Do small menu costs cause bigger and badder business cycles in monopolistic macroeconomies?

NICHOLAS ROWE Carleton Ufkersity

Ottawa, Ontario, Canada

Do Small Menu Costs Cause 5igger and Badder Business Cycles in Monopolistic Macroeconomies?*

I present a macroeconomic model of monopolistic and monopsonistic firms facing general demand and labor supply functions. Small costs of changing prices support large output fluctuations when nominal demand changes; this holds in both partial equilibrium and macroeconomic contexts, and for both monopolistic and competitive firms. A monopolistic economy may exhibit multiplier effects, but need not allow bigger business cycles than a competitive economy. Small menu costs have large welfare consequences only in monopolistic economies, but the welfare costs of symmetric fluctuations are on average trivial in any case.

1. Introduction Nominal aggregate demand fluctuations may affect real output

if nominal prices are assumed imperfectly flexible or “sticky.” The simplest way to justify assuming sticky prices is to point to the real resources needed to change them. That there are costs to changing price tags, printing new catalogues and menus, etc., is both plau- sible and requires no further explanation.

The reason such “menu costs” have not been more readily invoked to justify assuming sticky prices is probably the belief that small menu costs could only support equally small deviations of output from the frictionless equilibrium. This belief has been chal- lenged by Mankiw (1985) in a paper entitled “Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly.” He shows that “. . . sticky prices can be both privately efficient and socially inefficient” and also that “. . . small menu costs can cause large welfare losses.”

Despite the title, Mankiw’s model is not a macroeconomic model, but a partial equiZibrium model of a monopolistic firm faced with an exogenous change in its relative price. In a truly macro-

*I thank Randy Geehan, Steve Ferris, Tim Lane, and participants at a Carleton macroeconomics seminar for helpful discussions and comments. Opinions and errors are mine.

Journal of Macroeconomics, Winter 1989, Vol. 11, No. 1, pp. 25-48 25 Copyright 0 1989 by Louisiana State University Press 0164-0704/89/$1.50

Nicholas Rowe

economic context, if all firms held their prices constant, an increase in nominal aggregate demand would leave relative prices unchanged but would shift their demand curves outward (and perhaps also shift their cost curves upward as higher output and employment required higher real wages).

This distinction between partial equilibrium and macroeconomic contexts is made especially important by the possibility of “positive feedback’ (or “strategic complementarity”) whereby an exogenous increase in one firm’s output may cause other firms to raise their output, thus generating “multiplier effects” in which aggregate output changes by more than the sum of the firms’ partial equilibrium responses to some exogenous change. Such multipliers have been noted in macroeconomic models with monopolistic firms by, for instance, Startz (1986) and Cooper and John (1987).

Unlike Ma&w (1985), Akerlof and Yellen (1985), Par-kin (1986), and Blanchard and Kiyotaki (1987) do explore the consequences of small menu costs in models that are explicitly macroeconomic. Each paper shows that second-order menu costs can support first-order deviations of aggregate output from hictionless equilibrium, but each uses a model with rather special demand and cost functions. In par- ticular, each assumes the price elasticity of demand facing each firm to be a constant, independent of firm and aggregate output. This paper sets up a macroeconomic model of monopolistic firms with general demand and cost functions,’ and examines the consequences of costly price adjustment, comparing partial equilibrium with macroeconomic contexts, comparing monopolistic with competitive firms, and comparing the welfare costs of fixed prices to the private costs of changing them. The model, moreover, is provided with a graphical interpretation, using familiar concepts like marginal revenue and marginal cost, to help us see more clearly what is happening. The main findings are as follows:

(a) Second-order menu costs can support first-order output deviations for both competitive and monopolistic firms, and in both partial equilibrium and macroeconomic contexts.

(b) Though competitive economies must exhibit negative feedback, while monopolistic economies may exhibit positive feedback and multiplier effects, there is no proof, only a presumption, that small menu costs can support larger output deviations in dightly monopolistic macroeconomies rather than in competitive macroe-

‘The model is based on that in Rowe (1987) with a labor market added.

26

Business Cycles in Monopolistic Macroeconomies

conomies, and that presumption is reversed for highly monopolistic macroeconomies and for partial equilibrium contexts.

(c) The output deviations caused by small menu costs will have small welfare consequences in competitive economies and large welfare consequences in monopolistic economies, but the average welfare costs of symmetric output fluctuations will be small in either case-being less than the menu costs for competitive macroeconomies, though perhaps larger for monopolistic macroeconomies.

2. The Model Consider an economy with a large fixed number of firms, each

facing the same inverse demand function which firm i’s relative price, pi> depends negatively on its own output, yi, and positively on aggregate output per firm, Y.

