Environmental pollution, congestion, and imperfect competition on the car market Fran¸cois Combes March 7, 2006

Abstract The objective of this work is to study the way the fuel efficiency of cars is determined, and the impact of regulation tools on it. First, we describe the way car users choose the distance they drive. Then we study the car market : we assume it is an oligopolistic market. This framework allows us to assume car manufacturers compete both in price and in efficiency. At each of these steps, we study the impact of two different externalities, emissions or congestion, and the way to optimally regulate this market. One result of this study is that when there are two externalities (emissions and congestion) and when car manufacturers compete through two variables (car price and car efficiency), then using only an excise on gas is not sufficient to reach a social optimum. Furthermore, it is possible that when gas is heavily taxed, cars efficiency is higher than its socially optimal level. The intuition behind this result is that, as far as drivers do not take into account externalities, a tax on gas is necessary to have them take the socially optimal decisions. But car manufacturers will increase the fuel efficiency of the car they make, thus evicting a part of the effect of the fuel tax. A mean to correct that is to limit the efficiency of cars made by car manufacturers.

L’objectif de ce travail est l’´etude de la mani`ere dont la consommation d’essence des voitures est d´etermin´ee, ainsi que l’impact sur cette variable des outils de r´egulation. Nous d´ecrivons d’abord la mani`ere dont les usagers de v´ehicules choisissent la distance qu’ils parcourent. Nous ´etudions ensuite le jeu du march´e automobile, que nous supposons oligopolistique. La concurrence y est donc imparfaite. Ce cadre th´eorique nous permet de supposer ais´ement que les industriels se concurrencent simultan´ement en prix et en efficacit´e. Un r´esultat de notre ´etude est que sous l’effet de deux externalit´es (pollution et congestion), quand les industriels se concurrencent en prix et en efficacit´e, il est impossible d’atteindre l’optimum social uniquement en taxant l’essence. De plus il peut arriver, lorsque l’essence est lourdement tax´ee, que l’efficacit´e des voitures `a l’´equilibre soit plus ´elev´ee que son niveau socialement optimal. L’intuition de ce r´esultat est que si l’on taxe l’essence pour internaliser les coˆ uts sociaux de la conduite en voiture, les industriels, par le jeu du march´e, vont fabriquer des voitures plus efficaces, permettant aux usagers de conduire plus, et donc ´evin¸cant une partie de l’effet escompt´e de la taxe. Un moyen de corriger cet effet est de limiter l’efficacit´e des voitures fabriqu´ees par les industriels.

Contents 1 Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling distance choice 2.1 Framework of the model . . . . . . 2.2 Choice of distance with research . 2.3 Choice of distance without research 2.4 conclusion . . . . . . . . . . . . . .

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3 Competition in price on the car market 3.1 Modeling the car market . . . . . . . . . 3.2 Free-entry equilibrium . . . . . . . . . . 3.2.1 Short term equilibrium . . . . . . 3.2.2 Long term equilibrium . . . . . . 3.2.3 Welfare analysis . . . . . . . . . 3.2.4 Regulation . . . . . . . . . . . . 3.3 Environmental pollution . . . . . . . . . 3.3.1 Welfare analysis . . . . . . . . . 3.3.2 Regulation . . . . . . . . . . . . 3.3.3 Graphical analysis . . . . . . . . 3.4 Congestion . . . . . . . . . . . . . . . . 3.4.1 Free-entry equilibrium . . . . . . 3.4.2 Welfare analysis . . . . . . . . . 3.4.3 Regulation . . . . . . . . . . . . 3.4.4 Graphical analysis . . . . . . . . 3.5 conclusion . . . . . . . . . . . . . . . . .

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4 competition in price and efficiency on the car market 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Equilibrium in efficiency without externalities . . . . . . 4.2.1 Price equilibrium . . . . . . . . . . . . . . . . . . 4.2.2 Efficiency equilibrium . . . . . . . . . . . . . . .

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4.2.3 Free-entry equilibrium . . . . . . . . . . . . . . . 4.2.4 Welfare analysis . . . . . . . . . . . . . . . . . . Equilibrium in efficiency with emissions . . . . . . . . . 4.3.1 Welfare analysis . . . . . . . . . . . . . . . . . . 4.3.2 Regulation with environmental pollution . . . . . Equilibrium in efficiency with emissions and congestion 4.4.1 Equilibrium characterization . . . . . . . . . . . 4.4.2 Welfare analysis . . . . . . . . . . . . . . . . . . 4.4.3 Regulation with one excise tax on gas . . . . . . 4.4.4 Numerical example . . . . . . . . . . . . . . . . . 4.4.5 Regulation with two taxes . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .

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46 47 48 48 50 51 51 54 58 60 62 64

5 Conclusion 66 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 Limitations and possible extensions . . . . . . . . . . . . . . . 67

3

Chapter 1

Introduction 1.1

Introduction

Regulating the car market is a particular problem. On the one hand, this market can indeed be considered as a classical oligopol, where a limited set of firms propose their products and take into account the preferences of their customers to decide their prices, designs, qualities, and so on. It is in this respect a matter of industrial economics. On the other hand, it is strongly linked to the transport specific problems, since many features of cars are valuable for the customers because they will use them to drive. It is for that reason a matter of transport economics, the problematics of which are characteristic. The need of people for personnal means of transport is obvious and cannot be ignored, but car users take decisions that are far from being socially optimal. They do not take into account environmental pollution (at least not directly), and they only consider congestion as the inconvenience other drivers cause them, not at the inconvenience they cause to others. Furthermore, car manufacturers, unless forced to do so, take nothing into account but the wishes of their potential customers, with their only aim being the optimisation of their profit. They want to improve their market power, so as to increase the loyalty of their customers towards themselves in order to draw higher margins. The regulator is not able here, as in some situations (passenger transport by train in France for example), to plan or design himself cars, prices, distances driven by car users, and so on. Only through regulation tools is it possible to consider improving the social welfare. The efficiency of cars, that is to say the amount of gas they consume per kilometer, is an important decision variable for car manufacturers, since an increase of the efficiency of a car results in potentially important savings for car users, and thus makes the car more attractive. Car manufacturers may indeed draw some important margins from customers who drive long 4

distances and are ready to pay more for more efficient cars. It is for example empirically shown (Verboven 2002) that the difference between the prices of gasoline and diesel cars in Western Europe is much more due to the market power of firms on a particular set of customers than to the difference between the production costs of the corresponding car models. This market power follows indeed from the fact that some car users, who drive a lot, are ready to pay more for a car that consumes less. In the end, we can consider that the efficiency of cars is the result of a trade-off made by car manufacturers between the resulting production costs and attractivity on the car users. Efficiency is also a major concern for the regulator for many reasons. Nowadays, for several reasons, the public and political opinion is that cars must be as efficient as possible. Concerns about sustainable development, global warming and the more or less predicted depletion of the world fossile energies even lead people to think that gas engines ought to disappear shortly, replaced by electrical engines (or at least hybrid engines) for the sake of all. In this climate, car manufacturers are asked to make even more efficient cars (for example the CAFE norms in the U.S.A, (Harrington & McConnell 2003)), whereas they already are induced to make very efficient cars because of the high price of gas. The objective of our work is to demonstrate that the intuition according to which increasing the efficiency of cars is welfare improving is not necessarily true, especially when gas is already heavily taxed. A change in the efficiency of cars has a lot of effects, which are closely related. Our work will focus on three of them. First, larger efficiency allows car users to drive larger distances, resulting in larger levels of congestion. Second, larger efficiency results in lower gas consumption, so certainly in lower environmental pollution (for example, the quantity of CO2 emitted by a car in the atmosphere is more or less equivalent to the quantity of carbon in the fuel consumed, so a higher efficiency directly induces lower emission levels). Third, larger efficiency is achieved through the use of more expensive technologies necessarily preceeded by expensive research and development. The regulator, in order to improve the social welfare, has to carefully take these effects into account and balance them. The theoretical analysis we lead here may give some lightings on that question.

1.2

Framework

Our objective is to provide a simple theoretical frame where car manufacturers compete both in price and in efficiency, so that we may explore the effects of different regulations. Since, as exposed at the beginning of the introduction, the problem studied is both a matter of industrial economics and transports economics, we will use tools from both fields. 5

First, we set in chapter 2 an underlying microeconomic model predicting the expected distance driven by car users. We provide a framework where the expected distance covered by car users depends naturally and directly from the price of gas and the efficiency of their cars. We will expose two possible behaviours of the car users, each leading to very similar results. We will not model complicated exploring behaviours, as this is not the purpose of this work. Then, we assume in chapter 3 that the efficiency of cars is fixed. As a consequence, car manufacturers only compete in price. We will then study the effect of two externalities : environmental pollution and congestion. This allows us to come accross some classical results of transport economics. The fact that these results hold makes sure the model is not inconsistent. Finally, we assume in chapter 4 that car manufacturers compete both in price and in efficiency. Introducing first emissions next congestion, we will be able to draw some interesting conclusions about the problem of regulation of the car market.

6

Chapter 2

Modeling distance choice 2.1

Framework of the model

In order to modelize the way drivers react to prices changes, we will set up the framework in which these drivers evolve. We make a number of assumptions. First we assume that there are N drivers, thereafter indexed by k, real number between 0 and N . Second we assume that the universe of these drivers is a circular road of length L, on which their homes are uniformately distributed. On this road are located activities, with density δ. These activities are generic. They can refer to anything that is not in the homes of the drivers and that they want to reach (school, work, week-end, park, shopping, and so on). For example, if drivers all drive distance d, then the density of vehicles on the road is (N · d)/L vehicles per kilometer. But we also assume that this road is long enough to be of infinite length from the drivers’ point of view. Although drivers and activities should be indexed by discreet variables, we will use from now on continuous variables. We also assume drivers can only explore the part of the road located to the right of their homes (without loss of generalities thanks to the symmetries of the model). Figure 2.1 and 2.2 illustrate respectively the first and the second points of view. We must now modelize the way drivers choose the distance they drive. They can access a number of activities located on the road, but they have to drive to reach them, and that implies a cost. We consider that this cost is proportional to the distance covered. For each kilometer, car users consume some gas, the corresponding expense per kilometer is equal to pG ·θ, where pG is the cost of fuel per liter and θ the efficiency of the car in liter per kilometer. Using a car implies other operating costs like insurance, maintenance, inconvenience due to the time spend in the car and so on. We assume these other operating costs are proportional to the distance covered, and equal to pf . The generalised cost of covering distance d is then modelized 7

density of activites : į

length of the road : L

Figure 2.1: Model from a global point of view density of activites : į +d

-d length of the road : L

Figure 2.2: Model from a driver’s point of view by a function d 7→ c(d). c(d) = (pG θ + pf )d

(2.1)

Because of these costs drivers will be reluctant to cover large distances. They will choose the activity they want to reach taking into account the distance they have to drive to reach it. The way they choose it can be modelized from different ways, we will focus on two of them : with or without research.

2.2

Choice of distance with research

We here assume that drivers do not know the value of the different activities, so they have to parse them before choosing. We assume they decide a total distance d before leaving. First, they parse the first half of this limit distance. In the meanwhile, they observe the activities, located between their home, at distance 0, and the farthest one, at distance d/2. When they reach the farthest one, they have observed δ · d/2 activities. They therefore turn back, driving the remaining d/2, and on their way home, stop at one activity they have chosen. They choose d before choosing l (we do not consider the possibility for them to choose none, or two or more activities on their way back, nor do we consider the possibility that they stop before d/2 because they found an activity that provide so much utility they are virtually certain there is no better activity anywhere else). 8

We modelize the values of the different activities by centered1 , independant, identically distributed variables εl , of double exponential distribution function of heterogeneity parameter µa . Thus we are in the frame of a multinomial logit model. The driver does not know the realisations of these random variables before leaving, but he knows their distribution function. That is the reason why he must parse the activities. He must choose d before he chooses the activity. We know that (Anderson, de Palma & Thisse 1992a) his expected utility of choosing distance d is :  U(d) = E

 max εl

− c(d)

l∈[0;δd/2]

which is equivalent to (assuming δ · d/2 is an integer):   δd/2 X U(d) = µa ln  exp(0) − (pG θ + pf )d 0

 U(d) = µa ln

δ·d 2

 − (pG θ + pf )d

(2.2)

We no longer assume that (δ · d) is an integer, and keep this formula for next calculations. The driver will choose, if it exists, the distance d that mawimizes his expected utility U. As we can derivate U two times : µa U 0 (d) = − (pG θ + pf ) d mua U 00 (d) = − 2 < 0 d So : µa U 0 (d) = 0 ⇐⇒ d = (pG θ + pf ) As U is concave, this is equivalent to : d = arg max U(d) ⇐⇒ d = d

µa (pG θ + pf )

So : Proposition 1 Given all precedently made assumptions, a driver that has to parse the different activities before he chooses one of them will drive distance dmax before coming back, with µa (2.3) dmax = pG θ + pf 1

We could add a constant term to the utilities of the different activities, we dismiss it since it adds nothing to the model

9

density : 1/(G.d)

distance : d 

Figure 2.3: Demand from the searching driver As we are only interested in the distance the driver covers, we will study only briefly the demand profile. Once he has parsed δ · d of them, the probability density that he chooses any of them is 1/(δ · d). So the expected demand density from that driver for each activity l is 1/(δ · d) if (δ · d) < d, 0 else. The demand density is representated in figure 2.3. His expected utility of the driver is :   δµa U(dmax ) = µa ln − µa (2.4) 2(pG θ + pf ) Now we will study the case when the driver does not have to parse the activities before choosing one of them.

