Stochastic reaction strategies and a zero inflation equilibrium in a Barro-Gordon model Текст научной статьи по специальности «Математика»

Christian-Oliver Ewald1 and Johannes Geifiler2

1 University of St.Andrews,

School of Economics and Finance,

Department of Economics, University of St.Andrews,

St Salvator’s College, St Andrews, Fife KY16 9AL, Scotland, UK E-mail: [email protected] WWW home page: http://www.st-andrews.ac.uk/~ce16

2 University of St.Andrews,

School of Economics and Finance,

Department of Economics, University of St.Andrews,

St Salvator’s College, St Andrews, Fife KY16 9AL, Scotland, UK E-mail: [email protected]

Abstract We study a game theoretic model of the conflict which arises between a monetary authority and the private sector with regard to the inflation-rate. Building on the simple static Barro and Gordon (Barro and Gordon, 1983a) model we assume that rather than playing a one shot game the monetary authority and private sector react to each other repeatedly for an infinite number of times. Both, the monetary authority’s and the private sector’s reactions are assumed to be stochastic in the form of fixed behavioral transition probabilities. These probabilities are interpreted as strategies in a new game. We study the set of Nash-equilibira of this new game and how these correspond to the classical discretionary Nash-equilibrium identified by Barro and Gordon as well as the non-Nash low inflationary state. In contrast to Barro and Gordon we show that the low-inflationary state can be realized as a Nash-equilibrium in our model.

Keywords: Monetary Policy; Game Theory; Stochastic Reactive Strategies.

1. Introduction

Monetary policy has been discussed in many macroeconomic investigations by use of various techniques, models and assumptions. One direction of research leads into game-theoretic models and has been started by influential works such as Barro and Gordon (Barro and Gordon, 1983a) and Kydland and Prescott (Kydland and Prescott, 1977). These game theoretic models have at their center the conflict between a monetary authority and the private sector. The monetary authority is assumed to have at least some sort of control over the level of inflation, from which it may try to exploit the Phillips curve, while the private sector’s objective is to predict the inflation rate correctly in order to make the right decision with regards to current employment. Specifically the model by Barro and Gordon (1983a) is set up in the following way: First the private agents choose their expected rate of inflation ne <G R+ and announce it. Then the monetary authority private can choose actual inflation n <G R+. The cost to the monetary authority is given by

Z := —it2 — b(TT — 7re).

(1)

This cost illustrates the tradeoff between aversion toward inflation on the one hand, and benefits due to a lower level of unemployment caused by a surprise inflation. These effect are controlled by the parameters a and b. The private sector’s losses are given by

(n - ne)2 (2) which means that whatever the level n of inflation the monetary authority sets, it is optimal for the private sector to have expected exactly this level, i.e. ne = n. Assuming optimal behavior of the private sector, the game then essentially becomes a single player game and in fact a standard quadratic optimization problem which can be easily solved. In fact the equilibrium level of inflation is given by

*D = - (3)

a

The result is a single strict Nash-equilibrium (nD , nD) where the monetary authority delivers and the private sector correctly anticipates high inflation. The inflation level nD in equation (3) is referred to as discretionary inflation. Though the private agents’ optimal behavior is dependent on the bank’s choice of n, the opposite is not true for the bank at all - equation (1) is always optimized not taking into account any value for ne. Therefore the original character of the game (private agents choose ne and announce it) is technically equivalent to a simultaneously played game. We only have to ensure that the private sector cannot see the n when choosing the corresponding ne. The beneficial effect of high inflation for the monetary authority is however eradicated because the private sector correctly anticipates discretionary inflation, while the negative effect caused by the part |7r2 remains in the payoff to the monetary authority.

