New Keynesian monopolistic competition and objective demand

Journal of Mathematical Economics 43 (2007) 153–173

New Keynesian monopolistic competition and objective demand夽 Gerd Weinrich∗ Catholic University of Milan, Largo Gemelli 1, 20123 Milano, Italy Received 15 July 2005; received in revised form 5 October 2005; accepted 22 August 2006 Available online 4 January 2007

Abstract The New Keynesian model of monopolistic competition by Blanchard [Blanchard, O., Kiyotaki, N., 1987. Monopolistic competition and the effects of aggregate demand. American Economic Review 77 (4), 647– 666] is reformulated according to an objective demand approach making the behavior of all agents fully rational. The revised model is compared with the original model in terms of prices, quantities and welfare. Working with the revised model enhances the validity of the menu-cost argument and, different from the original model, implies that price rigidity is increasing in market concentration. © 2006 Elsevier B.V. All rights reserved. JEL classification: D43; D50; E12 Keywords: New Keynesian Economics; Monopolistic competition; Objective demand

1. Introduction The model of monopolistic competition as presented by Blanchard and Kiyotaki (1987) (BK for short) has been of central importance in New Keynesian macroeconomic theory. It has brought together, in a macroeconomic setting, monopolistic competition and menu cost so as to work out in a very clear and transparent way the relevance of monopolistic competition for an understanding of the effects of aggregate demand movements on economic activity. All the same, New Keynesian 夽 This paper has greatly benefitted from valuable and helpful comments and suggestions by anonymous referees. Of

course, the responsability for all remaining shortcomings is the author’s only. ∗

Tel.: +39 02 72342728, fax: +39 02 72342671. E-mail address: [email protected] (G. Weinrich). URL: http://www.unicatt.it/docenti/weinrich (G.Weinrich).

0304-4068/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.08.003

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economics has been criticized on the grounds that it be too much of a partial-equilibrium nature.1 In particular, two facts are not taken into account by the monopolistically competitive firms (on the goods markets) and households (on the labor markets) in that theory. First, they neglect that, if they change their price/wage, this will affect the income of the consumers and thus shift their demand curves. Second, they disregard the effect a change in their price/wage has on the price/wage level. However, for a model of such an importance as BK’s, it is crucial to know what are the implied changes if all agents are supposed to be fully rational or, in other words, if they maximize against objective demand curves. An objective demand structure in a general framework has been proposed by B´enassy (1988).2 Since prices are set individually and strategically by the agents in the economy, they are not necessarily compatible with market clearing on all markets. As a consequence, an allocation rule must be adopted which is capable of assigning feasible trades even when some agents are rationed. The corresponding appropriate concept is that of equilibrium with quantity rationing or fixprice equilibrium.3 Thus to any vector of prices and wages p and vector of exogenous parameters α is associated an equilibrium with rationing E(p, α) which specifies transactions and, inter alia, payoffs Ui (E(p, α)) for all agents i = 1, 2, . . .. Each agent i is assumed to know the mapping E(·, α) and, controlling the price pi , an equilibrium with price makers is a vector p∗ such that, for all i, p∗i maximizes Ui (E(·, p∗−i , α)). As the agents know the precise allocation corresponding to any p, they maximize against an objective demand. Thus agents are completely rational but the informational requirement is quite demanding and B´enassy himself does not give an example for his approach.4 In the BK-model, by contrast, each agent has a payoff-function Vi (pi , p, ¯ αi ), where p¯ = φ((pk )k ) is the aggregate price level and αi is a vector of parameters which is perceived as being exogenous from individual agent i’s point of view. Typical components of αi are the stock of money M and the aggregate wealth I. An equilibrium p∗ is obtained when, for all i, p∗i maximizes Vi (·, p, ¯ αi ) and p¯ = φ((p∗k )k ), but agents ignore their influence on p. ¯ This is common for Chamberlinian monopolistic competition (Chamberlin, 1933) but, in the general equilibrium framework with redistribution of profits of New Keynesian economics, a further element is neglected, namely that, to obtain again a feasible allocation after a change in pi , the component I in the vector αi has to be adjusted, too. Following up on Yang and Heijdra (1993) and Dixit and Stiglitz (1977, 1993), d’Aspremont et al. (1996), have shown that, when firms are supposed to take into account the above effects,

1 As B´ enassy (1993, p. 758)writes: “While many of these ideas are worth pursuing, a problem with this literature (noted by Gordon (1990)) is that for the most part these insights are of a partial equilibrium nature. But, as we have seen in this article, many interesting results (and certainly most of “true Keynesian” insights) come from “spillover” effects between several non-clearing markets in a full general equilibrium framework, such as the one we provided. Thus it would certainly be worthwhile to integrate the most relevant “New Keynesian” insights into the general equilibrium approach developed here.” 2 Earlier modelling of monopolistic competition in a general-equilibrium context includes Negishi (1961), Marschak and Selten (1974) and Nikaido (1975). These contributions suffer, however, from the one or the other shortcoming: Negishi works with subjective demand, Marshak and Selten assume that price makers do not sell to each other, and Nikaido assumes a Leontief economy. Stahn (1999) generalizes Nikaido’s technology assumptions but has to allow for randomization to avoid indeterminacies. 3 See e.g. B´ enassy (1975), Dr`eze (1975) and B´enassy (1988). 4 It is true that B´ enassy (1993, Section IV.C) contains an example of an economy with price makers but it works with subjective demands. B´enassy writes that “. . . the same results are obtained with objective demand curves (B´enassy, 1987, 1990)” but these contributions do not present worked-out examples.

