Monetary institutions, imperfect competition and employment outcomes

North American Journal of Economics and Finance 22 (2011) 131–148

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North American Journal of Economics and Finance

Monetary institutions, imperfect competition and employment outcomes George Chouliarakis a,b,∗, Mónica Correa-López a,c a b c

Department of Economics, School of Social Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK Centre for Growth and Business Cycle Research, University of Manchester, Oxford Road, Manchester M13 9PL, UK Economic Research Department, BBVA Research, P/ Castellana 81, 28046 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 9 March 2010 Received in revised form 28 December 2010 Accepted 5 January 2011 Available online 13 May 2011 JEL classification: D43 E24 E42 E52 Keywords: Monetary union Monetary policy Oligopoly Wage bargaining Employment

a b s t r a c t This paper explores the employment effects of strategic interactions between firms, trade unions and monetary institutions in the context of an imperfectly competitive macroeconomic model with right-to-manage bargaining. The results suggest that the employment effect of joining a monetary union is conditional upon the degree of monetary accommodation of the union-wide central bank, the degree of product market competition, and the relative bargaining power in wage-setting. In addition, the employment effect of a change in the degree of monetary accommodation of the domestic or the union-wide central bank is conditional upon the degree of product market competition and the distribution of bargaining power. © 2011 Elsevier Inc. All rights reserved.

1. Introduction It is widely accepted that the set of incentives and constraints that policy makers face when setting monetary policy has a significant effect on nominal variables such as mean inflation. In two famous papers, Kydland and Prescott (1977) and Barro and Gordon (1983) show that, under a discretionary

∗ Corresponding author at: Department of Economics, School of Social Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK. E-mail address: [email protected] (G. Chouliarakis). 1062-9408/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.najef.2011.01.001

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regime, the inability of policy makers to commit themselves to a low inflation rate leads to high average inflation with no gain in terms of employment. Furthermore, Rogoff (1985) demonstrates that the delegation of monetary policy to a conservative central banker can reduce the discretionary bias while Lohmann (1992) shows that the dynamic-inconsistency problem can be best resolved through an institutional arrangement in which the government grants partial independence to a conservative central banker while retaining the option to override the central bank’s decisions at a strictly positive but finite cost. The relevance of the design of monetary policy institutions for nominal variables, such as mean inflation, has been borne out by a number of empirical studies and, to a good extent, is beyond dispute.1 Yet relatively new and evolving research shows that the effect of monetary policy institutions is not confined to nominal variables. Building upon the insights of Bruno and Sachs (1985), Calmfors and Driffill (1988) and Soskice (1990) on the importance of wage-setting institutions on macroeconomic outcomes, a relatively new strand of the literature shows that, once one allows for strategic interactions between firms, trade unions and monetary policy institutions, a change in the degree of monetary policy accommodation, or the degree of central bank conservatism, or the delegation of monetary policy to a common central bank in the context of a monetary union, will have significant effects on real economic variables, such as equilibrium employment and output, too. In a seminal paper, Iversen (1998) shows that the relationship between the degree of centralization of wage bargaining and equilibrium unemployment is contingent upon the type of monetary regime. Cukierman and Lippi (1999) explore the macroeconomic effects of the interaction between the degree of central bank independence and the degree of centralization of wage bargaining in a framework in which sectoral unions are inflation averse and show that the relationship between real wages and the degree of centralization of wage bargaining depends upon the degree of central bank independence. Cukierman and Lippi (2001) argue that the creation of a monetary union may aggravate collective action problems in wage-setting as, due to the smaller weight that each union wage has in the determination of the union-wide inflation rate, trade unions become less aware of the inflationary consequences of their individual wage strategies. The aggravation of the collective action problem renders unions more aggressive in their nominal wage demands and, other things being equal, leads to higher inflation and higher unemployment. Coricelli, Cukierman and Dalmazzo (2001) examine the macroeconomic implications of the interaction between the degree of conservatism of the monetary policy maker and wage-setters. A more conservative central bank, which reacts to individual wage increases by tightening, induces wage restraint and increases employment. A straightforward implication of this result is that if the creation of a monetary union leads to an increase in the degree of central bank conservatism, joining the monetary union will reduce the country-level unemployment rate. Coricelli et al. (2001) have shown that the strength of this effect depends critically on the degree of centralization of wage bargaining as well as on country size. Hall and Franzese (1998) and Franzese and Hall (2000) show that an independent central bank can elicit wage moderation from trade unions only when collective action problems in wage bargaining are mitigated through union coordination. In the absence of union coordination, the signaling mechanism, through which an independent central bank conveys its anti-inflation commitment, fails to operate and the real economic costs of low inflation can be high. Soskice and Iversen (2000) show that, in the presence of a finite number of sectoral monopoly unions, a lower degree of monetary accommodation increases equilibrium employment. More particularly, a shift to a less accommodating policy on the part of the central bank, and the concomitant decline in real money supply, elicits wage restraint from wage-setters who dislike a fall in actual employment. Wage restraint in turn delivers a higher level of equilibrium employment. More recently, Holden (2005) explores the effects on equilibrium unemployment of the interaction between the monetary policy rule and the degree of coordination in wage-setting. Holden (2005) shows that a strict monetary policy rule disciplines the uncoordinated wage-setters as the central bank may react to excessive wages by reducing the nominal money stock. As a result, the strict monetary policy stance combined with uncoordinated wage-setting delivers lower equilibrium unemployment and reduces the wage-setters’ incentives to coordinate. In addition, Holden (2005) shows that an accom-

1

See, among others, Alesina and Summers (1993) and Cukierman, Webb, and Neyapti (1992).

