A rational expectations paradox

(1)

A RATIONAL EXPECTATIONS PARADOX

(2)

Mario Henrique Simonsen

1) The inflationary tax equilibriurn

A well known property of the Cagan's money demand function

is that it yields a Laffer curve for the inflationary tax. Money

market equilibrium is descrihed by equation:

B

-p

=

e -Cl'Ir

*

(a> O) (1)

where B stands for the monetary base,P for the general price leveI and

*

TI for the expected inflation rate. In steady state equilibrium

the latter should coincide with the actual inflation rate:

•

TI

=

_---p

P (2)

dots indicat1ng time derivatives. Hence, for a constant inf1ation

rate, the inf1at.ionary tax revenue will be given by:

•

B d /

Tinf

=

-p-

= dt (B P) + B TI =

-ª--

'It = TI e-a1T

-p

P (3)

increasing for TI< l/~ and decreasing thereafter, as indicated in

figure 1.. The inflationary tax revenue is bounded by its rnaximum:

1

-

_ex e -1 (4)

Now, let us assume that the real pub1ic sector deficit is

kept at a constant leveI k < __ 1_ e-I, being fully financed by the

ex

creation of high powered money. There i5 no other source of

expansion or contraction of the monetary base, which implies:

,

(3)

The equilibrium inflation rate must equal the inflationary

; tax revenue to k. Now, as indicated in figure 1, because of the

Laffer curve effect there are two different equilibrium inf1ation

rates w

1 and w2• Increasing the pub1ic sector deficit k,i.e, ~g

up the stralght 11ne AB, increases the low equi1ibrlum rate w_l and

decreases w₂' providing a prior i support to the idea that stable

equl1ibrium inf1ation rate shou1d be w

1•

Hyperinf1ations are exp1ained by k> __ l_

a

-1

e a

A

I

--~'I

I

-1

e

L-L-____

L---L---~~~---w

w1 l/a 'Ir 2 Figure 1

Oynamic hypotheses must be introduced to discuss the ~llty

of the two possible equl1ibria. Assuming adaptative expectations

according to Cagan's speclflcatlon:

. *

*

w

=

8 (n - w ) (8 > O)

one may easi1y prove that w₁ is the stab1e equl1ibrlum inflation rate

if and only if 0.8< 1, name1y, if and on1y if the Cagan's stabi1ity

condltion is meto

Now, let us turn to rational expectatlons. In the absence

*

of stochastlc disturbances thls means perfect foresight, w

=

w • In

this case, taking time derivatives in both sides of equation (1)

B

=

e -em (P- a P w)

combining with equations (2) and (5):

-aw •

(4)

This is to say that sign of ~ coincides with that of the right side

of the above presented equation, tnat is nothing but the Laffer ~

in figure 1 with the horizontal axis moved up to AB. As arrows

indicate, the stable equilibrium inflation rate is now the high

root 'TT

2 "

The conclusion appears to make the rational expectations

hypothesis inconsistent with both empirical evidence and common ~.

In fact it implies that cutting the public sector deficit (which is

assumed to be fully financed by money creation) leads to an acoelaraticn

of the inflation rate, since 'TT

2 increases when k declines.

2) Solving the paradox

What is wrong in the preceding analysis? Simply the :phase

diagram technique in figure 1 implicitly treats the inflation rate as

a backward looking variable, as in case of adaptative expectations.

The conclusion that 'TT

2 is the stable inflation rate equilibrium assumes that the inflation rate is subject to initial condition contraints.

Now, in rational expectations models both the price level and the

inflation rate are forward looking variables. Since they keep no

links with the past, their initial levels will jump, depending on

expectations. Hence, in a rationalexpectations framework, the phase

diagram in figure 1 can only be understood as a guide to forecasters.

It simply tells that economic agents may foresee either 7T

=

'Ir 1 ' 'Ir

=

'IT

2 or some variable inflation rate path either converging t07T 2 or

tending to minus infinity.

The diagram shows that forecasters face an indeterminacy

problem as often occurs in rational expectations models. In fact,

since forward looking variables are not constrained by fixed initial

conditions, difference or differential equation systems usually yield

(5)

usually removed by introducing boundness requirements on the

expected

paths of the variablE~s (or their time derivatives). The problem Df

the inflationary tax equ1libr1um requ1res an additional assumption,

as will be shown in section 3. It emerges naturally once the ~lem

is solved accordinq to the rational expectations methodology, namely,

once the inflation rate is expressed as a function of the ant1cipated

path of money supply.

