Fuel switching, gasoline price controls, and the leaded-unleaded gasoline price differential

JOURNAL OF ENVIRONMENTALECONOMICSAND MANAGEMENT8,287-302(1981)

Fuel Switching, Gasoline Price Controls, and the Leaded- Unleaded Gasoline Price Differential PAUL KOBRIN American Petroleum Institute, 2101 L Street, N. W., Washington,D.C. 20037 Received August 3,1979; revised December 27,1979 The Environmental Protection Agency and others have opposed gasoline price decontrol, alleging a wider posted price differential between leaded and unleaded grades would result, inducing more motorists to switch illegally to leaded gasoline fouling catalytic converters and hence increasing air pollution. EPA’s model must assume that only the unleaded price’ceiling is binding. It is shown that the resulting excess demand is shifted to a close substitute: leaded gasoline. Hence, controls cause more consumption of leaded fuel in new cars (switching) and more pollution. Thus, decontrol would have a palliative effect, contrary to EPA’s claim.

INTRODUCTION

A current public policy issue is the prospective decontrol of motor gasoline. One aspect of the issue is what effect, if any, such decontrol would have on the environment. Some, such as the Environmental Protection Agency, have argued that decontrol of gasoline prices would have an adverse environmental impact [6]. They reason that decontrol would widen the posted price spread between leaded and unleaded gasolines, inducing more motorists to switch illegally to leaded gasoline, fouling catalytic converters, and hence increasing air pollution. For this reason, the EPA has questioned gasoline price decontrol and has considered directly limiting the posted price differential between leaded and unleaded gasolines. The extent to which gasoline price decontrol would induce fuel switching by motorists is amenable to economic analysis. In Section I of this paper, it is demonstrated that under at least one set of plausible conditions which are consistent with EPA’s thinking concerning gasoline demand, the level of switching or cheating will be diminished-not increased-by decontrol. It is shown also that EPA’s contrary result is obtained by mistaking a controlled price for a market clearing price. Once this mistake is rectified, the thrust of the analysis is changed and the changed result follows. In Section II the effects of a policy to limit directly the posted price spread between leaded and unleaded gasoline are examined. This policy is not amenable to analysis without specification of additional assumptions and is not carried out here. I. PRICE CEILINGS

AND POLLUTION

Four price control situations can be distinguished. 1. Price controls are not binding. Decontrol would have no effect. This is clearly not the EPA case. 2. Only the leaded ceiling is binding. Decontrol would then narrow the differential. This is not the EPA case, either. 287 0095~06%/81/030287-16$02.00/O Copyright Q 1981 by Academic Press, Inc. Au rights of reproduction in any form resewed

288

PAUL KOBRIN

3. Both grades are depressed by controls. Decontrol would raise both and the differential could widen or narrow. This case has an uncertain effect on the differential but EPA is certain the differential would only widen. Thus, it is not the EPA case, either. 4. Only the unleaded ceiling is binding. Decontrol definitely widens the differential. This and only this case is consistent with EPA’s premise. Thus, the subsequent argument assumesfor purposes of analysis that only the unleaded ceiling is binding. A. A Formal Model We will prove rigorously that, in the context of the EPA’s assumptions, cheating will be diminished by decontrol. Then, a verbal-graphical rationale for this result will be provided which also explains the analytic error by which the opposite conclusion is reached. We first model a binding retail price ceiling on unleaded gasoline and prove that cheating is diminished by relaxing the control.’ The model is then revised to capture the case of a binding wholesale price control and it is proved that there, too, relaxing the constraint diminishes cheating. A possible complication is the importing of unleaded gasoline so as to clear markets even with price controls over domestic output. But under gasoline price regulation, firms are deterred from importing gasoline. Foreign gasoline is more costly than the domestic product manufactured from price controlled crude oil. Thus, to recover the higher cost of foreign material, higher selling prices at some levels of trade are required. However, higher prices are possible only at those levels of trade where ceiling prices are binding. The regulations, though, do not permit the recovery of the higher cost of imports solely from those levels of trade. Thus, full recovery is precluded, deterring firms from importing gasoline and leaving some markets uncleared. We now proceed with development of our formal model. The quantity of unleaded gasoline (4,) demanded for use in cars equipped with catalytic converters is a function of the prices of unleaded gasoline and leaded gasoline or