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modating monetary stance could also reduce equilibrium unemployment via the strengthening of the wage-setters’ incentives to coordinate. Despite the plethora of insights that the new literature has brought forth, a common feature of the above models is that labor markets are dominated by a small number of unions that act unilaterally as wage setters. Indeed, sectoral monopoly unions characterize the work in Cukierman and Lippi (1999, 2001), Soskice and Iversen (2000), Coricelli et al. (2001), Coricelli, Cukierman and Dalmazzo (2006) and Holden (2005), and a single economy-wide monopoly union is present in Grüner and Hefeker (1999). Yet, however analytically convenient this assumption may be, it is not realistic. In many of the advanced market economies, wages are set through bargaining between firms and unions and not by unions that act unilaterally. Also, while union membership has experienced a secular decline during the past three decades, the extent of collective bargaining coverage remains high.2 How would the key insights of the existing literature fare if one relaxed the assumption of monopoly unions in favor of the encompassing case of wage bargaining? The aim of this paper is to address this question by exploring the effects of changes in monetary policy institutions on labor market outcomes in the presence of nominal wage contracts that are negotiated in a right-to-manage fashion between trade unions and firms. In so doing, the paper shows that some of the results obtained in the literature to date may be qualified once the assumption of monopoly unions – an extreme case of the right-to-manage model that occurs when trade unions have full bargaining power – is relaxed. More particularly, the paper develops a general equilibrium model of unionized oligopoly in which sectoral wages are set through bargaining and the monetary policy maker monitors, and responds to, price developments at the sectoral or oligopoly level. This setup allows for the study of strategic interactions between the policy maker, oligopolies and trade unions under alternative degrees of product market competition and union bargaining power. In this context, the model examines the real economic effects of two distinct changes in the institutional design of monetary policy: first, the delegation of monetary policy to a monetary union-wide central bank and, second, a change in the monetary policy rule to which the policy maker adheres. In so doing, our model is able to replicate the results derived by Soskice and Iversen (2000) under the extreme cases of monopoly union and nonunion wage-setting. Yet, importantly, our model shows that, once wage bargaining is introduced in the analysis, the sign and size of the effect of an institutional change in monetary policy on equilibrium employment depends upon the degree of product market competition and the distribution of bargaining power between unions and firms. The remainder of the paper is organized as follows: Section 2 describes the setup of the model and outlines the strategic interactions between firms, trade unions and the policy maker. Section 3 presents the model’s predictions regarding the creation of a monetary union. Section 4 explores the impact on equilibrium employment of a shift from a tightening to an accommodating policy rule. Section 5 concludes. 2. A model of unionized oligopoly and monetary institutions Consider an economy that consists of a large number of price-setting oligopolistic sectors. Each firm in each sector produces one good that is a substitute to other goods of that sector, while goods produced in different sectors are independent. The number of firms in each sector is small, where Fik denotes firm i, i = 1, . . ., n, of sector k, k = 1, . . ., K, producing quantity of good xik . A large number of consumer–workers are fully unionized in firm-specific unions. Thus, the union Uik receives the demand for labor from the firm Fik and controls the supply of labor to it. There is no labor mobility across unions. Each consumer–worker demands goods from each sector and owns a fraction of every firm in the economy. Firms cannot significantly influence the income of their consumer–workers, hence they take income as given. The wage-setting process takes place at the sectoral level in rightto-manage negotiations between unions and employers. We assume symmetry across both sectors and firms within a sector, thus the analysis focuses on the representative firm Fik of the representative

2 In the mid-1990s, collective bargaining coverage was exceeding 80% of the labor force in several would-be EMU members such as Austria, Belgium, Finland, France, Germany, Italy and the Netherlands (see Soskice and Iversen (1998)).

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sector k. The underlying assumptions of the model imply that economic agents take aggregate income and the aggregate price level as given when taking their sectoral strategic decisions. We consider, alternatively, two types of monetary authority to which the conduct of monetary policy may be delegated. More particularly, monetary policy may be conducted either by a domestic central bank or set in the context of a monetary union formed by two countries. For analytical simplicity, macroeconomic equilibrium in a monetary union is obtained under the assumption of cross-country symmetry. In line with Soskice and Iversen (2000), we assume that the interactions among firms, unions, and the monetary authority can be described by the following sequence of events modelled by means of a four-stage game: Stage 1: The central bank sets its nominal money supply precommitting to a monetary policy rule defined by the parameter , where  takes values within the range [ − 1, 1]. Under a domestic central bank, we assume that the domestic nominal money supply exhibits the following structure: MS =

 K 

 ˛k

.

k

(1)

k=1

In a monetary union, the domestic nominal money supply is given by:



K ˛ k=1 k

S

M =

k

k

+

K

˛∗ k=1 k

∗ k

 .

2

(2)

In Eqs. (1) and (2), the parameter ˛k represents the sector k’s share of aggregate expenditure and is a sector-specific price index that takes the following functional form:

⎛ n n k

⎜ ⎜ ⎝

=⎜

2

i=1

i
n(n − 1)

pik pjk

⎞(1/2) ⎟ ⎟ ⎟ ⎠

,

(3)

where pik denotes the price of the good produced by firm i of sector k and pjk denotes the price of the good produced by firm j of the same sector. Correspondingly, ˛∗k and k∗ refer to foreign country variables such that the domestic and foreign country are assumed of equal size. Notice that the sectoral price index k captures the interaction of prices within sector k and that its functional form is obtained from the microfoundations of consumer choice that are introduced below. Eqs. (1) and (2) indicate that, in setting the nominal money supply, the monetary authority responds to a weighted average of sectoral price indices, domestic and foreign (when relevant). Eqs. (1) and (2) should be interpreted as reduced-form specifications of the corresponding central bank reaction function. Similarly, Holden (2005) sets the nominal money stock as a function of the aggregate wage level and Coricelli et al. (2006) show that this class of monetary policy rules can be derived as the optimal monetary policy of a central bank with a loss function that is quadratic in inflation and unemployment. The policy rule is able to admit a wide range of monetary policy responses to price-setting behavior through alternative values of the parameter . In particular, the parameter ,  ∈[− 1, 1], encompasses three possible monetary policy stances, namely, (i) nominal accommodation occurs when  ∈ (0, 1], (ii) nominal tightening occurs when  ∈[− 1, 0), and (iii) the nominal money supply is set exogenously when  = 0. Monetary accommodation (respectively, tightening) implies that the monetary authority increases (respectively, reduces) the nominal money supply in response to a sectoral price increase. Similarly, the monetary authority is not responsive to sectoral prices when  = 0, in which case MS equals the constant value of 1.3 Note that by restricting the value of  to  ∈[− 1, 1] the money supply

3

The choice of 1 owes to analytical convenience. Alternatively, one may model exogenous variations in nominal money supply 

through the parameter m, such that M S = m(Kk=1 ˛k k ) under a domestic central bank, thus possibly capturing exogenous changes in the volume of transactions that are met by the monetary authority via nominal money supply changes.

