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Lecture 5 Optimum path for CO2 emission

Based on Chapter 3

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Introduction

Global warming due to the emission of CO2 has attracted a lot of attention in recent years. Al Gore received a Nobel Peace Prize in 2007 because of his effort to call the world’s attention to this problem. On December 3, 2007, the UN Climate Change Conference opened in Bali, a resort island of Indonesia with participation of representatives of over 180 countries and regions. The UN adopted a Bali Roadmap on Dec. 15 after two weeks of consultations and negotiations. The Bali roadmap as a clear agenda for key topics is expected to launch negotiations up to the end of 2009 on a crucial climate change regime.

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The problemmany experts claim that the total amount of CO2 in the atmosphere E should not exceed a critical level of 600 GtC (gigatons of carbon equivalent) above which violent climate changes would occur, while its current level is 200 GtC. The current amount of annual emission is e = 8 GtC/yr. [NYT editorial p. A22, August 18, 2009: atmospheric concentration of greenhouse gases, now about 380 parts per million, should not be allowed to exceed 450 parts per million. But keeping emissions (accumulated?) below that threshold would require stabilizing them by 2015 or 2020, and actually reducing them by at least 60 percent by 2050. ??] To keep the steady state level of E at 600, we need to reduce e from 8 to 3. This is derived from the dynamic relation Et+1 = b1 Et + b2 et (1) where Et denotes the cumulated amount of CO2 at the beginning of year t and et denotes the amount of emission during year t. Empirically b2 = 0.5, 1-b1 = b2/200 and b1 = 0.9975. (The denominator 200 actually increases with time, roughly by 1% every year.) Equation (1) means that half of the emission in the current year will remain in the atmosphere at the beginning of the next year. Of the total accumulation in the atmosphere at the beginning of this year, 1-.5/200 or 99.75 percent will remain at the beginning of the next year. At the steady state, when Et+1 = Et = E and et = e, solving the above equation gives E = 200 e. If E = 600, e has to be 3, much lower than the current level 8.

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One important question in the study of global warming to be addressed in this

chapter is how to reduce annual CO2 emission from 8 GtC/yr to 3 GtC/yr. Should e

first increase from 8 and then decline later to reach 3, or should it start decreasing

immediately and gradually from 8 to 3? What is the most desirable time path of et

from here onward? Many scholars have provided answers to this question. The

economic approach suggested in this chapter is to evaluate the cost of having

amount E of accumulated CO2 in the atmosphere and the benefit that emission e

will bring to increase production at each period; the discounted values of the costs

and benefits for all future years will be summed up in order to judge whether one

path for e to go from 8 to 3 is better than another. Since the costs and benefits of all

future years can be summarized by a multi-period objective function, we need to

solve an optimal control problem to determine the optimum path of future CO2

emission e to maximize the value of the objective function.

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In this discussion I abstract from the problems of having capital and labor in

production, of capital accumulation and of estimation of economic models

incorporating these features. These problems will be discussed in the context of a

macro-economic model of one country in chapter 4. After setting up a simple model

of dynamic optimization (or optimal control, as the two terms can be used

interchangeably), I will present in section 3.2 the Lagrange method for solving

dynamic optimization problems that will be used in many applications of this book,

including a numerical method for solving optimal control problems. Section 3.3

deals with the optimum time path et of CO2 emission for a model that does not take

into account possible inertia of changing the value of e from year to year or of

changing the growth rate in e from ∆e(t-1) to ∆e(t) while the model in section 3.4

does. Section 3.5 presents numerical results of selected dynamic optimization model

for solving for the optimum path of CO2 emission. Section 3.6 discusses a model of

Socolow and Lam (2006) which solves the same problem using a different

framework that does not consider the tradeoff between the cost in utility and the

benefit in production of CO2 emission. It will be interesting to compare the optimal

path of their solution with ours.

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In much of the discussion in this chapter and in the technical parts of this book,

specific utility functions are employed and derivation of optimum control results is

worked through using the specific utility functions. There is much to be learned in

working through examples for particular parametric utility functions. First, the use

of parametric analysis in a specific model helps one think quantitatively about

environmental problems. Second, it facilitate statistical estimation of the parameters.