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the corresponding objective demands of households are less elastic than the demands in BK.5 However, these authors do not investigate households’ wage-setting behavior subject to objective labor demand by firms, and thus it is not clear whether their analysis is compatibel with full rationality of all agents in the economy. A central issue in BK’s model regards the allocation after a change in the money stock M but without adjustment of prices; it involves price rigidity due to menu cost and second-order versus first-order losses in utility and welfare. A question is whether, and possibly how, BK’s results regarding the relevance of menu costs change when demand is objective. From these considerations follow the main purposes of the present paper. The first one is to fit the BK-model into the structure of B´enassy by determining the behavior both of firms (as price setters) and households (as wage setters) subject to objective demands by households for consumption goods and by firms for labor services, respectively. The decisive observation here is that, because any price setter chooses his price above marginal cost, at that price he would like to expand his production and sales. He cannot do so since the demand function for his product is downward sloping and thus de facto acts as a rationing device. In short, a monopolist can be seen as a supply-rationed agent. This establishes the link to B´enassy’s approach. Next, the fact that BK’s model is different from the version with objective demand does not say how far both models are away one from the other. Confronting them yields of course that at the limit, as the numbers of firms and households tend to infinity, they coincide. For a finite number of agents, however, the difference between the two models and, as a consequence, between the corresponding allocations, profits and welfare levels, may be substantial. We also confirm d’Aspremont et al. (1996) in that the objective demand of households for consumption goods is less elastic than the corresponding demand in BK. It is true that, on the one hand, the income effect is negative: when pi is increased, revenue pi Yi decreases as the demand curve is elastic at the point chosen by the firm. Thus the reduction in quantity demanded tends to be more marked than without taking this effect into account. On the other hand there is the positive effect of the single firm’s price increase on the aggregate price level. It turns out that this price level effect dominates the income effect. More importantly, we investigate the impact of objective demand on the validity of the menucost argument. We show that it is to diminish the relative losses of non-maximizers versus maximizers, and thus to reinforce the relevance of menu costs. Intuitively this is due to the above mentioned reduction in the demand curve’s elasticity: by not adjusting a firm foregoes a smaller gain in profit when demand is less elastic. Thus BK’s main results involving price rigidities become more significant when objective demand is assumed in their model. Having derived the fully rational version of BK’s model allows us also to extend previous work by Stigler (1947), Carlton (1986) and Rotemberg and Saloner (1987) on the relationship between price rigidity and the market power of firms in an industry. We show that, as suggested by the above authors, this relationship is positively monotonic, but only in the version of the model with objective demand. An ultimate contribution of the present paper is that, as we adapt BK’s model-specification to the case of objective demand, we present a completely worked-out example (and, to our best knowledge, the first) of B´enassy’s approach.

5 This follows from Eq. (15) on p. 626 of d’Aspremont et al. (1996) by setting σ(q) = 1 which corresponds to the case considered in BK.

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The remainder of the paper is organized as follows. In Section 2 we present the model, referring mostly to BK. In Section 3, starting as a point of reference from an equilibrium with supply rationing as a most basic and weak allocation rule, we first derive households’ wage-setting behavior subject to objective labor demand by firms and then introduce successively three further equilibrium concepts such that the first one corresponds to BK’s and the third to the one by B´enassy. This allows us to work out precisely their similarities and differences. Section 4 deals with the derivation of households’ objective demand for consumption goods and its elasticity. In Section 5 we show how to determine prices and quantities for each of the equilibrium concepts introduced in Section 3 and rank these equilibria and the perfectly competitve equilibrium with respect to price, nominal and real wage, output, employment, profit and welfare. We illustrate all this by a simple, completely worked-out example. In Section 6 we show that using objective demand enhances the relevance of menu cost and permits to prove that price rigidity is monotonic in market concentration. Finally, Section 7 contains concluding remarks. All proofs are collected in Appendix. 2. The model Since the aim of this paper is to compare New Keynesian monopolistic competition a` la Blanchard-Kiyotaki with the objective demand curve approach of B´enassy, the elements of the model are specified as in BK. In particular, the utility function of household j = β 1, . . . , n is Uj = (m1/(1−θ) Cj )γ (Mj /P)1−γ − Nj , where m denotes the number of firms, Cj =  (θ−1)/θ θ/(θ−1) ) composite consumption, Mj money holding, Nj labor, P the aggregate ( m i=1 Cij price level, and θ > 1, γ ∈ (0, 1) and β ≥ 1 are parameters. Then households’ combined demand for good i = 1, . . . , m, Yid = nj=1 Cij , is, according to Eq. (A1) in BK, Yid

γ = mP



Pi P

−θ

I =: Di (Pi , P, I)

(1)

Under the assumption of constant marginal disutility of labor, utility can be written as  μ(Wj Nj + m i=1 Vij + Mj ) Uj (Nj , Wj ) = − Nj , μ = γ γ (1 − γ)1−γ P (BK, (A14),  (A15), with β = 1) where Wj is the wage rate for the type of labor that offers 6 household j, m i=1 Vij is its profit income and Mj its money endowment. From this follows that ∂Uj /∂Nj > 0 iff Wj > P/μ. To produce Yi units of its good, firm i incurs a cost of n1/(1−σ) WYiα (BK, Eq. (A8)), where σ > 1 and α > 1 are parameters of the firm’s production function fi ((Nij )j ) =  (σ−1)/σ (σ/(σ−1))/α ) and W is the aggregate wage level.7 Thus the firm’s marginal cost ( nj=1 Nij is MC(Yi ) = αn1/(1−σ) WYiα−1 .

β

m

(A14) and (A15) in BK state, respectively, Uj = μIj /P − Nj , β ≥ 1, and Ij = Wj Nj + V + Mj . BK (p. i=1 ij 652) consider β = 1 a convenient special case where the characterization of the equilibrium is much simpler. This is true also for the present paper’s purposes because the quantities on the goods’ markets will be determined by the price setters only. n 7 To be precise, (A8) in BK is W N = nσ/(1−σ) WYiα but this is a misprint and the formula should read as stated j=1 j ij here. 6

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3. Equilibrium concepts Denoting with M the total money stock, we can now give the following Definition 1. A fixprice equilibrium with supply rationing (FESR) is a list (P1 , . . . , Pm , P, W1 , . . . , Wn , W, Y1 , . . . Ym , N1 , . . . , Nn , I, M) such that: (i) (ii) (iii) (iv) (v) (vi) (vii)

γ Pi −θ Yi = mP ( P ) I, for all i = 1, . . . , m; I= m Y + M; i=1 P i i 1−θ 1/(1−θ) ) ; P = (1/m m i=1 Pi α−1 1/(1−σ) WYi , for all i = 1, . . . , m; Pi > αn  Nj = m nσ/(1−σ) (Wj /W)−σ Yiα , for all j = 1, . . . , n; i=1  W = ( n1 nj=1 Wj1−σ )1/(1−σ) ; Wj > P/μ, for all j = 1, . . . , n.