In addition, the quantity of leaded gasoline demanded for use in converter-equipped cars is a function, g, of the two prices. Cheating or switching, then, is represented by this function, g. We will later determine if g rises (environmentally undesirable) or falls upon a relaxation of price controls. Finally, the quantity of (leaded) gasoline demanded for use in old cars is a function, f, of the price of leaded gasoline. The total demand for leaded gasoline (q2) is, then,

‘Although this analysis is couched in terms of a retail ceiling price, it is equally applicable to a maximum dealer margin on unleaded gasoline. That policy sets a ceiling price indirectly. The margin control is an alternative considered by the Department of Energy [7].

FUEL SWITCHING AND PRICE CONTROLS

289

S p, -----

)L

x -----

------------

D

FIG. 1. The unleaded gasoline market.

Some clarification of P,, the price of unleaded gasoline, is required. Under binding price controls, the nominal controlled price is less than the value purchasers assign the commodity. “Value” may be thought of as the price per unit consumers would be willing to pay given the quantity brought forth by the controlled price. This is illustrated in Fig. 1. The controlled price, X, is less than P,, the price which consumers would be willing to pay. This gap can be described as the shadow value of the control.* In fact, freely traded ration coupons (for the quantity 4, which is supplied at X) would be worth P, - X, yielding a total price to consumersof P,. In the absenceof ration coupons, consumerswould be willing to incur a marginal cost of P, - X to purchase a gallon of gasoline at X, again yielding a total price of P,.3 This price may be paid by queuing, making illicit payments, etc. Nonetheless,the price which consumersare willing to pay (the demand price) exceeds the controlled price facing producers (supply price).4 The result of this market situation is that the quantity of the rationed commodity is supply-determined. That is, producers equate the controlled price with marginal cost.’ The resulting quantity, though sold for X, has a value to consumers of P, which is computed from the demand function (once the quantity has been set by producers).6 21n a programming problem, a constrained resource may have a nominal cost. Its shadow value, derived from the dual, will be higher than this nominal cost (if the resource is fully employed). This is the phenomenon considered here. 3Rothbarth [ 51and Graaf [ 31developed the concept of “virtual prices” for rationed commodities in the days before programming. In their papers, consumersbehaved as though the virtual prices prevailed. That is, the virtual prices are demand function arguments. P, is such a virtual price or shadow value in programming terminology. 4P, would be the unique demand price if consumers could trade the rationed commodity, ration coupons or the like among themselves.In fact, the rationing procedure may preclude trading in which case different consumers may assign different values to a (marginal) unit of the commodity. Then, P, becomes a statistic of a value distribution. ‘This notion is well established. Hirshleifer [4, p. 35) states “ students . . . expect the quantity traded to be some compromise between the larger quantity demanded . . . and the smaller quantity offered. . . . But the result is no compromise; it is the smaller of the desired transaction magnitudes that governs.” Bowden [I] utilizes a similar disequilibrium model, which he calls the short-side or Fair- Jaffee model, for econometric purposes. %me readers may prefer to see the controlled price, X, in the demand functions in place of the more abstract P,. We will show momentarily that such a formulation is mathematically intractable and assumes away the problem.

290

PAUL KOBRIN

On the supply side, total cost- marketing, refining, etc.-depends on the quantities of the two gasoline grades or c = ah

4.

We assume that all functions are continuous and twice differentiable with respect to each argument. There are four variables, P,, Pz, ql, and q2 and one parameter, X. Equilibrium, if it exists, results from the simultaneous solution of the following four equations.

