Macarena Lozano

Pilar Moreno

Methods and Programming < C -Mathematical and Quantitative Methods

Keywords: Balking, Discrete-time queue, Optimal control, Markov chain, Practical applications Balking, Discrete-time queue, Ergodicity, Markov chain, Optimal control, Practical applications, Unfinished work, Waiting/spending time

This paper studies a discrete-time single-server queue with finite and infinite buffer where the users have the option to leave the queue upon arrival (balking). We consider two variants of the model in accordance with the balking policies. Firstly, all the arriving customers balk with a constant probability. Secondly, arriving customers increase their balking probabilities as more customers join the system. Specifically, we find the ergodicity condition and closed form expressions for the stationary distribution of the system size, of the waiting/spending time in the FCFS system and of the unfinished work. The mathematical model is applied in order to resolve several real-life problems in the economic field; in this sense, practical applications in the secondary and tertiary sector are shown. We also develop a cost model to determine the buffer capacity that minimizes certain cost function and give some numerical examples.

Introduction

Recently, different continuous-time queueing systems have been applied to a wide class of economic activities, including dynamic market process [START_REF] Sattinger | A queuing model of the market for access to trading partners[END_REF], labor market [START_REF] Masters | Wage posting in two-sided search and the minimum wage[END_REF], holdups in markets with frictions [2], manufacturing and computer systems [START_REF] Köchel | Finite queueing systems-structural investigations and optimal design[END_REF], multiproduct monopoly [START_REF] Parra | Multiproduct monopoly: a queueing approach[END_REF], hospitals [START_REF] Smet | Multi-product costs and standby capacity derived from queuing theory: the case of Belgian hospitals[END_REF], manufacturing context [START_REF] Van Nyen | Simulation results on the performance of Albin and Whitt's estimation method for waiting times in integrated production-inventory systems[END_REF], job shop environment [START_REF] Haskose | Modelling flow and jobbing shops as a queueing network for workload control[END_REF] and repair shops [START_REF] Sleptchenko | Effects of finite repair capacity in multi-echelon, multi-indenture service part supply systems[END_REF]. However, the discretetime systems are more appropriate than their continuous-time counterparts for F o r P e e r R e v i e w modelling diverse productive processes, since the basic units in these systems are digital such as machine cycle time. The analysis of discrete-time queueing models has received considerable attention in the scientific literature over the past years, in view of its applicability in the study of many computer and communication systems in which time is slotted, see [START_REF] Bruneel | Discrete-time models for communication systems including ATM[END_REF][START_REF] Hunter | Discrete-time models: techniques and applications[END_REF][START_REF] Takagi | Queueing analysis: a foundation of performance evaluation[END_REF]28] and the references therein. Another important application stems from the secondary and tertiary sector since, for example, the current production systems of numerous factories operate on a discrete-time basis where events can only happen at regularly spaced epochs. Nonetheless, despite the wide applications in the aforementioned areas, no work seems to have been done concerning these last applications. That is why this paper makes efforts to fill this space and proposes a new approach modelling some economic activities in discrete-time.

On the other hand, in many queueing situations an arriving customer may balk, so there exists literature devoted to the design and applications of such models [START_REF] Abou-El-Ata | Point estimation and confidence intervals of the M/M/2/N queue with balking and heterogeneity[END_REF][START_REF] Artalejo | On the single server retrial queue with balking[END_REF][START_REF] Ikeda | M/G/1 queue with balking[END_REF][START_REF] Kapur | Numerical computations of discrete-time solutions for a multi-server queue with balking and reneging[END_REF][START_REF] Kumar | Transient solution of an M/M/1 queue with balking[END_REF][START_REF] Montazer-Haghighi | On a multiserver Markovian queueing system with balking and reneging[END_REF]27,29]. Intuitively, the introduction of the balking assumption makes the system less congested than if it was not present, thus the existence of balking provides a mechanism to control an excessive congestion at the system. There exist another queueing models where the customers can leave the system before completing his service, for instance, queues with negative customers [START_REF] Atencia | The discrete-time Geo/Geo/1 queue with negative customers and disasters[END_REF][START_REF] Atencia | A single-server G-queue in discrete-time with geometrical arrival and service process[END_REF], disasters [START_REF] Atencia | The discrete-time Geo/Geo/1 queue with negative customers and disasters[END_REF] or impatient customers [START_REF] Choi | M/M/1 queue with impatient customers of higher priority[END_REF]. Unlike what happens in our model, in the case of negative arrivals or disasters, the abandonments are generated by external causes. Our system could be thought as a related model with impatient customers; however, the balking supposition involves an automatic system abandonment upon arrival at the system, whereas impatience means to take the abandonment decision after some random time.

Several continuous-time queueing models with balking have been discussed during the last years. Nevertheless, to the best of our knowledge, the only work about balking in discrete-time can be found in [START_REF] Kapur | Numerical computations of discrete-time solutions for a multi-server queue with balking and reneging[END_REF], where the authors regard a discrete-time multi-server queue with balking and reneging under arbitrary initial conditions and finite waiting space. This work studies a discrete-time single-server queue with finite and infinite buffer, in which arriving customers balk with a certain probability: in a case constant and in the another one depending on the number of customers in the system upon arrival. Moreover, certain economic activities can be modelled by this queueing system; specifically, two examples are shown in the secondary and tertiary sector. Besides, a cost model is developed to determine the optimal buffer size in order to minimize the steady-state expected cost per unit time.

The rest of the paper is organized as follows. The mathematical model is described in the next section. Section 3 provides some examples of real systems which can be modelled by our queue. Section 4 investigates the stationary distribution and the stability condition of the model under study. Waiting/spending time in the FCFS system is analysed in Section 5. Section 6 shows the stationary distribution of the unfinished work. Section 7 develops a cost model to determine the optimum buffer capacity. Finally, in Section 8, some numerical results are presented to illustrate the effect of the parameters on several performance characteristics. We consider a discrete-time queueing system where the time axis is divided into intervals of equal length, called slots. It is assumed that all queueing activities (arrivals and departures) occur at the slot boundaries, and therefore they may occur at the same time. For mathematical clarity, we will suppose that the departures occur at the moment immediately before the slot boundaries and the arrivals occur at the moment immediately after the slot boundaries; that is, we will discuss the model only for the early arrival system policy (details on this and related concepts can be found in [START_REF] Gravey | Simultaneity in discrete-time single server queues with Bernoulli inputs[END_REF][START_REF] Hunter | Discrete-time models: techniques and applications[END_REF]).

F

We assume that the system conforms to the following assumptions:

• Customers arrive according to a Bernoulli process with probability p , thus p is the probability that a customer arrives at a slot and p = 1 -p is the probability that an arrival does not take place in a slot.

• If the system is free at the instant of an arrival, the service of the arriving customer commences immediately and the customer leaves the system after service completion.

• If the arrival finds the system busy, the arriving customer either with a probability r k , when the system size is k , joins the waiting line in order to be served (persistent customer), or with complementary probability abandons the system forever without service (non-persistent customer).

• The balking probability is 1 -r k if there are k customers in the system upon arrival.

• We will study two cases separately: the case of a fixed r k = r (constant rate policy) and the case of r k = r k (discouraged rate policy) where 0 ≤ r ≤ 1 .

• It is always assumed that services can begin only at slot boundaries and their durations are integer multiples of a slot duration.

• Service times are independent and geometrically distributed with probability s = 1 -s , where s is the probability that a customer does not conclude his service in a slot.

• It is supposed that the buffer capacity is N . We consider two cases singly: an infinite buffer N = ∞ and a finite buffer 2 ≤ N < ∞. For a finite buffer, an arriving customer who finds the system completely full is lost.

It appears reasonable to assume, in certain situations, that the balking probability is constant and is not dependent on the system length; such situations may arise when the system length is not observable by the customer and he has no knowledge of the system length. It is also logical to suppose that the balking probability increases as more customers enter the system; in fact, if the system size increases, then the waiting time is higher and the arrivals are discouraged to wait in order to receive their services. It is obvious that we introduce the balking probabilities as a mechanism to control the level of internal congestion in the system. It should be pointed out that the discouraged rate policy is more intelligent and rational than the constant one, since the first discipline has more information concerning the queueing system.

