Appendix I:  Policy making under uncertainty

Formal treatments of Bayesian decision theory are available in Cyert and DeGroot (1987), Hirshleifer and Riley (1992), Rhodes (1994), and Silvey (1975). Gorringe (1991, 1998) discusses the implications for policy making under uncertainty. The purpose of this Appendix is to outline the decision framework in a manner directly applicable to the qualitative analysis in this paper. The first part models the simplest case where policy is described by one variable. The second part generalises the model to the case where policy is described by two variables.

One dimensional policy

Consider the structure described in Section 3.2 where the true model of the economy is either s or i and the policy choice is either H or L. The policy maker does not know which is the true model but assigns probabilities p_s and p_i such that p_s + p_i = 1. The payoffs under each policy option are random variables Ṽ_TP, Ṽ_FP, Ṽ_TN, and Ṽ_FN. Equivalently, in Bayesian terminology, the policy option should be implemented if the “risk” of the policy option is less than the “risk” of the status quo.

The ex ante values of policies H and L are denoted ṽ_H and ṽ_L, respectively, and are given by:

(1a)     ṽ_H = p_sṼ_TP + p_iṼ_FP

(2a)    ṽ_L = p_iṼ_TN + p_sṼ_FN

Equations (1a) and (2a) may be rewritten as:

(1b)    ṽ_H = Ṽ - p_iL̃_FP

(2b)    ṽ_L = Ṽ - p_sL̃_FN

where

Ṽ = p_sṼ_TP + p_iṼ_TN

L̃_FP = Ṽ_TN – Ṽ_FP > 0

L̃_FN = Ṽ_TP - Ṽ_FN > 0

Ṽ is the weighted-average value of the true positive and true negative outcomes. It is the ex ante value obtainable if the policy maker could learn the true model prior to making the policy decision, thereby being sure of avoiding false positive and false negative errors. L̃_FP and L̃_FN are the losses associated with false positive and false negative errors that occur when policies H and L are inconsistent with the true model.

Equation (1b) says that the value of policy H, ṽ_H, is the ex ante value with learning (Ṽ) less the probability-weighted loss of policy H being “incorrect” (p_iL̃_FP). Similarly, equation (2b) says that the value of policy L, ṽ_L, is the ex ante value with learning (Ṽ) less the probability-weighted loss of policy L being “incorrect” (p_sL̃_FN).

Risk neutral policy maker

A risk neutral policy maker would prefer policy L to policy H if and only if the expected value of policy L exceeds the expected value of policy H:

(3)     E[ṽ_L] > E[ṽ_H]

    ⇔ p_sE[L̃_FN] < p_iE[L̃_FP],

where E[.] is the expectations operator.

Equation (3) indicates that a risk neutral policy maker would prefer policy L if and only if the expected loss is lower than for policy H. Hence, minimising expected losses is equivalent to maximising expected value.

Risk averse policy maker

A risk averse policy maker would consider both expected losses and risk levels. Bayesian decision theory assumes the policy maker has a preference ordering satisfying standard axioms so that there exists a cardinal utility function, U(.). A risk averse policy maker is assumed to maximise expected utility. Policy L would be preferred if and only if E[U(ṽ_L)] > E[U(ṽ_H)].

As noted in Section 3.2, a practical consideration in public policy is that the analyst is not the decision maker and under public service conventions would not presume to know the decision makers risk preferences. The proposal in this paper is that the policy analyst should present to the policy maker estimates of expected loss and various measures of risk such as variance, skew and kurtosis. The worst-case loss may be presented also. From the information presented the policy maker would choose the policy option that maximises his or her expected utility.