alternative price and quantity controls for regulation under uncertainty

Vol. 44 (1984), No. 2, pp. 103--115 Zeitschrlft f~r National~konomie Journal of Economics 0 b y Springer-Verlag 1984

Alternative Price and Quantity Controls for Regulation Under Uncertainty

By

John Bennett, Cardiff, United Kingdom*

(Received July 15, 1983; revised version received February 10, 1984)

1. Introduction

Under uncertainty, price controls do not generally lead to the same outputs as quantity controls. W e i t z m a n (1974) provides a framework for analyzing the conditions which favour the use of each type of control. In his model, cost and benefit functions are approximated quadratically in order to measure the performance of a control. This means that marginal cost ("supply") curves and marginal benefit ("demand") curves are assumed linear within a region of the optimum. Random shocks, which the regulator cannot predict, lead to vertical shifts in supply and demand curves, so that the actual optimum is unknown. The price control used by Wei t z - m a n in his comparisons is an announcement of price to the supply- ing firm, which is then assumed to choose the output which maxi- mizes its profit. However, as L a f f o n t (1977) points out, by restrict- ing his analysis to this type of price control W e i t z m a n is, in a sense, exploiting only half his model. For it might be possible, alternatively, to announce a price control to the consumer, the sup- plier of a good being instructed to produce whatever is demanded at that price. In a one-good model L a f f o n t shows that, ceteris paribus, a quantity control should never be chosen1: given the aim of maximizing expected net benefits, the choice is always between a price announcement to the producer and a price announcement to the consumer.

* I am grateful to a referee for very helpful comments. 1 Unless the slope of the supply curve equals the slope of the demand

curve, in which case the expected net benefits from each type of control are the same.

104 J. Bennett:

In the present paper we examine exactly how L a f f o n t ' s conclusion is affected when there is more than one good. One important aspect of this is the generalization of his result for "mixed" signals: simultaneously using different types of controls for different goods. In doing this we extend the model by allowing for non- separability of both cost and benefit functions. This enables us to examine the effects on the performance of the different controls of the interaction of complementarity/substitutability in consumption with complementarity/substitutability in production 2.

We begin by developing the model in Section 2 for pure signals, and in Section 3 we interpret the results. Then in Section 4 we bring mixed signals into the comparison. In Section 5 our conclusions are pulled together.

2. The Model for Pure Signals

Assume that there are n goods, all produced by one firm. Let q~ be the output of the i-th good, the cost function being C (q, _X), where q is the vector of outputs (ql, q2 , . . . , qn) and _X_ is a vector of random variables. Benefit is given by B (q, Y), where _Y is a vector of random variables. _X and _Y represent ex ante uncertainty from the point of view of the regulator (there is no uncertainty as to costs at the level of the firm or to benefits at the level of the consumer). We shall write C~ for 8 C/8 q~ and B~ for 8 B/8 q~; we shall also write Cij and B~j for 8 2C /Sq~Sq j and 8 2B/Sq~Sq~, respectively.

It is convenient to call Ct and Bl the "supply" and "demand" functions for q~, respectively; of course, these apply for given values of all qJ (j#i). For all _X, _Y and i we assume that C~ (q, _X) >0, Bi (q, Y) > 0, Cil (q_, X) > 0, and Bi~ (q, Y) < 0. Also, it is assumed that for large enough q~, B~ (_q, Y)<C~ (q, _X), while for qt=O, B~ (q, Y)>C~ (q, _X). These assumptions ensure that the upward- sloping supply function crosses the downwards-sloping demand function in the positive quadrant.

2 The many-good case is considered briefly by Wei tzman (1974) and Yohe (1977), but without reference to the interplay between substitutability/complementarity in consumption and that in production. Weitz- man (1978b) and Bennet t (1982) also examine many-good models, though under different informational assumptions to those in Weitzman's original article, and without allowing for joint costs. None of these authors, however, consider the price signal to the consumer discussed by Laffont .