In Figures 1 and2, the system is depicted for an infinite and finite buffer, respectively. In order to avoid trivial cases, we assume 0 < p < 1 and 0 < s < 1 . The traffic intensity is given by ρ = p/s .

At time m + (the instant immediately after the m-th slot), the system can be described by the random variable X m , which designates the number of customers in the system (including the one in service, if any).

To conclude this section it should be pointed out that if r = 1 , arriving customers always decide to enter the system in order to receive their service, hence we obtain a well-known special case: the classical discrete-time singleserver queue with geometrical arrivals and service times. In the particular case r = 0 , arriving customers only enter the system when their waiting time in the buffer is zero, i.e., when the system is totally empty (it is an extreme case of impatience).

Practical applications

This section gives two examples of productive processes which can be modelled as a discrete-time single-server queue with balking.

Secondary sector

This subsection applies the model under consideration to the industrial sector. The production system operates in the following way. It is considered a manufacturing plant in which the production process is divided into several phases. We will concentrate our attention on a specific stage, as appears in Figure 3. Customers are components or products that pass by different jobs on consecutive machines. Customers are transported in a conveyor belt with regularly spaced partitions (which, for example, move every thirty seconds). Each division of the conveyor belt contains a component/product with probability p and consequently the arrivals follow a Bernoulli process. In addition, a buffer is opposite the machine (server) in a lower level, which impedes empty positions in the waiting line (see Figure 3).

1-r k buffer ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ gate ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
The components are processed on a first-come-first-served (FCFS) basis. If the machine is empty at the moment of an arrival, the service begins immediately and the corresponding component/product leaves the system after concluding his service. When the system is busy, customers with a probability r k , if the system size is k , join the buffer (see the left-hand side of Figure 3). On the other hand, if the queue in front of the machine is "too occupied" (right-hand side of Figure 3), a gate prevents the passage to the waiting line; in this case, the customers with a probability 1 -r k abandon the studied system forever without receiving service (non-persistent customers) and pass to another phase of the productive process.

Components/products may be distinct and consequently manufactured at different times, that is, service times are not deterministic. For instance, a service finishes with probability s each half minute (if the duration of a slot is thirty seconds).

Tertiary sector

The mathematical model described in section 2 allows to analyse problems that take place in activities of the services sector. In this subsection the previous queueing system has been applied to a particular attraction of an amusement park, as you can see in the enlargement of Figure 4. A train with as many carriages as attractions covers all the amusement park so every five minutes, for example, an only carriage stops at each of the attractions. The group of people who get off the carriage in each attraction (for instance, one or twenty persons) will be called a visitor. In the attraction under study, a carriage arrives empty or busy each five minutes (the duration of a slot) with probabilities p and p , respectively. When the attraction is unoccupied, the visitor will join the attraction immediately. Otherwise, based on the characteristics of the queue, the visitor will choose either to stay in the carriage or to alight in order to enter the attraction. That is, when the attraction is occupied, there are two possible situations.

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(a) If the visitor has knowledge of the system length, he will join or leave the attraction (system) depending on the waiting time. In this case, the balking probability increases as more visitors enter the system, that is, it is an increasing function of the system size (discouraged rate policy). Two different possibilities are assumed: a finite queue if there is a limited space, or an infinite queue when the extension is so large that it may be considered infinite.

(b) If the visitor does not know the system size, the balking probability will be fixed (constant rate policy). For example, when there is an opaque fence with a gate and some instructions about the attraction. Like in the previous assumption, the buffer can be regarded finite or infinite.

We assume that visitors arrive at the attraction through a single waiting line and are served in order of arrival (FCFS). An only visitor can be served in each attraction and his service time is a multiple of five minutes. Every five minutes, the attraction concludes with probability s. Once the visitor is served leaves the attraction (system).

System size

This section studies the stationary distribution of the system size and its ergodicity condition, and it also links our discrete-time model to the continuous-time one.

It can be readily shown that {X m , m ∈ N} is the one-dimensional Markov chain of our queueing system, whose state space is {0, 1, 2, . . . , N + 1} . One of the main objectives is to find the stationary distribution

π k = lim m→∞ P [X m = k] , k = 0, . . . , N + 1 of the Markov chain {X m , m ∈ N} .
The state space and one-step transitions are illustrated in Figure 5.

The Kolmogorov equations for the distribution π k are

π 0 = p π 0 + s p π 1 π 1 = p π 0 + [s p + s p + s p (1 -r 1 )] π 1 + [s p + s p (1 -r 1 )] π 2 π k = s p r k-1 π k-1 + [s p r k-1 + s p + s p (1 -r k )] π k + +[s p + s p (1 -r k )] π k+1 , k = 2, . . . , N π N +1 = s p r N π N + (s p r N + s) π N +1 if N < ∞ and the normalization condition is N +1 k=0 π k = 1 .
The solution of these equilibrium equations is presented in the following theorems, whose proofs are easily obtained proceeding by recurrence. Theorem 1 (Constant rate discipline) The Markov chain {X m , m ∈ N} is ergodic if and only if ρ r δ N,∞ < 1 and its stationary distribution is given by the formulae:

π 0 = p (1 -ρ r) (1 -p r) N [p + ρ (1 -r)] (1 -p r) N -ρ (ρ s r) N +1 π k = ρ k (s r) k-1 (1 -ρ r) (1 -p r) N -k+1 [p + ρ (1 -r)] (1 -p r) N -ρ (ρ s r) N +1 , k = 1, . . . , N + 1 .
Theorem 2 (Discouraged rate discipline) The Markov chain {X m , m ∈ N} is ergodic if and only if ρ δ r,1 δ N,∞ < 1 and its stationary distribution is given by the formulae:

π 0 = 1 + N +1 k=1 ρ k s k-1 r (k-1) k 2 k-1 j=0 (1 -p r j ) -1 π k = ρ k s k-1 r (k-1) k 2 k-1 j=0 (1 -p r j ) 1 + N +1 l=1 ρ l s l-1 r (l-1) l 2 l-1 i=0 (1 -p r i ) -1 , k = 1, . . . , N + 1 .
Let us remark that if r < 1 , in case of infinite buffer and discouraged rate, the system is always ergodic independently of the parametric values. This fact was expected since as the system size increases, the effective rate of entrance diminishes, which constitutes a mechanism to impede an overloaded and unstable system.

We must emphasize the importance of finding the ergodicity hypothesis, which is related to the term "propagation of chaos" [START_REF] Borovkov | Propagation of chaos for queueing networks[END_REF]. Although chaotic systems obey certain rules that can be described by mathematical equations, chaos theory shows the difficulty of predicting their long-range behaviour. This queueing system could be tackled in an alternative way using the chaos theory [START_REF] Borovkov | Propagation of chaos for queueing networks[END_REF][START_REF] Friedman | Long-run dynamics of queues: stability and chaos[END_REF][START_REF] Haxholdt | Mode locking and chaos in a deterministic queueing model with feedback[END_REF]; specifically, we should follow the two crucial steps given by Borovkov [START_REF] Borovkov | Propagation of chaos for queueing networks[END_REF]. Of course, when the parametric values tend to the ergodicity condition, the system becomes unstable and tends to the disintegration. Furthermore, although the system is always ergodic under a finite buffer, it is seriously disturbed-unbalanced when the parametric values approach the ergodicity condition of the corresponding model under an infinite buffer (see the maxima of the graphics in Figure 8).

Remark 1 (Relation to the continuous-time system) This remark analyses the relation between our discrete-time model and its continuous-time counterpart.

We consider the continuous-time single-server balking queueing system where customers arrive according to a Poisson process with rate λ . The balking operation is exactly the same than in discrete-time. Service times are independent and exponentially distributed with mean µ -1 . Interarrival and service times are assumed to be mutually independent. , the previous continuous-time system can be approximated by our discrete-time model choosing the parameters as follows: p = λ ∆ and s = 1 -µ ∆ .

• For the constant rate policy, we have:

lim ∆→0 π 0 = 1 -λ µ r 1 + λ µ (1 -r) -λ µ λ µ r N +1 lim ∆→0 π k = λ µ λ µ r k-1 lim ∆→0 π 0 , k = 1, . . . , N + 1 .
• For the discouraged rate policy, we obtain:

lim ∆→0 π 0 = 1 + N +1 k=1 λ µ k r (k-1) k 2 -1 lim ∆→0 π k = λ µ k r (k-1) k 2 lim ∆→0 π 0 , k = 1, . . . , N + 1 .
These stationary probabilities can be checked since this continuous-time system is a special case of a birth-death process with state-dependent arrivals and departures [START_REF] Cooper | Introduction to queueing theory[END_REF].

