- Email: [email protected]
A note on the optimal choice of monetary and fiscal policy
NAOYUKI YOSHINO Saitama
University,
Japan
A Note on the Optimal Choice of Monetary and Fiscal Policy* This paper studies the instrument-target problem associated with a simple macroeconomic model with conditions of uncertainty. The purpose of the paper is to demonstrate that if a model is linear with nonstochastic coefficients and with additive disturbances and if the policy objective is to maximize the expected value of the quadratic loss function, then it is the choice of endogenous variables that alters the expected loss value. This contrasts with the findings of Poole, Sargent, the expected losses are compared for different instruTurnovsky, et al., in which ments while endogenous variables are changed at the same time. Thus, if basic model specifications are unchanged, an important cause for different expected losses under different policy instruments is seen to lie in the choice of endogenous variables.
In this paper, in regard to the conventional instrument-target control problem in macroeconomics, it will be shown that the expected value of the quadratic loss function is independent of the choice of instrument as long as endogenous variables are not altered. The problem has been previously examined by Poole, Sargent, Turnovsky, Friedman, et al. They conclude that the policies give rise to different values of the expected loss. In their studies, they did not merely change the instrument but they also changed the endogenous variables. Indeed the two models they compared, although the same in underlying structure, were quite different with respect to their choice of endogenous variables. And it was this, and this alone, that gave different values for their expected losses. The purpose of this paper is to demonstrate that if a model is linear with nonstochastic coefficients and with additive disturbances and if the policy objective is to maximize the expected value *I would like to thank the referee of this article for helpful comments. I am also indebted to Carl F. Christ, Alan Walters, Louis Maccini, Isaac Ehrlich, N. Revankar, and A. Horiuchi for their valuable comments. This paper was written while the author was an assistant professor at the State University of New York at Buffalo. Journal Copyright
of Macroeconomics, 0 1983 by Wayne
Fall 1982, Vol. State University
4, No. 4, pp. Press.
459-468
459
Naoyuki
Yoshino
of the quadratic loss function, then it is the choice of endogenous variables that alters the expected loss value. This principle will be shown in the framework of a simple linear nonstochastic-coefficients model with additive disturbance terms and a quadratic expectedloss function as the policy objective. The application of the principle to macro policies when the policy objective is to maximize the expected value of a quadratic loss function will be briefly discussed.
1. A Wide Class of Linear Stochastic Models The analysis in this section is based on a two-constraint model with additive disturbance terms, nonstochastic coefficients and a quadratic loss function as the policy objective. The following symbols will be used throughout. EL Y, yf 2,:
= = = =
r, = m, = A,, a,, bj = Uit
=
A,,, Bjt =
the expected value of the quadratic loss function; real income (the actual value of the missile); a target value of real income. sometimes endogenous and sometimes exogenous, (i = 1, 2, 3); the interest rate; the real stock of money; known nonstochastic coefficients, (i = 1, 2; j = 1, 2, 3); random disturbances whose mean and variance-covariances are 0, 4, and crij respectively, (i = 1, 2); stochastic coefficients with known means and variante-covariances, (i. j = 1, 2).
Suppose the policy authority wishes to choose that policy with the least (quadratic) loss between an actual value of y, and the target value y{ set by the policy maker. The problem will be formed as follows: Minimize EL(t) = Et-l(yt - y{)’ ,
(1)
subject to Fr(y,, Zr,, ZZt, Z,J + ult = 0 ,
(2)
where 460
Optimal
Choice of Policy
E(u,) = 0 (for i = 1, 2, and V t) ,
E(& = a;; (V t) , E(ui++) = uV; (V t) . Equation (1) is the quadratic loss function and Equations (2) and (3) are the linear constraints with additive disturbances. Here we assume two constraints, which may be regarded as the IS and LM curves of a macro model, with one target. This paper will analyze four policies (see Table 1) in which it will be shown that, where the same endogenous variables are chosen by the authorities and the disturbance terms are additive with nonstochastic coefficients in structural equations, the same expected value of the loss function will result whichever instrument is used. The intuitive reason for the result is as follows: As long as endogenous variables are identical, elimination of such variables always leads to identical reduced-form equations. Here, applying the certainty-equivalence theorem (whereby the values of instrument variables other than those in question are set at their mean values), the optimal value of the policy instrument is set to attain the target value. Substitution of these optimal values into reducedform equations shows that the actual value of the missile, y,, is expressed in terms of the target value of the missile, y$, plus the additive disturbance terms. Since the loss function is the expected value of the quadratic loss between the actual value of the missile and the target, the additive disturbance terms determine the value of the expected-loss function. The following briefly summarizes the two sets of policies. First, we compare Policy 1 with Policy 3 where a different set of endogenous variables is chosen by the authorities under two different policy instruments. Then we compare Policy 2 with Policy TABLE
1.