Alternative Price and Quantity Controls 105

The aim of the regulator is to maximize expected net benefits:

M=Z{B (~, y ) - C (q_, X)). (~)

One option for the regulator would be to calculate the quantity vector ~_=(~l, ~z , . . . , qn) which maximises (1) and order the firm to produce this vector. Actual net benefits would differ from expected net benefits according to what shocks Y and _X occur. For simplicity, we follow W e i t z m a n in making the restriction that the shocks enter the cost and benefit functions only through the zero- order and first-order terms of the quadratic approximation. The implications for our model of shocks to the second-order terms are discussed in footnote 13. In the present case, with joint costs and benefits, the approximations can be written

C (q, X_)~ a+.S (C't+~ ~) (q*_~t) + x/2 XXc,~ (q*-~*) (qS-~J)

(2) B (q, Y)~ fl+ S (B '* +a ~) (q*-c) *) + 1/2 X X blj (q*- c) *) (qJ-~J)

I I j

where o:, fl, y* and ~* are all random variables. ~ and fi represent the values of costs and benefits, respectively, when q =a/. It is assumed that E (y*)= E (~*)=0 and that E (y~ ~J)=0 for all i and ]. C "~, B 'i, c~j and bi~ are fixed coefficients (where C 'i, B '~ >0; cii> 0; hi, < 0 for all i). We assume that the values of these coefficients are known to the regulator a.

The approximations in (2) are more easily interpreted if we first differentiate partially with respect to qL Setting c,~ = cji and bij = bji, we obtain

C~ (q, _x)- (C'~ +~q + S c~j (qJ-~J) J

(3) Bi (q, Y) -'- (B '* +a t) + Z blj (ql - ~y).

J

(3) gives the supply and demand functions for good i. The quadratic approximations for cost and benefit mean that supply and demand are each approximated by straight lines in the region of g/. Day-to- day random shocks, y* and 8', cause vertical shifts in these curves. Neither ~* nor (~* (for all i) is assumed observable to the regulator

3 C,t and B '~ must be known since, as we shall see, the regulator uses them as pure price signals. It is not necessary that caj and b~j be known to the regulator in order for any of the signals discussed in this paper to be set. It is necessary, however, that the regulator know cij and btj in order to determine which signal yields the greater expected net benefits.

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ex ante. However, the producer knows before production takes place what the values of the y~'s are, while the consumers know before any purchases take place what values the ~'s take. Thus, all the "expectations" in the model to which we refer are held by the regulator. Returning to Eq. (3) it can be seen that when q~ = ~ for all i, E [C~ (_q, Y)] ~-C '~ and E [B~ (_q, _X)] ~B '~. This means that where expected supply equals expected demand (expectations held by the regulator), C '#--- B '~. (It is therefore implicit in the way the approximations are set up that C '~-~ B '~ for all i.)

Following W e i t z m an (1974), "Option 1" for the regulator would be to choose a quantity for each good and order the firm to supply it to consumers. By definition, the optimal signal (maximizing (1)) with Option 1 is 8/. "Option 2" suggested by W e i t z m a n is for the regulator to choose a price for each good and order the firm to supply its profit-maximizing output vector at that price vector. Under Option 2 the optimal signal is found to be C '~ (~-B '~) for all i 4. The alternative suggested by L a f f o n t , "Option 3", is to announce this same set of prices to consumers, who then choose the quantities produced. Thus, all three signals are taken from the inter- section of the expected supply and demand curves.

If Option 1 is used the value of the maximand is M 1 = E {B (~_,Y) - C (~/, X)} 5. Using (2) we therefore have that MI~-E (t-a). The value of the maximand under Option 2, M 2, however, is less straight- forward to find. We denote the price vector announced and the corresponding vector of quantities which come to be actually produced in this case by I~ and ~/, respectively. That is, given the value of "_P, the firm chooses ~/ to maximise profit, /5. ? / - C (_q, _x) and so sets

l~ ~ = C, (?/, _X) (i = 1, 2 . . . . , n). (4)

As we have seen, the price chosen by the regulator is that at which expected demand equals expected supply: ~ = C '~. Facing this price for each good the firm chooses _~ such that actual marginal cost equals C '~ for each good i. _~ is not then equal to _~, in general,

4 Although our quadratic approximations are different from Weitz- man's, it is a simple matter to show that the optimal signals he calculates also apply to our case.

