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Intertemporal resource allocation: distributive issues surrounding gasoline price hikes.

I. INTRODUCTION

When Iraq invaded Kuwait in August 1990, millions of barrels of daily oil production suddenly dried up. Because of delivery lags, however, this production shortfall did not actually materialize in downstream markets for several weeks. Knowing this, many consumers expressed outrage when retail gasoline prices rose immediately. They concluded that oil companies and service stations were exploiting the situation to make enormous windfalls by charging high prices for gasoline they had bought (or produced) at relatively low cost prior to the crisis. Talk in many quarters turned to collusion, and the Department of Justice, The Federal Trade Commission, several Congressional committees, and a majority of states' attorneys general began looking into antitrust violations.

A few media outlets and some economists pointed out the need for higher prices to encourage conservation. Reallocating supplies from the present to the future (where they would be more highly valued) would enhance efficiency. Furthermore, they claimed, the oil market may or may not have vestiges of monopoly, but no evidence of this could be garnered from the quick adjustment of prices. Regardless of market structure, anticipation of future price hikes often causes prices to rise immediately as arbitragers buy low and sell high.

Yet in the heat of the moment consumers did not seem to care about efficiency. They saw oil companies raking in profits selling gas at prices far exceeding their original costs, and they knew that these windfalls came at their expense. Nor did they seem to care about consumer surplus, which arguably could also be higher if quantities were reallocated from the present to the future. Their outrage focused upon a quick price adjustment that seemed to imply higher energy bills than a slower adjustment would have. Even those who advocated a rapid price increase often apologized for its distributive consequences. The Bush Administration pleaded for restraint, and oil companies cooperated, presumably from fear of public relations consequences or possible antitrust indictments.

The sad truth, however, was that whether consumers paid up front or down the road, they nonetheless paid. The blame for oil company windfalls lies not with the sudden price increase but rather with the supply reduction that gave rise to this increase. Failure to attach blame appropriately led to an improper assessment of distributive consequences and potentially disastrous policy prescriptions. Ironically, raising the price more quickly would in all likelihood have saved consumers money and increased their total surplus.

The issue can also be viewed another way. Suppose there is a fixed supply of gasoline. Oil companies can allocate it by charging a low price initially, thereby making supplies plentiful at the outset. If they do so, however, they will charge a higher price later when gas is far more scarce. How would oil companies allocate supplies so that they earn the maximum profits? How would we instruct them to allocate supplies if we want the market to clear and consumers to pay the least amount possible? How would we instruct them to allocate supplies so that consumers earn the maximum surplus? As it turns out, profits, consumer expenditures, and consumer surplus all depend upon demand elasticities and the rate at which these elasticities change when supplies rise and fall. Oil companies might prefer a slow price adjustment and consumer expenditures could be lowest and surplus highest if price adjusts immediately. Indeed, given the facts outlined below, consumer advocates would probably have served their constituency best had they prescribed a quick price adjustment and oil companies in all likelihood profited from their restraint.

II. TWO EXAMPLES: THE IMPORTANCE OF DEMAND ELASTICITIES

Consider the supply of gasoline in a hypothetical country. For simplicity, assume that production is perfectly inelastic: regardless of price, the country's domestic oil companies costlessly produce [X.sup.D] per month and Middle East suppliers provide [X.sup.M]. (1) Suddenly, however, Middle East production is temporarily disrupted. Due to delivery lags, the country has one month advanced notice that [X.sup.M] units of gasoline normally supplied from the Middle East will suddenly not be available. Let X = 2[X.sup.D] + [X.sup.M] represent the total two-period supplies. The two examples here illustrate the ramifications of the hoarding that may accompany this supply disruption, both during the period in which Middle East supplies dry up (t = 2) and in the period preceding it (t = l). In both examples [X.sup.D] = 3 and [X.sup.M] = 2, where quantities are expressed in millions of gallons per month. For simplicity, assume for now that Middle East suppliers sell all of their output upon delivery.

Although domestic production does not respond to price changes, individual oil companies may not supply consumers with all the gasoline produced each period. In particular, they may withhold some first-period production for sale in the second period. In doing so, they force the first-period price higher but by easing the second-period shortfall they drive the second-period price lower than it otherwise would be. This practice is widely observed and comes under many names, including arbitrage, price-gouging, hoarding, stabilizing or destabilizing speculation (where conditions are uncertain), and conservation. Let [Q.sup.D.sub.t] be the time-t sales of domestic oil companies, where [Q.sup.D.sub.1] + [Q.sup.D.sub.2] = 2[X.sup.D], and let [Q.sub.1] = [X.sup.M] + [Q.sup.D.sub.1] and [Q.sub.2]=[Q.sup.D.sub.2] represent total sales. The next two subsections discuss two types of demand.

