### 2.1 Estimate of Annual Gasoline Consumption

Below, we first formulate annual domestic gasoline consumption. If the total number of cars newly purchased in the year *t* is defined as {C}_{t} and the market share of the new cars of type *k* is defined as {\gamma}_{t,k}, the number of new cars of type *k* in the year *t* can be expressed as {\gamma}_{t,k}{C}_{t}. Accordingly, the annual driving distance of the type *k* cars can be obtained as {d}_{t,k}{\gamma}_{t,k}{C}_{t} (km), where {d}_{t,k} is the annual average distance driven per car. Dividing this annual driving distance {d}_{t,k}{\gamma}_{t,k}{C}_{t} (km) by the fuel economy of the type *k* cars, {e}_{t,k} (km/L), yields the annual gasoline consumption of the type *k* cars. Summing over all of the type *k* cars, we can estimate the annual gasoline consumption of new cars as follows (see Kagawa et al. [2011]):

{f}_{t,\mathrm{new}}=\sum _{k=1}^{3}\frac{{d}_{t,k}{\gamma}_{t,k}{C}_{t}}{{e}_{t,k}}.

(1)

We examined three types of cars: ordinary passenger cars (cars that exceeded any of the kei passenger car specifications are classified into this category), kei passenger cars (engine displacement ≤ 660 cc, vehicle length ≤ 3.40 m, vehicle width ≤ 1.48 m, vehicle height ≤ 2.00 m), and hybrid passenger cars (cars fitted with a gasoline engine and an electric motor), the respective market shares of which can be represented as {\gamma}_{t,1}+{\gamma}_{t,2}+{\gamma}_{t,3}=1.

To estimate the annual gasoline consumption of vintage passenger cars, we assume that if {K}_{t,k}^{t-y} is the number of newly registered vehicles of type *k* during the year t-y, then {r}_{t} is the proper value indicating that the oldest vehicles existing in the year *t* were registered during the year t-{r}_{t}, and the stock of the vintage passenger cars can be obtained as

{K}_{t,k}=\sum _{y=1}^{{r}_{t}}{K}_{t,k}^{t-y},

(2)

where the range of *y* is 1\le y\le {r}_{t}. Here, assuming that the annual average driving distance of a vintage car of type *k* is equivalent to the annual average driving distance of a new car of type *k*, we can estimate the annual gasoline consumption of vintage cars by dividing the annual driving distance of vintage cars, {d}_{t,k}{K}_{t,k}^{t-y} (km), by the fuel economy of vintage cars, {e}_{t,k}^{t-y} (km/L) (see Kagawa et al. [2011]).

{f}_{t,\mathrm{stock}}=\sum _{k=1}^{3}\sum _{y=1}^{{r}_{t}}\frac{{d}_{t,k}{K}_{t,k}^{t-y}}{{e}_{t,k}^{t-y}}.

(3)

By combining Eqs. (1) and (3), we can estimate the annual gasoline consumption of passenger cars (new passenger cars and vintage passenger cars) as follows:

{f}_{t,\mathrm{total}}=\sum _{k=1}^{3}\frac{{d}_{t,k}{\gamma}_{t,k}{C}_{t}}{{e}_{t,k}}+\sum _{k=1}^{3}\sum _{y=1}^{{r}_{t}}\frac{{d}_{t,k}{K}_{t,k}^{t-y}}{{e}_{t,k}^{t-y}}.

(4)

Here, the first term on the right-hand side of Eq. (4) represents the annual gasoline consumption of new cars and the second term represents the annual gasoline consumption of vintage passenger cars.

### 2.2 Decomposition Analysis

From Eq. (4), the change in the annual gasoline consumption between the year t-1 and the year *t* can be written as

\mathrm{\Delta}{f}_{k,\mathrm{total}}={f}_{t,k,\mathrm{total}}-{f}_{t-1,k,\mathrm{total}}.

(5)

Using the structural decomposition analysis, we can empirically examine the sources of the change in annual gasoline consumption. We will first present the structural decomposition analysis of the gasoline consumption of new cars, i.e., the first term on the right-hand side of Eq. (4). From Eq. (1), the annual gasoline consumption in the year *t* and t-1 can be written as

{f}_{t,k,\mathrm{new}}={d}_{t,k}{\gamma}_{t,k}{C}_{t}{e}_{t,k}^{-1},

(6)

and

{f}_{t-1,k,\mathrm{new}}={d}_{t-1,k}{\gamma}_{t-1,k}{C}_{t-1}{e}_{t-1,k}^{-1},

(7)

respectively. From Eqs. (6) and (7), the change in gasoline consumption between the year t-1 and *t* can be obtained as follows:

\mathrm{\Delta}{f}_{k,\mathrm{new}}={d}_{t,k}{\gamma}_{t,k}{C}_{t}{e}_{t,k}^{-1}-{d}_{t-1,k}{\gamma}_{t-1,k}{C}_{t-1}{e}_{t-1,k}^{-1}.

