- Email: [email protected]
The efficient policy frontier under parameter uncertainty and multiple tools
DOUGLAS W. MITCHELL West Morgantown,
Virginia
University West
Virginia
The Efficient Policy Frontier under Parameter Uncertainty and Multiple Tools* This paper considers optimal policy design for arbitrary (that is, not necessarily quadratic) loss functions. There are a single target variable and an arbitrary number of policy tools with coefficient uncertainty. Under normality, an efficient policy frontier is derived. Any efficient policy vector is a weighted average of two arbitrary efficient vectors. If a tool with a known coefficient is introduced, the vector of optimal values of risky tools is independent of the loss function. Implications are discussed for policy advising when the policy maker’s loss function is imperfectly understood, and for numerical policy optimization when the loss function is analytically intractable.
1. Introduction There has been great interest over the years in the general topic of policy optimization in the context of a wide variety of models. Often, for the sake of mathematical simplicity, it is assumed that there is only additive uncertainty and no uncertainty about the values of parameters that enter the model multiplicatively with respect to policy tools. In reality, of course, there is almost always such parameter uncertainty, and various authors have dealt with this complication. ’ Generally, quadratic loss is assumed. However, as several authors have recognized, this assumption is very restrictive’: for one thing, it imposes symmetry, and, even within the class of symmetric loss functions, it is restrictive because of its specific form. Unfortunately, no one has ever presented a nonquadratic loss function that is tractable in the context of parameter uncertainty.
*I am grateful to two referees for their helpful comments. ‘See, for instance, Brainard (1967); Ch ow (1975, ch. 10); Deissenberg (1987); Gordon (1976); Holbrook and Howrey (1978); Kareken (1970); Mitchell (1979, 1983); Norman (1984); Tumovsky (1974); Waud (1976); and young (1975). *See Brandsma (1986); Caravani (1986); Hughes Hahett (1984, 1987); Horowitz (1987); Kunstman (1984); Mitchell (1979, 1983); and Waud (1976).
Journal of Macroeconomics, Winter 1990, Vol. Copyright 6 1990 by Louisiana State University 0164..0704/90/$1.50
12, No. Press
1, pp.
137-145
137
Douglas W. Mitchell The present paper addresses the issue of policy optimization under multiplicative parameter uncertainty when loss is not necessarily quadratic. In Section 2, it is assumed that there are n policy tools whose marginal effect on the single target variable is uncertain, and there is also additive uncertainty. Under a normality assumption, an “efficient policy frontier” is derived to characterize all policy vectors that could be optimal for some loss function. A direct analogy is made to Merton’s (1972) mutual fund theorem. Then, in Section 3, it is assumed that there is one further policy tool whose marginal effect is known. In this case, efficient policies are characterized in a way that is partially analogous to Merton’s separation theorem. Section 4 discusses the importance of the results for the design of policy when the policy maker’s loss function (that is, its functional form and parameterization) is not explicitly known, and for the numerical optimization of policy when the loss function is analytically intractable.
2. The Case of No Risk-Free Tools THEOREM 1. Assume that loss is a convex function of one target variable with a unique minimum, that there are n policy tools (n 2 2) with random multipliers, and that there is an additive shock; and assume that the additive shock and the multipliers are jointly normally distributed. Then, regardless of the specific form of the loss function, the optimal policy vector is a weighted average of two given policy vectors. Proof and Discussion. The value of the single target variable is y, and the vector of values of the policy tools is P.3 The economic structure is given by y = A’P + u , where A is the vector of random additive shock. While the simplest context, one can also view (1) as dynamic system, provided loss is
(1)
policy multipliers, and u is the interpretation of (1) is for a static derived from the final form of a still a function of a single scalar
‘Of course, this approach permits more than one target variable if they can be aggregated into a single index, y, which appears in the loss function. An example would be the “economic discomfort index,” widely alluded to in the late 197Os, which equals the sum of the inflation rate and the unemployment rate.
138
Efficient variable.
