Solutioil approach

Nearly all models of climate and the econolny involve optimization over a finite horizon as a solutioil technique. This has the advantage of being computationally efficient. However, direct optimization is very difficult t o use when there is a stocllastic element. All variables depend on the possible realizations of the shock ^{( E ) .} A more elegant, though not easy t o implement, approach involves rewriting the model using the Bellman equation and then solving that system nunierically (see Stokey and Lucas, 1989).

To describe the approach concisely, without getting bogged down in notation, suppose we have a vector of controls (Ct) ^- variables that are adjusted to maximize the objective ^- and a vector of state variables (St), which evolve over time, based on what controls are chosen. Assume there is

Table 1. Parameter values.

Parameter \ra.lue

P 0.7441"

b l 0.0686

b2 2.887

el

^0.01478

0 2 2

Y 0.25

6~ 0.6513

6~ 0.083

a 0.5819

4 0.0944

W 0.02

\'ar ( E ) 0.11

"Equivalent t o 3% annually.

Note: O n e unit of time = 10 years.

some stochastic shoclc every period, e, and a parameter that is imperfectly known,

p.

We can write this problem as

The Bellman principle of optimality states that the net present value of the objective [equation (9a)l must obey a dynamic consistency condition, known as the Bellman equa.tion. Let F(,S) be the value of the objective in equation (9), starting a t any sta,te S . The Bellman equation in this case is

The Bellman equation [equation (lo)] states that the maximum attainable net present value of the objective sta.rting a t 5' must be equal to today's one- period objective ( f ) plus tomorrow's maximum attainable net present value of the objective, assuming the optimal control is chosen today. Because

p

and E are random variables ex ante, one must take the expectation of F on the right-hand side with respect t o the distributions of ,O and E . In our case, the distribution of

P

is defined by two state variables (the prior mean and variance).

Note t h a t if one kllows F , tllell it is easy t o calculate the optimal action t o take a t any point in time: simply solve the right-hand side of equation (10) for C*. However, the real problem is finding an F ( ) that satisfies equation (10). Equation (10) is really a fullctional equation with, as an unknown, t h e function F.

So the problem is how t o find an F ( ) that satisfies equation (10). This is a problem of some concern in macroeconomics and a number of numerical techniques have been developed.3 T h e basic idea is t o define a family of functions of which F is a member. The family must be parameterized by some parameter vector, X. Thus the family is defined by the function

@(S;

x ) where t h e solution t o equation ( l o ) , F ( S ) , corresponds t o some particular value of the parameter,

x*.

We can rewrite equation (10) as

We may not be able t o have equation (10) hold exactly because we are using a restricted set of value functions, parameterized by X, instead of t h e universe of real-valued functions. T h a t is the reason for the error term, 7 , appended t o equation ( l l b ) . T h e task is t o find the

x*

for which 7 in equation (1:Lb) is as close t o zero as possible over relevant values of t h e state variable, 5'.

An obvious nor111 is least squares: i.e., find the

x

t h a t minimizes the sum of squared 7 over a finite set of va.lues of S , which span t h a t portion of the state space t h a t is of interest. This defines a, recursive algorithm for finding X'S

given some

xj,

evaluate equation ( l l a ) . Then find yj+l t h a t solves equation (1 1 b), minimizes some norm of 11 over S .

There are a number of alternative pa.ra.metric families t h a t can be used t o define t h e appropriake set of value fuilctions for c ~ n s i d e r a t i o n . ~ T h e primary requirements are t h a t any real-valued function can be approximated t o any degree of precision and t h a t the parameterization be computational

3 T h e January 1990 issue of the Journnl of Btrsiness and Economic Statistics was de- voted t o such techniques. In particular, see the review article by Taylor and Uhlig (1990) in t h a t issue.

Judd and Guu (1993) describe the familiar Taylor approximation (a series of polyno- mials) and the Pad6 approximation (quotient of polynomials). Judd (1991) argues for the use of a series of Chebysher polynomials as more computationally efficient. Hornik et al.