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does not ‘overreact’ to a change in any sectoral price index. Thus, for example, in the limiting case of  = − 1, the fall in MS induced by a sectoral price increase is determined by the size of the increase in, say, k once the central bank accounts for the weight of k in MS . In terms of the central bank’s ultimate goal, note that a value of  = 1 indicates a central banker who fully accommodates shocks to the model’s composite price index, i.e. a central banker whose only objective is employment stability, while a value of  = − 1 indicates a central banker who adjusts the supply of money to fully offset shocks on the composite price index, i.e. a central banker whose single ultimate goal is price stability. Finally, it is worth noting that in the symmetric equilibrium solution, where all prices (domestic and foreign, if relevant) are identical, the specifications in Eqs. (1) and (2) are reduced to Soskice and Iversen’s (2000) specification where MS = P˛ and ˛ ∈ [0, 1]. The difference being that Soskice and Iversen (2000) do not consider the possibility of a nominal contraction. We find this an unwarranted restriction. Empirical evidence suggests that some central banks, such as the Austrian and the German ones, pursued monetary tightening in response to what they considered excessive wage agreements (see Cukierman, Rodriguez and Webb (1998)). In their analytical frameworks, Coricelli et al. (2001) and Holden (2005) also account for the possibility of nominal contraction on the part of the monetary authority as a reaction to high nominal wage growth. Stage 2: In the knowledge of the policy reaction function and the value of the monetary rule parameter precommitted to by the monetary authority, sectoral unions and employers’ confederations throughout the economy simultaneously and independently bargain over the corresponding sectoral nominal wage wk , k = 1, . . ., K. Stage 3: In the knowledge of the policy reaction function and the value of the monetary rule parameter of the central bank and in the knowledge of the sectoral wages that result from stage 2 negotiations, firms compete in their corresponding oligopoly by setting prices simultaneously and independently. Stage 4: Given the price vector resulting from stage 3 interactions, the central bank sets the domestic nominal money supply MS according to either Eq. (1) or Eq. (2), depending upon the prevailing institutional arrangement. Therefore, the central bank fulfils its precommitment.4 The game of complete and quasi-perfect information is solved by backward induction. Stage 4 is automatic due to the central bank’s adherence to the precommitted monetary rule in stage 1. As a result, the attention is confined to stages 2 and 3 of the game. Before proceeding to the solution of the four-stage game, we describe the microfoundations of consumer choice and the assumptions regarding the objective functions of unions and firms. 2.1. The microfoundations of consumer choice The utility of the representative consumer–worker is increasing with the consumption of goods and the accumulation of real money balances and decreasing with work. Consumer choice can be modelled as occurring in two sequential steps. First, the representative individual allocates income between consumption and money holdings and, second, the representative individual allocates consumption across specific goods. The individual delegates the labor supply decision to the union. Since preferences are homothetic over consumption and real money balances, we can extrapolate and deal with the aggregated individual, who firstly solves the following optimization problem: max

1

{X, M} c c (1 − c)

1−c

Xc

M 1−c P

− N e , (4)

s.t. PX + M = I where, in utility, X is the total quantity demanded of the nK goods, M represents nominal money balances held by individuals, P is an aggregate consumer price index, and the parameter c, c ∈ (0, 1), weights consumption and real money balances in utility. The term Ne denotes the disutility from supplying N units of labor, where N ≤ T and T is the total number of labor units available, ,  > 0, is

4

The foreign country simultaneously experiences an identical set of events.

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a parameter reflecting the marginal disutility from work, and e, e > 1, determines the elasticity of the marginal disutility with respect to work, which is given by e − 1. In the budget constraint, PX is aggregate nominal expenditure on the nK goods, M denotes nominal money balances, and I is nominal income where I = WL + ˝ + Ms . Hence, nominal income is the sum of total rents from labor, WL, total profits, ˝, and the initial level of nominal money balances, Ms . Following the assumptions of the model, the individual takes P and I as exogenously given. The solution to the optimization problem set out in Eq. (4) constitutes the basic macroeconomic framework, where PX = Y = cI and M = (1 − c) I. Thus, the income-expenditure identities yield Y = (c/(1 − c))Ms . In the second step, the aggregated individual allocates optimally the consumption budget cI across the nK goods produced in the economy, such that PX = Y = cI holds. In order to solve for the optimal allocation of nominal expenditure, we model consumer choice using the ‘extended linear-homothetic’ preferences. Based on the expenditure function approach, the preferences are an extended version of the ‘linear-homothetic’ preferences introduced in Datta and Dixon (2000). It is worth noting that the expenditure function is a companion measure to the indirect utility function in that it summarizes consumer’s market behavior in an equally neat and powerful way (see Jehle and Reny (2001)). The expenditure function is a minimum-valued function that captures the minimum level of money expenditure that the consumer must incur in order to achieve a certain level of utility given an exogenous, fixed set of prices.5 In particular, we assume that the aggregated individual’s expenditure function is described by: E(p, u) = b(p) u, where p ∈ nK + is the price vector of the nK goods and u denotes utility units or utils. The unit cost function b(p) : nK + → + , i.e. the extra minimum level of money expenditure necessary to obtain an additional util u, takes the following form: b(p) = (1 − ı) + ı +  [ − ] ,

(5)

where ı, ı ∈ (0, 1], and ,  > 0, are parameters of the model. The function b(p) is composed of the following price indices: n K  k=1

=

K k=1

n

nK

K ;

nK

  =

pik

i=1

k=1

 =

p2 i=1 ik

K

k

 n n ;

k

=

2

i=1

p p i
n(n − 1)

(1/2) ;

(1/2) ,

(6)

where  is the arithmetic average of individual prices,  is the arithmetic average of sector-specific price indices such that the sector-specific price index k captures the interaction of prices within sector k or ‘within-sector effects’, and  is the standard deviation of prices from zero. Note that b(p) replicates Datta and Dixon’s (2000) ‘linear-homothetic’ preferences when ı = 0 and Leontieff preferences when both ı = 0 and  = 0. The ‘extended linear-homothetic’ preferences imply that the minimum expenditure to be incurred in order to obtain an additional util u is a weighted average of the arithmetic mean price and the arithmetic average of within-sector price effects, plus the difference between the average of prices and their standard deviation from zero. Proof of the validity of the unit cost function defined by Eqs. (5) and (6) is shown in Appendix A. Applying the Shephard’s lemma to Eq. (5) yields pik xik /Y =(∂ b/∂ pik )(pik /b) ≡ ˛ik , that is the share of aggregate nominal expenditure allocated to good xik equals the elasticity of the unit cost function

5 On the other hand, indirect utility seeks the maximum level of utility that the consumer can achieve for a given fixed level of income and given prices.