It shouldbe noted that a large group of rational expetat10ns

difference or differential equation systems have no stabl-e solut1ons.

There are pri\Tileged solutions interms of boundness or oonvergence,

usually the meaningful ones from the economic theory standpoint, but

they are a180 unstable. Forward 100k1ng variables are supposed to

jump whenever necessary so as to meet these solutions, disconnecting

rational expetations from stability analysis.

As an example, let us take Cagan' s monetary equatioo expressed

in 10gs and assuming perfect foresight:

-b-p'

= -

a p (6)

where b=ln B, p=ln P and, as a consequence, p

=

n. The price leveI,

.

according to the rational expectations hypothesis, is determined by

solvtng forward equat10n (6):

p(t)

=

t-T a b (T) dT

t

-+ce Ct (7)

where c 1s a constant. lf the monetary base expands at a constant

rate b = r, then beT)

=

b(t) + r(T-t). Introduc1ng this expression

in the riqht side of (7) and calculating the integral:

t

-p (t) = b (t) + a r + ce a ( 8 )

Taking time derivatives, the inflation rate will be given by:

p

=

r +

r

1

- c e Ct

t

(6)

The usual indeterminacy prob1em is displayed, since there are

infinitely many perfect foresiqht paths for the inflation rate~ one

for each va1ue of c. ·It is removed by addinq a boundary CXI1ditioo, the

expected rate of price chanqe cannot race to infinity. This forces •

c=o, yieldinq p

=

r. The inflatinn rate equals the constant rate of

expansion of the monetary base, a sensible economic resulto Taking

c=O in equation (8) leads to an important conclusion: if r chanqes, the price level will jump.

,

Now, it should be made clear that p

=

r is an unstable

inflation rate equilibrium, asshown in figure 2. In fact, takinq

time derivatives in (6) and assuminq b

=

r: •

r-'If

=

-(1 'If (10)

a differentia1 equation that has no stable solution: •

'If

Another example is provided by the popular 2x2 saddle-point

converqence models, where backward lookinq variable Xl interacts with

the forward lookinq X

2 accordinq to the linear dynamic system:

(11)

where Xl and X

2 startd for the riqht side time derivatives of Xl and X

2 and Et for conditiona1 expectation at time t. Equations (11) read as followa: i)X

2 is determined by ita expected riqht aide time

.

derivative and by Xl (thia ia poasib1e, since a 22!-O): 1i) Xl ia 8.11near function of Xl and X

(7)

inf1uence the path of the backward 100king one, which requires

8+2

Jl

O.

Moreover, matrix

A =

is assumed to have a negative determinant. Hence, its eigenva1ues are

r 1 < O < r _{2 • Since a 12 ; O, there are two corresponding eigenvectors}

Yl = (l;kl ) , Y2=(1;k2), where k1

1

k

2•

• •

In the absence of stochastic dlsturbances, E

tX2

=

X2 i.e,

rationa1 expectations yie1d perfect foreslght, 1eading to the

conventional linear system:

•

X

=

AX + b (12)

(13)

where constants c 1 and c 2 meet the initlal condition constraint:

-1

+ c_{2Y2 - A b}

or equlva1ent1y, if -A-1b

=

(h_l ;h

2)":

(14)

In the above equatlons XIO ls glven by hlstory, but X20 can jump to any leveI, slnce X2 ls a forward-Iooklng varlable. To remove the indeterminacy, that aIIows for infiniteIy many perfect foreslght

paths, one for each X20 ' the conventlonal extra assumptlonis ~,

(8)

.

are supposed r t

r 1c1e 1 Y1 +

to rule out the possibility of any oa~t of EtX

=

X ~

,

r t

r₂c₂e 2 Y2 racing to infinity. This requires c₂=0,

locating the initial value of the forward looking variable at:

and leading to the unique perfect foresight convergent path:

(16) -1

where X converges to the sadd1e-point -A b, since rI < O.

Now, once again, it should be stressed that all solutions of

the differential equation system (12) are unstable, including the

perfect foresight convergent path. In fact, any slight deviation

between X

20 and the expression in the right side of equation (15)wi1l

move X indefinitely away from the convergent path.

Economic literature often refers to stationary or convergent

perfect foresight paths as "s table paths". The semantic confusion

provides, perhaps, the heuristic explanation for the rational

expectations paradox under discussion.

3) The tnflationary tax equilibrium reconsidered

Let us now provide the correct rat10nal expectat10ns solutian

to the problem discussed in section 1. AlI economic agents share the

sarne informations, that can be summarized by equations:

b-p

₌

-a p (6)

•

b

=

(17)

Equation (6) is the already presented log-linear translatian

(9)

Equation (17) was obtained dividing (5) by (1) and a1so assuming

i

perfect foresight.