Condition (i) states that households’ demands for all goods are satisfied whereas (ii) requires that aggregate wealth equal the value of total production plus the money stock. This is an accounting identity which must be satisfied in any feasible allocation. Conditions (iii) and (vi) define the price and wage levels while (iv) and (vii) express that, at prevailing prices, firms would be willing to sell more goods and households to sell more labor. Thus there is supply rationing on all markets. Finally, Eq. (v) specifies the transaction levels on the labor markets as the aggregates of labor demands of firms. It follows from the last equation in BK preceding (A8) which we report here for further reference: Nij = nσ/(1−σ) (Wj /W)−σ Yiα

(2)

A FESR fulfills the minimum requirements for an allocation to be reasonable: it is feasible, consistent and such that all agents behave rationally according to their preferences, given prices and wages. Since due to (iv) prices exceed marginal cost, firms are willing to produce and sell the quantities Y1 , . . . , Ym . Symmetrically, since at a FESR households’ marginal utility of working is positive (condition (vii)), they are willing to exchange the amount of labor demanded by firms. Thus, at the prevailing prices, no agent has an incentive to deviate from his action described by the FESR. A limiting case of a FESR is the symmetric perfectly competitive equilibrium which is obtained replacing the inequalities (iv) and (vii) by equalities and setting Pi = P and Wj = W for all i and j. In particular this determines a firm’s competitive output as 1/(α−1)  μ Yi∗ = (3) αn1/(1−σ) which, inserted in (ii) and (i), yields the competitive price  1/(1−σ) 1/(α−1) αn M γ ∗ (4) P = 1−γ μ m A FESR is not necessarily an equilibrium when prices are allowed to be chosen by firms and wages by households. For both, several possibilities can be envisioned, depending on the level of rationality with which firms and households are supposed to act. We here consider three levels. The first and lowest level assumes that firms and households consider the direct effects only. This means that firms maximize profit taking into account the effect of Pi on Yi as expressed

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by condition (i) whereas households maximize utility caring only for the effect of Wj on Nj as given by condition (v). In that case we shall speak of a FESR as a New Keynesian monopolistic competition equilibrium (NK). It is the equilibrium concept used by BK. At the second level firms in addition take into account the effect of their price choice, via condition (ii), on aggregate wealth I while households are also aware of the fact that their wage choice influences firms’ profits and thus their profit income, in addition to their labor income. That situation will be called monopolistic competition equilibrium with objective demand (MC) and corresponds to the general-equilibrium version with objective demand of Chamberlinian monopolistic competition, i.e. where agents neglect their influence on the price level and on the wage level. Finally, in a situation of oligopolistic equilibrium with objective demand (OL) firms and households will be completely rational considering the two above effects plus the third one of the choice of their individual price/wage on the price level/wage level, respectively. This is corresponding to B´enassy (1988). More precisely, starting with households their behavior is summarized as follows, where the case NK is BK’s wage formula (A16). Lemma 1. In a symmetric FESR of type T ∈ {NK, MC, OL} household j’s wage is WjT = λT P where λNK = λOL =

σ σ > λMC = (σ − 1)μ (σ − 1 + 1/n)μ

(5)

When a household takes into account the negative effect of an increase of its wage rate on firms’ profits, it behaves more moderately, i.e. WjMC < WjNK . When, in addition, it is aware that the wage increase also increases the wage level, it knows that the relative wage Wj /W increases by less than it would do otherwise, and therefore WjOL > WjMC . It turns out that both effects cancel each other, i.e. WjOL = WjNK . Regarding firms, observe that, in a neighborhood of a given FESR, condition (i) defines a map D=

m 

Di : ((P1, P, I), . . . , (Pm , P, I)) −→ (Y1 , . . . , Ym )

i=1

Conditions (i) and (ii) together define, under appropriate regularity conditions, a map E = (E1 , . . . , Em , Em+1 ) : (P1 , . . . , Pm , P, M) −→ (Y1 , . . . , Ym , I) (the m + 1 variables Y1 , . . . , Ym and I are determined as the solution of the m + 1 equations given in (i) and (ii)). This map takes account, in addition to the direct price effects represented by the map D, of the income effects of price changes as I is determined endogenously. Finally, conditions (i) to (iii) define, under appropriate regularity conditions, a map F = (F1 , . . . , Fm , Fm+1 , ) : (P1 , . . . , Pm , M) −→ (Y1 , . . . , Ym , I, P) where is given by condition (iii).8 Therefore, Fi (P1 , . . . , Pm , M) = Ei (P1 , . . . , Pm , (P1 , . . . , Pm ), M) ∀i = 1, . . . , m + 1 8 The regularity conditions will be seen to be satisfied in the special cases we will be considering below; thus there is no need to spell them out explicitly at the present stage.

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and F embodies the direct price effect, the income effect and the price-level effect. Now denote with Vi (Pi , Yi ) = Pi Yi − n1/(1−σ) WYiα

(6)

firm i’s profit. Then a FESR (P1 , . . . , Pm , P, W1 , . . . , Wn , W, Y1 , . . . Ym , N1 , . . . , Nn , I, M) is of type • NK iff Wj = WjNK for all j = 1, . . . , n and, for all i = 1, . . . , m, Pi = arg max Vi (Qi , Yi ) s.t. Yi = Di (Qi , P, I); Qi

• MC iff Wj = WjMC for all j = 1, . . . , n and, for all i = 1, . . . , m, Pi = arg max Vi (Qi , Yi ) s.t. Yi = Ei (Qi , P−i , P, M); Qi

• OL iff Wj = WjOL for all j = 1, . . . , n and, for all i = 1, . . . , m, Pi = arg max Vi (Qi , Yi ) s.t. Yi = Fi (Qi , P−i , M). Qi

Equilibria of type NK are the ones considered in BK and in B´enassy (1993, IV.C). When in BK the stock of money is varied but prices are not adjusted, the resulting allocation corresponds to none of the above types of price-setting equilibria. However, it still is a FESR – at least for not too large variations in M – which in turn ensures its feasibility (and which is a further reason why we have introduced that concept). In B´enassy (1988) and B´enassy (1993, III.C) models with an OL-type equilibrium are studied. Our next aim is to compare the three types of price-setting equilibria and the perfectly competitive equilibrium with respect to prices, quantities and welfare. The first step towards this end is to determine objective demand as faced by firms. 4. Objective demand Inserting (ii) of the definition of a FESR into (i) and solving for Yi yields ⎛ ⎞ (γ/m)(Pi /P)−θ ⎝ 1

M⎠ Yi = , for all i = 1, . . . , m P k Yk + 1 − (γ/m)(Pi /P)1−θ P P k =i

Although we shall limit ourselves to the consideration of symmetric equilibria, we must keep firm i, when investigating its optimal pricing policy, conceptually distinct from the other firms. This means that we set Pk = Pl and Yk = Yl for all k, l = i. Then the above system of m equations can be written as a system in two equations, namely   1 (γ/m)(Pi /P)−θ M Yi = (m − 1)P (7) Y + k k 1 − (γ/m)(Pi /P)1−θ P P   Pi (γ/m)(Pk /P)−θ M Yi + (8) Yk = 1 − ((m − 1)γ/m)(Pk /P)1−θ P P