5 When the quantities are supplied, it is assumed that there are no transactions or rationing costs. The same assumption is made under Op- tion 2 where, of course, the amounts the firm decides to supply do not necessarily equal the amounts that would be demanded at the price quoted by the regulator to the firm.


because of the shocks, y6. 8/ is found by combining p~ = C '~ with (3) and (4) to obtain

y~ + X c~j ( ~ - 0J) -'- 0 (i = 1, 2 , . . . , n). (5) J

Here there are n simultaneous equations to determine the n-vector 3- For simplicity, and to get some insight into the solution, we assume that n =2. Thus,

Cii Cl2~ qi--Ol __),i

Denoting the determinant of the matrix in (6) by D, we have

D = c i l c22-c122 (7)

which must be positive if ((~1, ~2) is to represent a profit maximum. The solution to (6) is

~_~Oi i : _~2c12] -~-[? c~2 (8)

~2 .,,- 02 1 2 __~,1 C12]. - ~ [ ~ ' cll

(8) shows the difference between the outputs which actually come about under Option 2, _~, and the outputs the regulator expects will come about, 0. ~/ is unknown to the regulator ex ante, but substituting q = q into (1) for n = 2 we obtain the value of expected net benefits with Option 2 (taking into account the shocks which may possibly occur):

M 2 ~ E ( ~ - ~)

+ ~E{[C222 (bll + Cll) + (b22 - c22) c12 ~ - 2 c22 bi2 c12] a l l (9)

+ [Cll 2 (b22 + c22) + (bll - Cll) ci22 - 2 Cll bi2 c12] (r2z

+ 2 [Cll c22 (ble - ci2) + ci~ 2 (ci2 + big) - c19~ (bil c22 q- b22 c11)] ff12}

where a~i ---- E ()J~)pJ) = coy (),~,)~J) (i, i = 1, 2). (10)

6 Note that it would, in general, be incorrect to argue that marginal cost for i equals C '~ when q = g/. If there is a non-zero shock ~, marginal cost does not equal C '~ even when q=~. Nevertheless, there is a set of outputs at which marginal cost equals C 'i (for all i); this set is shown by Eq. (5).

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We also wish to find the value of the maximand under Option 3, M 3. The reasoning is parallel to that for Option 2. We denote the price vector this time by ~ and the corresponding vector of quantities actually chosen by consumers by 67. Consumers maximize B (q_, _Y)-_P. q_- and so choose _q such that

~r = Be (q_-, _Y) (i = 1, 2, . . . , n). (11)

Again, the regulator chooses the price at which expected demand equals expected supply: i F~ = B '~ (-'-C'q. Consumers therefore choose 67 such that actual marginal benefit equals B '~ for each good i. _~ does not generally equal ~ because of the shocks 8. q is found by combining i5r 'i with (3) and (11) to obtain

3t + X b,;' (qr - 0 j) -'- 0 (i = 1, 2 , . . . , n). (12) i

Again assuming n =2, we get

where

ql _ ~,1 _ ~- [31b22 _ ~ 2 b12]

q2 --,- @2 __ ~- [32bl 1 _ 31b12]

H = b l l b ~ 2 - b 1 2 ~

(13)

(14)

which, to satisfy the sufficient conditions for the maximisation of B (q, Y)-P__--_q, must be positive. Substituting q = q into (1) for n = 2 we find

M a ~ E (fl - a) 8

+ ~ - {[ - b222 (bll + c11) + (b22 - c22) b12 ~ + 2 b22 c12 b12] sll (15)

+ [ - - bll 2 (b22 + c22) + (h i1 - - C11) bl~ 2 + 2 bll c12 b12] s22

+2 [bll b22 (b12-c1~)-b122 (b12 +cl~) +b12 (bll c22 +b22 cn)] s12}

where s~j - E @ ~J) = c o v @ , @ (i, j = 1, 2). 06)

