Transcript Lecture 5 Analysis of Pollution by Macro

Lecture 2 A macroeconomic
model assuming pollution to be
proportional to output
Based on first part of Chapter 4. Pollution is assumed
to be proportional to output. The model explains
consumption growth in China in recent years but not in
the long-run. Parameters of the utility function are
estimated. A measure of Green GDP is provided.
Possibility of cleaning up (scrubbing) is ignored.
1
Modeling philosophy
I build a macroeconomic model under the assumption that the market economy in
China is efficient and use it to describe the Chinese macro-economy and to explain
Chinese macro-economic data. This model is constructed by assuming that a central
planner is maximizing an objective function for China. Deriving a macroeconomic
model for China by optimization has had a long history , including the work of
Kwan and Chow (1996) and Chow and Kwan (1996), among others.
In our model we assume a Cobb-Douglas production function. Yt = atKtγLt1-γ where
at , Kt, and Lt denote respectively total factor productivity in period t, capital stock
at the beginning of period t and labor in period t. As is fairly customary in
macroeconomic modeling the economy is composed of a number of representative
consumers and the same number of representative firms. This construction assumes
away the problem of aggregating the behavior of heterogeneous consumers and
firms, but has been found useful in modeling certain important features of a macroeconomy. It enables us to use the same symbol to denote a variable for one consumer,
one producer or for the aggregate economy.
2
I begin by assuming that a central planner maximizes a utility function in each period t sub
to a budget constraint: national saving Kt+1 – (1-d)Kt, d being the rate of depreciation of the
capital stock at the beginning of period t, equals income Yt minus consumption Ct. For each
period t this constraint for Kt+1 is introduced by using a Lagrange multiplier λt+1 to form a
Lagrange expression as given below. The utility function is log Ct + θ log(M-et) where et
denotes emission or pollution in period t and M is the maximum amount of pollution that ca
be tolerated. It thus has two parameters θ and M, with θ measuring the relative importance
clean environment M-e as compared with consumption. In chapter 1 and the old chapter 4
assumed output Y to be generated by a Cobb-Douglas production function atKtγLt1-γ eδ with
emission e as a factor of production. In this chapter we choose a different production functi
by assuming emission to be proportional to output, namely, et = cYt, and Yt = atKtγLt1-γ. Un
these assumptions δet/δKt = c γYt/Kt = γet/Kt.
The problem of the central planner is to maximize the sum of discounted future
utilities in all future periods subject to the above constraint for each period, β being
3
the discount factor.
Assumptions of the macromodel
• Assume that a central planner maximizes
a utility function in each period t subject to
a budget constraint:
• national saving Kt+1 – (1-d)Kt = Yt - Ct
• Emission e not in production function,
• et = cYt, and Yt = atKtγLt1-γ, implying
• δet/δKt = cγYt/Kt = γet/Kt.
4
Brief explanation of the Lagrange method for
dynamic optimization – 2 steps
• 1. Start with the constrained maximization
problem max r(x,u) subject to x=f(u).
• Set up the Lagrange expression
• L = r(x,u) –λ[x-f(u)].
• Differentiate L with respect to x, u and λ to
obtain three first-order conditions.
• Solve these equations for the three variables.
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step 2 - Generalize above
procedure to many periods
• Objective function is a weighted sum of
r(x(t),u(t)) over time t.
• Constraints are x(t+1) = f(x(t),u(t)).
• We call x the state variable and u the
control variable.
• Set up the Lagrange expression
• L = Σt βt{r(x(t),u(t)) –λt+1[x(t+1)- f(x(t),u(t))]}
and differentiate to obtain first-order
conditions to solve for the u’s and x’s.