For example,
Policy Frontier
suppose GNP in time t is given by rt in xt
=
ax,-,
+
PPt
+
6
>
(2)
where p, is a vector of policy tools. Suppose further that loss is simply a function of a weighted average of GNP over a finite horizon. Then (2) can be solved for a final form for each t, and then the weighted average can be written in the form of (l), where P = (pi p; . . .) ’ is a stacked vector of policy values of all time periods. Then, the discussion of this paper applies to open-loop optimization Since open-loop control is known to be suboptimal under multiplier uncertainty, and, in general, dynamic loss cannot be expressed in terms of a scalar argument, an important area for future research is to identify a frontier, or some similar’problem-simplifying device, in the context of closed-loop control with multivariate loss. We now use the assumption that the additive shock u and the policy coefficients in A are jointly normally distributed. This assumption implies that y is normally distributed. It is known that, in this circumstance, expected loss can be expressed as a function of the mean and variance of y, with higher ai giving higher expected loss (Chipman 1973). Since our loss function is not monotonic in y, expected loss is not monotonic in the mean of y, and the mean-variance indifference curves are not monotonic (Waud 1976). Thus, we can identify an efficient policy frontier, in which, for any given mean of y, the variance of y is minimized. Every loss function will lead to an optimally chosen policy vector that is on this frontier. This is analogous to the mean-variance efficient frontier of portfolio theory, though with two differences. First, in portfolio theory, the analog of the policy vector-the vector of portfolio weights-must be such that the elements sum to unity; there is no such requirement in our policy context. And second, in the portfolio context, one must limit the focus to the upward sloped region of the frontier because expected utility is everywhere increasing in the mean of portfolio return, but, in our context, expected loss (the negative of expected utility) is not monotonic in the mean of y because loss is not monotonic in y, so the entire frontier is relevant. Using (l), we express the mean and variance of y as’
can
41f there are any be incorporated
linear restrictions in (3) without
that apply affecting fiti
across policy vector settings, substance of the results.
these
139
Douglas W. Mitchell Ey = /i’P + ii,
(3)
and
tg = P’VP + of + 2P’C ,
(4)
where A and ii are the means of A and u, V is the variance-covariance matrix of the elements of A, and C is a vector whose ith element is the covariance between u and the ith element of A. The parameters A, I!& V, and C are known and fixed moments. The problem then is to minimize 2, in (4) subject to Ey = Y, where Y is a constant that will parameterize the efficient frontier. The Lagrangian is
L = P’VP + cr: + 2P’C + 2A(Y - A’P - ii) , where are
2X is the Lagrangian
multiplier.
The first-order
(5) conditions
(1/2)dL/dP
= VP + c - AA = 0 )
(6)
(1/2)dL/dh
= Y - A’P - ii = 0.
(7)
and
We obtain the efficient P contingent on any Y by solving (6) provisionally for P in terms of A, substituting into (7’), solving the resulting equation for A, and substituting for A in the provisional expression for P. The result is
P* = P, f br ,
(8)
where
and
b = (Y - ti + AT-‘c)/(A’v-‘A) 140
3 0.
w
Effzcient
Policy Frontier
It can be shown that b, which is contingent upon Y, is one half of the reciprocal of the slope of the mean-variance frontier (with the mean plotted vertically) at the height Y. The efficient policy vectors in (8) can be rewritten as P* = kP, + (1 - k)P, , where PI and Pz are any efficient
vectors given by
PI = P, + b,I’ ,
(114
and Pz = PO+ b,I- , for arbitrary
parameters
b, # b, >
OW
b, and bp, and where
k = (b - b2)/(bl - b,) 52 0 .
WC)
Equation (10) above shows that every efhcient policy vector is a weighted average of any two arbitrarily specified efficient vectors. The set of all efficient points is found by varying k in (10). For all policy makers, regardless of loss function, the optimal point is a mean-variance efficient point by Chipman’s (1973) theorem. Therefore, all policy makers, regardless of loss function, will pick a point characterized by (lo)--that is, a weighted average of two arbitrary efficient points. The value of k at the policy maker’s optimal point will, of course, depend on the specific loss function employed. This result and its proof are essentially identical to those of Merton’s (1972) mutual fund theorem for portfolio theory.
