François Lescaroux

Denis Babusiaux

Car ownership in relation to income distribution and consumers' spending decisions

Les cahiers de l'économie -n° 67

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Car ownership in relation to income distribution and consumers' spending decisions

François LESCAROUX Address for correspondence: Dr François Lescaroux, Direction des Etudes Economiques, Institut Français du Pétrole, 1 et 4, avenue de Bois-Préau -92852 Rueil-Malmaison -France.

The author wishes to thank Olivier Rech for helpful discussions and comments.

Introduction

The increasing needs for mobility at a global level have raised many concerns about the sustainability of our way of living.

From the standpoint of the energy economist, about one half of total oil consumption is used in the transportation sector, where the substitution possibilities are very limited (at least in the medium term). This share should increase in the future as the demand for passenger cars in important emerging countries like China or India is rapidly growing. The surge in oil consumption in the last few years is directly responsible for the oil price hike that has occurred. Moreover, the structural nature of the upward movement in oil demand makes an oil price fall unlikely, unless supply develops even more rapidly. Further, the growing demand for motor fuel puts pressure on the refining sector where the upgrading capacities tend to saturate.

From the environmental economics' point of view, about 25 per cent of global CO 2 emissions from fossil fuels come from the transportation sector alone. The projected growth of this sector raises worries about global warming and clean air because of the limited possibilities of substitution and because of the impossibility to capture the pollutants.

Other important aspects include the needs for infrastructure to develop and for intra-city traffic to adapt and avoid congestion.

As a consequence, a lot of projections of passenger cars' stocks are performed by national and international institutes (IMF, 2005), as well as individual researchers to help politics and industrials in their decision making. Quite all of them rely on S-shaped curves estimated on pooled or panel data, with or without country-specific effects. The most widely used is the Gompertz function [START_REF] Dargay | Vehicle ownership to 2015: implications for energy use and emissions[END_REF][START_REF] Babusiaux | Mondialisation et formes de concurrence sur les grands marchés de matières premières énergétiques : le pétrole[END_REF][START_REF] Dargay | Vehicle ownership and income growth, worldwide: 1960-2030[END_REF].

Letting Car denote the long-run equilibrium level of the passenger car ownership rate and letting Y denote real per-capita income, it can be written as:

( ) ( ) ( ) t t Y Car Car exp exp exp 2 1 ⋅ ⋅ ⋅ = ∞ β β , (1)
where Car ∞ is the saturation level and β 1 and β 2 are negative parameters defining the shape of the curve.

Other S-shaped curves have been proposed, like the log-logistic function used by [START_REF] Lescaroux | The impact of automobile diffusion on the income elasticity of motor fuel demand[END_REF]. It can be written as:

( ) [ ] { } θ λ - ⋅ - + = ∞ t t Y Car Car ln exp 1 , (2)
where λ and θ are two positive parameters defining the shape of the curve 1 .

1 [START_REF] Button | Modeling vehicle ownership and use in low income countries[END_REF] modelled the vehicle ownership rate as a logistic function of time. As the logarithm of percapita income can be approximated locally -and roughly -by a linear time trend, the log-logistic function of per-capita income is closely related to the logistic function of time.

Finally, the car ownership rate can be modeled with a truncated log-quadratic specification, as in [START_REF] Medlock | Car ownership and economic development with forecasts to the year 2015[END_REF]. It is given by2 :

( ) ( ) ( ) (3)

The main limit of the forecasts obtained by simulating these various functions has been expressed very clearly -and crudely -by [START_REF] Greenman | The car park: diffusion models revisited[END_REF]: "Forecasting results were found [...] to be very sensitive to the functional form used. The development of these models therefore tended to become an exercise in sophisticated curve fitting without any theoretical guidance as to the appropriate form to use." To correct for this problem, Greenman proposed a model of car ownership relying on income and automobile diffusion densities which was fitted to data from the UK and Japan.

Another limit of the previous models (that the model proposed by Greenman overcomes) is that they rely just on the mean of the income distribution and do not take into account the shape of its density. The importance of income distribution on passenger car demand has since been confirmed by [START_REF] Storchmann | Long-run gasoline demand for passenger cars: the role of income distribution[END_REF], who showed that "high inequality leads to a higher car stock in poor countries, while it leads to a smaller car stock in rich countries".

Nonetheless, the detailed data on income distribution and households expenditures for transportation that Greenman' s model requires make it difficult to use at a large scale, particularly for emerging countries. On the other hand, the model proposed by Storchmann can be implemented practically for a wide set of countries but, for the sake of simplicity and comparability, it is linear in logarithms, which implies constant elasticities. Such a formulation is at odds with most of the recent literature on passenger car demand and Storchmann acknowledge that this simplification "will be at the expense of consistency" with an individual model that he constructs first.

In this paper, we adopt an alternative approach to overcome the problem pointed out by Greenman: we propose a formal (non-linear) model of passenger car ownership based on income distribution and consumers' spending decisions which can be estimated at a large scale.

In the next section, we present the theoretical model, derive some properties of its dynamics and propose a practical approximation to the formal relation; we also mention some advantages of our model compared to the main alternatives and comment on the influence of some determinants of the car ownership rate, such as income inequality or car purchase cost. Then, in the third section, we concretely apply the model and explain econometrically the car ownership rate in 64 countries between 1986 and 1998 as a function of average per-capita income, income' s dispersion and a country-specific indicator of the "cost/utility" ratio of owning a car. The fourth section presents the results of the projections of car ownership to the year 2030. Finally, the fifth section summarizes our results and consider some implications of the research.

A formal model of car ownership based on income distribution and consumers' spending decisions

We consider a given population and assume that its income distribution is well approximated at time t by the Log-Normal law, with mean t Y and standard deviation For each individual i in the population, we also consider that owning a car procures a utility U i . Given y, the income level of the individual and P, the smallest amount of money needed to own and use a car, individual i chooses to buy a car if P/y ≤ U i (for each individual, there exists a "Cost/utility" ratio which acts as an income threshold beyond which he chooses to buy a car). Let U, the associated real-valued random variable, have a cumulative distribution function, F α , and a density function, f α . Then the probability for an individual to own a car is:

( ) - = ≤ = y P F U y P P y α α 1 . ( 5
)
α is a strictly increasing function of y, whose value in y = 0 is 0 and which tends towards α ∞ when y tends towards infinity, with the saturation level α ∞ being3 :

( ) +∞ ∞ ⋅ = 0 du u f α α . ( 6
)
Therefore, the car ownership rate as a function of average per-capita income is

( ) ( ) ( ) +∞ ⋅ ⋅ ⋅ = 0 , , dy Y Y y f y Y Car t t Y t β α (7)
In fact, the utility of owning a car depends on many factors, like population densities or access to other transport alternatives, which evolve through time as well as car purchase costs do.

Consequently, the function α is not time invariant. We do not indicate this inconsistency to lighten the equations but we keep it in mind.

Empirically, it is possible to estimate the function α when households expenditures surveys are available. [START_REF] Greenman | The car park: diffusion models revisited[END_REF] analyzes the cases of the UK and Japan for example. But for most of the developing countries, these statistics do not exist. Nonetheless, another practical approach is possible. There exists a sequence of simple functions, (α n ), that converges towards α and the convergence is uniform. The car ownership rate can therefore be expressed as:

( ) ( ) ( ) ( ) ( ) ( ) . , , lim , , lim 0 0 ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ∞ + +∞ → +∞ +∞ → dy Y Y y f y dy Y Y y f y Y Car t t Y n n t t Y n n t β α β α (8)
The function α n (y) can be described by a sequence of real numbers, α 0 , ..., α n-1 , Y 0 , ..., Y n-1 :

( ) [ ] [ ] +∞ - - = - + ⋅ + ⋅ = ; 1 2 0 ; 1 1 n i i Y n n i Y Y i n y 1 1 α α α , ( 9
)
where 1 denotes the indicator function (Y 0 > 0 and α n-1 = α ∞ ). Define τ i as:

( ) ( ) ( ) ( ) t i U Y t t Y t i Y g F dy Y Y y f Y i - = ⋅ ⋅ = +∞ 1 , , β τ (10) where ( ) ( ) ( ) ( ) (
)

1 ln 1 ln 5 . 0 ln ln + + ⋅ + + - = β β i t t i Y Y Y g . ( 11
)
With γ i = (α i -α i-1 )/α ∞ , γ 0 = α 0 /α ∞ and by substituting ( 9) and ( 10) in (8), we obtain:

( ) ( ) ( ) ( ) ( ) ( ) ( ) . lim , , , , lim , , , , lim 1 0 0 1 1 1 1 2 0 0 1 1 ⋅ ⋅ = ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ - = ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ = - = +∞ → ∞ ∞ + - = ∞ + - +∞ → ∞ + - - = +∞ → - + n i t i i n Y t t Y n i Y t t Y i i n Y t t Y n n i Y Y t t Y i n t Y dy Y Y y f dy Y Y y f dy Y Y y f dy Y Y y f Y Car i n i i τ γ α β α β α α β α β α (12)
So, the car ownership rate appears as the limit of a sequence of functions Car n :

( ) ( ) . 1 0 - = ∞ ⋅ ⋅ = n i t i i t n Y Y Car τ γ α (13)
As a first approximation, Car n can be approached by:

( ) ( ) ( ) ( ) . 1 ln 1 ln 5 . 0 ln ln 1 1 0 + + ⋅ + + - - ⋅ ≈ ∏ - = ∞ β β α γ n i i t U t n i Y Y F Y Car (14)
Let U * be the truncation of U such that its probability mass is restricted to the subinterval [0; +∞[. When n increases, ( )

∏ i i i Y γ ln
tends towards the expected value of ln(P/U * ), denoted ln(P * ). Consequently, the equation relating the car ownership rate to average per-capita income can be approximated by:

( ) ( ) ( ) . 5 . 0 ln ln 1 2 * ⋅ + + - - ⋅ = ∞ σ σ α P Y F Y Car t U t (15)
This means that, on average, a share α ∞ of the population buys a car when its income level exceeds an income threshold P * . As discussed before, P * depends on many factors and it varies both through time and from one country to another. Intuitively, we could expect a positive correlation between P * and average per-capita income and a negative correlation between P * and time. We will explore this possibility further in the empirical part of our work.

The cumulative log-normal distribution model has an important advantage against the logquadratic model and the Gompertz model: its coefficients are directly interpretable. The dispersion of the income distribution acts on the slope of the S-shaped curve while the income threshold acts on the position of the curve along the income axis: the more unequal the distribution of income, the more gentle the slope and the higher the income threshold (that is, the higher the car purchase cost and/or the lower the utility it gives), the most right-translated the curve. Figure 1 shows these influences.

As we will see in the next section, this allows to introduce heterogeneity between countries in a very simple way. On the contrary, the introduction of heterogeneity in the two other models is rather arbitrary. Indeed, it would not be sensible to estimate a separate equation for each country because the estimation procedure needs observations over the whole range of income; as a consequence, some parameters need to be common for all countries. But the parameters in these models are linked and have no sense individually. For example, [START_REF] Medlock | Car ownership and economic development with forecasts to the year 2015[END_REF] let only the intercept vary and [START_REF] Dargay | Vehicle ownership to 2015: implications for energy use and emissions[END_REF] let only the parameter which determines the curvature at low-income levels vary, whereas [START_REF] Dargay | Income' s effect on car and vehicle ownership, worldwide: 1960-2015[END_REF] let only the parameter which determines the curvature at high-income levels vary. According to our formal model, such an approach is artificial because the curvatures at high-and lowincome levels are linked and determined only by the dispersion of the income distribution (as formulated by Storchmann, "income inequality tends to move car demand from both sides").

Estimation procedure and results

We now turn to the empirical application of the formal model. Up to this point, our formal model was close to the "individual model" of [START_REF] Storchmann | Long-run gasoline demand for passenger cars: the role of income distribution[END_REF]. Differences arise in the practical approximation to the theoretical model. Storchmann uses a formulation which is linear in logarithms whereas our approach is non-linear. We consider that Equation ( 15) describes the long-run equilibrium level of the car ownership rate4 , denoted Car * ,

( ) ( ) ⋅ + + - - ⋅ = ∞ σ σ α 2 * * 5 . 0 ln ln 1 P Y F Car t U t , ( 16
)
and assume a partial adjustment mechanism depending on the rate of change of per-capita income5 :

( ) ( ) ( ) [ ] ( ) ( ) [ ] 1 * 1 2 1 1 ln ln ln ln - - - - ⋅ ⋅ + + = t t t t t t Car Car Y Y h Car Car δ δ , ( 17
)
where h is a function to be determined and t Y is the average per-capita income level (2000US$).

We chose to restrict α ∞ , δ 1 and δ 2 to be common for all countries. Considering the dispersion parameter σ, it does not need to be estimated. Indeed, in the case of a log-normal distribution, there is a one-to-one relation between the Gini coefficient and the standard deviation of the underlying normal distribution:

( ) 1 2 2 - ⋅ = σ U F Gini . ( 18
)
Finally, we allow the income threshold parameter to be country-specific. As this parameter reflects the ratio of the cost of owning a car to the utility it procures, it depends of course on the car purchase costs (including taxes), but it depends as well on less easily quantifiable factors, as the access to other transport alternatives or sociological and cultural factors. Therefore, for the first application of the model, we prefer to consider it as a time-invariant country-specific parameter.

The model to be estimated econometrically thus becomes:

( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ), ln 5 . 0 ln 1 ln ln 1 ln 1 , , 2 1 , 2 , * , 1 , , 2 1 1 , 1 , , 2 1 , ∞ - - - - ⋅ ⋅ + + ⋅ + + - - ⋅ ⋅ + + ⋅ ⋅ - - = α δ δ σ σ δ δ δ δ t i t i t i t i i t i U t i t i t i t i t i t i Y Y h p Y F Y Y h Car Y Y h Car (19)
where the subscripts i and t refer respectively to country i and date t and p i * stands for ln(P i * ).

The most demanding variable in Equation ( 19) is σ because the Gini coefficients are available only for some countries and for a few years. We were nonetheless able to construct a database covering 64 countries between 1986 and 1998 (see Appendix B for more details). Because of its non-linear nature, Equation ( 19) was estimated using maximum likelihood methods by postulating successively a set of different functions h.

After various experiments, the best specification that we estimated is as follows:

( ) ( ) [ ] ( ) ( ) ( ) , 5 . 0 53 . 8 ln 1 ln 07 . 0 ln 07 . 0 1 ln , 2 , , 2 1 , , 1 , 2 1 , , , ⋅ + + - - ⋅ ⋅ + ⋅ ⋅ - = - - - t i t i t i U t i t i t i t i t i t i Y F Y Y Car Y Y Car σ σ (20)
where 8.53 is the median value of ln(P i * ). It appeared that the choice of the function h is not crucial for the other parameters and, notably, α ∞ was not statistically different from 1 in quite all the cases (and still close to 1 when it was significantly different from 1). The detailed estimates of this preliminary specification are reported in Table 1.

The median income threshold is about $5,500 per-capita (2000US$), but the dispersion is rather large, with a standard deviation of about 7,000. We made a first step towards explaining the level of the income threshold. We evaluated the cross-correlation between the estimated parameters, P i * , and the countries' population densities (at the middle of the time sample, in year 1995). When data were available, we also evaluated the cross-correlation with car prices (in year 2003). The coefficients obtained are respectively 0.34 and 0.41. This confirms that concentration of population and car purchase costs are important determinants of the car ownership rate. The construction of a cross-section time-series database covering these two indicators and some others (measures of urbanization or access to other transport alternatives for example) would enable to model explicitly the "Cost/utility" ratio of owning a car. Nonetheless, we found that the evolution of the "Cost/utility" ratio as the economy develops is pretty well synthesized, as a first approximation, by the evolution of per-capita income. We evaluated the cross-correlation between the income threshold parameter and average per-capita income: the coefficient obtained is 0.81 (Figure 2 shows the scatter of points). This is not really surprising because the cost of owning a car tends to be positively correlated with percapita income (the rise in standards of living leads to buy more expensive cars and the greater needs for infrastructure lead to more taxes) while the influence of wealth on the utility of owning a car is ambiguous (the urbanization process and the development of public transportation systems tend to lower it but the demand for leisure and travel increases it).

We tried to take into account in the model the link between the income threshold parameter and per-capita income. From Equation (20), we tried a variety of specifications6 . The best relationship that we found is as follows:

( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) , 5 . 0 ln ln 1 ln ln 1 ln , 2 , , , 2 1 , , 2 1 , 2 1 , , 2 , ⋅ + + ⋅ + - - ⋅ ⋅ + ⋅ ⋅ - = - - - t i t i i t i t i U t i t i t i t i t i t i Y Y F Y Y Car Y Y Car σ σ β γ δ δ (21)
where the β i are country-specific. Table 2 reports the result of the estimation.

