ outputs, the money supply, and the price level, which relationship we could call an aggregate demand function. Section IV describes one environment in which demand functions of this sort could be derived, but it is a rather uninteresting environment since there is no direct feedback in that environment from firms’ sales to agents’ realised incomes to consumers’ expenditure.

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Having stated the model’s assumptions, we now discuss the determination of equilibrium prices and output.

With zero costs, maximizing profits is equivalent to maximizing total revenue. The firm thus sets its price to yield that level of output where its marginal revenue is zero, or to yield capacity output, whichever is greater. We define the individual firm’s marginal revenue function as

Note that this defines the firm’s real marginal revenue. (Maximizing real and nominal revenues is the same if the firm takes the price level as given.)

Notice that the firm’s marginal revenue is a function both of aggregate demand and of its relative price. A firm’s marginal revenue curve, as normally depicted in partial equilibrium contexts, shows how marginal revenue varies with the firm’s relative price, holding the general price level and hence aggregate demand constant. The second order condition for a unique global maximum to the individual firm’s problem is satisfied if we assume that this marginal revenue curve is downward sloping, so that marginal revenue increases as the firm’s relative price increases, holding aggregate demand constant:

Rz = 2 - DzzD/Dg > 0. (7)

The assumption of a monotonically decreasing marginal revenue curve also enables us to rule out asymmetric equilibria. For a given level of aggregate demand each firm has a unique profit maximizing relative price, and since all firms face the same demand curve, this price must be the same for all firms, so that the equilibrium relative price must be one, and all firms will therefore produce the same level of output in equilibrium.

To solve for the equilibrium level of output, we notice that a firm will be maximizing profits only if its marginal revenue is zero, or else its marginal revenue is positive and it is producing at the capacity level of output. The first type of equilibrium we call a “monopolistic equilibrium,” and the second a “competitive equilibrium.” A monopolistic equilibrium would be at a level of aggregate output, Y (equal to aggregate demand), at which the firm’s marginal revenue, evaluated at the relative price of one, is equal to zero?

R(Y, 1) = 0

6. An alternative way of characterizing a monopolistic equilibrium is that the own-price elasticity of the firm’s demand function, evaluated at the equilibrium level of aggregate output and at a unit relative price, is equal to minus one. Notice that the elasticity of aggregate demand, given our assumed aggregate demand function, is everywhere minus one. This may bother the reader who is used to analyzing a monopolistically competitive industry where it is normally assumed that the individual firm’s demand curve is everywhere more elastic than the industry demand curve. This paradox is dispelled if we remember that the demand curve for a particular industry, which shows how the demand for a particular subset of goods vanes with the average price of those goods, holding real income and the price level fized, is conceptually quite distinct from an aggregate demand curve along which both the price level and real income are varying.

ROWE: MONOPOLISTIC MACROECONOMICS a7

or equivalently, recalling (6):

[ m y , l ) / D d Y , 111 + 1 = 0 (9)

or, substituting for D(Y, 1) from (3):

[Y/nD,(Y, l ) ] + 1 = 0.

Any value of aggregate output, Y, less than aggregate capacity output, y, which satisfies equation (8) yields a monopolistic equilibrium. If R(Y, 1) is strictly positive for all values of output less than capacity, then the only equilibrium is the competitive one, with output at capacity. (If R(Y, 1 ) is strictly negative for all levels of output up to capacity, then no interior equilibrium exists.)

An example of a demand function which yields a unique monopolistic equilibrium for any values of capacity greater than unity is

D(Y, P , / P ) = Y / n + 1 - P , / P

R(Y, PJP) = -Y/n - 1 + 2Pi/P.

R(Y, 1) = 1 - Y / n

(11)

(12)

(13)

Y = n ( 1 4 )

which implies that each of the n firms produces one unit of output in equilibrium, provided that capacity output exceeds one unit per firm.

