Paris School of Economics - L'Agence Française de Développement

Furthermore, in the above equation, shadow prices are measured in units of utility per unit of
capital. This is not convenient for empirical use. One type of capital that has an observed market
price can be taken as numeraire. Let us suppose it is i=1. This posited price is equal to 1. The
prices of other types of capital, expressed in this numeraire, become price indexes. Let us call W
the total value of wealth expressed in this price system:
n
n dW
Wt = �� pit Kit and =�� pit Iit
i =1
dt i = 1
The condition of sustainability is dW/dt �� 0. This can be used to measure weak sustainability
where the range of the different types of capital is the most extensive that can be measured
and where prices are the best possible approximations of shadow prices.
Technological progress and population growth
The condition of sustainability can be defined in a slightly different way. By dividing both terms
of the equation by the value of the first type of capital, one obtains:
1
p2 K2
pn Kn
( dV /dt ) ( ) = ( 1 /K 1 ) ( dK1/dt ) + ( ) ( 1 /K 2) ( dK2/dt ) +.+ ( ) ( 1 /Kn ) ( dKn/dt p1 K1
p1 K1
p1 K1
)
The economy is sustainable if the sum of the growth in the volume of the different types of
capital, weighted by their elasticity of substitution to the one type chosen as numeraire, is
non-negative.
Now let us suppose that there is Hicks-neutral technological progress. This can be interpreted
as the “knowledge” growth rate taken as the numeraire. Its rate of growth is the growth rate (�)
in total factor productivity (TFP). With neutral technological progress, the elasticity of output
to knowledge is 1. Therefore, the growth rate of real wealth becomes simply the sum of TFP
growth and the growth rates of the other types of capital:
n
1 dW
1 dK i
= � +��
W dt
i = 2 K i dt
The growth in the volume of the different types of capital, including TFP growth is measured
and the values obtained are added together.
The formula is valid if the population is constant. If the population is growing at rate g, the sustainability
criterion must be applied to calculate the growth of real wealth per capita, with the
caveat of constant population growth and of a wealth distribution that is independent of population
change:
1
W
dW
dt
n
- g = � +��
1= 2
1
K i
dK i .
dt
December 2011 / Measure for Measure / How Well Do We Measure Development? / © AFD [ 193]