5 Waiting/spending time in the FCFS system

This section presents the stationary distribution of the waiting/spending time in the FCFS system of an arriving customer. Firstly, we define the waiting time W (measured in slots) in the FCFS system in the m+1-th slot as the time that a customer would wait in the corresponding system with FCFS discipline if he arrived in the m + 1-th slot.

Theorem 3 The stationary distribution of the waiting time W in the FCFS system is given by:

P [W = 0] = π 0 + s π 1 P [W = j] = min{j,N +1} k=1 j -1 k -1 sk s j-k+1 π k + + min{j+1,N +1} k=2 j -1 k -2 sk s j-k+1 π k , j ≥ 1 .
Proof: The waiting time of a customer who arrives in the m + 1-th slot will be zero if X m = 0 or (X m = 1 and the current service finishes in the m + 1-th slot). Let us remember that the remaining service time follows a geometrical distribution with probability s due to the memoryless property of the geometrical law.

If X m = k ≥ 1 and the current service does not conclude in the m + 1-th slot, the waiting time will be the sum of k -1 complete service times plus the remaining service time of the customer currently being served; then the waiting time will be the sum of k independent and geometrically distributed random variables with parameter s .

If X m = k ≥ 2 and the current service ends in the m + 1-th slot, the waiting time will be the sum of k -1 complete service times, that is, the waiting time will be the sum of k -1 independent geometrical distributions with parameter s .

We now define the spending time S (measured in slots) in the FCFS system in the m+1-th slot as the time that a customer would spend in the corresponding system with FCFS discipline if he arrived in the m + 1-th slot. It is equal to the waiting time in the m + 1-th slot plus the service time of the arriving customer in the m + 1-th slot.

Theorem 4

The stationary distribution of the spending time S in the FCFS system is given by:

P [S = j] = s s j-1 (π 0 + s π 1 ) + (1 -δ j,1 ) min{j-1,N +1} k=1 j -1 k sk+1 s j-k π k + +(1 -δ j,1 ) min{j,N +1} k=2 j -1 k -1 sk+1 s j-k π k , j ≥ 1 .
The proof of this theorem follows the steps given in the proof of the previous theorem.

Unfinished work

This section finds the stationary distribution of the unfinished work, that is, the stationary distribution of the remaining number of slots needed to empty the system of all the customers present. Specifically, the unfinished work U (measured in slots) immediately after the m+1-th slot is the sum of the service times of all customers in the queue and the remaining service time of the customer being served at that time. It is equal to the waiting time in the m + 1-th slot plus the service time of a possible arrival in the m + 1-th slot.

P [U = 0] = p (π 0 + s π 1 ) P [U = j] = p s s j-1 (π 0 + s π 1 ) + +(1 -δ j,1 ) min{j-1,N +1} k=1 (1 -δ k,N +1 ) p r k j -1 k sk+1 s j-k π k + + min{j,N +1} k=1 [1 -(1 -δ k,N +1 ) p r k ] j -1 k -1 sk s j-k+1 π k + +(1 -δ j,1 ) min{j,N +1} k=2 p r k-1 j -1 k -1 sk+1 s j-k π k + + min{j+1,N +1} k=2 (1 -p r k-1 ) j -1 k -2 sk s j-k+1 π k , j ≥ 1 .
Proof: If X m = 0 or (X m = 1 and the current service finishes in the m + 1-th slot):

• if an arrival occurs in the m+1-th slot (with probability p), the unfinished work will be the service time of the arriving customer.

• if an arrival does not occur in the m + 1-th slot (with probability p), the unfinished work will be zero.

If X m = k ≥ 1 and the current service does not conclude in the m + 1-th slot:

• if a customer enters the system in the m + 1-th slot (with probability (1 -δ k,N +1 ) p r k ), the unfinished work will be the sum of k complete service times plus the remaining service time of the customer currently being served; then the unfinished work will be the sum of k+1 independent and geometrically distributed random variables with parameter s .

• if a customer does not enter the system in the m + 1-th slot (with probability 1 -(1 -δ k,N +1 ) p r k ), the unfinished work will be the sum of k -1 complete service times plus the remaining service time of the customer currently being served; then the unfinished work will be the sum of k independent and geometrically distributed random variables with parameter s .

If X m = k ≥ 2 and the current service ends in the m + 1-th slot:

• if a customer joins the system in the m+1-th slot (with probability p r k-1 ), the unfinished work will be the sum of k complete service times, that is, the unfinished work will be the sum of k independent geometrical distributions with parameter s . • if a customer does not join the system in the m+1-th slot (with probability 1 -p r k-1 ), the unfinished work will be the sum of k -1 complete service times, that is, the unfinished work will be the sum of k -1 independent geometrical distributions with parameter s .

Optimal control

In practice of designing concrete systems (computer and communication networks, manufacturing systems,. . . ), a question arises to the manager: How is the buffer size chosen to minimize the cost function? This type of matters motivates that the optimal control of the buffer size is investigated in this section. We develop a steady-state expected cost function per unit time, in which N is a decision variable. Our aim is to determine the optimum value of the control parameter N , say N * , so that this function is minimized. The following costs are considered: The performance measures, such as the expected number of customers in the system µ s and the expected number of customers in the queue µ q , can be obtained from the steady-state probabilities and are given by:

µ s = N +1 k=1 k π k , µ q = N +1 k=2 (k -1) π k .
Since the probability that a customer balks in the system is 1 -r k if the system size is k = 1, . . . , N , the balking probability is given by:

α = N k=1 (1 -r k ) π k 1 -π 0 -π N +1
.

Using the definitions of each cost element listed above, the total expected cost function per unit time is given by:

C(N ) = C 1 µ q + C 2 π 0 + C 3 α + C 4 π N +1 .
We now carry out a sensitivity analysis on the optimum value N * based on changes in specific values of the system parameter r under the two balking disciplines. The following cost elements are used: The lightest cases (C 1 and C 2 ) are due to the costs induced per buffer size and per maintaining the server inactive, respectively; albeit holding customers in the buffer is costlier than having an unoccupied server (C 2 < C 1 ) . The highest costs (C 3 and C 4 ) are presented when a customer does not receive his service, though it is less serious when the omission of the service is a decision of the customer (C 3 < C 4 ) . Obviously, the values of the costs can change as function of the decisor priorities.

The analytical study of the behaviour of the expected cost function is an arduous task since the decision variable N is discrete and appears in a complex and non-linear expression. Hence, the optimum value N * will be the first integer by satisfying the inequality Tables 1-2 present results for constant and discouraged rate policy, respectively. The results are rounded to six significant digits. We first fix (p, s) = (0.5, 0.3) and choose several values of r . The minimum expected cost C(N * ) and the values of various system performance measures µ q , π 0 , α and π N * +1 at the optimum values N * are shown in Tables 1-2 for different values of r . One sees that N * , as function of r , starts increasing to a maximum and then decreases until 2. From the last five columns of Table 1, N * does not change at all when r varies from 0.4 to 0.95. From the last three columns of Table 2, the optimum values N * are the same even though r changes from 0.8 to 0.95. As the intuition tells us, the mean buffer size µ q increases with increasing values of r , while π 0 and α reduce as r increases.

C(N * -1) ≥ C(N * ) ≤ C(N * + 1) . r = 0.05 r = 0.1 r = 0.2 r = 0.4 N * 5
Let us briefly compare the data in Table 1 with those in Table 2. It is observed that C(N * ) is slightly superior with a constant rate policy and the balking probability is moderately higher with a discouraged rate discipline. As was expected, the costs are minimized with the discouraged policy since, under this rule, the balking probability increases as function of the system size. Moreover, the probability that the system is empty (respectively, the loss probability) is smaller (respectively, bigger) in the constant case than in the discouraged one.

Numerical work

This section shows some numerical results regarding the stationary distribution of our systems. Besides, as performance measures, we choose the mean system size µ s , the balking probability α , the variance σ 2 s of the random variable "system size", the mean waiting/spending time and the mean unfinished work. The arrival and departure probabilities p and s in a slot are assumed to be equal to 0.5 and 0.3, respectively. The following graphics and tables corroborate what the analytical results and intuition tell us.