3. The Case of One Risk-Free Tool We now consider the case in which, in addition to the risky policy tools assumed in Section 2, there is also one tool whose coefficient is known with certainty. The intuition of this case is straightforward. The policy optimization for any loss function can be conceptualized as a two-stage process. In the first stage, the vector P of levels of policy tools whose coefficients are random is chosen so as to unconditionally minimize 2,. Then, in the second stage, the level 4 of the risk-free tool is chosen so as to bring the mean of y to that value which is considered optimal, given the value of ai 141
Douglas W. Mitchell achieved in the first stage.5 This separation is possible because the risk-free tool only affects one moment of y (the mean). Note that, because Ey can be set exactly in the second stage, any additional risk-free tools would be purely redundant. Notice also that the efficient frontier is now vertical at a fixed value of a:--namely, the lowest value of c$ that can be achieved using the vector P of risky tools. The choice of q in the second stage, and thus the choice of a height on the efficient frontier, does not affect ai, and thus ensures that the frontier is vertical. The above intuition suggests the following separation theorem: THEOREM 2. Retain the assumptions of Theorem 1, but assume further that there is one policy tool whose multiplier is known with certainty. Then, the optimal vector of tools with uncertain multipliers is independent of the loss function, as is the variance of the target variable at the optimum.
and Discussion. This separation theorem is analogous to that of Merton (1972) in the portfolio context, though it differs somewhat in that Merton’s efficient locus, unlike ours, is not vertical. The derivation of the efficient policy frontier is as follows. The economic structure, instead of (l), is now
Proof
y = A’P + aq + u , where a is a known multiplier. are
(12)
The mean and variance of the target
Ey = A’P + aq + ii,
(13)
a2Y = P’VP + 4 u + 2P’C .
(14)
and
To derive the efficient frontier, the expression in (14) for cri is to be minimized subject to Ey having the arbitrary value Y, so that the Lagrangian is
L = P’VP + at + 2P’C + 2X(Y - A’P - aq - ii) .
not
142
‘Waud (1976) shows that, with an asymmetric loss function, be independent of the achieved value of IJ:.
this
value
(15) of Ey will
Efficient
Policy Frontier
The first-order condition for q immediately implies A = 0. The interpretation is that the constraint of a specified value for Ey imposes no added variance on y because q can be adjusted to obey the constraint without influencing a:. Using X = 0, the first-order condition for P implies p* = -v-‘c In addition,
the first-order
condition
.
(16)
for h implies
q* = (Y - A’P - 2%)/a )
(17)
or, using (16), q* = (Y + A’V-‘C
- ti)/a
(17’)
The set of efficient points is found by using P* from (16) and q* from (17’) while varying Y. Now (16) shows that P* is independent of Y. Since, by Chipman (1973), the expected loss minimizing point is in the set of efficient points (regardless of the loss function), the expected loss minimizing value of P is that given in (16), regardless of loss function. Thus, the optimal vector of risky policy magnitudes is independent of the loss function. The set of efficient values for q is found by varying Y in (17’). By Chipman, the optimal q, given some loss function, is in the efficient set. By (17’), q* varies with Y, so the optimal q is dependent on the loss function. All of these results and their proofs are essentially identical to those of Merton’s (1972) separation theorem for portfolio theory. Note further that P* in (16) is indeed, as previously asserted, the policy vector that unconditionally minimizes ai. This policy vector is zero if the additive shock is uncorrelated with all the multiplicative shocks (so C is the zero vector), but not in general otherwise. The optimal ai, found by using (16) in (14), is crz - C’V’C (which is independent of Y). Th is expression gives the location of the vertical efficient frontier.