It is worth noting that, provided the equilibrium is monopolistic, the level of capacity output has no effect whatsoever on the actual level of output, which is determined solely by the form of the individual firms’ demand functions. In this sense, Say’s Law is wholly invalid in this model. An increase in productive capacity will simply result in an equal increase in unemployed resources. Since there are no costs of producing output, this unemployment is also inefficient. Whether unemployment is voluntary or involuntary depends on how one wishes to define those terms. The owners of firms would like to sell more output at the going price, and cannot; but could sell more output by cutting price, and choose not to do so. These results, of course, are exactly what one would expect from studying monopolistic power in a partial equilibrium setting. All we have shown is that the same results carry over to a general equilibrium setting.

Although the above discussion of unemployment might seem to have a “Keynesian” flavor, such an impression would be quite misleading. The equilibrium level of output has been derived solely from individual jirms’ demand functions-the aggregate demand function has played no role at all. With a flexible price level, aggregate demand adjusts to equal equilibrium output,

which yields a marginal revenue function:

Evaluating the marginal revenue function at a relative price of one yields

and since marginal revenue must be zero at a monopolistic equilibrium,

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not vice versa. The equilibrium price level is determined by substituting the level of output determined by (8) into the aggregate demand function (5), or else by substituting (5 ) into (8) yielding

R ( M V / P , 1) = 0. (15)

Provided that the equilibrium level of output is unique and stable, a change in the quantity of money (or of velocity) causes an equiproportionate change in all prices, with no effect on real output. Monopoly power per se does not invalidate the classical macroeconomic doctrines of the Quantity Theory and the Neutrality of Money, nor does it cause inflation or an indeterminate price level. The aggregate supply curve (in price level-output space) remains vertical. The only difference between monopolistic and competitive equilibria is that the level of output is lower and the price level is higher.

111. STABILITY AND UNIQUENESS

Under what conditions will a monopolistic equilibrium (assuming it exists) be both unique and stable?

We have assumed (see equation (7)) that the individual firm’s marginal revenue curve is downward sloping-that marginal revenue, for a given level of aggregate demand, is monotonically increasing in the firm’s relative price. This assumption ensures a unique maximum to the individual firm’s problem-given the prices set by other firms, and hence given the level of aggregate demand, there is only one relative price at which the individual firm’s profits are maximized. A higher (lower) relative price would imply positive (negative) marginal revenue and so the firm would reduce (increase) its price to increase (reduce) its sales. And since all firms are identically situated, all have the Same profit maximizing relative price, which must therefore be unity in equilibrium. The fact that there is a unique equilibrium relative price, however, does not entail that the monopolistic equilibrium price level and aggregate output level are unique and stable.

To explore uniqueness and stability of the price level and aggregate output, let us conduct a stability experiment, in which we arbitrarily displace the economy from an initial monopolistic equilibrium, and examine the forces, if any, which would tend to return the economy to its initial equilibrium.

Start in a monopolistic equilibrium. Now reduce all firms’ prices by the same amount. With a lower price level, aggregate demand, and hence the output of each firm, will increase in the Same proportion as the price level has been reduced. Would individual firms now wish to raise their prices back towards the initial equilibrium levels? The answer to this question depends on whether each firm’s marginal revenue, evaluated at the new higher level of aggregate demand, is negative, positive, or zero. If negative, each firm would wish to increase its relative price. Now it is not possible for all firms to increase their relative prices, but the attempt by each to do so would result in an increase in the general price level and a fall in aggregate demand and


output back towards the original level, and so the equilibrium would be (locally) stable. If instead marginal revenues were positive, each firm would wish to reduce its relative price, leading to a further fall in prices and increase in output, meaning that the initial equilibrium was unstable. If zero, no firm would wish to change its price, so we are in a new equilibrium, hence the old equilibrium was not unique.

To repeat this analysis more formally, define a “marginal revenue locus” (as distinct from a marginal revenue curue), which shows how the representative firm’s marginal revenue, evaluated at the equilibrium relative price of one, varies as aggregate demand varies:

L(Y) = R(Y, 1). (16)

We assume a dynamic process in which (1) each firm sets the same price P, (2) customers choose the quantity of output Y = M V / P , and (3) firms react to the demand signal by raising P (each attempting to raise its relative price) whenever L(Y) < 0, and lowering P whenever L(Y) > 0. Thus the disequilibrium behavior of P is determined by a differential equation of the form

?r = dP/dt = F [ L ( M V / P ) ] (17)

where F’ < 0 and F ( 0 ) = 0. Differentiating (17) with respect to P yields

d?r/dP = - (F’L‘MV)/P2 (18) which must be negative, and hence L‘ negative, for stability.