(1) (2) (3) (4) Equations (1) and (2) are the equilibrium conditions on the demand side while Eqs- (3) and (4) si ve a supply equilibrium; price equals marginal cost for each gasoline grade. As described earlier, the controlled price, X, affects the quantity supplied which, then, through the demand side (Eq. (1)) affects P,. If X appeared in the demand functions (g and h) in place of P,, the model would consist of one parameter (X), three variables (Pz, qi, and q2) and the four equations. With fewer unknowns than equations, a solution typically would exist only if X were set to clear the unleaded gasoline market. But with the price control nonbinding (and nothing left in the model to represent excessdemand), the problem is assumed away. When P, appears in the demand function however, the model is mathematically tractable and concomitantly economically meaningful because (the shadow value of) excessdemand then is reflected in the difference between P, and X. We assume that the controls are binding: P, > x. We also assume unleaded is the more expensive (valuable) gasoline: P, > P*. This assumption underlies conditions (ii), (iii), and (v), presented momentarily, which deal with substitution between grades. A number of conditions are imposed on the demand and cost functions. The old car demand for leaded gasoline is downward sloping: (i)f’ = df/dP, < 0. Increasing the shadow price of unleaded gasoline increases cheating, holding all else constant: (ii) ag/ap, > 0.

FUEL SWITCHING AND PRICE CONTROLS

291

The new car (cheating) demand for leaded gasoline is downward sloping: (iii) ag/i3P, < 0.7 The demand curve for unleaded gasolineis downward sloping: (iv) ah/aP, < 0. Increasing the price of leaded gasoline shifts demand (from g), increasing h: (v) ah/aP* > o.8 An increasein the price of either grade does not increasethe demand for gasoline in new cars, or (vi) a( h + g)/i3P, I 0 and (vii) a( h + g)/3P, 5 0. The two gasoline grades are joint products (certainly in the refining stage). They compete for the same resources(e.g., cracking capacity) and are thus not complements in production. Therefore, increasing the quantity supplied of either does not reduce the marginal cost of the other, at least in the short run, or (viii) i32c/aq,i3q2L 0. Marginal cost of either grade is more affected by an increment of that grade than by an increment of the other grade, or

(ix) avia+

a2c/aq,aq2>o,

tx) a2c/aq,2-a2c/aqlaq2>0. Finally, there are the three second-order conditions for a profit maximum which are implied by conditions (viii), (ix), and (x):

(xi)a2c/a++o (xii)azc/a+o

(increasing marginal cost of unleaded gasoline), (increasing marginal cost of leaded gasoline),

(xiii) (a2c/aq;)(a2c/a&)- (a2c/aq,aq,)'>o

(conve~tyofCl.

‘We prove shortly that conditions (ii) and (iii) capture EPA’s assumptions concerning the illegal gasoline demand and are also consistent with other formulations of the cheating function. *The reduction in g need not equal the increase in h. See conditions (vi) and (vii).

292

PAUL KOBRIN

A caveat must be added to conditions (xi) and (xii); i?*C/aqf and Zl*C/aqz must be finite. That is, the elasticities of supply of the two grades must be strictly positive although they may be very small. In Section II, it is clear that (some) price responsivenessmotivates the result that relaxing controls diminishes pollution. We justify the assumption of strictly positive supply elasticities even in the very short run as follows. For the industry supply elasticity to be zero, every one of the approximately 270 domestic refineries would have to be operating such that the gasoline supply elasticity of each refinery was zero. That is a difficult assumption to accept in light of the flexibility which refineries have.’ Thus, the decontrol-pollution reduction result developed shortly does not require an interval long enough for capacity to be increased. Bather, the positive (though perhaps slight) short-run supply elasticity permits the result to occur even with given refining facilities.‘o Only 8 of the 13 conditions are independent. Five conditions-(xi), (xii), (xiii), (iv), and (iii), for example- are redundant (derivable from the other eight) and are provided only to elucidate the model and to facilitate the subsequent proofs. In the implicit model underlying EPA’s contention concerning the effects of decontrol, illegal demand is a function of the price differential, i.e., g = g(P, - P2), such that cheating increases as the differential widens or dg/d(P, - P2) > 0. We have specified the g demand function in the usual, general form g = g( P,, P2) rather than as g = g( P, - P2). However, if g( P,, P2) is assigned the more specific form g(P, - P2), then conditions (ii) and (iii), tJg/aP, > 0 > ag/aP,, imply dg/d(P, - P2) > 0. By demonstrating this, we show that any result which holds for the general form of g also holds forthe more specific form. That is, results derived herein are applicable to the special case sketched by EPA and to other cases. Symmetry of the arguments in the specific function g( P, - P2) implies - ag/i3P, = ag/aP,. Substituting this relationship in the total differential of g yields dg=$dP,+ 1