4. Summary
and Implications
This paper has considered the design of policy with arbitrary loss function, under multiplicative as well as additive uncertainty, with jointly normally distributed shocks, and with a single target 143
Douglas W. Mitchell variable. With no risk-free policy tool, it has been shown that all policy makers will choose a weighted average of any two arbitrary efficient policy vectors. This is an important result, in part, because it suggests a way of presenting policy options to a policy maker whose loss function is not fully understood: all efficient policy vectors and the resultant means and variances of the target can be expressed parameterized by a single scalar choice parameter k. Then, the policy maker is simply presented with a series of efficient (Ey, ai) combinations and is asked which one is most satisfactory. Further, if the loss function is known but is analytically intractable (as appears inevitable with nonquadratic loss and multiplier uncertainty), numerical optimization can be performed more inexpensively with the problem expressed in terms of a single choice parameter k. It has also been shown that, in the presence of a single riskfree tool, the choice of an optimal vector of risky tools is independent of the loss function. The only choice to be made, by the policy maker or by the numerical optimization, can again be expressed in terms of a single choice parameter, Y (= Ey), contingent on the fixed optimal cri. Received: May 1988 Final version: May 1989
References Brainard,
American
William.
“Uncertainty
Economic
and the Effectiveness
of Policy.”
Review 57 (May 1967): 411-25.
Brandsma, Andries. “Implications of Risk-Sensitive Decision-Making for the Design of Economic Policies.” Journal of Economic Dynamics and ControE 10 (June 1986): 301-06. Caravani, P. “On Extending Linear-Quadratic Control Theory to NonSymmetric Risky Objectives.” Journul of Economic Dynamics and Control 10 (June 1986): 83-88. Chipman, John. “The Ordering of Portfolios in Terms of Mean and Variance.” Review of Economic Studies 40 (April 1973): 167-90. Chow, Gregory. Analysis and Control of Dynamic Economic System-s. New York: Wiley, 1975. Deissenberg, C. “On the Minmax Lyapunov Stabilization of Uncertain Economies.” Journal of Economic Dynamics and Control 11 (June 1987): 229-34. Gordon, Roger. “An Interpretation of the Costs on the Instruments 144
Efficient
Policy Frontier
in Deterministic Linear-Quadratic Control.” Znternational Economic Review 17 (October 1976): 779-81. Holbrook, Robert, and E. Phillip Howrey. “A Comparison of the Chow and Theil Optimization Procedures in the Presence of Parameter Uncertainty.” Znternational Economic Review 19 (October 1978): 749-59. Horowitz, Ann. “Loss Functions and Public Policy.” Journal of Macroeconomics 9 (Fall 1987): 489-504. Hughes Hallett, A. J. “On Alternative Methods of Generating Risk Sensitive Decision Rules.” Economic Letters 16 (1984): 37-44. -. “International Competitiveness and Economic Recovery: Examples of the Risk-Ambition Trade-Off in Dutch Economic Policies.” The Manchester School 55 (March 1987): 38-59. Monetary Instrument Variable.” Kareken, John. “The Optimum Journal of Money, Credit, and Banking 2 (August 1970): 38590. Kunstman, A. “Controlling a Linear Dynamic System According to Asymmetric Preferences. ” Journal of Economic Dynamics and Control 8 (September 1984): 261-81. Merton, Robert. “An Analytic Derivation of the Efficient Portfolio Frontier.” Journal of Financial and Quantitative Analysis 7 (September 1972): 1851-72. Mitchell, Douglas. “Risk Aversion and Optimal Macro Policy.” Economic Journal 89 (December 1979): 913-18. -. “A Note on Two-Stage Monetary Policy Under Multiplier Uncertainty.” Eastern Economic JonrnaZ 9 (April-June 1983): 10709. Norman, Alfred. “Information Structure and Stochastic Control Performance. ” Journal of Economic Dynamics and Control 8 (November 1984): 137-50. Turnovsky, Stephen. “The Stability Properties of Optimal Economic Policies.” American Economic Review 64 (March 1974): 136-48. Waud, Roger. “Asymmetric Policymaker Utility Functions and Optimal Policy Under Uncertainty.” Econometrica 44 (January 1976): 53-66. Young, Richard. “Certainty Equivalence, First Order Certainty Equivalence, Stochastic Control, and the Covariance Structure.” Econometrica 43 (May 1975): 421-30.