(1989) s l ~ o w t h a t neural networks, approximations involving transcendental functions, can approximate any Bore1 measurable mapping arbitrarily well.

e f f i ~ i e n t . ~ We choose to use neura.1 network approximations t o the value function. As we have implemented this, it is a close relative of the Fourier network. Our approximation is

where Xze is a vector and other x s are scaler components of the parameter vector X, and S is the state vector.

T h e next step is t o define a compact region of the state space where equation (11) will be required to hold. For instance, if the capital stock is one of the state variables, a lower limit would be zero and an upper limit would be any stock for which it is optimal for investment t o be less than depreciation, thus causing tlle stock t o shrink over time. Having defined the relevant compa.ct region of the state space, choose a finite set of points in that region, S;, i = 1,

. .

.

,

I. The finer the mesh covering the region of interest, the more accurate the approximation, although the computations will also be more intensive. We then recursively generate a sequence of

xj,

^{{xo, XI,}

. . . , x J } ,

starting from some initial guess xo. If one knows

xj,

then

xj+l

is the error-minimizing solution t o equation ( l l b ) where the rhs in equation ( l l b ) is evaluated at ,xi:

These

x,

converge to an

x*,

which defines the approximate solution,

@(S; x*), t o the Bellman equation (10). Convergence criteria are discussed in the next section.

This approach is essentially the same as the contraction mapping con- structive proofs of existence of solution to dynamic program (see, for exam- ple, Stokey and Lucas, 1989). These proofs define an operator ( T ) from the space of continuous functions t o the space of continuous functions. Choosing a n arbitrary continuous function (v) on the value function on the right-hand side of equation (lo), the left-hand sets of the equation define T(v). If T is

51n other words, for X. of sufficiently large dimension, @ spans the space of continuous functions. To be more precise, define @(S, x.) : R n x R m -r R where n is the dimension of the s t a t e space and rn is the dimension of the parameter vector X. For any C' function G: A -+ R where A is a compact subset of R n and any X > 0, 3m,,

x

^E Rn, 3 IIG(S) -

@(S, x)

Il.

^{< A.}

Table 2. Region of interest, in st,atci space and 198.5 values of state variables.

Nuinber of Grid 1985

State Units grid point,s values value

11' (10' 1987 US$) 5 10,40,100,190,310 51.26

M (10' tonnes) 4 600,1000,1400,1800 730

T (OC from 1950) 4 0.2,1.7,3.2,4.7 0.45

0 (OC from 1950) 3 0.1,0.75,2 0.11

r 3.07 w/m2 3 0.72,1.1,1.4 0.81a

v

^- 3 0.1,0.5,1 0.75

t (Decades from 1905) 6 1,3,7,12,20,50 3

"Equivalent t o 2.S°C temperature rise from doubling of greenhouse gas stock.

a contraction mapping, then multiple applications of T t o v eventually con- verge t o a fixed point, a solution t o equation (10). In our case, we are dealing with a restricted set of functions, @ ( S ,

x).

Iteration on

x

as described above is essentially the same as multiple applications of T.

This defines the solution t o model (9). While this is much more com- putationally intensive thail solving a deterministic finite horizon version of equation (9) using standard optilnization software, two clear advantages of this approach are: a ) stochasticity is represented; and b) once solved, the solution (optimal actions) for all values of the state vector is also known

- no further coinputatioils are necessary for other values of the state vec- tor. This is a very inlportant advantage of dynamic programming. It allows straightforward comparative statics analysis as well as great flexibility in policy analysis.

3.3. T h e solutioil

We now turn t o the specific solution of the model defined in equations (4) and (5). Recall that there are seven state variables: t (time), T (atmospheric temperature), 0 (ocean temperature),

Ii

(capital stock), M (greenhouse gas stock), r (estima.ted meail of IS), and V (estimated variance of

p).

T h e range of interesting values for these states is shown in Table 2, along with the discrete values of each state that are used to make up the points that span the state space for approsinlation purposes. The state variable t (time) was truncated a t the point where exogenous labor and technical change virtually stop changing.

Finding a good grid over the state space requires considerable trial and error. The idea is t o put grid points in the regions where the value function

has significant curva.ture, t o improve the fit. Also, the grid must have the stationary state in the interior. Tlle grid we used contained 12,960 points.