The equilibrium outcome of the classical Barro Gordon game must therefore be regarded as inefficient. In fact, both monetary authority and private sector would be better off choosing n = ne = 0. While not applicable, inefficiencies like this can arise in the economic context. On the other hand there is also empirical evidence that the inflation rate (3) predicted by the classical Barro and Gordon model appears to be too high. In practice there should be mechanisms that lower the equilibrium inflation rate, which are not present in this model. Within the context of efficiency many of these mechanisms have to do with reputation and trust and are generally studied within the framework of repeated games. In a repeated game players have the opportunity to punish their opponents if they divert from a particular strategy. A general problem however is the credibility of these punishment strategies. Barro and Gordon introduced the so called loss of reputation framework in (Barro and Gordon, 1983b). However it has been argued by al-Nowaihi and Levine (1994) that under the assumption that punishments only hurt the central bank but not the private sector, the only remaining equilibrium remains the high inflation discretionary one. In this article we argue that the low- or zero-inflationary state can be realized as a Nash-equilibrium within a different sort of repeated game setting in which players are assumed to react stochastically to each other according to fixed behavioral transition probabilities. In this way we show that low inflation should not a priori be ruled out based on evidence coming from the classic and repeated Barro and Gordon models. In our model low inflation and high inflation are equally reasonable outcomes, and in furthermore under a simple behavioral assumption the low-inflation level becomes more realistic. The type of strategies which we are using in our model have been introduced by Hofbauer in (Hofbauer and Sigmund) to

explain positive levels of cooperation in the context of the prisoners’ dilemma. Our adaptation of this idea to the Barro Gordon game diverts from Hofbauer in the way that we apply it to asymmetric games and in addition to that give an interpretation of the realized payoffs as accumulated payoffs, while taking discounting into account. Even though we assume that the game is repeated infinitely often, our model is finite time, which means that the time between two periods converges to zero. Effectively our model is then a continuous time model. The time horizon is intended to correspond to a time unit of economic significance, possibly a financial year or a business cycle and it is assume that agents can not change or update their behavioral strategies in this period. In a Companion paper we investigate the change of behavioral strategies over a long term time scale under an adaptive dynamic and in this way investigate dynamic stability aspects of the Nash-equilibria presented in this article. The remainder is organized as follows. In section 2 introduce stochastic reactive strategies and adapt Hofbauer’s approach to asymmetric games. In section three we give an interpretation of payoffs as discounted payoffs which are accumulated over time. We apply this framework to the Barro-Gordon game in section 4. In section 5 we investigate how the discretionary Nash-equilibrium of the original Barro Gordon game can be realized as a Nash equilibrium in our game. In section 7, following a similar analysis as in section 6, we demonstrate that zero-inflation can also be realized as a Nash-equilibrium of our game. Section 8 contains the conclusions.

2. Stochastic Reactive Strategies : Classical Setup

Stochastic reactive strategies have been introduced by Hofbauer in (Hofbauer and Sigmund) to explain a certain degree of cooperation within the prisoners dilemma game. The general idea is that, while the original game is repeated infinitely many times, players act according to fixed behavioral strategies which are represented by certain transition probabilities. If the original game has two pure strategies for each player these transition probabilities are given by the conditional probabilities of playing pure strategy 1 after the opponent played his pure strategy 1 in the previous round and playing pure strategy 1 after the opponent played his pure strategy 2 in the previous round. The unconditional probabilities can be computed using Bayes theorem and it can be shown that these unconditional probabilities converge and can in fact be interpreted as mixed strategies in the original game, leading to a well defined payoff. It has to be said clearly however that the strategies in the new game are the stochastic reaction strategies, consisting of a pair of conditional probabilities (p, q)T and that the space of pure strategies in the new game is therefore given by [0,1] x [0,1]. We do not consider mixed strategies, which would be probability distributions on [0,1] x [0,1] in what follows, but focus on pure Nash-equilibria of the new game. The setup has originally been used for symmetric games, but it is not hard to adapt it to asymmetric games. To see this assume that the payoffs for player one and two are given respectively by the two matrices

Pure strategies can be represented

stochastic

reaction multi strategy then consists of a pair (v,v') <G ([0,1] x [0,1])2 where

v

and v'

Here p,p', q, q' denote the following conditional probabilities:

p = probability player 1 plays ei given player 2 played ei in the previous round

q = probability. player 1 plays e1 given player 2 played e2 in the previous round

p' = probability player 2 plays ei given player 1 played ei in the previous round

q' = probability player 2 plays ei given player 1 played e2 in the previous round

The game is now repeated over and over again and players react stochastically according to the probabilities identified above. If we define

as the action chosen in period n by players 1 and 2 respectively and define

then (Xn) can be formally considered as a discrete time Markov chain with four states, for which transition probabilities can be computed in terms of p,p'q,q'. In principal the following construction can be realized assuming an arbitrary number of strategies ej. However, we stick to the case of two strategies and assume a finite time horizon T, which could possibly correspond to a financial year, a business cycle or a different unit of time, but will essentially assume that within this period the game is repeated infinitely many times. This interpretation is inessential for the classical setup, but crucial in our adaptation in the next section. Hofbauer now proceeds by introducing the unconditional probabilities

and considering their limits for n ^ rc>. These limits can be computed as follows. Note that it follows from Bayes formula that

and by cross-substitution

Cn+2 =p(p'c„ + q'(1 - c„)) + q(1 - p'c„ - q'(1 - c„))