Example 1: Linear Demand

Suppose inverse demand in each period is as follows (See Figure 1):

(1) [P.sub.t] = A-B[Q.sub.t],

[FIGURE 1 OMITTED] where [P.sub.t] is denominated in millions of gallons (e.g., [P.sub.t] = 1 means $1 million per one million gallons, or simply $1 per gallon). As with all linear demand curves, the demand elasticity, [eta], rises as quantity fails:

[eta] = (d[Q.sub.t]/d[P.sub.t])(Pt/[Q.sub.t]) = -(1 / B)([P.sub.t] / [Q.sub.t])

= [A/(B[Q.sub.t])] - 1

Consumer surplus (C[S.sub.t]) equals the area above price but below the demand curve. Hence, C[S.sub.t] = (1/2)(A - [P.sub.t])[Q.sub.t]= (B/2)[Q.sup.2.sub.t]. Assuming costless production, producer surplus ([[PI].sub.t]) equals consumer expenditures. Hence, [[PI].sub.t] = [P.sub.t][Q.sub.t] = A[Q.sub.t] - B[Q.sup.2.sub.t]. Let [[PI].sup.D.sub.t] represent domestic oil companies' profits. Since Middle East producers supply no time-2 output, [[PI].sup.D.sub.2] = [[PI].sub.2]. At time t = l, however, domestic oil companies split the producer surplus with Middle East suppliers in proportion to their market shares.

Suppose A = 9.33 and B = 1.66. (2) Then [eta] = 0.12 when [Q.sub.t] = 5; [eta] = 0.4 when [Q.sub.t] = 4; and [eta] = 0.87 when [Q.sub.t] = 3. With [Q.sub.t] = 5 initially, a two-million gallon drop in quantity causes a seven-fold increase in the demand elasticity, and a one-million gallon drop results in a tripling of the elasticity. In short, though demand is inelastic, quantity reductions of these magnitudes generate dramatic changes.

As a first scenario, suppose domestic oil companies do not withhold any first-period production. With no output disruption and no hoarding, first-period supplies are plentiful: [Q.sub.1] = 5 and [P.sub.1] = $1. Hence, [[PI].sub.1] = $5 million, [[PI].sup.D.sub.1] = $3 million, and C[S.sub.1] = $20.8 million. This scenario is summarized in the first row of Table I.
TABLE I
Linear Demand with A = [??] and B = [??]

 Price/ [P.sub.t][Q.sub.t]
 Gallon

[Q.sub.t] = 5 ([eta] = 0.12): $1.00 $5 million
 No Disruption
 No Hoarding

[Q.sub.t] = 4 ([eta] = 0.40): $2.67 $10.67 million
 Hoarding [right arrow]
 [Q.sub.1] = [Q.sub.2] = 4

[Q.sub.t] = 3 ([tau] = 0.87): $4.33 $13 million
 Disruption
 No Hoarding

 Price/ [[pi].sup.d.sub.t]
 Gallon

[Q.sub.t] = 5 ([eta] = 0.12): $1.00 $3 million
 No Disruption
 No Hoarding

[Q.sub.t] = 4 ([eta] = 0.40): $2.67 [[pi].sup.D.sub.1] =
 Hoarding [right arrow] $5.33 million
 [Q.sub.1] = [Q.sub.2] = 4 [[pi].sup.D.sub.2] =
 $10.67 million

[Q.sub.t] = 3 ([tau] = 0.87): $4.33 $13 million
 Disruption
 No Hoarding

 Price/ C[S.sub.t]
 Gallon

[Q.sub.t] = 5 ([eta] = 0.12): $1.00 $20.8 million
 No Disruption
 No Hoarding

[Q.sub.t] = 4 ([eta] = 0.40): $2.67 $13.3 million
 Hoarding [right arrow]
 [Q.sub.1] = [Q.sub.2] = 4

[Q.sub.t] = 3 ([tau] = 0.87): $4.33 $7.5 million
 Disruption
 No Hoarding


But if oil companies withhold no first-period production, there are only three million gallons available for second-period consumption. The price rises to $4.33, consumer expenditures and producer surplus rise to $13 million, and C[S.sub.2] drops to $7.5 million. This outcome is summarized in the third row of Table I.