Since the estimated value of α is smaller than 1, this is, from Lemma 1 (Appendix A), a logconcave function of ln( t Y ) (as well as Equation ( 20)), which means that the elasticity of car ownership with respect to average per-capita income is decreasing (see Appendix A for an approximation of the elasticity). This result might be disturbing at first sight because the Sshaped pattern implies that the diffusion process accelerates and then decelerates, but this means that the ratio of the changes first increases and then decreases, not the elasticity.

As before, α ∞ was estimated to be statistically not different from 1. This value corresponds to the saturation level estimated by [START_REF] Lescaroux | The impact of automobile diffusion on the income elasticity of motor fuel demand[END_REF] but it is bigger than the ones estimated by Dargay andGately (1997, 1999) and [START_REF] Medlock | Car ownership and economic development with forecasts to the year 2015[END_REF]. The reason for this discrepancy might be the use of different data or it could result from the different S-shaped functions used.

The average adjustment coefficient is 8.6 per cent (with a minimum of 3.9 per cent and a maximum of 12.5 per cent), which means that 90 per cent of full adjustment takes about 26 years. Nonetheless, in a country which is not developing, the adjustment process can be more than two times slower. Our adjustment coefficient is much smaller than the value of 0.205 estimated by [START_REF] Medlock | Car ownership and economic development with forecasts to the year 2015[END_REF], close to the value of 0.09 estimated by [START_REF] Dargay | Income' s effect on car and vehicle ownership, worldwide: 1960-2015[END_REF], close to the values of 0.095 (for rising income) and 0.084 (for falling income) estimated by [START_REF] Dargay | Vehicle ownership and income growth, worldwide: 1960-2030[END_REF] and higher than the value of 0.03 estimated by [START_REF] Lescaroux | The impact of automobile diffusion on the income elasticity of motor fuel demand[END_REF] 7 .

The mean and the median income thresholds, γ.ln( t i Y , ) + β i , are about $8,000 and $5,600 percapita (2000US$) respectively and the standard deviation is about 7,100. The lowest value is about $600 in Ethiopia (maybe because of an important second-hand market) 8 and the biggest values are about $30,000 in Singapore (because of 1) the quotas established and the booming market for Certificates of Entitlement and 2) the incredibly high population density) and in Japan.

The model performs remarkably well in (in-sample) dynamic simulation (see Appendix C). Notably, we are able to reproduce the particular pattern that characterizes countries in Eastern Europe (decrease of average per-capita income in the aftermath of the fall of the Berlin Wall and simultaneous increase of car ownership) and that results partly from the increase in income inequality.

Projections of passenger car stocks to the year 2030

On the basis of assumptions concerning per-capita income growth and demographic changes to the year 2030 (see Appendix B), we used our model of car ownership to project how the car stock could evolve in each country according to different hypothesis regarding variations in income inequality and changes in the "Cost/utility" ratio.

The reference case corresponds to a "business-as-usual" scenario. On one hand, we assumed that the estimated relationship between the income threshold level and per-capita income would remain stable over the whole forecast sample (this means that we do not take into account the possibility that automakers could propose cheap cars designed for developing countries, which would accelerate the expansion of their car stocks). On the other hand, we also considered that the structures of income distribution would stay the same in each country9 , except in Europe where we imagined that a standardisation of fiscal policies should cause a kind of partial convergence in the Gini coefficients. On the basis of past data, we postulated 5 sub-groups of European countries: very egalitarian Scandinavian countries (Denmark, Finland, Norway and Sweden), very un-egalitarian Anglo-Saxon countries (Ireland, UK), egalitarian Continental countries (Austria, Bulgaria, Estonia, Germany, Hungary, Latvia, Luxembourg, Netherlands), un-egalitarian Southern countries (Greece, Portugal, Spain) and mixed countries (Belgium, France, Switzerland), halfway between the Continental and the Southern models. The assumptions and results of this projection are reported in Table 3 and Figure 3 summarizes the forecasts 10 . As would be expected, the projections indicate that the growth rates in both car ownership and car stock should be much larger in poor countries than in rich countries. Particularly, Asian developing countries should enter a period of strong acceleration in the diffusion process of private automobile, with average annual ownership' s growth rates reaching 11.6 per cent in China, 11.0 per cent in Bangladesh, 9.7 per cent in India and 5.8 per cent in Indonesia and in Pakistan. Given the weight of these five countries in the present and future world population, they will account for an increase of almost 0.5 billion of cars (that is, more than one half of the total increase for the 64 countries). By 2030, the top six countries for the passenger car stock should be the United States (299 millions), China (288 millions), India (178 millions), Japan (78 millions), Germany (60 millions) and France (44 millions). From 1998 to 2030, the top six countries for the increase in the car stock should be China (+282 millions), India (+173 millions), the United States (+95 millions) 11 , Korea (+29 millions), Japan (+28 millions) and Indonesia (+25 millions).

Nonetheless, as can be seen in Figure 3, the assumptions underlying the projections imply almost a tripling in the total car stock for the 64 countries that we consider, from 529 millions of passenger cars in 1998 to 1,420 millions of passenger cars in 2030 12 . Given the actual level of technology and the expected innovations, such an increase seems hardly sustainable in terms of CO 2 emissions and motor fuel demand.

It is not easy to compare our projections with the projections from other sources because of differences in starting date, geographical coverage and type of vehicles covered. Nonetheless, Table 4 shows for some regions the projected ratios of ownership growth to per-capita income growth which synthesize our results (second column) and the results of Dargay, Gately and Sommer (DGS), the IEA and the Sustainable Mobility Project (SMP) 13 . Overall, our projections indicate a more elastic demand for passenger cars than those of the other sources 14 . Dargay, Gately and Sommer explain that the relatively low ratios projected by the SMP result from assumption of low income-elasticity of ownership. The lack of information about the model used by the IEA prevents from explaining the reason for the discrepancy with their projections. The divergence is very important for non-OECD countries and the world in the 2004 edition. The projections seem closer in the 2006 edition but a deeper look indicate that they are not: the IEA project that the light-duty vehicle stock should climb to 100 millions in China (our value is 288 millions) and 56 millions in India (our value is 178 millions) in 2030. For an unknown reason, their projected number of light-duty vehicles in use worldwide is close to the passenger car stock that we project for our sample (which excludes Russia and Iran notably). The gap between our values and those of Dargay, Gately and Sommer comes mainly from different data sources (as discussed in Lescaroux and Rech) and from the different S-shaped curves used: the Gompertz function implies a lower incomeelasticity at low levels of per-capita income, which implies a lower ratio for many non-OECD countries (it is nonetheless difficult to compare them as they do not refer to the same stock of vehicles). Similarly, the IMF (2005) projects a growth in total vehicle stock which seems rather weak compared with our projections: for most of the countries, the vehicle stock that it projects is just slightly greater than the passenger car stock that we project (they estimate that the world vehicle stock should reach approximately 1,660 millions by 2030 and our projections indicate that the passenger car stock for a subset excluding notably Russia and Iran should reach 1,420 millions by 2030). These low projections of transportation demand result probably from the calibration of the saturation level at 850 vehicles per 1,000 people and from the a priori assumption of low income-elasticities at low levels of per-capita income. Moreover, our projections already include (at least partially, on the basis of recent experience) the rise in car purchase cost associated to the diffusion of more efficient cars15 and they neglect the possibility that automakers could propose cheap cars designed for developing countries. This would accelerate the automobile' s diffusion in countries like China or India. For these two countries alone, the passenger car stocks would reach respectively 370 and 265 millions if the elasticity of the "Cost/utility" ratio with respect to per-capita income was lowered by 2.5 percentage points (γ' = γ -0.025 = 0.346) over the projection sample. This would imply an increment of approximately 170 millions of vehicles compared to the reference scenario.

Another issue that could affect the development of private transportation in China is the evolution of income distribution. As a consequence of the very fast economic development, it is actually very un-egalitarian, but this seems to be at odds with the nature of the Chinese society. If we were to assume a linear decrease of the Gini coefficient to 0.35 in 2030, the car stock would be greater by about 20 millions (approximately the size of the car stock in the United Kingdom in 1998). If an equalization of income and a relative decrease of the "Cost/utility" ratio were to occur at the same time, the Chinese car stock should reach 406 millions of vehicles. Therefore, our modelling of the passenger car stock imply a growth in vehicle ownership, and hence in motor fuel demand, stronger than the growth projected by other sources. The gap is not dramatic in the reference scenario but the alternative simulations suggest that demand could be accelerated by many factors. This raises concerns about the balance between supply and demand and about the related CO 2 emissions.