Third, understanding of the general case often follows from first working out the

solution of specific cases. Mathematicians who prove general theorems often start

working with specific examples that provide them with intuition and insights for the

general case. They later summarize the results as a general theorem. Finally this

book is devoted to solving particular problems rather than proving general

mathematical propositions. We introduce particular mathematical models and

methods in order to analyze environmental problems

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The utility function we have chosen is log(Aet

δ(M-Et)θ) = logA + δlog(et) + θ log(M-Et)

Aet

δ is output produced by a production function with capital and labor taken as given. The variable (M – Et) allows E to have a negative effect on utility and the effect is detrimental when E is close to M; its logarithm approaches minus infinity as E approaches M. The parameter θ measures the relative importance of having more current emission et for the increase in output and the damage caused by having a larger quantity Et of accumulated CO2 in the atmosphere. We rescale this utility function by dividing it by δ or by setting δ =1. The redefined utility function Aet(M-E t)

θ can be interpreted as an index of world output net of the harmful effect of the amount E of accumulated CO2 in the atmosphere. In the literature on global warming there are estimates of the percentage change of world output resulting from an increase in mean temperature caused by a change in M-E by one percentage point, namely estimates of the parameter θ. An estimate of θ will be used to calculate the optimum path of et numerically.

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Given this utility function and the dynamic equation (1) which determines the state

variable Et+1 as a function of Et and the control variable et we can set up a dynamic

optimization problem to determine the optimum time path for the emission et.

Since for each period t we maximize a given utility function subject to a dynamic

constraint, when we maximize a weighted sum (weighted by the discount factor βt)

of the utility functions each subject to the constraint for that period we can apply

the Lagrange method using a Lagrangean expression of the form

L = Σ { βt [log (et ) + θ log (M - Et) ] – βt+1 λt+1[Et+1 – b1 Et – b2 et ] } (2)

A set of first-order conditions for an optimum can be obtained by differentiating (2)

with respect to the control and state variables for each period. The resulting

equations are functional equations for the control variable et and the Lagrange

multipliers λt as functions of the state variable Et. In general these functional

equations do not have analytical solutions.

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A numerical solution to an optimum control problem formulated as the

maximization of a Lagrangean such as expression (2) can be obtained by

approximating the dynamic equation for its state variable by a linear function and

its objective function by a quadratic function. To pursue this approach we

formulate a “linear-quadratic” optimum control problem using the Lagrangean

L = Σt=1T { βt(xt –at}’Rt(xt –at}/2– βt λt’[xt – b - A xt-1 – C ut-1 ] }

(3)

Treating a more general case we let xt be a column vector of p state variables, ut be a

column vector of q control variables, at be a vector of specified targets for xt to

approach, Rt be a given symmetric matrix in the utility function r and λt’ be a row

vector of Lagrange multipliers for period t, with prime denoting transpose. In the

dynamic equation for xt, b is a given column vector of p components, A is a given p

by p matrix and C is a given p by q matrix. Since we are dealing with a vector of p

state variables, we need a vector of p Lagrange multipliers for each period. Our

objective is to maximize L by choosing the optimum policies u0 to uT-1.

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3.3 Optimum path for CO2 emission without considering the inertia of change

If we solve the dynamic optimization problem specified by the Lagrangean expression (2), our solution may be questioned by those who conceive the problem for CO2 emission as a problem of finding some optimal path for e(t) from the present value of 8 to a steady state value for e that guarantees E to equal 600. They may ask how the optimum path of et obtained by solving our optimization problem will guarantee a steady state value of E equal to 600. We respond by challenging the premise of this question that the steady state of E has to be 600. Why not somewhat above or below 600? Being above it may mean more serious climate problems but the world might accept the consequences if there are tremendous gains in output. Our viewpoint is that a steady state level for E close to 600 is justified only in the context of a solution to a dynamic optimization problem that allows for the proper balancing of the costs and benefits along the way and at the steady state. If the costs and benefits are described properly by our utility function we have shown that the optimal level of E is M/(1+θ) where M is the maximum tolerable value for E and θ is a parameter in the utility function log(e) + θ(M-E). The larger the value of θ the more we want the steady state value for E to be below M. If the steady state level of E turns out to be different from 600, it means either that the level 600 is not really the optimum, or that we have chosen incorrect parameter values for M and θ and should change them to reflect the tradeoff between more output and more CO2 in the atmosphere more accurately. An advantage of our solution is that it takes into account the costs and benefits of emission in all future years up to time T.