The slope of the marginal revenue locus, L‘, is given by

L’(Y) = Rl(Y, 1) = Dl/D, - DDzl/D$. (19)

In a monopolistic equilibrium, marginal revenue is zero, and so D, = -D. Evaluating the slope of the marginal revenue locus at a monopolistic equilibrium thus yields

L’(Y) = -(Dl + Dzl ) /D . (20)

A level of aggregate output is a monopolistic equilibrium if the marginal revenue locus is zero at that level of output, and the competitive (capacity) level of aggregate output is an equilibrium if the marginal revenue locus is positive at capacity output. A monopolistic equilibrium is stable only if the marginal revenue locus is downward sloping (L‘ < 0) at that equilibrium level of aggregate output. Inspecting equation (20) to determine the sign of L‘, note that D1 is positive (and, from equation (3), equal to l/n). The second term in the numerator is Dzl, which we cannot sign a priori. D, is just the inverse of the slope of an individual firm’s demand curve, so a positive (negative) value of Dzl tells us that the firm’s demand curve, evaluated at the equilibrium relative price of unity, gets steeper (flatter) as an increase in aggregate demand causes it to shift to the right. If Dzl is positive then L’ must be negative and so the equilibrium is stable. On the other hand, Dzl may be

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negative, and may exceed D, in absolute value, in which case L‘ is positive and the equilibrium is unstable. As a “knife-edge” possibility, Dzl may exactly equal minus D1 for some range of aggregate output, in which case there is a continuum of equilibria in that range.

The same analysis can also be conducted in terms of the own-price elasticity of demand, rather than marginal revenue. All firms are simultaneously in profit maximizing equilibrium at a level of aggregate demand such that individual firms’ demand curves, evaluated at a relative price of one, are unit elastic. An equilibrium is stable (unstable) if an increase in aggregate demand causes firms’ demand curves to become less (more) elastic, leading each firm to wish to raise (lower) its relative price above (below) one. Either case seems equally likely a priori.

The particular example of a demand function in equation (11) yields a monopolistic equilibrium which (it can be verified) is unique and stable. To illustrate the possibility of a continuum of equilibria (the borderline between stable and unstable equilibrium) consider the following example of a demand function for which all values of output from zero to capacity are monopolistic equilibria:

D(Y, P i p ) = (Y/n)(2 - Pip). (21) The associated marginal revenue function is

R(Y, P i p ) = 2(P,/P) - 2

and evaluated at the relative price of one yields

R(Y, 1) = 2 - 2 = 0

so that the marginal revenue locus is zero for all values of aggregate output. The individual firm has a unique profit maximizing price (or output) given the aggregate price (or output) level, but that aggregate price or output level is indeterminate.

This particular example of a demand function, though something of a curiosus, nevertheless illustrates the fact that the existence of a unique maximum to the indiuidual firm’s problem, taking aggregate demand as given, does not ensure a unique, stable aggregate equilibrium. This can also be seen by noticing the absence of any similarity between the slope of the marginal revenue curue in (7), and the slope of the marginal revenue locus in (20).

Although a continuum of equilibria is in some a priori sense “rare,” it cannot be said that a unique and stable monopolistic equilibrium is “com- mon.” Economists of ten take homothetic preferences as a “standard” case. One of the properties of demand functions derived from homothetic preferences is that relative quantities demanded depend only on relative prices, and are independent of income. In this context, this means that homothetic preferences yield demand functions of the form

(24) D(Y, PiIP) = ( y / n ) G(P, IP)

and thus yield marginal revenue functions:


R(Y, P J P ) = G ( P , / P ) / G ’ ( P , / P ) + P , / P (25)

so that marginal revenue is a function of the relative price only and is independent of aggregate demand, which implies that the slope of the marginal revenue locus is everywhere zero. Thus homothetic preferences entail that either none or all levels of output are monopolistic equilibria. Homothetic preferences also represent the dividing line between preferences which yield stable and those which yield unstable monopolistic equilibria. Relative to homotheticity, if goods became less substitutable as income increased, the equilibrium would be stable. If more substitutable, the equilibrium would be unstable. Since either case seems equally likely a priori, we cannot say that a unique and stable monopolistic equilibrium is most likely a priori.