Contrast this with a second scenario under which oil companies hoard one million gallons of their first-period output. This implies only four million gallons available for first-period consumption. The price immediately jumps to $2.67, so that despite the drop in sales consumer outlays more than double to $10.67 million and consumer surplus drops to $13.33 million. From a short run perspective, oil companies and Middle East suppliers appear to profit at consumers' expense.

But the story does not end here. The hoarded gas does not evaporate but rather reappears one period later. The three million gallons of domestically produced gasoline at time t = 2 are supplemented by one million gallons withheld from the market at the outset. As such, the second-period price remains at $2.67--in contrast to the $4.33 consumers would have paid had oil companies not engaged in price-gouging at the outset. Hoarding caused the first-period price to rise by $1.67 but reduces the second-period price by the same amount. Second-period consumption is one third higher with hoarding than without it, yet consumers' second-period outlays are $5.8 million less. This information is also summarized in the second row of Table I.

As is apparent from Table I, the gains and losses associated with hoarding are not entirely offsetting--except for domestic oil companies. Hoarding lowers first-period consumer surplus by more than it raises second-period surplus, and aggregate consumer expenditures are higher with hoarding than without it. Ironically, all of the additional first-period producer surplus accrues to Middle East suppliers--the price of their first-period supplies rises by 167%. Domestic oil companies also enjoy a higher price, but they reduce their initial sales by one million gallons. More important, by raising second-period supplies these domestic oil companies receive a second-period price of $2.67 rather than $4.33, and it is at this time that they sell most of their output. Undiscounted domestic oil company profits are exactly the same with hoarding and without it. If we assume a zero interest rate (an appropriate simplifying assumption considering the few short weeks between periods), then on the basis of this demand curve and these short run considerations hoarding helps Middle East suppliers, hurts customers, and leaves oil company profits unaffected.

These conclusions are not inconsistent with the enormous windfalls earned by domestic oil companies. Although oil companies' profits balloon, none of this is attributable to hoarding per se. Moreover, the overall impact of hoarding is remarkably small. Despite enormous price and quantity differences under the two scenarios, total (undiscounted) consumer expenditures are only 16% higher with hoarding than without it, hoarding reduces aggregate consumer surplus by only 6%, and hoarding offers no benefits to oil companies. In short, the effects are of a much smaller order of magnitude than the changes in prices and quantities. It makes much less difference than consumers may think whether they absorb the impact of the anticipated supply shock immediately or wait to bear its full consequences in the second period. Either way they feel its full force.

Example 2: Constant Demand Elasticity

All linear demand curves have the property that demand elasticities rise as quantities fall, and given the magnitude of the supply shock in the first example the elasticity changes are especially dramatic. This rising elasticity helps to blunt the impact of the supply shock. If the elasticity did not rise, consumers might suffer even more as a consequence of any disruption in supplies and hoarding may have different implications for consumer and producer surplus. To investigate this possibility, consider the following demand curve with a constant demand elasticity of [eta]:

(2) [Q.sub.t] = [gamma][P.sup.-[eta].sub.t].

Suppose also that if the price of gasoline rises above [bar.P] then consumers costlessly switch to some alternate fuel (e.g., ethanol) that is available in infinitely elastic supplies. (3) This gives rise to the second demand curve in Figure 1.

To illustrate, suppose [gamma] = 5, [eta] = 0.40, and [bar.P] = 8. (4) Table II provides information about this demand curve under the "hoarding" and "no hoarding" scenarios in a manner analogous to Table II. (5) The tables have several similarities. For instance, when supplies are plentiful ([Q.sub.t] = 5), price equals $1 in both cases and consumer and producer surplus are the same. However, the demand elasticities are equal only where [Q.sub.t] = 4. Elsewhere they vary markedly.
TABLE II
Constant Elasticity ([eta] = 0.4) with [gamma] = 0.5
and [bar.P] = 8

 Price/ [P.sub.t][Q.sub.t]
 Gallon

[Q.sub.t] = 5: $1.00 $ 5 million
 No Disruption
 No Hoarding

[Q.sub.t] = 4: $1.74 $7.0 million
 Hoarding [right arrow]
 [Q.sub.1] = [Q.sub.2] = 4

[Q.sub.t] = 3: $3.59 $10.8 million
 Disruption
 No Hoarding

 [[pi].sup.D.sub.t]