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with respect to the individual price pik .6 Thus, we can derive the Marshallian demand function for the representative good as follows:

⎛

xik =

Y bnK

⎞

⎜ ⎟ n  ⎜ pjk pik ⎟ ⎜a + ı ⎟, −  ⎜  ⎟ (n − 1) k ⎝ ⎠ j= / i

(7)

j=1 where a = 1 +  − ı. An innovation of the ‘extended linear-homothetic’ preferences is the presence of a sector-specific price index in product demand which provides an additional channel through which the individual firm’s price may affect demand. As a result, the price-setting firm will take into account the effects of both its strategy and its competitors’ strategies on the sector-specific price index . In Eq. (7), the parameter ,  > 0, characterizes the product demand elasticity in symmetric equilibrium, and the parameter ı, ı ∈ (0, 1], captures the degree of substitutability of goods within a sector in the sense that the cross-price elasticity increases in ı, i.e. a higher value of ı implies closer substitutes.7 For analytical convenience, and due to the symmetry of the model’s economy, we can work with a simplified version of the direct demand function in Eq. (7). This is given by:

⎛

xik =

cM S nK(1 − c)P

⎞

⎜ ⎟ n  ⎜ pjk pik ⎟ ⎜a + ı ⎟, −  ⎜ P ⎟ (n − 1) k ⎝ ⎠ j= / i

(8)

j=1 where P represents the (anticipated) aggregate price level in symmetric equilibrium, such that  = 1 = 2 = . . . = K =  =  = P and b = P, and where the macroeconomic income-expenditure identity, Y = (c/(1 − c))MS , has been introduced.8 Notice that the parameter c, c ∈ (0, 1), is the proportion of nominal income that the aggregated individual allocates to consumption in the first step of consumer’s choice. Overall, the assumption of a large number of sectors in the economy implies aggregate price taking by economic agents, that is P is taken as given in the product and labor market games. On the other hand, the assumption of a small number of firms operating in each oligopoly implies that price and wage setters take into account the effect of their individual strategies on the sector-specific price index and, thus, on product demand. Given the knowledge of the central bank’s monetary policy rule, this assumption also implies that firms and unions internalize the central bank’s response to their individual price- and wage-setting strategies. 2.2. Price and wage formation In stage 3 of the game, each firm in each sector maximizes nominal profits subject to product demand. Assuming that the firm’s short-run technology exhibits constant unitary returns to labor, that is xik = lik , the objective function of the representative firm Fik can be written as: ˝ik = (pik − wk )xik ,

(9)

6 Summing up across all goods produced in the economy implies that the aggregated individual has optimally allocated the total consumption budget or nominal expenditure, thus PX/Y = 1 holds. 7 The product demand elasticity in symmetric equilibrium introduces the symmetric solution, where pik = pjk ∀i, j, ∀ k, to the standard expression of product demand elasticity. Note that  > ı by assumption, ensuring that, in absolute terms, the own-price effect on product demand is greater than the sum of the cross-price effects. 8 In Eq. (8), the anticipated symmetric equilibrium outcome cannot be assumed with regard to the sectoral price index k since the assumption of a small number of firms in each oligopoly implies that firms internalize the effect of their respective pricing strategy on k .

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such that the firm maximizes Eq. (9) subject to Eq. (8) in order to choose pik , taking the competitors’ prices and the sectoral nominal wage wk as given.9 The partial equilibrium outcome of stage 3 interactions yields the firm’s derived demand for labor lik (wk ). The objective function of the representative union is described by the following expression: Uik =

wk e , l − lik P ik

(10)

that is obtained from the utility function in Eq. (4) once the aggregated individual has allocated wealth between consumption and real money balances. For analytical convenience, we set e = 2 which implies that the elasticity of marginal disutility with respect to work, e − 1, equals 1. The union utility function in Eq. (10) assumes that the union cares about the total surplus of its representative worker, namely the expected real wage bill minus the disutility from supplying lik units of labor. In stage 2 of the game, sectoral unions and firms bargain over the sectoral nominal wage wk . In doing so, they take into account the derived demands for labor that stage 3 interactions yield. Once the sectoral nominal wage is agreed, each firm in the oligopoly unilaterally chooses the employment level. In the representative sector k, the bargained nominal wage is the outcome of the maximization of the following Nash bargaining function: Bk =

 n  i=1

Uik

ˇ  n 

1−ˇ

˝ik

,

(11)

i=1

where the parameter ˇ, ˇ ∈ [0, 1], reflects the relative strength of each party in the negotiation. Notice that ˇ = {0, 1} captures two extreme cases of the labor market, namely, the nonunion and the monopoly union scenarios, respectively. In Eq. (11), the sectoral union and the employers’ confederation care about the sum of the utilities of their respective members. In case of disagreement, the payoffs are assumed equal to zero.10 The backward-induction solution to the four-stage game in general equilibrium is presented next. First, the model is solved under the assumption of a domestic central bank that conducts monetary policy, then we present the solution for the case of a single monetary union authority. 2.3. General equilibrium under a domestic central bank From Eqs. (1), (3) and (8), we obtain the own-price elasticity of product demand written, in absolute value, as:

   ∂xik pik  =+ ı −  , n nK ∂pik xik 

|ε|DCB =  

(12)

where the symmetric general equilibrium condition, pik = pjk = P ∀i, j ∈ k ∀k and k = P ∀k, has been introduced. In Eq. (12), the elasticity of product demand is greater than 1 if and only if  >  DCB where  DCB = 1 − (ı/n) + (/(nK)). Similarly, we derive the cross-price elasticity of product demand, given by  DCB =(∂ xik /∂ pjk )(pjk /xik ) ≡ (ı/(n(n − 1))) + (/(nK)) in symmetric equilibrium, noting that it increases in ı, i.e. a higher ı implies closer substitutes. The first order condition of profit optimization, given by pik (1 − (1/|ε|DCB )) = wk , yields the expression of the equilibrium real wage as follows: wk DCB 1 , = 1 − DCB = 1 − P  + (ı/n) − (/nK) where DCB = 1/|ε|DCB denotes the equilibrium markup and (wk /P)

(13) DCB

< 1.

9 We assume throughout the analysis n and K fixed. To address the issue of entry in a sector (n endogenous), one might assume that  = f(n), where is an exogenous constant and f ( ·)> 0. In symmetric equilibrium, the unit cost of utility would be still invariant to n. 10 The parameter ˇ may depend on a number of labor market factors, such as the extent of bargaining coverage, employment protection legislation, and the unemployment insurance system (see Layard, Nickell and Jackman (1991)).

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From the functional forms in Eqs. (9) and (10) and from the first order condition of the right-tomanage negotiation described in Eq. (11), we derive the expression for aggregate employment and output in symmetric general equilibrium: E=

nK 



(1 − (wk /P))(1 − ) wk −ˇ P (1 − ˇ)(1 − (∂pik /∂wk )) − (1 + ˇ)((P/wk ) − 1)



,

(14)

where  is the wage elasticity of labor demand, that is  = (∂xik /∂wk )(wk /xik ). The derivative of the firm’s price with respect to the wage, ∂pik /∂wk , is obtained from the first order condition of the product market game, once we have introduced the simplification of industry level bargaining, that is pik (wk ) = pjk (wk ) ∀i, j ∈ k. In symmetric equilibrium, the derivative is given by: ∂pik ∂wk

DCB

 + (ı/n) − (/nK)

=

2 + (ı/n) − (/nK) + (1 − (wk /P)

DCB

)((/nK) + (/nK 2 ) − (ı/n))

.

