Comparisons of price and quantity controls: A survey

JOURNAL

OF COMPARATIVE

parisons

ECONOMICS

1,

213-233 (1977)

of Price and Quanti GARY WYNN Yoan

Department of Economics, Wesleyan University, ~idd~etow~~ Connecticut 06457 Received January 11, 1977; revised May 6,1917 Y&e, Gary W.-Comparisons of Price and Quantity Controls: A Survey The recent comparisons of price and quantity controls under uncertainty are reviewed with an eye toward discovering their unifying concepts. Prom the original comparison conducted by Weitzman through the more sophisticated studies of multiple producers, multiple goods, intermediate goods, and pollution, output variation emerges as that unifying link. When the control choice is important, the control that creates the better distribution of outputs in terms of magnitude and of correlation with the stochastic elements of the social loss function is preferred. J. Comp. Econ., Sept. 1977, l(3), pp. 213-233. Wesleyan University, MiddIetown, Connecticut. Jourtzal ofEconomic Literature Classification Number: 324.

The relative merits of single-valued price and quantity controls in an uncertain economic environment have begun to attract considerable analytic attention. The results of that attention have underm~ed the co~ve~t~o~ai wisdom of Western economists that price controls should be imposed in ah but the few cases in which structural constraints preclude their use. Despite wellknown theorems that assert the certainty equivalence of prices and qu~t~ties~ it had been thought that a price control would achieve an efficiency that wouk! be sacrificed, were a quota imposed. It has now een observed, however, that uncert~~ty destroys the ability of either type of ontrol to produce a precise outcome; each can create only a distribution of possible outcomes that de upon the states of nature that might occur. A comparison of price and qu controls that are specified before the true state of nature is observed must therefore weigh the welfare losses and gains associated with the corr ~str~b~t~o~s, Circumstances do exist in which the distribution achi quota is far superior, ceteris paribus, to the distributive achiev certainty-equivalent price. Unilateral impositio of price controls is thus an inferior strategy. The mu-pose of this paper is to review the progress that has been made in a~pl~~~g and understanding this observation since its initial ~~bIicati~~ by 1The author wishes to acknowledge the %sistance and encouragement offered by J. M. Montias in his preparation of this review and in his earlier work on price-quantity comparisons. Copyrighr @ 1977 by Academic Press, Inc. 213 Ail rights of reproduction Printed in Great Britain

in any form reserved.

PSSN 0147-5967

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Martin Weitzman in 1974. Section 1 records Weitzman’s original contribution. His single-firm, single-good model will provide an intuitive framework within which the subsequent results can be presented and understood. Several of the technical objections to this model are explored in the second section before we turn to more substantive extensions. Section 3, for example, extends the analysis to the multiple-firm case; mixed controls and the effect of changes in the number of firms producing the regulated good are subjects of study. The fourth section records a similar extension to multiple goods, while the fifth presents the comparison for the case in which the good in question is an intermediate good or even a pollutant. It is, however, in the final section that we meld the various cases. We conclude there by confronting two conjectures drawn by Weitzman from his original analysis with the more subtle insights of the subsequent work. His conjectures are remarkably robust.

1. THE BASIC COMPARISON2 The fundamental comparison, published by Weitzman (1974), involves a central regulatory authority that attempts to maximize social welfare by controlling the output (4) of a single profit-maximizing firm. The firm faces a cost schedule that depends explicitly upon a vector of random variables (0, r), as well as on the production of 9. These variables are meant to reflect the random shocks that can influence costs on a day-to-day basis. Thus, C = C(q, 0, 0. It is assumed that in response to any price order, the firm reads the current values of (0, 0 and selects its output by setting actual marginal costs equal to the specified price. In response to a quantity order, on the other hand, the firm stays on the cost schedule in an effort to produce exactly the specified amount. The center meanwhile selects the mode of control, with full knowledge of the firm’s reactions, by maximizing benefits minus costs. The benefit side is assumed to depend not only upon 4, but also upon another vector of random variables denoted by q. These variables reflect imprecise knowledge of the actual benefit schedule as well as random shocks that can be neither controlled nor foreseen. Thus, B = B(q, q). It is finally presumed that while the center must issue its orders before the values of any of the random variables become known, the benefit and cost functions are shaped to guarantee that the optimal output for any (19,<, q) is positive. The center’s first-best control options are contingency ordersp*(& [, q) and 4*(0, <, q) which ensure that actual marginal costs equal actual marginal benefits in all possible states of nature; that is,

p*(e, 6 q)= c,(~*(e, r,d,e,0 =w*(e, 6 4,td ZIn presenting this analysis, we will closely follow interpretation of the results will, however, differ slightly.

the notation

of its original

author.