Policy 1
Policy 2
Policy 3
Yj Missile Instrument
Y 2, 52
z3
z*
zy z:
z3
Status in original Policy 4 reduced form
Yf
-
z” 2
Endogenous Exogenous Endogenous Exogenous
z,
461
Naoyuki
Yoshino
4 where the same set of endogenous variables is chosen under two different policies. When the two sets of policies are compared, it will be found that the value of the expected-loss function is identical under Policies 2 and 4. Policy 1 and Policy 3 Eliminating an endogenous variable Z,, for Policy 1 from Equations (2) and (3), yt will be derived in terms of Zr,, Z,, and the additive disturbance terms. The problem is formed as Minimize
EL = Etel(y, - y {)” ,
subject to yt = fr(Z1,, Z3J + Aiur, + ALust .
(4)
Here, invoking the certainty-equivalence theorem, the optimal value of the policy instrument ZZ is determined such that Y: = .mT,, Z,,) 9
(5)
where
Et-,h3 = Et-,(4 = 0; (V t) . Substituting the optimal value of the policy instrument reduced-form equation (4) yields yt = y ; + A&, Therefore
the expected
Z;i; into the
+ A;u,, .
loss for Policy 1 is
EL, = (A;)“& + (A;)%; + 2A;A;cr,,
.
Eliminating an endogenous variable Zr, for Policy 3 from Equations (2) and (3), yt will be derived in terms of Zzt, Z,, and the additive disturbance terms. The problem is formed as Minimize
EL = Etwl(y, - yp,
subject to yt = frrl(Zzt, Z,,) + A:“ur, + A!/u,, Using the certainty-equivalence 462
theorem,
.
the optimal
(4’) value of the
Optimal policy instrument
Zg is derived
Choice of Policy
such that
where
Et-h,) = Et-h,) = 0; ( t) . Substituting Zzt derived Equation (4’) yields
in Equation
yt = y ; + A:“un Therefore
the expected
(5’) into the
4 A:“u,,
reduced-form
.
loss for Policy 3 is
EL,,, = (A:“)“$
+ (A;‘)20; + 2A:‘1A;%12
Thus Policies 1 and 3 yield different
values of the expected
loss.
Policy 2 and Policy 4 Eliminating an endogenous variable Z,, from Equations (2) and (3), yt will be derived in terms of Z1,, Z,, and the additive disturbance terms in both Policies 2 and 4. Therefore, the problem is formed as follows: Minimize
EL = E,-,(y, - y{)’ ,
subject to yt = f(&,,
Z,,) + A,u,, + A2uet .
(6)
The optimal value of the policy instrument 2;” under Policy 2 is derived as follows using the certainty-equivalence theorem:
Substituting Zr,* derived Equation (6) yields yt =
in Equation
Y:
+ Alult
(7) into
+
A2u2,
the reduced-form
.
The optimal value of the policy instrument 4 is derived as follows:
Z,*, under
Policy
463
Naoyuki
Yoshino Yi = ff&t,
Substituting 2; derived Equation (6) yields
a)
’
in Equation
(7’) into the reduced-form
Therefore the value of the expected-loss 2 and 4 is expressed as EL = (AJ%; + (A&
(7’)
function
for both Policies
+ 2A1Azo12 .