[Q.sub.t] = 5: $3 million
 No Disruption
 No Hoarding

[Q.sub.t] = 4: [[pi].sup.D.sub.1] = $3.5 million
 Hoarding [right arrow] [[pi].sup.D.sub.2] = $7.0 million
 [Q.sub.1] = [Q.sub.2] = 4

[Q.sub.t] = 3: $10.8 million
 Disruption
 No Hoarding

 C[S.sub.t]

[Q.sub.t] = 5: $20.7 million
 No Disruption
 No Hoarding

[Q.sub.t] = 4: $17.4 million
 Hoarding [right arrow]
 [Q.sub.1] = [Q.sub.2] = 4

[Q.sub.t] = 3: $11.1 million
 Disruption
 No Hoarding


As is apparent from Table II, hoarding now lowers first-period consumers' surplus by less than it raises their second-period surplus, and total expenditures are lower with hoarding than without. The increase in first-period producer surplus falls short of the drop in second-period surplus, and oil company profits drop markedly. With a zero interest rate this demand curve implies that hoarding helps Middle East suppliers, helps customers, and hurts oil companies. As before, the total impact of hoarding remains far smaller than the wild price and quantity swings would indicate.

III DISCUSSION

The two previous examples suggest that the welfare consequences of hoarding depend upon demand elasticities--and even more important, the rate at which these elasticities change when quantities rise and fall. Although linear and constant elasticity demand curves represent special cases, it is nonetheless interesting to see whether the conclusions from the numerical examples generalize. Proposition 1 shows that the distributive properties outlined in the first example hold true for all linear demand curves (see the Appendix for a formal proof):

PROPOSITION 1: With linear demand and a zero interest rate, it follows that:

i) consumer surplus is minimized when [Q.sup.D.sub.1] = [X.sup.D] - ([X.sup.M]/2), i.e., when [Q.sub.1] =[Q.sub.2] and [P.sub.1] = [P.sub.2];

ii) producer surplus and consumer expenditures are maximized when [Q.sup.D.sub.1] = [x.sup.D] - ([X.sup.M]/2);

iii) domestic oil companies' profits are the same when [Q.sup.D.sub.1] = [X.sup.D]- ([X.sup.M]/2) as they are when [Q.sup.D.sub.1] = [X.sup.D]; and

iv) domestic oil companies' profits are maximized when [Q.sup.D.sub.1] = [X.sup.D] - ([X.sup.M]/4), i.e., when [X.sup.D] + [X.sup.M] > [Q.sub.1] >[ Q.sub.2] and [P.sub.1] <[P.sub.2]'

When oil companies (or others) hoard first-period output to take full advantage of any difference between first- and second-period prices, they allocate their two-period output so as to minimize consumer surplus, maximize consumer expenditures, and maximize producer surplus. Yet oil companies receive no more profits when they equalize total supplies in this way than they do when they sell all output as it is produced, i.e., they receive the same profits when they hoard ([Q.sup.D.sub.1] = [X.sup.D]- [X.sup.M]/2) and when they do not [Q.sup.D.sub.1] = = [X.sup.D]).

The last part of Proposition 1 explains why this outcome provides no benefit to domestic producers. If they could collude, domestic oil companies would indeed hoard: they would maximize their collective profits by selling [X.sup.M]/4 units less than they produce in the first period. But they get too much of a good thing. When [Q.sup.D.sub.1] = [X.sup.D] - ([X.sup.M]/4), arbitrage opportunities remain unexploited (i.e., [P.sub.2] > [P.sub.1]) and individual oil companies or other arbitragers have incentives to cut back further on their first-period sales. But in doing so they increase the supply of gasoline in the second period still further, reduce the second-period price, and thereby eliminate any profits other domestic oil companies might have gotten from hoarding themselves. By the time arbitragers exploit all profit opportunities the oil companies are no better off than if they had not hoarded at all.

Proposition 2 shows that the second numerical example also generalizes:

PROPOSITION 2: With constant demand elasticity and a zero interest rate, it follows that:

i) consumer surplus is maximized when [Q.sup.D.sub.1] = [X.sup.D] - ([X.sup.M]/2), i.e., when [Q.sub.1] = [Q.sub.2] and [P.sub.1] = [P.sub.2]; and

ii) producer surplus and consumer expenditures are minimized when [Q.sup.D.sub.1] = [X.sup.D] - ([X.sup.M]2).