Pi = D(Yi, y, > D1 < 0, D, > 0 (1)

(For the origins and interpretation of such a demand function, see Appendix B).

By symmetry, if all firms set the same price (p = l), then all sell the same level of output (y = Y); so the following restriction holds identically in. Y:

1 = D(Y, Y) . (2)

Differentiating the firm’s (real) total revenue, yipi, with respect to its output defines its (real) marginal revenue function, R(y,, Y).

d(YiPJ/dYi = R(Y, Y) = D(Y, Y) + YDl(Y, Y) ; (3)

where D,(y, Y) refers to the partial derivative of D(y, Y) with respect to its first argument, etc., and subscripts have been suppressed.

Each firm faces the same inverse production function, which gives its labor requirements, n,, as a function of its output.

ni = F(Yi) T F’ > 0, F’ > 0 . (4)

Each firm faces the same inverse labor supply function, which specifies the real wage, oi, it needs to pay to attract labor as a

27

Nicholas Rowe

function of its own employment and aggregate employment per firm, N.

~)i = S’(ni, N) 9 s1 > 0, sz > 0. (5)

If the number of firms is large and if all firms other than i produce the same level of output (as they will in equilibrium), then F(.) also defines the aggregate production function, relating aggregate employment per firm to aggregate output per firm.

N = F(Y) . (6)

Substituting (4) and (6) into (5) enables us to express the supply price of labor as a function of the firm’s own output and aggregate output per firm.

Oi = s[F(Yi), F(Y)I . (7)

Multiplying real wages by employment defines the firm’s (real) total variable costs, which, when differentiated with respect to its output, defines its (real) marginal cost function, M(yi, Y).

= F’(Y)CWY)> F(Y)1 + F(y)S,[F(y), WI) . (8)

Assume that each firm sets its price (or output) to maximize profits, taking aggregate output as given. (Appendix C shows that when the number of firms is large this is the same as taking other firms’ prices as given.)

Each firm’s (real) profit function is defined as

VY, Y) = YD(Y> Y> - F(y)Wy)> F(Y)1 .

Assuming it exists and is unique, identically situated firms will produce the same equilibrium level of output, y* = Y*, at which marginal profit is zero, or marginal revenue equals marginal cost.

Tl(y*, Y*) = qy*, Y*) - M(y”, Y*) = 0. (10)

Henceforth, for brevity, we adopt the convention that suppressing the arguments of a function indicates that the function is evaluated

28


at the equilibrium point (y*, Y*), and take it as understood that aggregate output or employment always refers to per firm values.

The second-order condition for a maximum requires the firm’s marginal revenue curve cut its marginal cost curve from above.

T,, = RI - MI < 0 (11)