6
Dynamic optimization problem
L = ∑t { βt [ log Ct + θ log (M – et) – β λt+1 [ Kt+1-(1-d )Kt - Yt + Ct]}
(1)
Differentiation of (1) with respect to the control variable Ct and the state variable Kt
period t yields
Ct-1 = βλt+1
(2)
-θ γetKt-1/(M - e) – λt + (1-d) βλt+1 + γYK-1βλt+1 = 0
(3)
Using (2) to substitute C for λ in (3) gives
-θγetKt-1/(M- e) – β-1Ct-1-1 + Ct-1[(1-d) + γYK-1] = 0
(4)
which can be rewritten as
Ct = [1-d + γYK-1]/ [θ γ etKt-1/(M- e) + β-1Ct-1-1]
(5)
7
Model without pollution overestimates
consumption growth rate in China
If the pollution term does not appear in the utility function or if θ = 0, equation (5) will
reduced to
Ct = [1-d + γYtKt-1] Ct-1
(6)
Let us examine whether this model without pollution can explain the evolution of consu
in China. Empirically the ratio of output Y to capital K for China has a mean of .2768 f
period 1978-2005 (See Table 3.2 below). If γ is about 0.6 and d is 0.04 (see Chow(2007, c
5) for estimates of γ and the depreciation rate d), the coefficient in square brackets on t
right-hand side of (6) is 0.96 + 0.6 times 0.2768 = 1.126. This means that consumption w
by about 12.6 percent per year. In fact, historically the mean growth rate of consumptio
the period 1979-2005 is only 9.068 percent. This is an indication that the macro-econom
model of this section without taking pollution into account does not fit the data well.
8
Pollution term explains decline in
rate of consumption growth Ct/Ct-1
if the disutility of pollution does not matter or if θ = 0, equation (5) is reduced to equation (6).
Since the contribution of the pollution term in the denominator of equation (5) is positive, the
disutility of pollution makes consumption smaller than it would be otherwise. To put this poin
in terms of the ratio Ct/Ct-1 we divide both sides of equation (5) by Ct-1 to obtain
Ct/Ct-1 = [1-d + γYK-1]/ [θ γ et Ct-1Kt-1/(M- e) + β-1]
(7)
Equation (7) shows that the rate of growth of consumption is made smaller than otherwise by
the positive pollution term θ γ et Ct-1Kt-1/(M- e) in its denominator if we assume the ratio Y/K
in its numerator to be given. In the course of economic development there is a tendency for th
pollution term in the denominator to increase because of the increase in e, unless this effect is
somehow offset by a reduction of the ratio Ct-1/Kt. The data for China to be presented in the
next section will show that the ratio Ct/Ct-1 has been indeed declining. Our model provides an
explanation of this decline, although there are other reasonable models that can also provide
9
an explanation.
Model pinpoints the importance of controlling
pollution for sustainable economic development
• Because et cannot exceed the limit M, under the
assumption Yt = et/c, Yt cannot exceed M/c.
Thus economic growth eventually stops
according to this model, unless we revise this
assumption and allow technological innovation
to lower the ratio et/Yt. In the framework of this
model economic development can be sustained
only by solving the environmental problems, or
by reducing the ratio c = e/Y so as not to allow e
to reach the limit M. Thus this model pinpoints
the importance of controlling pollution for
sustainable economic development.
10
Measuring the damage to environment
in the production of GDP
• This model provides a measure of the disutility of pollution
associated with a given increase in consumption, as given by the
utility function for specific values of the parameters θ and M.
• This measure is related to the measurement of Green GDP. The
latter nets out from GDP the cost of productive resources used to
repair the damage to the environment. Green GDP has limited use
because knowing the cost of repairing environmental damage in the
production of a given amount of output one still does not know
whether the environmental cost is worth paying for.
• Our measure nets out the disutility of a polluted environment from
the utility derived from consuming a given output. Our framework
can be used to measure the change in net utility when consumption
changes from C1 to C2 while pollution changes from e1 to e2.
• The measure is logC2 + θlog(M-e2) –[ logC1 + θlog(M-e1)].