(15)

Labor demand elasticity is evaluated from Eq. (8), once the objective function of the monetary authority has been introduced. In symmetric equilibrium, it takes the form:



DCB = −  −

 K

∂p

ik

DCB

∂wk

wk DCB , P

(16)

note that xik = (Y/P)/(nK) = cP−1 /(nK(1 − c)) is the output share of the individual firm in symmetric general equilibrium. Finally, equilibrium employment under a domestic central bank (EDCB ) is obtained by substituting Eqs. (13), (15) and (16) into Eq. (14).11 2.4. General equilibrium under a monetary union In a monetary union, the price elasticity of product demand is derived from Eqs. (2), (3) and (8). It is given, in absolute value, by: |ε|MU =  +

ı  − , n 2nK

(17)

such that we assume cross-country symmetry in equilibrium, that is P = P∗ . The assumption on  for product demand to be elastic is stated as |ε|MU > 1 ↔  >  MU where  MU = 1 − (ı/n) + (/(2nK)). Similarly, for the case of a monetary union, the cross-price elasticity of product demand is given by  MU = (ı/(n(n − 1))) + (/(2nK)) in symmetric general equilibrium. The first order condition of profit optimization, given by pik (1 − (1/|ε|MU )) = wk , yields the following equilibrium real wage: 1 wk MU , = 1 − MU = 1 − P  + (ı/n) − (/2nK)

(18) MU

< 1. where the equilibrium markup is denoted as MU = 1/|ε|MU and (wk /P) The derivative of the firm’s price with respect to the wage is now given by: ∂pik ∂wk

MU

=

 + (ı/n) − (/2nK) 2 + (ı/n) − (/2nK) + (1 − (wk /P)

MU

)((/2nK) + (/4nK 2 ) − (ı/n))

.

(19)

From Eqs. (8) and (2), we derive the expression of the labor demand elasticity in general equilibrium:



MU = −  −

11

 2K

∂p

ik

∂wk

MU

wk MU , P

(20)

We define, when necessary, a lower-bound limit, ¯ DCB , on parameter  – such that  > ¯ DCB and ¯ DCB >  DCB – in order to

exclude corner solutions (hence, negative outputs). Specifically, the lower-bound limit, ¯ DCB , is derived by setting EDCB = 0 when DCB < 1 for the assumed ˇ = 1. It yields a lowest ceiling for the markup of 33% approximately. Finally, note that 0 < (∂pik /∂wk ) range of parameter values.

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note that xik = c((P + P ∗ )/2) /(nK(1 − c)P) = cP −1 /(nK(1 − c)) is the output share of the individual firm in the cross-country symmetric general equilibrium solution. Finally, to obtain equilibrium employment in a monetary union (EMU ), we introduce Eqs. (18), (19) and (20) in Eq. (14).12 Once the general equilibrium model of unionized oligopoly has been solved, we turn our attention to the examination of the effects of the interaction among monetary institutions, union bargaining power, and product market competition on labor market variables. 3. Employment, the real wage, and the monetary authority This section assesses the effects of a change in the type of monetary authority on employment and the real wage. More particularly, we examine the effects of the creation of a monetary union for any given value of the monetary rule parameter . The first set of results is summarized in Proposition 1. Proposition 1. The formation of a monetary union brings about the following effects on the equilibrium MU DCB MU MU DCB real wage: (i) (wk /P) > (wk /P) for  > 0; (ii) (wk /P) < (wk /P) for  < 0; (iii) (wk /P) = DCB (wk /P) for  = 0. Proof.

See Appendix B.



Intuitively, the results established in Proposition 1 can be explained by the different size of the ‘monetary rule effect’ that is experienced under each monetary authority. The ‘monetary rule effect’ captures the elasticity of the domestic money supply with respect to the individual price, as given by:



∂M S pik ∂pik M S

DCB

=

 ; nK



∂M S pik ∂pik M S

MU

=

 . 2nK

(21)

In Eq. (21), note that the larger the value of  :  > 0, the larger the percentage increase in MS induced by a one percentage increase in pik . Thus, the higher the firm’s incentive to increase pik relative to other prices, since monetary accommodation cushions the adverse impact on product demand of a price increase. Accordingly, the smaller  :  < 0, the larger the percentage fall in MS yielded by a one percentage increase in pik . Hence, the higher the incentive to restrain in the price-setting game, since monetary tightening intensifies the adverse impact on product demand of a price increase. Notice that the ‘monetary rule effect’ underpins the product demand elasticity in symmetric equilibrium, as Eqs. (12) and (17) show. If the monetary authority accommodates price increases ( > 0), the formation of a monetary union reduces the incentive of the individual firm to increase its relative price. In a monetary union, the monetary expansion induced by an individual ‘domestic’ price increase is smaller. This is because the monetary authority reacts now to individual prices across the union, and each individual price has a smaller weight on monetary policy. As a result, the firm perceives a more elastic product demand when the monetary union is established, and a lower (respectively, higher) markup (respectively, real wage) is obtained. Accordingly, if the monetary authority tightens the money supply in response to a price increase ( < 0), the formation of a monetary union reduces the incentive to restrain of the individual firm. In a monetary union, the monetary contraction induced by an individual ‘domestic’ price increase is smaller. As a result of the establishment of a monetary union, the firm perceives a less elastic product demand. Thus, a higher (respectively, lower) markup (respectively, real wage) is obtained. Finally, for  = 0, that is the money supply is an exogenous constant and strategic interactions are assumed away, the ‘monetary rule effect’ vanishes and the establishment of a monetary union does not affect the real wage. In Eq. (21), notice that the larger the number of firms n and the larger the number of sectors K, the smaller the effect of an individual price increase on the domestic nominal

12 As in the case of a domestic central bank, we define, when necessary, a lower-bound limit, ¯ MU , on parameter  – such that  > ¯ MU and ¯ MU >  MU – in order to exclude corner solutions. In particular, the lower-bound limit, ¯ MU , is obtained by setting

EMU = 0 when ˇ = 1. It produces a lowest ceiling for the markup of 33% approximately. Note also that 0 < (∂pik /∂wk ) the assumed range of parameter values.