As shown in Rowe (1987), stability of equilibrium requires that if all firms were to increase output together, each would find its marginal profit negative (or marginal revenue below marginal cost), so each would choose to reduce output, restoring equilibrium. Thus stability requires

~~~ + T,, = (R, + R2) - (M, + M2) < 0. (12)

Equilibrium is depicted in Figure 1. Equations (l), (3), and (8) show that the individual firm’s demand price, marginal revenue, and marginal cost depend both on the firm’s own output, y, and on aggregate output, Y. The middle diagram’s demand curve, marginal revenue curve, and marginal cost curve show how its relative price, marginal revenue, and marginal cost vary as the firm’s own output varies holding aggregate output constant at its equilibrium, level Y*. The lower diagram’s (in my terminology) marginal revenue locus and marginal cost locus show how a firm’s marginal revenue and cost vary when both the firm’s own output and aggregate output expand together (as would happen, for instance, if aggregate demand increased and all firms held their prices fixed at the same level).

Intersection of marginal revenue and cost loci determines (f t’ 1 ) q ‘l’b ric ion ess e ui 1 rium aggregate output and the position of the aggregate supply curve in the top diagram. The latter is vertical since the nominal price level, P, does not affect firms’ profit-maximizing choices of output. To determine that price level, however, and close the model, we need to add an aggregate demand function, but providing it yields us a determinate downward sloping aggregate demand curve, the exact source of that function does not matter. The standard IS/LM model would be as good as any, for instance.

The above model describes an economy with firms that are both monopolistic (Di < 0, so they face downward sloping demand curves) and also monopsonist (S, > 0, so they face upward sloping labor supply curves). But nothing prevents us from thinking of

29

\ ,-----------.

AS

%,

LAD

Y’ Y

pi/p

1

D(y,Y*)

I

Y* Y

I

R(Y,Y)

Y

Figure 1.


economies in which firms are perfectly competitive in output or labor markets, or in both, as limiting cases in which D, = 0 and S, = 0. The firm’s demand curve, marginal revenue curve, and marginal revenue locus then all become horizontal at a relative price of one, and its marginal cost function becomes

WY, Y) = F’(y)S[F(y), F(Y)1 . (13)

The marginal cost curve defined by (13) will slope up (M, > 0) if we assume diminishing returns (F” > 0), and the marginal cost locus will slope up (M, + Mz > 0) due to both diminishing returns and an upward sloping labor supply curve for the economy as a whole (S, > 0). Equilibrium output for an economy competitive in both output and labor markets is where marginal revenue (=l) equals marginal cost (F’S), implying the real wage equals the marginal product of labor.

3. Adjustment Costs Provided the equilibrium is unique and stable, changes in

nominal aggregate demand have no real effects in the above model. Suppose an increase in the money supply causes the aggregate demand curve to shift rightward. At existing prices consumers demand more output from all firms (and, provided that the price re- mains above marginal cost, as it will for small demand shifts and monopolistic firms, those firms will willingly produce the extra goods to meet that demand). The higher aggregate demand and output cause a movement along the marginal revenue locus and marginal cost locus, but cause a shift in each firm’s marginal revenue curve and marginal cost curve. If the stability condition is satisfied, marginal cost will now exceed marginal revenue, and each firm will find it individually profitable to increase its price and reduce its output. If they do so, the price level will rise, causing a movement back along the aggregate demand curve and returning output and all real variables to their original equilibrium values. Money would then be neutral. But now suppose there are small fixed costs of changing prices. If these menu costs exceed the foregone profits (defined ex- clusively of menu costs) from having marginal revenue not equal to marginal cost, it will benefit each firm individually to hold its price fixed so that output will remain above the frictionless equilibrium.

Could small menu costs allow changes in aggregate demand to have big real effects? To address this question we define a sec-

31

Nicholas Rowe

ond-order approximation to the firm’s profit function (and hence its underlying functions) in the neighborhood of equilibrium:

T(y, Y) = T + T,dy + T&Y + (1/2)T,,dy2

+ (1/2)T2,dY2 + T,,dydY ;

where dy = y - y* and dY = Y - Y* denote the deviations of firm and aggregate output from frictionless equilibrium, and the suppressed arguments indicate the function is to be evaluated at the equilibrium point (y*, Y*).

The Size of Output Deviations First suppose only one firm has menu costs and deviates from

equilibrium. The maximum deviation, dy, that can be supported by menu costs, C, is where the menu costs equal the lost profits:

C = T(y*, Y*) - T(y* + dy, Y*)

= -T,dy - (1/2)Tudy2. 05)

Since T1 = 0 (by the first-order condition), the maximum deviation is

dy = (-2C/T,,)“2 . (16)

The second-order condition is that TI1 < 0, but square roots can be positive or negative, reminding us that menu costs can support levels of output both above and below the frictionless equilibrium. Note also that the output deviation varies with the square root of the menu cost, so the ratio of output deviation to menu cost becomes infinite in the limit as the latter becomes zero. This means that a first order of small output deviation needs only a second order of small menu cost.