Finally, it will prove useful to derive the "true" optimal outputs: that is, the outputs which, ex post, in the light of the shocks which actually occur, maximize the difference between actual benefits and actual costs. To find these we set Cl (q_, _X)= Be (q, Y) in


Eq. (3) for i = 1 and 2 7. Using the fact that C ' ~ B '~ ( i=1, 2), we obtain two simultaneous equations, which can be solved to yield the optimal outputs

qr ~1 1 + X- [(b~2

q~.2~__[t2 1 b + ~ - [ ( 11

- c 2 2 ) ( ~ 1 _ 81 ) _ ( b i 2 - c i 2 ) (yg. _ 8 2 ) ]

- c 1 1 ) (~-'~ - 82 ) - - ( b i ~ - c i 2 ) (~,1 __ 81)]

(17)

where K = (b l l -c11) ( b 2 2 - c 2 2 ) - (b12-818) 8, which, for stability, is assumed positive. It should be remembered that, given the values of C '~, B "~, ctj and blj (for all i, j) and given that E (y~)= E (8 i) = 0 (for all i), the point at which the expected supply and demand curves cross is given. This means that the optimal quantity signal, @~, as well as the optimal price signals, C '~ (or B 'r can be regarded as constants. In contrast, q~l and q~,2 are variable, depending on what stochastic shocks actually occur.

3. Interpretation

Continuing in the same way as W e i t z m a n (1974) we compare the performance of the different options by examining (M 2 - M 1) and (M 8 - M 1) 8. Since M 1 = E (fl - ~) we must therefore concentrate on the signs of the terms in the curly brackets of (9) and (15). By assumption, we already have that c11, c22 > 0 and bli , b22 < 0, while a11, a22, sl l and s22 are, by definition, non-negative. The terms remaining wi thout specific signs are b12, c12, ax2 and s12.

The coefficient bi2 is a measure of complementari ty/substi tut- ability in consumption between goods 1 and 2. If b12 > 0 we refer to 1 and 2 as "complements in consumption"; if b12<0 we refer to them as "substitutes in consumption". The coefficient c12 plays a similar role in production. If c12 < 0 we call 1 and 2 "complements in product ion"; if c12 > 0 we call them "substitutes in product ion" 9

v That is, we set actual marginal cost (including the v*-shock) equal to actual marginal benefit (including 6~) for each good i. The resulting outputs do not, of course, equal the expected outputs 3, unless all y~- and ~l-shocks are zero.

8 Separate consideration of (M 3 - M 2) can be omitted since this equals [(M 3 - M 1) - (M 2 - M1)].

9 Notice that there is an asymmetry. Consider, for example, what happens when the production and consumption of 2 increase. If, as a result, the demand (marginal benefit) curve for 1 shifts up, there is complementarity in consumption; if it causes the supply (marginal cost) curve to shift up, there is substitutability in production.

8 Zeitschr. f. National6konomie, Vol. 44, No. 2

110 J. Bennett:

The term ~12 is the covariance of the errors made in estimating the supply curves of goods 1 and 2, while s12 is the covariance of the errors made in estimating the demand curves of 1 and 2. We shall not assume that either o12 or sl~ takes any particular sign 1~

Suppose, first, that there are neither supply nor demand interactions (c12=b12=0). It is then easily checked from (9) and (15) that we have obtained a two-good replica of L a f f o n t ' s results. If the demand curve is steeper than the supply curve for each good (]b~i] >ci~ for i = 1 and 2) then M a > M I > M 2 ; while if each supply curve is the steeper, M S > M 1 > M a. In neither case is the quantity signal chosen. But when b12 and/or c12 are non-zero, any shock to a supply or demand curve (via yt or (~*) is found, through supply and/or demand interactions, to generate further deviations unex- pected by the regulator. These deviations mean that an uncondi- tional ranking of the three options is not possible; each case must be examined separately ~1. In this section we give an intuitive example, concentrating on the effects of a stochastic shock on the performance of Opt ion 1. We leave until the Conclusion a listing of the most general (conditional) conclusions that can be made about (9) and (15).