=q + p(q' + (p' - q')c„) - q(q' + (p' - q')c„)

=q + (p - q)(q' + (p' - q')c«)

and similar for cn+2. Under the assumption -1 <p - q< 1 or -1 < p' - q' < 1 convergence of these sequences is guaranteed. Substitution of

(4)

cn+i =p'cn + q'(1 - cn)

cn+2 =pcn+i + q(1 - cn+i)

c = lim cn

n

c' = lim c'n

n

above gives

c(1 - (p - q)(p' - q')) =q + q'(p - q) q + q'(p - q)

c

1 - (p - q)(p' - q')

In a similar way we get an expression for c'. In summary we obtain the following expressions for c and c' as functions of p, q,p' and q':

q+q'(p-q) i ~(p-q)(p'-q')

q'+q(p'-q')

i-(p-q)(p'-q')'

(5)

The vectors (c, 1 - c)4op and (c', 1 - c')4op can be interpreted as (long run) mixed strategies, and Hofbauer argues, that if the payoff in the first round is rather insignificant, a payoff for the whole period can be defined as

vh

PH

= (c1 - c)^ 1 - c' =(c, 1 - c)^ 1 c_c,

where the superscript H indicates Hofbauer-payoffs. Strategies spaces in this setup are given by [0,1] x [0,1]. While not expressed in Hofbauer, it is natural to think of the strategies in terms of (p, q) and (p', q') as geno-typic strategies, while ei and e2or alternatively on the level of mixed strategies c and c', are their pheno-typic realizations.

c

c

p

3. The Model

In this section we apply the the formal setup developed in the previous sections to the original Barro Gordon game as it is stated as a one shot game in (Barro and Gordon, 1983b). In contrast to (Barro and Gordon, 1983b) we assume that this particular game is played repeatedly for infinitely many times within the original period [0,T] according to some fixed stochastic reaction strategies. Whenever the game was played a payoff for each player arises. These infinitely many payoffs are discounted and accumulate over time to produce a payoff for the whole period as described in the previous section. If one only believe in finitely many sub periods (e.g. four) our concept is still reasonable by the fast convergence as shown in the previous table. We now have to specify the matrices A and B that correspond to the original Barro and Gordon game. To be consistent with game theoretic standards we would like to deal with welfare functions rather than cost functions. Furthermore, we set the parameters a and b equal to 1. Hence by (1) and (2) we get

Zt := (trt - 7rte) - ^tt2 (6)

for the central Bank and

-(nt - ne)2

(7)

for the private agents. Under these assumptions, the discretionary rate, obtained when optimizing equation (6) is given by 1. The payoff the monetary authority gets then equals | if the private agents choose irf = 0 respectively — ^ if the private sector chooses nf = 1. If the central bank chooses n = 0 it gets 0 (for nf = 0) respectively —1 given nf = 1. The private agents on the other hand get 0 if they were right and —1 if not. In particular we have:

Strategy Payoff

Bank Private Agents Bank Private Agents

zero zero 0 0

zero discretion -1 -1

discretion zero i -1

discretion discretion l 2 0

For convenience we multiply the payoffs of the monetary authority by two. As this is a positive affine transformation it does not affect the Nash-equlibria at all. Hence suitable payoff matrices for the bank and the private agents are given by

A :=

0 —2 11

and B :=

0 —1 10

(8)

Though the game is started by the private agents, in the following we think of the bank to be player one and the private agents to represent player two (i.e. the monetary authority’s strategy is denoted by v, whilst v' is the strategy of the private agents). Let us re-emphasize however, that for the original one-shot game it does in fact not matter whether the game is started by the private agents or played simultaneously. The concrete interpretation of p,q,p',q' in this case is given as follows: (MA=monetary authority, PA= private agents)

p = MA sets inflation low given PA expected low inflation in the previous round

q = MA sets inflation low given PA expected high inflation in the previous round

p = PA expect low inflation given MA delivered low inflation in the previous round

q' = PA expect low inflation given MA delivered high inflation in the previous round

For a fixed set of probabilities, i.e. fixed strategies v = ^p^ and v' = ^p^, the payoffs are given by

and

Pi (v, v') = (c, 1 — c)

P2(v,v') = (c, 1 — c)

0 —2 11

0 —1 10

c

1 c'

c

1 c'

= 2c' c 1

= 2cc' c c'

(9)

(10)

where c and c' are given as functions of p, q,p' and q' as in (5). It is worth noting that the private agents’ optimal payoff is given by 0.