The relation between dy and C can be shown graphically. If the firm deviates from the frictionless equilibrium, its lost profits are given by the area of the triangle between its marginal revenue and cost curves. The height of the triangle is given by dy times the difference in slope of the two curves (Mi - Ri = -TJ, so its area is (1/2)dy2(M, - RJ, which will just equal the menu cost, C,

32


for the maximum deviation. The closer together the slopes of the two curves, the smaller the area between them will be, and the menu costs needed will be the smaller. If we assume a linear demand function (setting second derivatives and cross partials to zero), we can show that the slope of the marginal revenue curve is negative but becomes closer to horizontal as the firm becomes more competitive. Similarly, a linear labor supply function implies a flatter marginal cost curve as the firm becomes less monopsonistic. This means that the marginal revenue and cost curves of competitive firms are closer together in slope, and small menu costs have bigger output consequences for competitive firms than for monopolistic and monopsonistic firms.

In partial equilibrium contexts then, small menu costs tend to cause smaller fluctuations for monopolistic firms than for competitive firms (though in both cases of an order of magnitude greater than the menu costs themselves). Does this result change in macroeconomic contexts, where positive feedback may reinforce and multiply the original output deviation?

Suppose every firm deviates from frictionless equilibrium by some amount, dy. How large can dY be before each firm finds it individually profitable to pay the menu cost and change its output to some new level, y* + dy?

First, suppose one firm did decide to change its output. Its problem is to choose dy to maximize T(y* + dy, Y* + dY), which requires

T1(y” + dy, Y* + ClY) = T, + T,,dy + T&Y = 0, (17)

and since T, = 0,

dY = (-~w.I~,W . 08)

The term ( -T12/Tll) in (18) measures the effect on one firm’s profit-maximizing output of an exogenous unit increase in the output of every other firm. Let us grant it the symbol b to draw a parallel to the marginal propensity to consume in Keynesian multiplier models.

The largest deviation of aggregate output from equilibrium, dY, that can be supported by given menu costs is when the menu costs just equal the individual firm’s extra profits in moving from y* + dy to its optimum, y* + dy.

33

Nicholas Rowe

c = 7yy* + dy, Y* + dY) - T(y" + dY, Y* + dY)

= T,(dy - dY) + (1/2)T,,(d# - cnr”)

+ T,,dY(dy - dY) . (1%

Remembering that Ti = 0 and substituting from (18) yields

dY = [l/(1 - b)](-2C/T,,)“2 f (20)

I have expressed the “macroeconomic deviation,” dy, as the product of two terms. The second term is identical to the “mi- croeconomic deviation” of Equation (16) for an individual firm in a partial equilibrium context. The first term is the “multiplier,” l/(1 - b), or TdTll + T12).

The stability and second-order condition require both the denominator and numerator be negative, so the multiplier must be positive. If Ti2 is positive, then (18) indicates positive feedback and a multiplier greater than one.

Differentiating the profit function with respect to y and then Y, we find

T12 = R2 - A42 = (II2 + Y*D12) - W2@2 + J-12) * (21)

The first term measures the upward shift in the firm’s marginal revenue curve, and the second term measures the upward shift in its marginal cost curve, when aggregate output increases. If marginal revenue shifts upward by more than marginal cost, there is positive feedback and a multiplier exceeding one. But we cannot tell which one shifts upward the most (or indeed if both do shift upward) with- out exactly specifying the demand, labor supply, and production functions. In a perfectly competitive economy, for instance, marginal revenue equals one everywhere, so R2 = 0, and the firm’s own employment does not affect its wage, so Si = 0 and hence Si2 = 0, giving

T12 = -(F’)2S, < 0 . (22)

Thus in a competitive economy there is negative feedback and a multiplier less than one since an exogenous increase in aggregate output and employment pushes up real wages, reducing the individual firm’s profit-maximizing level of output. In a monopolistic

34


economy, this source of negative feedback from the supply side will likely be offset by positive feedback from the demand side. Mo- nopolistic economies may or may not have multipliers. Competitive economies cannot.

Will small menu costs support bigger macroeconomic deviations in monopolistic economies than in competitive economies? We cannot say in general because a presumably larger multiplier for monopolistic economies is countered by a presumably smaller mi- croeconomic deviation. However, if we consider the special case of of a linear demand function (and use the symmetry restriction II1 + II2 = 0 from differentiating [2]), we can rearrange (20) to yield

C = (1/2)[(0, - Ml - MJ2/(M1 - 2DJ]dy2 . (23)

Differentiating (23) with respect to D1, we find the effects of making firms more competitive in the output market (flatter demand curves):

g = [(Dl - Ml - M,)(-D, - M,)/(M, - 2DJ21dy. (24) 1