(8)

Since we have the four sources of changes in gasoline consumption, i.e., change in the annual average driving distance, \mathrm{\Delta}{d}_{k}={d}_{t,k}-{d}_{t-1,k}, change in the market share, \mathrm{\Delta}{\gamma}_{k}={\gamma}_{t,k}-{\gamma}_{t-1,k}, change in the total number of new passenger cars, \mathrm{\Delta}C={C}_{t}-{C}_{t-1}, and change in the reciprocal of fuel economy, \mathrm{\Delta}{e}_{k}^{-1}={e}_{t,k}^{-1}-{e}_{t-1,k}^{-1}, the 4!=24 equivalent decomposition formulae can be obtained as follows (see Ang et al. [2003]; Dietzenbacher and Los [1998, 2000]).

(9)

where the first term on the right-hand side of Eq. (9) is the effect of the change in the annual average driving distance on gasoline consumption of new cars, the second term is the effect of the change in the market share of new passenger cars, the third term is the effect of the change in the total number of new passenger cars, and the forth term is the effect of the change in the fuel economy of new passenger cars. Following Dietzenbacher and Los ([1998, 2000]) and Ang et al. ([2003]), for example, taking an average of the first terms on the right-hand side of Eq. (9) yields the average effects of the annual average driving distance.^{Footnote 2}

Ang et al. ([2003]) and de Boer ([2009]) elegantly showed the similarity between the index decomposition analysis proposed by Sun ([1998]) and the Shapley value widely used in game theory (Shapley [1953]). Following Ang et al. ([2003]), the average effects of the annual average driving distance can also be written as follows:

{E}_{\mathrm{new}}(\mathrm{\Delta}{d}_{k})=\sum _{u=t-1,t}f(|{s}_{1}|)\mathrm{\Delta}{d}_{k}{\gamma}_{u,k}{C}_{u}{e}_{u,k}^{-1},

(10)

f(|{s}_{1}|)=\frac{|{s}_{1}|!(N-|{s}_{1}|-1)!}{N!},

(11)

where *N* denotes the number of determinants (i.e., sources examined in the study) and |{s}_{1}| is the number of determinants with a subscript *t*. Similarly, using the expression shown in Eq. (10), the average effects of the changes in the market share of new passenger cars, the changes in the total number of new passenger cars, and the change in the reciprocal of the fuel economy on gasoline consumption can be formulated as

{E}_{\mathrm{new}}(\mathrm{\Delta}{\gamma}_{k})=\sum _{u=t-1,t}f(|{s}_{2}|){d}_{u,k}\mathrm{\Delta}{\gamma}_{k}{C}_{u}{e}_{u,k}^{-1},

(12)

{E}_{\mathrm{new}}(\mathrm{\Delta}C)=\sum _{u=t-1,t}f(|{s}_{3}|){d}_{u,k}{\gamma}_{u,k}\mathrm{\Delta}C{e}_{u,k}^{-1},

(13)

and

{E}_{\mathrm{new}}\left\{\mathrm{\Delta}\left({e}_{k}^{-1}\right)\right\}=\sum _{u=t-1,t}f(|{s}_{4}|){d}_{u,k}{\gamma}_{u,k}{C}_{u}\mathrm{\Delta}\left({e}_{k}^{-1}\right),

(14)

respectively, where

f(|{s}_{i}|)=\frac{|{s}_{i}|!(N-|{s}_{i}|-1)!}{N!}\phantom{\rule{1em}{0ex}}(i=2,3,4),

(15)

which gives us the following decomposition formula for new passenger cars:

\mathrm{\Delta}{f}_{k,\mathrm{new}}={E}_{\mathrm{new}}(\mathrm{\Delta}{d}_{k})+{E}_{\mathrm{new}}(\mathrm{\Delta}{\gamma}_{k})+{E}_{\mathrm{new}}(\mathrm{\Delta}C)+{E}_{\mathrm{new}}\left\{\mathrm{\Delta}\left({e}_{k}^{-1}\right)\right\}.