Our

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for all (0, <, q). There is obviously nothing to chsose between these two alternatives, since they both transform ex ante uncertainty into ex post certainty; the certainty equivalence of price qu~t~ties is preserve administrative constraints usually preclude e of such co~ti~ge~~ies, however, and attention is focused on the extreme second-best problem i the center must select either a single-valued price order @,I or a single-valved quantity order (4) that maximizes expected benefits minus costs. Except ’ cases of negligible probability, we then know that 4 f 4*( inal costs and marginal benefits therefore almost always diverge t-9Qh unde r type of control, and the camp n of price and quantity ~0~~~~~s is reduced to determining which control, on the average, comes closer ts the optimum. To answer that question, Weitzman defined a summary index, the comparative advantage of price control over quantity control:

where @(it?,Q is the firm’s response to the optimal price order simply the difference between expected social welfare under p and quantity control by $. When it is positive, price control is preferred; wbc~ it is negative, quantity control is better. The comparative advantage becomes mathematic y tractable when costs enefits are both represented by second-order Taylor a~~roxirnat~~ns cx~a~ded around @4 G?, 0, Q z w, 8 0 + Cl@> 6, m - 41 + QC,,@, e, 5x4 - 4)” = a(@ 0 + (C’ f a(& 5))(q - 4) + 4C& - a2,

(1)

B(q, Q) z b(Q) + @’ + t&?)>(q - ci>+ (3 bserve that the stochastic effects have been confined, by assumption, to the intercepts of the marginal schedules. These intercepts are also dissected into means and disturbances around those means so that the expected values of a(& 0 and /I(q) are both zero. Despite their formal justification (see ~arn~eIso~~ 1970), some readers may suspect that these technical restrictions may jeopardize the economic validity of any subsequent conclusions. Extensive investigation of this suspicion has shown, however, that it is substantively unfounded; alternative assumptions complicate the mathematics but leave the economic content of the results virtually unchanged. We report briefly on these investigations in the next section. 3 In a certain economic environment, there always exists a quantity (price) control that elicits precisely the same response as an arbitrarily selected price (quota). For example, issuing a command for i such that C’(i) =j produces the same output response from a profit maximizer as the price order j; and this is true for any& 4 To conform to the Taylor theorem, we have made the following definitions: a(@ 0 = C(& 8, 0 > 0; b(y) sz B($, q) > 0; C’ = EG,(q^, 8, 0 > 0; B’ EzEB,(& r) > 0; a@, 0 = C,(& e, a C; /3(q) = B(Q, q) - 23’; C,, E C,,(& 0, Q > 0; and B,, - B,,(q^, n) < 0. The signs given hold fer all possible (0, c, n).

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The profit-maximizing firm sets the optimal price (p = C’) equal to actual marginal costs from (1) in each state of nature and produces a& 0 = 4 - (a(4 B~C,,). (3) The optimal quantity order is meanwhile simply 4, and it is clear that E(g(0, r)) = $ Figure 1 illustrates the determination of these optimal orders, as well as the output produced under p” in a costly state of nature. Figure 1 can also be

?M,E) qO’p+ (&C,,)

4

q

FIGURE 1

used to illustrate the geometric genesis of the comparative advantage that is derived from (3): 4 = WI1 + Cl,) VhX+% 5)>+ Cov(S, 0; P(r)>. (4) While neither the center nor the firm reacts to random variation on the benefit side, the optimal output appears beneath the intersection (point E) of C,(q, 0, 0 and the actual marginal-benefit schedule. The area of triangle ABE therefore represents the loss in welfare created by producing g(8, 0 rather than @Pt(B, 5, q) in response to j. The area of ECD similarly reflects the welfare costs of producing 4. The comparative advantage of price control is then simply the expected value of (area ECD) minus (area ABE) as (8, 0 varies over its domain. When it is positive, price control is favored because, on the average, losses under 4 are larger. The economic genesis of A, is equally straightforward. Observe that the variable output $0, Q produced under 5 plays a crucial role in determining the sign of A,. The variance reflects the magnitude of this output variation, while the covariance term indicates its correlation with the randomly shifting marginal-benefit schedule. Such variation causes the level of expected benefits achieved under ~7 to fall below the level achieved under a constant output @. This loss increases with both the curvature of the benefit schedule and the

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variance in output (see Fig. 2), and is reflected by the first term in (4); it is always negative. A similar loss is felt on the cost s but it is by the efficiency gain of always having price eq to actual rnargi~~~ co Output therefore always decreases (increases) in nsive (inexpensive) states of nature; the combined effect is C,, Var(qj 2 0. The final term in d, is covariance between output under p” and the randomly shifting marginal-Beirut schedule. If it is positive, for example, output tends to increase umder when the marginal-benefit schedule is shifting upwar his is the correct direction, and a positive bias for price control is obse

FIG. 2. Figure 2a compares the level of benefits achieved by 4 with the level of expected benefits produced by a lottery that yields either (4 + L) or ($ - L) with equal probability. Distance AB represents the loss if the benefit schedule is B’(q); AC is larger and indicates the loss associated with the more highly curved schedule. Figure 2b shows that the loss also increases, for a given schedule, as L, and thus the variance ofthe return, increases.