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Inserting Yk into the expression for Yi yields the following result: Proposition 1. The objective demands for firm i’s output when all other firms are among themselves symmetrical are  −θ γ M/m Pi MC (9) Yi = 1−θ 1−θ P 1 − (γ/m)(Pi /P) − (m − 1/m)γ(Pk /P) P and, with  1 1−θ m − 1 1−θ 1/(1−θ) P= P Pk + =: φ(Pi , Pk ) m i m −θ  M/m γ Pi · YiOL = φ(Pi , Pk ) 1 − γ φ(Pi , Pk ) 

(10)

From (9) it is clear what are the regularity conditions alluded to before: γ, m, Pi , Pk and P must be such that the denominator in (9) is positive. However, as is shown in the proof of Proposition 1, at an equilibrium with Pk = Pl for all k, l = i, it becomes 1 − γ and thus is positive anyway as 0 < γ < 1. Comparing (9) with the demand derived in BK, namely  −θ γ Pi M/m NK · (11) Yi := Di (Pi , P, M/(1 − γ)) = P 1−γ P (see Eqs. (A5) and (A17) in BK or, equivalently, (1)), it is evident that BK’s demand is different. However, since limm→∞ YiOL /YiNK = 1, the indirect general-equilibrium effects disappear as the number of firms tends to infinity. At this point it is natural to ask which of the demand curves is more elastic. Writing, for T ∈ {NK, MC, OL}, the elasticities as εT = −(∂YiT /∂Pi )/(YiT /Pi ), the answer is as follows: Proposition 2. The elasticities of the demand curves at a symmetric equilibrium derive from ε=θ+

1 1 1 γ (θ − 1) − (θ − 1) 1−γ m 1−γ m

(12)

where the first term represents the direct price effect, the second the income effect and the third the effect via the price level. Therefore εMC = θ +

γ 1 1 (θ − 1) > εNK = θ > εOL = θ − (θ − 1) 1−γ m m

(13)

The expressions in the above proposition can be interpreted as follows. The objective demand MC is more elastic than the NK-demand. This can be explained from the fact that in case MC the firm takes into account that a price rise, say, decreases the quantity demanded Yi and also the aggregate wealth I = Pi Yi + k =i Pk Yk + M. This is in turn due to the fact that Pi Yi decreases as the demand curve has elasticity bigger than one. When the effect of the price rise on the aggregate price level is taken into account, too, the overall effect is positive: the demand in case OL is less elastic than in case NK. The fact that an increase in its own price Pi raises the price level P means that the increase in firm i’s relative price is less big than in case P remains constant. Thus the decrease in quantity demanded is less accentuated. This effect turns out to dominate the negative income effect.

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5. Equilibrium prices and quantities The elasticities found in the previous section consent to determine the corresponding equilibrium prices and quantities. Eq. (6) yields the first-order condition Yi +

∂Yi {Pi − αn1/(1−σ) WYiα−1 } = 0 ∂Pi

¯ = M/m, from (3), (4), (9) and (11) at a symmetric Writing the money stock per firm as M equilibrium YiT (P T ) =

¯ γ M 1 − γ PT

(14)

for all T ∈ {∗, NK, MC, OL} whereas ¯ ∂YiT γM = −εT ∂Pi (1 − γ)(P T )2

(15)

Therefore, and using Lemma 1, the equilibrium price P T is determined by

   ¯ ¯ ¯ α−1 γ M γ M γM T T 1/(1−σ) T T P − αn =0 −ε λ P 1 − γ PT (1 − γ)(P T )2 1 − γ PT

(16)

Solving for P T yields Lemma 2. In a symmetric FESR of type T ∈ {NK, MC, OL} the firms’ price is 1/(α−1)  T γ ε 1/(1−σ) T ¯ M λ PT = αn 1 − γ εT − 1

(17)

Denoting with w = W/P the real wage we can now establish the following result. Proposition 3. Prices, nominal wages, real wages, quantities, profits and utilities are ranked across the T-equilibria, T ∈ {NK, MC, OL}, and the perfectly competitive equilibrium as follows: P ∗ < P MC < P NK < P OL ,

W ∗ < W MC < W NK < W OL ,

w∗ < wMC < wNK = wOL ,

YiOL < YiNK < YiMC < Yi∗ ,

Vi∗ < ViMC < ViNK < ViOL ,

UjOL < UjNK < UjMC < Uj∗

The above ranking of utilities confirms the standard wisdom that imperfect competition is welfare-worse than perfect competition. Nevertheless, in the present context this is not immediate since there are opposing effects at work: on the one hand, profit and wage incomes are highest in the OL-equilibrium and lowest in the perfectly competitive one. Moreover, less work in the OL-case reduces disutility. On the other hand, high prices for the consumption goods reduce consumers’ real wealth which in turn depresses market demands, production and consumption. It turns out that the latter effect dominates the former ones. Note also that the relationship between rationality and welfare is not monotonic: in both the cases MC and OL agents are more rational than in case NK but in MC their utility is higher than in NK whereas in OL it is lower.

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Table 1 The values of economic variables in different market forms T

P

W

W/P

Yi

Nj

Vi

Uj

∗ MC NK OL

2 6.35 8 12

4 24.19 32 48

2 3.81 4 4

5 1.57 1.25 0.83

0.125 0.012 0.008 0.003

5 6.99 7.5 8.31

0.38 0.15 0.12 0.08

Example 1. Suppose α = 2, γ = 1/2 (and thus μ = 1/2), θ = 2 and σ = 2. Then at a symmetric equilibrium YiT (P T ) = (M/m)/(P T ) for all T and PT =

ε T λT 2 M εT − 1 mn

with εNK = 2, εMC = 2 + 1/m, εOL = 2 − 1/m, λNK = λOL = 4 and λMC = 4/(1 + 1/n). Therefore an equilibrium of type NK yields the equilibrium price and quantity 16M n , YiNK = mn 16 For T = MC follows 8(2 + 1/m)M P MC = , (1 + 1/m)(1 + 1/n)mn P NK =

YiMC =

(1 + 1/m)(1 + 1/n)n 8(2 + 1/m)

while for the oligopolistic case one obtains P OL =

8(2 − 1/m)M , (1 − 1/m)mn

YiOL =

(1 − 1/m)n 8(2 − 1/m)