(16)

We will now proceed with the decomposition analysis of the gasoline consumption of vintage passenger cars. The gasoline consumption of vintage passenger cars of type *k* in the years *t* and t-1 can be estimated by using Eq. (3) as follows:

{f}_{t,k,\mathrm{stock}}=\sum _{y=1}^{{r}_{t}}{d}_{t,k}{K}_{t,k}^{t-y}{\left({e}_{t,k}^{t-y}\right)}^{-1}

(17)

{f}_{t-1,k,\mathrm{stock}}=\sum _{y=1}^{{r}_{t-1}}{d}_{t-1,k}{K}_{t-1,k}^{t-1-y}{\left({e}_{t-1,k}^{t-1-y}\right)}^{-1}.

(18)

Accordingly, the change in the gasoline consumption of vintage passenger cars is

\mathrm{\Delta}{f}_{k,\mathrm{stock}}=\sum _{y=1}^{{r}_{t}}{d}_{t,k}{K}_{t,k}^{t-y}{\left({e}_{t,k}^{t-y}\right)}^{-1}-\sum _{y=1}^{{r}_{t-1}}{d}_{t-1,k}{K}_{t-1,k}^{t-1-y}{\left({e}_{t-1,k}^{t-1-y}\right)}^{-1}.

(19)

For example, if we set t=2000 and y=5 in Eq. (19), the fuel economy of the vintage passenger cars newly purchased in 1995 and owned in 2000 can be expressed as {e}_{2000,k}^{1995}. If we set t=2001 and y=6 in Eq. (19), then the fuel economy of the vintage passenger cars newly purchased in 1995 and owned in 2001 can also be expressed as {e}_{2001,k}^{1995}. In this study, we assumed that if the vintages (superscripts) are the same, then the fuel economy of passenger cars will also be the same, irrespective of year. For the above example, this means that {e}_{2000,k}^{1995}={e}_{2001,k}^{1995}, or more generally that {e}_{t,k}^{v}={e}_{t-1,k}^{v}, where *ν* represents the car vintage. It should be noted that the fuel economy across car ages is different as shown Table 6.

Using the relationship {r}_{t}={r}_{t-1}+1, in the sense that car age in the year *t* is equal to car age in the year t-1 plus 1, Eq. (19) can be further transformed into an algebraic form as follows:

\begin{array}{rcl}\mathrm{\Delta}{f}_{k,\mathrm{stock}}& =& {d}_{t,k}\underset{{\mathbf{K}}_{t,k}}{\underset{\u23df}{\left[\begin{array}{cccccc}{K}_{t,k}^{t-1}& {K}_{t,k}^{t-2}& {K}_{t,k}^{t-3}& \cdots & {K}_{t,k}^{t-{r}_{t-1}}& {K}_{t,k}^{t-{r}_{t}}\end{array}\right]}}\underset{{\mathbf{e}}_{t,k}}{\underset{\u23df}{\left[\begin{array}{c}{({e}_{t,k}^{t-1})}^{-1}\\ {({e}_{t,k}^{t-2})}^{-1}\\ {({e}_{t,k}^{t-3})}^{-1}\\ \vdots \\ {({e}_{t,k}^{t-{r}_{t-1}})}^{-1}\\ {({e}_{t,k}^{t-{r}_{t}})}^{-1}\end{array}\right]}}\\ -{d}_{t-1,k}\underset{{\mathbf{K}}_{t-1,k}}{\underset{\u23df}{\left[\begin{array}{cccccc}0& {K}_{t-1,k}^{t-2}& {K}_{t-1,k}^{t-3}& \cdots & {K}_{t-1,k}^{t-{r}_{t-1}}& 0\end{array}\right]}}\underset{{\mathbf{e}}_{t-1,k}}{\underset{\u23df}{\left[\begin{array}{c}{({e}_{t-1,k}^{t-1})}^{-1}\\ {({e}_{t-1,k}^{t-2})}^{-1}\\ {({e}_{t-1,k}^{t-3})}^{-1}\\ \vdots \\ {({e}_{t-1,k}^{t-{r}_{t-1}})}^{-1}\\ {({e}_{t-1,k}^{t-{r}_{t}})}^{-1}\end{array}\right]}}.\end{array}

(20)

