Oil production and institutions

5.1.1 The dynamic system

Under a managed exchange rate regime I, the policy rule followed by the foreign country is assumed to react to the level of the nominal exchange rate, while the home country just follows a simple Taylor-type feedback rule

^{^H_t = ^H_t + y_t^H; (10)

^{^F_t = ^F_t + y_t^F ₁S^_t;

Equations (3), (4), (5), (6), and (7), together with the interest rate rule (10), characterize the log-linear equilibrium in the managed regime I.

If again denote ^R_t = ^F_t ^H_t , and ^W_t = n ^H_t + (1 n) ^F_t, following the same procedures as above, the equilibrium conditions are

8>

in which the matrices are given in Appendix A.5.

Under the managed exchange rate regime I, the complete separation be-tween world variables and relative variables again disappears due to the ad-ditional reaction to the international variable, the nominal exchange rate, in the policy rules of the foreign country.

5.1.2 Determinacy

Under this managed regime, ^R_t, ^St, ^W_t , y^W_t are the nonpredetermined vari-ables, while ^T_{t 1} and ^S_{t 1} are the predetermined variables. If introduce the notation ^T_t^l = ^T_{t 1}, ^S_t^l = ^S_{t 1}, add them to the system, and then replace expectations by their true values, we can rewrite the full equilibrium system (29) as

L^{M I}₁₁ 0 L^{M I}₂₁ L^{M I}₂₂

! Y_1;t+1^{M I} Y_2;t+1^{M I}

!

= M₁₁^{M I} 0 M₂₁^{M I} M₂₂^{M I}

! Y_1;t^{M I} Y_2;t^{M I}

!

+ other^{M I}; (30) in which we de…ne Y_1;t^{M I} = ^R_t S^t T^_t^l S^_t^{l 0} and Y_2;t^{M I} = ^W_t y^W_t ⁰, and the matrices are given in Appendix A.5. The matrix L^{M I}₂₁ and M₂₁^{M I} are not null under the managed regime I, due to the reaction to the nominal exchange rate in the policy rule followed by the foreign country, where the foreign policymaker starts to concern international variables.

The reduced form of full system (30) is Y_1;t+1^{M I}

Y_2;t+1^{M I} = J^{M I} Y_1;t^{M I} Y_2;t^{M I}

!

+ other^{M I};

where

J^{M I} = L^{M I}₁₁ ¹M₁₁^{M I} 0

L^{M I}₂₂ ¹L^{M I}₂₁ L^{M I}₁₁ ¹M₁₁^{M I} + L^{M I}₂₂ ¹M₂₁^{M I} L^{M I}₂₂ ¹M₂₂^{M I}

!

and the matrix J^{M I} is lower block triangular. As the discussion in section 2.5.1, to obtain the condition for equilibrium determinacy for the full system (30) is therefore equivalent to calculating the determinacy condition for the

following two simpli…ed subsystems

Y_1;t+1^{M I} = J₁₁^{M I}Y_1;t^{M I} + other_1;t^{M I}; (30-1) Y_2;t+1^{M I} = J₂₂^{M I}Y_2;t^{M I} + other_2;t^{M I}; (30-2) where

J₁₁^{M I} = L^{M I}₁₁ ¹M₁₁^{M I}; J₂₂^{M I} = L^{M I}₂₂ ¹M₂₂^{M I}:

Since there are two nonpredetermined variables and two predetermined vari-ables in subsystem (30-1), and there is no predetermined variable in subsys-tem (30-2), the determinacy condition for full syssubsys-tem (30) is that exactly two of four eigenvalues of J₁₁^{M I} lie outside the unit circle, and all of two eigenvalues of J₂₂^{M I} lie outside the unit circle. Notice that the subsystem (30-2) is in the same form as under the ‡oating regime (23-4), which implies the condition (MI2). Therefore, following BB (2006), I get the following proposition.

