Download The Environmental Kuznets Curve

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The Environmental Kuznets Curve
Kuznets conjectured that economic growth is characterised by an increasing inequality of
the income distribution, starting from a low level (developing countries) and then after a
certain level of per capita income is achieved the inequality will decrease. In the early
90ies this bell-shaped pattern of development was also conjectured for the relationship
between the level of pollution and per capita income, the EKC. The main reasons for this
development were the same as for the income inequality change; structural change during
economic growth from agriculture to industry to services and change in income
elasticities. Such effects are investigated in the literature both in static models, dynamic
models and steady state paths.
A specific argument for the EKC is based on the nature of the abatement function.
Consider the following model (Andreoni and Levison, 2001):
U  U (C , P),U C  0,U P  0
P  P(C , E ), PC  0, PE  0
(1)
CEM
U = utility (representative consumer)
C = consumption
P = net pollution
E = abatement effort
M = resource endowment (exogenous)
Primary pollution is directly proportional to consumption.
Maximising utility (the social planning problem)
Eliminating P and M using the resource constraint:
Max C{U  U (C, P)  U (C, P(C, E ))  U (C , P(C , M  C ))}
(2)
Necessary first order condition:
U C  U P ( PC  PE )  0 
U C
 PC  PE  0
U P
The rate of substitution between consumption and environmental quality (negative
pollution) involves the partial derivatives of the abatement function. Decreasing
(3)
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marginally the abatement effort leads to a similar marginal increase in consumption due
to the marginal transformation rate of the resource constraint being 1, but then net
pollution increases because pollution increases with consumption and abatement
decreases due to the reduction of the abatement effort.
Growth and pollution
Growth is simulated by changing the exogenous income or resource M. Differentiating
the pollution generation function yields:
dP
dC
dE
dC
dC
dC
 PC
 PE
 PC
 PE (1 
)
( PC  PE )  PE
dM
dM
dM
dM
dM
dM
(4)
We see that we have increasing pollution with growth interpreted as dC / dM  0 if
dC
dC U C
( PC  PE )   PE 
  PE
dM
dM U P
(5)
(Notice that dC / dM  1 when abatement is positive.)For a given increase in
consumption an increase (decrease) in pollution is associated with a high (low) absolute
value of the marginal rate of substitution between consumption and pollution and a low
(high) value of the marginal productivity of abatement effort.
Simplified functional forms
Consider the following specification of the model:
U CP
P  C  A(C , E )  C  C  E 
(6)
Consumption generates primary pollution that can be abated through a function
depending both on consumption and abatement effort. The necessary condition (3)
becomes:
U C
 PC  PE 
U P
1


 1   C  1 E    C  E  1  1  A(C , E )  A(C , E ) 
1
C
E

C


E
(7)
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Substituting for E in the last equation we can solve for the optimal amounts of C, E and
P as functions of M:

C

C

E


M C

 
  (M  C )   C 
M,
E  M C  M 

M

M,
 
 



P  C  C E  
M (
M ) (
M ) 
 
 
 

     
M (
) (
) M
 
   
(8)
The last equation gives the relationship between pollution and income as income growths
over time. We see that the bell shape appears when he returns to scale in the abatement
function A(C,E) has increasing returns to scale;     1. Differentiating the last
equation in (6) once and twice w.r.t M yield:
P

       1

 (   )(
) (
) M
,
M   
   
2 P
       2
 (    1)(   )(
) (
) M
2
M
   
(9)
We have that the first derivative is positive for small values of M, and then turns negative
when M is large enough. From the second derivative we have that the first derivative is
decreasing. Thus, a bell shaped curve is generated.
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P
M
Figure 1. The environmental Kuznets curve
Reference:
Andreono, J. and A. Levinson (2001): “The simple analytics of the environmental
Kuznets curve,” Journal of Public Economics 80, 269-286.