$dp2=gdP,-gdP2=$$d(p,-p2), 2

dg - ag

d( P, - P,)

1

1

1

ag

tlP, = - =

2

’ ”

ag ag ap>o>-. 1

ap2

Thus, the general demand function and its conditions (ii) and (iii) capture the specific demand form and its condition as a special case. The general form, however, also admits casesin which illegal demand depends upon the prices of each of the two substitutes rather than just upon their difference. Given this model (Eqs. (l)-(4) and the 13 conditions), we show that a relaxation of controls reduces cheating, that is, 9Two examples are provided of elasticity creating flexibility. Even in the short run, maintenan~ can be deferred. Another means of increasiq gasoline output is to change catalysts in cracking aad reforming units more frequently. Both of these possibilities permit increasing gasoline production at high costs even after the low cost options of “setting the valves wide open” have been exhausted. ‘OA supply elasticity which is positive at all marginal cost (price) levels is consistent with fiite capacity. The upward sloping marginal cost curve just approaches the vertical line representing capacity asymptotically.

FUEL

SWITCHING

AND

293

PRICE CONTROLS

dg/dX < 0. This is the central result of this section. Also, we show that dP, /dX < 0, dq, )dX > 0 and dq, /dX < 0. THEOREM.

Proof. The total differentials of Eqs. (l)-(4) are 0

ah aP,

0

g

i3h ap, f+z

=

-1

0

0

-1

dX

0

0

a*c a4:

PC 34, aq2

0

0

-1

a*c a41aq2

a*c

dP, dP2

(5)

41 42

2

Let [J] represent the matrix in Fq (5). Solving for dP1, dP,, dq, , and dq2 and dividing each by dX yields ah aP, f-tap,

-1

0

0

-1

a*c aqlaq2

a*c 2

%

-1 dP, -= dX

IJI ah aP,

-l

-ap, ag 0 dP, _ YE--

O

0

-1

a% aql aq2

a*c

g

dql YE-

= ah aP, f+z

0

-

a4:

IJI ah ap,

IJI

=

--ah a*c + --ag a2c a4 aq,2 1 i ap, aqla42

IJI

’

0 -1

-1

azc ad

IJI 03)

294

PAUL KOBRIN

ap,

ah aP,

0

-1

i3h

-

a*c 34, h2

(9)

lJ,=[$(ff+$)-&f$$g$-

i

(g-)

--ah a*c + --ag a*c ap, aq,aq2 1 . apI aq:

The sign of I JI is positive. ” The numerators of Eqs. (6), (8), and (9) are negative, positive, and negative, respectively.‘2 It then follows that dP,/dX

< 0,

(10)

dq, /dX > 0,

(11)

dq2/dX

(12)

c 0.

We distinguish the two possible casesdP2/dX L 0 and dP2 /dX < 0. If dP2 /dX 2 0, it is apparent from the total differential of g divided by dX, dg/dx=

ag4 --agde --ap, dX + ap, dX

that dg/dX

< 0

“By conditions (ii) and (vi), 0 c 8g/W, c: -31/8Z’, and similarly by conditions (v) and (vii), 0 < CVt/iW, I -3g/W,. Taking the product of the above two expressions and transposing yields 0 5 (iM/i3P,)(~g/CJP2) - (Sr/8P2)(8g/W,). Addingf@h/W,), which is positive by conditions(i) and (iv), yields 0 < (8h/@,)(/’ + 3g/W,) - (ah/W’,)@g/M,). (This expression also appears in Eqs. (8) and (9).) Similar analysis using conditions (ii), (vi), (viii), and (ix) shows the last expression in parentheses to be negative. The middle, bracketed term is positive by conditions (xiii). “The numerator of Eq. (6) can be shown to be negative by the type of analysis in the preceding footnote using conditions (i), (v), (vii), (viii), (x), and (xii). The signs of the numerators of Eqs. (8) and (9) are readily determined from the signs of the expressions ascertained in the previous footnote.