We begin our intuitive example by assuming, temporarily, that ~ 2 = s , 2 = 0 . We suppose that goods are substitutes in production (c12 >0) and complements in consumption (b12 >0). The only stochastic shock which is assumed to occur is a negative yX. With Opt ion 1, the signals given, ~1 and ~ , are, as always, determined by the intersections of (expected) demands and supplies in the ab- sence of shocks. The ex post-optimal outputs, q,~ and q.~~ however, change due to yl in the way shown by (17). The direct impact of the shock is to cause a d o w n w a r d movement in the supply curve of good 1, so that the optimal output , q~,l=ql-)'l/(Cll-b11), ex-

ceeds the quantity signal, 91 . The indirect impact of the shock is that, since q~l has increased, the "true" supply and demand curves

l0 It does, however, seem likely that if the regulator knows that 1 and 2 are substitutes in consumption, then an overestimate of the marginal benefit (demand) for one good will tend to be associated with an overestimate in the marginal benefit for the other. That is, if (b1~+b21)< 0 then it seems more likely that s12 > 0.

11 If ~l~=sl~=0 and [bt~]>c~ for i=1 and 2 then we still find that M 8 > M 1 > M s if c1~ ~ 0, provided bl~ = 0. While if crl~ = s12 = 0 and Ibis] < c~ for i=1 and 2 we still find that M~>MI>M 8 if b lz#0 , provided c12=0. See Section 4 for the case of ]bit] exceeding cr for one good, while [b~] is less than c~ for the other.


for good 2 shift upwards. If, for example, clz = b12, the two curves shift up by an equal amount and so, as can be seen from (17), q~2=92. Thus, Option 1 in this case leads to offsetting errors, to an output of good 2 equal to q.:2, and an output of good 1 which deviates f rom q~.l according to the size of (cll-bll). If, however, c12 and b12 had been unequal, but with the same sign, the errors for good 2 would have been only partially offsetting; while if c1~ and b12 had differed in sign the errors would have been cumulative.

Now assume that sl2 > 0 ((r12 being irrelevant in this example). There is now an additional type of repercussion to be taken into account. With s12 positive, the negative 71 means that there is likely to be a negative y2. By itself, this would mean that the supply curve of good 2 being used by the regulator were above its actual value, so that ~ would be set "too low". This type of repercussion (due to s12) combines with the type considered in the previous paragraph (due to c12 and b12). Suppose, for example, that c19. > hi2 > 0. Then, when c) is set "too low" (due to yl), the supply curve for 2 rises by more than the demand curve does (this is the repercussion due to c12 and b12). By itself, this would mean that q,~' would be less than ~2. Putting the two types of repercussion together, we see that they are offsetting in this example, leading to relatively close values of q,2 and ~2 12. In other examples, the errors may be cumulative ~3.

18 There are, of course, further interactions between the demand and supply curves which can be intuitively explained in a similar way. Eq. (17) shows the ultimate values of q~l and q.:~

1~ It is convenient to note here what happens to our analysis if we allow for uncertainty in the second-order terms in (2). The effect is that the value of M 1 is unchanged, but that the equations for M 1 and M s must be rewritten to allow for the new set of random variables. In outline, however, the existing equations, (9) and (15), still show what happens. Sup- pose, for example, we place a (multiplicative) random variable, ~/j, in front of c~j in (2), while a similar random variable, r is placed in front of b~j. Assume that e~j and r are independent of each other and of the random variables we have already written in (2). It would be natural to assume, however, that the different eij's are mutually dependent, and similarly for the r Eqs. (3) and (5)--(8) must be amended, replacing c~j with el3" cij. The same change is required in Eq. (9); but, in addition, we must place

8 the expectation operator, E, in front of the term b- ~ { . . . } . Remembering

that D is now a random variable, it can be seen that (9) would now con- tain the expected values of much more complicated random variables than those in our original presentation. Similar factors apply to (15) and the

8*

112 J. Bennett:

4. Mixed Signals

Instead of using the same option for both goods, "mixed signals" can be given. If Option I is used for good 1 and Option J for good 2, then we denote the value of the maximand by M xJ. We shall consider M 12 and M 13. We shall not, however, consider M 23. For this would involve transmitting a price for good 1 to the producer and a price for good 2 to the consumer. Before determining the (expected) profit-maximizing output of good 1 the producer would need to estimate the demand for good 2. Before determining the (expected) utility-maximizing demand for good 2 the consumer would need to estimate the supply of good 1. Such estimation by producer and consumer involves further problems beyond the scope of this paper.