2.3

Choice of distance without research

We consider now another assumption : the driver knows the values of the different activities, but the observer does not. For this reason we also modelize here these values by centered, independant, identically distributed variables εl , of double exponential distribution function of heterogeneity parameter µa . But going to activity l, located at distance l/δ from the driver’s home, would cost the driver c(l/δ) way and back. So the utility of activity l is :   2l Ul = εl − c (2.5) δ which is equivalent to : Ul = εl −

2l · (pG θ + pf ) δ

We know (Anderson et al. 1992a) that the probability that the driver goes to activity m then is : Um

e µa

Pm = P ∞

l=0 e

Ul µa

2m − δµ ·(pG θ+pf )

e

Pm = P ∞

a

2l − δµ ·(pG θ+pf ) a l=0 e

10

density : f(d)

distance : d 

Figure 2.4: Demand from the searching driver Which gives the following density of probability if we use continuous variables : 2m ·(pG θ+pf ) − δµ

e

f (m) = R +∞ 0

a

2l − δµ ·(pG θ+pf )

e

a

dl

As choosing activity m is strictly equivalent to driving distance d = 2m/δ, the density of probability g that d is chosen is : δ g(d) = · f 2



δ ·d 2



So : −

d

·(p θ+p )

δ e µa G f g(d) = R 2 +∞ e− δµ2la ·(pG θ+pf ) dl

(2.6)

0

But : Z

+∞



e

2l(pG θ+pf ) δµa

 dl = −

0

Z 0

+∞

2l(pG θ+pf ) δµa − δµa e 2(pG θ + pf )

+∞ 0

  2l(pG θ + pf ) δµa exp − dl = δµa 2(pG θ + pf )

The probability density that the driver chooses to drive distance d, that is to say that he chooses the activity located at distance d/2 from his home is : g(d) =

pG θ + pf − µd ·(pG θ+pf ) ·e a µa

The shape of this following density is represented in figure 2.4. We can now calculate the expected distance d chosen by the driver. Z

+∞

d · g(d)dd

E(d) = 0

11

Z

+∞

E(d) = 0

d(p θ+p ) pG θ + pf − Gµ f a ·d·e dd µa

Integrating by parts :   Z d(p θ+p ) +∞ − Gµ f a + E(d) = −d · e

+∞



e

d(pG θ+pf ) µa

dd

0

0

The first term of the RHS being zero, E(d) =

µa pG θ + pf

Proposition 2 Given precedently made assumptions, the expected distance driven by a driver knowing the values of the different activities is : E(d) =

µa pG θ + pf

(2.7)

As we will work on a large number of drivers, our attention will mainly focus on this expected value from now on. The expected utility of the driver is : Z +∞   2l(pG θ + pf ) U(θ) = µa ln exp − δµa 0  U(θ) = µa ln

δµa 2(pG θ + pf )

 (2.8)

This expected utility depends on the efficiency of cars θ. The better the efficiency of cars, the lower they consume per kilometer, the lower θ, and as a consequence, the higher the expected utility U(θ). We observe the expected utility in the case where the driver has the information before leaving is strictly larger than in the case where he does not have it. In this framework, car users would be willing to pay up to µa to have information at their disposal before leaving. This can be considered as the value of information, since apart from that point, the two cases are entirely identical.

2.4

conclusion

These two different ways of modeling the choice of distance give the same results. This gives a good reason to keep this result for further study, since it seems rather consistent. Indeed, whether drivers’ behaviour is best predicted by the first model, or the second, or even if one part of the drivers’ population behaves along the first model and the other along the second one, the expected distance will be the same. 12

But there are limits : for example we cannot imagine a larger model, with a previous step where the driver chooses whether he behaves along the first way, or the second. This is due to the fact that those different behaves are imposed to the driver : in the first case, he does not know the values of the different activities, whereas he knows them in the second one. That is a situation he faces, not a choice he makes. Eventually, if we assumed that the driver is given the choice of consulting the information before leaving or not, he would certainly decide to consult it, since as stated in the previous section, the global expected utility in the second case (2.8) is higher than in the first case (2.4). Another point worth noting is that our modelization of the way drivers parse the different activities when they do not know their values is not very realistic. A more credible model would have been one where each time he observes an activity, the driver decides whether he has found a sufficiently high value and goes home or he searches a little bit more. This model, however, seems more difficult to explore than the two previous ones, so it will not be studied here. From now on we will use the second model.

13

Chapter 3

Competition in price on the car market 3.1

Modeling the car market

Before studying the case where car manufacturers compete simultaneously in quality and price, we will explore the case were they compete only in price. Remind there are N consumers on the car market. We assume the demand for car is totally rigid : each consumer buys exactly one car. We will set ourselves in the framework of discrete choice models and product differentiation for the following reasons. First, it is one way to handle competition in both price and quality, as will be done later. Second, our work will thus be easier to extend. The car market is described as an oligopolistic market, where firms play a two stages game : first they decide whether they enter the market or not, second they decide the price they will charge the customers. Their objective is to make an optimal profit. We are looking for an equilibrium in pure strategy. Denote n the number of firms. The firms are indexes by i ∈ {0, n − 1}. Each firm proposes only one car model, of annualized purchase price pi , and quality θi (remind that quality is here defined as fuel consumption, in liter per kilometer : the higher θi , the worse the fuel efficiency). Consumers, indexed by k, choose the model they buy among those available on the market. This choice depends on the purchase price of the car model pi , the expected utility of driving this car U(θi ), and an idiosyncratic value represented here by a random variable εk,i . Valuation Vk,i of car model k by consumer i is : Vk,i = −pi + U(θi ) + εk,i

(3.1)

with U(θi ) = µa ln(δµa /(pG θi + pf )). We assume the εk,i to be centered, independant across k and i, and identically distributed of double exponential 14

distribution function of heterogeneity parameter µ. As a consequence we will have logit model demand functions (Anderson et al. 1992a). Producing a car costs a marginal cost mc to the car manufacturer. Entering the market costs a fixed cost K to the firms. We assume mc and K do not depend on the firm. Thus the car market is symmetric. This model set up, we will first study the free-entry equilibrium, then the effects of the different externalities, finally its regulation.

3.2

Free-entry equilibrium

The free-entry equilibrium is the result of a two stages game that we will study using backward induction. We will first concentrate on the choice of prices given the number of firms and then on the number of firms itself. These two steps can be more or less considered as short term and long term equilibria.

3.2.1

Short term equilibrium

Number of firm n is given. The firms have N customers, they want to optimize their profit, they choose the price they charge customers. Profit function of firm i is : ei (pi , p−i ) − Ki πi = (pi − mci ) · X

(3.2)

where pi is the price charged by firm i, and p−i = {pj }j6=i the set of prices ei (pi , p−i ) stands for the expected demand for charged by the other firms. X model i given the set of prices. Ki and mci are identical for all i and equal to K and mc. The expected demand for model i is :   exp µ1 (−pi + U(θi )) fi (pi , p−i ) = N ·   X (3.3) Pn 1 j=0 exp µ (−pj + U(θj )) We consider only price competition here, therefore we assume that the θj are identical among all firms and equal to θ. Then :   exp µ1 (−pi + µa ln(δµa /(pG θ + pf ))) fi (pi , p−i ) = N ·   X Pn 1 exp (−p + µ ln(δµ /(p θ + p ))) j a a G f j=0 µ which simplifies itself in : 

−pi µ



exp fi (pi , p−i ) = N ·   X Pn −pj exp j=0 µ 15

Then we know (Anderson et al. 1992a) that there is a unique equilibrium in prices, which is symmetric. At this equilibrium, no firm can unilaterally change its price witout decreasing its profit. The equilibrium margin is : n ∀i, p∗i − mc = µ (3.4) n−1 The expected demand for all models is : ei = N ∀i, X n

(3.5)

Firms make the following profit : ∀i, πi =

3.2.2

µN −K n−1

(3.6)

Long term equilibrium

Given the precedent results, the long-term equilibium is easy to resolve. Each firm decides to enter the market knowing the price it would charge if it entered. If the firm makes a positive profit by entering, then it enters. As a consequence, firms’ profits are null at the equilibrium. So the long-term free-entry equilibrium nimber of firms is deduced from : ∀i, π = 0 µ ⇔ =K n−1 The free-entry equilibrium number of firms and margin are : ne = 1 +

Nµ K

pe − mc =

(3.7)

K +µ N

(3.8)

We will now procede to the welfare analysis of this problem.

3.2.3

Welfare analysis

The welfare analysis concern two groups of actors and their decisions : firms and consumers. The firms decide to enter the market or not, and the price they will charge. The consumers decide the car model they will buy, and the distance they will drive. Denote n the number of firms on the market, charging prices (pi )i∈[0;n] . Then the expected total amount paid by consumers to firms is : n X

fi (pi , p−i ) pi · X

i=1

16

The expected consumer surplus is :   Z N  n X fi (pi , p−i ) + N µa ln E max Vk,i dk − pi · X 0

i∈{0;n−1}

i=1

µa δ p G θ + pf



The first term of the RSH represents the valuation of heterogeneity by the car market by consumers. The second term is the amount paid for the cars by consumers. The third term stands for the utility drivers draw from driving their cars (equation 2.8). We know (Anderson et al. 1992a) that the first term is equal to N µ ln(n). The expected total profit of firms (thereafter denoted FS, which stand for firms’ surplus), is equal to the total gross profit minus the entry costs : n−1 X

fi (pi , p−i ) − nK (pi − mc) · X

i=0

The social welfare, equal to the expected consumer surplus and the expected firms surplus, has the following value :   n−1 X µa δ fi (pi , p−i ) − nK + N µa ln W = N µ ln(n) − mc · X pG θ + pf i=0

which is equal to :  W = N µ ln(n) − N mc − nK + N µa ln

µa δ pG θ + pf

 (3.9)

We observe that neither the prices, not the distance appear in the social welfare expression. The prices do not appear because the demand for car being rigid, prices have no influence on it. They only determine the amount of a tranfer from consumers to car manufacturers, which does not change the global social welfare. The reason why distances do not appear here is that they are random variables, varying for each driver. The classical welfare analysis, which consists in taking the best decision for each actor, is valid though. Since there are no externalities or distortion here, the optimal decisions are the one taken by drivers, as described in section (2.3), since they maximize their own utility0 without affecting others. This leads to the expected utility (2.8). The only decision variable in (3.9) is the number of firms. A rapid analysis summarized thereafter leads to the optimal number of firms. W (n) = C t + N µ ln(n) − nK where C t stand for a constant in n term. W 0 (n) =

Nµ −K n 17

Nµ <0 n2 The socially optimal number of firms therefore is : W 00 (n) = −

Nµ = ne − 1 K The social welfare is not at its optimal level at the free-entry equilibrium, which demonstrates that there is a distortion on this market. This distortion is a consequence from the way competition is modelized in this model. Products are differentiated by consumers, which means that for a readon the modeler cannot observe, they prefer one product more than the other. The reason can be the brand, the design, some other subjective or objective characteristics that are not measured by the modeler. They value one product more than the others, and so are ready to pay more than its marginal production cost for it. In other words, they are attached to that product. This loyalty results in market power for the car manufacturers, which use the resulting loyalty of consumers to charge prices that are strictly higher than the perfect competition prices, even at the free-entry equilibrium. They draw an extra margin which not only allows them to cover their sunk costs but also implies a too high number of firms on the market. The equilibrium number of firms is quiet similar to the optimal number of firms though. We will now focus on the question of regulation : how can we reach the social optimum through regulation tools? no =

3.2.4

Regulation

As explained previously, there are three kinds of decisions taken by actors in the frame of our model. First, do firms enter the market or not? Second, what car model will car users buy? Third, what distance will car users cover? These three decisions can be influenced by the three following regulation tools : a tax on capital, a tax on cars, a tax on distance. As the demand for cars is rigid, a tax or subsidy of any kind on cars would only be a revenue-raising instrument, letting the social welfare unchanged. In the same way, as distances chosen by the consumers are socially optimal, a tax on distance could not improve the social welfare. Remains the problem of firms number. There are too many firms at the free-entry equilibrium. We can imagine that a tax on capital, making entry more expensive for firms would diminish the number of competitors on the market, thus potentially increasing the social welfare. Denote τK the tax on capital. The equilibrium number of firms with this tax is : Nµ ne (τK ) = 1 + K(1 + τK ) 18

The equilibrium price is : pe (τK ) = mc +

K(1 + τK ) +µ N

The equilibrium expected utility of drivers is :   δµa U(θ) = µa ln 2(pG θ + pf ) So the consumer surplus is :  e CS (τK ) = N µ ln 1 +  + N µa ln

Nµ K(1 + τK )



δµa 2(pG θ + pf )

− N · mc − K(1 + τK ) − N µ 

The firms surplus is zero, and the total tax revenue is :   Nµ e G (τK ) = τK · K · 1 + K(1 + τK ) Therefore the social welfare is :  e W (τK ) = N µ ln 1 + +

N µ τK 1 + τK

 Nµ − N · mc − K − N µ + K(1 + τK )   δµa + N µa ln 2(pG θ + pf )

(3.10)

The first derivative of W e with respect to τK is :   ∂W e N µτK − K(τK + 1) = −N µ ∂τK K(1 + τK )3 + N µ(1 + τK )2 The social welfare function is quasi-concave in τK if and only if N µ > 1, that is to say if the optimal number of firms is larger than 1. We will assume this is true. Then the optimal tax on capital is : ∗ τK =

K Nµ − K

The equilibrium number of firms when the capital tax is set to its optimal level is : ∗ ne (τK )=

Nµ K

When the tax is set to its optimal level, the number of firms reaches the optimal level, ensuring that the social welfare reaches its optimal level. 19

Proposition 3 Under imperfect competition as modeled here, there is overentry on the car market at the free-entry equilibrium : ne = no + 1

(3.11)

It is possible to correct this distortion with a tax on capital τK set at the optimal level : ∗ τK =

K Nµ − K

(3.12)

Then the number of firms and the social welfare reach their optimal values. The distortions and regulation problems we will focus on later concern mainly the distance choice of the drivers. Since the demand for car is totally rigid, the distortion due to imperfect competition on the car market is not linked to the distortions in the way car users choose the distance they drive. As a consequence, the regulation solution to this distortion is valid in our whole study, hence it will not be raised again. We will now consider the case with one externality : environmental pollution.