In Figure 6, the stationary probabilities π k are plotted versus r when the buffer size is 3. The curves corresponding to the constant and discouraged rate policies show a similar behaviour. We can observe that π 0 and π 1 decrease with increasing values of r , which also agrees with the intuition. On the other hand, we see that π 2 and π 3 are increasing and then decreasing according to the rises of r . As is to be expected, the loss probability π 4 is increasing as function of r .

Figure 7 considers the effect of the parameter r on the mean system size and the balking probability.

Figure 7(a) illustrates the development of µ s as function of r . We present six curves which correspond to N = 10, 20, 30 subject to the two disciplines. As was expected, the mean system size µ s grows with increasing values of r and N . Under the constant rate rule, this increase is moderate when r varies from 0 to 0.5, while the increment is considerable when r is superior to 0.5. An analogous behaviour is observed for the discouraged rate policy, since the growth is controlled if r < 0.9 and is heightened if r approaches 1. Moreover, assuming r inferior to 0.5, µ s is scarcely affected by the discipline and the buffer capacity. We also note that, for any value of r , the mean system size µ s is bigger in the constant case than in the discouraged one; although the difference between both rules, which increases with N , is intensified when r exceeds 0.5.

In Figure 7(b), the balking probability α is plotted against r for the two disciplines. The curves are decreasing as function of r , in accordance with the analytical expression of α . As is to be expected, the highest curves correspond to the discouraged rate policy. Under the discouraged case, the balking probability α rises with increasing values of N , but these differences are only appreciated when r is close to 1.

Figure 8 depicts the variance σ 2 s of the random variable "system size" against r in relation to three values of N and the two considered schemes. Obviously, σ 2 s increases with N , i.e., the concentration of the stationary distribution of the system size reduces as the buffer capacity grows. It can be seen that all the curves, as function of r , increase to a maximum and then decrease. In the constant rate discipline, observe Figure 8(a), the maxima are got when r takes the value of the ergodicity condition in the case of infinite buffer (r = 1/ρ) . In the discouraged rule, see Figure 8(b), the values of r where the maxima are reached tend to 1 as N increases. Moreover, the maxima of σ 2 s are smaller in the discouraged case than in the constant one, what confirms that the discouraged rate policy continually receives and processes information in order to decide the entrance in the system or not of the arriving customers. In both policies, when r belongs to certain intervals, the variance σ 2 s is only lightly affected by the buffer size, whereas the parameter N influences considerably in the complementary of such intervals.

Tables 345present the mean waiting/spending time and the mean unfinished work for different values of N and r . The results are rounded to four decimal digits. As was expected, these performance measures increase with increasing values of r . In the constant case, for r = 0.05, 0.2 the values do not change when N varies from 10 to 30, whereas for r = 0.4, 0.6, 0.8, 0.95 the values are increasing as function of N . In the discouraged case, for r = 0.05, 0.2, 0.4, 0.6 the values remain constant when N varies from 10 to 30, while for r = 0.8, 0.95 the values grow with N . As is to be expected, these performance characteristics are bigger for the constant policy than for the discouraged one; though the differences, which increase with N , are very small when r = 0.05, 0.2 and are intensified when r = 0.4, 0.6, 0.8, 0.95 .

Conclusions

It should be pointed out that the economic importance of this paper resides in the multiple applications to productive processes, since most of them operate on On the other hand, the mathematical importance is in the direct resolution of the systems of equations, since we find explicit formulae for the stationary probabilities of the system size, of the waiting/spending time in the FCFS system and of the unfinished work. From the solutions we can obtain, among others things, all the moments for the stationary distributions of the number of customers in the system and in the waiting line, the measures of effectiveness,. . . Moreover, the formulae of the finite buffer can be used for the approximate calculus of the stationary features of the respective queueing systems with infinite buffer capacity (for values of N large enough). . . .

Figure 1 :

 1 Figure 1: Mathematical model for the queueing system with infinite buffer.

Figure 2 :

 2 Figure 2: Mathematical model for the queueing system with finite buffer.

4 Page

 4

  Figure 3: Industry.

5 Page

 5

Figure 4 :

 4 Figure 4: Amusement park.

6 Page

 6

  If time is divided into intervals of length ∆ ∈ 0, min 1 λ , 1 µ

10 PageTheorem 5

 105 The stationary distribution of the unfinished work U is given by:

C 1 ≡

 1 holding cost per unit time per customer in the waiting line, C 2 ≡ cost incurred per unit time for keeping the server idle, C 3 ≡ cost per unit time per balked customer, C 4 ≡ cost per unit time per lost customer.

C 1 = 12 Page

 112 2 euros/slot , C 2 = 1 euro/slot , C 3 = 3 euros/slot , C 4 = 4 euros/slot.

  05 r = 0.2 r = 0.4 r = 0.6 r = 0.8 r = 0

  05 r = 0.2 r = 0.4 r = 0.6 r = 0.8 r = 0

[ 29 ] 19 Page

 2919 Yang, T., Posner, M.J.M. and Templeton, J.G.C. The M/G/1 retrial queue with nonpersistent customers. QueueingSystems 7, 209-218 (1990).

Figure 5 : 20 Page

 520 Figure 5: One-step transition probabilities diagrams.

Figure 6 : 21 Page

 621 Figure 6: The buffer capacity is N = 3 .

Figure 7 : 22 Page

 722 Figure 7: The effect of the parameter r .

Figure 8 : 23 Page 23 of 23 Editorial

 82323 Figure 8: The variance σ 2s of the random variable "system size" versus r .

Table 1 :

 1 +1 5.62163 • 10 -7 2.38219 • 10 -6 0.000184056 0.153029 Constant rate policy.

	6	6	2

*

Table 2 :

 2 +1 1.87364 • 10 -8 1.37687 • 10 -10 1.37509 • 10 -7 7.89711 • 10 -9 Discouraged rate policy.

	4

*

Table 3 :

 3 The mean waiting time.

	.95

Table 4 :

 4 The mean spending time. a discrete-time basis. In fact, this paper presents two examples applied to the industrial and services sectors. Furthermore, the optimal control of the buffer size has been investigated in order to minimize a cost model, what represents a main objective from the enterprise point of view.

Table 5 :

 5 The mean unfinished work.Journal of Mathematical and ManagementSciences 15, 35-55 (1995).[2] Acemoglu, D. and Shimer, R. Holdups and efficiency with search frictions.International Economic Review 40, 827-849 (1999).[27] Wang, K.-H. and Chang, Y.-C. Cost analysis of a finite M/M/R queueing system with balking, reneging, and server breakdowns. Mathematical Methods of Operations Research 56, 169-180 (2002). [28] Woodward, M.E. Communication and computer networks: modelling with discrete-time queues. IEEE Computer Society Press, Los Alamitos, California (1994).

	.95

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

supported by the MEC through the project MTM2005-01248.


Ralf ; Fendel
email: ralf.fendel@whu.edu.

Michael Frenkel
email: michael.frenkel@whu.edu.

Otto Beisheim

Inflation Differentials in the Euro Area: Did the ECB Care?

Keywords: European Central Bank, Inflation Differentials, Monetary Policy Rules JEL classifications: E31, E43, E52, E58, E65

Compared to inflation differentials among regions in the United States, EMU inflation differentials are larger and more persistent. Based on augmented monetary policy reactions functions, this paper addresses the question whether the presence of pronounced inflation differentials in combination with low average inflation rates has influenced monetary policy decisions of the ECB. The paper finds statistical evidence that the ECB took inflation differentials into account which may reflect the fear of deflation in low inflation countries like Germany.

Introduction

Six years after the launch of the euro, inflation differentials among participating countries still exist. Although they converged in the run-up to European Monetary Union (EMU), they diverged around 1999, before they somewhat narrowed down again thereafter. Nevertheless, stylized facts show that inflation differentials within EMU remain larger and more persistent than those between regions of the United States and during the deutschmark era before 1999 between the different Länder of Germany. The European Central Bank (ECB) emphasizes that it pays due attention to the presence of inflation differentials within the euro area (ECB, 2003 and 2005). It can also be anticipated that the accession process further increases possible inflation differentials.

Potential reasons for the observed inflation differentials are manifold and have been extensively discussed in the literature (see, for example, Altissimo et al., 2005).