If the conditions for a unique, stable, monopolistic equilibrium are satisfied, then the macroeconomic predictions of this model are similar to those of a competitive macroeconomic model with similar assumptions of flexible prices and full information. The “real” side of the economy determines real output and relative prices independently of the money supply and velocity, which affect only the general price level. The only difference is that the level of output is lower, and the price level higher, than they would be for competitive firms (which, having marginal revenue equal to the relative price of one for all levels of aggregate demand, would produce at capacity).

However, once we recognize that uniqueness and stability cannot so safely be assumed, we recognize the possibility of very non-classical macroeconomic predictions. With a continuum of equilibria, for instance, an increase in the money supply might cause an increase in the price level and no change in output, but it might equally well cause an increase in output and no change in the price level. Even with a multiple but nevertheless finite number of equilibria, an increase in the money supply, with a purely temporary imperfect flexibility of prices, might cause a shift to another equilibrium-a permanent increase in real output. Like alternative models of multiple macro- equilibria, such as Diamond [ 19821, Hahn [ 19851, Howitt [ 19851, Negishi [ 19791, and Woglom [ 19821, this model offers a possible explanation of the severity and, in particular, the persistence of major depressions.

Figure 1 illustrates a monopolistic equilibrium. The bottom diagram shows a downward sloping marginal revenue locus, ensuring stability and uniqueness, having a value of zero at the equilibrium level of aggregate output Yo. The middle diagram shows the representative firm’s demand and marginal revenue curves, each drawn holding aggregate demand constant at the equilibrium level YO. The horizontal axis of the middle diagram is scaled up by a factor of n, relative to the bottom and top diagrams, to provide comparability between the individual firm’s equilibrium output and equilibrium aggregate output. The top diagram shows the vertical monopolistic aggregate supply curve intersecting the aggregate demand curve to determine the equilibrium price level.

The arrows on the marginal revenue locus depict the disequilibrium behavior of the economy. With a downward sloping marginal revenue locus, as

FIGURE 1

drawn, an increase in aggregate demand above the equilibrium level YO would 3ause firms to have negative marginal revenue. Each would then attempt to increase its price relative to other firms’, and the resulting rise in the general price level would reduce aggregate demand, and so the equilibrium is stable. [f instead the marginal revenue locus were upward sloping at equilibrium, then that equilibrium would be unstable. If the marginal revenue locus in-


tersected zero at several levels of aggregate output, there would be multiple equilibria, alternatively stable and unstable, and hence multiple aggregate supply curves. If the marginal revenue locus were everywhere above zero, then equilibrium would be at the capacity (competitive) level of output.

For simplicity we have assumed that Brms have zero costs up to the capacity level of output. If we relaxed this assumption we would need to posit a marginal cost function, with the individual firm’s output and aggregate output as arguments. The middle diagram would then also contain a marginal cost curue, showing how marginal cost varies with an individual firm’s output holding aggregate output constant, and the bottom diagram would then also contain a marginal cost locus, showing how marginal cost varies with aggregate output when all firms produce the same level of output. Equilibrium aggregate output would then be determined where the marginal revenue locus intersects the marginal cost locus, and would be stable if it intersected from above.

IV. AN ALTERNATIVE REPRESENTATION

This section can be skipped without loss of continuity. Its purpose is simply to provide an alternative way of looking at the same model, at the deeper level of consumer preferences.

Consider a two-generation model. The young produce output which they sell for money, which they spend on consumption goods when old.