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From our approach to finding an optimum path of CO2 emission

through dynamic optimization three comments can be made. First,

while we consider global warming a serious problem we note that if we

slow down the growth trend of emission in the near future, there is loss

in output that is associated with the slow increase in e. No wonder some

developing countries do not want to worry about controlling CO2

emission today or in the near future in order to obtain more output.

There will be time in the future to do so. For every year that these

countries are asked to help control the emission of CO2, they lose the

benefit of more production and capital accumulation to enable them to

have a developed economy.

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Second, like many other recommendations for solving the global warming problem the above solution has not allowed for the possibility of important technological innovations in the use of clean energy. If such innovations occur, efforts to reduce output today to solve the problem of global warming shall be regrettable. Without knowing the prospect of great innovations with regard to clean energy, one can point out that historically pessimists have often been wrong than right, beginning with the prediction by Malthus that food production could not catch up with the increase of population and famine would occur. Historically food was produced mainly by the use of land and labor and land was a limited resource just like natural resource in the discussion of environmental problems. Land as a resource was certainly limited in supply just like oil, and yet technological innovations have enabled the output of food per unit of land to increase many fold.

Third the specific formulation of the optimal control problem as given by the Lagragean (11) can be improved to deal with possible inertia in changing the quantity of emission from year to year, or in changing the trend of emission, as will be discussed in the next section.

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3.4 Accounting for the inertia in the quantity of emission and in the trend of emission.

To account for possible inertia in changing the amount of emission from year to year, and to insure that the starting value of emission in the optimum path is not too far from the historical value in the previous year ( e = 8 in our example) we can introduce a cost for changing the level of emission e each year. This is done by defining the change ct = et-et-1 and incorporating this variable in the utility function as follows

Σ1T { βt [log (et ) + βθ log (M - Et+1)-τ ct

2/2] (25)

where the second term in the utility function is simply βt+1θ log (M - Et+1) that is included in the sum.

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The dynamic model for the state variable xt is given by the following matrix equation

Et+1 = b1 0 0 Et + b2 et

et 0 0 0 et-1 1

ct 0 -1 0 ct-1 1

The vector of state variables so defined includes all variables in the above utility function and the algorithm of section 3.2 can be applied after this utility function is approximated by a quadratic utility function, or after the derivative of this utility function is linearized, and the dynamic model becomes a first-order system in the form to be used in the Lagrangean expression (3), the vector Lagrangean multiplier λ having three components.

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In the notation of our algorithm the derivative of the utility function is R(xt-at). In the notation of (25) the derivatives of the utility function with respect to the vector x’ = (Et+1, et, ct are ( –βθ(M-Et+1)

-1, et-1, -τct).

We linearize –βθ(M-Et+1)-1about E*(which is close to 200 when t is small

and close to 600 when t is close to T) to get βθ(M-E*)-2 (Et+1- E*). We linearize et

-1 about e* (which is close to 8 when t is small and close to 3 when t is close to T) to get -e*-2(et- e*). Using these derivatives we can write R(xt-at) as

βθ(M-E*)-2 0 0 Et+1 E*

0 -e*-2 0 et - e*

0 0 -τ ct 0

Now we have a quadratic utility function and a linear dynamic model to be used to find the optimum path of CO2 emission et using the algorithm of section 3.2.

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To insure that the initial e1 for the determination of E2 is closed to 8 we can further introduce a target a1 with its second component equal to 8 and apply some severe penalty in the 2-2 element of the matrix R1 to prevent large deviation from 8. We can introduce the second difference or the rate of change in ct which equals st = ct – ct-1 as given in the model below to insure that the path of et does not decrease immediately at the beginning because of the momentum of the past increasing trend.

Et+1 = b1 0 0 0 Et + b2 et

et 0 0 0 0 et-1 1

ct 0 -1 0 0 ct-1 1

st 0 -1 -1 0 st-1 1

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3.5 Numerical Solutions to the optimal path for CO2 emission

We now consider the numerical solution of the dynamic optimization

problem represented by the Lagrange expression (25) that penalizes

large change in emission from year to year. One problem is to find an

optimum path of CO2 emission e beginning with the current value of 8

to reach a steady state given by M/ 200(1+ θ) with the corresponding

steady state of E equal to of M/(1+ θ) beginning with the current level of

200. Illustrative numerical solutions to the optimum path et of CO2

emission and the associated value for Et for selected values of the

parameters β, θ and M are given below.