3.3

Environmental pollution

We will here consider an externality we will model very simply : environmental pollution. This is a complex problem in itself, oversimplified by the assumptions we will do later. We indeed assume that the environmental damage due to vehicle use is proportionnal to the distance covered. The reality is much more complicated than that : emissions depend on the type and age of the vehicle, but also on the way it is driven. A significant part of the emissions occurs when the engine starts or stops, independantly from the distance driven (Harrington & McConnell 2003). Anyway, we introduce here environmental pollution as an example of externality, our objective is not accuracy in the way it is modeled. We assume that vehicle use results in emissions, proportionnally to the distance driven. We denote E the social cost emissions generate. We denote η the social cost related to the use of one liter of gas. Therefore, if a driver drives the distance d, he will generate the social cost ηθd. The expected total pollution damage will therefore be : E = −N η θ E(d)

(3.13)

where E(d) is the expected distance driven by a consumer. Remind that in the frame of model developed in 2.3, this expected distance is identical for all drivers. 20

The free-entry equilibrium is the same than in 3.2. There is now an environmental cost. Knowing the expected distance (prop. 2), this expected cost is : E e = −N η θ

µa pG θ + pf

So the expected social welfare at the equilibrium is : W

e



e

= N µ ln(n ) − N mc − K − N µ + N µa ln −N η θ

µa pG θ + pf

µa δ 2(pG θ + pf )



(3.14)

The social welfare is not optimal at the free-entry equilibrium. There are two reasons for that : imperfect competition and environmental pollution. The first problem has been delt with in section 3.2, and is independant from the second one, on which we will now focus.

3.3.1

Welfare analysis

We have seen in section 2.3 the way drivers chose the distance they drive. They consider their valuations of the different activities, taking into account the travel cost incurred. The valutation of activity l, located at distance l/δ from the home of driver k is : Ul = εk,l −

2l · (pG θ + pf ) δ

But if the driver considered the full social cost of its travel, including the environmental cost, he would valuate differently activity l, and Ul would become : Ul = εk,l −

2l · ((pG + η)θ + pf ) δ

(3.15)

This is the valuation we have to consider to determine the social optimum. Indeed, we reach the social optimum if all actors take all the external costs while taking their decisions. We verify easily that the expected distance from the driver would then be : E(d) =

µa (pG + η)θ + pf

and the expected utility of the driver :   δµa E(U) = µa ln 2((pG + η)θ + pf ) 21

So the optimal expected social welfare is :   δµa o t W = C + N µa ln 2((pG + η)θ + pf )

(3.16)

where C t stands for the part of the social welfare formula corresponding to the number of firms, which we are not concerned with here. The environmental damage does not appear in this result like in 3.14, because it is already taken into account by the drivers in 3.15. We will now study the problem of regulation.

3.3.2

Regulation

As the environmental pollution is not taken into account by drivers, thus causing a distortion and reducing the expected social welfare, regulation tools are necessary. We will try here to reach the social optimum using an excise tax on gas. The expected distance covered by car users if an excise tax on gas τG is set up is : E(d)(τG ) =

µa (pG + τG )θ + pf

Their expected utility is :   δµa U(τG ) = µa ln 2((pG + τG )θ + pf ) The expected tax revenue is : G(τG ) = N τG θ

µa (pG + τG )θ + pf

The expected environmental damage is : E(τG ) = −N η θ

µa (pG + τG )θ + pf

The expected social welfare therefore is :   µa δ µa e t W (τG ) = C + N µa ln − Nη θ 2((pG + τG )θ + pf ) (pG + τG )θ + pf + N τG θ

µa (pG + τG )θ + pf

Its derivative with respect to τG is : ∂W e N µa θ2 =− (τG − η) ∂τG ((pG + τG )θ + pf )2 22

Which ensures that W e is quasi-concave in τG , and that the optimal tax level τG∗ is strictly positive and equal to η. When the tax on gas is set to this value, the social welfare is equal to :   µa δ e ∗ t = Wo W (τG ) = C + N µa ln 2((pG + η)θ + pf )

Proposition 4 When there are emissions, it is possible to reach the social optimum with an excise tax on gas τG , set at the level : τG∗ = η

(3.17)

This result corresponds to our intuition : if the social cost of driving is internalised, car users will naturally shift their behaviour towards the social optimum.

3.3.3

Graphical analysis

We will here provide an illustrative graph. This approach, although giving a intuition of the phenomenon, is quiet dangerous in the frame of this model. In classical transport economics, all car users drive the same distance d for which the marginal utility and marginal cost are equal. In the frame of our model the distance actually covered by a driver is the realisation of a random variable, resulting from a choice among a set of possibilities. But the expected distance (2.7) is the mathematical solution of the following equation : µa = Cg d where Cg stands for the generalised marginal cost of driving. If we replace Cg in this formula by pG θ + pf , the solution of the equation is indeed the expected distance predicted by our model. It remains the case if we replace Cg by (pG + η)θ + pf for the social welfare analysis, and by (pG + τG )θ + pf for the regulation analysis. The graph (3.1) illustrates this fact that : without regulation, drivers drive too much; whereas if environmental costs are nternalised, the regulator can force the equilibrium to reach the social optimum. Those very simple results correspond to our intuition. We introduced a negative externality on distance travelled by drivers. Without any regulation, drivers do not take into account all the costs. A tax on gas is necessary and sufficient to correct the subsequent distortion. We will now study the case with two externalities : emissions and congestion. 23

Social marginal cost

η

Private marginal cost

Private marginal utility Equilibrium distance

Optimal distance

d

Figure 3.1: Equilibrium and social optimum with emissions

3.4

Congestion

We will add an externality to the previous model. We will assume from now on that the road on which car users drive is subject to congestion. This means that the presence of each driver on the road lowers the speed of all drivers. We will not model explicitely the effect of congestion. We will only assume that the cost generated by congestion for a driver is proportional to the distance he covers and the density of vehicles on the road. Denote d(k) distance driven by consumer k. The density of vehicles on the road is : Z 1 N d(k)dk L 0 The driving cost function will have the same expression that in (2.1) with an additional term related to congestion, depending linearly from the density of vehicles on the road.   Z N c(d) = pG θ + pf + α d(k)dk d (3.18) 0

We will first study the free-entry equilibrium, then proceed to the welfare analysis, and finally study the regulation problems.

3.4.1

Free-entry equilibrium

The first thing to determine is the distance drivers cover. The question is somewhat different than in the previous sections, since each driver’s decision depends on all the others. We will try to find a Nash equilibrium in mixed 24

strategies, that is to say a set of behaviours so that if all drivers behave along it, no one will find it profitable to behaves differently. Assume such a set ofR behaviours exists. Each car user takes the expected N global density of cars, 0 E(d(k))dk, as given . The expected distance he will drive, calculated as in section 2.3, will therefore be : E(d) =

µa RN pG θ + pf + α 0 E(d(k))dk

Since this equation is true for all car users, then they all drive the same expected distance, solution of the following equation : E(d) =

µa pG θ + pf + αN E(d)

(3.19)

which is equivalent to : (E(d))2 +

pG θ + pf µa E(d) − =0 αN αN

This equation has a single positive solution : r pG θ + pf (pG θ + pf )2 µa E(d) = − + + 2 2 2αN 4α N αN which can written this way : E(d) =

µa pG θ+pf 1 2

+

q

1 4

+

µa αN (pG θ+pf )2

If a set of behaviours is a Nash equilibrium in mixed strategies, then the expected distance covered by all car users is necessarily the solution of equation(3.19). Conversely, if all car users behave such that the expected distance they cover is equal to the solution of equation (3.19), then each car user will drive as modeled in section 2.3, and its expected distance will be equal to the solution of equation (3.19). So this equation indeed describes a Nash equilibrium in mixed strategies. Proposition 5 At the equilibrium, the expected distance driven by any driver is : E(d) =

µa pG θ+pf 1 2

+

q

1 4

+

(3.20)

µa αN (pG θ+pf )2

Not surprisingly, the equilibrium distance is the same than (2.7), with a correction term corresponding to congestion. The higher α, the more distance is reduced. Concerning firm numbers and prices, the equilibrium is the same as described in section 3.2. 25

3.4.2

Welfare analysis

We do not focus on the over-entry problem. At the equilibrium, the valuation of activity l, located at distance l/δ from the home of driver k, takes into account the expected distance driven by all other drivers, {E(d(k))}k∈[0;N ] , that we will note thereafter {d(k)}k∈[0;N ] . This valuation is :   Z N 2l d(k)dk Ul = εk,l − pG θ + pf + α δ 0 A valuation taking into account all costs should take into account the social cost of driving due not only to emissions, but also to congestion. The total expected social cost of congestion is :  Z N Z N Cg = α d(k2 )dk2 d(k1 )dk1 0

0

Denote sc(d) the social cost of driving distance d. Then the marginal social cost of driving distance d is sc0 (d). Imagine η > 0 drivers driving the same expected distance decide to increase it of the value dx 1 . The new distribution of expected distance is {d(k) + ν(k)}k∈[0;N ] , where ν is the variation distribution. Then the new congestion cost is :  Z N Z N Z N Z N α d(k2 )dk2 d(k1 )dk1 + 2 α ν(k)dk Cg + dCg = 0

0

Z

N Z

0 N

+

0

 ν(k2 )dk2 ν1 (k1 )dk1

0

0

which is equivalent to : 

Z

Cg + dCg = Cg + 2α

N

 d(k)dk ηdx + o(ηdx)

0

so :  dCg =

Z 2α

N

 d(k)dk ηdx + o(ηdx)

0

The variation can be considered as the additionnal social cost sc0 (d) caused by η drivers : dCg = sc0 (d)ηdx 1

We won’t focus here on the problem of the existence of a sufficiently large set of drivers driving similar distances. It is possible, but it will not be demonstrated here, to construct from the distribution {d(k)}k∈[0;N ] a distribution where these values are ranked from the lowest to the largest. This new distribution is equivalent to the old one for our present problem (since the congestion level does not depend on the way drivers are indexed) and is almost everywhere continuous. Then the following calculations can be easily generalized to every point where the distribution is continuous.

26

As a consequence :  Z 0 sc (d)ηdx = 2α

N

 d(k)dk ηdx + o(ηdx)

0

which, if we make η and dx tend towards 0, implies that the social marginal cost implied by one car user increasing the distance he covers is : Z N sc0 (d) = 2α d(k)dk (3.21) 0

This implies that in order to do the socially optimal choice, drivers must valuate the different activities as precised here :   Z N 2l d(k)dk (pG + η)θ + pf + 2αN Ul = εk,l − δ 0 The expected distance driven by driver k at the social optimum is (we dismiss the E operator) : µa

d=

(pG + η)θ + pf + 2α

RN 0

d(k)dk

This is true for all drivers. The expected distance they drive is therefore identical among all drivers, and solution to the equation : µa d= (3.22) (pG + η)θ + pf + 2αN d This equation is equivalent to d2 +

(pG + η)θ + pf µa d− =0 2αN 2αN

This equation has a single positive solution : r (pG + η)θ + pf ((pG + η)θ + pf )2 µa d=− + + 2 2 4αN 16α N 2αN which can written this way : d=

µa (pG +η)θ+pf 1 2

+

q

1 4

+

2µa αN ((pG +η)θ+pf )2

This distance defines an equilibrium : no driver can draw benefit from behaving differently from what the others expect him to do. Proposition 6 The socially optimal expected distance is : E(d) =

µa (pG +η)θ+pf 1 2

+

q

1 4

+

2µa αN ((pG +η)θ+pf )2

27

(3.23)

The optimum distance appears here as the optimum distance when there are only emissions, with a correction due to congestion, like in the equilibrium case, the difference being that the correction is higher : this was to be predicted since car users only take into account half of the social cost of congestion when taking their decisions. The optimum social welfare is :   o

δµa (pG +η)θ+pf

t

W = C + N µa ln 

1+

q

1+

2µa αN ((pG +η)θ+pf )2



(3.24)

We will now examine how to reach the social optimum with regulation tools.