145
Virginia
University West
Virginia
The Efficient Policy Frontier under Parameter Uncertainty and Multiple Tools* This paper considers optimal policy design for arbitrary (that is, not necessarily quadratic) loss functions. There are a single target variable and an arbitrary number of policy tools with coefficient uncertainty. Under normality, an efficient policy frontier is derived. Any efficient policy vector is a weighted average of two arbitrary efficient vectors. If a tool with a known coefficient is introduced, the vector of optimal values of risky tools is independent of the loss function. Implications are discussed for policy advising when the policy maker’s loss function is imperfectly understood, and for numerical policy optimization when the loss function is analytically intractable.
1. Introduction There has been great interest over the years in the general topic of policy optimization in the context of a wide variety of models. Often, for the sake of mathematical simplicity, it is assumed that there is only additive uncertainty and no uncertainty about the values of parameters that enter the model multiplicatively with respect to policy tools. In reality, of course, there is almost always such parameter uncertainty, and various authors have dealt with this complication. ’ Generally, quadratic loss is assumed. However, as several authors have recognized, this assumption is very restrictive’: for one thing, it imposes symmetry, and, even within the class of symmetric loss functions, it is restrictive because of its specific form. Unfortunately, no one has ever presented a nonquadratic loss function that is tractable in the context of parameter uncertainty.
*I am grateful to two referees for their helpful comments. ‘See, for instance, Brainard (1967); Ch ow (1975, ch. 10); Deissenberg (1987); Gordon (1976); Holbrook and Howrey (1978); Kareken (1970); Mitchell (1979, 1983); Norman (1984); Tumovsky (1974); Waud (1976); and young (1975). *See Brandsma (1986); Caravani (1986); Hughes Hahett (1984, 1987); Horowitz (1987); Kunstman (1984); Mitchell (1979, 1983); and Waud (1976).
Journal of Macroeconomics, Winter 1990, Vol. Copyright 6 1990 by Louisiana State University 0164..0704/90/$1.50
12, No. Press
1, pp.
137-145
137
Douglas W. Mitchell The present paper addresses the issue of policy optimization under multiplicative parameter uncertainty when loss is not necessarily quadratic. In Section 2, it is assumed that there are n policy tools whose marginal effect on the single target variable is uncertain, and there is also additive uncertainty. Under a normality assumption, an “efficient policy frontier” is derived to characterize all policy vectors that could be optimal for some loss function. A direct analogy is made to Merton’s (1972) mutual fund theorem. Then, in Section 3, it is assumed that there is one further policy tool whose marginal effect is known. In this case, efficient policies are characterized in a way that is partially analogous to Merton’s separation theorem. Section 4 discusses the importance of the results for the design of policy when the policy maker’s loss function (that is, its functional form and parameterization) is not explicitly known, and for the numerical optimization of policy when the loss function is analytically intractable.
2. The Case of No Risk-Free Tools THEOREM 1. Assume that loss is a convex function of one target variable with a unique minimum, that there are n policy tools (n 2 2) with random multipliers, and that there is an additive shock; and assume that the additive shock and the multipliers are jointly normally distributed. Then, regardless of the specific form of the loss function, the optimal policy vector is a weighted average of two given policy vectors. Proof and Discussion. The value of the single target variable is y, and the vector of values of the policy tools is P.3 The economic structure is given by y = A’P + u , where A is the vector of random additive shock. While the simplest context, one can also view (1) as dynamic system, provided loss is
(1)
policy multipliers, and u is the interpretation of (1) is for a static derived from the final form of a still a function of a single scalar
‘Of course, this approach permits more than one target variable if they can be aggregated into a single index, y, which appears in the loss function. An example would be the “economic discomfort index,” widely alluded to in the late 197Os, which equals the sum of the inflation rate and the unemployment rate.
138
Efficient variable.