The opposite sign is similarly recorded when the covariance is negative and output tends to move incorrectly with respect to benefits. Table 1 records the values registered by d, as the two ~~~~at~re parameters reach their extremes. It is clear that output variation under m-ice control is beneficial only when the cost-side efficiency gains dominate the ~o~~aris~~. TABLE

1

THE EXTREMES

Condition

FOR A,

Interpretation

4

Reason

Nearly linear benefits

(+)

Highly curved benefits

-m

Cost efficiency of output variation dominates. Welfare losses of the benefit side dominate. Price and quantity controls both produce 4. Output variation blows up under p and the benefit side dominates.

Highly curved costs

0

Nearly linear costs

--co

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This is unlikely, however, when 4 is found in a highly curved region of the benefit function or in a nearly linear region of the cost schedule. The issuance of even the optimal price could be disastrous in such circumstances. In all of these extreme cases, increases in output variation only increase the magnitude of the comparative advantage of the preferred control. Weitzman has subsequently employed these observations to support two rather far-reaching conjections; they have been identified and summarized in a recently released volume by Montias (1976, pp. 226-227). In the first conjecture, Weitzman considers the adaptation of a market economy to the needs of a wartime emergency. He surmises that since goods are needed in precise amounts and proportions during wartime, the appropriate benefit schedules are highly curved around the critical levels. He concludes, therefore, that the planning hierarchy should be built within a structure of quantity controls that would produce the required outputs with more certainty. His second conjecture compares two separate productive hierarchies; one ultimately faces a competitive market while the other is simply a single step in a larger, centrally regulated process. Since the first faces a linear “benefit” schedule (“benefits” are really total revenues for a competitive profit maximizer, and the resulting constant price mirrors a constant marginalrevenue curve for any state of nature), he argues that a price control is more appropriate. If, on the other hand, a constant flow of output is required of the latter firm to keep the process moving, Weitzman appeals to d, to conclude that a quota is the better choice. While the Weitzman suggestions have intuitive appeal in the light of this simple comparison, we stop short of embracing them at this point. Discussions in subsequent sections will, for the most part, uphold their validity, but the present model is simply too restrictive to capture the conditions under which they falter. The intricacies of multiple firms, substitution within a production function, and the simultaneous control of multiple goods must all be understood before we are able to confront the conjections directly in Section 6. 2. TECHNICAL

EXTENSIONS

Since the appearance of the original analysis, concerns have been expressed that the simplicity of the Weitzman model may have obfuscated some significant economics. Most of these concerns have since been investigated, and we devote the remainder of this paper to reporting the conclusions. In the present section, three technical caveats are explored before we turn to more substantive extensions in Sections 3 through 5. A. The Quadratic Approximations Some readers have observed that quadratic approximations impose a rather strong restriction upon the objective function: a surplus in q reduces social welfare by precisely the same degree as a shortage of an equal amount. They argue further that allowing those losses to be different (by using cubic

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approximations, for example) would have an immediate impact on b&h the average welfare losses caused by output vocation and the ~es~~~~~ comparative advantage of prices5 They are correct on two counts. First of ah, the optimal price would no longer necessarily be defined expected marginal costs and benefits; cubic costs would sk C” so that E (3 # 4. A deadweight Bossis thereby created, against prices in the comparison with certainty-equivalen also directly effects the loss in expected welfare created by a given output variation. Were such variation skewed toward a more highly curved (relatively Batter) region, for example, that loss would be increased (decreased). oth of these effects are, however, quite intuitive. I f~thermore, a simple matter not only to identify the cases in which third are ~~e~~ to play an important role, but also to deduce the stren ction of that role. Even when cubic approximations are allowed, output variation is t dominant force in the comparison, and the primary focus of our unchanged. The considerable mathematical difficulty that is treat buys little in the way of new economics. Ah of the subsequent anal be ted with quadratic representations of sig third-order effects will simply be necessary. B. Stochastic Second-Order Terms

The assumption that confines uncertainty to the intercepts of both marginal costs and marginal benefits has also been criticized. Since no one reacts to random changes on the benefit side, however, we need nly be concerned with making CI1(@, 0, Q stochastic. Weitzman (1914, p. 58) as observed that if the first- and second-order cost effects are independent, then increases in the variance of C,,(q^, 0, Q tend to reduce the comparative advantage of controls. It is difficult, however, to visualise situations in which independence is, in fact, a reasonable presumption. We will concentrate on two al ative possibilities. ppose, first of all, that c,a 4>4) 2 CA27 421r2> for tws arbitrary states of nature if and only if6 j Cubic approximations of benefits and costs can be represented by B(q, q) = b(r) + (23’ -+ /X11)&1 - 4) + lDB,,(q - $Y + MB,,,(q - 413,and W&5) = 4%4’) + (Cl + u@,OXq4)+ 1/2C,,(q - 4))’ + 1/6C,,,(q - 4))“. The slopes of the marginal schedules then clearly depend npon 9. While it is difficult to place a sign on B,,, and C,,, taken individually, it can be shown that the sign of (B,,,-C,,J has some economic significance. Simple analysis of the geometry of quadratic marginal cost and benefit curves reveals that (Blll-Cin) is positive (negative) if and only if a surplus is always less (more) harmful than a shortfall of the same amount. 6 This is, of course, a very special case of positive correlation. Not only is the expected value of the product positive, but the product (C,(q^, 8, k) - C’)(C,,(g, 8, r) - EC,,((I^, @ 5)) is positive for ali possible (0, r). The figures will illustrate the case where (8: Q aEd TJ are independent.