3.4.3

Regulation

As for the case where we only considered emissions, we will try to internalize the congestion and emissions externalities through a tax on gas τG . If this tax is set up, the expected distance driven by drivers will be : E(d)(τG ) =

µa (pG +τG )θ+pf 1 2

+

q

1 4

+

µa αN ((pG +τG )θ+pf )2

Their expected utility is :  E(U)(τG ) = µa ln 

δµa (pG +τG )θ+pf

1+

q

1+

4µa αN ((pG +τG )θ+pf )2

 

The expected tax revenue is : G(τG ) = N τG θ · E(d)(τG ) The expected environmental damage is : E(τG ) = −N ηθ · E(d)(τG ) Denote here f the following function : f : τG 7→

µa (pG +τG )θ+pf 1 2

+

q

1 4

+

µa αN ((pG +τG )θ+pf )2

The expected social welfare therefore is (remind that the congestion cost is totally contained in the expected utility of the drivers):   δ W e (τG ) = C t + N µa ln f (τG ) − N ηf (τG ) + N τG θf (τG ) 2 28

So its derivative with respect to τG is : ∂W e N µa f 0 (τG ) = + (−N ηθ + N τG θ) f 0 (τG ) + N θf (τG ) ∂τG f (τG ) f (τG ) is solution of the following equation : f 2 (τG ) +

(pG + τG )θ + pf µa f (τG ) − =0 αN αN

which implies, if we derivate both members with respect to τG : 2f (τG )f 0 (τG ) +

(pG + τG )θ + pf 0 θ f (τG ) + f (τG ) = 0 αN αN

As a consequence : f 0 (τG ) =

2αN θ f (τG )

−f (τG ) + (pG + τG ) +

pf θ

So :  ∂W e = A −µa + ηθf (τG ) + pG θf (τG ) + pf f (τG ) + 2αN f (τG )2 ∂τG where A is a positive expression. This equation is equivalent to :   (η + pG )θ + pf ∂W e µa 2 = 2αN A − + f (τG ) + f (τG ) ∂τG 2αN 2αN

(3.25)

we can observe the term in the parenthesis is similar to equation (3.22), which solution is the socially optimal distance. It implies that it is zero when f (τG ) is a solution of (3.22). As f is an decreasing function of τG , it ensures that the maximizing τG is the one so that E(d)(τG ) is equal to do . So when the regulator forces car users to drive the socially optimal distance, the social welfare reaches its optimum. Translating this into an equation gives : µa (pG +τG )θ+pf 1 2

+

q

1 4

+

µa αN ((pG +τG )θ+pf )2

=

µa (pG +η)θ+pf 1 2

+

q

s

1 4

+

2µa αN ((pG +η)θ+pf )2

((pG + τG )θ + pf )2 +1 4µ2a αN s (pG + η)θ + pf ((pG + η)θ + pf )2 √ + +2 = 4µ2a αN 2µa αN √ Equation x + x2 + 1 = a has a single solution in x if a ≥ 1 which √ is x = 1 2 2 (a − 1/a). The preceding equation has the following form: x + x + 1 = (pG + τG )θ + pf √ ⇔ + 2µa αN

29

p y + y 2 + 2. Thepsolution of this specific form of the preceding equation always exists (y + y 2 + 2pbeing always greater than 1, as we consider only y ≥ 0) and is: x = 34 y + 41 y 2 + 2, , which allows us to deduce : s (pG + τG )θ + pf 3 (pG + η)θ + pf 1 ((pG + η)θ + pf )2 √ √ = + +2 4 2µa αN 4 4µ2a αN 2µa αN 1 τG = η + 4

r



! pf pf 2 8µ2a αN − (pG + η + ) pG + η + + θ θ2 θ

We can interpret in this formula the first term as the part of the tax corresponding to emissions, and the second term as a correction due to congestion. Proposition 7 The social optimum is reached when drivers drive the socially optimal distance. Drivers drive the socially optimal distance when an excise tax on gas is set at the following value : ! r  2 8µ2 αN p p 1 f f τG∗ = η + pG + η + (3.26) + a 2 − (pG + η + ) 4 θ θ θ An interesting observation is that this tax is not merely the sum of a tax that would correct emissions, and another one that would correct congestion. Indeed, if we denote τG∗ (η, α) the optimal tax for emission coefficient η and congestion coefficient α, then we have :  pf  pf  1  τG∗ (η, α) − (τG∗ (η, 0) + τG∗ (0, α)) = f pG + η + − f pG + 4 θ θ with function f defined as follows : r 8N αN f : x 7→ x2 + −x θ2 The derivative of f is : q x − x2 + 8NθαN 2 f 0 (x) = q x2 + 8NθαN 2 which is negative. f is a decreasing function, and as a consequence : ∀ η > 0, α > 0

τG∗ (η, α) < τG∗ (η, 0) + τG∗ (0, α)

(3.27)

This means that if the two externalities are handled separately, and two taxes on gas are designed, one targetted against environmental pollution and the other against congestion, and assuming that these taxes have been 30

correctly designed, then the resulting taxation level would be too high. This is due to the fact that a tax targeted on one of the two problems, reducing the expected distance drivers drive, would partly correct the other problem. Another point worth noting is that the optimal tax level now depends on pf , which was not the case when emissions were the only externality considered (3.17). The higher pf /θ, the lower the second term of (3.26). This RN results from the fact that if pf is large compared to pG θ + α 0 d(k)dk, then gas and congestion costs are insignificant before other operating costs of the car. As a consequence, those othe operating costs are the main determinant of expected distance driven. Thus, distance at the equilibrium is not far from the socially optimal distance, in so far as congestion is not the main concern of drivers.

3.4.4

Graphical analysis

As in the case where emissions are the only externality taken into account, we will provide here some graphs that can give some intuition of the phenomenon. These graphics are very similar to the ones of classical transports economics. In the first one, we illustrate the difference between the equilibrium and the social optimum. It is based on the same reasoning than in section 3.3.3, on the “equivalent equation” : µa = Cg d where Cg stands for an equivalent of the generalised marginal cost of driving. Cg can be the private marginal cost, then its formula is : (pG )θ+pf +αd, and it is represented on figure 3.2 by the plain black line. It can be the marginal social cost, then its formula is : (pG + η)θ + pf + 2αd, and it is represented on the graph by the thick dotted line. The thin dotted line stands for the private marginal cost plus the social cost of emissions, in order to distinguish the effects of the two externalities. The graph illustrates the fact that users underestimate the marginal social cost, and thus drive too much. Cg can also stand for the private marginal cost under optimal regulation : (pG + τG )θ + pf + αd.It appears as a thick plain line in figure 3.3. Under regulation, the private marginal cost is shifted upwards, thus reducing the distance covered by drivers. It appears that the regulator does not really internalize the two externalities : the marginal cost perceived by the users is not identical to the marginal social cost. The graphs illustrate the fact that one tax is sufficient to correct the distortion due to two externalities, the reason being that there is only one variable to correct : the distance. As soon as the distance is at its socially optimal subject, then the social welfare is optimal. 31

Social marginal cost

Private marginal cost Private marginal utility Optimal distance

Equilibrium distance

d

Figure 3.2: Equilibrium and social optimum with congestion

Social marginal cost

Tax on fuels Private marginal cost Private marginal utility Equilibrium distance

d

Figure 3.3: Optimal regulation with congestion

32

3.5

conclusion

In the very simple model presented here, three imperfections have been studied : imperfect competition, environmental pollution, and congestion. These imperfections can be separated into two categories : imperfections over the car market, and imperfections over what we could call the “distance market”. These two categories are independant from each other, mainly because the demand for car being totally rigid, the overall distance driven by car buyers is independant from the car prices. It also appears that although they are not perfectly correlated, the emissions and congestion externality can be corrected with a single regulation tool. Indeed, if drivers drive the socially optimal distance, then the social welfare is optimal. One regulation tool is sufficient to bring them to that socially optimal distance. The results of this chapter are very classical ones in transport economics. The fact that our framework is consistent with these results is a good point concerning the consistency of our model. Our next step is the introduction of competition in efficiency. Car manufacturers decide the efficiency of their engines depending on what customers want and what their competitors do. In that framework, one regulation tool will no longer be efficient enough to reach the social optimum.

33

Chapter 4

competition in price and efficiency on the car market 4.1

Introduction

In section 3.2.1, we had assumed that efficiencies were identical among all customers and equal to θ. We will now release this assumption. This should have significant consequences on the previous results, mainly because it links the car market and what we could denote the “distance market”. Car manufacturers will indeed consider the “distance demand” to decide the efficiency of the car they will make. They will of course not take anything into account but their customers. As we will see later, this will cause some distortion on the way efficiency is chosen, and has to be carefully taken into account by the regulator. Better efficiency is costly for car manufaturers. We now assume that the marginal cost of producing a car, mc, depends on θ. Nevertheless, we assume that the cost function mc(θ) is identical for all firms, keeping the symmetry of our market. We do not assume for the moment that higher efficiency leads to higher research and development costs (which corresponds to K depending on θ). We will first consider the equilibrium without any externality, then we will introduce emissions and congestion. For each of these steps we will consider the question of regulation.

4.2

Equilibrium in efficiency without externalities

We study a three stages game : first, firms decide whether they enter the market or not. Second, they decide the efficiency of the car model they make. Third they decide the price they will charge customers for the cars they sell. They take into account the consequences of their decisions before making them, their objective being profit maximisation. We will first consider the 34

price equilibrium given efficiencies, then the equilibrium in efficiency, and finally the free-entry equilibrium.

4.2.1

Price equilibrium

The following reasonings are strongly inspired by (Anderson, de Palma & Thisse 1992b). We first determine the price equilibrium for a given number of firms on the market n and for a given set of efficiencies {θi }i∈[0;n−1] , not necessarily symmetric. Firm’s i profit function is : ei − K πi = (pi − mc(θi ))X ei , demand for model i, is : where X   i) exp −pi +U(θ µ ei = N   X Pn−1 −pj +U(θj ) exp j=0 µ Its derivative with respect to pi are : ei 1 e ∂X ei ) =− Xi (1 − X ∂pi Nµ The profit’s derivative is :   ∂πi ei fi 1 − pi − mc(θi ) + pi − mc(θi ) X =X ∂pi Nµ Nµ So it is increasing as long as : pi ≤ mc(θi ) +

Nµ ei N −X

ei ) decreases in pi , there is a unique pi so that this Given that µ/(1 − X inequality is an equality, and it maximises the profit. The following equation implicitly defines i’s best-reply function : pbr i = mc(θi ) +

Nµ ei N −X

(4.1)

An equilibrium in pure strategies {p∗i }i∈[0;N ] verifies the following equations, which define the best-reply functions : ∗ ∗ ∀i, p∗i = pbr i (p1 , .., pn )

in other words, it is a fixed point of the following function : Rn → Rn br (p1 , .., pn ) 7→ (pbr 1 , .., pn )

(4.2) 35

The following lemma will be useful (x denotes a vector of dimension n, and x−i denotes the vector of dimension n − 1 consisting of all the coordinates of x but the ith): Lemma 1 Let f be a continuous real function defined over a compact B of dimension n. Assume that for all {xj }j6=i , xi 7→ f (xi , x−i ) has a single maximum in xi . Denote gi the following function : gi : B → R x 7→ arg max f (xi , x−i ) xi

For all i, gi is continuous. Proof : Assume ∃ i so that gi is not continuous. Then ∃ {xn }n∈N a sequence of B so that : lim xn = xl

n→+∞

and

lim gi (xn ) 6= gi (xl )

n→+∞

The sequence {xn }n∈N as an adherence value s in B. In other words, there exists σ a strictly increasing function of N to N so that : lim xσ(n) = s

n→+∞

Since g(xl ) maximises f (·, x−i,l ), necessarily : f (s, x−i,l ) < f (g(x−i,l ), x−i,l ) But for all n, g(x−i,σ(n) ) maximises f (·, x−i,σ(n) ), this implies : ∀n,

f (g(x−i,l ), x−i,σ(n) ) < f (g(x−i,σ(n) ), x−i,σ(n) )

If we pass on to the limit in both sides of this inequality it implies : f (s, x−i,l ) ≥ f (g(x−i,l ), x−i,l ) which is absurd. So g is a continous function. Q.E.D. The best-reply function defined by (4.1) verifies all the hypothesis of lemma 1. So it is continuous on any compact. The next part of the proof of the existence of the equilibrium will not be done here. Roughly, the Brouwer fixed point theorem (any continuous function of a compact in itself has a fixed point) is to be used on the function defined by (4.2). The last important point is the unicity of the equilibrium. We can demonstrate that the gradient of the best-reply function (4.1) is so that there cannot be two equilibria. Indeed, the derivative of pbr i with respect to j, for j 6= i, verifies, from the implicit function theorem :   ∂pbr 1 ∂pbr i i e e e e = −Xi (N − Xi ) + Xi Xj ei )2 ∂pj ∂pj (N − X 36

This implies that : ej ei X X ∂pbr i = ei ) ∂pj N (N − X Therefore the sum of all the derivatives is : 0<

X ∂pbr i

j6=i

∂pj

=

ej X <1 N

This implies that the norm of the gradient is always lower than 1, and as a consequence that the function defined by (4.2) cannot have two fixed points. The equilibrium is unique. We can summarize these results in the following proposition. Proposition 8 For each set of qualities {θi }i∈[0;n−1] , there is a unique equilibrium p∗ in pure strategies, determined by the following system of implicit equations : ∀ i ∈ [0; n − 1],

p∗i = mc(θi ) +

Nµ ei (p∗) N −X

(4.3)

This equilibrium has no reason to be symmetric when the efficiencies are not equal among all car manufacturers. Now that we know the reactions of car manufacturers to a given set of qualities, we will study the way they choose quality of the car they produce.