11
ndustrial
Waste Air
Emission1
uels Burning
roduction
rocess
ulphur Dioxide
Emission
4.3 Estimation of the macro-model
incorporating pollution for China
100 million
cu.m
100 million
cu.m
100 million
cu.m
10 000 tons
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
113375
121203
126807
138145
160863
175257
198906
237696
268988
330992
70918
72985
75919
81970
93526
103776
116447
139726
155238
181636
42457
48218
50887
56032
67337
71481
82459
97971
113749
149353
2346
2090
1857
1995
1948
1927
2159
2255
2549
2589
12
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
Y
3645.22
3922.25
4228.45
4450.81
4851.78
5380.34
6196.87
7031.62
7654.96
8540.74
9503.08
9889.48
10268.58
11212.69
12809.29
14595.45
16505.54
18309.93
20143.47
22013.47
23737.66
25549.33
27700.01
30000.14
32726.76
36007.46
39638.09
43695.22
K
13910.7
14769.03
15746.23
16691.12
17816.94
19160.45
20709.59
22709.03
24961.7
27470.07
30686.13
34351.8
37781.34
41306.76
45441.53
51172.44
57508.32
64716.96
72516.72
80523.36
88879.67
97719.5
106568
116207.5
126940.3
139605.6
153682.1
168794.5
C
2239.1
2542.7
2798
3058.6
3385.7
3723.4
4166.4
4668.7
5082.2
5527.8
6216
6497.5
6650.6
7254.2
8184.8
9046.2
10014.1
11067.9
12429.5
13419
14508.9
15851.2
17174.8
18297
19497.8
20532.3
21577.8
23130.6
e
20201.34
21736.6
23433.53
24665.81
26887.94
29817.15
34342.25
38968.33
42422.8
47331.68
52664.84
54806.23
56907.14
62139.29
70987.43
80886.1
91471.57
101471.3
111632.5
113375
121203
126807
138145
160863
175257
198906
237696
268988
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Ct = [1-d + γYK-1]/ [θ γ etKt-1/(M- e) + β-1Ct-1-1] (5)
For the purpose of estimation the values of d and γ are assumed to be .04 and .60
respectively from our knowledge of these parameters; the value of β-1 is assumed to be
1.02 as the value of the discount factor β is often assumed to be 0.98. Only parameters M
and θ in (5) are required to be estimated. We first use the sample from 1997 to 2005
when data on pollution are available as given in Table 4.2. For different assumed values
of M, the first four rows of Table 4.3 give estimates of θ obtained by the nonlinear
regression routine of STATA, together with its t statistic, root mean square error and Rsquare of the regression. All estimates of θ are highly significant and the values of Rsquare are very high.
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Period
1997-2005
1997-2005
1997-2005
1978-2005
1978-2005
1978-2005
1978-2005
1978-2005
1997-2005
1-d
.96
.96
.96
.9958
.96
.96
.96
.96
.95850/105.5
M
1000000
10000000
1100000
1100000
90000000
1100000
1000000
10000000
1000000
θ
.7468209
9.345796
.8430062
1.965221
85.72903
.814789
.7226108
8.95161
.6802665
t stat
7.12
7.37
7.16
13.35
7.38
7.29
7.28
7.31
2.47
R-quared
0.9999
0.9999
0.9999
0.9996
0.9998
0.9998
0.9998
0.9998
0.9998
RootMSE
188.9336
183.2962
187.9435
232.4537
183.1053
183.3053
183.4771
182.9961
187.0087
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Note that the value of θ increases as M increases. The reason is that for a larger M the
percentage change of M-e is smaller for the same change in e; this requires a larger value
of θ to yield the same percentage change in the term θlog(M-e) in the utility function.
When we vary the value of M substantially the goodness of fit of the regression as
measured by the RMSE remains almost the same. This fact is consistent with the fact that
if we try to estimate both θ and M simultaneously the standard errors of both are very
large or we cannot obtain reasonably accurate estimates of both parameters. In any case,
the positive and highly significant estimates of θ supports strongly our model of pollution.
Our theory of pollution would be rejected if the estimates of θ were statistically
insignificant, and equation (5) would be reduced to equation (6). I have tried to estimate
both parameters d and θ, given the value of M, and found that the estimate of d is almost
exactly equal to 0.04 and that the estimate of θ remains almost the same.
16
Using longer sample period 19782005
• I have also tried to estimate equation (5)
using a longer sample period from 1978 to
2005. To do so data for e before 1997
have been constructed them by multiplying
Y by 5.5419, the mean of the ratios C/Y for
the years 1997-2006 (Y in 2006 not shown
in Table 3.2). As shown in the lower half of
Table 3.4, all statements of the last
paragraph remain valid for the larger
sample.
17
After the successful estimation of the model using data for 1978 to 2005 it then occurred
to me that the variable e in our utility function can be replaced by national output Y or
any other variable proportional to it. To test this proposition I used the sample from 1997
onward when the data on pollution are available and estimated equation (5) after
substituting Y for e. The result, for a given value of M equal to four times the value of Y
in 2005 as M = 1100000 is about four times the value of e in 2005, is about as good as
the model using e. The estimate of θ is 4.109 with a standard error of .570, and a t ratio of
7.21 while he Root MSE equals to 187.0105 which is about the same as given in the top
half of Table 3.4, and R-sq is 0.9999. Thus, if we let Y instead of e enter the utility
function we will find the estimate of the parameter θ to be equally good and the resulting
equation to explain C equally well. As is often the case when a macro-economic
hypothesis is proposed, one finds the hypothesis to be sufficient in explaining the data but
not necessary. There are alternative hypotheses that will explain the data equally well.