The principle developed here will apply to the cases where the model is linear, the solution for the optimal policy instrument is interior, the coefficients are nonstochastic, and the costs of pursuing each policy are negligible. (In the Appendix, it is shown that the above principle can be applied if the coefficients are stochastic). 2. An Application of the Principle to a Macroeconomic Model Suppose the policy authority seeks to control the real income (y,) as close as its full-employment target level (y{) with a macro model of an IS-LM type. The problem is formed as follows just as shown in the previous section: Minimize EL(t) = Etml(y, - y{)” , subject to yt = a,, + a,r, + a&
mt =
(1’) + ult,
b, + b,y, + b,r, + b&r
(2’)
+ ~2t >
(3’)
where yt y{ r, m, Z,,
real income; a target value of real income; the interest rate [= Z,, of Equations (2) and (3)]; the real stock of money [= Zzt of Equations (2) and (3)]; a vector of other variables including taxes, government spending, and government bonds held by private sectors [= ZBt of Equations (2) and (3)]; ai, b, = known nonstochastic coefficients; ui, = random disturbances with means and variance-covariantes as stated in Section 1. 464
= = = = =
Optimal
Choice of Policy
Previous work compared the loss function under the interest rate policy with money supply as an endogenous variable (Policy 1 in Section 1) and under the money supply policy with the interest rate as an endogenous variable (Policy 3 in Section 1). As was shown in Section I, the expected losses in these two cases are different. However, when the same endogenous variable (Z,,) is considered with two different policies (money supply policy and the interest rate policy with fiscal policy endogenous-Policy 2 and Policy 4 in Section I), the value of the expected-loss function is identical as follows: EL = (a,)-2(b$:
+ aEo$ - 2a,b,a12) .
A full discussion of the previous principle can be applied to monetary and fiscal policy by using the government budget restraint in a macro model. The above principle can also be applied to a multi-period model. 3. Conclusion Previous work employed an IS-LM type of linear model with additive disturbances to show that different policy instruments lead to different expected losses. This analysis has shown that the expected value of the quadratic loss function is independent of the choice of instrument as long as endogenous variables are not altered in the context of the linear nonstochastic-coefficients model with additive disturbance terms. Received: February, 1981 Final version received: May,
1982
References Brainard, W. “Uncertainty
and the Effectiveness
of Policy. ” Amer-
Review 57 (May 1967): 411-25. G.C. Analysis and Control of Dynamic Economic
ican Economic
Chow, Systems. New York: John Wiley, 1975. Christ, C.F. “On Fiscal and Monetary Policies and the Government Budget Restraint. ” American Economic Review 69 (September 1979): 526-38.
Currie, D.A. “Optimal Stabilization Policies and the Government Budget Constraint. ” Economica 43 (May 1976):159-67. Friedman, B.M. “Targets, Instruments and Indicators of Monetary 465
Naoyuki
Yoshino
Policy.” Journal of Monetary Economics 1 (October 1975): 44% 73. -. “The Inefficiency of Short-Run Monetary Targets for Monetary Policy.” Brookings Papers on Economic Activity 2 (1977): 293335. Poole, W. “Optimal Choice of Monetary Policy Instrument in a Simple Stochastic Macro Model.” Quarterly Journal of Economics 80 (May 1970): 197-216. Revankar, N. and N. Yoshino. “Framework for Analysis of the Instrument Problem in Monetary Theory. ” Economics Letters 9 (1982): 49-55. Sargent, T.J. “The Optimum Monetary Instrument Variable in a Linear Economic Model, ” Canadian Journal of Economics 4 (February 1971): 50-60. Tumovsky, S. J. “Optimal Choice of Monetary Instrument in a Linear Economic Model with Stochastic Coefficients.” Journal of Money, Credit, and Banking 7 (February 1975): 51-80. Yoshino, N. “Optimal Choice of Monetary and Fiscal Policy under Uncertainty. ” Ph. D. dissertation, Johns Hopkins University, 1979. Appendix In the text, we found the Policies 2 and 4 yield the same value of the expected-loss function. The case of the stochastic coefficients of the reduced form (6) is analyzed here. Eliminating an endogenous variable 2, from Equations (2) and (3), the reduced form for yt is derived in terms of Z,, 2, and the additive disturbance terms in both Policies 2 and 4.