The neural network approximation [equation (12)] is assumed t o have 16 terms in the summation, resulting in 129 elements in X . 6 The maximization in equation ( l l a ) is solved using sequential quadratic programming.7 The expectations in equation ( l l a ) are evaluated using numerical integration based on 12-24 point Gaussian quadrature (Tauchen, 1990).

Equation (13) is solved using a quasi-Newton method with analytic first derivatives of 71 with respect to

x.

The nonlinear least squares is assumed t o converge when the objective is less than As indicated in the previous section, we iterate on

x

until convergence of

x

appears to be ~ b t a i n e d . ~ Our convergence criterion is that the difference in value function approximations [equation ( l l b ) ] , between one iteration and the next is less than or equal to

a t all grid points.

The problem was implemented using Matlab on a Sun Sparc 20 (75 mHz). Solution time was about 72

4. Results

The solution of the dynainic progra,nl is a va.lue function F(t,T,O,K,&l,r,V), and the corresponding policy functions giving optimal pollution control and investnlent as functions of the state variables: p*(t,T,O,Iir,M,r,V) and I*(t,T,O,lc,M,r,l/). Today's values of the state variables are sufficient to

6An important issue in the st.at,istics literature is the optimal choice for the number of parameters in the neural net ( . m ) . When observations are subject t o iid noise, one often gets the result t o set m such t h a t the number of parameters equals the square root of the number of observations. So given 13,000 observations, we might set m =

d m , -

^114.

This results in the number of terms [the Cs in equation (12)] being approximately 13. Given t h a t the actual value function is deterministic, we can use a few more terms t o increase computationally intensive parts of t,he process rather than using the Matlab interpreter exclusively.

determine today's optiinal a,ction. In order t o simulate the path of invest- ment, pollution control, or a.ny of the states over time, one must simulate the transition equations (5) and optimal control functions, using particular realizations of the random shock, E . It is important t o realize that, while in the model there is uncertainty over the climate response parameter,

/3,

the dynamic system requires a specific value of

/3

in order t o evolve. Thus, when we simulate the model, we must simulate it for a specific

/3,

even though learning is occurring about beliefs on

/3.

Starting in 1985 from initial values of the state variables given in Table 2 and assuming the shock is distributed rl(O,O.ll),10 we examine the evolutioil of the system for two different values of

/3,

a high value and a low value. These high and low values correspond respectively t o the IPCC high and low climate sensitivities of a 4.5OC and 1.5OC temperature change from greenhouse gas doubling (Lempert e t al., 1995).

Figure 3 shows how the estimate of the mean of the distribution on the true

/3

evolves over time for each of the two actual values of

/3.

(Actually, for clarity, climate sensitivity is shown, equal t o 3.07/3.) T h e curves are jagged because they involve the realizatioil of the random shock. A different set of realizations could very well have resulted in somewhat different paths.

Although the variance is not shown, it is virtually identical for the two values of

/3.

This is because the updating formula for the variance [equation (7b)l distribution changes. At a.ny point in time the agent can test the hypothesis t h a t

/3

= 0.86. We have run severa,l hundred Monte Carlo simulations ( t h e realization of the shocl; varies from one siillulatioil t o another) and for each determined the time period where we first reject t h a t null hypothesis. Table 3 shows t h e mean time t o rejection of the null under different assumptions about the true

/3

and different levels of confidence. Note t h a t rejection a t t h e 95% level takes less than 100 years if the true

/3

is high, but over 160

''The variance of the shock is computed from an analysis of the global annual tempera- ture data. We found the standard error t o be approximately 0.105, equivalent t o a variance of 0.011. Bassett (1992) and Nordhaus (1994) report standard errors for the interannual variation in t h e range of 0.1 t o 0.15. Although it is not strictly statistically correct t o d o so, we multiply our 0.111 variance by 10 t o yield a decadal variance of 0.11.

2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 2200 year

Figure 3. Evolutioil of the estimate of the mean of the distribution on the true ,kl over time for two actual values of

p.