Proposition 2 holds for any 0 < [eta] < [infinity]. In contrast to Proposition 1, hoarding now maximizes consumer surplus and minimizes producer surplus. Consumers prefer a rapid price adjustment and oil companies prefer a slow one.

Virtually all motorists can attest to the inelasticity of gasoline demand simply by observing that in scenarios such as this their total expenditures have increased sharply when gasoline prices rise. More fundamentally, demand does not appear to become appreciably more elastic at higher prices, at least not over the range of prices and time intervals described here. Gasoline expenses comprise a relatively small fraction of total driving costs, so if the supply disruption is short, motorists will not typically react by trading in their cars for more fuel-efficient models, by moving closer to work, or by converting to alternate fuels. This does not imply that motorists ignore price increases. On the contrary, they may reduce discretionary driving, consolidate shopping trips, check tire pressures, and take other measures to economize. They simply recognize both the limits to such measures and the modest and very transitory benefits, and in doing so they make no dramatic changes in their short run behavior. Across many commodities, including gasoline, demand elasticities respond far less to short-lived supply disruptions than they do to longer run changes. The frequency with which economists quote the elasticity of demand for gasoline irrespective of price confirms that they believe this elasticity is relatively unaffected by modest price changes. In short, it appears that the constant demand elasticity example--with all of its welfare implications--represents a reasonable description of real-world conditions. In the wake of Iraq's invasion of Kuwait, hoarding helped Middle East suppliers and consumers, and it hurt oil companies.

In light of this conclusion, why would oil companies hoard gasoline? Why would they engage in a practice that operates against their collective interest? The answer lies with competition. Suppose oil companies collude to keep first-period prices low, thereby allowing them to raise second-period prices all the more dramatically. In other words, suppose they agree not to hoard. Then the second-period price exceeds the first-period price and individual oil producers are tempted to hoard even though this is not in the industry's collective best interests. Ironically, oil companies may not only "cave in" to the pleas of government officials and consumer advocates who want them to exercise first-period price restraint, they may welcome these pleas, along with any accompanying penalties for "price-gouging."

Moreover, even if oil companies collude to keep first-period prices low and second-period prices high, other parties could sabotage the agreement. Middle East suppliers could withhold their first-period output. Retail service stations could also hoard. Other arbitragers could buy gasoline at the outset, store it, and then sell at the higher second-period price. And consumers could "top off" their gas tanks in the first period to take advantage of any price differential. This suggests that oil companies' efforts to "exercise restraint" can at best be only modestly successful, and any success whatsoever depends upon all of the following factors: i) Middle East suppliers must not hoard; ii) service stations must also anticipate sanctions for hoarding; iii) other arbitragers must lack the inexpensive storage facilities needed to profitably buy low and sell high (or they must face some other constraint); and iv) motorists must have only limited capacity (or patience) for topping off their tanks. Otherwise, equilibrium prices will equalize across the periods regardless of the supply decisions oil companies make.

IV. CONCLUSIONS

Consumers and their (alleged) advocates have confused the windfalls oil companies earn as a consequence of the cutoff of Middle East supplies and the profits they generate through hoarding and "price-gouging." This confusion is understandable: although the immediate consequences of hoarding are readily apparent even to the most unsophisticated observer, the subsequent effects are less obvious. Consumers see the first-period price increases that arise with hoarding but often do not realize that second-period prices are significantly lower as a result. This paper clears up the confusion by showing how gasoline that is hoarded at the outset reappears one period later to ease the anticipated supply disruption. Oil companies' profits may skyrocket, but these profits have little to do with hoarding per se. The blame for oil company windfalls lies not with sudden price increases but rather with the supply reduction that gave rise to these increases. The sad truth is that consumers foot the bill whether or not the price of gasoline rises quickly.

It may be tempting to conclude from the examples here that efforts to restrain price increases have had few harmful effects. Whether or not hoarding occurs, the aggregate effect on welfare is far less than the wild price and quantity swings would suggest, especially since the demand for gasoline is so highly inelastic. Given the short time periods involved in the recent Middle East crisis (other OPEC producers made up the production shortfall within weeks), given the relatively modest changes in supply, and considering the extent of consumer outrage, even those policymakers who fully appreciated the advantages of free market pricing must have figured that the benefits were insufficient to justify the political cost of supporting it.