Here we express the reduced form in an explicit linear form and B,, B1,, Bzt, Alt, and A, are assumed to be stochastic. Policy 2:
= 466
WY
-
yf)
41
;
Optimal
Choice of Policy
Policy 4:
= WY
- yf) %I .
Let us assume that there exists an optimal value of the policy instrument such that E(y) = yf. Provided that the stochastic coefficients and the additive disturbances are independent, the optimal value of the policy instrument is derived as follows: Policy 2:
zi+= ~l/W?N~ [EYE
- EWW,!
- W%A)I
;
Policy 4:
The values of the expected loss are Policy 2: 1 ELz = E { B” ’ Var(B,) + [E(B,)]2
[E@,)Y f - WVW, 2
- E(B,B,)]B,
+ B,Z, + A+,
+ A,u, -
.
Policy 4: 1 -
[W,)Y~
Var(B,)+ [E03,)12
WV32K
2
- E(B,B,)]B,
+ A+,
+ A,u, - yf
In the above, EL, and EL, yield different deterministic coefficient case in which
I
. values except for the
467
Naoyuki E&J
Yoshino = Bi (i = O,l,Z);
E(A,,A,,) = AiAj (ij Var(B,)
E(A,J = Ai (i = 1,2);
= 1, 2);
E&B,,)
and
= BIBj (i, j = 1, 2, 3) ;
= 0 = Var(A,J.
In the case of the fured coefficients,
EL2 = E(A,q
the expected
loss becomes
+ A&’
for Policy 2 and
EL4 = E(A,u, + A,u,)’ for Policy 4. Therefore, fixed-coeffkients case.
468
both values are identical
only under
the
University,
Japan
A Note on the Optimal Choice of Monetary and Fiscal Policy* This paper studies the instrument-target problem associated with a simple macroeconomic model with conditions of uncertainty. The purpose of the paper is to demonstrate that if a model is linear with nonstochastic coefficients and with additive disturbances and if the policy objective is to maximize the expected value of the quadratic loss function, then it is the choice of endogenous variables that alters the expected loss value. This contrasts with the findings of Poole, Sargent, the expected losses are compared for different instruTurnovsky, et al., in which ments while endogenous variables are changed at the same time. Thus, if basic model specifications are unchanged, an important cause for different expected losses under different policy instruments is seen to lie in the choice of endogenous variables.
In this paper, in regard to the conventional instrument-target control problem in macroeconomics, it will be shown that the expected value of the quadratic loss function is independent of the choice of instrument as long as endogenous variables are not altered. The problem has been previously examined by Poole, Sargent, Turnovsky, Friedman, et al. They conclude that the policies give rise to different values of the expected loss. In their studies, they did not merely change the instrument but they also changed the endogenous variables. Indeed the two models they compared, although the same in underlying structure, were quite different with respect to their choice of endogenous variables. And it was this, and this alone, that gave different values for their expected losses. The purpose of this paper is to demonstrate that if a model is linear with nonstochastic coefficients and with additive disturbances and if the policy objective is to maximize the expected value *I would like to thank the referee of this article for helpful comments. I am also indebted to Carl F. Christ, Alan Walters, Louis Maccini, Isaac Ehrlich, N. Revankar, and A. Horiuchi for their valuable comments. This paper was written while the author was an assistant professor at the State University of New York at Buffalo. Journal Copyright
of Macroeconomics, 0 1983 by Wayne
Fall 1982, Vol. State University
4, No. 4, pp. Press.
