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Optimal policy choice under uncertainty
Journal of Public Economics 4 (1975) 205-209. 0 North-Holland Publishing Company
OPTIMAL
POLICY
CHOICE
Geoffrey Australian
National
Universily,
Thomas
UNDER UNCERTAINTY
BRENNAN* Canberra,
A.C.T.
2600, Australia
McGUIRE*
Yale University, New Haven, COWL, U.S.A. Received June 1973, revised version received September 1974 Neutrality is shown to be the appropriate goal of policy in the class of realistic policy situation in which allocative decisions are taken with virtually no knowledge of the direction or magnitude of pre-existing resource misallocation. This follows as the principal corollary of the central theorem of the paper which states that with linear demand and constant marginal cost schedules, the appropriate excise tax under uncertainty is equal to the expected value of the market distortion. This theorem is proved and its major implications for corrective fiscal policy are discussed.
1. Traditionally, the case for fiscal neutrality’ has been taken to depend crucially on the absence of market distortions prior to policy imposition. The reason for this is clear: in the presence of an externality, monopoly or perhaps some government induced distortion elsewhere in the system, the appropriate government policy involves the explicit introduction of an exactly compensating distortion, so that the social benefits and costs of each product are equalized at the margin a non-distorting policy will simply sustain existing inefficiencies. Given the conspicuous abundance of distortions evident in the ‘freely operating’ market those arising, for example, from tariffs, from monopoly elements in product and factor markets, and from generalized external effects (some partially corrected, some no doubt over-corrected) - the case for fiscal neutrality seems extremely weak. T’he theoretical literature appears largely to have ignored this problem. The design of completely neutral (or minimally distorting) tax systems remains a *The authors are respectively Senior Lecturer in Public Finance, Australian National University, Canberra, and graduate student in the Department of Economics, Yale University. Gratitude is due to John Head for comments on an earlier draft. ‘Throughout the analysis, ‘fiscal’ is used in the widest possible sense to include all aspects of budgetary policy.
206
G. Brennan and T. McGuire,
Optimal policy choice under uncertainty
predominant concern in analytical public finance; the correction of externalities continues to be considered in an environment where the externality in question represents the only barrier to optimality; and so on. The question arises as to whether there is any conceivable justification for this? Is this, perhaps, simply another case where theory is of zero practical relevance? What we aim to show in this discussion is that the objective of fiscal neutrality is indeed the legitimate one in a broad class of realistic policy situations, of which perfect pre-policy allocation is a special case. This result emerges as a principal corollary of the central theorem of this paper. Our theorem states that in a simple world of linear demand and constant cost schedules, but with uncertain knowledge of the direction and magnitude of misallocation, maximization of the expected welfare gain from government policy requires the introduction of a distortion set so as to offset exactly the expected value of the preexisting distortion. For convenient proof, we assume (i) that all demand curves are linear and all marginal cost curves are constant; (ii) that there are two goods, X and a composite good ‘dollars’. The distortion, d, in the market for X is a random variable equal to the social marginal rate of transformation of X for dollars minus the dollar price of X and may take positive or negative values. The slope of the demand curve for X is S and is defined to be positive. The theorem states: (1) The appropriate excise tax for commodity X is given by t where t = E(d). (2) The expected welfare gain from imposing the appropriate tax is equal to what the gain would be if the expected value of the distortion occurred with certainty, i.e., E(W) = 3 l/S (t)‘, where W is the welfare gain. (3) If the actual excise rate for commodity X differs from t by amount r, the expected welfare cost of this deviation is the welfare cost of a tax of rate r when pre-tax allocation is perfect, i.e., -3 l/S*. Proof. For any d, the welfare gain or addition with a tax of rate t on Xis W=Aqd-;
. (
to consumer
surplus associated
(1)
>
Note that (1) is not an expression for the total excess burden due to the distortion and tax taken together, but shows the change in welfare from imposition of the excise tax. Now, Aq = t/S, so that W
=
d’_‘!t s
2s’
G. Brennan and T. McGuire, Optimal policy choice under uncertainty
E(W)
(3
= E(d);-;;,
aE(w) WI - at = ---9s =
t
(4
s if t = E(d).
0,
This is a maximum since When t = E(d),
207
a%(w)/at”
= - 1.
E(W) = t;-;;,
(5)
1 t2 = --_ 2s
(6)
Thus, the excise, optimally calculated, offsets the expected initial distortion, 4 E(d)‘/,!?; i.e., the expected distortion offset is the expected welfare gain. We note that if t = E(d) + r,
E(w> = ~[E(d)2+E(d)r-~E(~2+2E(d)r+r2)l 1 Ed = z-F-is
(7)
1 r2 (8)
So that an error of r in the optimum rate implies a welfare cost equal to the excess burden introduced by a tax of rate r in isolation.
2.