MU

< 1 for

G. Chouliarakis, M. Correa-López / North American Journal of Economics and Finance 22 (2011) 131–148

φ >0

wk P

wk MU P w k DCB P

l

d

DCB

φ <0

wk P

w k DCB P wk M U P

ld M U

141

l d DCB ld M U

lDCB

lM U

lM U

l

lDCB

l

Fig. 1. Equilibrium employment and the creation of a monetary union, ˇ = {0, 1}.

money supply. Thus, for  > 0, the incentive to increase a relative price is reduced. Accordingly, for  < 0, the incentive to restrain is reduced too. We next explore the effect of the creation of a monetary union on equilibrium employment considering, first, the nonunion and the monopoly union scenarios, where ˇ = {0, 1} respectively. Results are summarized in Proposition 2 and illustrated graphically in Fig. 1. Proposition 2. In the nonunion and the monopoly union models, the formation of a monetary union brings about the following effects on equilibrium employment: (i) EMU > EDCB for  > 0; (ii) EMU < EDCB for < 0; (iii) EMU = EDCB for  = 0. Result (iii) holds for any ˇ ∈ [0, 1]. Proof.

See Appendix C.



The difference in the size of the ‘labor demand elasticity effect’ under the two monetary authori-

 MU

ties underpins this result. In particular, 

 DCB

> 

 DCB

for  > 0, 

 MU

> 

 DCB

for  < 0, and 

=

 MU  for  = 0. For  > 0, any sectoral wage increase triggers monetary accommodation. When a mone-

tary union is established, the extent of monetary accommodation is reduced since a ‘domestic’ sectoral wage has a smaller weight on union-wide monetary policy. In other words, a one percentage increase in wk induces a larger percentage fall in the demand for labor of the individual firm. Hence, as a result of the creation of a monetary union, the monopoly union, or the employers’ confederation in the nonunion case, moderate their wages and deliver higher equilibrium employment. As Fig. 1 shows, other things being equal, the wage-setting agent picks a relatively low point on the labor demand schedule that produces higher equilibrium employment under a monetary union. For  < 0, the monetary authority reduces nominal money supply in response to a sectoral wage increase. In principle, monetary tightening should provide an incentive to exercise wage restraint. However, in a monetary union, this incentive is weakened as the extent of tightening following a sectoral wage increase is reduced. Thus, other things being equal, the wage setting agent picks a relatively high point on the labor demand schedule that produces lower equilibrium employment under a monetary union. (see Fig. 1). Finally note that, for  = 0, we assume away any strategic interaction between the central bank and price and wage setters. Thus, the switch from a domestic to a union-wide monetary authority has no effect on real economic aggregates. We next consider whether the results established in Proposition 2 are robust to the introduction of a right-to-manage negotiation over the sectoral wage, that is when ˇ ∈ (0, 1). In order to address this question, we use numerical simulation methods and examine the behavior of the equilibrium employment differential (EDCB − EMU ) with respect to parameters {, ˇ}. Results are plotted in Fig. 2.13

13 In so doing, we set K = 100, n = 2,  = 1, ı = 0.5 and  ={− 0.5, 0.5}. Note that alternative parameter values yield qualitatively similar results.

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φ =-0.5 E DCB -E MU

γ =5

φ =0.5 γ =10

E DCB -E MU 1

0 0.006 γ increasing

β

γ =25 -0.002

0.004

-0.004 γ =25

0.002

1 β

0

γ increasing

-0.006 γ =5

γ =10

Fig. 2. The equilibrium employment differential: EDCB − EMU .

Fig. 2 shows that the predictions summarized in Proposition 2 do not necessarily hold when there is a right-to-manage negotiation over the wage. In particular, for  = − 0.5 (respectively,  = 0.5), we find that the employment differential is positive (respectively, negative) for all ˇ-values if and only if  is sufficiently low. As  increases, the employment differential becomes negative (respectively, positive) for an increasing range of ˇ-values. In the limit, when product demand elasticity tends to infinity ( → ∞) the employment differential falls to zero.14 Therefore, the model predicts that the effect on employment of the establishment of a monetary union depends upon three factors, namely, the monetary rule parameter , the degree of product market competition , and the distribution of bargaining power in the wage-setting process ˇ. Similarly, the establishment of a monetary union has an effect on employment in imperfectly competitive environments, whereas it does not affect employment when competition is perfect. Intuitively, the behavior of the employment differential as shown in Fig. 2 can be explained by the relative dominance of the ‘labor demand elasticity effect’ and the ‘wage sensitivity effect’. The former effect is simply given by the elasticity of labor demand while the ‘wage sensitivity effect’ captures the extent of the change in the bargained wage induced by an (exogenous) change in parameter ˇ. For  < 0 and for a certain  value, Fig. 2 shows that as ˇ falls from ˇ = 1, EMU increases more than EDCB up until ˇ reaches a certain threshold value. Thereafter, further falls in ˇ increase EDCB more than EMU . From the labor demand elasticity effect, we expect that to an identical percentage fall in the bargained wage, the latter as a result of a fall in ˇ, EDCB increases more than EMU . Hence, in order to explain the behavior of the employment differential when  < 0, it must be that the bargained wage is more responsive to a change in ˇ in the context of a monetary union and a relatively powerful union (high ˇ). In this case, a fall in ˇ induces a larger fall in the bargained wage, that is the employers’ confederation pushes for a larger percentage drop in the wage, since the monetary expansion that follows a unitary percentage wage fall is smaller in a monetary union. Overall, as ˇ falls from ˇ = 1, the larger fall of the bargained wage in a monetary union yields a larger employment expansion: EMU is boosted more than EDCB . Thus, the wage sensitivity effect dominates the labor demand elasticity effect for a certain range of ˇ values, causing the reversal of the employment differential. The opposite occurs when ˇ is already below its threshold value, hence EDCB expands more than EMU as ˇ falls, and the labor demand elasticity effect dominates.

14

It is straightforward to check that lim (E DCD − E MU ) = 0 for all parameter values. →∞

G. Chouliarakis, M. Correa-López / North American Journal of Economics and Finance 22 (2011) 131–148

143

A similar intuitive explanation holds for  > 0, where the bargained wage is more sensitive to a change in ˇ in the context of a domestic central bank regime and a relatively powerful union (ˇ high). Finally, in Fig. 2, note that as the degree of product market competition increases, the employment differential becomes smaller at ˇ = 1, which implies that the fall in ˇ required to prompt the employment reversal is smaller. 4. Employment, the real wage, and the monetary rule Does a switch from a tightening monetary policy rule ( < 0) to an accommodating monetary policy rule ( > 0) matter for real economic activity? If yes, do the sign and size of the economic effects vary with the type of monetary authority and the product and labor market institutions? In Eqs. (12) and (17), we show that a switch from tightening to accommodation reduces the price elasticity of product demand and increases the markup. Monetary accommodation induces the individual firm to pursue a high relative price strategy. As a result, the aggregate price level rises and the real wage falls. On the other hand, other things being equal, the switch from tightening to accommodation brings about a reduction in equilibrium employment in the nonunion and the monopoly union models, as we show below. Proposition 3. In the nonunion and the monopoly union models, a switch from a tightening monetary policy rule to an accommodating monetary policy rule reduces equilibrium employment. This result holds under both types of monetary authority. Proof.