459-468
459
Naoyuki
Yoshino
of the quadratic loss function, then it is the choice of endogenous variables that alters the expected loss value. This principle will be shown in the framework of a simple linear nonstochastic-coefficients model with additive disturbance terms and a quadratic expectedloss function as the policy objective. The application of the principle to macro policies when the policy objective is to maximize the expected value of a quadratic loss function will be briefly discussed.
1. A Wide Class of Linear Stochastic Models The analysis in this section is based on a two-constraint model with additive disturbance terms, nonstochastic coefficients and a quadratic loss function as the policy objective. The following symbols will be used throughout. EL Y, yf 2,:
= = = =
r, = m, = A,, a,, bj = Uit
=
A,,, Bjt =
the expected value of the quadratic loss function; real income (the actual value of the missile); a target value of real income. sometimes endogenous and sometimes exogenous, (i = 1, 2, 3); the interest rate; the real stock of money; known nonstochastic coefficients, (i = 1, 2; j = 1, 2, 3); random disturbances whose mean and variance-covariances are 0, 4, and crij respectively, (i = 1, 2); stochastic coefficients with known means and variante-covariances, (i. j = 1, 2).
Suppose the policy authority wishes to choose that policy with the least (quadratic) loss between an actual value of y, and the target value y{ set by the policy maker. The problem will be formed as follows: Minimize EL(t) = Et-l(yt - y{)’ ,
(1)
subject to Fr(y,, Zr,, ZZt, Z,J + ult = 0 ,
(2)
where 460
Optimal
Choice of Policy
E(u,) = 0 (for i = 1, 2, and V t) ,
E(& = a;; (V t) , E(ui++) = uV; (V t) . Equation (1) is the quadratic loss function and Equations (2) and (3) are the linear constraints with additive disturbances. Here we assume two constraints, which may be regarded as the IS and LM curves of a macro model, with one target. This paper will analyze four policies (see Table 1) in which it will be shown that, where the same endogenous variables are chosen by the authorities and the disturbance terms are additive with nonstochastic coefficients in structural equations, the same expected value of the loss function will result whichever instrument is used. The intuitive reason for the result is as follows: As long as endogenous variables are identical, elimination of such variables always leads to identical reduced-form equations. Here, applying the certainty-equivalence theorem (whereby the values of instrument variables other than those in question are set at their mean values), the optimal value of the policy instrument is set to attain the target value. Substitution of these optimal values into reducedform equations shows that the actual value of the missile, y,, is expressed in terms of the target value of the missile, y$, plus the additive disturbance terms. Since the loss function is the expected value of the quadratic loss between the actual value of the missile and the target, the additive disturbance terms determine the value of the expected-loss function. The following briefly summarizes the two sets of policies. First, we compare Policy 1 with Policy 3 where a different set of endogenous variables is chosen by the authorities under two different policy instruments. Then we compare Policy 2 with Policy TABLE
1.
Policy 1
Policy 2
Policy 3
Yj Missile Instrument
Y 2, 52
z3
z*
zy z:
z3
Status in original Policy 4 reduced form
Yf
-
z” 2
Endogenous Exogenous Endogenous Exogenous
z,
461
Naoyuki
Yoshino
4 where the same set of endogenous variables is chosen under two different policies. When the two sets of policies are compared, it will be found that the value of the expected-loss function is identical under Policies 2 and 4. Policy 1 and Policy 3 Eliminating an endogenous variable Z,, for Policy 1 from Equations (2) and (3), yt will be derived in terms of Zr,, Z,, and the additive disturbance terms. The problem is formed as Minimize
EL = Etel(y, - y {)” ,
subject to yt = fr(Z1,, Z3J + Aiur, + ALust .
(4)
Here, invoking the certainty-equivalence theorem, the optimal value of the policy instrument ZZ is determined such that Y: = .mT,, Z,,) 9
(5)
where
Et-,h3 = Et-,(4 = 0; (V t) . Substituting the optimal value of the policy instrument reduced-form equation (4) yields yt = y ; + A&, Therefore
the expected
Z;i; into the
+ A;u,, .
loss for Policy 1 is
EL, = (A;)“& + (A;)%; + 2A;A;cr,,
.