For example,
Policy Frontier
suppose GNP in time t is given by rt in xt
=
ax,-,
+
PPt
+
6
>
(2)
where p, is a vector of policy tools. Suppose further that loss is simply a function of a weighted average of GNP over a finite horizon. Then (2) can be solved for a final form for each t, and then the weighted average can be written in the form of (l), where P = (pi p; . . .) ’ is a stacked vector of policy values of all time periods. Then, the discussion of this paper applies to open-loop optimization Since open-loop control is known to be suboptimal under multiplier uncertainty, and, in general, dynamic loss cannot be expressed in terms of a scalar argument, an important area for future research is to identify a frontier, or some similar’problem-simplifying device, in the context of closed-loop control with multivariate loss. We now use the assumption that the additive shock u and the policy coefficients in A are jointly normally distributed. This assumption implies that y is normally distributed. It is known that, in this circumstance, expected loss can be expressed as a function of the mean and variance of y, with higher ai giving higher expected loss (Chipman 1973). Since our loss function is not monotonic in y, expected loss is not monotonic in the mean of y, and the mean-variance indifference curves are not monotonic (Waud 1976). Thus, we can identify an efficient policy frontier, in which, for any given mean of y, the variance of y is minimized. Every loss function will lead to an optimally chosen policy vector that is on this frontier. This is analogous to the mean-variance efficient frontier of portfolio theory, though with two differences. First, in portfolio theory, the analog of the policy vector-the vector of portfolio weights-must be such that the elements sum to unity; there is no such requirement in our policy context. And second, in the portfolio context, one must limit the focus to the upward sloped region of the frontier because expected utility is everywhere increasing in the mean of portfolio return, but, in our context, expected loss (the negative of expected utility) is not monotonic in the mean of y because loss is not monotonic in y, so the entire frontier is relevant. Using (l), we express the mean and variance of y as’
can
41f there are any be incorporated
linear restrictions in (3) without
that apply affecting fiti
across policy vector settings, substance of the results.
these
139
Douglas W. Mitchell Ey = /i’P + ii,
(3)
and
tg = P’VP + of + 2P’C ,
(4)
where A and ii are the means of A and u, V is the variance-covariance matrix of the elements of A, and C is a vector whose ith element is the covariance between u and the ith element of A. The parameters A, I!& V, and C are known and fixed moments. The problem then is to minimize 2, in (4) subject to Ey = Y, where Y is a constant that will parameterize the efficient frontier. The Lagrangian is
L = P’VP + cr: + 2P’C + 2A(Y - A’P - ii) , where are
2X is the Lagrangian
multiplier.
The first-order
(5) conditions
(1/2)dL/dP
= VP + c - AA = 0 )
(6)
(1/2)dL/dh
= Y - A’P - ii = 0.
(7)
and
We obtain the efficient P contingent on any Y by solving (6) provisionally for P in terms of A, substituting into (7’), solving the resulting equation for A, and substituting for A in the provisional expression for P. The result is
P* = P, f br ,
(8)
where
and
b = (Y - ti + AT-‘c)/(A’v-‘A) 140
3 0.
w
Effzcient
Policy Frontier
It can be shown that b, which is contingent upon Y, is one half of the reciprocal of the slope of the mean-variance frontier (with the mean plotted vertically) at the height Y. The efficient policy vectors in (8) can be rewritten as P* = kP, + (1 - k)P, , where PI and Pz are any efficient
vectors given by
PI = P, + b,I’ ,
(114
and Pz = PO+ b,I- , for arbitrary
parameters
b, # b, >
OW
b, and bp, and where
k = (b - b2)/(bl - b,) 52 0 .
WC)
Equation (10) above shows that every efhcient policy vector is a weighted average of any two arbitrarily specified efficient vectors. The set of all efficient points is found by varying k in (10). For all policy makers, regardless of loss function, the optimal point is a mean-variance efficient point by Chipman’s (1973) theorem. Therefore, all policy makers, regardless of loss function, will pick a point characterized by (lo)--that is, a weighted average of two arbitrary efficient points. The value of k at the policy maker’s optimal point will, of course, depend on the specific loss function employed. This result and its proof are essentially identical to those of Merton’s (1972) mutual fund theorem for portfolio theory.