Consider now the mixture of Options 1 and 2. The producer is instructed to produce the quantity ~1 of good 1, and is also informed of the price, ~2= C,2, of good 2. Profit-maximization subject to the constraint q l = c)1 then yields the output chosen of good 2, ~2. We can find this by substituting Ca (q, X_) = D ~ = C '2 and ql = ~1 in Eq. (3) to obtain ~12=~-y2/c22. Thus, we find

M 12 ~-E (fl-~) + ~ (b~2+c~). (t8)

Similarly, if we combine Options 1 and 3, using the quantity signal ~1 for good 1, but giving the price signal ~2 =B,2 for good 2, we find that the quantity of 2 chosen is q2=~2_(52/b22 and the value of the maximand is

M 13 ~ E (fl - a) s2~ -- 2 b~2 ~ (b2~ +c22). (19)

Furthermore, M 21 and M 31 are found simply by changing the "2"- suffixes into "l"s on the right hand sides of (18) and (19) 14.

From (18) and (19), and bearing in mind that MI=E (fl-oc), we obtain the following results. Firstly, as c~2~162~] then M12N M I N M 1~. Secondly, as c11.~]b111 then M 2 1 ~ M l ~ M al. Thus, if we compare the various mixed signals with the pure quantity op-

equations leading up to it. We have excluded second-order uncertainty from the model because of the difficulty it leads to in interpreting the revised versions of (9) and (15).

14 It will be noticed that, if we exclude E (fi-~), Eq. (18) is the same as Weitzman's (1974) Eq. (20), which is the same as M 2 for the one- good case. (Weitzman excludes E (fl-~) for simplicity.) Similarly, our Eq. (19) corresponds to L1 (3/1) in Laffont (1977, p. 179).


tion, we find that except when C l l = I b l l l and c22 = [b22[ (in which case the value of the maximand is always zero), we would always pre[er a mixed signal to a quantity signal. If the supply curve is steeper than the demand curve for both goods, then our preferred mixed signal is a combination of Options 1 and 2. To decide which good should be given which signal we should have to examine further M 12 and M 21. Similarly, if the demand curve is steeper than the supply curve for both goods, our preferred mixed signal is a combination of Options 1 and 3. If, alternatively, the supply curve is the steeper for good 1, but the demand curve is the steeper for good 2, we would choose either a quantity signal for good 1 and a price signal to the consumer for good 3, or a price signal to the producer for good 1 and a quantity signal for good 2. While if the demand curve were the steeper for good 1, but the supply curve were the steeper for good 2, we would choose either a quantity signal for good 1 and a price signal to the producer for good 2 or a price signal to the consumer for good 1 and a quantity signal for good 215 .

The above result is easily generalized when there are n goods, where n >2. Suppose a quantity signal is set for the last ( n - 1 ) of these: 0 2, 03 , . . . , O n. If Cll > [bl~ I, then the value of the maximand will be greater if, for good 1, we set a price signal to the producer than if we set a quantity signal for good 1. Conversely, if cll < [bill, the value of the maximand will be greater if, for good 1, we set a price signal to the consumer than if we set a quantity signal for good 1. It follows that (except in the theoretically trivial case in which the slope of the supply curve equals the slope of the demand curve for every good) there is always at least one mixed signal 16 which is superior to a pure quantity signal.

15 It may be asked again what the implications are for our results if higher-order uncertainty is included in the approximations for cost and benefit. In the model of Section 2 such an amendment complicates the results considerably (see fn. 13). In the mixed signal model of this section there would also be complications; but we may apply the analysis of Weitzman (1978 a) here. Weitzman shows that his 1974 results, using only quadratic approximations, are accurate in the limit, provided that the stochastic terms of different orders within an approximation (i) are independent and (ii) have variances which go to zero together. He argues that while (i) is a restriction on the validity of his main results, (ii) will be satisfied for "'most' distributions of interest" (p. 210). The same reasoning applies to the mixed signals we discuss in this section.