Eliminating an endogenous variable Zr, for Policy 3 from Equations (2) and (3), yt will be derived in terms of Zzt, Z,, and the additive disturbance terms. The problem is formed as Minimize
EL = Etwl(y, - yp,
subject to yt = frrl(Zzt, Z,,) + A:“ur, + A!/u,, Using the certainty-equivalence 462
theorem,
.
the optimal
(4’) value of the
Optimal policy instrument
Zg is derived
Choice of Policy
such that
where
Et-h,) = Et-h,) = 0; ( t) . Substituting Zzt derived Equation (4’) yields
in Equation
yt = y ; + A:“un Therefore
the expected
(5’) into the
4 A:“u,,
reduced-form
.
loss for Policy 3 is
EL,,, = (A:“)“$
+ (A;‘)20; + 2A:‘1A;%12
Thus Policies 1 and 3 yield different
values of the expected
loss.
Policy 2 and Policy 4 Eliminating an endogenous variable Z,, from Equations (2) and (3), yt will be derived in terms of Z1,, Z,, and the additive disturbance terms in both Policies 2 and 4. Therefore, the problem is formed as follows: Minimize
EL = E,-,(y, - y{)’ ,
subject to yt = f(&,,
Z,,) + A,u,, + A2uet .
(6)
The optimal value of the policy instrument 2;” under Policy 2 is derived as follows using the certainty-equivalence theorem:
Substituting Zr,* derived Equation (6) yields yt =
in Equation
Y:
+ Alult
(7) into
+
A2u2,
the reduced-form
.
The optimal value of the policy instrument 4 is derived as follows:
Z,*, under
Policy
463
Naoyuki
Yoshino Yi = ff&t,
Substituting 2; derived Equation (6) yields
a)
’
in Equation
(7’) into the reduced-form
Therefore the value of the expected-loss 2 and 4 is expressed as EL = (AJ%; + (A&
(7’)
function
for both Policies
+ 2A1Azo12 .
The principle developed here will apply to the cases where the model is linear, the solution for the optimal policy instrument is interior, the coefficients are nonstochastic, and the costs of pursuing each policy are negligible. (In the Appendix, it is shown that the above principle can be applied if the coefficients are stochastic). 2. An Application of the Principle to a Macroeconomic Model Suppose the policy authority seeks to control the real income (y,) as close as its full-employment target level (y{) with a macro model of an IS-LM type. The problem is formed as follows just as shown in the previous section: Minimize EL(t) = Etml(y, - y{)” , subject to yt = a,, + a,r, + a&
mt =
(1’) + ult,
b, + b,y, + b,r, + b&r
(2’)
+ ~2t >
(3’)
where yt y{ r, m, Z,,
real income; a target value of real income; the interest rate [= Z,, of Equations (2) and (3)]; the real stock of money [= Zzt of Equations (2) and (3)]; a vector of other variables including taxes, government spending, and government bonds held by private sectors [= ZBt of Equations (2) and (3)]; ai, b, = known nonstochastic coefficients; ui, = random disturbances with means and variance-covariantes as stated in Section 1. 464
= = = = =
Optimal
Choice of Policy
Previous work compared the loss function under the interest rate policy with money supply as an endogenous variable (Policy 1 in Section 1) and under the money supply policy with the interest rate as an endogenous variable (Policy 3 in Section 1). As was shown in Section I, the expected losses in these two cases are different. However, when the same endogenous variable (Z,,) is considered with two different policies (money supply policy and the interest rate policy with fiscal policy endogenous-Policy 2 and Policy 4 in Section I), the value of the expected-loss function is identical as follows: EL = (a,)-2(b$:
+ aEo$ - 2a,b,a12) .