Lemma 1. The payoff for the private agents never exceeds 0. It will be equal to 0 if and only if either c = c' = 0 or c = c' = 1 holds.

Proof. Clearly c, c' <G [0,1] implies that

‘led — c — c' = c(c' — 1) + c' (c — 1)

<o <o

with equality if and only if c = c' = 0 or c = c' = 1. The statement therefore follows from equation 10.

So far we have motivated the use of stochastic reaction strategies and have developed a concrete model based on the original Barro and Gordon game. We will now focus on particular Nash-equilibria of this game.

4. The Discretionary Nash Equilibrium

We indicated before that we can think of the strategies v = (p, q)T and v' = (p', q' )T as being on the genetic level, while they relate to pheno-typic strategies c, c', which in fact relate to mixed strategies in the original Barro-Gordon game, i.e. mixed in discretion and zero-inflation. In this section we study under which conditions the discretionary phenotype, i.e. high inflation delivered by the monetary authority and high inflation expected by the private agents, can be realized as a Nash-equilibrium on the geno-typic level. We will show that this poses rather restrictive assumption on the behavior of the private agents. First of all we note that (pure) discretion is the case if and only if c = c' = 0. To start with, we need to identify the genotypes which belong to the phenotype discretion.

Lemma 2. Discretionary phenotypes always require q = q' =0. More precisely we state that c = c' = 0 implies q = q' = 0 and furthermore if p < 1 or p' < 1 then q = q' = 0 implies c = c' = 0.

Proof. Looking at the formulas for c and c' (5) the second statement is obvious. Now suppose c, c' and q are equal to zero. Then we conclude from pp' < 1 and (5)

0 =q' + q(p' — q') = q'..

Similarly. assuming that q' = 0 we find that q = 0. Hence suppose q = 0 and q' = 0. Then

and 0 = q' + q(p' — q')

A ' ' q>

and p — q =-----------

q

0 = q + q (p — q) ^ q =>p-q= - — q'

1

=>p-q

p' — q

The only two strategies satisfying this relation are (1, 0, 1, 0) and (0, 1, 0, 1). As can be easily verified, (0, 1, 0, 1) never delivers c = c' = 0. Hence q = q' = 0.

The following Corollary states that each players payoff under discretionary behavior is independent of the values p and p'. This fact will play an important part in the proof of our first main theorem.

Corollary 1. Assume that 0 < p + p, p' + p' and either p + p < 1 or p' + p' < 1. Then we have

Pi(p, 0,p', 0) = Pi (p + p, 0,p' + p', 0) = —1 P2(p, 0,p', 0) = P2 (p + p, 0,p' + p', 0) = 0.

Proof. By the previous Lemma c and c' are zero for any choice of p and p' as long

as q,q' = 0 and p,p' < 1 holds. The result then follows directly from (9) and (10).

An immediate consequence is that in our setup, discretion can no longer be realized as a strict Nash-equilibrium. The next theorem shows that there are in fact genotypic strategies which lead to discretion, but do not represent a Nash-equilibrium. This in turn gives scope for evolutionary drift away from a high inflation state on the pheno-typic level. The following theorem contains our first main result.

Theorem 1. Among all the strategies leading to discretionary behavior only those represent Nash equilibria, which satisfy that p' < ^.

Proof. We have to show the following:

Pi(p, 0,p', 0) >Pi(p,qp', 0) Vp,q e [0,1] (11)

P2(p,0,p',0) > P2(p,0,p',q') vp',q' e [0,1] (12)

if and only if p' < Equation (14) is a direct consequence of Lemma 6.1 and Lemma 5.1. Let us therefore turn to equation (13). It follows from Corollary 6.2 that

Pi(p, 0,p', 0) = Pi(p, 0,p', 0).