FUEL SWITCHING AND PRICE CONTROLS

295

by Eq. (10) and conditions (ii) and (iii). If dP,/dX < 0, then df/dX > 0 by condition (i). The total differential of Eq. (2) divided by dX is dq2/dX = df/dX + dg/dX. Given Eq. (12) and df/dX > 0, this implies dg/dX < 0. This exhauststhe possible cases.In all, dg/dX < 0, dP, /dX < 0, dq, /dX > 0 and dg,/dX < 0. Thus, relaxing (raising) the unleaded gasoline retail price ceiling reduces cheating and hence reducespollution, contrary to the assertion of the EPA. The model is now extended and revised to investigate the case of a binding wholesale price ceiling on unleaded gasoline. We prove that relaxing the constraint reduces pollution. The demand side of the model--s. (1) and (2) and conditions (i) through (vii) -remains the same. However, becausethe retail price is not bound by controls, P, is here a market clearing price as well as a shadow value. Previously, the supply side was represented by the total cost function C( ql, q2) which incorporated all costs-refining, marketing, etc. To model a price constraint which affects an intermediate stage, it is necessary to decompose C into two functions. Let M(q,, q2) represent the total cost of all stagesbeyond the point at which the control is binding. Let R(q,, q2) represent the total cost of all stagesprior to the point at which the control is binding. (R and M m ight be thought of as refining and marketing although this need not be the case.)Of course, C = R + M . Let Z represent the binding wholesaleprice ceiling on unleaded gasoline. Figure II portrays the unleaded gasoline market in this case. There, the refinery gate supply price is the marginal cost up to that point or i!lR/k?q,. It is at this point that the wholesaleprice control is assumedbinding. Thus, Z = aR/lIq,.13

(13)

The curve marked “service station supply price” is the marginal cost of marketing plus the wholesaleprice of purchasedproduct or &l4/~q, + Z.14 As Fig. 2 indicates, the quantity of unleaded, q,, is determined at the wholesale level by the price constraint, Eq. (13). When this quantity is put on the retail market, a market clearing price, P,, results which exceedsthe service station supply price by an amount Y or P=E+Z+Y ’ a41

*

(14)

I3 One of the classic motives for vertical integration is the avoidance of a market in which prices are controlled. Under vertical integration, the interstage controlled price is immaterial for allocation purposes. We assumethe controlled price is binding in a meaningful sense,i.e., it affects resource allocation, which implies that the transactions in question occur between unaffiliated parties. 14Becausethe wholesale price ceiling is assumedbinding, there must be excessdemand and some sort of rationing. When rationing is necessitatedby price ceilings in the wholesale petroleum products markets, federal allocation regulations dictate the method by which product is rationed to dealers. There is no reason to suspect that these regulations facilitate the equalization of marginal costs across dealers. Thus, industry marginal cost of marketing is a fuzzy concept. However, for any given set of allocation rules, there is a supply price of marketing servicesfunction. gM/aq, should be regarded as that function.

296

PAUL KOBRIN

I , 1 61

6

q1

FIG. 2. (a) The unleaded gasoline retail market. (b) The unleaded gasoline wholesale market.

Y may be interpreted as the shadow value of the constraint. The leaded gasoline market clears at both the wholesale and retail levels so the retail price equals combined marginal costs or

-aR+E p2- a42 392’

(15)

The model then consists of Eqs. (l), (2), (13), (14), and (15) or

4t= v,9

p2L

92

+

=f(P2)

0) dp19

p2>9

Z = aR/aq,,

(2)

(13)

p2 =

a0 + Mm,,

(15)

P, =

ah4/aq, + z + Y.

(14)

We assume that the restrictions placed on C by conditions (viii) through (xiii) apply to the components of C. That is, those conditions hold when C is replaced by R or M.

297

FUEL SWITCHING AND PRICE CONTROLS

We prove that relaxing the wholesale price constraint reducescheating or dg/dZ

THEOREM.

