Politica economica - Appunti

La politica economica in un contesto strategico (a.a. 2007-08)

1 (Provisional notes by N. Acocella and G. Di Bartolomeo).

• 1. Introduction 2
• 2. The classical theory of economic policy: statics 5
• 2.1. Principles of matrix algebra 6
• 2.2. The Tinbergen-Theil approach 20
• 2.3. Controllability and sub (partial) controllability 24
• 3. The classical theory of economic policy: dynamics 26
• 3.1. Dynamic structural form (p. 125) 27
• 3.2. Dynamic reduced form (127) 27
• 3.3. Multipliers (129) 28
• 3.4. Dynamic policy objectives (132) 28
• 3.5. State-space form or linear system representations 28
• 3.6. Stability and the instrument multiplier equivalence (p. 143) 31
• 3.7. The effects of instruments on the states (existence property) 35
• 3.8. Observability properties 35

1 Si è cercato di rendere continua la numerazione delle equazioni, ma potrebbe esservi qualche ripetizione o discontinuità o errore nel testo di riferimento alla equazione esatta, dato che il materiale contenuto in ciascun paragrafo è tratto da fonti diverse. La numerazione dei teoremi è sicuramente valida soltanto all’interno di ogni paragrafo. Potranno pertanto emergere discontinuità o duplicazioni nella numerazione da un paragrafo ad un altro.

• 3.9. Dynamic theory of stationary objectives. (205) 35
• 3.10. The target path existence problem (229) 36
• 4. The Lucas Critique 38
• 4.1. The essence of the critique 39
• 4.2. A Formal Exposition 40
• 4.3. Deep Parameters and Rational Expectations: A Solution? 41
• 4.4. Overtaking the Critique 42
• 5. Principles of game theory (see slides) 44
• 6. A brief analytic survey of the issue of neutrality in policy games 59
• 7. The theory of economic policy in a strategic context: statics 65
• 7.1. The policy game approach 66
• 7.2. Policy neutrality 70
• 7.3. Implications of the theory (model building & institution building) ??
• 8. The theory of economic policy in a strategic context: dynamics 73
• 8.1. The Basic Setup 74
• 8.2. The Golden Rule and the Equilibrium Properties 75
• 8.3. A Generalization: Sparse Economic Systems 78
• 8.4. An Example 79
• 8.5. RE ??
• 9. References 82

1. Introduction

Tinbergen (1952, 1956) first addressed the issue of the controllability of a fixed set of targets by policymakers endowed with given instruments in a parametric context. He stated some general conditions for policy effectiveness and the need for policymakers to have recourse to second best solutions – by maximizing the value of their preference function subject to the model representing the economy – in the case of a non-perfectly controllable system, an approach later developed by Theil (see Theil, 1964). More formal conditions for controllability both in a static and dynamic context were later asserted (see Preston and Pagan, 1982; Hughes Hallett and Rees, 1983).

Tinbergen, Theil, and the other founding fathers of economic policy were not concerned with analyzing the effectiveness of specific policy instruments. However, in the framework of the classical theory of economic policymaking it is not difficult to find the counterpart of the concepts of policy ineffectiveness and neutrality raised in the economic literature with reference to specific instruments, monetary policy, fiscal policy or others.

The classical theory of economic policy has been the object of fierce criticism from a number of points of view. The introduction of rational expectations led to an assertion of the ineffectiveness of monetary policy on income more forceful than that stated by Friedman (1968) (see Sargent and Wallace, 1975). In a similar way, Barro (1974) developed the argument of fiscal policy neutrality based on the assumption of ultra-rational agents. A proposition of policy neutrality or “invariance” was then stated. Apart from the critiques advanced with reference to the effectiveness of specific instruments, Lucas (1976) raised the more general and forceful argument according to which a Tinbergen-type decision model is inconsistent with the assumption of rational expectations.

In more recent years a new approach to the analysis of economic policy has been developed, that of policy games. Within this approach the questions of effectiveness and neutrality of specific policy instruments have been raised again, mainly with reference to monetary policy. More or less formal conditions leading to monetary policy ineffectiveness – or neutrality – have thus been investigated.