The.price leveI wi11 be a function of the projected path of

the monetary base. This invo1vese1iminating p and

p

from equations

(6) and (17). Simp1e and tedious calculations yield:

.

+ b =

•

-1-

In a

.

b k

or, indicating by r=b the rate of expansion of the monetary base:

2 -

₌

r

or equivalent1y:

r

-

=

r

where:

f(r)

=

1 r

In ~ - r

-

a

f(r) - 1 _ex

1

In r - r

-

_a

In k

(18)

(19.a)

(19.b)

The stationary values of r are the solutions of the equation;

f(r)

=

or equivaIently:

1

- I n _a r - r

re -ar = k (20)

=

-

1

In k

a

the roats of which, as one might expect, are the sarne ~l _{and rr 2 of}

section 1. The phase diagram in figure 3, a linear transformation of

that in figure 1, based on equations (l9.a) and (l9.b) is to be

understood as a simple forecaster's chart, since the projected rate

of growth of the monetary base is not bound to initial condition

(10)

í

• r

-

r

f (r) •

1

- - R.nk _a

~~~~---~+---r

Figure 3

The usual forecast1nq 1ndeterminacy prob1em in rationa1

expectations models emerqes aqain. What i8 new, is that it can no

lonqer be removed by a s1mple boundness assumption. An additions:l

hypothes1s is needed to solve the forecaster's dilema. The most

natural one is that the rate of qrowth of the monetary base should

eventually become positively correlated with the public sector

•

def1cit. Th1s requlres r-b,l/a, namely, that r should be projected

in the increasing reglon of the f(r} curve, which corresponds to the

increasinq part of the Laffer curve-in figure 1. Given the boundness

assumptions, forecasters have no other choice except to locate

•

b = r = ~l. The inflation rate, accordlng to equation (17) wi1l

(11)

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in~

~

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Caqan,Phi1ip (1956) - "The Monetany Oynamics of Hyperinflation",in

Milton Friedman (ed), studies in the Quantity Theory of Money

-University of Chicago Press.

sar"qent,Thomas and Ne"i1 wallace (1973) "Ra1;.ional Expectations and the Oynamics of Hyperinflation, International Economic Review, 14 (2)

(12)

I. ANALISE COMPARATiVA DAS ALTERNATIVAS DE POlrTICA COMERCIAL DE UM PAIS EM PRO-CESSO DE INDUSTRIALIZAÇ~O - Edmar Bacha - 1970 (ESGOTADO)

2. ANALISE ECONOM~TRICA DO MERCADO INTERNAC!ONAL DQ CAF~ E DA POLTTICA BRASILEI-RA DE PREÇOS - Edmar 8acha - 1970 (ESGOTADO)

3.

A ESTRUTURA ECONOMICA BRASILEIPA - MarIo Henrique SImonsen - 1971 (ESGOTADO)

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6. POLrTICA ANTI-INFLACIONARIA - A CONTRIBUIÇ~O BRASILEIRA - MarIo Henrique 51-monsen - 1973 (ESGOTADO)

7.

ANALISE DE S~RIES DE TEMPO E MODELO DE FORMAÇ~O DE EXPECTATIVAS - José Luiz Carvalho - 1973 (ESGOTADO)

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9. UMA NOTA SOBRE A POPULAÇ~O OTP-IA DO 8MS!L - Edy Luiz Kogut - 1973

10. ASPECTOS DO PROBLEMA DA ABSORÇ~O DE ~O-DE-OBRA: SUGESTOES PARA PESQUISAS José Luiz Carvalho - 1974 (ESGOTADO)