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Inexpensive (expensive) states of nature are thus assumed always accompanied by slowly (rapidly) rising marginal costs. Figure 3 illustrates the situation for arbitrary (~9,0. The losses associated with j and 4 are represented

nonstochastic

FIGURE

3

by the areas of triangles (1) and (2), respectively, when C,,(q^, 19,0 is nonstochastic. The random slope we envision increases the area associated with the price control from (1) to ABC, and the area associated with the quota from

C,

) with

q

FIGURE

4

(2) to DEC. Observe that both the height and base of ABC are larger, relative to (l), than are the height and base of DEC relative to (2). The loss created by p” is therefore relatively larger than before for all (0, 0, and the advent of

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stochastic second-order terms causes the comparative advantage of the pricecontrol mode to fall. Quantity control is consequentl ore likely to be favored, Were the firm operating near its physically deter ed capacity constraint, however, we might expect that the above correlation should be reversed. Inexpensive states of nature, since they encourage high levels of might exhibit rapidly rising costs. Figure 4 ihustrates this circum shows that the area of triangle ABC is smailer relative to (I) than relative to (2). Since (19, LJ is arbitrary, these stochastic second-order terms favor the price-control alternative. Beyond these simple observations, little can be said about the cases in w the second-order terms are random. Perhaps the best way of proceeding would be to construct a series of simulation experiments. In any case, all of the subsequent studies upon which we will report have assumed a constant secsndorder Taylor coefficient. C. Output Variation under Quantity Control The reader will observe that output under a quantity control has thus far been presumed to be precisely equal to the prescribed 4. This assumption introduces a disturbing asymmetry that has correctly suggested to some critics that the comparison reflected in d, may be systematically biased away from prices (since only prices create random output disturbances). In response to that criticism, we now relax the troublesome assumption by indexing those random events that can cause a quantity order to be filled imprecisely by the variable r; i.e.,

where qa(a is the actual output achieved when qp is ordered and state of nature < appears. Since the variables that disturb a firm’s response to a quota shouI also be expected to inf3uence costs, r has been lifted from the stochastic parts of the cost schedule. The optimal quantity order can then be shown to be

and actual output is fully adjusted to the distortion:

and E4,(0 = 4. A second comparative advantage r that #(LJ creates:

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The appearance of da(O in d, is perfectly analogous to the appearance of @((B, Q in d,, and reinforces the observation that output variation weighs heavily on the price-quantity comparison.7 TABLE

2

THE EXTREMES FORA, Condition

Cl, -0 c;;

+ w

B,,

+--co

B,, ‘--co B,, -+O

Interpretation Nearly Highly Highly Highly Nearly

linear costs curved benefits curved benefits curved benefits linear benefits

Qualifications

VarW, Vdq’(R

None None 0) > Vd%(t)) 0) < vmm None

4

A,

-w

-w

W

0

--co

w

A:b.

-,”

The extent of its impact is most easily uncovered when the curvature parameters are again allowed to approach their extremes. Table 2 contrasts the results with those for which 4 was produced with certainty. While the suspected bias is evident, there still exist circumstances in which a quota is overwhelmingly better. As costs become linear, for example, the variance in output under price control becomes arbitrarily large. The resulting level of expected benefits achieved by price control falls precipitously toward negative infinity and dominates the efficiency gain that renders the cost side increasingly positive. Quantity control is therefore favored. Conversely, when the cost function is highly curved, output variation under p” tends toward zero; since #(a is unaffected, expected costs under &, rise without bound and price control is the preferred mode. Furthermore, the comparison is seen to turn critically upon which control produces the smaller variation in output when the benefit schedule is highly curved. Such variation is then extremely deleterious and ’ It has been observed by a referee that this model may have destroyed the symmetry of the original model in the following sense. Weitzman has presumed that either marginal costs (under a price measure) or output (under a quota) could be met with certainty. We, on the other hand, have allowed only marginal costs to be met precisely. An interesting extension therefore questions the results when neither marginal cost nor output can be assured. To approach this possibility, suppose that < is the offending variable under either type of control. The quantity alternative is treated in the text, but the firm’s reaction to a price control is in doubt. It can be shown, however, that if the firm maximizes expected profits over <, then the optimal price remains $ = C’ and the firm will produce q(8) = L$- (Ela(B, a/C,,); Et(-) is the expected value operator with respect to only c. Writing a(@ <) = E,a(B, 0 + D(@, 4, we see that EtD(13, r) = E(E a(@ 0) = 0, and the new comparative advantage is simply A; = A, - Cov(D(B, Q;g (0)). The onI y new term reflects the simultaneous variation of output under price control and the part of marginal costs to which the firm cannot react. Its interpretation parallels the other covariance terms, and it has little impact on the comparison when the curvature parameters reach their extremes. Table 2 remains intact. While we will continue to use only the output distortion outlined in the text, the reader should not find it difficult to extrapolate the results to include this variant as well.