Moreover, the economic consequences -positive or negative -have been highlighted (Hofmann and Remsperger, 2005). There seems to be a consensus that monetary policy within EMU cannot offset inflation differentials between member countries because there is no room for regional or national considerations. However, it may well be that the ECB, in deciding on its monetary policy, takes inflation differentials into account.

Against this background, this paper examines the question whether the presence of relative pronounced inflation differentials has influenced the behavior of the ECB since the launch of the euro. With average inflation above target and considerable inflation differentials, some countries may experience quite low levels of inflations or even approach a situation of deflation. Then, although aggregate inflation calls for restrictive monetary policy, the ECB might have been reluctant because of the "fear of national deflation" with possible contagion effects spreading through the euro area. The rest of the paper has the following structure. Section 2 presents some stylized facts on inflation differentials in the euro area. Section 3 discusses possible causes of the observed inflation differentials. Section 4 includes measures of inflation differentials into interest rate reaction functions in order to find statistical evidence whether such differentials have influenced the interest rate policy of the ECB. Finally, section 5 concludes.

Inflation Differentials in the Euro Area: Some Stylized Facts

Inflation differentials have significantly declined in the euro area in the run-up to EMU.

Figure 1 shows that the unweighted cross-country standard deviation of annual national inflation rates measured by the Harmonized Index of Consumer Prices (HIPC) has decreased significantly. 1 Most of this process can be attributed to the process of convergence and reflects the fact that individual countries needed the reach lower rates of inflation in order to qualify for EMU. After establishing the EMU, the inflation differentials tended to increase again. As this was also a period when the aggregated rate of inflation in the euro area rose somewhat, one could expect that the increase in dispersion was mainly driven by an increase in levels. However, the data do not support this view. The correlation between the standard deviation of inflation rates and the aggregate euro area inflation is about 0.18. 2 Figure 1 also shows the inflation dispersion of the US measured by the unweighted standard deviation of inflation rates in the four US Census Regions Northeast, South, Midwest and West. The US standard deviation fluctuates around 0.5 percentage points, whereas the unweighted euro area standard 1 Alternative measures for the inflation dispersion are the weighted cross-country standard deviation, the maximum span of inflation rates, and the coefficient of variation. All these measures, however, display the same qualitative picture. See ECB (2003, Annex 1) for a discussion of possible divergence measures. 2 The ECB (2003, p. 46) also expresses the view that there is no comovement in inflation rate levels and its dispersion. Whether the inflation rate differentials within the EMU will converge to the levels observed in the US will depend on various structural features within the EMU. These are discussed in more details in the section 3.

< INSERT FIGURE 1 ABOUT HERE >

Inflation differentials in the euro area also appear to be more persistent than in the US in the sense that many countries have systematically maintained either a positive or a negative inflation gap against the euro area average since the introduction of the euro.

Table 1 shows the national inflation differentials against the euro area average. It reveals that, since 1999, seven out of twelve countries have inflation rates that are constantly either above or below the euro area average. Some of the differentials are of relative large size (i.e., larger than one percentage point) over prolonged periods of time. Looking at the US, inflation differentials larger than one percentage point and lasting for more than two years have occurred only in a few specific cases. This leads the ECB to the statement that "[t]he persistence of inflation differentials seems, thus, to be a specific feature of the euro area" (ECB, 2005, p. 63).

3 An alternative available measure for the US inflation dispersion is the unweigthed standard deviation of CPI inflation for 14 metropolitan areas. This measure is somewhat higher than for the four Census Regions (see ECB, 2005, Chart 1). However, since these metropolitan areas are much smaller than most euro area countries, the four Census Regions are probably the more appropriate base for a comparison.

< INSERT TABLE 1 ABOUT HERE >

The differences between inflation rates among euro area countries also involve an important sectoral dimension. The HICP consists of five main sub-components: services, non-energy industrial goods, energy, processed food, and unprocessed food.

Due to their high weights, about two thirds of the overall HICP consist of the first two sub-components. According to an analysis of the ECB (2005), the service sector shows 50 percent more inflation dispersion than the sector of non-energy industrial goods, which reveals that, within the euro area, cross-country inflation differentials in the tradable sector are much lower than differentials in the services sector. 4 However, the available evidence by and large indicates that there is no one single factor that explains the existence and the persistence of inflation differentials in the euro area. The following section provides a survey on the potential origins of the inflation differentials.

Causes of Inflation Differentials within a Currency Area

In this section, we briefly discuss possible causes of inflation differentials in a monetary union and, in turn, the potential economic consequences that arise from them.

Identifying the underlying causes of inflation differentials is not an easy task, because in a large monetary union like the euro area, a number of factors are likely to contribute to the observed inflation differentials. Three types of factors can be distinguished: transitory factors related to the process of convergence; permanent factors related to national economic structures; and policy-induced factors related to diverging national policies or to divergent regional responses to euro area-wide policies. Transitory factors related to the process of convergence among EMU members have contributed and are still likely to contribute in the near future to the dispersion of inflation rates in the euro area. This is the case for at least three reasons. First, the convergence of nominal and real interest rates among the countries to levels that previously were observed only in the low inflation countries before EMU together with the high degree of capital market integration within the euro area may has contributed to a surge in aggregate demand in the formerly high interest rate countries. This has exerted sustained upward pressure on prices, particular in the non-tradable and services sector (ECB, 2005). Second, the implementation of the Single European Market together with the introduction of a single currency has certainly contributed to a marked decline in price level dispersion, mainly in the tradable goods sector. This convergence is likely to have contributed to some degree to the inflation differentials at least in the first years of EMU. Rogers (2002) estimates that the price level dispersion in 1999 contributed to the observed annual HICP inflation dispersion at the end of 2002 and amounted to around 16 percent of the overall inflation dispersion. Third, the Balassa-Samuelson effect (BSE) might have also contributed to the inflation dispersion in EMU.

Even with tradable goods prices being similar in EMU member countries, productivity differences account for differences in nontradable goods prices and, thus, for differences in overall inflation rates. However, as productivity convergence has already been substantial in EMU, the size of the BSE for the current member countries is likely to diminish over time and recent calculations suggest that its size is fairly small. 5 5 See, for example, the annual report of the "Sachverständigenrat" -the German Council of Economic Advisors -for 2001. See additionally ECB (2003, p. 32 Table 5) for a summary of available studies. Those studies, however, consider periods prior to the EMU and tend to overstate to actual contribution of the BSE, since productivity convergence was considerably stronger prior to the EMU. On average, the studies imply that the BSE results in a standard deviation of 0.6. Another structural feature that is likely to contribute to the divergence of inflation rates is the degree of openness, and related to this the composition of EMU-external trade. According to this argument, countries within EMU face different inflationary pressure from exchange rate depreciation and/or from a price increase of imported goods and raw material. Indeed, Honohan and Lane (2003) find that country-specific exchange rate effects matter to some degree for inflation differentials. 7 More generally, F o r P e e r R e v i e w 8 the process of adjustment to changing economic conditions is a source of inflation differentials in the euro area. If the degree of price and wage rigidities which are responsible for the path of adjustment differs between EMU members the inflation dynamics will also differ. Empirical evidence of the quantitative importance of these rigidities is rarely available and is a focus of current research (see Angeloni et al., 2004). 8

The last group of reasons for inflation differentials consists of policy-related factors.

Both area-wide and regional policies might contribute to the degree of heterogeneity in the euro area. In particular, fiscal policy is such a source. For example, changes in administered prices, which account for about 6 percent of the HICP, and changes in indirect taxes can contribute to inflation differentials at least in the short run. In addition, the use of national fiscal instruments potentially reinforces inflation differentials. Finally, monetary policy, although centralized in EMU, can contribute to inflation dispersion via different transmission across member countries, e.g. through different structures of the banking sector. Clausen and Hayo (2006), for example, find evidence for asymmetries in the interest rate transmission within the EMU for France, Germany and Italy.

In sum, inflation differences can arise for a variety of reasons. However, it is widely recognized that assigning monetary policy the role of directly addressing these inflation differentials would clearly overburden a central bank. However, central banks might take into account the issue of inflation differentials more indirectly. The ECB (2003, p.