It seems accordingly that a high rate of technical progress will be needed to "meet the challenges of sustainability", or that a large set of countries will have to implement policies aimed at reducing the demand for private transportation; otherwise, the adjustment should come from the market in the form of a gasoline price surge.

Conclusion

In this paper, we have been able to build a formal model of passenger car ownership rate and to propose a practical approximation which works remarkably well in explaining past insample evolutions and meets in the long-run the desired theoretical properties of the car ownership rate.

Given the distribution of income in a particular population, the formal model explains the car ownership rate as a function of the average per-capita income level and a country-specific indicator of the "Cost/utility" ratio of owning a car. When applied over the 1986-1998 period to a set of 64 countries, the model is able to reproduce past changes in ownership rates. As we do not consider only the average per-capita income (as other models do) but also the standard deviation of the income distribution, the model can explain differences in the pattern of growth between egalitarian countries (like the Scandinavian countries) and less egalitarian countries (like South-American countries) or breaks in the trajectory followed by a particular country when changes occur in the distribution of income (like in the transition economies). In the present form of the model, the indicators of the "Cost/utility" ratio are estimated, countryspecific parameters. It could be desirable to express these parameters as a function of observable indicators in order to evaluate the effects of alternative transportation policies. As a first step in this direction, we showed that the ratios are strongly correlated with car purchase costs and population densities.

We used the estimated model to perform projections to the year 2030. The reference scenario assumes that the relationship between the "Cost/utility" ratio and average per-capita income observed in the near past still holds into the future. We project that the total car stock should be about 3 times greater in 2030 than it was in 1998 (1,420 millions of passenger cars, excluding notably Russia and Iran), but the growth should not be uniform. For example, the Chinese and Indian car stocks should be respectively 45 times and 35 times greater than they are and China and India should become respectively the second and the third countries for the number of passenger cars. On the other hand, the growth of the car stock in the OECD should come mainly from the USA, and it should be triggered there by population growth only.

We performed some alternative projections to highlight the effects of some socio-economic evolutions allready in the tube, like the diffusion of cheap cars in developing countries, or likely to occur in the coming decades, like an equalization of the income distribution in China. These evolutions would lead, more or less, to a stronger growth of the car stock. In particular, a relative fall in car purchase cost would considerably accelerate the diffusion of automobile in middle income populations.

Nonetheless, in the alternative scenarios as well as in the reference scenario, the projected growth of the total car stock seems hardly sustainable in terms of oil demand and related CO 2 emissions. In the absence of important technical evolutions improving fuel efficiency and sociological evolutions reducing vehicle usage, the adjustment should come from the market in the form of a gasoline price surge. [START_REF] Bagnoli | Log-concave probability and its applications[END_REF], it follows that 1 -F Y (y) is a log-concave function of ln(y).

1 1 1 ln ln 2 0 2 y g F dv v f dv v du s m u s u y F U y g U s m y y Y - = ⋅ - = ⋅ ⋅ - ⋅ ⋅ - = ⋅ - ⋅ - ⋅ ⋅ ⋅ ⋅ - = - ∞ - - ∞ - π π Since 1 -F U (y) is a log-concave function of y
Approximation of the car ownership elasticity. From Equation ( 22), there is no exact formulation for the income elasticity of car ownership, ε. Nonetheless, a well-known approximation of the CDF of the log-normal distribution can be obtained based on the logistic distribution (with the notations in (4)):

( ) ( ) ( ) 3 exp 1 1 , , ⋅ + ≈ ⋅ σ π β y m Y Y y F t t Y .
Consequently, ε can be approached by the formulation:

( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) { } 3 5 . 0 exp 1 3 1 2 3 1 ⋅ ⋅ + ⋅ - ⋅ + ⋅ ⋅ - = ⋅ ⋅ - σ π β σ σ π γ ε σ γ π i t t Y Y ,
which is a positive (because γ < 1) and strictly decreasing function of t Y .

  logarithm of Y follows a Normal law with mean m and standard deviation σ: the density function of real per-capita income, F Y (y, , f U (u) the density function of the standard normal distribution and F U (u) its cumulative distribution function (CDF).

Figure 1 :

 1 Figure 1: Influence of the dispersion and threshold parameters on the S-shaped curve

Table 1 : Estimated parameters of the car ownership model (Equation (20))

 1

			Coefficient	z-statistic			Coefficient	z-statistic
	δ2		0.07	8.60
	ln(Pi * )	Algeria	8.15	4.31	ln(Pi * )	Japan	10.19	3.52
	ln(Pi * )	Argentina	9.18	16.83	ln(Pi * )	Jordan	8.33	40.39
	ln(Pi * )	Australia	9.15	8.67	ln(Pi * )	Kenya	7.70	29.29
	ln(Pi * )	Austria	9.44	1.57	ln(Pi * )	Korea, Rep. of	7.64	1.84
	ln(Pi * )	Bangladesh	7.16	65.73	ln(Pi * )	Latvia	7.95	66.45
	ln(Pi * )	Belgium	9.43	1.58	ln(Pi * )	Luxembourg	9.50	0.73
	ln(Pi * )	Botswana	9.00	44.38	ln(Pi * )	Mexico	8.89	12.04
	ln(Pi * )	Brazil	9.05	61.90	ln(Pi * )	Morocco	7.76	12.42
	ln(Pi * )	Bulgaria	7.38	10.65	ln(Pi * )	Mozambique	6.66	261.73
	ln(Pi * )	Burundi	6.58	142.04	ln(Pi * )	Netherlands	9.69	8.07
	ln(Pi * )	Cameroon	8.10	80.69	ln(Pi * )	Norway	10.18	5.89
	ln(Pi * )	Canada	9.57	5.55	ln(Pi * )	Pakistan	7.26	24.13

Table 2 : Estimated parameters of the car ownership model (Equation (21))

 2

			Coefficient	z-statistic			Coefficient	z-statistic
	δ2		0.08	7.92	γ		0.37	2.43
	βi	Algeria	5.38	3.01	βi	Japan	6.40	2.79
	βi	Argentina	5.91	4.16	βi	Jordan	5.54	4.81
	βi	Australia	5.56	3.37	βi	Kenya	5.47	5.79
	βi	Austria	5.86	1.46	βi	Korea, Rep. of	5.29	3.98
	βi	Bangladesh	5.09	5.89	βi	Latvia	5.15	4.34
	βi	Belgium	5.84	1.05	βi	Luxembourg	5.85	0.92
	βi	Botswana	6.18	5.17	βi	Mexico	5.78	4.08
	βi	Brazil	6.03	4.83	βi	Morocco	5.20	4.43
	βi	Bulgaria	4.73	3.92	βi	Mozambique	4.77	6.11
	βi	Burundi	4.71	6.31	βi	Netherlands	6.06	3.39
	βi	Cameroon	5.68	5.70	βi	Norway	6.38	3.11
	βi	Canada	5.91	2.89	βi	Pakistan	5.01	5.15
	βi	Chile	5.77	4.20	βi	Panama	5.85	4.50
	βi	China	5.17	6.17	βi	Paraguay	5.10	4.70
	βi	Costa Rica	5.56	4.10	βi	Peru	5.87	4.94
	βi	Denmark	6.37	3.22	βi	Portugal	4.93	1.56
	βi	Ecuador	5.71	5.14	βi	Sierra Leone	5.72	6.88
	βi	Egypt	5.31	4.39	βi	Singapore	6.63	4.19
	βi	El Salvador	5.90	5.23	βi	South Africa	5.75	4.35
	βi	Estonia	4.60	3.20	βi	Spain	5.59	1.42
	βi	Ethiopia	4.73	6.49	βi	Sri Lanka	5.05	5.07
	βi	Finland	6.04	3.47	βi	Swaziland	5.61	5.08
	βi	France	5.88	1.21	βi	Sweden	6.14	2.70
	βi	Germany	5.79	2.89	βi	Switzerland	6.15	1.77