3. The Case of One Risk-Free Tool We now consider the case in which, in addition to the risky policy tools assumed in Section 2, there is also one tool whose coefficient is known with certainty. The intuition of this case is straightforward. The policy optimization for any loss function can be conceptualized as a two-stage process. In the first stage, the vector P of levels of policy tools whose coefficients are random is chosen so as to unconditionally minimize 2,. Then, in the second stage, the level 4 of the risk-free tool is chosen so as to bring the mean of y to that value which is considered optimal, given the value of ai 141
Douglas W. Mitchell achieved in the first stage.5 This separation is possible because the risk-free tool only affects one moment of y (the mean). Note that, because Ey can be set exactly in the second stage, any additional risk-free tools would be purely redundant. Notice also that the efficient frontier is now vertical at a fixed value of a:--namely, the lowest value of c$ that can be achieved using the vector P of risky tools. The choice of q in the second stage, and thus the choice of a height on the efficient frontier, does not affect ai, and thus ensures that the frontier is vertical. The above intuition suggests the following separation theorem: THEOREM 2. Retain the assumptions of Theorem 1, but assume further that there is one policy tool whose multiplier is known with certainty. Then, the optimal vector of tools with uncertain multipliers is independent of the loss function, as is the variance of the target variable at the optimum.
and Discussion. This separation theorem is analogous to that of Merton (1972) in the portfolio context, though it differs somewhat in that Merton’s efficient locus, unlike ours, is not vertical. The derivation of the efficient policy frontier is as follows. The economic structure, instead of (l), is now
Proof
y = A’P + aq + u , where a is a known multiplier. are
(12)
The mean and variance of the target
Ey = A’P + aq + ii,
(13)
a2Y = P’VP + 4 u + 2P’C .
(14)
and
To derive the efficient frontier, the expression in (14) for cri is to be minimized subject to Ey having the arbitrary value Y, so that the Lagrangian is
L = P’VP + at + 2P’C + 2X(Y - A’P - aq - ii) .
not
142
‘Waud (1976) shows that, with an asymmetric loss function, be independent of the achieved value of IJ:.
this
value
(15) of Ey will
Efficient
Policy Frontier
The first-order condition for q immediately implies A = 0. The interpretation is that the constraint of a specified value for Ey imposes no added variance on y because q can be adjusted to obey the constraint without influencing a:. Using X = 0, the first-order condition for P implies p* = -v-‘c In addition,
the first-order
condition
.
(16)
for h implies
q* = (Y - A’P - 2%)/a )
(17)
or, using (16), q* = (Y + A’V-‘C
- ti)/a
(17’)
The set of efficient points is found by using P* from (16) and q* from (17’) while varying Y. Now (16) shows that P* is independent of Y. Since, by Chipman (1973), the expected loss minimizing point is in the set of efficient points (regardless of the loss function), the expected loss minimizing value of P is that given in (16), regardless of loss function. Thus, the optimal vector of risky policy magnitudes is independent of the loss function. The set of efficient values for q is found by varying Y in (17’). By Chipman, the optimal q, given some loss function, is in the efficient set. By (17’), q* varies with Y, so the optimal q is dependent on the loss function. All of these results and their proofs are essentially identical to those of Merton’s (1972) separation theorem for portfolio theory. Note further that P* in (16) is indeed, as previously asserted, the policy vector that unconditionally minimizes ai. This policy vector is zero if the additive shock is uncorrelated with all the multiplicative shocks (so C is the zero vector), but not in general otherwise. The optimal ai, found by using (16) in (14), is crz - C’V’C (which is independent of Y). Th is expression gives the location of the vertical efficient frontier.