Proposition 5 Under the managed exchange regime (I) with following in-terest rate rules

^{^H_t = ^H_t + y_t^H;

^{^F_t = ^F_t + y_t^F ₁S^_t;

with , and ₁ non negative as well as ₁ non zero, if the degrees of rigidity are equal across countries, the necessary and su¢ cient condition for equilibrium determinacy is

1 > 0; (MI1)

k_C( 1) + (1 ) > 0: (MI2)

Proof. See Appendix A.1.

It is easy to see that the Taylor principle is again su¢ cient for the

deter-necessary and su¢ cient condition for determinacy is the same as its closed economy counterpart. Compared with the ‡oating regime case, the region for determinate REE is therefore enlarged under this managed regime, due to the additional reaction towards the nominal exchange rate in the policy rule followed by the foreign country.²⁴

5.1.3 Learning stability

The full system (29) can be reduced to the following y^{M I}_1;t

y^{M I}_2;t

!

= P^{M I} Ety_1;t+1^{M I} E_ty_2;t+1^{M I}

!

+ Q^{M I} y^{M I}_{1;t 1} y^{M I}_{2;t 1}

!

+ K^{M I} !^{M I}_1;t

!^{M I}_2;t

!

; (31)

where

P^{M I} = A^{M I}₁₁ ¹B₁₁^{M I} 0

A^{M I}₂₂ ¹A^{M I}₂₁ A^{M I}₁₁ ¹B₁₁^{M I} A^{M I}₂₂ ¹B^{M I}₂₂

!

;

Q^{M I} = A^{M I}₁₁ ¹F₁₁^{M I} 0 A^{M I}₂₂ ¹A^{M I}₂₁ A^{M I}₁₁ ¹F₁₁^{M I} 0

!

;

K^{M I} = A^{M I}₁₁ ¹C₁₁^{M I} 0

A^{M I}₂₂ ¹A^{M I}₂₁ A^{M I}₁₁ ¹C₁₁^{M I} A^{M I}₂₂ ¹C₂₂^{M I}

! :

It is in the same form as system (28) under the …xed regime, and therefore the discussion of learnability for system (31) is equivalent to the discussion of learnability for two subsystems

y^{M I}_1;t = P₁₁^{M I}E_ty^{F I}_1;t+1+ Q^{M I}₁₁ y_{1;t 1}^{F I} + K₁₁^{M I}!^{F I}_1;t; (31-1) y^{M I}_2;t = P₂₂^{M I}E_ty^{M I}_2;t+1+ K₂₂^{M I}!^{M I}_2;t ; (31-2)

24The condition happens to be the same as under the …xed regime, since we assume the identical parameters in the policy rules with = , and = .

where

P₁₁^{M I} = A^{M I}₁₁ ¹B₁₁^{M I}; Q^{M I}₁₁ = A^{M I}₁₁ ¹F₁₁^{M I}; K₁₁^{M I} = A^{M I}₁₁ ¹C₁₁^{M I}; P₂₂^{M I} = A^{M I}₂₂ ¹B₂₂^{M I}; K₂₂^{M I} = A^{M I}₂₂ ¹C₂₂^{M I}:

Supposing !^{M I}_1;t and !^{M I}_2;t follow vector AR(1) processes as before, the MSV solutions for the two subsystems are respectively in the form of

y^{M I}_1;t = a^{M I}₁ + b^{M I}₁ y^{M I}_{1;t 1}+ c^{M I}₁ !^{M I}_1;t ; y^{M I}_2;t = a^{M I}₂ + c^{M I}₂ !^{M I}_2;t :

Insert them into (31-1) and (31-2), and then the REE is solved for the man-aged regime I as

n

a^{M I}₁ ; b^{M I}₁ ; c^{M I}₁ o

and a^{M I}₂ ; c^{M I}₂ :

The corresponding E-stability conditions are that all of the eigenvalues of F₁₁^{M I} I₁₁ P₁₁^{M I}b^{M I}₁

1

P₁₁^{M I}, ^{M I}₁₁ I₁₁ P₁₁^{M I}b^{M I}₁

1

Q^{M I}₁₁ , and P₂₂^{M I} have real parts less than one. The subsystem (31-2) is again the same as (23-2) under the ‡oating regime, which implies that condition (MI2) is one necessary condition for the learnability of REE in the whole economy. Fur-thermore, the result of McCallum (2006) means that the condition (MI1), and (MI2) for the equilibrium determinacy is su¢ cient for the learnability of REE. Therefore, I obtain the following proposition.