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must be minimized. Only when benefits are nearly near do the secondary covariance terms have a bearing on the sign of A,; the fifth line of Table accurately reports the resulting ambiguity. In deference to these observa~~~s~ subsequent analysis will always admit this output disto~io~ lander the quantity mode. 3. THE SIMULTANEOUS REGULATION 0 The economist who casually applies these conclusions to a more str~~t~~~~ regulatory problem can overlook potentially significant forces that may be exerted by the structure itself. Consider, for example, the imposition sf homogeneous controls on an entire collection o s. Allowing each firm its own cost schedule, Ci(q, ~9~,&), we will see that t ost side of a price-quantity comparison is simply the sum of the previously erved cost effects that are now recorded at each firm. The benefit side will, however, be derived from the insertion of the totd output of all firms into the benefit s~~ed~le~ our conclusions will therefore require some modification. The optimal price and quantity controls for each firm are still ~e~~e~ by the intersections of the expected cost and benefits curves: and for all i. Quadratic approximations of B(Zq, q) around (Z$J and the C”(q, 8, CJ around 4, similar to those recorded above allow us to observe that c7ii(4,0 = 4i - bJ4,
is its response to &. We have, of course, defined Gil = Cfl(di, O,, ti) and ~~(8, &) = C,$ji, 8, + c0vC4,i; B(q))]. It should be clear that the benefit side depends upon the relative losses caused by the variations in total output allowed by p” and the LiPi.All of the previous observations are therefore applicable when they are couched in terms of total, s A more complete description of the analysis and resnlts reported in this section can be found in Yohe (1977b). The cartel described there is interested in maximizing profits, but the observations are perfectly analogous.

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and not individual output. Either a low (high) variance or a positive (negative) correlation with random marginal benefits would serve to increase (decrease) the comparative advantage of the control that creates it. The cost side is meanwhile seen to be the sum of the effects recorded at the individual firms. The previously observed efficiency gain is thus again created by j, and the impact of price-induced output variation on d, is unambiguously positive. Negative correlations between & and marginal costs similarly work in favor of quotas, and push d, in the opposite direction. Variation in the total output can be small for a variety of reasons. Were the production levels of all of the firms nearly certain, total output would, quite obviously, be fairly constant. Even if individual outputs vary widely over the states of nature, however, a negative correlation across firms can produce a constant total; the overproduction of one firm could be canceled by the shortfalls of another. Such cancellation produces the potential for diversification gains in expected benefits that can accrue to either type of control if the number of imperfectly correlated firms is large. To the extent that one control allows more diversification than the other, it is favored, on the benefit side, by increases in the scope of the control to more firms. A second influence must also be recorded before the effect of increasing the number of firms can be determined. Only a price control guarantees that marginal costs will be equal across the entire collection of firms. The resulting efficiency gain, reflected by the positive influence of ZCf, Var(qJ in d,, also increases with the number of firms. While the overall effect is ambiguous, an increase in the number of firms facing the controls is therefore more likely to favor the price mode. Dealing with a collection of firms also raises the possibility that a socially preferred mixed control-a scheme that assigns a price to some of the firms and quotas to the others-might exist. It has been shown that such a mix always exists whenever, under homogeneous control of either type, we can find an individual firm for which the comparative advantage of the opposite control is positive. We must be careful, though, to compute that advantage in the context of that firm’s position in the collection as otherwise regulated. Its relative size is, of course, a significant determinant of that position, but the correlations of its output with the outputs of the other firms are equally important. These correlations will amplify or reduce the impact of our single firm on total output. In any case, any single firm can influence only a fraction of total output. The importance of the benefit sides of these individual computations will therefore be diminished by a multiplicative factor that is roughly proportional to that fraction. Despite the obvious complications, several observations can be made about the likely compositions of a preferred mix. We can suggest that there are strong reasons to believe that large firms should usually face quotas rather than a price. We need only argue that large firms can be modeled as subcollections of

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ler firms. The resulting flatter marginal-cos supply curves and wider ranges of outp avoid such variation and the associate that our condition is suficient: but not rms exist such that le negative correlation when they bers favors the change on an indiv 9 LTIPLE G ea in which structure has been shown to exert an in~ue~~~ is in the sirn~~t~~eo~s regulation of compleme consider two goods (ql and &I that are pr each can thus be summarized by a distinct stochastic cost fun 2, l?,, &). If we suppose that le, we can represent their i efits minus costs subject to both goods the standard Taylor a~~roxirnat~o~s ca are of the form given by Eq. (I), the

and The sign of BP2, of course, depends upon er 4, and 4, are G ements n (>O) or substitutes ((0). It is consistent wi (2) to also define ‘i 1’ $29 E rk= 1,2. ;,+ The al price orders are again defane by the intersections of the expected marginal cost and benefit schedules; i.e.,

9 A thorough presentation of the multiple-goods case can be found in Yohe (19766). A S~COIIJ section of that paper extends the comparison to joint products, but the intuition developed here is suficient zo explain that case as well.