At the same time, however, the price controls, windfall profits taxes, and other regulations that followed previous energy crises had far reaching implications for economic efficiency. Few economists would advocate repeating these errors even if a Middle East war had caused drastic and permanent supply reductions. Yet the reaction of the press, Congress, the Bush Administration, and various states' attorneys general had much the same effect as price controls and started us down the slippery slope toward a repetition of past mistakes. Afraid of bad publicity and antitrust litigation, oil companies tried with only limited success to keep prices low. Unsatisfied with these efforts and unswayed by efficiency arguments in favor of price-induced conservation, many people expressed outrage and often called for more aggressive efforts to reign in oil company profits. By showing that hoarding by itself in all likelihood helped consumers and hurt oil company profits, this paper may blunt some of these calls for a return to the failed policies of the past.

APPENDIX

PROOF OF PROPOSITION 1: Since C[S.sub.t]-(B/2) [Q.sup.2.sub.t], it follows that

C[S.sub.1] + C[S.sub.2] = (B/2)[[Q.sup.2.sub.1] + [(X - [Q.sub.1]).sup.2]].

Hence, the first and second derivatives are as follows:

d(C[S.sub.1] + C[S.sub.2])/d[Q.sub.1] = B[2[Q.sub.1] - X]

and

[d.sup.2] (C[S.sub.1] + C[S.sub.2])/d[Q.sup.2.sub.1] = 2B

It follows that consumer surplus reaches a minimum where [Q.sub.1] = [Q.sub.2] = X / 2.

Similarly, since [P.sub.t][Q.sub.t] = A[Q.sub.t]- B[Q.sup.2.sub.t], it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[d.sup.2] ([P.sub.1][Q.sub.1] + [P.sub.2][Q.sub.2])/d[Q.sup.2.sub.1] + -4B.

Hence, consumer expenditures and producer surplus are maximized at [Q.sub.1] = X/2.

Domestic oil companies" profits are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking first and second derivatives gives

d([[PI].sup.D.sub.1] + [[PI].sup.D.sub.2] / d[Q.sup.D.sub.1] = 4B ([X.sup.D] - [Q.sup.D.sub.1]) - B[X.sup.M]

and

[d.sup.2] ([[PI].sup.D.sub.1] + [[PI].sup.D.sub.2] / d[Q.sup.D2.sub.1] = -4B. d2(FI1D * II~) / dQD2 = -48.

Hence, oil companies' profits reach a maximum where [Q.sup.D.sub.1] = [X.sup.D] - (X.sup.M]/4)).

Oil company profits when [Q.sup.D.sub.1] = [Q.sup.D.sub.2] = [X.sup.D] (i.e., with no hoarding) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Compare this outcome to oil company profits when [Q.sup.D.sub.1] = [X.sup.D] - ([X.sup.M]/2), i.e., when oil companies hoard to the point where [Q.sub.1] = [Q.sub.2] and [P.sub.1] = [P.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A comparison of these profits immediately reveals that oil company profits are the same when [Q.sup.D.sub.1] = [X.sup.D]-([X.sup.M]/2) as they are when [Q.sup.D.sub.1] = [X.sup.D].

PROOF OF PROPOSITION 2: Begin by rewriting equation (2) as follows:

(A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [alpha] = -(l/[eta]), [beta] = [[gamma].sup.-[alpha]], and [Q.sub.0] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Regardless of [eta], it follows that consumer surplus reaches a maximum where [Q.sub.1] = [Q.sub.2] = x/2.

Similarly, by (A1), [P.sub.t][Q.sub.t] = [beta][Q.sup.[alpha]+1.sub.t], so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, consumer spending and producer surplus are minimized at [Q.sub.1] = X/2.

(1.) Since the discussion revolves around the inter-temporal allocation of a fixed level of production, all results go through unchanged if production is costly, except that profits in each period would fall by a fixed amount.

(2.) These parameter values are chosen to simplify comparisons with the constant elasticity of demand example that follows.

(3.) Without such an assumption consumer surplus is unbounded.

(4.) Estimates of the short run demand for gasoline vary widely, but it is not unusual to see estimates below 0.1 and it is rare to see estimates above 0.4. In other words, this example offers a demand elasticity at the far upper range of estimates.

(5.) See the proof of Proposition 2 for a computation of consumer surplus.

DAVID A. BUTZ, Assistant Professor, University of California, Los Angeles. I would like to thank Bill Allen, Robert Moore and two anonymous referees for useful suggestions.
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Author:Butz, David A.
Publication:Economic Inquiry
Date:Jul 1, 1991
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