Table 3 : Assumptions and projections of passenger car ownership, 1998-2030

 3

		per-capita	income	Passenger cars per 1000	Passenger	car	stock	ratio of
														Population (millions)
		(thousands, 2000US$)	people aged 15 or more	(millions)			growth
												rates: car
			Average			Average			Average			Average
												ownership
			annual			annual			annual			annual
	Country	1998 2030		1998 2030		1998 2030			to	per-	1998 2030
			growth			growth			growth			growth
												capita
			rate (%)			rate (%)			rate (%)			rate (%)
												income
	OECD, North America
	Canada	21.2 35.6 1.6		572	757	0.88	13.9 25.1 1.9		0.54		30.1	39.1	0.8
	United States	32.7 64.1 2.1		943	997	0.17	203.2 298.6 1.2		0.08		279.0 366.2 0.9
	Mexico	5.5	13.5 2.8		157	299	2.04	9.8	30.6 3.6		0.72		96.6	128.1 0.9
	OECD, Europe
	Austria	22.7 41.1 1.9		587	850	1.16	3.9	6.3	1.5		0.62		8.1	8.6	0.2
	Belgium	20.9 38.1 1.9		534	802	1.28	4.5	7.3	1.5		0.67		10.1	10.8	0.2
	Denmark	28.3 51.4 1.9		418	687	1.56	1.8	3.2	1.8		0.83		5.3	5.6	0.2
	Finland	21.4 41.7 2.1		481	782	1.53	2.0	3.6	1.8		0.73		5.1	5.5	0.2
	France	21.1 38.5 1.9		566	790	1.05	26.8 44.0 1.6		0.55		58.8	66.6	0.4
	Germany	22.0 38.9 1.8		604	865	1.13	41.7 59.6 1.1		0.63		82.0	79.3	-0.1
	Greece	9.6	20.1 2.3		293	603	2.28	2.7	5.9	2.5		0.97		10.8	11.2	0.1

Table 4 : Comparison of projected ratios of ownership growth to per-capita income growth

 4

				1998-2030		DGS		(2007),	IEA	(2004),	IEA		(2006),	SMP	(2004),
								2002-2030		2002-2030		2004-2030	2000-2030
				Passenger vehicles		Total vehicles		Total vehicles		Total vehicles	Light duty vehicles
	OECD				0.57		0.42				0.57					0.40
	non-OECD		1.84		1.61				1.12					1.13
	China		2.03		2.20				1.38			1.96		1.42
	India		2.39		1.98				0.39			2.25		1.23
	World				1.13 *		0.94				0.61			0.86		0.59
	(*) 64 countries.
	Appendix C Figures
	Algeria					Argentina					Australia
	.105						.21							.8
	.100
								.20							.7
	.095
	.090						.19							.6
	.085	7.3	7.4	7.5	7.6	.18	8.6	8.8	9.0		9.2	.5	9.6	9.8	10.0	10.2
	Austria					Bangladesh					Belgium
	.60							.0012						.55
	.55							.0010
															.50
	.50							.0008
	.45							.0006						.45
	.40	9.7	9.8	9.9	10.0 10.1	.0004	5.5	5.6	5.7	5.8	5.9	.40	9.7	9.8	9.9	10.0

Medlock and Soligo also introduce in their panel model the retail price of motor fuel and country-specific effects.

The support of U is the extended real line: for some people, the utility of owning a car may be negative (for example, people who cannot drive). Otherwise, the saturation level should be 1 car per individual.

As in[START_REF] Lescaroux | The impact of automobile diffusion on the income elasticity of motor fuel demand[END_REF], the car ownership rate is defined as the number of passenger car per people aged 15 or more.

[START_REF] Lescaroux | The impact of automobile diffusion on the income elasticity of motor fuel demand[END_REF] suggest that the speed of adjustment towards the equilibrium level increases with per-capita income but they were not able to identify a statistically significant relationship.[START_REF] Dargay | Vehicle ownership and income growth, worldwide: 1960-2030[END_REF] argue that it is the sign of the growth rate which matters. We pursued in this direction using a continuous function of the rate of change of per-capita income.

We also tried to take into account the temporal decrease of car purchase cost that results from productivity growth in the automobile sector by introducing a time trend in the approximation of ln(P i * ), but we could not find a satisfactory specification. This results probably from the shortness of the temporal dimension of our sample.

Their very slow adjustment process results from the estimation of a common equation for all countries.

Technically, the slowest income threshold is approximately $450 in the United States. Nonetheless, this coefficient should be biased: as discussed in Lescaroux and Rech, the UN data overestimates the car stock in the United States because they consider since 1987 all the pick-ups, minivans and SUVs as passenger cars. To our knowledge, other data sources do not consider any pick-up, minivan or SUV as passenger car. So, the UN data might be the least biased.

There is a huge literature on income inequality. According to[START_REF] Kuznets | Economic growth and income inequality[END_REF], it should first rise and later fall as per-capita income increases. Nonetheless, more recent work indicates that this relationship could have weaken or that its capacity to explain variations in income inequality across countries and over time is very limited[START_REF] Barro | Inequality and Growth in a Panel of Countries[END_REF] or that is does not reflect a "natural" evolution but a purely "accidental" evolution[START_REF] Piketty | The Kuznets' curve, yesterday and tomorrow[END_REF]. Because of this lack of consensus, we prefer keep the Gini coefficients constant for the base case.

We checked that the practical implications of this assumption are very limited: for the EU-15 area, in year 2030, the car stock would be smaller by 4 millions of cars (about 1.5 per cent of the European stock) with stable Gini coefficients.

Although the USA are the most mature market, they will keep being one of the most dynamic markets because of the sustained population growth.

If we do not consider an increase in the "Cost/utility" ratio as per-capita income rises (Equation (20)), the total car stock quadruples, to 1,900 millions of passenger cars, because of the low actual income threshold levels in most of the developing countries. Our approximation of the relationship between the income threshold and percapita income is rather rough, but it should provide a better idea of what might happen in the future.

 13 As they are reported in[START_REF] Dargay | Vehicle ownership and income growth, worldwide: 1960-2030[END_REF]

).14 For the sake of comparison, ownership rates were expressed as a fraction of total population (and not the

population aged 15 or more) in this table. Consequently, the reported values do not correspond to the values in Table3.

When Equation (20) is used, with constant "Cost/utility" ratios, the total car stock reaches 1,900 millions in 2030.

Appendix A: Mathematics Lemma 1. Let Y be a random variable with a Log-Normal distribution (with mean m and standard deviation s).

i The cumulative distribution function, F Y (y), is a log-concave function of ln(y).

ii The reliability function, 1 -F Y (y), is a log-concave function of ln(y).

Proof: Result (i) immediate because F Y (y) is a log-concave function of y [START_REF] Bagnoli | Log-concave probability and its applications[END_REF], so it is a log-concave function of ln(y). For Result (ii), notice that for any y positive, we have:

Appendix B: Data sources

We estimated the model using all the countries for which the data on income, population and car ownership were available between 1986 and 1998. This sample results from arbitration between the temporal and the spatial dimensions of our model: the longer the time sample, the lower the number of countries in the sample. Some OECD countries and some big markets are not taken into account for the following reasons. Russia, Poland, the Czech Republic and the Slovak Republic were excluded from the sample because of a lack of some data before 1991. New-Zealand was excluded because of a discontinuity in the car stock series. Iran was excluded because no information was available concerning the distribution of income.

The data sources are various issues of the Statistical Yearbook (United Nations) for the passenger cars in use and the World Bank online database (http://devdata.worldbank.org/dataonline/) for real per-capita GDP, total population, the part of the population in the 0-14 years age group and population density. The Gini indexes come from the World Bank as well but because of the scarcity of Gini index time series, we had to construct approximate measures when the data were unavailable. When we had many observations for a country, we interpolated the missing data linearly. When just one observation was available, we considered the Gini index as constant. When no observation was available between 1986 and 1998, we used the closest value if there was one. Consequently, the income dispersion measures that we use are very rough. The car prices in year 2003 come from the Union Bank of Switzerland' s annual survey "Prices and Earnings Around the Globe".

The demographic data used in the projections are from the online UN Population Database (http://esa.un.org/unpp/index.asp?panel=2). Data on per-capita real income growth using Purchasing Power parity come from the DOE' s International Energy Outlook, 2006; we assumed the same rate of change for the growth in per-capita real income.

Appendix C: In-sample dynamic simulations

In the following graphs, the ownership rate (on the left) is plotted against the logarithm of percapita income (at the bottom). The line is the true path and the dotted line is the simulated path.