Proposition 6 Under a managed exchange rate regime I de…ned by the rules of the following form

^{^H_t = ^H_t + y_t^H;

^{^F_t = ^F_t + y_t^F ₁S^_t;

with and non negative as well as ₁ non zero, if the degrees of rigidity are equal across countries, the necessary and su¢ cient condition for learnability of REE is

1 > 0; (MI1)

k ( 1) + (1 ) > 0: (MI2)

0.2 0.4 0.6 0.8 1 f 1

2 3 4 5

y Figure3

MI2 (FL2) FL1

Given the calibration in Section 2.6, the …gure 3 plots the conditions of determinacy and learnability as a function of and . Given any positive value of ₁, the lower line (MI2) is the condition for determinacy and learn-ability of REE under this managed regime, which is the same condition as (FL2) for the ‡oating regime. The numerical results suggest that the con-ditions for determinacy and learnability coincide again under this managed regime.

In particular, the condition (MI2) is the necessary and su¢ cient condition for determinacy and learnability of REE for the second subsystem of the world average variables (30-2) and (31-2), while additional condition (FL1) is required for the determinate and learnable REE of the …rst subsystem for relative variables (30-1) and (31-1) when ₁ = 0. It implies that an additional reaction to the level of nominal exchange rate in the policy rule of the foreign country will generally enlarge the region for determinacy and learnability of REE and therefore improve the trade-o¤ between parameters and for the central banks, compared with the ‡oating regime. The enlarged region is the part of northeast of line (MI2), which implies the same condition for determinacy and learnability of REE in the closed economy case. Therefore, even though the conditions for determinacy and learnability could become more stringent due to the open economy considerations of the central bank, the restriction for policymakers is not necessarily stricter than the closed economy case when there is additional reaction towards the nominal exchange rate in the policy rules. The enlarged region is due to the terms of trade e¤ects over the output gap. To see this more explicitly, we

recall the equation (22), which is derived from the de…nition of the output gap and aggregate demand functions (2)

^

y_t^H y^_t^F = ^Tt T~t:

Substitute the terms of trade de…nition equation (4) into it, we can get

^

y_t^H y^_t^F = ^T_{t 1}+ ^S_t S^_{t 1}+ ^F_t ^H_t T~_t;

which implies

S^_t = ^y_t^H + ^H_t y^_t^F + ^F_t T^_{t 1}+ ^S_{t 1}+ ~T_t:

Therefore, the policy rule followed by the foreign country becomes

^{^F_t = ^F_t + y^F_t + ₁ y^_t^F + ^F_{t 1} y^_t^H + ^H_t T^_{t 1}+ ^S_{t 1}+ ~T_t : This is a more aggressive rule than the Taylor-type rule with the same para-meters and for the foreign country, due to an additional reaction towards domestic in‡ation rate and output gap by size of ₁. Equivalently, it implies the condition for determinacy and learnability of the foreign country is less stringent. Intuitively, it is because the home central bank not only in‡uence the domestic demand of households but also the foreign demand through the movements of the terms of trade. The condition for determinacy and learn-ability of the home country and therefore the world economy is less stringent, which implies explicit or implicit monetary interdependence across countries.

Finally, Taylor Principle is again su¢ cient for determinacy and learnability under this managed regime. In particular, given = 0, it is easy to see that

> 1 is the necessary and su¢ cient condition to guarantee the determinacy and learnability of REE.