Since we assume the relationship {\mathbf{e}}_{t,k}={\mathbf{e}}_{t-1,k} in Eq. (20), then the change in the gasoline consumption of vintage passenger cars can be decomposed into changes in the annual average driving distance of vintage passenger cars, \mathrm{\Delta}{d}_{k}={d}_{t,k}-{d}_{t-1,k}, and changes in the stock of vintage passenger cars, \mathrm{\Delta}{\mathbf{K}}_{k}={\mathbf{K}}_{t,k}-{\mathbf{K}}_{t-1,k}. The effects of these changes on the gasoline consumption of vintage passenger cars can then be formulated as follows:

{E}_{\mathrm{stock}}(\mathrm{\Delta}{d}_{k})=\sum _{u=t-1,t}f(|{s}_{5}|)\mathrm{\Delta}{d}_{k}{\mathbf{K}}_{u,k}{\mathbf{e}}_{u,k},

(21)

{E}_{\mathrm{stock}}(\mathrm{\Delta}{K}_{k})=\sum _{u=t-1,t}f(|{s}_{6}|){d}_{u,k}\mathrm{\Delta}{\mathbf{K}}_{k}{\mathbf{e}}_{u,k},

(22)

where

f(|{s}_{i}|)=\frac{|{s}_{i}|!(N-|{s}_{i}|-1)!}{N!}\phantom{\rule{1em}{0ex}}(i=5,6).

(23)

Consequently, the following structural decomposition analysis of vintage passenger cars can be represented as

\mathrm{\Delta}{f}_{k,\mathrm{stock}}={E}_{\mathrm{stock}}(\mathrm{\Delta}{d}_{k})+{E}_{\mathrm{stock}}(\mathrm{\Delta}{\mathbf{K}}_{k}).

(24)

Combining Eq. (16) with Eq. (24) yields the structural decomposition analysis of the gasoline consumption associated with driving new passenger cars and vintage passenger cars.

\begin{array}{rcl}\mathrm{\Delta}{f}_{k,\mathrm{total}}& =& \mathrm{\Delta}{f}_{k,\mathrm{new}}+\mathrm{\Delta}{f}_{k,\mathrm{stock}}\\ =& {E}_{\mathrm{new}}(\mathrm{\Delta}{d}_{k})+{E}_{\mathrm{new}}(\mathrm{\Delta}{\gamma}_{k})+{E}_{\mathrm{new}}(\mathrm{\Delta}C)\\ +{E}_{\mathrm{new}}\left\{\mathrm{\Delta}\left({e}_{k}^{-1}\right)\right\}+{E}_{\mathrm{stock}}(\mathrm{\Delta}{d}_{k})+{E}_{\mathrm{stock}}(\mathrm{\Delta}{\mathbf{K}}_{k}).\end{array}

(25)

Or alternatively, summing over car types, we have the following relationship:

\begin{array}{rcl}\mathrm{\Delta}{f}_{\mathrm{total}}& =& \sum _{k=1}^{3}\mathrm{\Delta}{f}_{k,\mathrm{total}}=\sum _{k=1}^{3}\mathrm{\Delta}{f}_{k,\mathrm{new}}+\sum _{k=1}^{3}\mathrm{\Delta}{f}_{k,\mathrm{stock}}\\ =& \sum _{k=1}^{3}{E}_{\mathrm{new}}(\mathrm{\Delta}{d}_{k})+\sum _{k=1}^{3}{E}_{\mathrm{new}}(\mathrm{\Delta}{\gamma}_{k})\\ +\sum _{k=1}^{3}{E}_{\mathrm{new}}(\mathrm{\Delta}C)+\sum _{k=1}^{3}{E}_{\mathrm{stock}}(\mathrm{\Delta}C)+\sum _{k=1}^{3}{E}_{\mathrm{stock}}(\mathrm{\Delta}{\mathbf{K}}_{k}).\end{array}

(26)

In this study, we abbreviated the effect of the change in the annual average driving distance on gasoline consumption of new and vintage cars (the first term of Eq. (25) (or Eq. (26)) plus the fifth term of Eq. (25) (or Eq. (26))), the effect of the change in the market share of new passenger cars (the second term of Eq. (25) or (26)), the effect of the change in the total number of new passenger cars (the third term of Eq. (25) or (26)), the effect of the change in the fuel economy of new passenger cars (the fourth term of Eq. (25) or (26)), and the effect of the change in the stock of vintage cars (the sixth term of Eq. (25) or (26)) as AD-effect, MS-effect, NR-effect, FM-effect, and the NM-effect, respectively.