The Optimal Timing of Transition to
New Environmental Technology in
Economic Growth
2009 International Energy Workshop
17-19 June, 2009
Fondazione Giorgio Cini, Venice Italy
Akira MAEDA and Makiko NAGAYA
Kyoto University
Background: Growth and the Environment

Natural resource use along w/ economic growth has been studied
since 1970s.



In 1980s, the environment began to be recognized as another
important factor that determines the trajectories of growth.
At the same time, the theory of economic growth began to change.


Dasgpta and Heal (1974), Stiglitz (1974), Solow (1974), etc.
“endogenous change” comes to a central issue in the economic
literature, inc. Romer (1990)，Lucas (1988), etc.
These two streams converge to a bunch of studies after late 1990s.


Endogenous technological change plays central roles in sustainable
development.
Barbier (1999), Tahvonen and Salo (2001), Bovenberg and Smulders
(1995), Schou (2000), Groth and Shou (2002), Cunha-e-Sá and Reis
(2007)
2
Background: Optimal Timing (1)

Cunha-e-Sá and Reis (2007). “The Optimal Timing of Adoption of a
Green Technology.” Environmental and Resource Economics 36.


Their focus:
 As the economy grows, new pollution-abating technology becomes
indispensable;
 Such a technology requires investment efforts at the time of
deployment;
 The optimal timing will be determined in an endogenous manner.
Their model
 is based on a typical Ramsey-type model, incorporating
“environmental quality” and “clean technology.”
 The level of clean technology may change discontinuously once
“grade-up” happens.
 Consideration for the net benefit determines the optimal timing of
grade-up.
3
Background: Optimal Timing (2)

Cunha-e-Sá and Reis contributed to the literature in that:


they underlined the significance of “the optimal timing” in the framework
of the environment and economic growth.
Some debatable issues remain on the other hand:


Technological change in clean technology is represented as a jump of
the level.
 Allowing such a sudden change of the level implies that the
technology is a kind of flow, not stock.
 This nature contrasts to the fundamental idea of modern endogenous
growth theory: their treatment of technological change for clean
technology thus seems old-fashioned in this respect.
Their treatment of investment-related costs seems strange.
 It may degrade the clear cut of their analysis.
4
The Purpose

is to examine the choice of timing for technological change vis-à-vis
environmental quality in economic development.

Based on the spirit and mathematical treatment of Cunha-e-Sá and
Reis, we develop an analytical model that addresses the optimal
timing of transition of environmental technology from old one to new
one.

In contrast to Cunha-e-Sá and Reis,
 we focus upon acceleration of technological progress, and
 we treat costs in a simpler manner so that our focus can become
clearer.
5
Analytical Frame (1)

Ramsey model w/ closed economy; constant population




Y: Production; K: Capital stock;
Q: Environmental quality
a: Environmental technology level
C: Consumption
Yt  AK t
dKt dt  AK t  Ct
A representative economic agent represents the household whose
time-additive utility is determined by not only consumption (C) but
also “environmental quality” (Q).
C Q 

 1
Ut
t
t
1
1
0    1,   0,   1
 1     1,  1     
6
Analytical Frame (2)

The environmental quality (Q) is a function of the consumption (C)
and “environmental technology level” (a)
Qt  at Ct

The progress of the environmental technology level itself may occur
as the economy grows.

the growth rates of a and K ( Y/A) are proportional to each other:
dat


 0
That is:
dt  at  dKt dt  K t
dat dt
dKt dt
 
at
Kt
The coefficient  may change discontinuously at time T.
7
Analytical Frame (3)

Discontinuous change in the coefficient :
dat dt K t  0

dK t dt at 1
for
t  T
for
t T
0  0  1
T  lim t
t T ,t T

Costs for the change


Typical costs: R&D and physical investments
Existing physical capital may become obsolete, and need to be
scrapped.
K T  K T 
0   1
KT   lim K t
t T ,t T
8
The Model
J K 0   Max

T
Ut e
T ,Ct  0

CQ 

 t
 1
s. t.
Ut
t
t
dt 


T
1
1
U t e   t dt
0    1,   0,   1
dKt dt  AK t  Ct
K T  K T 
0   1
Qt  at C 
 0
 a K t 0
at   
 a K t 1
for
t  T
for
t T
9
Solving the Problem
Control variables: T, Ct t0,
State variables:

Kt t0,
First step: Optimizing for the value function to a given capital stock

for after-transition.
  ( t T )
 KT   Max
Ct 

T
Ute
dt
Second step: Finding the optimal path until the transition.
V T   Max
Ct 



T
0
U t e   t dt   K T e   T
Third step: Solving for the optimal timing.
J K 0   Max V T 
T
10
First Step


Suppose that transition to the new technology occurred at time T.
The problem thereafter is:
 K T   Max
Ct 
s. t.