The first term in the numerator must be negative for stability, and the squared denominator must be positive, but the term (-Dl - M,) can be of either sign. When the economy is close to competitive, however, D, will be small, so (-Dl - M,) will be negative and dC/dD, positive, which means a more competitive economy needs bigger menu costs to support a given macroeconomic deviation. When the economy is highly monopolistic, on the other hand, D, will be large (and negative) and the reverse will be true. Even with a linear demand function we get no general result. As a competitive economy gets more and more monopolistic, small menu costs support first bigger fluctuations, but eventually smaller fluctuations, in aggregate output.

Figure 2 shows a deviation of output from frictionless equilibrium in a macroeconomic context. A rightward shift in the aggregate demand curve (not shown) causes all firms’ outputs to increase by dY at existing prices. This causes a movement along the marginal revenue locus and marginal cost locus in the lower diagram, but a (presumably) upward shift in the marginal revenue curve and marginal cost curve in the top diagram. (For consistency between the two diagrams, the points labelled r and m should have the same height on both diagrams. This means that if the slope of the locus

35

pi/p I

I

I I

Y”*dY y*+dY

i I

-R(Y,Y)

I I

Y*+dY Y

Figure 2.


exceeds that of the relevant curve, the curve must shift upward when we move rightward along the locus.)

If one firm did decide to increase its price and reduce its output while others held theirs fixed, the output it would choose would be at the intersection of the new marginal revenue and cost curves, labelled y* + dy. If it did so, its profits would increase by the area of the shaded triangle. The maximum deviation of aggregate output that given menu costs can support is therefore when the area of that triangle is just equal to those menu costs. When the angle between the marginal revenue and marginal cost loci is less than the angle between the curves, the new curves will intersect to the right of the frictionless equilibrium y*, implying positive feedback and hence a multiplier exceeding one. To get the biggest possible output fluctuation for the smallest possible menu cost, one needs the smallest possible angle between the loci, and the largest possible angle between the curves-the degree of monopoly power, or slope of demand curve per se is irrelevant.’

It is worth noting that Figure 2 can also be used to show the potential for multiple (frictional) equilibria, recognized by Blanchard and Kiyotaki (1987). Suppose the menu costs are just slightly larger than the shaded area between the two new marginal revenue and cost curves. This means that the individual firm will choose not to raise its price, and so keep its output at y* + dY, provided it expects other firms to do likewise. If instead it did expect other firms to raise their prices and return to the frictionless equilibrium Y*, then the relevant marginal revenue and cost curves are the old ones, and the profit it loses by not adjusting would be the triangle between the old curves from y* to y* + dY. With positive feedback, as drawn, the triangle between the old curves is bigger than the triangle between the new curves, and so can exceed the menu costs. Thus with positive feedback, moderate increases in aggregate demand will cause output increases if each firm expects output increases, and price increases if each firm expects price increases. With negative feedback, on the other hand, moderate increases in aggregate demand may cause self-denying rather than self-fulfilling

“With a constant elasticity demand function, as is commonly assumed, marginal revenue is a constant fraction of the firm’s price. The marginal revenue locus is therefore necessarily horizontal since the (relative) price stays constant at one when all firms expand output together, but the marginal revenue curve is downward sloping like the demand curve. An increase in monopoly power, or lower elasticity, then makes the marginal revenue curve steeper, while the marginal revenue locus stays horizontal, thus permitting larger output fluctuations for given menu costs.

35

Nicholas Rowe

expectations, as each firm will change its price if and only if it expects others not to. The assumption I have made, that firms expect other firms not to adjust prices unless that expectation can be shown to be irrational, permits multiplier effects to operate, and so allows small menu costs to support bigger output deviations when there is positive feedback.

The Costs of Fluctuations When it is costly for firms to change their prices, fluctuations

in aggregate demand may cause output fluctuations. Are the welfare costs of these fluctuations large relative to the menu costs which support them, and are they larger for monopolistic economies than for competitive economies?

We first ask this question in a partial equilibrium context, where a single firm departs from the frictionless equilibrium. The resulting changes in output and employment are valued at their demand price and supply price, respectively, giving the change in welfare, dW, as (see Appendix B for justification)

dW = Ddy + (1/2)Dldy2 - Sdn - (1/2)Sldn2. (25)

(Diagrammatically, welfare is the area under the firm’s demand curve minus the area under its labor supply curve, which is equivalent to consumer surplus plus profits, plus workers’ rents.) The change in employment is

dn = F’dy + (1/2)F”dy2, (26)

and substituting (26) into (25) gives

dW = (D - F’S)dy + (1/2)[D, - F”S - (F’)2S,]dy2 . (27)