In order to establish equation (13) it therefore suffices to show that the function

q ^ Pi(p,qp', 0)

is monotonic decreasing in q e [0,1]. Using the formulas (18) and (24) from the appendix it is easy to verify that

I'?, (p. a. o) = <2,/ -1) (1 _1(p:*')p,)2 ■

Note that neither 1 — pp' nor (1 — (p — q)p')2 can ever be negative. Under the assumption that p' < \ the derivative above is therefore negative for all q £ [0,1] and equation (13) holds. On the other side if p’ > a positive derivative at q = 0 implies that (13) can not hold, which concludes the proof.

Let us remind ourselves for the moment that p' denotes the conditional probability that private agents expect low inflation given that the monetary authority delivered low inflation in the previous round. The level of p' can therefore be interpreted as a trust parameter, which on a different time scale than considered here, may have been arisen from an effect of reputation of the monetary authority. The condition in Theorem 6.3 that p' <\ can then be interpreted, that discretion can only persist as a Nash-equilibrium in our setup, if the reputation of the monetary authority and hence the trust of the private agents in the monetary authority has been significantly damaged.

5. The Zero Inflation Equilibrium

In this section we study how low inflation can arise as a Nash-equilibrium on the pheno-typic level. It will turn out, that the analysis is very similar to the one carried out in the previous section. For matter of completeness and illustration though, we include all necessary arguments. First of all note that the zero inflation phenotype corresponds to the case c = c' = 1. In analogy to Lemma 2 the following Lemma helps us to identify those geno-typic strategies which correspond to the phenotype zero inflation and can be achieved if and only if p = p' = 1:

Lemma 3. Zero-inflation phenotypes require p = p' = 1. Furthermore if q > 0 or q' > 0 then p = p' = 1 implies c = c' = 1.

Proof. The second statement follows directly from the formulas for c and c' (see

(5)). In order to see that the first implication holds, let us assume for the moment that q > 0 or q' > 0 holds. Then c = 1 implies

1 — (p — q)(p' — q') = q + q' (p — q)

^ 1 — p'(p — q) = q

^ 1 — pp' = q(1 — p')

^ 1 — q = p'(p — q).

For p < q and p' > 0 the last expression has no solution, whereas p > q implies p' > 1 with equality if and only if p = 1. For p = q the above expression can only be fulfilled if p = q = 1. On the other hand c' = 1 yields

1 — q' = p(p' — q'),

which reduces to 1 = p' if p = 1 as required. To cover the case q = q' = 0 note that c and c' are equal to zero whenever pp' < 1. Hence p = p' = 1 must hold.

We will later need the following result which is proved in complete analogy to Corollary 6.2.

Corollary 2. Assume that q + q, q' + q' < 1 and either q + q > 0 or q' + q' > 0, then the following holds

Pi (1, q, 1, q') = Pi (1, q + q, 1,q' + q')

P2(1,q, 1,q') = P2(1,q + q, 1,q' + q').

The following Theorem includes our second main result.

Theorem 2. Among all strategies leading to zero inflation only those are a Nash equilibriums, which satisfy q' < ^ .

Proof. Similar as in the proof of Theorem 6.3 we have to show that

Pi(1,q, 1,q') >Pi(p,q, 1,q') Vp,q e [0,1] (13)

P2(1,q, 1, q') > P2(1,q,p',q') Vp',q' e [0,1] (14)

if and only if q' < Let us note that V2 (•) equals zero whenever p = p' = 1 and

that this is optimal for the private agents. Therefore the inequality (16) holds in

any case. Now, considering inequality (15), we note that from Corollary 7.2. we can conclude that

Pi(1, q, 1, q') = Pi(1,q, 1,q')

and that (15) would therefore hold, if the function

p ^'Pl(p,q, 1,q')

is monotonic increasing in p <G [0,1]. Now using equations (17) and (23) in the appendix it can be easily verified that

(p,q, 1 ,q') = (1 - 2q')

q' + q( 1 - q')

(1 - (p-q)(l — q'))2'

Note that nominator and denominator of the fraction are both positive and therefore that the derivative is positive as long as q' < which establishes inequality (15).

On the other side if q > | a negative derivative at p hold, which concludes the proof.