Table 3. Decades to reach confideilce levels on

P.

True beta

Coilfide~lce level

95% 99%

H 9.3 12.5

L 16.3 22.9

Note: Figures shown are in decades. Represents mean over Monte Carlo simulations of time to reach indicated confidence level for rejecting the null hypothesis that = 0.86.

years if it is low. This is son~ewhat surprising since most analyses of learning hypothesize that uncertainty is resolved in 20-60 years. Also note that the learning time is not symmetric.

T h e implication of this slow learning can be seen in the level of pollution control. Figure 4 shows the pollution control rate (p) as a function of time for both the low value of ,kl and the high value (solid lines). Also shown are the values of p t h a t would be obtained if ,kl were perfectly known (broken lines).

For low ,kl, where greenhouse gases have a smaller effect on temperature, pollution control starts low a,nd rises as uncertainty is reduced. This is a

year

Figure 4. Pollution control rate as a function of time for both the low and the high values of

P.

somewhat unusual result. Growth in control implicit in the underlying model donlinates learning which would tend t o reduce control t o the no-uncertainty case. Another esplanation is that learning causes the deferment of control until some of that uncertainty is eliminated. This is not as obvious when the true value of

p

is high. Pollution control still starts low but gradually builds t o 8-9% as the true value of /3 is learned.

We can provide additional insight as t o why control rises t o its peak around 2100 and then falls (when the true

P

is low).ll Much is changing over the next century; what is most important in driving emission control?

Assuming the true value of

P

is low, the control rate increases by 148%

([p2Og5 - p1985]/p1985 = 1.48). We know the values of each state variable in 1985 and 2095. We can set all state variables a t their 1985 value except one, which we set a t its 2095 value and see how much pollution control changes.

Let t h a t value be p;, where S; is the state that has been changed t o its

We are unable to offer an unambiguous explanation for why control levels drop after 2100.

Table 4. Proportion of change in control due to each state variable.

Variable I< M T 0 r V t

Pro~ortion 26% 70% 6% 1 % -40% -21% 57%

Note: Shown is

p , " d 9 ~ ~ ~ 8 ~ 8 s ,

assuming the true /3 is low. pi is the control level when all states take their 1985 vatue except state i , which takes its 2095 value.

Figure 5. Evolution of atmospheric and deep ocean temperatures over the very long run for the high value of

P.

2095 value. Table 4 shows the ratio of 11; - p1985 t o p2095 - p1985. Roughly speaking, these figures should sum t o loo%, though because of nonlinearities and other factors, they will not do so precisely.

Note in the table that

Ii,

M, and t have the biggest positive impact on p, whereas learning has the opposite effect. The change in T and V, which is due t o learning, tends to work in the opposite direction. On net, learning is dominated by the other variables and the control rates rise.

Figure 5 shows the evolution of atmospheric and deep ocean temperature over the very long run for the high value of

P.

Note the very long lags built into the temperature response of the deep ocean. It is clear that while the deep ocean temperature is not significant now, it will be eventually.

01 ' ^{0 9 I 4 I I}

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 year

Figure 6. Decadal emissioils for the high value of

/3

and a scaled plot of the product of three variables: technology, labor force, and emissions per unit GDP.

It should be pointed out tl1a.t 1llucl1 of the growth in emissions and greenhouse gas stock levels is due to growth in the labor force and growth in output-enhancing technology. Figure 6 shows a plot of decadal emissions (for the high value of

/3

^- a sinlilar result holds for the low value) and a scaled plot of the product of three variables: technology (A*), labor force (Lt), and emissions per unit of GDP ( a t ) . Higher A's and L's yield higher GDP;

a is the greenhouse gas emissions-GDP ratio. Thus the product of these three variables is proportional to ullcontrolled greenhouse gas emissions. The relationship between this product and emissions is striking. Since there is very little control ( < l o % ) , emissions and this triple product track each other well. Why is this significant? It is significant because the A, L, and a variables are all exogenous. An interpretation of Figure 6 is that exogenous variables determine how much is emitted and thus how serious the warming problem is. This is explored further in Kelly and Kolstad (1996).