2 Tratto da Acocella, N. and G. Di Bartolomeo (2006a), Equilibrium existence and policy neutrality in static LQ games, non pubblicato.

3 A good account of early contributions to the theory of economic policy is in Hughes Hallett (1989).

4 See Holly and Hughes Hallett (1989).

5 As is well known, a policy game can be informally defined as a situation in which two or more agents strategically interact. More precisely, a policy game is characterized by a set of players, strategies and payoffs that are linked to strategies.

Barro and Gordon (1983) deliver the well-known prediction of monetary neutrality as a result of the private sector expectations of the monetary policy. The private sector forms rational expectations of the money supply and acts to fully crowd-out monetary effects on real output by adjusting nominal wages, thereby creating a socially inefficient inflation bias. The same conclusion is reached if the Barro-Gordon problem is expressed in terms of a Stackelberg game between the central bank and a monopoly union, where the latter is the follower and trades off real wage and employment when setting the nominal wage rate.

Gylfason and Lindbeck (1994) suggest that monetary policy nonneutrality arises when the private sector (labor unions) shares the objective of price stability with the central bank. Acocella and Ciccarone (1997) generalize the above result by showing that monetary policy nonneutrality is the result of unions’ sharing some objectives (not necessarily price stability) with the monetary authorities. But this rule seems to loose ground when non-competitive markets are introduced into the picture: Soskice and Iversen (2000), Coricelli et al. (2000), Cukierman and Lippi (2001), and other studies show that non neutrality of monetary policy can derive from the interaction between imperfectly competitive goods and labor markets even when unions do not care for a common objective directly. Acocella and Di Bartolomeo (2004) show that in this case the rule is not violated, if reformulated in terms of unions’ sharing directly or indirectly some objective with the monetary authorities and state more general necessary and sufficient conditions for non-neutrality to hold.

6 See, among others, Stokey (1990) and Sargent (2002: Chapter 3).

7 See Cukierman (2004) for a survey.

The conditions derived by Acocella and Di Bartolomeo (2004), except for some conclusive hints (for which see section 6), are stated in terms apparently different from those of the classical analysis of policy effectiveness and controllability. One reason that could explain the difference refers to the contexts in which policy issues are examined in the two cases: a parametric one in the case of the analysis of Tinbergen and Theil; a strategic one in the case of policy games. The need then arises to:

• 1) Find conditions for policy controllability in a strategic context equivalent to those valid in a parametric context and clearly stated in the classical theory of economic policy; these should be ex ante conditions, i.e., conditions that allow us to know whether the game leads to a result of neutrality of economic policy before solving it, possibly by applying some simple counting rule of the kind stated by Tinbergen;
• 2) Check the corresponding conditions of those stated in the analysis of the policy effectiveness or neutrality of specific instruments.

There have been previous attempts at performing the task under 1). This is, e.g., the case of Holly and Hughes Hallett (1989), who consider conditions for controllability in a dynamic model with a private sector forming rational expectations. As far as this task is concerned, our analysis will have both a narrower and a wider coverage than theirs: in fact, on the one hand we will not consider differential games; on the other, we will not stick to the case of private agents forming rational expectations.

We restrict ourselves to a static context and consider the case of perfect information only, since it is well known that asymmetric information is itself a source of non-neutrality. However our simple logic can be extended to more complex frameworks.

2. The classical theory of economic policy: statics

2. Principles of matrix algebra

2.2. The Tinbergen-Theil approach

In this section we consider the optimization problem of a single decision-maker (from now on, without loss of generality, the Government). We assume that the Government aims to achieve certain given targets and, if this is not possible, to minimize deviations from them according to a quadratic function.