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If quantity orders are produced with certainty, the optimal orders are, of course, Ljl and &, and the comparative advantage of dual prices over dual quotas is simply

When B,, = 0, the two goods are independent and A, is reduced to the sum of the comparative advantages of prices for q1 and q2 taken individually. The two q2

5

FIG. 5. Consumption must occur roughly within the shaded region to preserve the proper consumption ratio for strong complements.

goods are complements, however, whenever B,, > 0. Indifference curves between complements are highly curved. This intuitively suggests that the control that maintains a constant ratio of consumption should be favored (see Fig. 5). Price controls will preserve this ratio only when Cov(g”,; &) > 0; the outputs of both goods would then tend to move in the same direction. Were that covariance negative, q1 and q2 would tend in opposite directions and the consumption ratio would have to change. The first circumstance, then, would work in favor of (pi,jQ); the second against it. The rigor behind this intuition is easily developed. In the case that B,, > 0 and Cov(q”,; &) > 0, increases in q1 raise the marginal benefits of qz precisely when the output of q2 is also increasing. lo Since this induced effect is in the correct direction, price controls are favored. The direction is also correct when both B,, and the covariance are negative; substitutes should optimally move in opposite directions. Only when one term is negative and the other positive will lo The marginal benefit schedule for q2 is simply pz + B,(v) + B,,(q, - dz) + B,,(ql - dl). Increases in q1do indeed shift the entire schedule around according to the sign of B,,.

PRICE AND

QUANTITY

CONTROLS

CQ~~A~~~

the induced changes in marginal benefits work against

22’7

e price esntrols that

tput variation under quotas can also be accommodated, md t are sm%lar. The optimal orders are familiar: 4pj = Bi - E$i(tJ;

i=

where $i(&) distorts the order from the actual out result, the comparative advantage must also refiect quantity regime

We still have the sum of individual compara new covariance term otherwise represents quotas. Table 3 summarizes the crucial role that its extremes. The underlying rule of thumb tells us to favor the control that induces changes in the correct direction to the greatest degree. TABLE 3 Ai FORTHEEXTREMEVALUESQFB~~

Condition”

4 1. CcvI>o>covII 2.

CovII>o>CovI

3.

cov. I > cov II > 0

4. covII>CovI>O 5. CovI
6. covII
Extreme substitutes (-co)

Extreme complementarity (F.z) Reason

co Only prices induce the correct changes. -cc Only quota3 induce the correct changes. co Prices induce larger, correct changes. -LX Quotas induce larger, correct changes. -cc Quotas induce smaller, incorrect changes. co Prices induce smaller, incorrect changes.

Reason

4

-cc Only quotas induce the correct changes. 03 Only prices induce the correct changes, --cc; Quotas induce smaller. incorrect changes. cc Prices icduce smaller.’ incorrect changes. c(: Prices induce larger, correct changes. w Quotas induce larger, correct changes.

a Note: Cov I = Cov(q,(B,, &); &,(B,, &)) and Cov II = Cov($,(c,);

&(&))~

The existence of a preferred mix across sim~lta~eo~s~y controlled goods is certainly an important possibility that we have not yet scussed. It should be no surprise, in the light of the previous section, that such a mix is w whenever we can find one good for which the opposite control is ind preferred. Much like the requirement that we take careful note of a firm’s

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position in a multifirm example, we must now take care to observe the changes in the induced effects that a switch in control can create; one set of effects will be abolished and a new set substituted for it, If the individual comparison of price and quantity controls still favors a switch when these newly induced effects are considered, then the switch will improve expected welfare. 5. REGULATION

OF AN INTERMEDIATE

GOOD

We have, thus far, considered only the regulation of a final good that enters the benefit function directly. We turn now to a discussion of the possibility that the good in question is an intermediate product that enters a benefit schedule only through the unregulated final good that it is used to produce. Surely the ability to substitute within the final production process will have an impact upon our comparison. Weitzman (1974) originally suggested that substitutability, or the lack of it, is reflected in the curvature of the benefit function.‘l While he is fundamentally correct, the impact of substitution is not as straightforward as his statement suggests. We can investigate the role of substitution by considering the control of an intermediate good (4) that is employed in conjunction with a second input (K) to produce a final good (X). A CES function is assumed to summarize that process: x = F(K, 4) = (Jm + (1 - y) qpp. It gives us a direct handle on the elasticity of substitution. The provision of 4 is meanwhile summarized by a stochastic schedule, C(q, S, 0, while K is presumed available at a constant price (r). The benefits from X are finally represented by B(X, 7). Maximizing

with respect to K and q produces a triple, (Z?, 4, X = F(K, @), upon which the standard approximations can be founded: and

WC d

= b(v) + @’ + P(r>>(x-m

ckh 0, r) = 4e, r) + cc’ + de, oh

+ &l(x-a2, - 4) + QC,,(~ - 32.

