## Thesis Working Paper n°1

*This year is definitely being bitchy all the way down to early spring. Can it get any worse? (I am masochist, so It’s a bit of wishful thinking)*

I’m getting pieces together for the thesis; it’s definitely going into game theory as a theoretical background in describing how a central bank should set the optimal interest rate, and how the fact that rate can be credible when a certain set of conditions are satisfied. I am having an enormous amount of fun in trying to get things going around… In an economic universe, the Central Bank has few objectives to fulfill: “*stabilizing inflation around an inflation target and stabilizing the real economy, represented by the output gap*“. (Svensson, 1999) the output gap is a reference to the gap between the actual GDP and the *optimal* GDP, or *potential* output, and can be roughly computed with the labour and net capital productivity, plus the total productivity factors, which can be proxy for technical innovation (or how to combine factors differently to get a higher output, or the same amount of output for lower level of capital and labour)

Because I claim to be a monetarist, and because I am fully in favour of an independent -but responsible before an elected body of representatives- central bank, I believe this institution is the one adequately geared to influence other players into accepting it as the best level of interest rate they should act upon. The game theory technique is there to prove that it can set a rate and an output target such that all players -that is economic actors- would stand by the targets as credible signals and would yield larger common welfare if they did not.

There is of course a strong assumption going on about the Central Bank’s motives: following institutional backgrounds, banks like the Federal Reserve has a triple objective to fulfill: inflation, growth and employment (Federal Reserve act, Section 2.a) the European Central Bank on the other hand, is mainly focused on inflation, and growth is a purely secondary objective, contrary to the Fed whose 3 objectives are of equal importance; these are even more stressed upon when put in perspective with regard to the kind of relationship their entertain with the political power; There is a wide-range consensus among economists about the virtues of an independent Central Bank, for all of the benefits it brings in terms of credibility, and thus efficiency in monetary policy-making. As for the Moroccan central bank (Bank Al Maghrib) things are certainly different, but that is another matter.

Another assumption is that other players are interest-rate sensitive: firms are likely to expand or restrain their investment; The assumption looks credible, even though there are occurrences of sub-optimal or indeed irrational decision-making regardless of what the levels of interest rates are. In a dynamic setting, some players, like the unions or households do not change their behaviour overnight, as indeed there is a certain delay (a lag variable that can empirically observed thanks to econometrics) and would therefore blur the bank’s decision;

– Starting/working assumptions:

The Central Bank handles interest rates setting and assigning to the economy a target output (or in most cases, a target output gap) both of which variables are set in a fashion such to maximize common welfare, namely by selecting a pair (r,Y) that yields to the Nash axioms (or at least, to start with, Pareto conditions). These decisions are taken subject to other players’ respective preferences sets. The Central Bank is considered benevolent, and pursues no agenda of its own (that is to say, the Nash pair is the Bank’s own preference).

The aim is to prove that in monetary policy, there exist a Nash pair (r,Y) for which common welfare is maximized, and that Game Theory techniques and findings would help describing mechanisms and strategies that would allow the Central Bank, under specified conditions, to reach this equilibrium set over time. The difference with the ‘regular’ Game Theory setting and the present attempt to model the monetary policy lies in lotteries and risk aversion. The first one is relegated to random events (as perhaps used in econometric study) the second one would be rather about time preferences, as we can assume that agents value time, and would rather reach an agreement sooner rather than later.

Therefore, the starting concepts for this paper are going to be related to bargaining issues: for each Player i there is a function [f(r,Y)] called a utility function, such that one lottery is preferred to another if and only if the expected utility of the first exceeds that of the second.

– Simple Model: We borrow elements from the bargaining model as specified by Osborne & Rubinstein with a simple monetary set: two players, Central Bank (CB) and a Business Firm (BF) in an economy, which have to reach agreement on a specific set (r,Y). Both have their own preferences (CB’s is exogenous to its own condition) We keep the definition 2.1. For the agreements pair (S,d) where S is the set of all feasible pairs (r,Y). Of all Nash axioms (SYM, PAR, IIA & INV) only the symmetry assumption has to be dropped, as CB and BF display different preferences. The feasible set of pairs (r,Y) is divided up between desirable pairs’ sets –to which both players want to yield- and worst outcome possible (WOP) which both players want to avoid at any price. (and is the primary component of bargaining cost)The border between both sub-sets is set by the economy’s own productive capacities (or indeed how far the output gap can be sustained without slipping down into recession) Let us start with the bargaining game of alternating offers, and define monetary policy setting as follow:

1/ CB announces a pair (r,Y) to BF. BF has three ways to go: accept the pair, refuse it or refuse it and submit to CB an alternative pair. If BF accepts or refuses point blank, the game is over, and both parties reach a pair (r,Y) that belongs to the disagreement pair, an outcome both of which are made worse off when reached.

2/ CB, in turns, accepts the pair, refuses it or re-computes another pair (r,Y) and submits to BF.

3/ The game is rolling as long as each players refuse the proposed pair and proposes another.

We have to assume beforehand that CB has access to complete information (a fair assumption as CB uses resources to obtain sufficient information to make an accurate decision- an assumption to be discussed later on) and therefore whatever decision made is necessarily optimal. We can also assume that BF has access to complete information as well, and knows why CB fields its strategy. We also assume both players to adopt an optimal strategy at each point of the bargain, that is, that they are able to order their preferred pairs and play them accordingly at each knot of the game.

This proves that an equilibrium set of pairs (r,Y) can be reached very quickly, as both players know each others’ respective preferences, and if there are resistances from one part or the other, the disagreement –that is, the delay in reaching agreement- does not go further than a couple of periods. In a sequential equilibrium however, ‘*We can interpret the equilibrium as follows. The players regard a deviation as a sign of weakness, which they “punish” by playing according to a sequential equilibrium in which the player who did not deviate is better off. Note that there is delay in this equilibrium even though no information is revealed along the equilibrium path*.’

This can be used to provide a first-hand punishment deterrent to hurry both CB and BF to reach an equilibrium pair. The time value can be used as well, as indeed the longer both parties reach agreement, the more painful –or indeed the less desirable- the equilibrium pair would be. In real life, that is the case when CB fails to convince other economic players that they will stick to their decision, or that their announcement was not credible. That compels the bank to come after with a much stringent, more constraining announcement, something that could have been avoided if they took their signal seriously in the first place. As mentioned before, the central bank has objectives to stabilize inflation by setting optimal interest rate, and computing optimal (resp. minimal) output (output gap). For the time being, we set up for the output gap, as described in the paper by Gaspar & Smets. Their starting assumption was that both the Central Bank and the private sector (in our case, BF) observe the potential output, an assumption that can be deemed to be realistic, in view of semi-perfect information universe they evolve in.

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