The formulas show that output is independent of the money supply (as it must be the case), and both price and output vary with the number of households and firms. For example for m = 2 and n = M = 20 one gets the numbers of Table 1. They illustrate clearly that strategic price- and wage-setting behavior leads to inferior welfare. It is true that households realize the highest real wage and the highest nominal profit income under oligopolistic competition but, since they are heavily rationed on the labor market and since the price level is high, this does not translate into high total real income and utility. Note finally that the case OL represents a completely worked-out example (and, to our best knowledge, the first) to B´enassy’s (1988)-model. 6. Objective demand, menu cost and market concentration An important issue in BK is that small menu cost may prevent firms from adjusting their prices after a change in the money stock. As originally pointed out by Akerlof and Yellen (1985a,b), this is due to the fact that, by not adjusting, a firm suffers a second-order loss only or, writ¯ the full equilibrium response to a change in M, ¯ ing profit as function Vi = Vi (Pi , P, W, M), ¯ ¯ dVi /dM, is equal to the partial response ∂Vi /∂M that comes forth when prices and wages are not adjusted. This reasoning holds also in the present context, independently of the type of equilibrium considered. Where there may be a difference, however, is in the magnitude of the loss because a non-infinitesimal change from M to (1 + δ)M, say, implies that the profit of nonresponders is smaller than that of responders. By how much it is smaller varies with the type of equilibrium.

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To explore this issue we assume that, before the shock occurs, all households set the equilibrium wage Wj = W and all firms charged the equilibrium price Pi = P and produced the equilibrium quantity Yi = Y . After the shock, all households increase their wage rate to (1 + δ)W while any firm behaves in one of two possible ways: either it adjusts its price to (1 + δ)P or it sticks with the old price P. Let ρ ∈ [0, 1] be the share of firms that adjust. Then by (iii) of Definition 1 the new aggregate price is Pnew = [ρ(1 + δ)1−θ P 1−θ + (1 − ρ)P 1−θ ]1/(1−θ) which implies Pnew = [ρ(1 + δ)1−θ P 1−θ + (1 − ρ)P 1−θ ]1/(1−θ) P −1 P = [ρ(1 + δ)1−θ P 1−θ P θ−1 + (1 − ρ)P 1−θ P θ−1 ]1/(1−θ) = [ρ(1 + δ)1−θ + (1 − ρ)]1/(1−θ) Next consider the quantities demanded for firms’ products. If a firm has not adjusted, it faces the demand −θ −θ   ¯ ¯ P γ (1 + δ)M γ M P P Yin = = (1 + δ) Pnew 1 − γ Pnew Pnew 1 − γ P Pnew 1−θ  P = (1 + δ)Y Pnew Therefore by (6) and Lemma 1 its profit will be Vin = PYin − n1/(1−σ) (1 + δ)WYinα 1−θ  (1−θ)α  P P = P(1 + δ)Y − n1/(1−σ) (1 + δ)λP(1 + δ)α Y α Pnew Pnew 1−θ  (1−θ)(α−1)   P P 1/(1−σ) α α−1 1−n = (1 + δ)PY λ(1 + δ) Y Pnew Pnew  −1 = (1 + δ)PY ρ(1 + δ)1−θ + (1 − ρ) × [1 − n1/(1−σ) λ(1 + δ)α Y α−1 [ρ(1 + δ)1−θ + (1 − ρ)]1−α ] Using (14) and (17),  n

1/(1−σ)

λ(YiT )α−1

=n

1/(1−σ)

λ

¯ γ M 1 − γ PT

α−1

=

εT − 1 αεT

(18)

and thus we obtain Vin = (1 + δ)PY [ρ(1 + δ)1−θ + (1 − ρ)]−1   εT − 1 1−θ 1−α [ρ(1 + δ) + (1 − ρ)] × 1 − (1 + δ)α αεT

(19)

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Similarly, a firm that does adjust its price faces the demand  1−θ   ¯ (1 + δ)P −θ γ (1 + δ)M P −θ 1−θ Yi = = (1 + δ) Yin = (1 + δ) Y Pnew 1 − γ Pnew Pnew 1−θ  P = Y (1 + δ) Pnew This yields the profit (1−θ)α P Vi = (1 + δ)PY −n (1 + δ)λPY (1 + δ) Pnew

1−θ  (1−θ)(α−1)   P P 1/(1−σ) α−1 1−n (1 + δ) = (1 + δ)PY (1 + δ) λY Pnew Pnew 

P (1 + δ) Pnew

1−θ



1/(1−σ)

α

Now (1 + δ)

P = (1 + δ)[ρ(1 + δ)1−θ + (1 − ρ)]1/(θ−1) Pnew = [(1 + δ)θ−1 ρ(1 + δ)1−θ + (1 + δ)θ−1 (1 − ρ)]1/(θ−1) = [ρ + (1 + δ)θ−1 (1 − ρ)]1/(θ−1)

This, together with (18), yields Vi = (1 + δ)PY [ρ + (1 + δ)

θ−1

(1 − ρ)]

−1

  εT − 1 θ−1 1−α × 1− [ρ + (1 + δ) (1 − ρ)] αεT (20)

Therefore, for any T ∈ {NK, MC, OL} 

Vin Vi

T

[ρ(1 + δ)1−θ + (1 − ρ)]−1 [1 − (1 + δ)α (εT − 1)/(αεT ) =

× [ρ(1 + δ)1−θ + (1 − ρ)]1−α ] [ρ + (1 + δ)θ−1 (1 − ρ)]−1 [1 − (εT − 1)/(αεT )

(21)

× [ρ + (1 + δ)θ−1 (1 − ρ)]1−α ] is the ratio of profits of a non-responder to those of responders if the latters’ share is ρ. It is easy to check that the right-hand side is decreasing in εT . In view of εOL < εNK < εMC this proves the following result. Proposition 4. The relative loss of not adjusting prices and wages is higher under monopolistic competition with objective demand than under New Keynesian monopolistic competition than under oligopoly with objective demand, i.e.  MC  NK  OL Vin Vin Vin < < Vi Vi Vi An immediate consequence is Corollary 1. In case of constant marginal disutility of labor (β = 1), introducing full rationality into the model by Blanchard and Kiyotaki (1987) enhances the validity of the menu-cost argument.