The coefficients are defined as before. In the context of these approximations, the optimal controls and the responses of the producers of both X and q are available.12 Under the optimal price control for q, p’ = C’, the deliveries of 4 are 4m 0 = 4 - (de, O/C,J; r1 Discussion of this very point begins on his page 60 and leads directly to the conjecture we have outlined above, which will be discussed more fully in the next section. ‘*The producer of X is presumed here to be a profit maximizer operating with a correct subjective distribution of the price of X. The possibility that these expectations are incorrect is treated in Yohe (1977a).

PRICE

AND

QUANTITY

CONTROLS

COMPARE

229

notice that the intersection of expected marginal costs an value product define jX The producer of X observes these deliveries an responds by purchasing I?(& (j = Zq”(B, Q = B - Z”(a(

IClA where o = (1 - p)-’ is the elasticity of substitution, and z = (yC’/( 1 - y)?$ The resulting production of X is finally ~(8,O=ACo)4”(8,5)=8-Abp)(cr($,$/C,,), where A(p) = (yZ”p + (1 - y))llp is the factor that translates variation in the production of q into variation in the production of X The optimal quantity order, L& = 4 - E#(Q, is similarly defined by expected marginal costs and vak product. It elicits similar responses:

s”,(<> = $ + $
= A @Ma40

The comparative advantage of price controls emerges from these reactions in a familiar form:

We see immediately that the impact of s~bstit~ti~~ in th oduction of the fiaal good is confined to the benefit side of the ~ornp~iso~~ n we recal: that A (p> simply translates variation in q into variation in X9 its rn~~t~p~~cat~v~ effect is entirely reasonable. By observing that -

= -@(I

- p))“(A(p)yZ”P)In(A(,o))

In(Z)

8P

we infer, in addition, that maximal translation occurs in case.13 Movement in either direction away from this extrem the magnitude of the benefit side, and the effect suggested monotonic with respect to the elasticity of substitution. I3 Deducing the impact of changes in p is not quite as changes usually alter the points around which the cost and factor Z would then depend upon p and Eq. (6) would however, to change p without changing the marginal rate of includes (K, 4). When this is accomplished, tbe expansion signs reported in Zq. (6) are accurate and rigorous.

easy as suggested here, since such ber,efit schedules are exparided. The be incomplete. Care can be taken, substitution along the input ray &at points are fixed, Z is fixed, and t&e

230

GARY

WYNN

YOHE

Before we examine the impact that this observation has on the Weitzman conjecture, one further remark is in order. The applicability of this analysis does not halt at the conventional boundary of intermediate goods. We need only observe that pollution behaves more like an input than an output to be able to compare eflluent charges with their certainty-equivalent quantity standards in the present context. l4 The advantages of this approach are twofold. To begin with, all of the previous pollution-control comparisons have had to presume that the pollutant and the positively valued output appear in fixed proportions (see, e.g., Adar and Griffen, 1976; Fishelson, 1976; Spence and Roberts, 1976). Only then can the benefit function reflect both the positive value of output and the negative value of the pollutant. With the pollutant as an input, however, varying proportions are certainly allowed. The elasticity of substitution between the pollutant and a second input simply reflects the ease with which production can be “cleaned up.” We can also accommodate the random weather distortions that affect the consumption of pollution, but not the consumption of the output; the two commodities are simply inserted into separate functions. The pollution-control analog of d, then supports similar conclusions. Increasing the ease with which pollution can be cleaned up does not, for example, have a monotonic effect on the comparison of charges and standards; the Cobb-Douglas case produces the most important benefit side. It is impossible, furthermore, to produce anything from only pollution. Isoquants must therefore cut the axes, and the Cobb-Douglas case must also represent the limit of complementarity between the pollutant and the other inputs. As a result, we can conclude that since the cost side almost always favors price controls, increases in the ease of cleansing (increases in the elasticity of substitution) favor effluent charges by diminishing the potentially negative side of the comparison. 6. WEITZMAN’S

CONJECTURES

We are now ready to explore the two Weitzman conjectures of Section 1. A study of wartime aircraft planning in Great Britain by Ely Devons (1950) can lead us through the first. Devons reports that wartime planners never even considered regulation by prices; they felt that quotas gave them an almost certain hold on the minimum production levels that would surely be sacrificed by a set of prices. These planners were also acutely aware that war materials had to appear in the proper proportions. Producing too many planes, for example, would not only create a fleet of planes for which there were no trained pilots, but also divert resources away from the production of other weapons. The i4 Most of the previous studies have modeled pollution as a joint product of a process that also yields a positively valued good. Compensation for pollution must, however, be paid by the polluter just as he must pay for the labor he employs. It makes sense,therefore, to consider the eflluent an input and treat it as such in a production function. A more thorough discussion is found in Yohe (1976a).