Here we see a major difference between competitive and monopolistic or monopsonistic firms. Competitive hrrns produce where wage equals value of marginal product, so the first-order effect vanishes. Monopolistic or monopsonistic firms have wage less than value of marginal product, so the first term in (27) is positive. By substituting for dy from (16), we can show that for a competitive firm dW = -C, which means tha.t private and social costs and benefits of price adjustment coincide.

We now consider the question in a macroeconomic context, where both individual firm and aggregate output increase by dY.

38


Again, evaluating output and employment changes at their demand and supply prices, a second-order approximation to the change in welfare is (see Appendix B for justification)

dW = dY - SdN - (1/2)(S, + S,)dN’ c-w

(Diagrammatically, welfare is the area under the demand locus D[Y, Y], which is horizontal at unity, minus the area under the labor supply locus S[N, N].) Substituting for dN from the production function yields

dW = (1 - F’S)dY - (1/2)[F”S + (S, + S,)(F’)‘]dY2. (29)

Again, we see a positive first-order effect of aggregate output on welfare for a monopolistic or monopsonistic economy, which vanishes, leaving only a (negative) second-order effect for a competitive economy. Small menu costs therefore, which can cause large output consequences in all economies, can cause large welfare consequences only in noncompetitive economies, where individual profit- and welfare-maximization do not coincide.

Though not incorrect, the above has missed one important distinction-changes in the level of output and changes in the vari- ability of output. Menu costs can support both positive and negative deviations of output from frictionless equilibrium. Now, in a competitive economy, this distinction is irrelevant, for only the second-order effect of squared output deviations affects welfare; so both increases and decreases in output equally reduce welfare. But in a noncompetitive economy, the distinction does matter. Output increases have a positive first-order effect on welfare, but output decreases an exactly equal negative effect. If both positive and negative output deviations are equally large and equally likely so that output equals the frictionless equilibrium level on average, then the first-order welfare effects cancel exactly, leaving only the second- order effects!

Small menu costs can cause large output fluctuations in both competitive and monopolistic economies. In competitive economies, both booms and slumps cause small misery. In monopolistic economies, slumps cause great misery and booms cause nearly as great joy; but on balance, booms and slumps together cause only small misery. If menu costs are trivial, so are the welfare costs of the business cycles they cause.

To compare the welfare costs of symmetric fluctuations to the

39

Nicholas Rowe

menu costs which create them, substitute A’ from (20) into (29) (remembering that the first-order effects cancel) to get

dw = {[(-F’S - (S, + S&q2

* WI-- R,)II(R, + R2 - MI - M2J2)C . (30)

The denominator in (30) is the squared difference in slope between the marginal revenue locus and marginal cost locus. The second term in the numerator is the difference in slope between the marginal revenue curve and marginal cost curve. The first term in the numerator can be thought of as the difference between the slope of the demand locus, D(Y, Y), which is zero, and the slope of the “competitive supply locus,” F’(Y)S[F(Y), F(Y)]. The area between these two loci is a measure of welfare, since the former shows the demand price of output and the latter shows the marginal social cost of output, which is the marginal labor requirement times the wage.

It is not generally possible to determine from (30) whether the welfare costs of fluctuations, dW, are greater or less than the menu costs, C. For a competitive economy, however, (30) simplifies to

dw = -{SF’/[SF’ + S2(Fy2]}C . (31)

The term in braces is less than one, so in a competitive economy, the welfare costs of the largest fluctuations that menu costs can support are smaller than those menu costs. This result occurs because of negative feedback-when other firms expand, wages rise (provided S2 > 0), which reduces each firm’s individual frictionless optimum output. The whole is less than the sum of the parts.

Under the restriction of linear demand and labor supply functions, we can examine the effect of the monopolization of an initially competitive economy by differentiating (30) with respect to II1 and evaluating the result where D1 = Sr = 0.

- = C[F’S + 2(F’)zS2]/[F”S + (F’)2S,]2 > 0 . dD,

(32)

Given our simplifying restrictions, an increase in D, (implying flatter demand curves and more competitive firms) will increase the (negative) welfare benefits from the largest output fluctuations that

40


pi/p

1

Y* yf+dY Y

I I I

I I

I I D(Y,Yl

t I

I I F’(Y) S(F(Y), F(Y))

I

Y*+ dY Y

Figure 3.

can be supported by given menu costs. This indicates a presumption that the monopolization of a competitive economy will, at least initially, tend to increase the welfare costs of business cycles which result from aggregate demand fluctuations in an economy with small menu costs.

41

Nicholas Rowe

Figure 3 illustrates the welfare implications of aggregate output fluctuations due to small menu costs in a monopolistic and monopsonistic economy. The horizontally shaded trapezoid shows the welfare benefits of an increase in aggregate output to Y* + dY. The vertically shaded triangle shows the welfare costs of symmetric fluctuations between Y* + dY and Y * - dY (equal to half the sum of the trapezoid shown and a similar negative trapezoid). The diago- nally shaded area represents the menu costs. Small menu costs cause the biggest welfare costs of symmetric fluctuations when the competitive supply locus is steep, the marginal revenue and marginal cost loci are nearly equal in slope, and the marginal revenue and marginal cost curves differ greatly in slope. The degree of monopoly power per se is irrelevant.