4.2.2

Efficiency equilibrium

Car manufacturers want to maximise their profits. Therefore they choose the efficiency of the car they make with respect to their profit functions. As in the precedent section, we will first consider the best-reply functions of the car manufacturers, then the existence and unicity of an equilibrium. Let {θi } denote a given set of efficiencies at which manufacturers produce their cars. Our objective is to determine whether it is an equilibrium or not. For that, we will consider the behaviour of the profit function of a manufacturer with respect to the efficiency variable θ. We have to take into account that if one manufacturer changes the efficiency of its car model, all the other manufacturers will react and as a consequence all the equilibrium prices will certainly change. Therefore we will first calculate the derivatives of the equilibrium prices with respect to the car efficiency. We will consider, without loss of generality, the first car manufacturer. We know, for proposition (8), that there is a single equilibrium in price, described by the following equation : ∀i,

p∗i = mc(θi ) + µ

N ei (p∗ ) N −X 37

In particular, the own price of the car manufacturer is : p∗1 = mc(θ1 ) + µ

N e1 (p∗ ) N −X

We will first study the first-order equilibrium condition (equation (4.15)), then the second order equilibrium condition (equation (4.21)), and finally give a sufficient condition on function mc(θ) for the existence and unicity of a symmetric equilibrium (assumption 1). First-order condition For the implicit function theorem, p∗1 is derivable in θ1 , and its derivative verifies :  ∂p∗1 N d e = mc0 (θ1 ) + µ X1 ((p)∗ e1 )2 dθ1 ∂θ1 (N − X But :  ∂X X ∂X e1 n−1 e1 ∂p∗ d e i X1 ((p)∗ = + dθ1 ∂θ1 ∂pi ∂θ1

(4.4)

i=0

The different terms of this equation are equal to : e1 ∂X 1 e e1 )U 0 (θ1 ) = X1 (N − X ∂θ1 µN e1 ∂X 1 e f1 ) =− X1 (N − X ∂p1 µN ∀i 6= 1,

e1 ∂X 1 e e = X1 Xi ∂pi µN

As a consequence : e1 X ∂p∗1 = mc0 (θ1 ) + e1 )2 ∂θ1 (N − X

∗ e1 ) U 0 (θ1 ) − ∂p1 (N − X ∂θ1



 +

X i>1

fi ∂pi ∗ X ∂θ1

!

where U(θ1 ) is the overall utility drawn by a driver using a car of efficiency θ1 . Remind that the higher θ, the more the car consumes gas, and thus the lower the overall utility of driving it. We therefore expect U 0 to be negative. The former formula is equivalent to : X ∂pi ∗ e1 e1  X X ∂p∗1 fi − mc0 (θ1 ) = U 0 (θ1 ) − mc0 (θ1 ) + X (4.5) f1 ) ∂θ1 N ∂θ1 N (N − X i>1

38

We can interpret the two terms of the previous formula as the effects of a variation of own efficiency on the margin of the car manufacturer. The first effect is due to the change of the indirect utility of owning a car made by the first manufacturer if efficiency changes, the second effect is due to the change of the prices of all the other car manufacturers, who adapt to the new market configuration. We similarly calculate the first derivative of the equilibrium price charged by another car manufacturer with respect to the efficiency of the first car manufacturer. We focus on the second car manufacturer, without loss of generality. The price charged by the second car manufacturer is determined by : p∗2 = mc(θ2 ) + µ

N e2 (p∗ ) N −X

So : N d e ∗  ∂p∗2 =µ X2 (p ) f2 )2 dθ1 ∂θ1 (N − X The cross-derivative of the market share with respect to efficiency is calculated as previously : X ∂X e2 n−1 e2 ∂p∗ d  e ∗  ∂X i X2 (p ) = + dθ1 ∂θ1 ∂pi ∂θ1

(4.6)

i=0

Replacing the derivatives by their expressions : e2 d e ∗  X X2 (p ) = dθ1 µN ⇔

e2 + X e2 N2 − NX 2 e2 )2 (N − X

!

∗ ∗ X ∂p∗ e1 U 0 (θ1 ) + X e1 ∂p1 − N ∂p2 + ei i −X X ∂θ1 ∂θ1 ∂θi

∂p∗2 = ∂θ1

!

i>1

!  X ∗ e1 X e2  ∂p∗ X ∂p 1 ei i − U 0 (θ1 ) + X e2 )2 ∂θ1 ∂θ1 (N − X i>1

which is equivalent to : ∂p∗2 ∂θ1

=

 ∗  e1 X e2 X ∂p1 0 − U (θ1 ) e2 + X e 2 ∂θ1 N2 − NX 2 +

X ∂p∗ e2 X ei i X 2 2 e e ∂θ1 N − N X2 + X2

(4.7)

i>1

Equation (4.5) for the first car manufacturer and (4.7) for the others form a system of n equations which are quiet uneasy to handle. For this reason we will now only consider equilibria that are symmetric in efficiency. We will first look for a candidate, then verify it is an equilibrium. From now on, we consider a set of efficiencies verifying ∀i, θi = θ. 39

If all efficiencies are equal, then, for a given θ, all the results of chapter 3 hold. All price are identical, equal to the sum of the marginal production cost and a margin of µn/(n − 1). Market shares are all identical, equal to N/n. Thus, equation (4.5) is equivalent to :  1 ∂p2 ∗ ∂p∗1 1 − mc0 (θ) = U 0 (θ) − mc0 (θ) + ∂θ1 n n ∂θ1 an equation (4.7) is equivalent to :  ∗  1 ∂p1 ∂p∗2 0 = 2 − U (θ) ∂θ1 n − 2n + 2 ∂θ1

(4.8)

(4.9)

Inserting (4.5) in (4.7) leads to :   1 1 0 ∂p∗2 1 0 1 ∂p∗2 0 0 = 2 mc (θ) + U (θ) − mc (θ) + − U (θ) ∂θ1 n − 2n + 2 n n n ∂θ1 ⇔

 (n − 1)(n2 − n + 1) ∂p2 ∗ n−1 = mc0 (θ) − U 0 (θ) 2 2 n(n − n + 1) ∂θ1 n(n − n + 1)

As a consequence :  ∂p∗2 1 mc0 (θ) − U 0 (θ) = 2 ∂θ1 n −n+1

(4.10)

This equation corresponds to our intuition : if U 0 (θ)−mc0 (θ) is positive, then the indirect utility of a user buying a car to the first manufacturer increases with θ. Therefore, the other manufacturers have to lower their prices in order to compensate the induced loss of market share if θ increases. So the derivative of p2 with respect to θ1 is the opposite sign of U 0 (θ) − mc0 (θ). In order to obtain ∂p∗1 /∂θ1 , we now replace (4.10) in (4.5) :     ∂p∗1 1 1 1 1 0 = 1− + 3 mc (θ) + − U 0 (θ) ∂θ1 n n − n2 + n n n3 − n2 + n which is equivalent to :  ∂p∗1 n−1 − mc0 (θ) = 2 U 0 (θ) − mc0 (θ) ∂θ1 n −n+1

(4.11)

We observe here an analogous behaviour : the sign of the derivative of the equilibrium price with respect to θ1 depends on the sign of U 0 (θ) − mc0 (θ). But as opposed to the previous case, ∂p∗1 /∂θ1 is here of the same sign as U 0 (θ) − mc0 (θ) : the car manufacturer can draw a higher margin if the indirect utility of users buying its car model increases. We can now easily calculate the first derivative of the profit function of the first car manufacturer with respect to θ1 : e1 (p∗ ) − K π1 = (p∗1 − mc(θ))X 40

so : ∂π1 = ∂θ1



   ∂p∗1 0 e1 (p∗ ) e1 (p∗ ) + (p∗1 − mc(θ)) d X − mc (θ) X ∂θ1 dθ1

(4.12)

but : d e ∗  X1 (p ) = dθ1

  N (n − 1) n−1 1 1− 2 U 0 (θ) − µn2 n − n + 1 n2 − n + 1  2   n − 2n + 2 1 0 + − 2 + 2 mc (θ) n −n+1 n −n+1

which is equivalent to :  d  e ∗  N (n − 1)(n2 − 2n + 1) 0 X1 (p ) = ⇔ U (θ) − mc0 (θ) 2 2 dθ1 µn (n − n + 1)

(4.13)

This equation shows that if an increase in θ results in an increase in the indirect utility of a driver buying car model 1, then the market share of the first car manufacturer increases. There was no staightforward intuition for this result. As we indeed observed in equation (4.11), the equilibrium price of the first car manufacturer also increases in this case, since the car manufacturers uses its higher market power to draw a higher margin from his customers. This could have had the effect of reducing the market share of the car manufacturer. Equation (4.13) demonstrates it is not the case. We can now replace (4.13) in (4.12), which leads to :    1 n2 − 2n + 1 ∂π1 n−1 0 0 + U (θ) − mc (θ) = ∂θ1 n(n2 − n + 1) n n2 − n + 1 The preceeding equation is equivalent to :  ∂π1 n2 − n = U 0 (θ) − mc0 (θ) 2 ∂θ1 n(n − n + 1)

(4.14)

These equations show that if an increase in θ results in an increase in the indirect utility of a user buying car model 1, then it also results in an increase of the profit of the firm. This is not a surprising result. Equations (4.11) and (4.13) showed that in this case, both the margin and the market share of the car manufacturer would increase. As a consequence, profit would increase. It is a necessary condition for θ to describe a symmetric equilibrium in efficiency that the derivative of the profit function of each manufacturer with respect to its own efficiency variable is null when ∀i, θi = θ. This implies, if we assume that n > 1, that : U 0 (θ) − mc0 (θ) = 0

(4.15)

It is straightforward that this condition implies that the derivatives of the equilibrium prices and demands of all manufacturers are null with respect to θ at the equilibrium. It is now necessary to study the stability condition, which will give us a second-order condition of an equilibrium in efficiency. 41

Second-order condition We know from equation (4.5) that : X ∂pi ∗ e1 e1  X X ∂p∗1 fi − mc0 (θ1 ) = U 0 (θ1 ) − mc0 (θ1 ) + X f1 ) ∂θ1 N ∂θ1 N (N − X i>1 This leads to :  ∂ 2 p∗1 1 d e ∗  00 = mc (θ1 ) + X1 (p ) (U 0 (θ1 ) − mc0 (θ)) N dθ1 ∂θ12 ∗ e1 X ∂p∗  X ∂p i i ei + + X 2 e e ∂θ ∂θ 1 (N − X1 ) i>1 N − X1 i>1 1

N

+

X

e1 ei X X e i ∂ 2 p∗ X X i (U 00 (θ) − mc00 (θ)) + 2 e N N ∂θ N − Xi i>1 1

At the equilibrium, all the expected demands are identical and equal to 1/n. As a consequence, the former expression reduces to (using equations (4.10), (4.11) and (4.13)): ∂ 2 p∗1 (n − 1)4 00 = mc (θ) + (U 0 (θ) − mc0 (θ))2 µn(n2 − n + 1) ∂θ12   1 ∂ 2 p∗2 00 00 + (U (θ) − mc (θ)) + n ∂θ12 We know for the preceeding part of our work that a necessary condition for θ to describe a symmetric equilibrium in efficiency is equation (4.15). As a consequence :   1 ∂ 2 p∗2 ∂ 2 p∗1 00 00 00 (U (θ) − mc (θ)) + = mc (θ) + (4.16) n ∂θ12 ∂θ12 Similarly, we know from equation (4.7) that : e2 ∂p∗2 X = e2 + X e2 ∂θ1 N2 − NX 2

 e1 X

!  X ∗ ∂p∗1 ∂p ei i − U 0 (θ1 ) + X ∂θ1 ∂θ1 i>1

Derivating the former expression leads to : !   X ∗ e2 ∂ 2 p∗2 N2 − X d  f ∗  e ∂p∗1 ∂p 0 2 i ei = X1 (p ) X1 − U (θ) + X e2 + X e 2 )2 dθ1 ∂θ1 ∂θ1 ∂θ12 (N 2 − N X 2 i>1 e2 X + e2 + X e2 N2 − NX 2

   2 ∗  d  f ∗  ∂p∗1 ∂ p1 0 00 e X1 (p ) − U (θ) +X1 − U (θ) dθ1 ∂θ1 ∂θ12 42

 ∗ X ∂ 2 p∗ X d  i ei ei (p∗ ) ∂pi + + X X dθ1 ∂θ1 ∂θ12 i>1

!

i>1

At the equilibrium, all the expected demands are identical and equal to 1/n. Furthermore, n X

ei = 1 ⇒ X

i=1

1 d e ∗  d e ∗  X2 (p ) = − X1 (p ) dθ1 n − 1 dθ1

As a consequence, the former expression reduces to (using equations (4.10), (4.11) and (4.13), and including the equilibrium condition (4.15)): ∂ 2 p∗2 1 = 2 n −n+1 ∂θ12



∂ 2 p∗1 ∂ 2 p∗2 00 − U (θ) + (n − 1) ∂θ12 ∂θ12



which is equivalent to : ∂ 2 p∗2 1 = 2 2 n − 2n + 2 ∂θ1



 ∂ 2 p∗1 00 − U (θ) ∂θ12

(4.17)