6) states:

analysis Cunningham/Haldane (2002) use calibrated simulations to demonstrate the impact of alternative specifications of import-price pass-through on the monetary policy transmission mechanism. 8 In a semi-structural modelling approach Clausen and Hayo (2006) find asymmetries in the effects of output gaps on inflation within EMU member countries. This can also be seen as an indication for the existence of structural differences in the EMU. This quote indicates that the ECB is indeed concerned about inflation differentials in the euro area and attributes a "secondary role" to them. Motivated by this observation, the subsequent section examines the ECB's monetary policy in order to identify statistically robust responses of the ECB to the observed inflation differentials.

The Reaction of the ECB to Inflation Differentials

All major central banks in industrial countries currently conduct monetary policy by using market-oriented instruments in order to influence the short-term interest rate (Borio, 1997). Since the seminal paper of Taylor (1993), it has become common to describe this interest setting behavior of central banks in terms of monetary policy reaction functions. In its plain form, the so-called Taylor rule states that the short-term interest rate which, in this analysis, represents the instrument of a central bank reacts to deviations of inflation and output from their respective targets. Clarida et al. (1998) proposed a forward-looking variant of the Taylor rule which takes into account the preemptive nature of monetary policy as well as interest smoothing behavior of central area can also satisfactory be explained by this kind of reaction functions. Among them are Fendel/Frenkel (2006), Fourçans/Vranceanu (2004), and Hayo/Hofmann (2006). 9

Following Clarida et al. (1998) the baseline policy rule takes the form:

(1) *) ( *) ( * 2 1 q t q t t k t t y y E E i i + + + + + = ,
where i* is the desired level of the nominal short-term interest rate, and i is its long-run equilibrium level. The second term on the right-hand side is the expected deviation of the k-period ahead inflation rate ( ) from the target rate ( *) which is assumed to be constant over time. The third term is the expected deviation of output (y) from its natural level (y*) q periods ahead (i.e., the output gap). Combining (1) and (2) leads to

(3) ( )

t t q t q t t k t t t i y y E E i i + + + + = + + + 1 2 1 *) ( *) ( ) 1 ( .
In order to arrive at a testable relationship, the unobservable terms in equation ( 3) have to be eliminated. Therefore, we rewrite equation (3) as:

(4) ( ) Following the common approach of the literature for testing reaction functions, we implement the GMM estimation methodology using the correction for heteroscedasticity and autocorrelation of unknown form. 11 In addition, we chose Bartlett weights to ensure positive definiteness of our estimated variance-covariance matrix. We set k =12 and q = 3 in equation ( 4) and use the constant as well as the lagged values of the output gap and the inflation rate and the short-term interest rate as the core set of instruments. 12 Specification (1) in Table 2 displays the forward-looking Taylor rule in its conventional form as shown in equation ( 4). All parameters are in the expected range. In particular, the reaction coefficient on inflation is greater than unity, indicating that the Taylor principle holds. We use this result as our benchmark.

t t q t q t k t t i y y i + + + + = + + + 1 2 1 0 * ) 1 ( ) 1 ( ) 1 ( ,
<INSERT TABLE 2 HERE> 10 We find that that monetary policy reaction functions for the ECB are robust against changes in detrending method. This confirms the result of Fendel/Frenkel (2005).

11 Huang/Lin (2006) propose an alternative estimation approach based on a dynamic ordered probit model with time-varying parameters carried out via recent advances in Bayesian simulation approach, namely, the Markov chain Monte Carlo.

12 More specifically, we use the first six lags as well as the ninth and the twelfth lag of the output gap and the first, the third, the sixth, the ninth and the twelfth lag of the short term interest rate and the inflation gap. This instrument set displays the core set of instruments that is present throughout all subsequent regressions. In the following we only report the instruments that we add the this core set of instruments in particular regressions. The J-statistics which are not reported here indicate that we chose a valid set of instruments for all subsequent regressions. We start with the first and indirect way of accounting for inflation divergence and calculate two different measures for the "divergence-adjusted inflation gap". The first measure is calculated by dividing the expected inflation gap of regression (1) by one plus the actual unweighted standard deviation of inflation. A standard deviation of zero would then mean that we divide the aggregate inflation gap by one, which leaves it unchanged. The higher the standard deviation is, the smaller is the modified inflation gap. Hence, higher inflation divergence then translates into a smaller inflation gap in the reaction function.

Why should larger inflation divergence have the same effect on the behavior of a central bank as a lower inflation rate? Imagine a monetary union in which all members have an expected inflation rate of, say, three percent. Given a target rate of inflation of, say, two percent, the aggregate inflation gap is one percent and the central bank then probably applies some restrictive monetary policy measures. Now imagine again an aggregate expected inflation rate of three percent but with the following distribution of inflation rates among member countries of the currency union: 50 percent of the member countries display an inflation rate of three percent, 25 percent a rate of 5 percent and the remaining 25 percent a rate of unity. Although the aggregate measure is identical as in the first case considered above, one could expect that the central bank is more reluctant to close the inflation gap in the second case, because applying restrictive 14 policy measures could lead to deflation in the low inflation countries. 13 Thus, the ECB cannot counteract inflation divergence directly through monetary measures but it could respond differently to aggregate inflation in the presence of inflation differentials.

Regression (2) in Table 2 reports the estimation results based on the modified inflation gap. The reaction coefficient of this estimate is still significant. This can be interpreted as support for the effect of inflation differentials on the behavior of the ECB as outlined above. We also calculate a second measure of the "divergence-adjusted inflation gap". This measure is calculated by dividing the expected inflation gap of regression (1) by one plus the coefficient of variation. The estimation of the reaction function that uses this modified inflation gap is shown as regression (3) in Table 2. The results show that the reaction coefficient remains significant indicating the robustness of the result.

Since the value of R 2 from regression (1), which is already very high, does not change significantly when including the modified inflation measures in regressions ( 2) and (3), it is not possible to finally judge whether the new inflation variables contribute to the explanatory power of the model. However, the high R 2 seems to be mainly driven by the presence of the smoothing term. Although including the smoothing term is consistent with the standard formulation for monetary policy reaction functions, we also examine alternative formulations that do not allow for interest rate smoothing. We instead use the change in the interest rate as the left-hand side variable which is then explained by a constant, the expected inflation gap and the expected output gap: 13 The Japanese experience points to adverse effects of deflation so that a "fear of deflation" seems justified. Among others, Morana (2005) argues that monetary policy played a pivotal role for Japanese deflation and that a more expansionary monetary policy stance of the Bank of Japan could have stopped the deflation in Japan. ) ( )

F

q t t q t k t t k t t t y E y E + + + + = 2 1 .
Table 3 presents the results. Since we consider the regressions in Table 3 as complements to the specifications in Table 2 we denote corresponding specifications with the same number but add an "a" to denote "alternative". All coefficients are again significant and of the expected sign, i.e., positive, for the inflation variable and the output variable. Due to the missing smoothing parameter, the R 2 values are much smaller compared to the standard formulations in Table 2. However, comparing their sizes shows that using the modified measures of the inflation gap in specifications (2a) and (3a) improves the overall explanatory power of the model. 14

< INSERT TABLE 3 HERE >

Judging whether the reaction to the expected inflation gap becomes indeed more reluctant given the information of a higher divergence of actual inflation rates is not possible from the regressions in Tables 2 and3. Given that the coefficient of the inflation is significant in both the benchmark and in the modified specification, we can 14 We also re-estimated the regressions based on formulation (4) and forced the smoothing parameter to be zero. As the R 2 values of the estimations (using the three alternative inflation measures) are significantly lower than for the specifications in Table 3, we do not present the results here. However, also in these cases an improvement in the R 2 when employing the modified measures for the inflation gap was also present. Taylor rule estimates. However, it does not yet show which way the ECB reacts to inflation differentials. In order to find out the sign of reaction of the ECB we need to proceed along the second route and add a measure of divergence as an individual explanatory variable. We therefore augment the benchmark regression equation ( 4) by adding a measure of inflation divergence (Div t+l ) as a separate variable according to:

(7) ( ) t t l t q t q t k t t i Div y y i + + + + + = + + + + 1 3 2 1 0 ) 1 ( * ) 1 ( ) 1 ( ) 1 ( , where ( ) ( ) ( ).
)

1 ( ) 1 ( ) 1 ( 3 2 1 l t t l t q t t q t k t t k t t t Div E Div y E y E + + + + + + =
In contrast to regressions (2) and (3), we allow for a forward-looking specification of the divergence measure. We use four different measures of inflation divergence and show the results in Table 4.