Approaching the problem in this way has the advantage to merge together the fixed and flexible target approach making the former a particular case of the latter (see Preston and Pagan, 1982). It implies that we implicitly assume that, if the fixed target approach fails, the Government sets the instruments according to a flexible approach. Although the flexible approach is not the only alternative to the fixed one, flexible targets seem to be the alternative more in line with our attempt at reformulating the classical theory of economic policy with reference to a strategic context.

8 Throughout the paper we use the following notation. All vectors are real column vector defined by their dimension; all matrices are real matrices defined by their two dimensions. Considering two vectors, a and b, (a, b) is a column vector; considering matrices A and B with the same number of rows, [A:B] is a matrix formed by merging the two matrices.

9 This and the following sections are drawn from Acocella, N. and G. Di Bartolomeo (2005), “Non-neutrality of economic policy: an application of the Tinbergen-Theil’s approach to a strategic context”, Dept of Public Economics, University of Rome ‘La Sapienza’, W.P. 82.

10 We will later relax the assumption of a given target by considering also the possibility of non-satiation.

We first derive the policy model in its structural form (sect 2.2.1); then we analize controllability (2.2.2) and subcontrollability (2.2.3).

2.2.1 Modello di decisione

The structural form is:

(1) Ay + Cz = Bu + Dwm

where is the vector of the relevant endogenous variables (Government’s Îy ¡ target variables), is the vector of irrelevant endogenous variables. rÎz ¡ is the vector of the Government’s policy instruments, n + ´( m r ) m ,Îu ÎA¡ ¡+ ´ and are parameter matrices (i.e. the target and instrument ( m r ) r + ´( m r ) nÎC Ρ B ¡ + coefficient matrices), and is a vector of constants, i.e. each m r= Îk Dw ¡ component is a linear combination of exogenous constants and/or white noise { } { }= Î = ÎM 1,2. m N 1,2n. ¡ ¡ shocks. We define and as the sets of policy targets and instruments, respectively.

By eliminating irrelevant variables z, we obtain the reduced form model

= +A * y B * u K *(1’) - - -1 1 1

where = - = - = -A* A C C A , B B C C B , k * ( D C C D ) w1 1 2 2 1 1 2 2 1 1 2 2

Box 1: passaggio da (1) a (1’)

In termini di matrici partizionate la (1) può essere scritta come:

y ué ù é ù
ê ú ê ú
[ ] [ ]⋮ ⋯ ⋮ ⋮=A C B Dê ú ê ú
ê ú ê ú
z wë û ë û

Risolviamo il modello eliminando le variabili irrilevanti:

+ + + + + = + + + + + ®a y ... a y c z ... c z b u ... b u d w ... d w riga 1ì 11 1 1m m 11 1 1r r 11 1 1n n 11 1 1 j jï + + + + + = + + + + + ®a y ... a y c z ... c z b u ... b u d w ... d w riga mà m1 1 mm m m1 1 mr r m1 1 mm n m1 1 mj jíïï + + + + + = + + + + + ® +a y ... a y c z ... c z b u ... b u d w ... d w riga m rî + + + + + + + +m r , 1 1 m r , m m m r , 1 1 m r , r r m r , 1 1 m r , n n m r , 1 1 m r , j j

Scrivo separatamente le prime m equazioni e poi le restanti r equazioni in forma compatta…

+ = + ´ ´A y C z B u D w con : A : m m; B : m n;1 1 1 1 1 1´ ´C : m r ; D : m j1 1

+ = + ´ ´A y C z B u D w con : A : r m ; B : r n;2 2 2 2 2 2´ ´C : r r ; D : r j2 2

+ = +( A y C z B u D w )

Risolvo per z…

2 2 2 2= + -C z B u D w A y2 2 2 2

se è possibile invertire C ( C è di rango pieno ρ(C ) = r)2 2 2

- -1 1= + -C C z C ( B u D w A y )2 2 2 2 2 2

 - - -1 1 1= + -z C B u C D w C A y2 2 2 2 2 2

inserisco questo valore di z in + = +( A y C z B u D w )...1 1 1 1

Controllo della conformabilità:

A * y + C * z = B * u + D *w

(m + r) x m m x 1 (m + r) x r r x 1 (m + r) x n n x 1 (m + r) x jj x 1 (m + r) x 1 (m + r) x 1 (m + r) x 1 (m +r) x 1

11 The linear reduced-form model can be written in matrix form as:

- -1 1(2) = + = +y A Bu A k Cu C

(provided A – remember that this is indeed A* in (1’) – is non-singular from our rank assumptions), or as

 =y Cu(3) y

(FORMA RIDOTTA CONSOLIDATA nel quale è il vettore degli obiettivi - 1 - 1 “consolidati” (y – il termine noto)) dove pongo ; , e y è il = -y y A k =C A B vettore degli obiettivi desiderati.

Vedi es. (file “Notes esercizio da lezione fulvimari”) y

D’ora in poi useremo la notazione y al posto di , pur volendo riferirci agli y obiettivi consolidati e indicheremo con un dato vettore di obiettivi consolidati.

2.2.2 Controllability of the model

Consideriamo il problema di un singolo decision-maker, il governo, che abbia determinati targets (valori desiderati) riguardo ai suoi obiettivi. Il governo può riuscire o non a raggiungere esattamente tali valori prefissati (target values). Il modo generale per affrontare il problema del governo è di dire che esso vuole minimizzare una funzione di perdita che dipende dai quadrati delle deviazioni dei valori effettivi degli obiettivi rispetto a tali targets values (quadratic loss function). Se il governo controlla il sistema, e riesce perciò esattamente a raggiungere i valori prefissati degli obiettivi, la perdita è uguale a zero. Se il governo non controlla il sistema, la perdita sarà positiva, ma, appunto, minima.

Affrontando il problema in questo modo fondiamo insieme il caso di obiettivi fissi e di obiettivi flessibili, rendendo il problema con obiettivi fissi un caso particolare di quello con obiettivi flessibili. Così facendo assumiamo implicitamente che, se fallisce l’approccio per obiettivi fissi (perché non si controlla il sistema), il governo userà gli strumenti a sua disposizione in modo tale da rendere minima la perdita.

Comunque, il problema con obiettivi flessibili non è solo un’alternativa al problema con obiettivi fissi. Infatti, la formulazione per obiettivi flessibili è più in linea con il tentativo di riformulare la teoria classica della politica economica in un contesto strategico.

Comunque, per il momento partiamo da un problema impostato in termini di obiettivi fissi e, quindi, dalla forma ridotta consolidata (3), dove è stata tolta la tilde alla y, per semplicità.

=y Cu(3)

Definiamo:

Efficacia di uno strumento di politica economica: uno strumento j u è Î efficace rispetto alla variabile obiettivo i y se cambiamenti dello strumento Î determinano cambiamenti nel valore dell’obiettivo. In caso contrario lo strumento è inefficace.

Neutralità (esogena) della politica economica: la politica economica è neutrale rispetto ad una certa variabile i y se tutti gli strumenti sono Î inefficaci ai fini del raggiungimento della variabile obiettivo stessa.

Quindi, con riferimento alla (2’):

• Se c = 0 (moltiplicatore dello strumento j per l’obiettivo i) un determinato ®ij strumento (j) è inefficace per raggiungere l’obiettivo i inefficacia. ®=j j
• Se ho una colonna di zeri (c = 0 e ) inefficacia dello strumento in i ®ij questione per il raggiungimento di qualsiasi obiettivo.
• Se ho una riga di zeri (c = 0 e ) tutti gli strumenti sono inefficaci ai =i ij ®ij fini del raggiungimento di quel preciso obiettivo neutralità delle politiche ® economiche (o della politica economica in generale).
• Quando usiamo 1 solo strumento, neutralità ed efficacia sono la stessa cosa.
• Quando usiamo più strumenti neutralità ed inefficacia non sono la stessa cosa.

Lo scopo del governo è controllare il sistema economico, Ay = Bu + k, e determinare i valori della variabile obiettivo Definiamo …® controllabilità: possibilità che manovrando un vettore di str