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We can use the analytical results derived here to deal with a further question which is whether price rigidity is monotone in industry concentration. Stigler (1947) and Carlton (1986) have suggested that this be the case and Rotemberg and Saloner (1987) have formally shown it to hold for a monopoly versus a duopoly. In the present general-equilibrium context the result holds only if we allow for full rationality, and then it is immediate: from (13) εT is decreasing/constant/increasing in the number of firms m for T = MC/NK/OL, respectively, while VinT /ViT is decreasing in εT . Since higher menu cost are needed to prevent adjusting the price when the loss from not doing so is higher, the following holds. Proposition 5. Under oligopoly with objective demand (i.e. with full rationality), the smaller is the number of firms, the more rigid are prices. The reverse holds under monopolistic competition equilibrium with objective demand whereas under New Keynesian monopolistic competition the number of firms is irrelevant for the degree of price rigidity. The above results are valid for any given ρ ∈ [0, 1]. Looking now in a more accurate way at the decision making of a firm as to whether or not to adjust its price, we have to acknowledge that it may take into account that its decision has an impact on ρ. Therefore let ρ now denote the share of adjusting firms if the firm under consideration adjusts its price, too. Consequently, if it does not, this share decreases to ρ − 1/m. Therefore, for evaluating the option of not adjusting, a firm under OL has to use (21) with, in the numerator, ρ − 1/m in place of ρ, whereas in case NK it ignores its impact on the price level, and thus on ρ. We can formally capture this by setting in that case m = ∞. Example 2. With the numbers of Example 1 and δ = 0.1 one gets from (21) and (13) for Vin /Vi the function v(ρ, m) =

[(ρ − 1/m)1.1−1 + (1 − ρ + 1/m)]−1 [ρ + 1.1(1 − ρ)]−1 1 − 1.12 [(1 − 1/m)/(2(2 − 1/m))][(ρ − 1/m)1.1−1 ×

+ (1 − ρ + 1/m)]−1 1 − [(1 − 1/m)/(2(2 − 1/m))][ρ + 1.1(1 − ρ)]−1

This is illustrated in Fig. 1 in ρ − (Vin /Vi )-plane. The four decreasing lines correspond, from top to bottom, to m = 2, m = 5, m = 22 (all OL-cases) and m = ∞ (i.e. NK). Specifically, v(1, 2) = 0.9915, v(1, 22) = 0.980053, v(1, 23) = 0.979992, v(1, ∞) = 0.9786 and v(0.915, ∞) = 0.98. Thus, if a firm’s menu costs are, say, 2% of profit, an increase in the money stock by 10% under NK causes it to adjust its price provided it expects that at least 91.5% of all firms do the same. Under OL, with up to 22 firms, it does not, even if all others did. The smaller is the share of firms that adjust, the smaller are menu costs needed. This is due to the fact that, with decreasing ρ, the average price Pnew decreases which is more harmful to the profits of adjusting firms than to those of non-adjusting ones. It is true, however, that it is always convenient for a firm to adjust when there are no menu costs at all, even if no other firm adjusts. This holds for NK as well as for OL.

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Fig. 1. Graphs of the function v(·, m) for, from top to bottom, m = 2, 5, 22 and ∞.

We can also calculate by how much, in percentages, menu costs needed to prevent an adjustment of prices are higher under NK than under OL: this is given by   1 − v(ρ, ∞) − 1 × 100% 1 − v(ρ, m) For example when ρ = 1, it is 151.7% for m = 2 and 7.3% for m = 22. 7. Concluding remarks The conclusions we can draw are as follows. The model presented in Blanchard and Kiyotaki (1987) can be modified so as to be consistent with objective demand as proposed by B´enassy (1988). In this way, all indirect equilibrium effects are taken into account by economic agents. The modified version shows that, although the equilibrium in BK can be approximated as a limiting case by an equilibrium with objective demand when the number of firms tends to infinity, for a finite number of firms there may be significant differences. We have been able to rank the four types of equilibrium considered – perfectly competitive, monopolistically competitive with BK-demand, monopolistically competitive with objective demand and oligopolistic with objective demand – with respect to prices, wages, output, employment, profit and welfare. When, starting from any equilibrium, the stock of money changes, the relative losses of firms which do not adjust prices in comparison to those which do are smaller with oligopolistic objective demand than with BK’s demand. Thus smaller menu cost are needed to prevent price adjustment than in BK’s model which in turn enhances the relevance of the menu-cost argument. Therefore all considerations valid in the BK-model – like the effects of aggregate demand movements on economic activity – not only remain so with the introduction of full rationality but they are reinforced. In addition we have shown that full rationality is needed, however, to obtain the result that price rigidity increases with the market power of firms. Finally, the oligopolistic version of BK’s model is a completely worked-out example of an equilibrium with objective demand as presented by B´enassy (1988). In this paper we have worked with specific forms of the utility and production functions. This, together with the symmetry assumptions, has enabled us to derive closed-form solutions and to prove existence and uniqueness of equilibria. On the other hand, to obtain results similar to the

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ones we have presented here in a general setting of imperfect competition in general equilibrium seems quite difficult if not impossible. In fact, the general definition of equilibrium as indicated in the Introduction involves the notion of fixprice equilibrium, E(p, α), as defined in B´enassy (1988). While existence of E(p, α) for any p may be assured under the regularity conditions given by B´enassy, uniqueness of it is a more problematic point. Stahn (1993) has argued that the conditions given in B´enassy (1988) may not be sufficient for uniqueness in which case it is not clear how to determine the choice of an optimal price by each single agent. The New-Keynesian specification of utility and production functions used in the present paper is one possibility to overcome this problem.9 A final point to note is that our use of the terminology “full rationality” refers to rationality within the class of utility and production functions assumed here. In particular, these functions are “one-shot” although they are employed for the study of intertemporal phenomena. Full rationality in a wider sense would require to set up a dynamic optimization problem.10 An explicit dynamic general-equilibrium model of an economy with menu costs and a continuum of NK-type monopolistically competitive firms, where each firm’s productivity and the money supply are exposed to random shocks, has been provided by Danziger (1999). Appendix Proof of Lemma 1. From Eqs. (A14), (A15), (A10) and (A11) in BK11 it follows that, for β = 1 and given money endowment Mj , a consumer’s utility is Uj =

μIj − Nj P

with Ij = Wj Nj +

m

i=1



Wj Vij + Mj , Nj = W

−σ

N n

and N = n1/(1−σ)

m

Yiα

(22)

i=1

Note that, since the production levels Yi are determined, for given P1 , . . . , Pm and M, by the first three conditions in Definition 1 only, they do not depend on wages, as long as these satisfy the inequalities (iv) and (vii). Thus also N is independent of Wj . 9

