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

2

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,

3

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

4

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

5

been developed, that of policy games. Within this approach the questions of

effectiveness and neutrality of specific policy instruments have been raised

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.

again, mainly with reference to monetary policy. More or less formal conditions

leading to monetary policy ineffectiveness – or neutrality – have thus been

investigated.

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

6

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

7

(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.

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

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

7 See Cukierman (2004) for a survey.

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. 8

2. The classical theory of economic policy: statics.

2. Principles of matrix al 9

3. 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

10

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 + Dw

m

where is the vector of the relevant endogenous variables (Government’s

Î

y ¡

target variables), is the vector of irrelevant endogenous variables.

r

Î

z ¡

is the vector of the Government’s policy instruments,

n + ´

( m r ) m ,

Î

u Î

A

¡ ¡

+ ´ and are parameter matrices (i.e. the target and instrument

( m r ) r + ´

( m r ) n

Î

C Î

¡ B ¡ +

coefficient matrices), and is a vector of constants, i.e. each

m r

= Î

k Dw ¡

component is a linear combination of exogenous constants and/or white noise

{ } { }

= Î = Î

M 1,2

. m N 1,2

n

.

¡ ¡

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 ) w

1 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 u

é ù é ù

ê ú ê ú

[ ] [ ]

⋮ ⋯ ⋮ ⋮

=

A C B D

ê ú ê ú

ê ú ê ú

z w

ë 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 1

ì 11 1 1

m m 11 1 1

r r 11 1 1

n n 11 1 1 j j

ï + + + + + = + + + + + ®

a y ... a y c z ... c z b u ... b u d w ... d w riga m

ï m

1 1 mm m m

1 1 mr r m

1 1 mm n m

1 1 mj j

í



ï

ï + + + + + = + + + + + ® +

a y ... a y c z ... c z b u ... b u d w ... d w riga m r

î + + + + + + + +

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 j

1 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 j

2 2

+ = +

( A y C z B u D w )

Risolvo per z…

2 2 2 2

= + -

C z B u D w A y

2 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 y

2 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 j

j 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

11 Questa è la forma del modello nella quale gli obiettivi sono espressi in termini soltanto di

strumenti e vi sono tante equazioni quanti gli obiettivi, ognuna per un obiettivo.

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