This leads to the obvious question of how the optimal policy is affected by the various state variables. The optimal policy function is nonlinear,

T a b l e 5. Sensitivity of enlission control t o 1% increase in value of states.

Ii' M T 0 ^T V t

0 . 4 3.8 0.1 0.02 1.7 0.2 1.2

Note: States evaluated at 1985 values; sho~vn is percent change in p.

which makes generalization difficult. However, we can ask specific questions.

Suppose we start with the value of the state variable in 1985 (see Table 2).

We can then investigate how a 1% increase in each of the states would affect the optimal C 0 2 control rate, assuming the true value of

P

is a t its high value. Table 5 shows the change in p from its base value of just under 3%

control of greenhouse gases.

We see t h a t the most significailt states are M , r , and t . T h e optimal policy puts little weight on T, because T is subject t o stochastic shocks and is not a very good predictor of future temperature. However, M and r are excellent predictors of future T values, and hf has much less stochastic fluc- tuation than T. Also i~llportant a.re the levels of the exogenous variables, represented by t (as mentioned earlier). So we see that learning is an impor- t a n t component of the optinla1 policy, although most analyses focus on A4 or T.

5. Conclusions

This paper is one of the first papers t o deal esplicitly with learning about the climate within a.n integrated-assessmlent framework. We have explored 1ea.rniag about the exa.ct relationship between elevated greenhouse gas levels and temperature rise. We lmve shown t h a t it can take a very long time t o resolve t h a t uncertainty, time during which significant suboptimal control can take place (relative t o perfect information).

It should be noted t h a t in this model, learning is from the point of view of the policy maker. We lmve assumed the agents within the model are perfectly informed. If agents also were learning then adaptation would be slower and t h e effect of learning more pronounced.

There are several drawbacks t o our approach, not the least of which is t h e computational complexity. A related problem is t h a t other parameters are also uncertain but the problem becomes too complex if learning about more parameters is included in the model. Finally, Bayesian leaning is not the only kind of learning that takes place. Clearly research and development also generates information; tha,t process is not represented here.

References

Balvers, R., and Cosiinano, T . , 1990, Act'ively Lea,rning About Demand and the Dynamics of Price Adjustment, The Economic Journal 100:882-898.

Bassett, G.W., J r . , 1992, Breaking Recent Global Temperature Records, Climatic Change 21:303-315.

Cunha-e-sa, M.A., 1994, Essays i11 Enviroilmeiltal Economics, Ph.D. dissertation, Department of Economics, University of Illinois, Urbana, IL, USA.

Cyert, R., and Degroot, M., 1974, Rational Espectations and Bayesian Equilibrium, Journal of Political Economy 82:521-535.

Edmonds, J . , and Reilly, J . , 1983, Globa.1 Energy and CO? to the Year 2050, Energy Journal 4(3):21-47.

Hammitt, J.K., Lempert, R.J., aiid Schlesinger, M.E., 1992, A Sequential-Decision Strategy for Abating Clinlate Change, Nature 357:315-318.

Hornik, I<., Stinchcombe, M., and White, H., 1989, Multilayer Feedforward Net- works are Universal Approximators, Neural Networks 2:359-366.

Judd, I<.L., 1991, Minimuin Weighted Residual Methods for Solving Dynamic Eco- nomic Models, unpublished manuscript, Hoover Institution, Stanford Univer- sity, Stanford, CA, USA.

Judd, I<.L., and Guu, Sy-Miiig, 1993, Perturbation Solution Methods for Economic Growth Models, in H.R. \'aria11 (ed.), Economic and Finailcia1 Modeling with Mathemaiica, Springer-Verlag/TELOS, Sa,ilta Clara, CA, USA.

Kelly, D.L., and I<olstad, C.D., 1996, A Note on the Importance of Population- Technology Growth i11 Integrated Assessment Models, mimeo, Economics De-

Kelly, D.L., and I<olstad, C.D., 1996, A Note on the Importance of Population- Technology Growth i11 Integrated Assessment Models, mimeo, Economics De-

Im Dokument Climate Change: Integrating Science, Economics, and Policy (Seite 152-167)