Consider an old agent j , holding a stock of money M,, and facing a vector of prices P for the n goods. His problem is to choose a consumption vector x, to maximize his utility U, subject to his budget constraint:

Max U,(x j ) s.t. Px, I M,. (26)

The result is his demand function for each good i:

X,,,(P> M,) . (27)

Aggregating across old agents yields the market demand function for each

x,(P, M ) (28)

good i

where M is the vector of old agents’ holdings of money. Now consider young agent i, who has a monopoly on the production of

good x,. His problem is to choose a price PI to maximize his total revenue Pix, (and hence to maximize his money holdings and hence utility when old) subject to the demand function for his good, and subject to a capacity constraint f,, taking as given all prices other than PI, and the vector of money holdings:

Max P,x,(P, M ) s.t. x, 5 Z,. (29)

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\

I I \;X,+PGX2=M

I x1

FIGURE 2

The result is an optimal price function for each young agent i :

PiW, M , f,). (30) An equilibrium is a price vector Po which satisfies each of the optimal price functions simultaneously:

W M , f ) (31)

where i is the vector of firms' capacity constraints. Substituting the equilibrium price vector into the individual consumers' and firms' demand functions yields the equilibrium consumption vectors:

Since the consumers' demand functions are homogenous of degree zero in P and M, the optimal price functions are homogenous of degree one in P and M, the equilibrium price vector is homogenous of degree one in M, and the equilibrium real output vector is homogenous of degree zero in M. Thus, provided equilibrium is unique and stable, a proportionate change in the vector of money holdings leads to an equiproportionate change in the price vector and no change in the output vector. We cannot say whether the equilibrium is unique, stable, and whether it is capacity constrained, without putting restrictions on agents' preferences.

The equilibrium can be depicted graphically if we assume there are only two firms and a single representative consumer. See Figure 2.


I

x1

FIGURE 3

The consumer is in equilibrium if his budget line is tangent to an indifference curve at a consumption point. Taking P , and M as given, varying PI traces out a price-consumption curve PCCl(P2, M). Firm one’s problem is to choose PI to maximize P,x , taking P , and M as given. Maximizing P,x, = M - P2x2 is equivalent to minimizing xz , so firm one will choose that point on PCC, which minimizes x,, which is where PCC, is horizontal. (We assume a unique minimum on any price consumption curve.)

Similarly, firm two will choose a price P,, given PI and M , at which PCC,(P,, M) is vertical. Therefore, both firms and the consumer are in equilibrium simultaneously at a pair of prices P? and P$ such that PCC,(P$, M) and PCC,(P?, M) intersect at a point in consumption space where PCC, is horizontal and PCC, is vertical. (We have assumed that the capacity constraints are not binding at this equilibrium.)

Questions of stability and uniqueness of equilibrium can be addressed as follows. For firm one, construct its price consumption curve for varying values of Pz and draw a line connecting the horizontal points on each price consumption curve. The resulting reaction curve 2, shows points at which the consumer and firm one are in equilibrium. Now construct a similar reaction curve Z, showing points at which the consumer and firm two are in equilibrium. Full equilibrium, if it exists, is at the intersection of Z1 and 2,.

Equilibrium is stable if 2, is steeper than Zz at their intersection, as in Figure 3. Consider a displacement from equilibrium point E to disequilibrium point A, caused, for instance, by an increase in the money supply holding

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both prices temporarily fixed. At point A the consumer is in equilibrium, but neither firm is. Firm one would want to increase its price to slide along PCC, towards Z1. Firm two would want to increase its price to slide along PCC, to Z2. As both prices increase, the consumer’s budget constraint moves inward, thus moving the consumption point back towards the equilibrium, which is thus stable. If instead Z, were flatter than Z, at their point of intersection, the equilibrium would be unstable.

Only if Z1 and Z, intersect within the capacity constraints will the equilibrium be monopolistic. If Z, lies everywhere below Z2, then one or both firms will produce at capacity. If the consumer’s preferences are homothetic, it can be verified that the firms’ reaction curves are rays from the origin, in which case there either exists no interior monopolistic equilibrium, or else a continuum of equilibria if both rays coincide.’

V. PRICE CONTROLS

In a competitive macroeconomic model, where the natural level of output is presumably efficient, the case for temporary price controls, to complement disinflationary aggregate demand policy in eliminating inflation more quickly and with less unemployment, is debatable. The case for permanent price controls is non-existent. In a microeconomic context, the case for permanent controls on the relative price of a particular monopolistic firm is familiar. This model, however, provides an “ideal case” argument in favor of permanent controls on the general price level.