< 0. Also, dP,/dZ

< 0, dq,/dZ

> 0 and dq2/dZ

< 0.

The total differentials of Eqs. (I), (2), (13), (14), and (15) are

Proof:

ah ap,

ah ap,

g

ff+g

-1

0

0

dP,

0

-1

0

dP2

a2R

a2R 34, h2

0

4,

ayR+M)

o

0 0

1

0

0

aa: ayR + M)

0

-I

3% aq2 a2M

0

-1

ad a2M 34, aq2

34:

dZ

=

0

42

1

-dZ

dY

J

L

06)

Let [L] represent the matrix in Eq. (16). Solving for dP, , dq,, and dq, and dividing each by dZ yields ah a2(R + M) + (f’+g)a2@q;MJ-l aP, a4,aq2 dP,/dZ

=

9

(17)

PI

dq,/dZ

=

-- ah w

, PI

dq,/dZ

=

Observe that

PI = [, ~(f~+~)-~~][9a’(~~“)-a2~,~~~)~ -

--ah a*R +ag ( w aq: w

aZR >() a4,h2 I *

(18)

298

PAUL KOBRIN

Also, the numerators of Eqs. (17), (18), and (19) are respectively negative, positive, and negative.I5 Thus, those equations imply dP,/dZ < 0,

(20)

dq, /dZ ’ 0, dq,/dZ < 0.

(21) (22)

We distinguish the two possible casesdP, /dZ L 0 and dP, /dZ C 0. If DP, /dZ 10, it is apparent from the total differential of g divided by dZ, dg/dZ = (ag/i3P, ) dP,/dZ + (ag/aP, ) dP,/dZ that dg/dZ < 0 by eq. (20) and conditions (ii) and (iii). If dP,/dZ < 0, then df/dZ > 0 by condition (i). The total differential of Eq. (2) divided by dZ is dq,/dZ = df/dZ + dg/dZ. Given Eq. (22) and df/dZ > 0, this implies dg/dZ -c 0. This exhausts the possible cases. In all, dg/dZ < 0, dP,/dZ -C 0, &,/dZ > 0, and dq,/dZ < 0. Thus, relaxing the unleaded gasoline wholesale price ceiling reduces cheating and hence reduces pollution. B. Discussion This section presents a verbal discussion of the quantitative result derived from the retail price ceiling model. In terms of that model, the EPA’s assertion is that cheating (g) is positively related to the spread in nominal prices between unleaded and leaded gasolines (X - Pz). As this spread widens, cheating purportedly increases. We discuss, as a pedagogical example, a simple case where the nominal price spread widens but cheating decreases.We assumein this example that production of either grade does not affect the cost of producing the other grade, that is, condition (viii) holds as a strict equality. This assumption simplifies the discussion because the supply curves of the two types of gasoline are now independent. However, the leaded and unleaded demand curves ( f +, g and h, respectively) are still coupled (substitution in consumption remains). Suppose that the controlled price, X, is raised to X’ (which is still binding in the new equilibrium). Three things will be shown to result. First, the quantity of “Noting that C = R + M, those numerators are equal to the numerators of Eqs. (6), (8), and (9), respectively, which were shown to be negative, positive, and negative.

299

FUEL SWITCHING AND PRICE CONTROLS

D, D; = a

q1

h

f+g

9;

FIG. 3. (a) Unleaded gasoline. (b) Leaded gasoline.

unleaded gasoline supplied and consumed rises. Second,the price of leaded gasoline (P2) falls, widening the spread of nominal prices. Third, cheating (g) falls.‘6 This example, then, constitutes a counterexampleto the EPA’s assertion. In Fig. 3, the increase in the controlled price brings forth more unleaded output (q; versus q,) which has a lower shadow value (Pi’). This lower value of P, shifts the demand curve for leaded gasoline downward (through condition (ii)), which depressesPz. The lower Pz then shifts the unleaded demand curve downward (through condition (v)) and so on. At the end of the process,the demand curves are 0; and 0;.