Déjà parus


Axel Pierru
email: axel.pierru@ifp.frdenis.babusiaux@numericable.fr

Denis Babusiaux

Ceg-7 I Cadoret

J.-L Karnik

Ceg-9 A Fauveau

P Chollet

F Lantz

Short-run and long-run marginal costs of joint products in linear programming

Keywords: JEL classification numbers: D20, C61 microeconomics, marginal cost, linear programming, energy economics. Déjà parus CEG-1. D. PERRUCHET, J.-P. CUEILLE, Compagnies pétrolières internationales : intégration verticale . CADORET, P. RENOU, Élasticités et substitutions énergétiques : difficultés méthodologiques. Avril 1991

La collection "Les cahiers de l'économie" a pour objectif de présenter des travaux réalisés à l' IFP et à l'École du pétrole et des moteurs, travaux de recherche ou notes de synthèse en économie, finance et gestion. La forme peut être encore provisoire, afin de susciter des échanges de points de vue sur les sujets abordés.

Les opinions émises dans les textes publiés dans cette collection doivent être considérées comme propres à leurs auteurs et ne reflètent pas nécessairement le point de vue de l'IFP ou de l'École du pétrole et des moteurs.

Pour toute information sur le contenu, prière de contacter directement l' auteur ; pour toute information complémentaire prière de contacter le Centre Économie et Gestion, Tél. 01 47 52 64 25 2 3

Introduction 1.1 The standard microeconomic theory

When one considers an activity involving the production of a single product, microeconomic theory tells us that with adjusted capacity, short-run and long-run marginal costs are equal under certain conditions. To take a specific example, consider a construction project for a facility of which the size (i.e. the production capacity) needs to be determined. We assume that this size can be represented by a continuous variable. We also assume that the annual production cost for a given size, i.e. the short-run annual cost, is a continuous function that can be derived with respect to the quantity of good produced over the year. If the marginal cost begins to increase at a certain level of production, the curves representing the annual cost for facilities of various sizes generally take the shape of those drawn by the dashed lines on the first graph of Figure 1. The curve representing the long-run annual cost is then the envelope curve (shown by a solid line) of the short-run cost curves. The second graph of Figure 1 reveals, in the case of a rising marginal cost over the long run (long dashed lines), the usual shape of average cost curves over the short run (dashed lines) and the long run (solid line), as well as curves representing short-run and long-run marginal cost (it should be noted that marginal cost can decrease over the long-run in the presence of economies of scale). Under the usual conditions of continuity and differentiability, these properties are well known (e.g. see Boîteux 1960).

LAC: long-run average cost; LMC: long-run marginal cost; AC k : short-run average cost with a production facility of size E k ; MC k : short-run marginal cost with a production facility of size E k .

Purpose of this paper

In this article, we analyze the case of the production of joint products using various types of interdependent equipment when the production system is modeled via a linear program. Our objective is to establish the relationship between short-run marginal costs and long-run marginal costs in this case.

A first example of such a production system is given by the petroleum refining industry: the main variables represent the quantities of crude oil to be processed and the various flows of intermediate products that characterize the operation of the production units (atmospheric and vacuum distillation, reforming, cracking, etc.) and the composition of the finished products (gasoline, automotive diesel, heating oil, heavy fuel oil, etc.). The main constraints are material-balance equations (which are the most common), product-quality specification equations (sulphur content, gasoline octane number, etc.) and demand equations for finished products (quantities produced, increased in some cases by imported quantities and decreased by exported quantities, greater than or equal to demand). In a short-term management model, there are also capacity equations limiting the feedstock to be processed in each of the units. The economic function only includes operating costs (decreased where applicable by revenues generated by the sale of certain products). In a long-term investment model, variables represent the capacities of the units to be built. If we are dealing with a dynamic model extending over several periods, (linear) investment costs must be associated with these capacity variables, and the economic function is then a total cost discounted over a long period. If we are dealing with a "static" long-term model, the model is representative of a given time horizon. An equivalent annual investment cost is then associated with each capacity variable (initial investment divided by the sum of the discount factors). The economic function to be minimized is then an equivalent annual cost, which results from the sum of the operating cost and equivalent annual investment costs.

Yet another example is supplied by an electrical production system using various types of power plants (nuclear, coal, gas turbine, etc.). The main variables are the powers supplied by the different types of plant for the various hourly/seasonal periods (peak periods during winter weekdays, off-peak periods during summer weekends, etc.). The main constraints are the demand equations expressing the requirement to supply the power demanded for each hourly/seasonal period (and for each outcome if the demand is randomized). [START_REF] Anderson | Models for determining least-cost investments in electricity supply[END_REF] offers a general description of this type of model.

In practice, the models used for short-term management (particularly in the very short run), are often profit maximization programs, with the price of products being fixed and assumed to be given by the market. It is always possible, however, by taking certain precautions where appropriate, to make the transition from such a profit-maximization formulation to a cost-minimization formulation, subject to the constraint of satisfying a demand for finished products. It is this latter formulation that will be used in the remainder of this article.

We will therefore consider a system composed of a variety of equipment, called production units, producing several joint products. The interdependencies between the units are represented via a linear programming model. We distinguish between a short-term program, for which the capacity of the units is assumed to be fixed, and a long-term program for which a certain number of capacities are variables to be determined. Equivalent annual investment costs are then associated with these variables. For purposes of simplification, we will present the analysis by referring to a static long-term model representative of a given time horizon. Over both the short-and long-term, we will consider a cost-minimization problem under demand constraints.

Preliminary remarks on marginal costs

We will begin with some general remarks. Marginal production costs are equal to the dual variables associated with demand constraints, at least if there is no degeneracy. When one varies the demand for a given product, while keeping the demand for other products unchanged, the marginal cost of that product is stable as long as the optimal solution corresponds to the same basic solution. The curve representing the marginal cost as a function of the quantity produced is therefore a stair-step curve. This is true for both the short-run marginal cost and the long-run marginal cost.

Let' s consider the optimal solution of a long-term problem, determined by taking into account a fixed demand i b for each product i. This solution gives the optimal value for the capacities to be built for the various units. Now set these capacities at their optimal values. The short-term program thus obtained supplies the values for the short-run marginal costs and, in more general terms, allows us to trace the short-run marginal cost of a product as a function of the demand for that product. The short-run marginal cost curve for a product i intersects the long-run marginal cost curve for a demand equal to i b , with the capacities consequently being adjusted. This is in line with standard microeconomic theory. The distinguishing feature of a production system represented by linear programming is related to the stair-step profile of the curves represented in Figure 2. In general, the long-run marginal cost is stable in the neighborhood of i b while one observes a discontinuity of the short-run marginal cost.

In fact, a long-term problem can be converted into a short-term problem by adding constraints setting the capacities of the units, requiring that the feedstock processed by a unit, or its production, be less than the capacity defined by the optimal solution of the longterm program. These constraints, or at least a certain number of them, are binding at the optimum of the short-term program, which is then degenerate. The objective-function value of the program is no longer differentiable in i b . Nevertheless, it has a sub-differential and directional derivatives, which explains the discontinuity of the short-run marginal cost observed at i b in figure 2.

For each finished product, the short-run marginal costs at the left side and right side of the anticipated demand for which the capacity is adjusted are then different and given by components of the dual variables vectors. The literature on degeneracy supplies the criteria for selecting these components in the presence of multiple dual solutions [START_REF] Aucamp | The computation of shadow prices in linear programming[END_REF][START_REF] Gal | Shadow prices and sensitivity analysis in linear programming under degeneracy[END_REF][START_REF] Greenberg | An analysis of degeneracy[END_REF]. Consequently, we must consider a lefthand marginal cost and a right-hand marginal cost for each finished product.

The objective of this article is to study the relationship between short-run marginal costs and long-run marginal costs. The analysis relies on the marginal values of capacities, equal to the dual variables associated with the capacity constraints introduced in the short-term model. In theory, these variables are negative: in the short run, an increase in available capacity should not generally result in an increase in the operating cost. In order to simplify the presentation of results, we assume throughout this article that the marginal values of capacities are all negative.

Breakdown of long-run marginal costs

We use the breakdown of long-run marginal costs in linear programming proposed by [START_REF] Pierru | Breaking down a long-run marginal cost of an LP investment model into a marginal operating cost and a marginal equivalent investment cost[END_REF]: the long-run marginal cost of a given finished product results from the sum of a marginal operating cost and a marginal equivalent investment cost. To obtain it, one simply breaks down the objective function into two elementary economic functions1 . The marginal operating cost (respectively the marginal equivalent investment cost) is equal to the variation of the elementary function representing the operating cost (respectively the equivalent annual investment cost) for an infinitesimal increase in finished product demand. It has to be noted that the marginal equivalent investment cost is equal to the marginal capacity adjustment times the equivalent annual investment cost per unit of installed capacity.