See Appendix D.

 

From Eqs. (16) and (20), we note that 

 

<0

> 

>0

, under both a domestic central bank and

union.15

For  > 0, the incentive to exercise restraint in wage-setting is weakened since a monetary the adverse effect on employment induced by a wage increase is partly accommodated by monetary expansion. Thus, the sectoral monopoly union, or the employers’ confederation in the nonunion case, locates itself at a high point on the corresponding labor demand schedule such that it delivers lower equilibrium employment under accommodation. Yet, the result established in Proposition 3 is not necessarily robust to the introduction of a rightto-manage negotiation over the sectoral wage. That is, unlike the nonunion and the monopoly union scenarios, the model predicts that switching to an accommodating monetary policy rule may bring about employment gains when ˇ ∈ (0, 1). For K = 100, n = 2,  = 1, ı = 0.5 and  ={− 1, 1}, we define the DCB − E DCB and DMU ≡ E MU MU . Fig. 3 shows how following employment differentials: DDCB ≡ E=−1 − E=1 =1 =−1 DDCB and DMU behave with respect to parameters {, ˇ}.16 The employment differential is positive for all ˇ-values as long as  is sufficiently low. As  increases, the employment differential turns negative for an increasing range of ˇ-values. In the limit, when product (and labor) demand elasticity tends to infinity ( → ∞), the character of the monetary policy rule does not have real effects. The latter is illustrated, for ˇ = 0.5, in the second row of graphs in Fig. 3. Furthermore, it is straightforward to evaluate that lim E DCB = (nK/)(1 − →∞

(ˇ/2)) and lim E MU = (nK/)(1 − (ˇ/2)), for all parameter values. Hence, under perfect competi→∞

tion, the monetary rule and the monetary authority are neutral. As in the previous section, the behavior of the employment differentials is explained by the relative importance of the labor

15 This is proven in Appendix D. Specifically, we show that ∂(∂pik /∂wk )/∂ < 0 under both monetary authorities. This inequality implies that, as  increases, an exogenous increase in the wage requires a smaller increase in firm’s price (thus, a smaller reduction in both output and the markup), in order to restore equilibrium in the product market. In other words, as  increases, the firm finds less costly, in terms of output and markup loss, to absorb a wage shock. This is because part of the wage shock is accommodated ( becomes positive) by the monetary authority. 16 The graphs in Fig. 3 are direct plots produced by Mathematica of the equilibrium employment differentials. Note that we have considered alternative parameter values obtaining qualitatively similar results.

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D DCB

γ =5

D MU

γ =10

0.05

γ =5

0.025

γ increasing 0.04

0.02

0.03

0.015

0.02

γ =10

γ increasing

0.01 γ =25

0.01 0

1

γ =25

0.005

β

1 β

0 -0.005

-0.01

D DCB

D MU

β =0.5

0.015

β =0.5

0.006 0.01 0.004 0.005

0.002 γ

0 200

400

600

800

γ

0 200

1000

400

600

800

1000

-0.002

-0.005

-0.004 -0.01 Fig. 3. Equilibrium employment under monetary tightening vis-à-vis accommodation.

demand elasticity effect and the wage sensitivity effect for different combinations of parameter values. 5. Conclusions This paper shows that the introduction of right-to-manage negotiations over wages sheds new light on the real economic effects of strategic interactions between firms, trade unions and monetary institutions. Whereas, in the presence of monopoly unions, the establishment of a monetary union under a tightening monetary rule reduces equilibrium employment, we show that this result does not hold when we allow for right-to-manage bargaining. More specifically, we show that the sign and size of the effect on employment of the creation of a monetary union depends upon the monetary rule parameter, the degree of product market competition, and the distribution of bargaining power in wage negotiations. Likewise, whereas a policy shift from tightening to accommodation reduces employment when unions set wages unilaterally, we show that, in the presence of right-to-manage negotiations, the sign and size of the employment effect of such a policy shift depends upon the degree of product market competition and the distribution of bargaining power in wage-setting, regardless of the type of monetary authority. These new insights suggest that a closer study of the interactions between monetary institutions, product market competition and union bargaining power may contribute to a better understanding of the factors underpinning the observed diversity in employment performance across the advanced market economies.

G. Chouliarakis, M. Correa-López / North American Journal of Economics and Finance 22 (2011) 131–148

145

Acknowledgements We would like to thank the Editor of this journal and two anonymous referees for valuable comments and suggestions. We are also grateful to Jan Boone, Huw Dixon, Robin Naylor, Neil Rankin and Michele Santoni for offering helpful comments on an earlier version of this paper. All errors remain our own.

Appendix A. Validity of the expenditure and unit cost functions The domain of function b(p) is defined by S ≡ {p ∈ nk + : pik > 0, i = 1, . . . , n, k = 1, . . . , K}. We check that b(p) exhibits the sufficient properties: (i) non-negative and non-decreasing in prices; (ii) homogeneity of degree one and concavity in p; (iii) continuous differentiability. As Datta and Dixon (2000) emphasize, property (iii) is not necessary for validity but for the application of Shephard’s lemma. Given K large, n ≥ 2,  > 0 and ı ∈ (0, 1], it is straightforward to conclude that b(p) is continuously differentiable and homogeneous of degree one. Concavity in p is proven by checking that b1 (p) and b2 (p) are concave, where b1 (p) = (1 − ı) + [ − ] and b2 (p) = ı . Specifically, concavity of b1 (p) implies that ¯ ≤ b1 (ϕp˜ + (1 − ϕ)p) ¯ where 0 < ϕ < 1 and {˜p, p} ¯ ∈ S. In order to assess the ϕb1 (˜p) + (1 − ϕ)b1 (p) concavity of b2 (p) we start by checking the concavity of the representative sectoral price index k , whose domain sk : sk ⊂ S is defined by sk ≡ {pk ∈ n+ : pik > 0, i = 1, . . . , n}. Denote H k as the Hessian matrix associated to k . Hence, given a n-size sector k, where   n ≥ 2, it is straightforwardto check  that all principal minors of H k exhibit the following signs: Hmk  < 0 for m odd and m < n, Hmk  > 0 for











m even and m < n and Hmk  = 0 for m = n (i.e. when Hmk  = H leading principal minors can be expressed as follows:

  Hmk  =



(−1)m (n − 1)

⎛⎛

m+1

nm+1

⎞2

m+2 k

2 k

−n(m − 1)(n − 1)