For the purpose of examining the macroeconomic implication of pollution, we know that
pollution is highly correlated with output Y. Hence it is difficult to distinguish between
the effect of pollution and of other variables that are highly correlated with Y.
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The failure of our model to distinguish between alternative variables to be used as e in
our utility function turns out to be a blessing in that it makes our theoretical framework
more general. Our general model implies that in the course of economic development the
increase in output enables the population to derive more utility from a higher level of
consumption but the increase in output itself reduces utility because it produces more
pollution, congestion, or whatever other negative side effect. Since our utility function is
identical with the formulation often used for the choice of labor hour where more labor or
hours of work (corresponding to our variable e now interpreted as “effort”) reduces utility
and this effect is measured by the difference between a maximum amount and the actual
amount. An important finding of this paper is that if pollution or any other variable
related to the increase in output asserts a negative effect on utility it should be
incorporated in a macroeconomic model and such a model explains the Chinese
macroeconomic data better than the one without using it. Although pollution is not a
necessary explanation of our empirical results, incorporating it has implications
supported by Chinese data. In the study of economic growth we suggest the consideration
not only of the positive effect of increased consumption but also the negative effect of
any variable associated with the increase in output itself. We are led to this proposition by
studying the disutility of pollution. (Pollution associated with consumption rather than
output can be modeled in our framework by defining consumption as ultimate
consumption net of all home production or productive work by the consumer herself.)
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Ct/Ct-1 = [1-d + γYK-1]/ [θ γ et Ct-1Kt-1/(M- e) + β-1]
(7)
To see how an increase in pollution contributes to a reduction in the rate of growth of
consumption, or in the ratio Ct/Ct-1 we divide both sides of equation (7) by Ct-1 and find
denominator on the right hand side to be θγCt-1etKt-1/(M- e) + β-1. If pollution is not mod
the first term involving θ would disappear. The importance of this pollution term is mea
by its ratio to the second term β-1. This ratio, with θγ = 1.37655, M = 1200000 and β-1 =
increases monotonically from .00338 in 1979 to .04984 in 2005 as the value of (M –e) in t
denominator of the first term decreases with the increase in emission e. Although the ra
small, by 2005 it is about 5 percent. Thus we find that the pollution term involving a non
θ reduces the ratio Ct/Ct-1 by about 5 percent in 2005.
20
The second effect of the pollution term is on the numerator [1-d + .6Y/K] of the expressio
explain the ratio Ct/Ct-1. The reduction in the ratio of Ct/Ct-1 as pollution increases which w
out in the last paragraph will cause a larger fraction of output Y to be used for capital form
This will lead to a reduction in [1-d + .6Y/K] in the numerator of the expression explainin
Ct/Ct-1. This is an additional factor in explaining why damage to the environment would m
economic growth unsustainable. Empirically the ratio Y/K for China decreases almost
monotonically from 0.311 in 1987 to 0.259 in 2005.
21
Measuring change of utility:
logC2 + θlog(M-e2) –[ logC1 + θlog(M-e1)]
•
•
•
•
•
•
•
Given M =1,000,000 and θ = 1.48, the change of e from 269000 in 2005 to
331000 in 2006 implies the change in utility due to increase in pollution by
1.48[ log 731000 – log 669000] = 1.48[.0886] = .1312.
Given Y in 2006 = 48000, the level of utility in 2006 adjusted for the damage
to the environment is log(48000) - .1312 = 10.779 - .1312 = 10.648 The
level of Green GDP in 2006 net of this adjustment is therefore exp(10.648)
= 42200.
The percentage reduction of Y from 48000 to 42200 is 5800/48000 = .121
or 12.1 percent.
Is this estimate too large? Not large as compared with the estimate of 16
percent by Shi Minjun.
Question is whether our index of pollution is a good approximation of the
overall index. Has a comprehensive pollution index increased faster than
our index? If so, our estimate of θ could be smaller.
A similar percentage should be subtracted from Y2005 as compared with
Y2004 and we should examine the percentage change in Green GDP from
2004 to 2005 as compared with the percentage change in Green GDP from
2005 to 2006.
•
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