The New Keynesian approach to monopolistic competition with specific utility and production functions and fixed costs (Blanchard and Kiyotaki (1987), Section 4) can also be seen as one of several approaches to imperfect competition and non-convex technologies in general equilibrium. Without any claim to completeness, other approaches can be related to: many firms that are or become small relative to the aggregate economy (Novshek and Sonnenschein (1978), Hart (1979)), a single monopolist embedded in an otherwise competitive economy (Heller (1993)), pricing rules (Bonnisseau and Cornet (1988)) and perceived demand with pricing rules (Dehez et al. (2003)). 10 As regards households, money as an argument of the utility function does not prove per se bounded rationality. As Grandmont (1983) has shown, under certain assumptions there exists an indirect utility index with money as an argument that summarizes the agent’s preferences for an n-period programming problem. m  m α m n 11 (A10) is N = −σ N/n, (A11) N = ( N = (W /W) W N )/W = n1/(1−σ) Y . j ij j i=1 i=1 j=1 j ij i=1 i

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Since ⎛ ⎛ ⎞⎞1/(1−σ)

1 W = ⎝ ⎝Wj1−σ + Wk1−σ ⎠⎠ =: W(Wj ), n k =j

⎛

Vij = νij ⎝Pi Yi − Wj Nij −



⎞ Wk Nik ⎠ =: νij Vi (Wj )

k =j

with Nij given by (2), inserting the above conditions into the consumer’s utility function it becomes     −σ −σ m Wj Wj N

N νij Vi (Wj ) + Mj /P − + Uj = μ Wj W(Wj ) n W(Wj ) n i=1

This yields as first-order condition μ P



+

N n



m

i=1

Wj W

−σ

dVi νij dWj

 − σWj



 +σ

Wj W

Wj W

−σ−1

−σ−1

W − Wj W  W2



W − Wj W  N =0 W2 n

which is equivalent to N n



Wj W

−σ 

μ σ + σW  (1 − σ) + P Wj



μ Wj 1 − P W W

 =−

m dVi μ

νij P dWj

(23)

i=1

If households ignore the influence of their wage choice both on the wage level and on firms’ profits, W  = dVi /dWj = 0, and (23) becomes Wj =

σ P = WjNK (σ − 1)μ

In a regime of monopolistic competition with objective demand households do take into account that their wage choice influences firms’ profits but they neglect the impact on the wage level. Thus W  = 0 whereas, from (2), dVi = −Nij = −nσ/(1−σ) dWj



Wj W

−σ

Yiα

Inserting in (23) and using νij = 1/n and (22) yields N n



Wj W

−σ 

μ σ (1 − σ) + P Wj



μ = nP



Wj W

−σ

N σ ⇔ Wj = P = WjMC n (σ − 1 + 1/n)μ

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169

In a regime of oligopoly with objective demand, using again (2), σNij W − Wj W 

σNik −Wk W  dVi = −Nij + Wj + Wk 2 dWj Wj /W W Wk /W W 2 k =j

    1 σ

= σ 1− Nik − 1 Nij − n n k =j

where we have also taken advantage of Wj = W and W  = 1/n which will be true in a symmetric equilibrium. Using moreover νij = 1/n, this yields

 m m n

σ

dVi 1

− νij = Nik + (σ − 1)Nij dWj n n i=1 i=1 k=1 

n m m

N σ

1 Nik + (σ − 1) Nij = − 2 − = n n n i=1 k=1 i=1 m where we have also used i=1 Nij = Nj = N/n for all j = 1, . . . , n. Therefore the right hand side of (23) becomes μN/(n2 P), and simplifying and multiplying it by Wj leads to μ σμ μ  σ (1 − σ) + − = −σ Wj P nP nP n which is equivalent to σ Wj = P = WjOL (σ − 1)μ It remains to show that the wages so determined give rise to supply rationing on the labor markets, i.e. satisfy condition (vii) of Definition 1. But this is immediate since, for T ∈ {NK,OL}, WT =

σ PT PT > (σ − 1)μ μ

as σ > 1 whereas W MC =

σ P MC P MC ≥ (σ − 1 + 1/n)μ μ



Proof of Proposition 1. From Eqs. (7) and (8) (γ/m)(Pi /P)−θ 1 − (γ/m)(Pi /P)1−θ     Pi (γ/m)(Pk /P)−θ 1 M M × + (m − 1)Pk Y + i P 1 − ((m − 1)γ/m)(Pk /P)1−θ P P P   (γ/m)(Pi /P)−θ ((m − 1)γ/m)(Pk /P)−θ Pi Pk ⇔ Yi 1 − 1 − (γ/m)(Pi /P)1−θ 1 − ((m − 1)γ/m)(Pk /P)1−θ P 2   Pk ((m − 1)γ/m)(Pk /P)−θ M (γ/m)(Pi /P)−θ M = + 1 − (γ/m)(Pi /P)1−θ P 1 − ((m − 1)γ/m)(Pk /P)1−θ P P   ((m − 1)/m)(γ 2 /m)(Pi Pk /P 2 )1−θ ⇔ Yi 1 − [1 − (γ/m)(Pi /P)1−θ ][1 − ((m − 1)γ/m)(Pk /P)1−θ ]

Yi =

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(γ/m)(Pi /P)−θ = 1 − (γ/m)(Pi /P)1−θ =

  ⇔

 ((m − 1)γ/m)(Pk /P)1−θ M +1 1 − ((m − 1)γ/m)(Pk /P)1−θ P

1 M (γ/m)(Pi /P)−θ 1 − (γ/m)(Pi /P)1−θ 1 − ((m − 1)γ/m)(Pk /P)1−θ P ⎧ ⎫ [1 − (γ/m)(Pi /P)1−θ ][1 − ((m − 1)γ/m)(Pk /P)1−θ ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − ((m − 1)/m)(γ 2 /m)(P P /P 2 )1−θ ⎬ i k ⇔ Yi ⎪ [1 − (γ/m)(Pi /P)1−θ ][1 − ((m − 1)γ/m)(Pk /P)1−θ ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

= YiMC



=  =

Pi P Pi P

Pi P

−θ −θ

−θ

M/m γ [1 − (γ/m)(Pi /P)1−θ ][1 − ((m − 1)γ/m)(Pk /P)1−θ ] P γ M/m 1−θ 1−θ [1 − ((m − 1)/m)γ(Pk /P) ][1 − (γ/m)(Pi /P) ] P − ((m − 1)/m)(γ 2 /m)(Pi Pk /P 2 )1−θ γ M/m 1 − (γ/m)(Pi /P)1−θ − ((m − 1)/m)γ(Pk /P)1−θ P

Finally, at an equilibrium with Pk = Pl for all k, l = i, using P = φ(Pi , Pk ) the denominator becomes    γ 1−θ m − 1 1−θ 1 1−θ m − 1 1−θ −1 1− + + =1−γ P γPk P Pk m i m m i m which yields the expression for YiOL .