There is little enough in this theorem that is particularly startling. All it states, after all, is that the appropriate policy is to correct for the expected distortion. The result is, however, of some policy significance. This is so because public decisions on allocative questions are customarily taken in a context of predominant ignorance. In general, the policy-member will not know which institutional constraints in factor markets are most significant or how the associated distortions are reflected in product prices. Me will not know which externalities and which monopolies are Pareto relevant; or which public goods are over-expanded and which under-expanded, and by how much. Nor does he
208
G. Brennan and T. McGuire, Optimal policy choice under uncertainty
possess full knowledge of the precise network of complement-substitute relationships and factor intensities which determine the pattern of net distortions. All that he can know is that given the prevalence of these individual ‘misallocations’, virtually all goods will be affected - and that in general the economy will lie somewhere off the utility-possibilities frontier. 2 Predictions about the direction, let along the magnitude, of the aggregate distortions can rarely be accepted with great confidence : ignorance is extreme. If this is accepted, the case for fiscal neutrality follows more or less immediately, and the reason for the precise correspondence between the situation of perfect pre-tax allocation and that of zero knowledge as to the direction of the aggregate distortion becomes clear - in both cases, E(di) is zero. In the former case, there is only one value for di (viz. di = 0) which obtains with probability one. In the latter, there is a range of possible di (the expected magnitude of the distortion may vary) but since the direction is unknown, there is for each positive di a corresponding negative di occurring with equal probability: hence, E(di) is zero. And our central theorem assures us that when E(di) is zero, neutral (or minimally distorting) tax policy is called for. Simply stated, the reason for this is that, when there is complete uncertainty about the direction of the pre-existing distortion, any distortion introduced by government action is just as likely to be in the wrong direction as the right. But the welfare cost (allocative loss) from moving in the wrong direction vastly exceeds the gain from moving in the right, and hence it is better for the government not to introduce a distortion. This result has application in two sorts of policy contexts. In the environment in which policy-makers usually find themselves when suggesting major tax reforms, for example, wholesale application of the ‘textbook’ approach, with its characteristic assumption of perfect pre-tax allocation, is surprisingly enough completely proper. Neutral tax policy is an appropriate goal. But what of explicitly corrective allocative policy, where the government seeks expressly to correct some ‘observed’ distortion? In this context, a number of points should be noted. Firstly, our case for fiscal neutrality is completely consistent with the observation of marginal social benefit / social cost discrepancies for individual products - indeed the prevalence of such observations is an essential ingredient in the predominant ignorance formulation. What needs to be emphasized is that the observation of one such distortion when there are a very large number of others present carries very little information with it: it may still be appropriate to do nothing. In many cases, of course, the expected value of the observed distortion will not be zero. What the foregoing analysis indicates is that there is a precise relationship between the chances of correct action and the ambition of the 2The special case where all distortions cancel one another out seems too happy a coincidence to form the basis of policy choice.
G. Brennan and T. McGrrire, Optimal policy choice under uncertainty
209
corrective policy. Suppose, for example, that the policy-maker observes a monopoly of particularly conspicuous proportions operating freely in the market and decides to impose the subsidy apparently required. While he may admit, when pressed, that there is a reasonable chance that the observed distortion is not Pareto relevant, he may discount the possibility of exacerbating an existing distortion almost completely: so that there is, say, a probability of one-half associated with zero distortion, and of one-half with an underexpansion of output of amount dq, . If a subsidy of V is required to achieve expansion of output by dq,, then there is as much to be said for imposing this subsidy as doing nothing - both involve an expected welfare gain of zero. Clearly, then, while imposing the subsidy to offset all of the ‘observed’ distortion achieves no allocative gain, there is no particular case here for fiscal neutrality. The appropriate policy, in fact, is a subsidy of V/2, which yields an expected welfare gain of + I/S (V/2)’ (i.e., of $1/S V’), unequivocally positive. If the policy-maker is only half sure that the observed distortion is Pareto relevant, then he should only aim to offset half of it. More generally, he should design policy on the basis of the quality of information he has available. One final point is worth emphasizing. The welfare-refecant probability distribution of the d is the objective one, calculable perhaps from the policy-maker’s past record in making such choices. It is, of course, inherent in the nature of the problem that this record is often unknown: it may well be impossible to tell ex post as well as ex ante whether policy works in the right or wrong direction. Hence, the expectations in use are the policy-maker’s own - and it is clear from experience that the policy-maker can readily evince ‘subjective certainty’ in the face of overwhelming ignorance. Thus, in many cases, the policy-maker appears to feel confident of doing the right thing even when, to lesser mortals, his poiicy choices may seem completely arbitrary. While it would be singularly inappropriate in this context to suggest that such policy choices are in fact often enough in the wrong direction, the scarcity of the relevant information seems to suggest that they well might be. If this is accepted, then a prima facie case for unambitious corrective policy seems to be established.