Equations (4.16) and (4.17) form a system whose unique solution is :  ∂ 2 p∗1 n−2 00 00 00 − mc (θ) = U (θ) − mc (θ) n(n − 1) ∂θ12  ∂ 2 p∗2 1 = − U 00 (θ) − mc00 (θ) 2 n(n − 1) ∂θ1

(4.18) (4.19)

The profit function, its first and second derivatives with respect to the own efficiency of the car manufacturer are : e1 − K π1 (θ1 ) = (p∗1 − mc(θ))X    ∂p∗1 e1 + (p∗ − mc(θ)) d X e1 (p∗ ) − mc0 (θ1 ) X 1 ∂θ1 dθ1  2 ∗   ∗  ∂ 2 π1 ∂ p1 ∂p1 d e ∗  00 0 e = − mc (θ ) X + 2 − mc (θ ) X1 (p ) 1 1 1 ∂θ1 dθ1 ∂θ12 ∂θ12 ∂π1 = ∂θ1



+

(p∗1

d2  e ∗  − mc(θ)) 2 X1 (p ) dθ1

At the equilibrium, the value of the second derivative of the expected demand of the first car manufacturer with respect to its own car efficiency is :  d2  e ∗  N (n − 1)2 X1 (p ) = U 00 (θ) − mc00 (θ) 2 3 µn dθ1 43

So :  ∂ 2 π1 N (n2 − n − 1) 00 = U (θ) − mc00 (θ) 2 2 µn (n − 1) ∂θ1

(4.20)

θ defines a local symmetric equilibrium in efficiency if equation (4.15) holds and if the second derivative of the profit is negative in θ. If we assume n ≤ 2, the second condition is equivalent to : U 00 (θ) − mc00 (θ) < 0

(4.21)

We can summarize the preceeding analysis in the following proposition. Proposition 9 A necessary condition for θ to describe a local symmetric equilibrium in efficiency is that : U 0 (θ) − mc0 (θ) = 0 U 00 (θ) − mc00 (θ) ≤ 0 If the second inequality is a strict inequality, then the two conditions are sufficient. If it is an equality, then it is possible that θ does not describe a local equilibrium. We will now try to find some simple conditions of existence and unicity of a global symmetric equilibrium in efficiency. Existence and unicity of an equilibrium Necessary conditions for θ to describe a local symmetric equilibrium in proposition 9 are quiet general, and cover almost all cases (as we saw previously, the only case when we do not know whether θ describes an equilibrium is when U 00 (θ) − mc00 (θ) = 0, which will most probably not happen). We will now use U(θ) calculated in section 2.3, and use the following assumption on mc(θ) : Assumption 1 The marginal production cost function mc(θ) verifies the two following properties :    d 1 mc >0 ∀q ∈]0; +∞[, dq q    d2 1 ∀q ∈]0; +∞[, mc >0 dq 2 q We assume indeed that mc(θ) is a convex, increasing function of 1/θ. This is quiet intuitive if we remind that 1/θ is the miles per gallon efficiency of the car. This assumption, as we will see later, gives good results concerning existence and unicity of a global symmetric equilibrium in efficiency, which 44

would not have been the case if we had simply assumed that mc(θ) was decreasing and convex in θ. Observe that :      d 1 1 0 1 mc = − 2 mc (4.22) dq q q q d2 dq 2



      1 2 1 0 1 00 1 mc = 3 mc + 4 mc q q q q q

(4.23)

which implies that mc(θ) is decreasing and convex. We know from equation (2.8) that :   δµa U(θ) = µa ln 2(pG θ + pf ) Changing variables :     1 δµa q U = µa ln q 2(pG + pf q) It implies :    d 1 µa pG U = >0 dq q pG q + pf q 2 d2 dq 2

   µa pG (pG + 2pf q) 1 U =− <0 q (pG q + pf q 2 )2

(4.24)

(4.25)

So d/dq(U(1/q)) is a positive, decreasing function of 1/q. It decreases from +∞ to 0. Under assumption 1, d/dq(mc(1/q)) is a positive, increasing function of 1/q. As a consequence :       d 1 d 1 e ∗ ∃!q ∈ R+ , U − mc e dq e =0 dq q q q=q q=q     1 1 1 1 ⇔ ∃!q e ∈ R∗+ , − e 2 U 0 = − e 2 mc0 e (q ) q (q ) qe     1 1 ⇔ ∃!q e ∈ R∗+ , U 0 = mc0 e q qe ⇔ ∃!θe ∈ R∗+ , U 0 (θe ) = mc0 (θe ) And we also deduce from assumption 1 that : U 00 (θe ) − mc00 (θe ) < 0 So θe describes a local symmetric equilibrium in efficiency. We can demonstrate that under assumption 1, this equilibrium is a global one. 45

Proposition 10 Under assumption 1, there exists a single θe so that ∀i ∈ {1, .., n}, θi = θe describes a global symmetric equilibrium in pure strategies. Proof : assume all car manufacturers but the first choose to make cars of efficiency θe . Assume the first one chooses to make cars of efficiency θ 6= θe . e1 > X e2 . This implies (from the demand expression (3.3)): Assume X  exp

   1 1 ∗ ∗ (U(θ) − p1 ) > exp (U(θ) − p2 ) µ µ

⇔ U(θ) − p∗1 > U(θe ) − p∗2 Replacing p∗1 and p∗2 by the equilibrium equations from proposition 8 : ⇔ U(θ) − mc(θ) −

µN e1 N −X

> U(θe ) − mc(θe ) −

⇔ U(θ) − mc(θ) − U(θe ) + mc(θe ) >

µN e1 N −X



µN e2 N −X µN e2 N −X

Assumption 1 implies that θe is a maximum of funtion U − mc. As a consequence : ⇒

µN e1 N −X

<

µN e2 N −X

e1 < X e2 X This is absurd. So the expected demand of the first car manufacturer is smaller than the expected demands of its competitors. Thus it is smaller than if it made cars of efficiency θe . In the same way, we deduce from proposition 8 that so is its margin. If the margin and the expected demand are lower than if the car manufacturer made cars of efficiency θe , so is its profit. As a consequence, θe maximises the first car manufacturer’s profit if all other car manufacturers make cars of efficiency θe . ∀i ∈ {1, n}, θi = θe describes a global equilibrium in quality. Q.E.D. We observe that θe does not depend on the number of firms n on the market. Now that the study of the second stage of our game is complete, we will proceed to the study of the third stage. From now on we will consider that assumption 1 holds.

4.2.3

Free-entry equilibrium

The remain step of the three-stages game to analyse is the entry decision. Remind that we only consider symmetric equilibria. For a given number of 46

firms n, the equilibrium efficiencies are identical and equal to θe , solution of the following equality : mc0 (θe ) = U 0 (θe ) Similarly, prices are all identical and equal to : pe = mc(θe ) + µ

n n−1

As a consequence, the expected profits of the car manufacturers are all identical and equal to : π=

N e (p − mc(θe )) − K n

⇔π=

µN n−1

This equation is equivalent to (3.6) : the analysis made in section 3.2 concerning the number of firms holds here : the equilibrium number of firms is ne = N µ/K + 1, one firm larger than the socially optimal number of firms. The fact that the number of firms at the free-entry equilibrium is the same here as previously, when we considered that efficiencies were fixed, is not surprising. The two questions are indeed not correlated as far as each consumer buys one and only one car. We could imagine that if the demand for cars was flexible, car manufacturers would have a tendancy to lure them to the market, and the choice of efficiency would be biased. This is one possible extension of this work. It is also due to the fact that we did not consider research and development costs. In this case, the number of firms on the market would also depend on the efficiency level, that could also lead to distortion. It is another possible extension of this work. We will now proceed to the welfare analysis of the game.

4.2.4

Welfare analysis

Our analysis will be limited to symmetric sets of efficiency. The welfare analysis here is similar to the one in section 3.2.3. The consumer surplus is : N µ ln(n) −

n X

ei + N U(θ) pi · X

i=1

the total profit of firms is : n X

ei − N mc(θ) − nK pi · X

i=1

47

The social welfare is therefore equal to : W = N µ ln(n) + N (U(θ) − mc(θ)) − N µ ln(µ) − nK

(4.26)

It depends on two variables : n and θ. It appears, as previously, that the social welfare does not depend on prices. The behaviour of the welfare function with respect to n is exactly as described in section 3.2.3. Its behaviour with respect to θ is the following : ∂W = N (U 0 (θ) − mc0 (θ)) ∂θ and : ∂2W = N (U 00 (θ) − mc00 (θ)) ∂θ2 Under assumption 1, there exists a single θo verifying these two conditions, and it is equal to θe . Proposition 11 The efficiency of cars made by the car manufacturers when they compete both in prices and efficiency is socially optimal at the equilibrium. We observe that the problems of the number of firms and the efficiency level are not linked. As a consequence, the problem of efficiency regulation and entry regulation are not linked. As noted in the previous section, this is due to the fact that the demand for cars is totally rigid. Releasing this assumption would certainly lead to distortions in the way efficiency is chosen and in the number of firms on the market. It would also certainly lead to a much greater complexity in the regulation problem.

4.3

Equilibrium in efficiency with emissions

We reintroduce here an externality already studied before, environmental pollution. We keep the modelization presented in section 3.3 : environmental damage due to emissions is proportional to distance driven. We will not study the equilibrium, which is exactly the same than in the preceeding section, since users do not take the environmental costs into account when they choose the distance they travel. We will directly proceed to the welfare analysis of the problem, then we will focus on the regulation question.

4.3.1

Welfare analysis

The welfare analysis is very similar to the case without externalities. The only difference is the environmental pollution term. The private expected utility function is : U(θ) =

δµa 2(pG θ + pf ) 48

As calculated in section 3.3.1, the social expected utility function is : U s (θ) =

δµa 2((pG + η)θ + pf )

The consumer surplus, including the environmental cost, is : N µ ln(n) −

n X

ei + N U ∫ (θ) pi · X

i=1

the total profit of firms is : n−1 X

ei − N mc(θ) − nK pi · X

i=0

The social welfare is therefore equal to : W = N µ ln(n) + N (U s (θ) − mc(θ)) − N µ ln(µ) − nK

(4.27)

As a consequence, the socially optimal efficiency of cars is the single solution of: mc0 (θo ) = −

2(pG + η) 2((pG + η)θo + pf )

(4.28)

From assumption 1, we deduce that θo maximises the profit function. Furthermore : ∀η > 0, −

2(pG + η) 2pG <− 2((pG + η)θo + pf ) 2(pG θo + pf )

As a consequence, under assumption 1, we can deduce the following property : Proposition 12 The socially optimal efficiency level with environmental pollution is the single solution of the following equation : mc0 (θo ) = −

2(pG + η) 2((pG + η)θo + pf )

At the equilibrium, the efficiency level is too high, which means that cars consume too much gas : θe > θo Figure(4.1) illustrates this property. The expected distance driven by car users is : E(d)e =

µa pG θ e + p f 49

θo

θe

θ mc’(θ) U’(θ) Us’(θ)

Figure 4.1: Welfare analysis with congestion when at the social optimum it should be : E(d)o =

µa (pG + η)θo + pf

It is not clear whether he expected distance is higher at the social optimum than at the equilibrium, since two effects compete. First, drivers tend to drive less since they take into account the environmental cost of driving. Second, they tend to drive more since the efficiency of cars is higher. We will now examine how possible it is to reach the social optimum thanks to regulation tools.

4.3.2

Regulation with environmental pollution

The point of this section is to verify whether it is possible or not to reach the social optimum using regulation tools. We will consider the use of an excise tax on gas τG . If this tax is applied, the equilibrium expected distance is : E(d)(τG ) =

µa (pG + τG )θe (τG ) + pf

where the equilibrium car efficiency is the unique solution of : mc0 (θe (τG )) = −

2(pG + τG ) 2((pG + τG )θe (τG ) + pf )

This equation imples that θe (τG ) decreases with τG . This is conform to our intuition : the higher the price of oil, the more users want efficient cars, the more efficient the cars made by the manufacturers. The social welfare is equal to : W e (τG ) = N (U s (θe (τG )) − mc(θe (τG ))) + C t 50

where C t stands for a term not depending on θ. Therefore : ∂θe ∂W e = N (U s 0 (θe (τG )) − mc0 (θe (τG ))) ∂τG ∂τG As θe (τG ) decreases with τG , the tax level which allows to reach the social optimum is the one that verifies : θe (τG ) = θo There is a unique such τG , equal to η. Proposition 13 When there is environmental pollution, it is possible to reach the social optimum by setting up an excise tax on gas equal to the marginal environmental cost of driving. τG∗ = η

(4.29)

With this tax, both expected distance and car efficiency level are at their socially optimal levels. The reason why it is possible to correct two decisions with only one tax is the fact that there is only one externality to correct, and that the efficiency level chosen by the car manufacturers is optimal for the users. Users would indeed choose the same efficiency level if they could decide in the place of the car manufacturers. As a consequence, if a tax internalizes the social cost of driving, then the efficiency level is socially optimal, because social costs are taken into account in its choice, through the preferences of the customers. As we will see in the next section, this not the case anymore when there are two externalities which are imperfectly correlated.

4.4

Equilibrium in efficiency with emissions and congestion

In this section, we will introduce another externality which has already been studied in section 3.4. We will first study the equilibrium, then the social welfare, and finally the regulation problem, in which some interesting and quiet counter-intuitive results are to be exposed.