16 The "best" signal may, of course, be a combination of the quantity option for several goods and the price option for the remaining goods.

114 J. Bennett:

5. Conclusion

In this paper we have examined more general cases in the type of comparison of alternative regulatory controls initiated by W e i t z m a n (1974). The alternative "pure" signals we have looked at are a quantity instruction for each good, a price signal to the producer for each good and a price signal to consumers for each good. We have also considered "mixed" controls, combining both the quantity signal and one of the price signals. Throughout, we have used a model with at least two types of good and we have assumed that all production takes place in one firmlL In the results an important r61e is played by substitutability/complementarity in both consumption and production.

Looking at our equations in the two good case for pure signals, the most general conclusions that can be obtained are the following.

(i) If the supply curve is steeper than the corresponding demand curve for each good then a sufficient condition for a quantity signal to be superior (in terms of expected net benefits) to a price signal to consumers is that either (a) the errors in demand curve estimation for the two goods

be non-negatively correlated and that the two goods be complements in production, but substitutes in consumption, the former quality being stronger (in the sense that clz be numerically greater than b12 in the cost and benefit functions (2));

or (b) the errors in demand curve estimation be non-positively correlated and that the two goods be complements in consumption, but substitutes in production, the former quality being stronger.

(ii) If the supply curve is flatter than the corresponding demand curve for each good then a sufficient condition for a quantity signal to be superior to a price signal to the producer is that either (a) the errors in supply curve estimation for the two goods

be non-negatively correlated and that the two goods be substitutes in production, but complements in consumption, the former quality being stronger;

or (b) the errors in supply curve estimation for the two goods be non-positively correlated and that the two goods be substitutes in consumption, but complements in production, the former quality being stronger.

iv For the case of a separate firm for each good we simply set c1~ = 0 in our equations.


These results give an idea of the role played by the interaction of substitutability/complementarity in consumption and production and of errors in demand and supply curve estimation in determining the relative performance of the different signals under other com- binations of conditions. A signal performs better when the errors that it generates are offsetting, rather than cumulative.

For the one-good case L a f f o n t (1974) has shown that one of the price signals will always yield greater expected net benefits than the quantity signal 18. Our conclusions above indicate that with pure signals there is no simple generalization of L a f f o n t ' s result without making restrictions on parameter values. When we intro- duce the possibility of mixed signals, however, there is a simple generalization. It is found that in the n-good case, a mixture of one price instruction with ( n - l ) quantity instructions is always superior to the pure quantity control for all n goods. (Other mixed signals may, of course, be better still.) However, we cannot eliminate the pure price signals in the same way. In the framework of this model, the choice of regulatory control lies between pure price signals and mixed signals.

References

J. Ben n et t (1982): Contingent Pricing and Economic Regulation, Bell Journal of Economics 13, pp. 569--574.

J. J. Laffont (1977): More on Prices vs. Quantities, Review of Eco- nomic Studies 44, pp. 177--182.

M. L. Weitzman (1974): Prices vs. Quantities, Review of Economic Studies 41, pp. 477--491.

M. L. Weitzman (1978 a): Reply to "Prices vs. Quantities: A Critical Note on the Use of Approximations" by James M. Malcomson, Review of Economic Studies 45, pp. 209--210.

M. L. Weitzman (1978b): Optimal Rewards for Economic Regula- tion, American Economic Review 68, pp. 683--691.

G.W. Yohe (1977): Comparisons of Price and Quantity Controls: A Survey, Journal of Comparative Economics 1, pp. 213--233.

Address of author: Dr. John Bennett, Department of Economics, University College, Cardiff, P. O. Box 78, Cardiff CF1 1XL, United King- dom.

18 With the proviso in footnote 1. A similar point holds for our conclusions in the n-good case.

alternative price and quantity controls for regulation under uncertainty

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