OPTIMAL
POLICY
CHOICE
Geoffrey Australian
National
Universily,
Thomas
UNDER UNCERTAINTY
BRENNAN* Canberra,
A.C.T.
2600, Australia
McGUIRE*
Yale University, New Haven, COWL, U.S.A. Received June 1973, revised version received September 1974 Neutrality is shown to be the appropriate goal of policy in the class of realistic policy situation in which allocative decisions are taken with virtually no knowledge of the direction or magnitude of pre-existing resource misallocation. This follows as the principal corollary of the central theorem of the paper which states that with linear demand and constant marginal cost schedules, the appropriate excise tax under uncertainty is equal to the expected value of the market distortion. This theorem is proved and its major implications for corrective fiscal policy are discussed.
1. Traditionally, the case for fiscal neutrality’ has been taken to depend crucially on the absence of market distortions prior to policy imposition. The reason for this is clear: in the presence of an externality, monopoly or perhaps some government induced distortion elsewhere in the system, the appropriate government policy involves the explicit introduction of an exactly compensating distortion, so that the social benefits and costs of each product are equalized at the margin a non-distorting policy will simply sustain existing inefficiencies. Given the conspicuous abundance of distortions evident in the ‘freely operating’ market those arising, for example, from tariffs, from monopoly elements in product and factor markets, and from generalized external effects (some partially corrected, some no doubt over-corrected) - the case for fiscal neutrality seems extremely weak. T’he theoretical literature appears largely to have ignored this problem. The design of completely neutral (or minimally distorting) tax systems remains a *The authors are respectively Senior Lecturer in Public Finance, Australian National University, Canberra, and graduate student in the Department of Economics, Yale University. Gratitude is due to John Head for comments on an earlier draft. ‘Throughout the analysis, ‘fiscal’ is used in the widest possible sense to include all aspects of budgetary policy.
206
G. Brennan and T. McGuire,
Optimal policy choice under uncertainty
predominant concern in analytical public finance; the correction of externalities continues to be considered in an environment where the externality in question represents the only barrier to optimality; and so on. The question arises as to whether there is any conceivable justification for this? Is this, perhaps, simply another case where theory is of zero practical relevance? What we aim to show in this discussion is that the objective of fiscal neutrality is indeed the legitimate one in a broad class of realistic policy situations, of which perfect pre-policy allocation is a special case. This result emerges as a principal corollary of the central theorem of this paper. Our theorem states that in a simple world of linear demand and constant cost schedules, but with uncertain knowledge of the direction and magnitude of misallocation, maximization of the expected welfare gain from government policy requires the introduction of a distortion set so as to offset exactly the expected value of the preexisting distortion. For convenient proof, we assume (i) that all demand curves are linear and all marginal cost curves are constant; (ii) that there are two goods, X and a composite good ‘dollars’. The distortion, d, in the market for X is a random variable equal to the social marginal rate of transformation of X for dollars minus the dollar price of X and may take positive or negative values. The slope of the demand curve for X is S and is defined to be positive. The theorem states: (1) The appropriate excise tax for commodity X is given by t where t = E(d). (2) The expected welfare gain from imposing the appropriate tax is equal to what the gain would be if the expected value of the distortion occurred with certainty, i.e., E(W) = 3 l/S (t)‘, where W is the welfare gain. (3) If the actual excise rate for commodity X differs from t by amount r, the expected welfare cost of this deviation is the welfare cost of a tax of rate r when pre-tax allocation is perfect, i.e., -3 l/S*. Proof. For any d, the welfare gain or addition with a tax of rate t on Xis W=Aqd-;
. (
to consumer
surplus associated
(1)
>
Note that (1) is not an expression for the total excess burden due to the distortion and tax taken together, but shows the change in welfare from imposition of the excise tax. Now, Aq = t/S, so that W
=
d’_‘!t s
2s’
G. Brennan and T. McGuire, Optimal policy choice under uncertainty
E(W)
(3
= E(d);-;;,
aE(w) WI - at = ---9s =
t
(4
s if t = E(d).
0,
This is a maximum since When t = E(d),
207
a%(w)/at”
= - 1.
E(W) = t;-;;,
(5)
1 t2 = --_ 2s
(6)
Thus, the excise, optimally calculated, offsets the expected initial distortion, 4 E(d)‘/,!?; i.e., the expected distortion offset is the expected welfare gain. We note that if t = E(d) + r,
E(w> = ~[E(d)2+E(d)r-~E(~2+2E(d)r+r2)l 1 Ed = z-F-is
(7)
1 r2 (8)
So that an error of r in the optimum rate implies a welfare cost equal to the excess burden introduced by a tax of rate r in isolation.
2.