From the right side of Fig. 3, it is apparent that P2 has fallen (as a result of the increasein X). Thus, the nominal price spread, X - Pz has risen to X’ - Pi. There are two ways to indicate that cheating declines. If “switching” implies only that owners of new cars use either leaded or unleaded gasoline depending on the price spread but that the total quantity of gasoline used in new cars is not affected by switching, then the increasein unleaded gasolineconsumption (see the left side of Fig. 3) must be accompaniedby a decreasein leaded gasoline consumption in new cars, that is, a reduction in cheating. The second demonstration of the decline in cheating is more general. From the right side of Fig. 3, the demand curve for leaded gasoline has shifted downward. Thus, at least one of its components-old car consumption (f) or cheating (g)- must fall. The decline in Pz increases old car consumption (by condition (i)) so cheating must decline. In this counterexample,then, the increasein X, the controlled price, has led to a decreasein g, cheating, although X - P2, the nominal price spread, has widened. The defect in EPA’s analysis (which led that agency m istakenly to expect dg/dX > 0) is that the controlled price is confused with a market clearing price. With an unchanged demand structure, an increase in the market clearing price corresponds to a decrement in quantity of unleaded demanded. And under the simple “switching” model, with fixed total consumption by new car owners, these owners merely substitute leaded gasoline for the decrement in unleaded gasoline. That is, cheating increases. ‘6The fiit and third results (dq, /dX > 0 and dg/dX < 0) were just proved. The second (dP, /dX < 0) results from Eq. (7) when condition (viii) holds as an equality.

300

PAUL KOBRIN

However, from Fig. 1, it is clear that an increase in the controlled price has the opposite effect- the quantity of unleaded gasoline supplied and consumed rises rather than falls. Hence, under the simple switching model, there must be a corresponding decreasein cheating. In sum, with effective controls, it is the quantity supplied which limits consumption, and an increase in such supply will allow more consumption of the environmentally more favorable good. A verbal justification of the wholesale price ceiling result is straightforward. Relaxing such a price ceiling brings forth more unleaded gasoline. Marketers, who had been earning a profit (Y) at the margin, willingly retail the higher quantity of unleaded gasoline. Of course, the higher quantity corresponds to a lower retail price. The lower price diminishes cheating. II. LIMITING

THE PRICE DIFFERENTIAL

In addition to favoring the maintenance of ceiling price controls, some have suggested a means to limit the price spread more directly: a maximum permissible price differential (differential control) [2]. We show that the effect of this policy cannot be ascertained without considerable further assumptions. The idea underlying controls over the price differential between leaded and unleaded gasolines is that there would be less use of leaded gasoline in converter equipped cars. We indicate that analytic inconsistencies arise when the spread control is modelled, suggesting that the model employed so far is not rich enough to generate a stable equilibrium. Alternative, superior models do not suggest themselves. We are thus left with a policy whose effect cannot be predicted without specification of a more complicated model (i.e., one incorporating further assumptions regarding the behavior of gasoline sellers and buyers). The imposition of an effective differential control prevents the two markets from clearing simultaneously. To demonstrate this, we reproduce the four equations of the model (with P, in place of X) and a fifth equation which gives the price differential, D. 41= wk 4>, 42 =fW + dp,, P, = ac/aq,, p2 = qaq, 9 D = P,- Pz.

a

There are four variables (P,, P2, ql, q2) and five equations. (D is a policy parameter, not a variable. The fifth relation holds as an equality in meaningful cases.)Thus, no solution generally exists in which both markets clear. One market or the other or both will not clear. There are only two possibilities in a market which fails to clear: excessdemand or excesssupply. There are two markets and three states for each: excess demand, excess supply and clearing. That makes nine cases. In addition, when the unleaded market experiences excessdemand, the differential control may or may not be binding. The total of 12 cases are shown in Table I. In case 12, both markets clear; the differential control is thus not binding. In the other eleven cases,at least one of the markets fails to clear. Which of these casesare candidates for a stable equilibrium?