A full discussion of the previous principle can be applied to monetary and fiscal policy by using the government budget restraint in a macro model. The above principle can also be applied to a multi-period model. 3. Conclusion Previous work employed an IS-LM type of linear model with additive disturbances to show that different policy instruments lead to different expected losses. This analysis has shown that the expected value of the quadratic loss function is independent of the choice of instrument as long as endogenous variables are not altered in the context of the linear nonstochastic-coefficients model with additive disturbance terms. Received: February, 1981 Final version received: May,
1982
References Brainard, W. “Uncertainty
and the Effectiveness
of Policy. ” Amer-
Review 57 (May 1967): 411-25. G.C. Analysis and Control of Dynamic Economic
ican Economic
Chow, Systems. New York: John Wiley, 1975. Christ, C.F. “On Fiscal and Monetary Policies and the Government Budget Restraint. ” American Economic Review 69 (September 1979): 526-38.
Currie, D.A. “Optimal Stabilization Policies and the Government Budget Constraint. ” Economica 43 (May 1976):159-67. Friedman, B.M. “Targets, Instruments and Indicators of Monetary 465
Naoyuki
Yoshino
Policy.” Journal of Monetary Economics 1 (October 1975): 44% 73. -. “The Inefficiency of Short-Run Monetary Targets for Monetary Policy.” Brookings Papers on Economic Activity 2 (1977): 293335. Poole, W. “Optimal Choice of Monetary Policy Instrument in a Simple Stochastic Macro Model.” Quarterly Journal of Economics 80 (May 1970): 197-216. Revankar, N. and N. Yoshino. “Framework for Analysis of the Instrument Problem in Monetary Theory. ” Economics Letters 9 (1982): 49-55. Sargent, T.J. “The Optimum Monetary Instrument Variable in a Linear Economic Model, ” Canadian Journal of Economics 4 (February 1971): 50-60. Tumovsky, S. J. “Optimal Choice of Monetary Instrument in a Linear Economic Model with Stochastic Coefficients.” Journal of Money, Credit, and Banking 7 (February 1975): 51-80. Yoshino, N. “Optimal Choice of Monetary and Fiscal Policy under Uncertainty. ” Ph. D. dissertation, Johns Hopkins University, 1979. Appendix In the text, we found the Policies 2 and 4 yield the same value of the expected-loss function. The case of the stochastic coefficients of the reduced form (6) is analyzed here. Eliminating an endogenous variable 2, from Equations (2) and (3), the reduced form for yt is derived in terms of Z,, 2, and the additive disturbance terms in both Policies 2 and 4.
Here we express the reduced form in an explicit linear form and B,, B1,, Bzt, Alt, and A, are assumed to be stochastic. Policy 2:
= 466
WY
-
yf)
41
;
Optimal
Choice of Policy
Policy 4:
= WY
- yf) %I .
Let us assume that there exists an optimal value of the policy instrument such that E(y) = yf. Provided that the stochastic coefficients and the additive disturbances are independent, the optimal value of the policy instrument is derived as follows: Policy 2:
zi+= ~l/W?N~ [EYE
- EWW,!
- W%A)I
;
Policy 4:
The values of the expected loss are Policy 2: 1 ELz = E { B” ’ Var(B,) + [E(B,)]2
[E@,)Y f - WVW, 2
- E(B,B,)]B,
+ B,Z, + A+,
+ A,u, -
.
Policy 4: 1 -
[W,)Y~
Var(B,)+ [E03,)12
WV32K
2
- E(B,B,)]B,
+ A+,
+ A,u, - yf
In the above, EL, and EL, yield different deterministic coefficient case in which
I
. values except for the
467
Naoyuki E&J
Yoshino = Bi (i = O,l,Z);
E(A,,A,,) = AiAj (ij Var(B,)
E(A,J = Ai (i = 1,2);
= 1, 2);
E&B,,)
and
= BIBj (i, j = 1, 2, 3) ;
= 0 = Var(A,J.
In the case of the fured coefficients,
EL2 = E(A,q
the expected
loss becomes
+ A&’
for Policy 2 and
EL4 = E(A,u, + A,u,)’ for Policy 4. Therefore, fixed-coeffkients case.
468
both values are identical
only under
the