We will adopt this breakdown with the comment that the equivalent annual investment cost taken into account can result from the sum of the investments in the various units whose capacities are to be optimized. In addition to the operating cost, we will therefore associate an elementary function with each capacity variable. The long-run marginal cost of a product can therefore be broken down into a marginal operating cost and as many marginal equivalent investment costs as there are capacity variables. To eliminate any ambiguity, we specify that for a given finished product the term "marginal operating cost" always refers in this article to a change in the operating cost determined in the long-term model.

The following section introduces the mathematical formulation of the model and the notations used. The results are then presented in the form of propositions. For reasons of clarity, in section 3, we first state and prove the propositions when the model only includes a single capacity variable. We then state and prove those pertaining to the general case in section 4.

Notations and mathematical formulation of the model

To illustrate the problem under study, we will consider a static long-term linearprogramming model and the corresponding short-term linear-programming model. The long-term model is used to define the optimal capacities to be built to meet a demand vector b and provides the long-run marginal cost of each finished product, as well as its breakdown into marginal operating cost and marginal equivalent investment costs. The short-term model determines the optimal production program for demand b, assuming that capacities are set at the values defined by the long-term model. This short-term model is used to calculate the marginal values of the capacities, as well as the short-run marginal costs (on the left-hand and right-hand of demand i b ) for each finished product i. Although the problem studied relates to a linear programming model, we will adopt a formulation that takes its inspiration from the lagrangian duality (which facilitates the formal application of the envelope theorem).

Long-term model

We consider a capacity variable and a feedstock variable for each equipment. A feedstock variable has to be smaller or equal to the corresponding capacity variable (plus a possibly preexisting capacity). With this formulation, the operating cost (associated with the feedstock variable) can be distinguished from the equivalent annual investment cost (associated with the capacity variable) in the objective function.

We adopt the following notations, considering that the long-term problem includes m capacity variables: b: vector formed from the demanded quantities of n finished products (the quantity of product i demanded is notated i b ); x : vector of size q formed from the variables which appear in both long-run and short-run primal problems; The long-term problem is written as follows:

, , .

.

f Min v x e k x X s t Ax b x k c + ∈ = -≤
Where: A is an n q × matrix, Ax b = represents the set of demand constraints; X is a closed convex set defined as the intersection of linear constraints (equalities and inequalities), corresponding to material balance equations, product-quality specifications, requirements on the signs of variables ... c is a vector whose components are the preexisting capacities of the m types of equipment considered; v is a vector of size q whose components are the unit costs associated with the variables in x; .,. denotes the inner product of two vectors.

The value of this long-term problem, considered as a function of the demand for finished products, is notated ( ) V b . We assume that the optimal solution is non-degenerate; the basic solution is thus unchanged in the neighborhood of b. ( )

V b is therefore continuously derivable. We will let ( ) i l b represent the long-run marginal cost of the product i:

( ) ( ) i i V l b b b ∂ = ∂
. It is equal to the dual variable associated with the demand constraint for that product.

The value of k at the optimum of the long-term problem is denoted ( )

k b , with ( ) ( ) ( ) ( ) ( ) 1

Short-term model

The short-term problem, derived from the long-term problem introduced in the previous subsection, is written as follows:

( )

, . . f Min v x x X s t Ax b x c k b ∈ = ≤ +
We introduce two vectors of dual variables, associated with two sets of constraints in this short-term problem:

( ) ) is the dual variable associated with the p th -capacity constraint).

Construction of a "short-run cost function with continuously-adjusted capacity"

Let W be a function correctly defined in a neighborhood of b, representing the value of the short-term program when the capacity is always adjusted. W is equal to V (the long-term and short-term problems have the same optimal solution) less the sum of the equivalent annual investment costs:

1 m p p p W V e k = = - (1
) W, which we will call a "short-run cost function with continuously-adjusted capacity", is everywhere equal to the part of the objective function representing operating costs in the long-term program (i.e. the elementary function associated with operating costs). As W is equal to a difference of differentiable functions (assuming that the long-term solution is non-degenerate), W is itself differentiable in b. This comment is particularly important for the proofs of the propositions stated in the article.

Lagrangian formulation of the problem

Where , , , ( )

f L v x y b Ax u x c k b = + - + --
denotes the usual Lagrangian constructed from the short-term program, we have:

( ) ( ) , 0 max min , , , ( ), x X y u W b L x y u k b b ∈ ≥ = (2) If we set ( ) ( ) ( ) ( ) ( ) ˆ, , , ,
, min , , , , ( )

( ) ( ) ( ) ( ) * * , , max max i i y u Y b y u Y b i i L L s b y b b + + ∈ ∈ ∂ ∂ = = = ∂ ∂ ( ) ( ) ( ) ( ) ( ) * * , ,
min min

i i y u Y b y u Y b i i L L s b y b b - - ∈ ∈ ∂ ∂ = = = ∂ ∂ Similarly, if the vector µ (with ( ) 1 2 , ,..., m µ µ µ µ =
) is formed from the marginal values of the capacities in the short run, as defined in the introduction, we have:

( ) ( ) ( ) ( ) ( ) ( ) * * * , , , min min max ˆp p p y u Y b y u Y b y u Y b p p L L u u k k µ - ∈ ∈ ∈ ∂ ∂ = = = -= - ∂ ∂ where p µ ( { } 1, 2,..., p m ∈
): marginal value of the p th capacity set at ( )

p p c k b + in the short-term problem.
3 Specific case: model with a single capacity variable

Statement of the propositions

Our results can be summarized by the following propositions, stated for a given finished product i:

(i)
the left-hand short-run marginal cost is equal to the marginal operating cost if the marginal capacity adjustment is positive; (ii) the right-hand short-run marginal cost is equal to the marginal operating cost if the marginal capacity adjustment is negative; (iiia) the difference between the two short-run marginal costs is equal to the absolute value of the product of the marginal value of capacity by the marginal capacity adjustment;

(iiib) if one considers two distinct finished products: the ratio of the differences between right-hand short-run marginal cost and left-hand short-run marginal cost is equal to the absolute value of the ratio of the marginal equivalent investment costs (or, which amounts to the same thing, to the absolute value of the ratio of the marginal capacity adjustments).

These propositions are proved in the following section. Proposition (iiib) follows directly from (iiia). We would remark that the first two propositions are quite intuitive, since, in the long-term model, the marginal operating cost represents the change in the operating cost when an additional ton of the product in question must be produced.

Proof of the propositions

The long-term model includes only a single capacity variable: k, ( ) As W is differentiable in b, its left-hand and right-hand partial derivatives with respect to i b are equal. We will distinguish the two following cases.

A product with a positive marginal capacity adjustment

Consider a finished product i for which the marginal capacity adjustment is positive, i.e., ˆ0 i k b ∂ ≥ ∂

. By applying the envelope theorem to the equation ( 3) we obtain the two following equations:

( ) ( ) ˆ( ) ˆî i i i i i W W L L k L k b b s b b b b b b k k + + - + - + ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ (4) ( ) ( ) ˆ( ) ˆî i i i i i W W L L k L k b b s b b b b b b k k - - + - + - ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ (5)
Equation ( 4) can be analyzed as follows: since . Equations ( 4) and ( 5) thus give us:

ˆ0 i k b ∂ ≥ ∂ ,
( )

( ) ( ) i i i i W k b s b s b b b µ + - ∂ ∂ = + = ∂ ∂ (6)
Furthermore, by differentiating equation ( 1) we obtain:

( ) ( ) ( ) ( ) ( ) ˆî i i i i W V k k b b e b l b e b b b b b ∂ ∂ ∂ ∂ = - = - ∂ ∂ ∂ ∂ (7) Equation (7) indicates that ( ) i W b b ∂ ∂ is equal to the marginal operating cost.
By combining equations ( 6) and ( 7), we obtain:

( ) ( ) ( ) ( ) ˆî i i i i k k l b s b e s b e b b µ + - + - ∂ ∂ = + + = + ∂ ∂
This result is crucial since it explains the transition from the long-run marginal cost to shortrun marginal costs. And finally:

( ) ( ) ( ) ( )

ˆˆî i i i i i k k k s b l b e l b e b b b µ µ + ∂ ∂ ∂ = -+ = - - ∂ ∂ ∂ (8) ( ) ( )
î i i k s b l b e b - ∂ = -∂ (9)
Equation ( 9) proves the proposition (i): the left-hand short-run marginal cost is equal to the marginal operating cost (i.e., the long-run marginal cost less the marginal equivalent investment cost).