Proof of Proposition 2. The claim regarding εNK is immediate from (11). Regarding εMC and εOL , write (9) as ¯ γ M Yi = A(Pi , P)−θ B(Pi , Pk , P) P γ Pi 1−θ Pk 1−θ (P ) − m−1 . Then with A(Pi , P) = Pi /P and B(Pi , Pk , P) = 1 − m m γ( P )    ¯ ∂Yi Pi ∂A γ M ∂A ∂φ ε=− = − −θA(Pi , P)−θ−1 + + A(Pi , P)−θ ∂Pi Yi ∂Pi ∂P ∂Pi B(Pi , Pk , P) P       ¯ ¯ ∂B ∂φ γ M γ ∂B ∂φ M + − × − + B(Pi , Pk , P)2 ∂Pi ∂P ∂Pi P B(Pi , Pk , P) P 2 ∂Pi

× =

A(Pi

Pi −θ , P) (γ/B(P

¯ i , Pk , P))(M/P)

     1 1 ∂φ θPi ∂A ∂B ∂A ∂φ ∂B ∂φ + Pi + + + A(Pi , P) ∂Pi ∂P ∂Pi B(Pi , Pk , P) ∂Pi ∂P ∂Pi P ∂Pi 

At a symmetric equilibrium Pi = Pk = P and thus ∂A 1 ∂φ ∂A ∂φ 1 + = − ∂Pi ∂P ∂Pi P P ∂Pi

G. Weinrich / Journal of Mathematical Economics 43 (2007) 153–173

and

171

  ∂B m−1 γ 1 1 ∂φ 1 ∂φ ∂B ∂φ + = − (1 − θ) − γ(1 − θ) + ∂P ∂Pi m P P ∂Pi m P ∂Pi ∂Pi

Therefore, and using A(x, x) = 1 and B(x, x, x) = 1 − γ ∀x > 0,     ∂φ ∂φ γ ∂φ 1 ∂φ + ε = θ 1− + + (m − 1) (1 − θ) −1 + ∂Pi ∂Pi 1−γ m ∂Pi ∂Pi =θ+

1 1 ∂φ γ (θ − 1) − (θ − 1) 1−γ m 1−γ ∂Pi

(24)

In case MC ∂φ/∂Pi is zero which yields εMC in (13). In case OL, φ(Pi , Pk ) = ((1/m)Pi1−θ + ((m − 1)/m)Pk1−θ )1/(1−θ) , and thus    −θ 1 1 1−θ m − 1 1−θ θ/(1−θ) 1 − θ −θ ∂φ 1 Pi Pi + Pk Pi = = ∂Pi 1−θ m m m P m which is 1/m at a symmetric equilibrium. Therefore (24) becomes (12) which, simplifying, yields εOL in (13).  Proof of Lemma 2. Eq. (16) is equivalent to   ¯ α−1 γ M T T 1/(1−σ) T 1 − ε + ε αn λ =0 1 − γ PT 1/(α−1)    ¯ α−1 ¯ γ M εT − 1 γ M εT − 1 ⇔ = ⇔ = εT αn1/(1−σ) λT 1 − γ PT εT αn1/(1−σ) λT 1 − γ PT ⇔ (17). To show that there is supply rationing on the goods’ markets note that this is equivalent to P T > αn1/(1−σ) W T (YiT )α−1     ¯ α−1 γ M γ ¯ α−1 T α−1 1/(1−σ) T M = αn1/(1−σ) λT P T ⇔ (P ) > αn λ 1 − γ PT 1−γ   α−1 1/(α−1)   T  γ ¯ α−1 γ ε 1/(1−σ) T 1/(1−σ) T ¯ M αn ⇔ λ > αn λ M 1 − γ εT − 1 1−γ ⇔

εT >1 −1

εT

Since εT ≥ 1 for all T the claim follows.



Proof of Proposition 3. From (13) and (5) 1 εMC λMC > εMC − 1 μ which implies P ∗ < P MC by (4) and (17). From (13) follows: εNK εOL εMC < < εMC − 1 εNK − 1 εOL − 1

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which together with (5) and (17) yields P MC < P NK < P OL . The ranking of wages is an immediate consequence of Lemma 1 and W ∗ = P ∗ /μ whereas the one of real wages follows from (5) and the one of quantities from (14). By (6) equilibrium profits are  α γ γ ¯ 1/(1−σ) T T α 1/(1−σ) T T 1−α T T T ¯α M M−n W (Yi ) = λ (P ) Vi = P Yi − n 1−γ 1−γ  1/(α−1) 1−α   T α γ γ γ ¯ ε 1/(1−σ) T 1/(1−σ) T ¯ ¯α M M = λ λ M−n αn 1−γ 1 − γ εT − 1 1−γ  T −1 ε γ ¯ γ ¯ 1/(1−σ) T λ M − n1/(1−σ) λT αn M 1−γ εT − 1 1−γ   T −1  ¯ γ M γ ¯ γ ¯ ε 1 = M− α M = α−1+ T T 1−γ ε −1 1−γ ε 1−γ α

=

From (13) follows ViMC < ViNK < ViOL . By (4) the competitive profit is similarly calculated as Vi∗ = (α − 1)

¯ γ M 1−γ α

which implies Vi∗ < ViMC . Regarding utilities, UjT = μIjT /P T − NjT where T ∈ {∗, NK, MC, OL},   γ M 1 T 1 1 T T T M+M = Ij = I = (mP Yi + M) = n n n 1−γ n(1 − γ) and, by (2), NjT

=

 mNijT

=n

This yields UjT

μM = n(1 − γ)

σ/(1−σ)



1 PT

m(YiT )α

=n

σ/(1−σ)



 −n

σ/(1−σ)

m

m

¯ γ M 1 − γ PT

γ M 1−γ m

α 

1 PT

α

α

1/(α−1)

The function Ax − Bxα , A, B > 0, α > 1, assumes its maximum for xˆ = (A/(αB)) whereas for x < xˆ it is increasing. Thus the ranking of utilities follows from that of prices provided it can be shown that 1/(α−1)  1 μM/(n(1 − γ)) ≤ P∗ αnσ/(1−σ) m((γ/(1 − γ))M/m)α But this follows from (4), γ < 1 and 1/(α−1)  μM/(n(1 − γ)) αnσ/(1−σ) m((γ/1 − γ))M/m)α 1/(α−1)  μ(1/(1 − γ))(M/m) 1 = = α/(α−1) ∗ P αn1/(1−σ) γ α ((1/(1 − γ))M/m))α γ



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