Suppose the economy is initially at a stable, unique, monopolistic equilibrium. A price controller could then prohibit price increases and increase the money supply until aggregate demand were equal to capacity output. Pro- ducing at capacity, each individual firm has negative marginal revenue and would like to increase its relative price to increase its profits. It is prohibited from doing so, however, and taking its relative price as given, has positive marginal revenue and maximizes profits by producing at capacity. Thus price controls may force monopolistic firms to act as if they were competitive.

Interestingly, aggregate real profits are higher in this model when firms

7. An example of a homothetic utility function which yields a continuum of equilibria is

u = -x; - x; + 9x:x, + 9X%X,.

Facing a consumer maximizing this utility function, firm one maximizes revenue by setting its price equal to firm two’s price. The Same is true for firm two. Though the equilibrium relative price is determinate, the price level is not.

With a symmetric Cobb-Douglas utility function,

u = x,x,, since each firm’s revenue is independent of its price, both relative prices and the price level are indeterminate.

An example of a utility function which yields a unique, stable monopolistic equilibrium (provided capacity is sufficiently high) is

u = x, + x, - x: - x;


behave competitively under price controls than if each is free to maximize its profits by behaving monopolistically! This follows immediately from the fact that, in this model, aggregate real profits equal aggregate output, which is higher at capacity than at a monopolistic equilibrium.

Thus price controls (accompanied by an increase in the money supply) lead to an increase in output which is not only permanent but also efficient.

Obviously, a realistic case for permanent, general price controls would have to assess the degree to which the economy is monopolistic, the costs of en- forcing and evading price controls, the information available to the price controller, and the degree to which he can be trusted not to abuse his power. Demonstrating a theoretical possibility that controls can be desirable is not to advocate controls.

VI. PRICE-RIGIDITY

One of the more puzzling stylized facts of the business cycle is that the money supply is procyclical. Most macroeconomists have sought to explain why changes in the money supply might cause changes in real output. The “Non Market Clearing” or “Keynesian” approach attempts to explain this by assuming, for various reasons, that prices (or wages) are imperfectly flexible. There are two problems with this approach. The first is to explain why prices are imperfectly flexible, and the second is to explain the determination of quantities when prices are fixed. Assuming that firms are monopolistic rather than competitive sheds a little light on the first problem, and solves the second problem.

We have seen that the assumption of monopolistic firms per se does not imply that prices are in any sense inflexible or that firms suffer from money illusion. The only exception to this is the “knife-edge” possibility of a continuum of equilibria, where it is possible that an increase in the money supply would lead to an increase in real output and no change in prices. This curiosus aside, one still needs to introduce “frictions” into a monopolistic as into a competitive economy, to get imperfect price flexibility. The difference between monopolistic and competitive economies, however, may lie in the size of price adjustment costs one needs to assume to get a given deviation of output from the natural rate.

Consider a competitive economy in a fixed price excess supply equilibrium. Each firm could remove its sales constraint completely by a fractional cut in its price below its competitors’ prices. In terms of this model then, with zero costs up to capacity, the cost of changing prices would have to exceed the value of lost output in order that firms not cut prices, which seems empirically implausible. Mankiw [ 19851 however, has noted that with monopolistic firms there is an externality in price adjustment, and that small costs of changing prices can support, under some parameter values, large deviations of output from its natural rate.

The problem with Mankiw’s analysis is that it is conducted in a partial

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equikbrium framework. Drawing the individual firm’s demand curve in relative price and output space, Mankiw treats a fall in nominal aggregate demand, holding the firm’s nominal price fixed, as an increase in the firm’s relative price with no change in the firm’s demand curve. If all firms were identically situated, however, a fall in nominal aggregate demand (MV) , holding all firms’ prices fixed, would cause real aggregate demand to fall and cause each firm’s demand curve to shift left, leaving relative prices unchanged. One cannot analyze whether individual firms would have an incen- tive to change their prices following a fall in aggregate demand, without knowing their marginal revenue at an unchanged relative price, but at a lower level of real aggregate demand; and it is exactly this information which my marginal revenue Zocus conveys, and which the standard marginal revenue curve does not.