1 implies that (15) can not

As in the previous section let us briefly elaborate on the meaning of the condition in Theorem 7.3. We have that q' is given by the conditional probability that private agents expect low inflation given that the monetary authority delivered high inflation in the previous round. Therefore q' can be interpreted as some kind of ignorance of the private agents with respect to observed behavior of the monetary authority. The interpretation of Theorem 7.3 is therefore, that as long as the level of ignorance is sufficiently low q' < zero-inflation can very well be realized as a Nash-equilibrium. Combining this result with Theorem 6.3, we can state that under the assumption that the reputation of the monetary authority resp. the private agents trust in the monetary authority is not significantly damaged while on the other side, the private agents are not blind and ignorant toward the monetary authorities action, zero inflation is realized as a Nash equilibrium, while the classical Barro-Gordon high inflation, discretionary policy is not.

Figurel. On the left the discretionary equilibrium: q = q' = 0 and p' < On the right the zero inflation equilibrium: p = p' = 1 and q' <

Finally we remark, that we can not exclude further equilibria on the genotypic level, which in fact correspond to mixed strategies in the original Barro-Gordon game. The analysis in this article was mainly motivated by establishing the

impossibility of discretion and the possibility of zero-inflation as a Nash equilibrium in a (modified) Barro-Gordon framework. Further studies which include the dynamic aspect and stability properties will follow.

6. Conclusion

In the original Barro Gordon game (the one shot game version) discretion is the only one Nash equilibrium. Furthermore it is a strict Nash equilibrium. As we have seen in the last two sections this is not longer the case for our continuous time version of the game. Furthermore we have seen that there are strategies p, q and p', q' that lead to discretion resp. to zero inflation. Among those there are strategies such that discretion (resp. zero inflation) are a Nash equilibrium (for p' < \ resp. q' < 5). In this sense discretion and zero inflation are now of the same quality (see Figure 1).

Discretion as a Nash equilibrium becomes even more unlikely if one assumes a strategy more likely to Tit For Tat for the private agents. In particular the private agents tend not to believe in zero inflation again when they just see discretion. On the other hand they are more likely to believe in zero inflation if zero inflation is the case right now. In other words the private agents would always have a strategy such that q' < | and p' > Under this restriction by Theorem 1 we know that discretion is never a Nash equilibrium, whilst by Theorem 2 we know that zero inflation is always a Nash equilibrium.

Acknowlegments. Christian-Oliver Ewald gratefully acknowledges support from the research grant Dependable adaptive systems and mathematical modeling, Rhein-land-Pfalz Excellence Cluster as well as travel grants from the Royal Society and the Deutsche Forschungs Gemeinschaft. Johannes Geifiler acknowledges support from the Center of Dynamic Macroeconomic Analysis (CDMA) at St.Andrews University. Both authors are very thankful to Charles Nolan for very useful comments and suggestions.

Appendix

C,C’

1 - (p - q)(p' - q') q' + q(p' - q')

1 - ip- q)ip' - q')

c, = (16)

Derivatives 1

dc q' + q(p' — q')

dp (1 - (p - q)(p' - q'))2

dc 1 - q' - p(p' - q') dq (1 - ip - q)(p' - q'))2

(17)

(18)

dc' q + q' (p - q)

dp' (1 - (p - q)(p' - q'))2

dc' 1 - q - p' (p - q)

dq' (1 — (p — q)(p' — q'))2

Derivatives 2

n

1 - (p - q)(p' - q')

(19)

(20)

<21)

<22)

<23)

- = (v'- a') 1 ~ l'~ P(P'~ l') =(p>_Q>)^ (24)

dq (1 — (p — q)(p' — q'))2 dq

Proofs

Proof (or equation (5)). We have seen that

cn+2 =q + (p - q)(q' + (p' - q')cn)

Hence for k £ M we get

cn+2k =q + (p - q)(q' + (p' - q')[q + (p - q)(q' + (p' - q')[

... (q' + (p' -q')cn)...])])

k-1

=q^3 ((p - q)(p'- q'))l

i=0

k-1

+ (p - q)q' ((p - q)(p' - q'))i + ((p - q)(p' - q'))k cn

i=0

k-1

=(q+q'(p - q)) 53 ((p - q)(p' - q'))i + ((p - q)(p' - q'))k

i=0

=(’+«'(>’ -'iXrq’ic■-?r+((p ■■,,,,v

k^^ q + q'(p - q)

where * such as the last convergence is true by the assumption that -1 < (p -q)(p' - q') < 1 holds. It it is easy to check that the very same can be done for c'. Hence we are done.

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