4. Conclusion Could a macroeconomic model based on monopolistic firms

and costly price adjustment provide a suitable basis for Keynesian views on business cycles. 2 We have shown that small menu costs can permit fluctuations in nominal aggregate demand to cause large fluctuations in real output and employment, but this holds for competitive as well as for monopolistic economies. What the assumption of monopolistic or monopsonistic firms does add is that it makes booms good and slumps bad since the frictionless equilibrium level of output is less than optimal in these noncompetitive economies, but optimal in competitive economies. Even so, if Keynesian demand management policies can hope at best only to stabilize the level of output at its frictionless equilibrium level and not keep output permanently or on average above that level, the welfare gains from successfully pursuing such policies would be of the same order of magnitude as the costs of adjusting prices. Eliminating the fluctuations caused by small menu costs can hardly be called a national priority.

Received: October 1987 Final version: April 1988

References Akerlof, G.A., and J.L. Yellen. “A Near Rational Model of the

Business Cycle with Wage and Price Inertia.” QuarterZy Journal of Economics C (1985): 823-38.

Blanchard, 0. J., and N. Kiyotaki. “Monopolistic Competition and

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the Effects of Aggregate Demand.” American Economic Review 77, no. 4 (1987): 647-66.

Cooper, R., and A. John. 1987. “Coordinating Coordination Fail- ures in Keynesian Models.” University of Iowa. Mimeo.

Mankiw, N. G. “Small Menu Costs and Large Business Cycles.” Quarterly Journal of Economics C (1985): 529-37.

Parkin, M. “The Output-Inflation Trade-Off When Prices Are Costly to Change.” Journal of Political Economy 94, no. 1 (1986): 200- 24.

Rowe, N. “A Simple Macroeconomic Model With Monopolistic Firms. ” Economic Inquiry 25, no. 1 (1987): 83-102.

Startz, R. 1986. “Monopolistic Competition as a Foundation for Keynesian Macroeconomic Models.” University of Washington. Mimeo.

Appendix A List of Variables

Pi = nominal price at firm i. P = nominal price at other firms. pi = Pi/P = firm i’s relative price. yi = output at firm i. Y = output at each other firm (or, approxi-

mately, aggregate output per firm). p, = D(y,, Y) = f urn i’s inverse demand func-

tion. R(yi, Y) = marginial revenue function.

ni = employment at firm i. N = employment at each other firm (or, ap-

proximately, aggregate employment per firm).

n, = F(y,) = inverse production function. vi = real wage at firm i. V = real wage at other firms. vi = S(ni, N) = inverse labor supply function.

M(y,, Y) = marginal cost function. T(yi, Y) = profit function.

C = price adjustment (menu) cost. dy, dY, dn, dN, etc. = deviation of variable from its frictionless

equilibrium value. W(y, Y, n, N) = nonmonetary utility function of represen-

tative household (also welfare function).

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Nicholas Rowe

U(M/P) = monetary utility function. M = nominal money supply. m = number of firms.

Appendix B The demand function should be interpreted as follows: If firm

i is producing, and consumers are just willingly purchasing, an amount yi, and if each other firm is producing, and consumers are just willingly purchasing, an amount Y, then a necessary condition for the consumers to be maximizing utility, given their budget con- straints, is that pi = D(y,, Y).

First, note that under this strict interpretation Y refers to the output of each firm other than firm i rather than the looser interpretation of aggregate output per firm. But when the number of firms is large, the effect of excluding firm i’s output from the average is negligible.

Second, provided we restrict our attention to symmetric Nash equilibria, the demand function need only be defined for cases where all firms but one produce the same level of output because in equilibrium all firms do produce the same level, and each firm, in as- sessing whether its profits are indeed maximized by its choice, con- templates departing from equilibrium under the Nash assumption that all other firms’ outputs will remain unchanged. We thus escape all index number problems!

Third, though it is more convenient to call pi = D(y, Y) an inverse demand function, some may prefer to think of it as an equilibrium condition, and it is indeed only a necessary condition for consumer’s individual optimization. The function D(y, Y) should be interpreted as defining the representative consumer’s marginal rate of substitution between firm i’s output and the output of one of the other firms, which depends on the consumer’s consumption vector (y, Y). A necessary condition for consumer’s equilibrium is that the relative price of firm i’s output to that of other firms’ equals the marginal rate of substitution. A second necessary condition is that the marginal rate of substitution between labor at firm i and the consumption of other firms’ outputs equals the real wage. A third necessary condition is that the consumer’s desired total expenditure equals actual expenditure. Full equilibrium requires consumers to be on their demand functions for individual goods, and on their labor supply functions, and on their aggregate demand (or money demand) functions, and also for firms to be maximizing profits.