C Q 
T
 1
t
t
1
e   t T dt
dKt dt  AK t  Ct
tT
Qt  at C 
at  a Kt1

FONCs:
1 1  1
H C  1   a  1  K t  1 Ct
Ct  t  0
1
H K  1 a  1  K t  1 Ct1 1  K t1  t A  t 
1
lim e  t t K t  0
t 
dt
dt
11
Solution for the First Step
Ct
Kt

1
0
M1
where
t T
(constant)
1   1  

M

0
 1
1  


    A1   1   1    0
 1
1  1   1   
dCt dt dK t dt
A1  1    A   


1  1   1   1  1   
Ct
Kt
We obtain:
a  1  

 KT 
1
t T
1 1 
1   1 

KT

M1 
1

1   1
 1   
1
12
Second Step

Finding the optimal path to time T
V T   Max 
C 
t
s. t.
T
C Q 
0
 1
t
1
t
1
dKt dt  AK t  Ct
K T  K T 
e   t dt   K T e  T
t  T
0   1
Qt  at C 
at  a Kt0

FONCs:
1 1  1
H C  1   a  1  K t  1 Ct
Ct  t  0
0
H K   0 a  1  K t  1 Ct1 1  K t1  t A  t 
0
T  
 ( K T ) K T

K T
K T 
dt
dt
13
Solution for the Second Step
Ct
0 M 0

K t 1  e  0 t T    1
where
t  T
 0   1  

M

0
 0
1  


    A1    0   1    0
 0
1  1   1   
1 1 

1  1      A1    0   1    11 1 


 0   1      A1   1  1   
1 1 
CT  CT  1 1 1 


KT  KT
We obtain:
 1  e   0 (T t )   1 
Kt  K0  e   t

  0 (T  t )
o


e

e


1


At
1
M0
t  T
14
The Final Step

Optimizing it w.r.t. T
J K 0   Max V T 
T
where
V T   Max
Ct 

T
0
U t e   t dt   K T e  T
KT has already been obtained analytically.

Thus, we obtain:
 1 

dV (T ) a
   1 


e  T  0  
dT
1

 M 0   


1 1 
1 1   1   


1    11 1 

1  KT 





1 0  1  
KT 
1
1
0
Examining its sign, we know the properties of the solution.
dV T  dT T T  0
if
0 T*  
dV T  dT T T  0
if
T*  0
*
*
15
Proposition 1
For a finite T* to exist, that is, T*< ,
the following condition is necessary:
1    A   0
M0   0
or, equivalently
1   and A   0 M 0

1   and A   0 M 0
1    A   0
M0   0
Conversely, if
holds, then the transition of environmental technology never occurs.
(i.e. T*= )
16
Interpretation of Prop. 1 (1)

The existence of a finite transition time is determined by:
1   and A   0 M 0

1   and A   0 M 0

Notice:
 
A   0 M 0
 
 
 1    CT  AK T   CT  YT 
 
This represents average consumption
propensity just before the transition.

 represents the magnitude of the elasticity of marginal utility.

It determines the sign of the cross second derivative of the utility
function:  2U
  1   C  Q  1 1
CQ


If 1> (1<), marginal utility of “environmental quality” U/Q is
increasing (decreasing) in consumption.
That is, the society prefers more (less) environmental quality as the
consumption grows.
17
Interpretation of Prop. 1 (2)

For the transition to occur in a finite time horizon, the economy must
be in either of the following two situations:

The economy highly prefers environmental quality, and its current
average consumption propensity is lower than a certain value (1/).
(The economy has a higher saving rate.)

The economy rather prefers consumption to environmental quality, and
its current average consumption propensity is higher than a certain value
(1/).
(The economy has a lower saving rate.)
18
Proposition 2
Suppose that the following conditions hold.
1    A   0
M0   0
1 
And, either
and
1 
or
and



1 
1  1 1 1  1  


1  0  


1 
1  1 1 1  1  


1  0  
 
K0  
 
K0  
Then there exists an optimal transition time T* such that 0 < T*< , and
it is obtained
 by solving the following equations:

1
M0
K

T
e
e
 oT 
AT 

  1
1
M0

K T 

K0


1 
1  1 1 1  1  


 
1  0  
19
Interpretation of Prop. 2
Given the existence of a finite T*,
1   case: Capital stock Kt increases, and reaches the fixed level KT from below.


1   case: Capital stock Kt declines, and reaches the fixed level KT from above.


1
1
20
Proposition 3
Suppose 1    A   0 M 0   0
Also, suppose the following approximations hold.
1    0
1  0
Then, we have following sensitivities.
dK T 
d 1  0 
0
1   case:
1
Note:
case:
dT 
d 1  0 
dT 
d 1  0 
dat dt K t  0

dK t dt at 1
0
0
for
t  T
for
t T
0  0  1
21
Interpretation of Prop. 3
1  0 represents the degree of the innovation. Intuitively, it tells how much the
transition to the new environmental technology is beneficial to the economy.
The increase makes the barrier to transition lower.
1
1
22
Conclusion



We investigated the optimal timing of transition in environmental
technology from old one to new one along with economic growth.
Whether transition to new technology is needed or not is determined
by two factors: the degree of complementarity between
environmental quality and consumption, and the average
consumption propensity (or saving rate).
Given the necessity, transitions to new technology may occur in two
possible ways.



An environmentally developing economy would accumulate social
capital and invest it to the development of the new technology.
A matured economy holding a sufficiency of capital and enjoying a high
consumption rate would realize the need for environmental quality
improvement in some day.
A higher degree of innovation would lead to earlier or later
transitions, respectively.
23