A product with a negative marginal capacity adjustment

Now consider a product j for which the marginal capacity adjustment is negative, i.e., ˆ0 j k b ∂ ≤ ∂ . An increase in the demand for this product results in the capacity constraint being no longer binding in the short run. Conversely, a decrease in this demand would keep this constraint binding. By applying the envelope theorem, we thus obtain:

( ) ( ) ˆ( ) ˆĵ j j j j j W W L L k L k b b s b b b b b b k k + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ (10) ( ) ( ) ˆ( ) ˆĵ j j j j j W W L L k L k b b s b b b b b b k k - - - - - - ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ (11)
By proceeding as in the earlier instance, we finally obtain via equations ( 7), ( 10) and ( 11):

( ) ( )

ĵ j j k s b l b e b + ∂ = -∂ (12) ( ) ( ) ( ) ( ) ˆˆĵ j j j j j k k k s b l b e l b e b b b µ µ - ∂ ∂ ∂ = -+ = - - ∂ ∂ ∂ (13)
Equation ( 12) proves the proposition (ii). Equations ( 8), ( 9), ( 12) and ( 13) are summarized in Table 1. For a given finished product, the difference between the two short-run marginal costs is therefore proportional to the marginal equivalent investment cost assigned to this product.

We do in fact find that for any product i: ( ) ( )

i i s b s b + - ≥ .
If we consider two finished products i and j, we have:

( ) ( ) ( ) ( ) ˆmarginal equivalent investment cost of product ˆmarginal equivalent investment cost of product i i i j j j k s b s b b i s b s b j k b + - + - ∂ - ∂ = = - ∂ ∂
This equation proves the proposition (iiib): the ratio of the differences of short-run marginal costs is equal to the absolute value of the ratio of the marginal equivalent investment costs.

General case: model with several capacity variables

Statement of the propositions

Let us consider a given finished product i. Let i Z + denote the set of capacity variables for which the marginal equivalent investment cost associated with product i is positive. i Z - then denotes the set of capacity variables for which the marginal equivalent investment cost associated with product i is negative. We can state the following propositions:

(j) the left-hand short-run marginal cost is given by the following formula: (jjj) the difference between the two short-run marginal costs is equal to:

marginal capacity adjustment marginal value of capacity

i i Z Z + + ×

Proof of the propositions

The long-run marginal cost of a given finished product i is broken down into the sum of 1 m + terms:

-the marginal operating cost, m marginal equivalent investment costs.

Without loss of generality, we will assume that the marginal equivalent investment costs corresponding to the first z capacity variables are positive and that those corresponding to the other m z are negative. In other terms, the set of the first z capacity variables forms i Z + . The marginal capacity adjustments are as follows:

( )

ˆ0 p i k b b ∂ ≥ ∂ for { } 1, 2,..., p z ∈ ( ) ˆ0 p i k b b ∂ ≤ ∂ for { } 1, 2,..., p z z m ∈ + +
Following the same line of reasoning as was used in the specific instance of a single capacity constraint, we obtain:

( ) By combining equations ( 14) and ( 15), we obtain:

1 1 ˆ( ) ( ) z m p p i p i p p p z i i i k k W b s b s b b b b µ µ + - = = + ∂ ∂ ∂ = + = + ∂ ∂ ∂ (
( )

1 1 ˆ( ) m m p p i i p p p p z i i k k s b l b e b b µ - = = + ∂ ∂ = - - ∂ ∂ (16)
( )

1 1 ˆ( ) m z p p i i p p p p i i k k s b l b e b b µ + = = ∂ ∂ = - - ∂ ∂ (17)
Equations ( 16) and ( 17) prove the propositions (j) and (jj). By combining these two equations, we obtain:

1 1 ˆ( ) ( ) m z p p i i p p p z p i i k k s b s b b b µ µ + - = + = ∂ ∂ - = - ∂ ∂
Or (assuming that the marginal values of capacities are negative):

1 ( ) ( ) m p i i p p i k s b s b b µ + - = ∂ - = ∂
(18) Equation ( 18) proves the proposition (jjj).

Conclusion: economic interpretation

In conclusion, we will emphasize the economic interpretation of the results obtained, formulated by the propositions in section II. We will first consider the specific case of a model with only one capacity variable.

Take the example of a finished product for which the marginal equivalent investment cost is positive. The right-hand short-run marginal cost of this product is given by the right-hand term of equation ( 8) which represents the sum of two terms. The first term is equal to the marginal operating cost (i.e. the cost of the optimal change over the long run in inputs which remain variable over the short run). The second term is equal to the marginal capacity adjustment (i.e. the additional capacity required in the long run to produce an additional unit of the finished product) multiplied by the marginal value of the capacity in the short run. Since one cannot have this additional capacity in the short term, a cost premium, equal to this additional capacity multiplied by the marginal value of the capacity, is generated. Conversely, the left-hand short-run marginal cost of this product is just equal to the marginal operating cost. In fact, in the long run, increasing the production of the finished product, to the point of meeting anticipated demand, requires having additional capacity. In the short run, however, this additional capacity is available (since the capacity was set in relation to anticipated demand). As a result, no cost premium is generated.

A similar analysis can be made for a product with a negative marginal equivalent investment cost. Thus, the left-hand short-run marginal cost is less than the marginal operating cost since, in the short run, producing more entails a relaxation of the capacity constraint thereby saving cost.

It is interesting to note that in the specific case in which the absolute value of the marginal value of capacity is equal to the equivalent annual investment cost per unit of installed capacity, each finished product has a short-run marginal cost equal to the long-run marginal cost (the right-hand cost for products for which the marginal equivalent investment cost is positive, and the left-hand cost for the others).

This type of analysis is also applicable in the general case of models with several capacity variables. The marginal operating cost is then increased by the sum of additional costs incurred (for the right-hand short-run marginal cost) or decreased by the sum of costs saved (for the left-hand short-run marginal cost).

In addition to their theoretical interest, these results, which complement traditional microeconomic theory, can be quite useful in practice. There are numerous constraints and variables in the field of oil refining. Apart from extreme situations, the number of steps comprising marginal cost curves is such that the curves are "smoothed" and are not far removed from those of traditional microeconomic theory. However, for runs with highlybinding capacities, the "jumps" in marginal costs can be significant. This is all the more true for peak demand in the electricity sector, where power cannot be stockpiled. A better comprehension of marginal costs should thus serve as a valuable aid for market power analysis.

Figure 1 .

 1 Figure 1. Equality of short-run and long-run marginal costs in standard microeconomic theory.

Figure 2 .

 2 Figure 2. Short-run and long-run marginal costs of product i in linear programming, with capacity adjusted to demand b i .

  f x : subvector of x whose components are the m feedstock variables; .., m k k k k = : vector formed from the m capacity variables in the long-term problem; ( ) , x k thus represents the entire set of variables of the long-term problem;

  the p th capacity to remain adjusted. This marginal capacity adjustment generates a marginal equivalent investment cost equal to ˆp

  not a singleton due to the degeneracy of the optimal solution (there are several vectors of dual variables and thus several saddle points). L thus possesses partial directional derivatives in i b and each product has two short-run marginal costs. Where run marginal cost, we encounter the traditional results of the degeneracy (see for instance[START_REF] Milgrom | Envelope theorems for arbitrary choice sets[END_REF]:

  k b , e and µ are therefore scalars. A change in the demand for finished product i requires a

  an increase in demand for product i keeps binding the capacity constraint in the short-run, with ˆL k

  short-run marginal cost left-hand short-run marginal cost = marginal value of capacity marginal capacity adjustment -×

  hand short-run marginal cost is given by the following formula:

  the marginal operating cost.

Table 1

 1

	: short-run marginal costs
		Left-hand short-run marginal cost	Right-hand short-run marginal
						cost
	marginal equivalent investment cost 0 ≥	marginal operating cost	marginal operating cost	+ ×	marginal value of capacity marginal capacity adjustment
	marginal equivalent investment cost 0 ≤	marginal operating cost -	×	marginal value of capacity marginal capacity adjustment	marginal operating cost
	For any finished product i, the proposition (iiia) is immediately deduced from the preceding
	equations:
	( ) ( )

With each elementary objective function these authors associate a vector composed of "elementary dual variables"; we will not review the method for breaking down long-run marginal costs here.