General Equilibrium Approach to Evaluate Real Option Value of Reserved Environments

Transcription

1 General Equilibrium Approach to Evaluate Real Option Value of Reserved Environments Iain Fraser Katsuyuki Shibayama University of Kent at Canterbury Fall 2010

2 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 2 1 Introduction 1.1 Key Features Real Option for Environments It becomes increasingly important. Environmental research is the most suitable area to apply RO analysis. Irreversibility: Once you convert old forest to farmland, then you cannot recover it anymore. Uncertainty: The importance of environments was not really understood a couple of decades ago, which implies that people could not predicted the future value of environments very well. General Equilibrium It reveals the source of the change in environments' value. It helps to identify some key parameters.

3 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Literature The model classication here is based on how to convert reserved land to farmland. All or Nothing Model: - Reserve all land or develop all land. Bang-Bang Model: - Conversion rate v must be positive but the maximum speed is limited: 0 v v. - In this case, the optimal choice is v = 0 or v. Barrier Control Model: - Conversion rate v must be positive and its speed is unlimited: 0 v. Research Area Matrix (cited papers are not exhaustive) DP(1994) Partial Equim General Equim All or Nothing Ch.5 e.g., Conrad (1996) Bang-Bang Ch.10 e.g., Leroux et al. (2009) Will be here(?) Barrier Control Ch.11 e.g., Bulte (2002) We are here! NB: DP is the textbook by Dixit and Pindyck (1994) "investment under uncertainty".

4 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 4 2 Previewing the Model 2.1 Previewing the Model: Constraints total land: 1 = A + R production: Y = W A resource const: Y = C + W v dw tech growth: W = 0 dt + 0 dw da conversion rate: dt = dr dt = v 0 Total land supply is normalized to be 1 and it can be used as farmland A or reserved environment R. The only production factor is A. W is technology level, which follows a GBM. Output Y is consumed or used as the conversion cost of land from W v. Marginal cost of land conversion W is proportional to W. Irreversibility comes from non-negative conversion rate v.

5 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Utility and Value Function Z 1 value function : F (R; W ) = max E 0 e t u (R; W; v) dt Value function F is the expected PV of ow utility u with discount rate. The state variables are reserved environment R and technology W in production. The only choice variable is the speed of land conversion v. ow utility : u (R; W; v) = 0 1= C1 1 1= + (R R min) 1 1= 1 1= 1st term implies (i) constant elasticity of intertemporal substitution (1=) and (ii) constant relative risk aversion (). 2nd term implies (iii) constant elasticity of substitution () between consumption C and reserved land R. is the relative importance of the service ow from R, and it also absorbs the problem of different measurement units. Assume that R min = 0; R min only shifts the lower bound of R.

6 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments The Model in Short Form Change of variable: Z := W 1 1= with := (1 1=) 0 02 =2 and := (1 1=) 0 In short form, Z 1 F (R; Z) = max E 0 e t u (R; Z; v) dt 1= (1 R v)1 u (R; Z; v) = Z 1 1= dz = dt + dw Z da = dr dt dt = v 0 0 R1 1= + 1 1=

7 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Intuition behind the Model We claim: - An old forest now has been the same old forest since several 100yrs ago. This is totally different from, say, a PC, which is much faster than 10yrs ago. - The value of an old forest has increased, because we have changed. - In most existing works, the values of R and A follow independent exogenous processes (if not constant), which we disagree with. These value should be determined endogenously. In our mode: - The value of R increases, because (i) it is non-reproductive and (ii) the productivity of alternative use A increases. - As W increases, people become richer. Hence, due to income effect, the demand for the service ow from R increases. - The increase in W also implies the cost reduction in producing Y, meaning that, given demand level, higher W implies lower demand for A. - This idea is related to Baumol's curse.

8 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Myopic Version Value function F inherits several features of ow utility u. Myopic version only can explain much of the intuition above. Assume no dynamics or no irreversibility constraint with = 0. Optimality Condition: max u (R; W; v) = W 1 1=(1 R)1 1= 1 1= R1 1= + 1 1= 1 optimal reserve land: R = W R shadow price of R: P = = W C - As W increases, the shadow price of reserved environment R increases - As W increases, optimal R increases if < 1, and vise versa; i.e., if R is irreplaceable, the demand for R increases as people become richer. - Obviously, plays an important role in the following.

9 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 9 Non-Reproductive Nature of R - If the supply of R is limited, the shadow price (slope of the indifference curve at the optimum) increases as people become richer. - The ow utility is quite at for the middle value of R.

10 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 10 Optimal R depends on - These graphs show the main reason why is important. - Even this myopic version captures the fact that the people in developed countries want to preserve environment more than those in developing countries, despite the fact that they destroyed the environment in the past.