<INSERT TABLE 4 HERE>

Regression (4) uses the 12-month ahead unweighted standard deviation of the inflation rates of the euro area countries as a measure of divergence. The reaction coefficient of the divergence measure is significantly negative. The inflation reaction coefficient stays significantly positive, but displays a far larger value compared to the benchmark regression. The negative coefficient for the divergence measure can be interpreted in the following way: higher expected inflation divergence makes the ECB more reluctant to raise interest rates for any given level of expected inflation gap. This result turns out to be robust to changes in the divergence measure. In specification (5), we use the squared standard deviation in order to account for possible non-linearity in the ECB's reaction towards divergence. Specifications ( 6) and ( 7) include the coefficient of variation in linear and squared form, respectively. Again, the "reluctance-effect" is significant. 15 We also re-estimate the specifications (4) through ( 7) by excluding the smoothing term and use the change in the interest rate as the left-hand side variable instead. As before, excluding the smoothing term allows us to better proof the additional explanatory power of the inflation divergence term. The results are shown in Table 5.

All parameters are again significant and of the expected sign. The R 2 values indicate again that the inflation differentials have additional explanatory power, particularly in squared form.

<INSERT TABLE 5 HERE>

We also examine whether the reluctance effect described before has been present from the very beginning of EMU. To this end, we re-estimate the previous specifications for the sub-period of 1999:01 to 2003:05. The endpoint of this period coincides with the announcement of the redefinition of the ECB`s strategy, whichaccording to the statement of the ECB -should address the issue of inflation divergence 15 On top of the core set of instruments we instrument the expected standard deviation of inflation rates and the expected coefficient of variation rates by their first six lags, respectively. For the squared forms of the two divergence measures we add a one before squaring them because they contain values smaller than unity. We also tried the contemporaneous divergence measures in some additional regressions but the effects were not significant in those cases and we, therefore, abstain from reporting them. 6 presents the estimation results based on the coefficient of variation as the measure of inflation divergence. 16 <INSERT TABLE 6 HERE> Regression (8) corresponds to the benchmark regression (1) and shows a significant response to the expected inflation gap according to the Taylor principle. However, employing the "divergence-adjusted inflation gap" in specification (9) shows a less significant effect for the sub-sample. The direct inclusion of the divergence measure in a linear as well as in a quadratic form in specifications ( 10) and ( 11) turns out to be statistically insignificant. 17 These results provide some indication that the "reluctance effect" in the ECB's interest rate setting behavior was not present from the very beginning but emerged in the course of declining aggregate inflation rates. 18

Conclusions

Our study examines whether inflation differentials may have had an effect on how the ECB responded to inflation gaps in the euro area. We use different ways of how inflation differentials could be taken into account in a central bank's reaction function and also examine whether the behavior of the ECB has been the same from the beginning. The results point indeed to an influence of inflation differentials on monetary policy in the euro area. With higher inflation divergence, the ECB was more reluctant to 16 We also performed the regressions based on the unweighted standard variation but results turned out to be quite identical. For the sake of brevity, these results are not presented. 17 Again the first six lags of the divergence measure were employed as additional instruments in regressions ( 10) and ( 11). 18 We also tested more explicitly for the presence of structural breaks over the whole sample period but did not get statistically robust results of a regime shift. The same results were obtained when we examined the specification in the alternative form with interest rate differences on the left-hand side. 19 fight an overall inflation gap. A possible reason could be that the ECB feared that restrictive monetary policy could easily lead to deflation in low-inflation countries if it took a very tough stance on fighting overall inflation. This view is supported by our finding that the ECB displayed this "reluctance effect" particularly in more recent years during which some member countries of the euro area had inflation rates that, once adjusted for the causes of a bias (e.g., quality effects and the influence of new products) were already close to zero.

A task for future research is to develop a theoretical macro model that starts from a central bank loss that explicitly incorporates to inflation differential to derive an optimal interest rate rule that also includes such a variable. Along the same lines, calibrated exercises could be performed to verify how a greater or a smaller differential affects the reaction coefficients. Instruments in all regressions are a constant, the first six lags as well as the ninth and the twelfth lag of the output gap and the first, the third, the sixth, the ninth and the twelfth lag of the inflation gap. The following instruments were used in all regressions: a constant, the first six lags as well as the ninth and the twelfth lag of the output gap and the first, the third, the sixth, the ninth and the twelfth lag of the short term interest rate and the inflation gap. Regressions (4) and ( 5) additionally contain the first six lags of the standard deviation of inflation rates. Regressions ( 6) and ( 7) additionally contain the first six lags of the coefficient of variation of inflation rates. The following instruments were used in all regressions: a constant, the first six lags as well as the ninth and the twelfth lag of the output gap and the first, the third, the sixth, the ninth and the twelfth lag of the inflation gap. Regressions (4a) and (5a) additionally contain the first six lags of the standard deviation of inflation rates. Regressions (6a) and (7a) additionally contain the first six lags of the coefficient of variation of inflation rates. The following instruments were used in all regressions: a constant, the first six lags as well as the ninth and the twelfth lag of the output gap and the first, the third, the sixth, the ninth and the twelfth lag of the short term interest rate and the inflation gap. Regressions (10) and ( 11) additionally contain the first six lags of the coefficient of variation of inflation rates.

F

Page 29 of 28 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

  twice as high since 1999.3 Regarding projections of the evolution of the inflation rate differentials, no clear cut picture emerges from fitting the data with detrending methods like a Hodrick-Prescott filter. Due to the usual endpoint problems of such methods, projections heavily depend on the chosen endpoint of the data series.

  may become more relevant in the future when new EU member countries join EMU.Permanent factors related to national economic structures may have also contributed to inflation differentials in EMU. One such structural factor relates to national differences in the consumption preferences among households. This heterogeneity in preferences is reflected in the fact that the shares of the various goods and services in national consumption and value added differ between member countries. Consequently, they have different weights in the various sub-indices of the national HICPs.6 Since the same product has different weights in the respective national HICP, a pure 'composition effect' can, thus, lead to different inflation rates even if individual product prices show the same dynamics. However, empirical evidence indicates that this factor constitutes only a relatively minor reason for the inflation dispersion. The German Council of Academic Advisers (Sachverständigenrat, 2001) calculated that the 'composition effect' explains only 2 percent of the measured inflation variance in the euro area in 2000.

  concern for the ECB`s monetary policy strategy is to avoid that the presence of unavoidable inflation differentials […] could push lowerinflation regions within the euro area towards inflation levels that could complicate the process of economic adjustment in the presence of downward nominal rigidities. Therefore, the ECB's monetary policy strategy attributes a secondary role to inflation differentials when calibrating the safety margin for admissible inflation in the euro area. In this respect, the ECB's explicit aim of maintaining euro area inflation below but close to 2% is regarded as sufficient to address those concerns."

  particular type of reaction function has become very popular in applied empirical research on Taylor rules. A number of studies demonstrated that the ECB's monetary policy over the course of the first years of single monetary policy in the euro

  benchmark regression, we include in additional estimations a measure of inflation differentials. To this end, we proceed along two alternative avenues. The first one is to modify the EMU average inflation gap measure in equation (4) when inflation differentials occur. The second avenue is to include an explicit measure of divergence as an individual explanatory variable in the reaction function.

(

  including inflation diversion in the inflation gap does not deteriorate the

  fear of (national) deflation. Table

Figure 1 :

 1 Figure 1: Dispersion of Inflation Rates in the euro Area and the US, 1996 -2004

  ) describes the degree of interest rate smoothing behavior and t represents an i.i.d. exogenous random shock to the interest rate.

	where the parameter		(with	0	<	1
				F o r
									P
												e
													e r
															R
																e
	leads to														v i e w
	(2)	i	t	=	1 (	i )	t	*	+	i	t	1	+	t	,
																11

The coefficients 1 and 2 represent the intensity with which the desired interest rate of the central bank reacts to the inflation and the output gap. The assumption of interest rate smoothing behavior

9

Altavilla/Landolfo (2005) use a Markov-switching approach in order to analyze whether the ECB reacts asymmetrically. The authors do indeed find that the phase of the business cycle is an important matter in the ECB's monetary policy decisions. Furthermore,

Fendel/Frenkel (2005)

employ augmented monetary policy reaction functions and show that the ECB also takes into account information from the term structure of interest rates.