It is easy to see how, under certain parameter values, vanishingly small costs of price adjustment may lead to indefinitely large deviations of output from equilibrium, and indefinitely large welfare losses. Simply consider the case of a marginal revenue locus which is zero for all levels of aggregate demand, leading to a continuum of aggregate output equilibria. Here an infinitesimally small cost of adjusting prices means that a fall in the money supply will cause a proportionate fall in real output. In general, the nearer the slope of the marginal revenue locus to zero, the smaller the costs of price adjustment needed to support a given deviation of output from the frictionless equilibrium.

The second problem with the “Keynesian” approach to business cycles concerns the assumption that is made about quantity determination when prices are not at market clearing levels. The assumption that quantity traded equals quantity demanded, adopted for instance by Fischer [1977] and Gray [1976], when used in a competitive setting, entails that quantity exchanged exceeds quantity supplied in the event of a positive demand shock. But how can, for instance, unemployed workers be forced to sell their labor when they do not want to? The alternative is to assume voluntary exchange, so that quantity is determined by the short side of the market. This assumption seems more plausible, and yet leads to perverse macroeconomic predictions. An increase in the money supply, starting at equilibrium, would then lead to a fall in output and employment, as shown for instance by Barro and Grossman [1971].

Assuming a monopolistic rather than a competitive economy enables us to escape this dilemma. Take a monopolistic firm in profit maximizing equilibrium, with price exceeding marginal cost. Hold his price fixed and increase demand, and the monopolist will quite willingly increase output and sales, at least up to the point where marginal ,cost equals price. Thus output can increase considerably above equilibrium, with fixed prices, before supply constraints become operative. In other words, the assumption of “short side rules” yields the predictions of “demand side rules” if supply (the quantity a firm


would like to sell at (I given price) exceeds demand in the neighborhood of equilibrium.

Assuming that firms’ outputs are imperfect rather than perfect substitutes therefore yields a far more hospitable environment for theories of the business cycle based on imperfect price flexibility.8 A further nice property of such a model is that a boom caused by an increase in the money supply with prices temporarily fixed causes an increase in welfare by increasing output towards the efficient competitive level. This property contrasts strongly with that of theories where the equilibrium or natural rate of output is efficient. In “New Classical” models such as Lucas [1972], an unanticipated increase in the money supply merely seems to make agents better off -an illusion that is dispelled later when agents realize their price perceptions were mistaken. In standard Keynesian models with a competitive economy an increase in output above the natural rate also leads to a fall in efficiency that is perceived as such, as is argued by Barro [1977]. The predictions of a monopolistic model better fit my preconceptions of the business cycle-booms are seen to be a ‘‘good thing,” are remembered as such, and really are such!

VII. FISCAL POLICY

Provided that the monopolistic equilibrium is unique and stable and that prices are perfectly flexible, we have seen that monetary policy cannot affect real output. An increase in the money supply merely shifts the aggregate demand curve up along an unchanged vertical aggregate supply curve, and so causes only a proportional rise in the price level with no change in real variables. Insofar as fiscal policy only affects the aggregate demand curve it too will affect only prices and not real output. However, there are other channels through which fiscal policy might affect the real economy-by shift- ing the aggregate supply curve. A change in government purchases may change the shape of the demand functions facing individual firms; by making them more or less elastic it may make the firms behave more or less competitively, and so move the economy towards or away from the competitive (capacity) level of output.

To explore the effect of government expenditure on output, let us include aggregate real government expenditure, G, as an argument in the individual firm’s demand function:

Yi = my, P J P , GI. (34)

As before, we assume that each firm faces the same demand function, so that when all charge the same price, each gets an equal share of aggregate demand:

(35) D(Y, 1, G ) = Y/n. 8. I am not the first to argue that Keynesian macroeconomics is better suited to a monopolistic

than to a perfectly competitive environment. See, for instance, Burstein [I9751 and Sweeney [1974].

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This means we can continue to analyze the economy in terms of the representative firm.

In a monopolistic equilibrium (assuming it exists), all firms charge the same price, and aggregate output is such that marginal revenue is zero:

R(Y, 1, C ) = D/Dz + 1 = 0. (36) (It is understood from now on that all functions are evaluated at a relative price of one.)