301

FUEL SWITCHING AND PRICE CONTROLS TABLE I Unleaded marketed Case number 1 2 3 4 5 6 I 8 9 10 11 12

Excess SUPPlY

Excess demand

Control binding

Leaded market Market clears

X

Excess demand

EXWSS

SUPPlY

Market clears

X X

X X

X X X

X X

X X

X X

X

X

X X

X X

X X X X

X X X

Supposecase 11 initially prevailed. There the unleaded market clears but there is excess supply of leaded gasoline. This equilibrium m ight or m ight not be stable depending on what assumptionsare incorporated in the adjustment mechanism.It is entirely plausible that excesssupply in only the leaded market would bring down the nominal price of leaded gasoline despite the fact that the unleaded price, which is market clearing, would decline also (so as to maintain the specified differential). As both prices fall, case 8 may be reached. There, excess demand prevails in the unleaded market and the leaded market clears. Symmetry in the adjustment mechanism would then force both prices back up until the original equilibrium was reestablished.But there the cycle starts over again. Other examplesof instability or oscillation could be developed from the table. Rather than accept this bizarre result, it may be appropriate to dismiss the underlying model of price-quantity determination as inadequate and to admit a broader class of possibilities as potential solutions. By analogy, the quadratic X2 + X + 1 = 0 has no solution among the real numbers but roots will be found among the broader class of complex numbers. No systematic method of determining these alternative solutions is apparent. In fact, the broader solution spaceis not even obvious. However, one solution suggests itself and is worth describing although it is not contended that it will occur or that it is unique. All gasoline sellers may comply with the differential control but nominal prices may vary among sellers.For example, supposea binding 3$/gal. maximum differential is imposed. One set of sellers may post prices of 6Ot/gal. for leaded and 63$/gal. for unleaded while a second set posts 7O+/gal. and 73$/gal., respectively. The first set sell all of the leaded gasoline becausethe second set are unable to sell any. Regarding unleaded gasoline, the first set offer none for sale or a rationed, token amount which constitutes m inimal compliance with the requirement to offer that product. Purchasersare then forced to buy unleaded from the second set of sellers. The price of the latter is the real or shadow value of the product (which determines consumption). Thus, the real price spread is 13$ (73+-60$), not the

302

PAUL KOBRIN

nominal 3$. I7 This solution, then, drops the requirement for uniform prices across sellers, broadening the class of admissible solutions.” There are numerous other outcomes which can be imagined. For example, the manufacture of leaded gasoline may be discontinued, premiums may be offered with the purchase of leaded gasoline, etc. Without further specification of assumptions, however, the outcome of the policy is unpredictable. III. CONCLUSION We have proved that, under plausible conditions, lifting the ceiling price of unleaded gasoline induces less, not more, cheating because it reduces the shadow value of that product (Eq. (10)) while increasing its supply (Eq. (11)). Controlled prices and market (shadow) prices are different, and analyses which confuse them can reach erroneous results. We considered narrowing the differential by regulation. This led to results which are unpredictable without further assumptions regarding seller and buyer behavior. REFERENCES 1. R. J. Bowden, “The Econometrics of Disequilibrium,” North-Holland, Amsterdam (1978). 2. C. M. Ditlow III, Director, Center for Auto Safety, statement before the Economic Regulatory Administration, U.S. Department of Energy, Washington, D.C. (July 14, 1978). 3. J. Graaf, Rothbarth’s “virtual price system” and the Slutsky equation, Rev. Econ. Studies 15, 91-95 (1fTb

4. J. Hirshleifer, “Price Theory and Its Applications,” Prentice-Hall, Englewocd Cliffs, N.J. (1976). 5. E. Rothbarth, The measurement of changes in real income under conditions of rationing, Reu. Econ. Studies 8, 100-107 (1941). 6. N. Shutler, Deputy Assistant Administrator, U.S. Enviromnental Protection Agency, statement before the Federal Energy Regulatory Commission, U.S. Department of Energy, hearing on Gasoline Price Decontrol (November 30, 1977). 7. U.S. Department of Energy, Draft environmental impact statement on motor gasoline deregulation (November 1978).

“This situation might evoke a regulatory response intended as a corrective. Still other, even less predictable reactions would then occur. ‘*Present price differences across sellers reflect differences in service, location, etc. The text refers to additional differences caused by the maximum spread constraint.