⎛

⎞2

⎛

k

 ). More specifically, for m < n, the

⎞2 ⎞

⎜⎜ n ⎟ ⎜ n ⎟ ⎜ n ⎟ ⎜⎜  ⎟ ⎜  ⎟ ⎜  ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ pjk ⎟ + ⎜ pjk ⎟ + . . . + ⎜ pjk ⎟ −(m − 2) ⎜⎜ ⎟ ⎝⎝ j =/ 1 ⎠ ⎝ j =/ 2 ⎠ ⎝ j =/ m ⎠ j=1

⎛

j=1

⎟ ⎟ ⎟ ⎟ ⎠

j=1

⎜ n n n n n n n n        ⎜ p p p p p p p pjk + +2 ⎜ + + . . . + + jk jk jk jk jk jk jk ⎜ ⎝ j =/ 1 j =/ 2 j= / 1 j= / 3 j= / 1 j= / m j= / 2 j= / 3 j=1 n 

j= / 2 j=1

pjk

n 

j= / 4 j=1

j=1

pjk + . . . +

j=1 n 

j= / 2 j=1

j=1

pjk

n 

j= / m j=1

j=1

pjk + . . . +

j=1

n 

j= / (m − 1) j=1

pjk

(A.1)

j=1

⎞⎤j = 1

n 

j= / m j=1

⎟⎥ ⎟⎥

⎥ pjk ⎟ ⎟⎥ , ⎠⎦

such that the long term in brackets in Eq. (A.1) equals zero for m = n. Overall, we conclude that the Hessian matrix associated to k is negative semidefinite, hence k is concave. Note that the sectoral price indices – { 1 , 2 , . . ., K } – have the same functional form as k in their corresponding subset of S. Thus, they are also characterized by negative semidefinite Hessian matrices whose principal minors exhibit the pattern of signs described above.

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Finally, given  = (

⎛H

K

0 H 2 0 ... 0

1

⎜ 0 H = ⎜ 0 ⎝ ... 0

k )/K,

k=1

the Hessian matrix associated to  is given by:

... ... ... ... ...

0 0 H 3 ... 0

⎞

0 0 0 ... H K

⎟ ⎟, ⎠

(A.2)

where the positive constant term (1/K) is omitted for simplicity. The Hessian matrix H is nK × nK, where 0 stands for a n × n null matrix. From the structure of (A.2) and the analysis of H k it can be shownthat H is negative semidefinite,  more  particularly, all its principal minors  exhibit the following   ≤ 0 for m odd and m < nk, H   ≥ 0 for m even and m < nk and H   = 0 for m = nk (i.e. where signs: Hm m m

    H m  = H  ). Overall, we can conclude that  and, hence, b2 (p) are concave.

Given the domain defined by S, property (i) is re-written such that b(p) has to be positive and non-decreasing in prices. Property (i) implies that demands are non-negative; it also implies that an additional unit of utility is costly. This property is met given the nature of firms operating in imperfectly competitive markets. Oligopolistic firms competing in substitutes will not, in general, find profitable to set a price above the choke-off price such that demands become negative. Finally, from the analysis derived above we note that the expenditure function is homothetic. Appendix B. Proof of Proposition 1 DCB

MU

> (wk /P) ↔ |ε|DCB > The comparison of Eqs. (13) and (18) yields the following results: (wk /P) MU DCB |ε|MU ↔ nK < 0 which holds for  < 0; (wk /P) > (wk /P) ↔ |ε|MU > |ε|DCB ↔ nK > 0 which DCB MU = (wk /P) ↔ |ε|DCB = |ε|MU ↔ nK = 0 which holds for  = 0. holds for  > 0; (wk /P) Appendix C. Proof of Proposition 2 Equilibrium employment in each monetary regime is derived from Eq. (14), once we introduce the corresponding expressions for the real wage, the labor demand elasticity and the derivative of the firm’s price with respect to the wage. For ˇ = 0, equilibrium employment takes the following form: E=

nK 

w k

P

.

(C.1)

Hence, the comparison of employment across monetary regimes yields: E DCB > E MU ↔ E DCB < E MU ↔ E DCB = E MU ↔

w DCB k

>

k

<

k

=

wP DCB wP DCB P

w MU k

↔ nK < 0

which holds for

 < 0;

k

↔ nK > 0

which holds for

 > 0;

wP MU

wP MU k

P

↔ nK = 0

which holds for  = 0.

In the monopoly union model, where ˇ = 1, equilibrium employment is given by: nK E= 2

w k

P



1−



1

  

.

(C.2)

Thus, we evaluate the employment differential F ≡ EDCB − EMU , and it follows that: nK F 0↔ 2



wk DCB P



1−



1

 DCB 

w MU − k P



1−



1

 MU 

 0.

G. Chouliarakis, M. Correa-López / North American Journal of Economics and Finance 22 (2011) 131–148

147

It is straightforward to conclude that F  0 for   0 respectively, once we note that:



−

∂pik ∂wk −

 K

DCB

|ε|DCB |ε|MU





−

∂pik  ∂wk



MU

 2K

for   0,respectively;





↔  |ε|MU − |ε|DCB +

  ı + − 2nK n 4nK 2



|ε|MU



|ε|DCB

ı   − + n nK nK 2



 0 which holds for   0, respectively;

 DCB

 MU

  for   0, respectively. given the comparison of equilibrium real wages, it holds that:  Finally, from the expression of employment in general equilibrium, that is Eq. (14), and from (wk /P)

DCB

= (wk /P)

MU

for  = 0, (∂pik /∂wk )

DCB

= (∂pik /∂wk )

MU

 DCB

for  = 0, and 

 MU

= 

for  = 0,

we can conclude that EDCB = EMU for  = 0 and ∀ˇ ∈ [0, 1].

Appendix D. Proof of Proposition 3 In the nonunion model (ˇ = 0), equilibrium employment is given by Eq. (C.1). In Eqs. (13) and (18), we find that a switch from  < 0 to  > 0 reduces the real wage, hence employment falls in general equilibrium under both monetary regimes. In the monopoly union model (ˇ = 1), equilibrium employment takes the form in Eq. (C.2). Hence, given the effect of an increase in  on the real wage and given Eqs. (16) and (20), we need to show that ∂(∂pik /∂wk )/∂ < 0 in order to conclude that an increase in parameter  reduces equilibrium employment. From Eqs. (13), (15), (19) and (18), we obtain that: DCB

∂(∂pik /∂wk ) < 0 ↔ −n(K(n + ı) − )(K(n + ı) +  + 2K 2 ((n + ı) − ı)) < 0; ∂ MU ∂(∂pik /∂wk ) < 0 ↔ −n(2K(n + ı) − )(2K(n + ı) +  + 8K 2 ((n + ı) − ı)) < 0; ∂ the above holds for the assumed range of parameter values in each regime. Hence, an increase in

 DCB

parameter  reduces 

 MU

and 

.

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