4.4.1

Equilibrium characterization

Congestion is modeled exactly as in section 3.4. Remind from section 2.3 that there is a density of activities on the road, indexed by l, each of them valuated :   Z N 2l Ul = εl − pG θ + pf + α d(k)dk δ 0 51

As in the preceeding sections, we calculate the expected distance driven by any consumer given the expected distances driven by all other consumers : µa

d=

pG θ + pf + α

RN 0

d(k)dk

It will prove useful to consider it is the d that maximises :   Z N µa ln(d) − pG θ + pf + α d(k)dk d

(4.30)

0

This equation means that we can consider the expected distance is the distance a driver would chose if he wanted to optimize a private utility function consisting of two parts : the utility of driving distance d, equal to µa ln(n), RN and the marginal cost of driving, equal to pG θ + pf + α 0 d(k)dk, times the distance itself. Keeping in mind this alternative interpretation will help in the next part of our work. Proposition 14 With emissions and congestion, there is a unique symmetric equilibrium in efficiency. The equilibrium expected distance de and efficiency level θe are the unique solution of the following system of equations : 

pG θe + pf + αN de = µdea mc0 (θe ) = −pG de

(4.31)

Proof : unicity of the solution. We know that the first equation has a single positive solution in de (θe ). This solution is : de (θe ) =

µa pG θe +pf 1 2

+

q

1 4

Which implies :   pG e 1 d = (q e )2 qe

+

µa αN (pG θe +pf )2

µa pG q e +pf (q e )2 2

+ qe

q

(pG +pf q e )2 4

(4.32) + µa αN (q e )2

which is a positive and decreasing function of q e . The second equation of system (4.31) is equivalent to :      d 1 pG e 1 mc e = e 2d dq q (q ) qe Under assumption 1, the LHS of this equation is a positive and increasing function of q e , whereas we just saw that the RHS is a positive and decreasing function of q e . So there exists a unique q e so that the former equation is 52

verified. This is equivalent to the fact that there is a single θe so that the second equation of system (4.31) holds. We must now demonstrate that θe , solution of this system, describes indeed an equilibrium in efficiencies, and that de , solution of this system, is the corresponding expected distance covered by car users at the equilibrium. We will set up a new hypothesis for the following of our study : Assumption 2 Car manufacturers do not take into account the change of the overall level of congestion when they decide the efficiency of the car they make. This hypothesis is quiet restrictive, but not unrealistic. It is indeed reasonable to assume that car manufacturers study the transport expense of their customers to determine the car efficiency, not to assume that they also take into account the effect on the overall congestion level of the release of a new, more efficient car model. Furthermore, they have little incentives to consider that effect, as far as users themselves certainly do not take into account the decrease of the overall congestion level when they buy a more efficient car. Assume then that all car manufacturers provide car of efficiency θe . Then all car users drive the same expected distance, solution of the following equation : pG θe + pf + αN de =

µa de

Assume a driver uses a car of efficiency θe . Then the expected distance he will drive is de , solution of the former equation. Assume now that he considers buying a car of efficiency θ. Then the expected distance d(θ) he expects to drive with this car is the solution of equation : pG θ + pf + αN de =

µa d(θ)

and the utility he expects from this car would be :   δµa U(θ) = µa ln 2(pG θ + pf + αN de )   δ ⇔ U(θ) = µa ln d(θ) 2 The derivative of d(θ) with respect to θ verifies (change in congestion not taken into account) : pG = −

µa 0 d (θ) d2 (θ)

⇔ d0 (θ) = −

pG d2 (θ) µa 53

So the derivative of the utility function with respect to θ is : U 0 (θ) = µa

d0 (θ) d(θ)

U 0 (θ) = −pG d(θ) From equation (4.32), we therefore know that U(θ) verifies the two following conditions :    1 d U >0 dq q    d2 1 U <0 2 dq q which ensures that propositions 9 and 10 hold here, and as a consequence that the solution of system of equation (4.31) describe the equilibrium level of efficiency and expected distance driven by car users. Q.E.D. The characterisation of the equilibrium given in proposition 14 is not the most natural in our framework, since d, the expected distance variable, is not a decision variable. It is just the expected distance covered by car users, the actual distance they cover being the realisation of a random variable. But it is easier to handle, and more intuitive. The first equation of system (4.31) can indeed be considered as the equalisation, by the car user, of the marginal cost of driving (gas, other operating costs, and congestion) and the marginal utility (more activities available). The second equation can be considered as the equalisation, by the car manufacturer, of the marginal cost of increasing the efficiency of the cars they make and the resulting marginal decrease of the cost of driving for the drivers due to this increase of efficiency, equal to the variation of the marginal cost of driving times the distance. This makes sense : the marginal production cost is supported by the car user in the end, so he is ready to buy a more efficient car only if it reduces the overall cost for him. As the driver covers distance d, and pays (pG θ + pf + αN d) for each kilometer he drives, the marginal change in his expense due to a marginal change in θ is pG d. Through competition, the preference of the driver is taken into account by the car manufacturers. We will now focus on the welfare analysis.

4.4.2

Welfare analysis

We will now proceed to the welfare analysis as if there were two decision variables : θ and d. We will only consider symmetric sets of distances and efficiencies. We will first define a working function that will prove useful later, then we will characterize the social optimum, and compare it with the equilibrium. 54

As noted in section 4.4.1, the expected distance can be considered as the distance maximising the following function :   Z N µa ln(d) − pG θ + pf + α d(k)dk d 0

And the value of this function when d equals the expected distance driven by car users is equal to the utility they draw of driving a car of efficiency θ :   Z N µa ln(E(d)) − pG θ + pf + α d(k)dk E(d) = µa ln(E(d)) − µa 0

to a constant µa . By an analoguous reasonment, we will now consider a working function denoted W and that we will refer as the “pseudo-welfare function”, function of two variables, θ and d, defined as follow.   δ W(d, θ) = −N mc(θ)+N µa ln d −N ((pG +η)θ+pf +αN d)d+C t (4.33) 2 For a given θ, the partial derivative of this function with respect to d is : ∂W N µa = − N ((pG + η)θ + pf + 2αN d) ∂d d

(4.34)

There exists a unique do , so that the pseudo-welfare increases if d is smaller than do and decreases if it is higher. This do is the solution of : µa = ((pG + η)θ + pf + 2αN do ) do

(4.35)

We know from section 3.4.2 that for a given θ, the expected distance driven by car users is indeed the unique solution of this equation. As a consequence, when d is equal to do for a given θ, W is indeed equal to the social welfare for a given θ (to a constant independant from d and θ). So, if W has a maximum in a point (do , θo ), then θo is the socially optimal level of efficiency. The derivative of W with respect to θ is : ∂W = −N mc0 (θ) − N (pG + η)d ∂θ

(4.36)

so it is null if the following equation is verified : mc0 (θ) = −(pF + η)d

(4.37)

As it will be demonstrated in the proof of the next proposition, the system of the two equations (4.35) and (4.35) has a single solution (do , θo ), which maximises the pseudowelfare function. As we saw preceedingly, the pseudowelfare function behaves like the welfare function when d = do . So θo maximises the social welfare. 55

Proposition 15 With emissions and congestion, there is a unique social optimum. The equilibrium expected distance do and efficiency level θo are the unique solution of the following system of equations :  (pG + η)θo + pf + 2αN do = µdoa (4.38) mc0 (θo ) = −(pG + η)do Proof : We must prove that there is a single solution to system (4.38). This is done in a very similar way than in the proof of proposition 14. We know from equation (3.23) that the single solution in d of the first equation of system (4.38) is : µa (pG +η)θ+pf

o

d (θ) =

1 2

+

q

1 4

+

2µa αN ((pG +η)θ+pf )2

The reasonment led in the beginning of the proof of proposition 14 is valid here, there is a unique θo so that : mc0 (θo ) = −(pG + η)do (θo ) and : ∀θ < θo , ∀θ > θo ,

mc0 (θ) pG + η mc0 (θ) do (θ) > − pG + η

do (θ) < −

We will not prove here formally that W has a single optimum, but we will rather give a graphical intuition of it. Considering that system (4.38) is equivalent to the following system : d =

µa (pG +η)θ+pf 1 2

d = −

+

q

1 4

+

2µa αN ((pG +η)θ+pf )2

mc0 (θ) pG + η

Considering also that the first of these two equations is the locus of the plan {(θ, d)/ θ > 0, d > 0} where the partial derivative of W with respect to d is null and the second one is the locus of the plan where the partial derivative of W with respect to θ is null, and considering the former observations on the behaviour of the pseudo-welfare functions with respect to its variables, the isowelfare lines behave like in figure (4.2). W decreases in both variables in the upper-right zone, increases in both variables in the lower-right zone, increasing in θ and decreasing in d in the upper-left zone, and decreasing in θ and increasing in d in the lower-right zone. Thus we observe that the point (θo , do ) is the single, global optimum of the pseudo-welfare function, 56

d

∂W =0 ∂d

do

∂W =0 ∂θ

θ

θo

Figure 4.2: Pseudo-welfare behaviour with congestion and, as a consequence, θo is indeed the socially optimal level of efficiency of cars, and do the expected distance driven by car users at the social optimum. Considering that the equilibrium efficiency θe and expected distance de are the unique solution of system of equations (4.31), that : ∀θ > 0, de (θ) > do (θ) ∀θ > 0, −

1 1 mc0 (θ) > − mc0 (θ) pG pG + η

we can conclude that the equilibrium point (θe , de ) is in the upper right zone of figure (4.2), where both partial derivatives of the pseudo-social welfare function are negative. Whereas intuition tells us that at the equilibrium without regulation, drivers drive too much and cars are not efficient enough, it is important to note that the model does not guarantee that. Indeed, the zone where the equilibrium point can be is slightly larger than the mere rectangle defined by {(θ, d)/ θ > θo , d > do }. Proposition 16 The equilibrium point verifies at least one of the two properties, but not necessarily both : θe > θo d e > do Illustration of this result is given in figure (4.3). The social welfare analysis demonstrates without surprise that the equilibrium point is not at the social optimum. We will therefore focus on the regulation question : is it possible to reach the social optimum thanks to regulation tools? If it is possible, how many tools are necessary? 57

d de

θe

θ

Figure 4.3: Equilibrium localisation

4.4.3

Regulation with one excise tax on gas

The regulation question is not as simple in this case as in the other ones. Our intuition is that as far as there are two decisions variables and two imperfectly correlated externalities at stake, two regulation tools will be necessary to reach the social optimum. However, it seems that the accent is currently on regulation through one tool, which is the excise tax on gas. We will therefore study regulation with one tool, and try to find out what is the socially optimal point we are able to reach with only excise taxes on gas. The excise tax level is noted τG . We consider assumptions 1 and 2 hold. The application of the excise tax on gas just results in a change of price of gas from pG to pG + τG in the view of the car users and the car manufacturers. Therefore property 14 holds, and the equilibrium is described by the following system of equations : 

(pG + τG )θe (τG ) + pf + αN de (τG ) = deµ(τaG ) mc0 (θe (τG )) = −(pG + τG )de (τG )

The only positive solution in de of the first of these equations is : µa (pG +τG )θe (τG )+pf

e

d (τG ) =

1 2

+

q

1 4

+

µa αN ((pG +τG )θe (τG )+pf )2

So the second equation of system (4.39) is equivalent to : (pG +τG )µa (pG +τG )θ+pf

0

mc (θ)) = −

1 2

+

q

1 4

+

µa αN ((pG +τG )θ+pf )2

58

(4.39)

This equation has a single solution in θe (τG ). When θ > θe (τG ) the LHS is lower than the RHS, it is the contrary when θ > θe (τG ). The LHS does not depend on τG , whereas for any given θ, the RHS decreases in τG . So θe (τG ) is a decreasing function of τG . Illustration of this demonstration is provided in figure (4.4) : the LHS is the curve that does not move. The RHS is slightly displaced towards the bottom, therefore shifting the intersection point to the left. Note that the vertical coordinate is not d on this figure.

θe(τG)

θ

Figure 4.4: Cars are more efficient if gas is more taxed For any tax level τG , the social welfare is equal to the te value of the pseudo social welfare function defined in section 4.4.2 : W = W (θe (τG ), de (τG )) As a consequence, the only way to reach the first best social optimum is to find the level of tax τG∗ so that θe (τG∗ ) = θo and de (τG∗ ) = do . As stated in the following proposition, it is impossible : Proposition 17 There exists no value of τG so that :  e ∗ θ (τG ) = θo de (τG∗ ) = do Proof : Assume there exists τG∗ so that the former system of two equations hold. Then, from proposition 15 : mc0 (θe (τG∗ )) = (pG + η)de (τG∗ ) but : mc0 (θe (τG∗ )) = (pG + τG )de (τG∗ ) this implies : τG∗ = η 59

and therefore : µa e d (η)

= (pG + η)θe (η) + pf + 2αN de (η)

µa e d (η)

= (pG + η)θe (η) + pf + αN de (η)

which is false unless α = 0. Q.E.D. As predicted at the beginning of this section, it is impossible to reach the first best social optimum with only one regulation tool if there are two decision variables and two externalities. We observe that if α = 0, that is to say if there is no congestion, then it is possible to reach the social optimum with only one tool, as stated in proposition 13. The model gives no guarantee that de decreases. Its behaviour depends indeed on the shape of the mc(θ) curve. For this reason, it is difficult to analyse any further the one-tool regulation problem from a theoretical point of view. We will prefer here a numerical approch, which will be the subject of the next section, before we finally study the two regulation tools problem.