The simplest underlying framework from which the demand

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and labor supply functions could be derived would seem to be the following consumer choice problem:

max W(yd, Yd, d, N”) + U(Md/p) , s.t.

byd + (m - Wdl - 1 -c unS + (m - l)VN”] + [(M/P) - (Md/P)]

+ [py + (m - l)Y - VN - (m - l)VN]

Assuming real money balances, M/P, yield utility is a defensible way to introduce money into a model which was never intended to explain why people hold money. Assuming utility to be separable in real balances makes the following simpler, but is not necessary.

The first term in the budget constraint is (real) consumption expenditure on firm i’s output, pyd, and on the other (m - 1) firms’ outputs, Yd, which are taken as the numeraire good. The second term is wage income, the third is dishoarding of real money balances, and the fourth is income from profits, which are exogenous to the individual household. Choice variables are yd, Yd, nS, N”, and Md.

The first-order conditions for yd, Yd, d, and N” are

W,(yd, Yd, d, N”) = hp , W

W,(yd, Yd, d, N”) = A(m - 1) , 032)

W3(yd, Yd, d, N”) = --ho , 033)

and

W,(yd, Yd, d, N”) = -h(m - 1)V.

Dividing (Bl) by (B2), and (B3) by (B2) yields

WlIW2)(m - 1) = p

and

-(WJWJ(m - 1) = 21

034)

035)

036)

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Nicholas Rowe

The marginal rates of substitution on the left-hand sides of (B5) and (B6) are functions of yd, Yd, n”, N”. If we assume that these desired quantities are equal to their actual counterparts (which means we are talking about equilibrium conditions, and also implies that Md = M from the budget constraint), we can then use the production functions n = F(y) and N = F(Y) to rewrite the marginal rate of substitution in (B5) as a function of y and Y only, and in (B6) as a function of n and N only. Equation (B5) then becomes our demand function:

(W,lW,)(m - 1) = WY, Y) = p ; (B7)

and (B6) becomes our labor supply function:

-(W,/W,)(m - 1) = S(n, N) = o . (Bf9

(With utility nonseparable in money, the marginal rates of substitution would be functions of money balances also, and we would need to solve the choice problem completely for M/P and substitute the solution into the marginal rates of substitution to eliminate M/P as an argument.)

An appropriate social welfare function is the utility of our representative consumer. If we ignore the utility yield on money balances, a second-order approximation to the change in welfare that results when just one firm expands output and employment is

dw = Wldy + W,dn + (1/2)W,,dy’

+ (1/2)W33dn2 + W,,dydn

= Wldy + W,dn + (1/2)(Wu + W13F’)dy2

+ (I/~)(w, + W1JF’)dn2. W)

If we normalize the utility function by setting h = 1, we can use the identities (B7) and (BB) to show that W, = D, Wu + W,,F’ = D1, W, = -S, and W, + WJF’ = -Si, and substituting these into (B9) we can express the change in welfare as

dW = Ddy - Sdn + (1/2)D,dy2 - (1/2)&dn2,

which justifies Equation (25) in the main text.

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Similarly, a second-order approximation for a change in welfare when all firms expand output and employment together so that dy = DY and dn = dN is

dw = (w, + W&Y + (W, + w,)dN + (l/z)d(W, + WJdY

+ (1/2)d(w, + W,)dN WO)

Normalizing now by setting h = l/m and noting that in a symmetric expansion all firms charge the same price and wage so p = 1 and v = V, we use the identities (B7) and (B8) to show that W, + W, = 1, d(W, + W,) = 0, W, + W, = -S, and d(W, + W,) = -(S, + S,), and substituting these into (BlO) we can express the change in welfare as dW = dY - SdN - (1/2)(S, + !&)dN, which justifies Equation (28) in the main text.

Appendix C The model formally assumes a Coumot-Nash equilibrium. Each

firm chooses its output (y,) taking all other firms’ outputs (Y) as given. When there are costs to adjusting prices, however, the Bertrand- Nash equilibrium is more sensible. Each firm chooses its price (P,) taking all other firms’ prices (P) as given. Fortunately, Coumot and Bertrand equilibria coincide in the limit as the number of firms becomes large.

To see this, consider the inverse demand function:

Pi/P = D(y,, Y) . KU

The Cournot firm, assuming Y constant, faces a trade-off between output and (relative) price given by

@‘i/P) ~

dyi 7 = WY, Y) . Ka

The Bertrand firm, on the other hand, assuming P constant, faces a trade-off:

d(PiIP) 4i p = WY> Y) t WY> Y) 2, .

1 P (C3)

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Nicholas Rowe

The difference between (C2) and (C3) depends on the term &‘/&/P-how a change in the output of one firm will affect the output of other firms on average if they hold their prices constant. This term should approach zero as the number of firms becomes large. To see this, assume there are m firms, and that the aggregate demand function is given by

(m - l)PY + Pigi = MV , (C4)

where V is the (fixed) velocity of circulation of money. Rearranging (C4) yields

Y = [l/Cm - l)I[MVIP - yJ%i, VI * (C5)

Differentiating (C5) yields

which approaches zero in the limit as m approaches infinity. Thus Bertrand and Cournot equilibria are the same with a large number of firms.

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Do small menu costs cause bigger and badder business cycles in monopolistic macroeconomies?

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