  In order to estimate equation (4), we employ the following data. The sample period is 1999:01 to 2005:06. For euro area output, we use monthly data on industrial production taken from the IMF International Financial Statistics. The short-term interest rate is the EONIA (European Overnight Index Average) as reported by the ECB.Inflation is measured by the HCPI provided by Eurostat. The output gap is based on natural output calculated by a linear trend.10

				F o r
								P
									e
	where									e r
	0 = i			1	*	and	t	=	t	1	1 (	( R k t + ) e t E	t	+	k	)	2	1 (	)	(	y	t	+	q	E	t	y	t	+	q	)	.
	Furthermore, we assume that			v i e
	(5)	E	t	= [ ] 0 t t I	.							w
	Here I t is the central bank's information set available at time t. Equation (5) simply
	states that the central bank uses its best possible guess about future inflation and output
	in its interest rate decisions.
													12

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	Submitted Manuscript
	2.5
	1.5 0.5 F unweighted standard deviation o r
	P	Time
	e	US	Euro Area
	e r
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Table 1 :

 1 Differentials in Annual HICP Inflation Relative to Euro Area Average Instruments in all regressions are a constant, the first six lags as well as the ninth and the twelfth lag of the output gap and the first, the third, the sixth, the ninth and the twelfth lag of the short term interest rate and the inflation gap.

			(in percentage points)
		1999 2000 2001 2002 2003 2004 2005
	Belgium	0.0	0.6	0.1	-0,7	-0.6	-0.3	0.3
	Germany	-0.5	-0.7	-0.4	-0.9	-1.0	-0.4	-0.3
	Greece	1.0	0.8	1.3	1.7	1.4	0.9	1.3
	Spain	1.1	1.4	0.5	1.3	1.0	0.9	1.2
	France	-0.6	-0.3	-0.6	-0.3	0.1	0.2	-0.3
	Ireland	1.3	3.2	1.6	2.5	1.9	0.2	0.1
	Italy	0.5	0.5	0.0	0.3	0.7	0.1	0.1
	Luxembourg	-0.1	1.7	0.1	-0.2	0.5	1.1	1.6
	Netherlands	0.9	0.2	2.8	1.6	0.2	-0.8	-0.7
	Austria	-0.6	-0.1	0.0	-0.6	-0.8	-0.2	-0.1
	Portugal	1.0	0.7	2.1	1.4	1.2	0.4	0.0
	Finland	0.2	0.8	0.3	-0.2	-0.8	-2.0	-1.4
	Source: European Central Bank

Notes: Standard errors are in parentheses. (*), (**) and (***) indicate significance at a level of 10%, 5% and 1%, respectively.

Table 3 :

 3 Regressions Based on the Change in the Short-term Interest Rate

	Left-hand side variable:	Regression	Regression	Regression
	change in the interest rate	(1a)	(2a)	(3a)
	0 (constant)	-0.372 **	-0.336 ***	-0.143 **
		(0.164)	(0.107)	(0.058)
	1 (inflation t+12 )	0.163 **
		(0.075)
	1 (inflation t+12 /(1+ std.dev t )		0.280 ***
			(0.098)
	1 (inflation t+12 /(1+ coef. of var t )			0.256 **
				(0.115)
	2 (output gap t+3 )	0.021 **	0.031 ***	0.018 **
		(0.010)	(0.007)	(0.009)
	No. of observations	66	66	66
	Adj. R 2	0.221	0.262	0.263
	DW	2.029	2.117	2.031

Notes: Standard errors are in parentheses. (*), (**) and (***) indicate significance at a level of 10%, 5% and 1%, respectively.

Table 4 :

 4 Results of Augmented Regressions

	Coefficient	Regression (4) Regression (5) Regression (6) Regression (7)
	(smoothing parameter)	0.911 ***	0.920 ***	0.909 ***	0.911 ***
		(0.014)	(0.019)	(0.019)	(0.020)
	0 (constant)	-3.253 **	3.995 ***	2.872	13.913 **
		(1.588)	(0.946)	(4.164)	(6.367)
	1 (inflation t+12 )	3.919 ***	3.938 ***	2.96 **	2.701 **
		(0.966)	(1.428)	(1.32)	(1.371)
	2 (output gap t+3 )	0.703 ***	0.708 ***	0.604 ***	0.625 ***
		(0.100)	(0.109)	(0.118)	(0.124)
	3 (std.dev t+12 )	-3.196 ***
		(1.186)
	3 ([1+std.dev t+12 ] ^2)		-1.157 ***
			(0.438)
	3 (coef. of var. t+12 )			-16.963 **
				(8.644)
	3 ([1+coef. of var. t+12 ]^2)				-6.719 **
					(3.275)
	No. of observations	66	66	66	66
	Adj. R 2	0.965	0.975	0.968	0.968
	DW	1.650	1.699	1.426	1.430
	Notes: Standard errors are in parentheses. (*), (**) and (***) indicate significance at a level of 10%, 5% and
	1%, respectively.

Table 5 :

 5 Results of Augmented Regressions Based on the Change in the Short-term

			Interest Rate
	Left-hand side variable:	Regression	Regression	Regression	Regression
	change in the interest rate	(4a)	(5a)	(6a)	(7a)
	0 (constant)		-0.131	-0.128	-0.171	0.056
			(0.204)	(0.204)	(0.178)	(0.266)
	1 (inflation t+12 )		0.156 ***	0.156 ***	0.191 ***	0.179 ***
			(0.063)	(0.063)	(0.051)	(0.061)
	2 (output gap t+3 )		0.031 ***	0.031 ***	0.019 ***	0.021 **
			(0.011)	(0.011)	(0.007)	(0.009)
	3 (std.dev t+12 )		-0.227 ***
	3 ([1+std.dev t+12 ] ^2) o r 3 (coef. of var. t+12 )	(0.088)	-0.057 *** (0.021)	-0.707 **
	3 ([1+coef. of var. t+12 ]^2) No. of observations	P e 66	66	(0.307) 66	-0.247 ** (0.113) 66
	Adj. R 2 DW		0.251 e r 2.237	0.273 2.252	0.311 2.179	0.303 2.181
			R
			e
			v i e
				w
			27

Notes: Standard errors are in parentheses. (*), (**) and (***) indicate significance at a level of 10%, 5% and 1%, respectively.

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Table 6 :

 6 Results for Sub-Sample 1999:01 to 2003:05

	Coefficient	Regression (8) Regression (9) Regression (10) Regression (11)
	(smoothing parameter)	0.822 ***	0.879 ***	0.853 ***	0.855 ***
		(0.041)	(0.014)	(0.029)	(0.029)
	0 (constant)	-2.402	1.180	0.331	2.163
		(2.162)	(0.886)	(2.849)	(3.706)
	1 (inflation t+12 )	2.688 ***		2.211 **	2.197 **
		(1.015)		(0.985 )	(0.999)
	1 (inflation t+12 /(1+coef. of var. t )		0.962 *
			(0.549)
	2 (output gap t+3 )	0.305 ***	0.645 ***	0.409 ***	0.413 ***
		(0.081)	(0.059)	(0.066)	(0.0672)
	3 (coef. of var. t+12 )			-5.043
				(3.521)
	3 ([1+coef. of var. t+12 ]^2)				-1.947
					(1.278)
	No. of observations	53	53	53	53
	Adj. R 2	0.916	0.964	0.934	0.934
	DW	1.072	2.176	1.397	1.404

Notes: Standard errors are in parentheses. (*), (**) and (***) indicate significance at a level of 10%, 5% and 1%, respectively.

See alsoECB (2003, p. 6).

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

For example, the category 'meat' has a weight of 3 percent in the Austrian HICP and 5.5 percent in the Spanish HICP.

Employing pass-through regressions Campa/Gonzalez/Minuez (2006) find that differences in the degree of transmission of a common exchange rate movement into consumer prices among the euro area countries do exist. For example, they calculate an average pass-through rate to consumer prices for the euro area to be about 0.5 with country-specific rates that range from 0.2 for Italy to 1.5 for Ireland. Furthermore, they demonstrate that most of the differences among member countries are due to the distinct degree of openness rather than to the heterogeneity in the structure of imports. In an applied