There is little enough in this theorem that is particularly startling. All it states, after all, is that the appropriate policy is to correct for the expected distortion. The result is, however, of some policy significance. This is so because public decisions on allocative questions are customarily taken in a context of predominant ignorance. In general, the policy-member will not know which institutional constraints in factor markets are most significant or how the associated distortions are reflected in product prices. Me will not know which externalities and which monopolies are Pareto relevant; or which public goods are over-expanded and which under-expanded, and by how much. Nor does he
208
G. Brennan and T. McGuire, Optimal policy choice under uncertainty
possess full knowledge of the precise network of complement-substitute relationships and factor intensities which determine the pattern of net distortions. All that he can know is that given the prevalence of these individual ‘misallocations’, virtually all goods will be affected - and that in general the economy will lie somewhere off the utility-possibilities frontier. 2 Predictions about the direction, let along the magnitude, of the aggregate distortions can rarely be accepted with great confidence : ignorance is extreme. If this is accepted, the case for fiscal neutrality follows more or less immediately, and the reason for the precise correspondence between the situation of perfect pre-tax allocation and that of zero knowledge as to the direction of the aggregate distortion becomes clear - in both cases, E(di) is zero. In the former case, there is only one value for di (viz. di = 0) which obtains with probability one. In the latter, there is a range of possible di (the expected magnitude of the distortion may vary) but since the direction is unknown, there is for each positive di a corresponding negative di occurring with equal probability: hence, E(di) is zero. And our central theorem assures us that when E(di) is zero, neutral (or minimally distorting) tax policy is called for. Simply stated, the reason for this is that, when there is complete uncertainty about the direction of the pre-existing distortion, any distortion introduced by government action is just as likely to be in the wrong direction as the right. But the welfare cost (allocative loss) from moving in the wrong direction vastly exceeds the gain from moving in the right, and hence it is better for the government not to introduce a distortion. This result has application in two sorts of policy contexts. In the environment in which policy-makers usually find themselves when suggesting major tax reforms, for example, wholesale application of the ‘textbook’ approach, with its characteristic assumption of perfect pre-tax allocation, is surprisingly enough completely proper. Neutral tax policy is an appropriate goal. But what of explicitly corrective allocative policy, where the government seeks expressly to correct some ‘observed’ distortion? In this context, a number of points should be noted. Firstly, our case for fiscal neutrality is completely consistent with the observation of marginal social benefit / social cost discrepancies for individual products - indeed the prevalence of such observations is an essential ingredient in the predominant ignorance formulation. What needs to be emphasized is that the observation of one such distortion when there are a very large number of others present carries very little information with it: it may still be appropriate to do nothing. In many cases, of course, the expected value of the observed distortion will not be zero. What the foregoing analysis indicates is that there is a precise relationship between the chances of correct action and the ambition of the 2The special case where all distortions cancel one another out seems too happy a coincidence to form the basis of policy choice.
G. Brennan and T. McGrrire, Optimal policy choice under uncertainty
209
corrective policy. Suppose, for example, that the policy-maker observes a monopoly of particularly conspicuous proportions operating freely in the market and decides to impose the subsidy apparently required. While he may admit, when pressed, that there is a reasonable chance that the observed distortion is not Pareto relevant, he may discount the possibility of exacerbating an existing distortion almost completely: so that there is, say, a probability of one-half associated with zero distortion, and of one-half with an underexpansion of output of amount dq, . If a subsidy of V is required to achieve expansion of output by dq,, then there is as much to be said for imposing this subsidy as doing nothing - both involve an expected welfare gain of zero. Clearly, then, while imposing the subsidy to offset all of the ‘observed’ distortion achieves no allocative gain, there is no particular case here for fiscal neutrality. The appropriate policy, in fact, is a subsidy of V/2, which yields an expected welfare gain of + I/S (V/2)’ (i.e., of $1/S V’), unequivocally positive. If the policy-maker is only half sure that the observed distortion is Pareto relevant, then he should only aim to offset half of it. More generally, he should design policy on the basis of the quality of information he has available. One final point is worth emphasizing. The welfare-refecant probability distribution of the d is the objective one, calculable perhaps from the policy-maker’s past record in making such choices. It is, of course, inherent in the nature of the problem that this record is often unknown: it may well be impossible to tell ex post as well as ex ante whether policy works in the right or wrong direction. Hence, the expectations in use are the policy-maker’s own - and it is clear from experience that the policy-maker can readily evince ‘subjective certainty’ in the face of overwhelming ignorance. Thus, in many cases, the policy-maker appears to feel confident of doing the right thing even when, to lesser mortals, his poiicy choices may seem completely arbitrary. While it would be singularly inappropriate in this context to suggest that such policy choices are in fact often enough in the wrong direction, the scarcity of the relevant information seems to suggest that they well might be. If this is accepted, then a prima facie case for unambitious corrective policy seems to be established.