4.4.4

Numerical example

We consider only simple hypothesis in this example. We do not aim at realism, this is not an econometric work. We just tried to provide roughly consistent parameters, in order to give an intuition of what can happen. θ is considered as the consumption of gas in liter to drive 1 kilometers. We took a simple marginal production cost function corresponding to assumption 1 : mc(θ) =

13, 57 + 3230 θ2

so that the marginal production cost of a car that consumes 0,07 liters per kilometer is approximately 6000 e and the marginal production cost of a car that consumes 0,06 liters per kilometer is approximately 7000 e. We gave the following values to the different parameters : pG = 0, 3e per 1 liter, η = 0, 2e per 1 liter, α · N = 0, 0000001e per kilometer (as this term is multiplied by the expected distance driven by the other car users in the driving marginal cost, this coefficient has to be very low in order not to overwhelm the other costs), and µa = 5000, so that the expected distance driven by car users is of the order of 105 kilometers. The excise tax on gas varies from 0 to 1e per liter of gas. Figure 4.5 illustrates both the equilibrium system of equation (4.31) and the social optimum system of equations (4.38). At the equilibrium, cars consume 9 liters of gas per 100 km, which more than the socially optimum efficiency, equal to 8,7 liters per 100 km. The distance car users drive at the 60

450000 400000 350000 300000 250000 200000 150000 100000 50000 0 0,05

0,06

Equibrium efficiency

0,07

0,08

Equilibrium distance

0,09 Optimum efficiency

0,1

0,11

Optimum distance

Figure 4.5: Equilibrium and social optimum, numeric example equilibrium, more than 120 000 km, is far greater than the socially optimal one : 81 000 km. Figure 4.6 illustrates the locus described by the different equilibrium points when the excise tax on gas increases. In this case, both θ and d decrease, which indicates that at the second-best social optimum (that is to say, when the excise tax on gas is set at its optimal level) cars are too efficient with respect to the social optimum (consider figure 4.2 for a graphical proof of this statement : if both d and θ decreases, the curve can be tangent to isowelfare lines only in the upper left quarter of the graph). Figure 4.7 is important because it illustrates one of the main reason why overtaxing gas is not welfare-improving. Marginal production cost indeed increase, and this increase weights on car users. Finally, figure 4.8 exposes the behaviour of the social welfare with respect to the gas tax level. We observe the optimal tax level is about τG∗ = 0, 3e per liter. When this tax is applied, the efficiency of cars is approximately 0,8 liters per 100 km, which is lower than the socially optimal efficiency. In this case, increasing the car efficiency level any further is no mean to enhance the social welfare, as we will demonstrate in the next section. This numeric example is far from being realist. Anyway, its merit relies in its ability to illustrate some effects that go against current common intuition about cars’ efficiency regulation. In particular, it indicates the possibility that in many countries where gas is already heavily taxed, cars are already too efficient and no measure should be taken in order to increase their efficiency any further. We will now study the case with two regulation tools. 61

300000

250000

200000

150000

100000

50000

0 0,05

0,06

0,07

0,08

Optimum efficiency

0,09

Optimum distance

0,1

0,11

Equilibrium loci

Production cost tau_G

Figure 4.6: Equilibrium locus when excise tax on gas varies 5800

Marginal cost (€)

5600 5400 5200 5000 4800 4600 4400 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Excise tax on gas Marginal production cost Page 1 Figure 4.7: Marginal production cost

4.4.5

Regulation with two taxes

We will here consider the use of another regulation tool, in addition to the excise tax on fuels, focused on the efficiency of cars. We will assume the regulator knows the optimum level of efficiency θo , and charges or subsides the efficiency of cars which are different from the optimal level. As a consequence, the car manufacturer making a car of efficiency θ should pay or 62

Welfare tau_G

47400 47200

Social welfare

47000 46800 46600 46400 46200 46000 45800 45600 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Excise tax on gas Social welfare level Page 1 Figure 4.8: Social welfare

receive the following amount : τθ (θ − θo ) for each car he sells. Our objective here is to design the socially optimal τG and τθ . This regulation tool only appears as a change in the marginal production cost from the car manufacturers’ point of view. The marginal production cost becomes mc(θ) + τθ (θ − θo ), and thus its first derivative with respect to θ is mc0 (θ) + τθ 1 . Proposition 4.31 still holds here, and the equilibrium efficiency θe (τG , τθ ) and expected distance de (τG , τθ ) is the single solution of the following system of equations :  (pG + τG )θe (τG , τθ ) + pf + αN de (τG , τθ ) = de (τµGa,τθ ) mc0 (θe (τG , τθ )) + τθ = −(pG + τG )de (τG , τθ ) As we saw before, whatever the values of τG and τθ , the social welfare is equal to the pseudo-social welfare function (defined by equation (4.33)) evaluated in (θe (τG , τθ ), de (τG , τθ )) : W = W (θe (τG , τθ ), de (τG , τθ )) The consequence of this statement is the same as in the preceeding section : the social welfare is optimal if and only if θe (τG , τθ ) = θo and de (τG , τθ ) = do , 1

It appears clearly here that the constant term τθ · θo of the regulation tool plays no role. We introduced it only because it seems reasonable not to charge the car manufacturer if he makes a car whose efficiency is socially optimal.

63

which is equivalent to :  (pG + η)θe (τG , τθ ) + pf + 2αN de (τG , τθ ) = de (τµGa,τθ ) mc0 (θe (τG , τθ )) = −(pG + η)de (τG , τθ ) This combined with the system of equations describing the equilibrium, implies the following result :  (τG − η)θe (τG , τθ ) = αN de (τG , τθ ) (4.40) τθ = (η − τG )de (τG ) The first equation implies that the socially optimal excise tax on gas τG∗ is higher than η. Thus, the second equation implies that the socially optimal coefficient τθ∗ is negative, which means that car manufacturers have to pay if they want to make cars more efficient than θo . Proposition 18 It is possible to reach the social optimum using two regulation tools : an excise tax on fuel τG∗ , and a tax on efficiency τθ∗ . These tools verify :  ∗ τG > η τθ < 0 This result looks counter-intuitive, whereas in fact it is not. In order to reach the social optimum, one solution is indeed to apply a tax on fuel, in order to internalize the environmental and other external costs as precisely as the regulator can. But, applying a tax on car users results in a high increase of the transport expense. Car users are therefore ready to buy more expensive cars if it allows them to save money on fuel. By the game of the market, this willingness to buy more efficient cars is taken into account by the car manufacturers, which will make more efficient cars, thus allowing car users to drive further. In this way, part of the intended effects of a tax on gas are evicted. It is possible for the regulator to correct this trend towards too efficient cars, by intervening directly in the efficiency choice by the car manufacturers, for example thanks to the tool we precedently exposed. Of course, since the car manufacturers have a tendancy to make too efficient cars, the regulator will want to charge them if they make too efficient cars, not at all subside them to make more efficient cars.

4.5

Conclusion

Internalizing the choice of efficiency quiet complicates the analysis, but leads to interesting results, particularly concerning regulation. Car manufacturers indeed take into account the preferences of their customers to design their cars, particularly to decide the efficiency of the model they make. As we 64

did not take into account any research and development costs, there are no distortions in the choice of car efficiency : car users would choose the same efficiency if they decided for the car manufacturers. For this reason the efficiency chosen by the car manufacturers is socially optimal when there is no distortion. But neither car users nor car manufacturers take into account the external costs. As a consequence, if there are some externalities, the decisions they will make will not be socially optimal; the distance car users drive and the efficiency of the cars made by the car manufacturers will not be at their socially optimal levels. This can be corrected by the regulator, through the use of regulation tools. We took the examples of two externalities : congestion and environmental pollution. When there is only environmental pollution, it is possible for the regulator to reach the social optimum by setting an excise tax on gas up. Thanks to this tax, correctly calibrated, the environmental cost is internalized in the transport expense of the car users, who, through the game of the market, then lead the car manufacturers to make cars with the socially optimal level of efficiency. This is no longer possible when there are two imperfectly correlated externalities. In this case, an excise tax on gas is not sufficient to reach the social optimum. The intuition of this statement is that by applying a tax on gas, the regulator induces an increase in the car users transport expense. Therefore, car manufacturers will make more efficient cars, since car users will be willing to pay for them, in order to buy less fuel. Car users, having at their disposal more efficient cars, will drive farther, thus evicting a part of the intended effect of the tax on fuel, and inducing undesirable environmental and congestion costs. As we show, it is therefore necessary to consider using two regulation tools. As the problem was the partial eviction of the intended effect of the tax because of a shift in the efficiency level of cars, we considered intervening directly on this efficiency level. The result of our work, consistent with the intuition just exposed, is that it is necessary for the regulator to charge car manufacturers if they want to make too efficient cars. In that way, is is possible to reach the social optimum.

65

Chapter 5

Conclusion 5.1

Conclusions

This work was mainly focused on the questions of efficiency of cars and its regulation. Its first step was to introduce an underlying microeconomic model explaining why people would buy and use cars. By using discrete choice models, we have been able to set a quiet general framework, potentially explaining different driving behaviours, and taking into account a potentially large heterogeneity among all car users, which is more satisfying than if we had assumed they all drove exactly the same distance. As we exposed it later on, we could still use the expected distance as a pseudo decision variable, which proved to be rather useful in the analysis of our problem. We then set up the frame of the car market model. We chose an oligopolistic framework, introducing imperfect competition and heterogeneity among the car users so as to introduce easily competition not only through prices but also through efficiency. We first studied the model assuming that the level of efficiency of cars was fixed. This allowed us to demonstrate that some very classical results of transport economics were still valid. Finally, we assumed that the car manufacturers competed through prices and efficiencies. This allowed us to expose some interesting and counterintuitive results concerning the regulation of car efficiency. It is indeed most often considered nowadays that cars, as a main source of environmental pollution, through the emission, among others, of green-house effect gas, are not efficient enough. The current political and public opinion concerning this problem, when it does not reduces to the mere prohibition of cars, is that we should focus on designing more efficient, less polluting cars. Our work tends to give a opposite intuition. We indeed showed that from quiet reasonable premises, in order to reach the social optimum, the regulator must combine a tax on gas and a tax on efficient cars, thus preventing car manufacturers 66

from making too efficient cars. It appears indeed that the social cost of making efficient cars (production and research and development costs) is not taken into account.

5.2

Limitations and possible extensions

Our work is of course subject to a set of limitations. We considered a symmetric oligopolistic market, whereas if the car market is indeed oligopolistic, it is not symmetric at all. Each firm proposes different car models, corresponding to different customers, each with its own price and fuel efficiency. We also considered a symmetric set of customers, whereas different people, for any reason, will not have the same willingness to travel; and the heterogeneity of the customers is large enough to induce a large heterogeneity in the efficiency of the cars, all other parameters identical (observe for example that in France, for virtually each car model, customers can choose between a gas engine and a diesel engine). We did not take into account research and development costs, and we considered a totally rigid demand for cars. This explains why there was no distortion in the way car manufacturers choose the efficiency of the cars they make. And finally we considered a close economy, which is of course much easier to handle for a regulator, which does not have to cope with international competition. All these limitations should be relaxed to test the consistency of our results. The possible extensions of this work, in link with what has just been exposed, are the following. First, elastic demand of cars should be considered. This may lead to distortions in the way efficiency is chosen by the car manufacturers, since it is a way to make their model more attractive. Second, research and development costs are also to be considered. Multiproduct competition would also give a much better representation of the car market. Finally, non symmetrical cases should be considered, potentially numerically.

67

List of Figures 2.1 2.2 2.3 2.4

Model from a global point of view Model from a driver’s point of view Demand from the searching driver Demand from the searching driver

. . . .

8 8 10 11

3.1 3.2 3.3

Equilibrium and social optimum with emissions . . . . . . . . Equilibrium and social optimum with congestion . . . . . . . Optimal regulation with congestion . . . . . . . . . . . . . . .

24 32 32

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Welfare analysis with congestion . . . . . . . . . . Pseudo-welfare behaviour with congestion . . . . . Equilibrium localisation . . . . . . . . . . . . . . . Cars are more efficient if gas is more taxed . . . . Equilibrium and social optimum, numeric example Equilibrium locus when excise tax on gas varies . . Marginal production cost . . . . . . . . . . . . . . Social welfare . . . . . . . . . . . . . . . . . . . . .

50 57 58 59 61 62 62 63

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Bibliography Anderson, S., de Palma, A. & Thisse, J.-F. (1992a), Discrete Choice Theory of Product Differentation, MIT Press, Cambridge, Massachussetts, London, England, pp. 39–45. Anderson, S., de Palma, A. & Thisse, J.-F. (1992b), Discrete Choice Theory of Product Differentation, MIT Press, Cambridge, Massachussetts, London, England, pp. 161–162. Harrington, W. & McConnell, V. (2003), Motor vehicles and the environment, Technical report, Resources for the Future, Washington. Verboven, F. (2002), ‘Quality-based price discrimination and tax incidence, evidence from gasoline and diesel cars’, RAND Journal of Economics 33(2), 275–297.

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Environmental pollution, congestion, and imperfect ...

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