11. A FORÇA DO TRABALHO MO BRASIL - Mario Henrique Slmonsen - 1974 (ESGOTADO)

12. O SISTEMA BRASILEIRO DE INCENTIVOS FISCAIS - Mario HenrIque Slmonsen - 1974 (ESGOTADO)

13. MOEDA - AntonIo Maria da SIlveira - 1974 (ESGOTADO)

14. CRESCIMENTO DO PRODUTO REAL BRASILEIP~ - 1900/1974 - ClaudIo LuIz Haddad

(13)

16. ANALISE DE CUSTOS E BENEFTclOS SOCIAIS I - Edy Luiz Kogut - 1974 (ESGOTADO)

17. DISTRIBUiÇÃO DE RENDA: RESUMO 'DA EVIDENCIA - Carlos Geraldo Langonl - 1974 (ESGOTA.DO)

18. O MODELO ECONOMrTRICO DE ST. LOUIS APLICADO NO BRASIL: RESULTADOS PRELIMINA RES - A.ntonio Carlos Lemgruber - 1975

19. OS MODELOS CLAsSICOS E NEOCLAsSICOS DE DALE W. JORGENSON - Eltseu R. de An-drade Alves - 1975

20. DIVID: UM. PROGRAMA FLEXTvEL PARA CONSTRUçAO DO QUADRO DE EVOLUÇÃO DO ESTUDO DE UMA DTvlDA - Clovis de Faro - 1974

21. ESCOLHA, ENTRE OS REGIMES DA TABELA PRICE E DO SISTEMA DE AMORTIZAÇDES CON5TAN TE5: PONTO-DE-VISTA DO MUTU~RIO - Clovis de Faro - 1975

22. ESCOLAR,IDADE, EXPERIENCIA NO TMB,ALHO E SALARIOS NO BRASIL· José Julio

Sen-na - 1975

23. PESQUISA QUANTITATIVA NA ECONOMIA - Luiz de FreItas Bueno - 1978

24. UMA ANA,LISE EM CROSS-SECTION DOS GASTOS FAMILIARES EM CONEXAo COM NUTRiÇÃO, SAODE, FECUNDIDADE E CAPACIDADE DE GERAR RENDA - José Luiz Carvalho - 1978

25'. DETERMINAÇAo DA TAXA DE JUROS IMPLTCITA EM ESQ.UEMAS GENrRICOS DE

FINANCIA-MENTO: COMPARAÇÃO ENTRE OS ALGORrTIMOS DE WILD E DE NEVTON-tAPHSON - Clovis de Faro - 1978

26. A URBANIZAÇAO E O CrRCULO VICIOSO DA POBREZA: O CASO ~ CRIANÇA URBANA NO BRASIL - José Luiz Carvalho e Urlel de Magalhães - 1979

27. MICROECONOMIA - Parte I - FUNDAMENTOS DA TEORIA DOS f'REÇOS • Mario Henrique SI monsen - 1979

(14)

30. MICROECONOMIA - Parte 2 - FUNDAMENTOS DA TEORIA DOS PREÇOS - Mario Henrique Slmonsen - 1980 (ESGOTADO)

31. A CORREÇ~O MONETARIA NA JURISPRUOENCfA BRASILE:IRA - Arnold Wald - 1980

32. MICROECONOMIA - Parte A - TEORIA DA DETERMINAÇAo DA RENDA E DO NTvEL DE PRE

ÇOS - José Julio Senna - 2 Volumes - 1980

-33. ANALISE DE CUSTOS E BENEFTclOS SOCIAIS II I - Edy Luiz Kogut - 1980

34. MEDIDAS DE CONCENTRAÇ~O - Fernando de Holanda Barbosa ~ 1981

35. CREDITO RURAL: PROBLEMAS ECONDMICOS E SUGESTOES DE MUDANÇAS - Antonio Sala-zar Pessoa Brandão e Urlel de Magalhães - 1982

36. DETERMINAÇ~O NUMERICA DA TAXA INTERNA DE RETORNO: CONFRONTO ENTRE ALGORTTl MOS DE BOULDING E DE WILD - Clovis de Faro - 1983

37. MODELO DE EQUAÇOES SIMULTANEAS - Fernando de Holanda Barbosa - 19B3

38. A EFICIENCIA MARGINAL DO CAPITAL COMO CklTERIO DE AVALIAÇ~O ECONO~ICA DE PR~ JETOS DE INVESTIMENTO - Clovis de Faro· 1983 (ESGOTADO)

39. SALARIO REAL E INFLAÇAo {TEORIA E ILUST~çAo EMPTRICA} - Raul José Ekerman - 1984

40. TAXAS DE JUROS EFETIVAMENTE PAGAS POR TOMADORES DE EMPREsTIMOS JUNTO A B~

COS COMERCIAIS - Clovis de Faro - 198~

41. REGULAHENTAÇ~O E DECISOES DE CAPITAL EM BANCOS COMEkCIAIS: REVIS~ DA LITE RATURA E UM ENFOQUE PARA O BRASIL - Urlel de Magalhães - 1984