PRICE AND QUANTITY

CQNTROIS

C~~~A~~~

231

goods produced by a wartime economy are both strong ~orn~~erne~~sin consumption and strong substitutes as joint products. We can, therefore, submit that while Weitzma~ was car not capture all of the intricacies of a wartime situation. observe the total output effect; in terms of A;, magnitude. We have now also seen, bowever, that least in the an-craft industry,

and finally, The direct curvature effects clearly favor quotas; variation is small in situations where variation is harmful. The sixth line of Table 3 is, in addition, directly applicable and reveals that the induced effects also favor quotas; price controls would allow disastrously large variations in the wrong directions. The structural impact of wartime therefore amplifies all of the observations derived from the basic model. Weitzman’s second proposition compares two production hierarchies, and does not fare as well. The first hierarchy ~timateiy faces a titive market. If the entrepreneur who manages this bierarc~y were s to maximize profits, he would face a linear “benefit” schedule. As a rest& the cost side would almost certainly dominate the intermediate-good c~~~~r~so~~ and price control should be favored regardless of the degree of substitution that is available.15 Were the entrepreneur risk-adverse, however~ his effective benefit schedule could be highly curved. The benefit side would then assu importance that depends both upon its curv d the e~asti~ity of effect is substitution. We argued in the previous section not monotonic, and a careful case-by-case study is uired to complete the control comparison. It may be the power of this management that explains the dearth of large com~a~i subsidiaries with efficiency prices.16 The contrasted hierarchy in the second conjecture is simply a single sector of a larger production process. The benefit side must thus be considered in the light of two possible sources of substitution. If the final output of the sector in question has an available substitute in its role as an input, ted ~ambi~u~usly reduce the necessity of a relatively constant Of i5 Only if the cost schedule for producing the intermediate good is nearly linear can there be a chance for the cost side to be dominated by a covariance effect of the quota alternative. This is the only way that the quota could emerge preferred. I6 Prior to making his conjecture, Weitzman recorded the results of several, experiments in price controls conducted by Ford and General Motors in the early 196%; they were origin&y reported by Whinston (1962). All of the experiments were terminated in abotit 1 year, and quotas have been in place ever since.

232

GARY

WYNN

YOHE

Potential substitution for the regulated input in producing this output meanwhile influences the amount of variation that is passed on; we have seen that when one input is regulated, a maximal transfer occurs when the elasticity of substitution is unity. While the sharpness of this result is undoubtedly reduced when both (all) inputs are to be controlled, it does cast further doubt on the conjecture. Surmising, as Weitzman does, that quotas should always be preferred in planned hierarchies therefore requires the belief that (1) little substitution is possible above the sector’s final good, (2) the production of that good holds few possibilities for substitution, and (3) variation in the deliveries of the regulated inputs are larger under price control. The avenues for future research are, from the present juncture, wide and untraveled. Intermediate-contingency orders that increase expected social welfare should be possible in a wide variety of circumstances; the best structure for each should be characterized. The amount and quality of the information passed back to the center for revision of the control specifications need not be the same for both price control and quotas in all situations. Since all controls need adjustment from time to time, the comparative informational advantage should be explored. Direct application of the techniques to several macrocontrol problems, like regulating the money supply and the amount of foreign trade, should also be attempted. Perhaps the most important lesson of the existing analysis is that the answers to these questions will not be as simple as they were once thought to be. REFERENCES Adar,

Zvi, and Griffen, James M., “Uncertainty and the Choice of Pollution Control Instruments.” J. Environ. Econ. Mngm. 3,3 : 178-188, Oct. 1976. Devons, Ely, Planning in Practice-Aircraft Planning in War-time. Cambridge: Cambridge Univ. Press, 1950. Fishelson, Gideon, “Emission Control Policies under Uncertainty.” J. Environ. Econ. Mngm. 3, 3 : 189-198, Oct. 1976. Montias, John M., The Struchue of Economic Systems. New Haven, Conn.: Yale Univ. Press, 1976. Poole, William, “Optimal Choice of Monetary Policy Instruments in a Simple Stochastic Model.” Quart. J. Econ. 84, 1 : 197-216, Feb. 1970. Samuelson, Paul A., “The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments.” Rev. Econ. Stud. 31,4 : 537-542, Oct. 1970. Spence, Michael, and Roberts, Mark, “Efiluent Charges and Licenses under Uncertainty.” J. Publ. Econ. 5,3 : 193-208, May 1976. Weitzman, Martin L., “Prices vs. Quantities.” Rev. Econ. Stud. 41, 1 : 5&65, Jan. 1974. Whinston, Andrew, “Price Guides in Decentralized Institutions.” Ph.D. Dissertation, Carnegie Institute of Technology, 1962. Yohe, Gary W., “Substitution and the Control of Pollution-A Comparison of Effluent Charges and Quantity Standards under Uncertainty.” J. Environ. Econ. Mngm. 3,4 : 3 12-325, Dec. 1976a. Yohe, Gary W., “Uncertainty and the Simultaneous Regulation of Complements, Substitutes, and Joint Products.” Albany Discussion Paper Number 71, 1976b.

Yohe, Gary W., Single-vaiued Control of an Intermediate Good under Uncerta&y.” Uohe, Gary W., “Single-valued Control of a Cartei under Uncertainty.” 1: Spring 191%. Yohe, Gary W., ‘“Toward a General Comparison cf Price Csntrois and Rev. Econ. Stud. i9llc, in press.

in~ztar.

BelE Jo Ecxx.