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we provide illustrative numerical solutions for the optimum path et* of

CO2 emission and the associated accumulation path Et* for selected

values of the parameters β, M, θ and τ. For β = .98, M = 800, θ = 2.4 and τ = 0, Figure 1 shows both the optimal paths for et and Et. Note that the initial et may differ from the current observed value 8 because we do not penalize the annual change in et by setting the value of τ equal to zero in the Lagrange expression L. The optimum path of e in Figure 1 is different from being a straight line downward and drops faster than a straight line because a discount factor less than 1 makes the present reduction in e worth more than future reductions. If β is reduced from .98 to .97 the future will become even less important and present reduction in e becomes more steep as shown in Figure 2. In order to insure that the initial value of e be close to the historical value 8 and to demonstrate that the path can start by increasing first before declining we set e0 =8, β =.98, M = 800, θ = 5.5 and τ = .3. The result is given in Figure 3. The result shows that the optimal path will begin with a value close to 8. Furthermore, the optimum path of e begins by increasing even when we give a high value for θ in the second term θlog (M – E) of the utility function。We do not mind too much the damage of increasing e because E is still very far from an intolerable M. In the mean time we benefit from increasing output.

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Figure1

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Figure 2

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Figure 3

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3.6. Comparison with the Socolow-Lam Optimum Path

• Socolow and Lam (2006) present an optimum path for CO2 emission for the years 2005 onwards that begins at the current observed value of 8 and reaches a steady state value of 3, such that the maximum change in et in any period is minimized. This implies that the rate of reduction in et must be constant. Hence, the emission path from e0 = 8 to the steady-state value eT = 3 must be a straight line because any deviation from a straight line would cause the rate of reduction to be larger than the minimum achievable. See Socolow and Lam (2006, Figure 7).

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The approach to solving the CO2 reduction problem presented in this

paper differs from the Socolow-Lam solution by considering the cost and

benefit of both quantities et and Et in each period along the way towards

the steady state. We found that the optimal path of et may not be a

downward sloping straight line, as in the Socolow-Lam solution.

Furthermore, there is no guarantee that the steady-state value of E has

to be 600. In our solution the steady-state value of E depends on the

parameters of the objective function and can be somewhat below 600 or

perhaps above it, depending crucially on the value of M and other

parameters in the utility function. In other words, the approach

presented in this paper takes into account the economics of the problem

in all periods and is flexible by allowing the specification of different

utility functions to reflect a set of assumptions chosen by the policy

maker.

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3.7. Concluding CommentsWe begin this paper by referring to a UN Climate Change Conference to discuss an optimum path for future CO2 emission. This paper will have no relevance if no consensus can be reached. If a consensus can be reached this paper may be useful in providing a framework under which the consensus can be summarized by a utility function for the world community and an optimum path for CO2 emission can be found.

It may be suggested that a consensus is unlikely because different nations have different utility functions. For example the utility function of the developing nations will have a smaller value for θ, giving less importance to the harm of global warming than the increase in output. Such a situation can be studied by postulating two sets of nations, the developing and the developed, as players of a two person dynamic game, each having its utility function and the action eit of player i affects the utility of player j because

Et = .9975 Et-1 + .5 (e1t + e2t)

Each will solving its own dynamic optimization problem as suggested in this paper with its optimum control path for eit depending on ejt.

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It is also suggested that uncertainty regarding the size of the damage of climate change makes the discussion more complicated than suggested in this paper in which the disutility of a large E is known and given in the utility function. (See the references by Nordhaus, Stern, Weitzman, Zedillo and Dyson). That general discussion even raises the question as to whether we should devote resources to mitigate climate change at all. The present paper assumes that we have sufficient knowledge as given by the model and will devote resources to mitigate climate change by reducing e for the benefit of production and at the same time increasing utility from its second term θlog(M-Et). We have tried to answer the question as to how much resource in terms of the reduction in output should be devoted to the reduction of carbon emission.