-42. INDEXAÇAo E AMB1ENCIA GERAL DE NEGOCIOS - Antonio Maria da Silveira - 1984

(15)

\

~,

1t4u SOBRE O NOVO PLANO DO BNH: "SIMC"*- Clovis de Faro - 1984

45.

sU8sToIOS CREOITfclOS ~ EXPORTAÇ~O - Gregório F.lo Stukart - 1984

46. PROCESSO DE DESINFlAÇ~O - Antonio C. Porto Gonçalves - 1984

47.

INDEXAÇ~O E REALIMENTAÇ~O INFLACIONARIA - Fernando de Holanda Barbc)sa -

1984

48.

SALARIOS M~DIOS E SALARIOS INDIVIDUAIS NO SETOR INDUSTRIAL: UM ESTUDO DE DI

FERENCIAÇAO SALARIAL ENTRE FIRMAS E ENTRE INDIVrnUOS - Raul José EI(erman

e

UrJe1 de Magalhães - 1984

49. THE DEVELOPING-COUNTRY OEBT PROBLEM - Mario HenrIque Sfr.10nsen - 1984

50. JOGOS DE INFORMAÇAo INCOMPLETA: UMA INTRODUÇÃO - S5rgio Ribeiro da Costa

Werlang - 1984

51. A TEORIA MONETARIA MODERNA E O EQUfLrBRIO GERAL WAlRASIANO COM UM HOMERO

INFINITO DE BENS - Ao Araujo - 1984

52. A INDETERMlNAÇAo OE MORGENSTERN - Antonio Maria da Silveira - 1984

53. O PROBLEMA DE CREDIBILIDADE EM POLrTlCA ECONOMICA - Rubens Penha Cysne

-.1984

54.

UMA ANALISE ESTATfSTICA DAS CAUSAS DA EMISsAo DO CHEQUE SEM FUNDOS:

FORMU-LAÇ~O DE UM PROJETO PILOTO - Fernando de Holanda Barbosa, Clovis dce Faro e

A1ofslo Pessoa de Araujo - 1984

55. POLfTICA MACROECONOMICA NO BRASIL: 1964-66 - Rubens Penha Cysne - 1985

56 c EVOLUÇ~O DOS PLANOS BAs I COS DE F I NANC lAMENTO PAR4 AQL" S I ç~O DE CASA PROPRI A

DO BANCO NACIONAL DE HABITAÇAO: 1964 - 1984. - Clo'"is cc'Fêro - 1985

57. MOEDA INDEXADA - Rubens P. Cysne - 1985

58.

INFLAÇ~O E SALARIO REAL: A EXPERI~NCIA BRASilEIRA - Raul José Ekerman - 1985

, TULlO VARGAS i'UNDAÇAO GE"F.~1UQ\JESlMONS'

,mu01ECAMAR\O ..

(16)

59. O ENFOQUE MONETARIO DO BALANÇO DE PAGAMENTOS: UM RETROSPECTO - Valdir Ramalho de Melo - 1985

60. MOEDA E PREÇOS RELATIVOS: EVIDENCIA EMPTRICA Antonio Sa1azar P. Brandão -1985

61. INTERPRETAÇAO ECONOMICA. INFLAÇAO E INDEXAÇAO Antonio Maria da Silveira -1985

62~ MACROECONOMIA - CAPITULO I - O SISTEMA MONETARIO - Mario Henrique Slmonsen

e Rubens Penha Cysne - 1985

63. MACROECONOMIA - CAPTTULO II - O BALANÇO DE PAGAMENTOS - Mario Henrique

Simonsen e Rubens Penha Cysne - 1985

64. MACROECONOMIA - CAPTTULO III - AS CONTAS NACIONAIS - Mario Henrique Simonsen

e Rubens Penha Cysne - 1985

65. A DEMANDA POR DIVIDENDOS: UMA JUSTIFICATIVA TEORICA - Tommy Chin-Chiu Tan e

Sergio Ribeiro da Costa Werlang - 1985

66. BREVE RETROSPECTO DA ECONOMIA BRASILEIRA ENTRE 1979 e 1984 - Rubens Penha

Cysne - 1985

67. CONTRATOS SALARIAIS JUSTAPOSTOS E POLrTICA ANTI-INFLACIONARIA - Mario Henrique Si monsen - 1985

68. INFlAÇAO E POlrTleAS DE RENDAS - Fernando de Holanda Barbosa e Clovis de

\ Faro - 1985

----\

69 BRAZIL INTERNATIONAL TRACE AND·ECONO~C GROWTH - Mario Henrique

Simonsen - 1986

70. CAPITALIZAÇAO CONTtNUA: APLICAÇÕES - Clovis de Faro 1986

71. A RATIONAL EXPECTATIONS PARADOX - Mario Henrique Simonsen - 1986

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