1. 1 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.

2. 2 Modeling philosophy

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4. 4 Assumptions of the macromodel Assume that a central planner maximizes a utility function in each period t subject to a budget constraint: national saving K t+1 – (1-d)K t = Y t - C t Emission e not in production function, e t = cY t, and Y t = a t K t γ L t 1-γ, implying δe t /δK t = cγY t /K t = γe t /K t.

5. 5 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.

6. 6 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 = Σ t β t { r(x(t),u(t)) – λ t+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.

7. 7 Dynamic optimization problem

8. 8 Model without pollution overestimates consumption growth rate in China

9. 9 Pollution term explains decline in rate of consumption growth C t /C t-1

10. 10 Model pinpoints the importance of controlling pollution for sustainable economic development Because e t cannot exceed the limit M, under the assumption Y t = e t /c, Y t 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 e t /Y t. 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.

11. 11 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 C 1 to C 2 while pollution changes from e 1 to e 2. The measure is logC 2 + θlog(M-e 2 ) –[ logC 1 + θlog(M-e 1 )].

12. 12 4.3 Estimation of the macro-model incorporating pollution for China

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17. 17 Using longer sample period 1978- 2005 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.

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22. 22 Measuring change of utility: logC 2 + θlog(M-e 2 ) –[ logC 1 + θlog(M-e 1 )] 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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