How Inclusive Is Abenomics?; by Chie Aoyagi, Giovanni Ganelli, and Kentaro Murayama; imf working Paper No. 15/54; March 1, 2015

 25 

In order to address inclusiveness (or the equality aspect of growth), we need additional 

information about the distribution of the opportunity curve. Since the poor are often 

constrained in available opportunities, a notion of inclusive growth should capture the pro-

poor redistribution of such opportunities. Hence, we require an inclusive growth measure to 

i) be increasing in its argument and ii) satisfy the transfer property (Ali and Son, 2007; 

Anand et al. 2014). These properties imply that i) the function is increasing in the level of 

income and thus captures the growth dimension; and ii) any income transfers from those with 

more income to those who with less income will increase the value of the function. 

For this objective, consider a cumulative distribution of the opportunity available for the 

bottom i-th percentiles. The obtained vector, call it opportunity curve, is expressed as: 

,

2

,

3

, … ,

∑

, … ,

∑

 

Note that the last term in the opportunity curve is equal to the population mean of the (index 

of) available opportunities. 

Finally, by considering a special case of the opportunity curve, where the opportunity 

considered is the income level, we can define: 

,

2

,

3

, … ,

∑

, … ,

∑

Furthermore, we can redefine the index to represent the proportion to the population: 

1/ , 2/ , …  1. We call this sequence, y , a Social Mobility Curve (SMC), as it 

represents the ability for the bottom percentiles in the income distribution to escape into the 

higher income groups. Again, the last term in the SMC is simply a (population) mean of the 

income. 

Note that, with the rescaling of population index, SMCs are comparable across time and 

space (i.e. social welfare/opportunity is not scaled by the population size). Hence, we can 

compute a “growth” of such measures. Empirically, we can define a continuous piecewise 

linear function, given a percentile of income distribution (e.g. quartiles, deciles, quintiles, and 

so on). 

To summarize the SMC and its changes across time and space, it is convenient to define an 

index for each distribution. Define a Social Mobility Index as the normalized area under the 

SMC: that is, 

y

∗

y . The relation of the SMI to the SMC is analogous to that of the 

Gini coefficient to the Generalized Lorenz curve: it is defined both by the mean value and the 

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distribution of household income.

 7

 Furthermore, the SMI allows us to decompose economic 

growth into the change in average income and the change in income distribution, as in the 

seminal work of Ravallion and Datt (1992) who decomposed income growth into mean and 

distribution. 

The interpretation of the SMI becomes clearer and intuitive when we divide it by average 

income: 

∗

/y 

And we get an expression whose value is equal to one when the income distribution is totally 

equal (i.e. everyone possesses the same income,

y) and zero when it is totally unequal (i.e. 

one person possesses the entire income). Recall that population size is normalized when 

computing SMI, and thus this ratio,  , never exceeds the value of one. We call this measure 

the Equity Index (EI). Furthermore, by rearranging the terms, we obtain: 

∗

y 

which can be ‘decomposed’ by total differentiation:  

d

∗

y

dy 

By words, the change in the SMI is a weighted average of the change in the EI and the 

change in average income, whose weights are the level of the counterpart: when the average 

income (equity) is high, the contribution of change in equity (income) is higher, and vice 

versa. 

Alternatively, the percentage changes (growth) of SMI can be expressed as: 

d

∗

dy

y 

That is, the growth of SMI is the sum of the 

growth (percentage change) in the equity 

index and the growth in the average income. 

An interesting aspect of SMI, as a measure 

of inclusive growth is that, even when 

equity is decreasing (that is,

0), it can 

be balanced out with the increase in the 

                                                 

7

 Generalized Lorenz curves are simply Lorenz curves (uniformly) scaled by the average income. See for 

example, Kleiber (2005). 

0

10

20

30

40

50

60

70

80

1

2

3

4

5

6

7

8

9

Original

Equity Growth

Average Income Growth

Two  Types of Inclusve Growth: 

Growth in Average Income and Equity Index

Cummulative Average  of Income

Decile

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average income. As examples of SMC and SMI, hypothetical distributions are depicted in the 

text charts. Relative to the blue line, the red (broken) line represents the increase in the 

average income while holding the equity index constant

8

. Note that the area under the curve 

increases even though the equality did not change. Next, the green (dotted) line represents the 

higher income equality while holding the average income constant. Again, the area under the 

curve increases. 

Furthermore, the case depicted is also a case of inclusive growth, although less intuitive. The 

average income of the lower end 

of population has decline, but the 

average income increased more 

than proportionally to the 

increasing inequality (as 

measured by Equity Index). Since  

SMI increases, we call still 

consider this case as one of 

“inclusive growth”, but with 

deteriorating income equality. 

By using he change in SMI as a 

measure of inclusive growth 

measure, we can consider all the three cases discussed above as examples of inclusive 

growth. The difference in the way in which inclusive growth is achieved can be highlighted 

by the decomposition of the MSI growth. 

 

Appendix C. Quadratic form of inflation, simulation of the estimated effects 

 

Interpretations of estimated coefficients are more complicated when the estimated equation 

involves polynomials. Unlike for the case of linear terms, the magnitude of marginal effects 

depends not only on the difference, but also on the levels of the variable to be evaluated. 

First, we compute the optimal level of inflation (conditionally on other explanatory variables) 

based on the equation below: 

 

 .

–   .

 

 

where Y stands for conditional expected value of inclusive growth and x for inflation. 

Coefficients are based on the all-sample results of the main model (Table 1); and   is an 

intercept term. Then, by taking the first derivative  

′

 .

–   .

 

                                                 

8

 These two curves are not parallel owing to the fact that the lower income population is weighted more than 

higher income population by the construction of SMC. 

0

10

20

30

40

50

60

70

80

90

1

2

3

4

5

6

7

8

9

SMI=270

SMI=285

Less Obvious Inclusve Growth: 

Average Income and Equity Index

Cummulative Average  of Income

Decile

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