4. Summary
and Implications
This paper has considered the design of policy with arbitrary loss function, under multiplicative as well as additive uncertainty, with jointly normally distributed shocks, and with a single target 143
Douglas W. Mitchell variable. With no risk-free policy tool, it has been shown that all policy makers will choose a weighted average of any two arbitrary efficient policy vectors. This is an important result, in part, because it suggests a way of presenting policy options to a policy maker whose loss function is not fully understood: all efficient policy vectors and the resultant means and variances of the target can be expressed parameterized by a single scalar choice parameter k. Then, the policy maker is simply presented with a series of efficient (Ey, ai) combinations and is asked which one is most satisfactory. Further, if the loss function is known but is analytically intractable (as appears inevitable with nonquadratic loss and multiplier uncertainty), numerical optimization can be performed more inexpensively with the problem expressed in terms of a single choice parameter k. It has also been shown that, in the presence of a single riskfree tool, the choice of an optimal vector of risky tools is independent of the loss function. The only choice to be made, by the policy maker or by the numerical optimization, can again be expressed in terms of a single choice parameter, Y (= Ey), contingent on the fixed optimal cri. Received: May 1988 Final version: May 1989
References Brainard,
American
William.
“Uncertainty
Economic
and the Effectiveness
of Policy.”
Review 57 (May 1967): 411-25.
Brandsma, Andries. “Implications of Risk-Sensitive Decision-Making for the Design of Economic Policies.” Journal of Economic Dynamics and ControE 10 (June 1986): 301-06. Caravani, P. “On Extending Linear-Quadratic Control Theory to NonSymmetric Risky Objectives.” Journul of Economic Dynamics and Control 10 (June 1986): 83-88. Chipman, John. “The Ordering of Portfolios in Terms of Mean and Variance.” Review of Economic Studies 40 (April 1973): 167-90. Chow, Gregory. Analysis and Control of Dynamic Economic System-s. New York: Wiley, 1975. Deissenberg, C. “On the Minmax Lyapunov Stabilization of Uncertain Economies.” Journal of Economic Dynamics and Control 11 (June 1987): 229-34. Gordon, Roger. “An Interpretation of the Costs on the Instruments 144
Efficient
Policy Frontier
in Deterministic Linear-Quadratic Control.” Znternational Economic Review 17 (October 1976): 779-81. Holbrook, Robert, and E. Phillip Howrey. “A Comparison of the Chow and Theil Optimization Procedures in the Presence of Parameter Uncertainty.” Znternational Economic Review 19 (October 1978): 749-59. Horowitz, Ann. “Loss Functions and Public Policy.” Journal of Macroeconomics 9 (Fall 1987): 489-504. Hughes Hallett, A. J. “On Alternative Methods of Generating Risk Sensitive Decision Rules.” Economic Letters 16 (1984): 37-44. -. “International Competitiveness and Economic Recovery: Examples of the Risk-Ambition Trade-Off in Dutch Economic Policies.” The Manchester School 55 (March 1987): 38-59. Monetary Instrument Variable.” Kareken, John. “The Optimum Journal of Money, Credit, and Banking 2 (August 1970): 38590. Kunstman, A. “Controlling a Linear Dynamic System According to Asymmetric Preferences. ” Journal of Economic Dynamics and Control 8 (September 1984): 261-81. Merton, Robert. “An Analytic Derivation of the Efficient Portfolio Frontier.” Journal of Financial and Quantitative Analysis 7 (September 1972): 1851-72. Mitchell, Douglas. “Risk Aversion and Optimal Macro Policy.” Economic Journal 89 (December 1979): 913-18. -. “A Note on Two-Stage Monetary Policy Under Multiplier Uncertainty.” Eastern Economic JonrnaZ 9 (April-June 1983): 10709. Norman, Alfred. “Information Structure and Stochastic Control Performance. ” Journal of Economic Dynamics and Control 8 (November 1984): 137-50. Turnovsky, Stephen. “The Stability Properties of Optimal Economic Policies.” American Economic Review 64 (March 1974): 136-48. Waud, Roger. “Asymmetric Policymaker Utility Functions and Optimal Policy Under Uncertainty.” Econometrica 44 (January 1976): 53-66. Young, Richard. “Certainty Equivalence, First Order Certainty Equivalence, Stochastic Control, and the Covariance Structure.” Econometrica 43 (May 1975): 421-30.
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