Totally differentiating (36) yields the government expenditure multiplier:

(37)

(38) For stability, we know that the marginal revenue locus must have a negative slope at equilibrium:

-(Dl + D Z W < 0 (39) which implies that if this equilibrium is stable (and this comparative statics exercise is meaningful) then the denominator of (38) must be positive, so that the government expenditure multiplier is positive if and only if 0% is negative.

This result is intuitively reasonable. Dz is just the inverse of the slope of an individual firm’s demand curve, so that if 0% is negative this means that an increase in government expenditure makes the individual firms’ demand curves more elastic (at a given level of aggregate demand and unit relative price) and leads firms to behave more competitively, reducing prices and increasing output. It is impossible to determine the sign of DS, however, without placing restrictions on private and government demand functions.

One restriction we might impose is to assume that private and government demand functions are quite independent, so that the disposition of aggregate private expenditure (Y - G ) is unaffected by the disposition of aggregate government expenditure, and vice versa:9

dY/dG = -(D3 + O B ) / ( D 1 + DZl) .

dY/dG = -DB/(D1 + DZl) .

From (35) we see that D, is zero, and so

D(Y, P , / P , G ) = C(Y - G, P i / P ) + H(G, P J P ) (40)

where C and H are respectively private and government demand functions for the output of an individual firm. To find D,:

0 2 = Cz + H2

Dm = -Czi + Hz1. (41) (42)

9. In assuming that the diseition of private expenditure ishdependent of the disposition of government expenditure we are in effect assuming that private agents get no utility from government expenditure. If we made the extreme opposite assumption, that private agents treat government and private expenditure as perfect substitutes, then government expenditure would have no effect.


C,, tells us how the slope of the private demand curve facing an individual firm varies as aggregate private expenditure changes. It is not possible to determine the sign of this parameter a priori, let alone determine its magnitude relative to its counterpart for government demand. Thus the sign of the government expenditure multiplier cannot be determined a priori even grant- ed our restriction that private and government demand functions are independent. This largely negative result demonstrates that any prediction about the effects of government expenditure on output in a monopolistic economy, as is made for instance in Hart [1982], must be based on special assumptions about private and government demand functions. In general, though government expenditure will almost certainly affect aggregate output in a monopolistic economy, the sign and magnitude of that effect are a priori indeterminate.

VIII. CONCLUSION

The central aim of this paper has been to construct a monopolistic counterpart to the simple competitive classical macroeconomic model, a counterpart simple enough to be understood intuitively and hence useful as an aid in thinking about the macroeconomic implications of imperfect substitutabil- ity between firms’ outputs.

The model has demonstrated that relaxing the classical model’s assumption of perfect competition does not per se invalidate the predictions of that model. Given stability and uniqueness of a monopolistic equilibrium, the Quantity Theory and Neutrality of Money still hold, but at a lower level of output and higher price level than under perfect competition. However, as Patinkin [1965] has argued, no equilibrium experiment is complete without a stability experiment, and the stability experiment in a monopolistic model is qualitatively different from a competitive model. In the latter, a general price level above (below) the equilibrium will cause an excess supply (demand) of goods, hence causing a fall (rise) in prices. Monopolistic firms, however, are never on their supply curves in this sense-they would always like to sell more at existing prices-and so we cannot talk about disequilibrium price levels causing excess supply or demand in this same sense. Instead we must consider how a move- ment along the aggregate demand curve away from equilibrium (an equiproportionate change in prices and inverse change in real output) would affect the marginal revenue equals marginal cost condition of profit maximizing monopolistic firms. It is exactly this information which my marginal revenue locus conveys. (In a model with positive costs, we would also need to define a similar marginal cost locus, showing how an individual firm’s marginal cost varies with aggregate output.) Since the stability experiment is qualitatively different, a monopolistic economy contains possibilities of instability and mul- tiplicity of equilibria of a type quite different from those of its competitive counterpart.

Quite apart from the crucial issue of the stability experiment, the model has also been used to explore some interesting macroeconomic implications

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of monopolistic power concerning price controls, imperfect price flexibility, and fiscal policy. Never intended to be a comprehensive study of macroeconomics with monopolistic firms, based on a formal and rigorous microfoun- dation, the model may nevertheless serve its purpose if it promotes further exploration of this topic by providing a useful initial crutch for our imperfect intuition.

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