11 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 11 3 Solution Value Function: In recursive form, In differential form, Ito's Lemma for df : Z 1 F (R; Z) = max E 0 e t (R; Z; v) dt F (R; Z) = max u (R; Z; v) dt + e dt E 0 [F (R F (R; Z) = max df (R; Z) = F R dr + 0 vdt; Z + dz)] u (R; Z; v) + E 0 [df (R; Z)] dt F Z Z + F ZZ 2 2 Z2 where the last term in RHS disappears in expectation. dt + F Z Zdw

12 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 12 Using dr=dt = v, E 0 [df (R; Z)] dt = F Z Z + F ZZ 2 2 Z2 F R v 3.1 Value Function with Optimal Conversion Rate v Hence, the following PDE governs the value function. F (R; Z) = Z (1 R v ) 1 1= 1 1= - Use Z, rather than W, for derivation. 1= R = + F 2 ZZ + F ZZ 2 Z2 F R v

13 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Non-Conversion Region (v = 0) In the region with v = 0, the PDE is not really a PDE but is merely an ODE. 1= F 0 (1 R)1 (R; Z) = Z 1 1= The analytical solution is available in this region. R1 1= + 1 1= + F ZZ 0 + FZZ Z2 F 0 = Z (1 R) 1 1= + R 1 1= 1 1= 1 1= + B (R) Z (1) = a 1 a 2 > 1 where a 1 = 1 r 2 2 and a 2 = a B (R) is an integration constant wrt Z, which is determined by the free boundary conditions. - The rst 2 terms show the value if you x R at the current level forever; the same formula as DDM. - The last term shows the value of the possibility that you can change R in the future, we call this option value. - Option value takes place only in non-conversion region.

14 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Conversion Region (v > 0) For the conversion region, since R jumps to R immediately, the value is the value of optimal land allocation at R minus conversion cost. F 1 (R; Z) = F 0 (R (Z) ; Z) = F 0 (R (Z) ; Z) Z = F 0 (R (Z) ; Z) + Z conversion cost Z R (1 dr 0 ) 1 R (Z) ( (1 R) 1 1= 1 1= ) (1 R (Z)) 1 1= 1 1= (2)

15 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Free Boundary Conditions The conditions that must be satised on the boarder. 1. Level matching: Used already for Conversion Region. 2. Slope matching: The marginal benet of conversion is equal to the marginal cost of conversion. 3. Smooth pasting: FR 0 (R ; Z) = u v (R ; Z; v) Z (1 R ) 1= + R 1= + B 0 (R ) Z = Z (1 R ) 1= (3) FRZ 0 (R ; Z) = u vz (R ; Z; v) 1 (1 R ) 1= + B 0 (R ) Z 1 = k (1 R ) 1= (4)

16 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 16 Computation: step1: From (3) and (4), we can nd: optimal R as func of Z (W ) : R 1 (Z(W )) = 2 Z + 1 = 1 2 W Z R integral const in F 0 x 1 : B (R) = 3 dx 0 (1 x) where 2 = 1(1=( ) + ) and 3 = step2: From (1), we can nd F 0 for non-conversion region. step3: From (2), we can nd F 1 for conversion region.

17 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 17 4 Numerical Examples Parameters: 0:04 discount rate 0 0:02 trend growth rate of Z := (1 1=) 0 02 =2 0 0:20 variance of Z growth := (1 1=) 0 5:0=0:7 elast. of subs. btw C and R: 0:1 relative importance of R in 2 = 1 (1=( )+) 0:0 W is MC of land conversion - We have not yet seriously discussed realistic parameter values... But, some brief discussion is given later. 4.1 Optimal vs. Myopic Barrier (Boarder):

18 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 18 myopic: R m 1 (W ) = W where 2 = 1(1=( ) + ) > optimal: R 1 (W ) = 2 W for small enough φ 2 /φ ρ/σ α/σ 2 3 4

19 Technology Level, W Technology Level, W General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 19 Myopic choice leads to too little reserved environment R Optimal vs. Myopic R for η=5 stochastic case (eqn( )) myopic case (eqn( )) Optimal vs. Myopic R for η=0.7 stochastic case (eqn( )) myopic case (eqn( )) Conversion Region Conversion Region Non Conversion Region 6 4 Non Conversion Region Reserved Environment, R Reserved Environment, R - The mistake when = 5:0 is larger than that when = 0:7.

20 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Option Value Option value is zero in conversion region; conversion is exercising an option. Option value is high near the boarder, since the possibility of future conversion is more likely. Option value is small. Have a look at the unit of vertical axis.

21 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Value Function in Both Regions Given W (i.e., along R axis), the value function is at. Total Value in Combined Regions: η=5 Total Value in Combined Regions: η= Technology Level, W Reserved Environment, R Technology Level, W Reserved Environment, R - This at shape is largely inherited from the ow utility (or myopic model).

22 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments Implication A small option value can causes the large gap btw R m and R. (Exactly speaking, not only option value causes the gap, though.) Why? => Answer: Because the value function is at. However, A at value function implies that a large mistake is not very painful. 5 Consumption Equivalence for Myopic Loss: η=5 Horizontal Gap btw R m and R * (as % of Optimal R) Consumption Equivalent (as % of Actual C) Consumption Equivalence for Myopic Loss: η=0.7 Horizontal Gap btw R m and R * (as % of Optimal R) Consumption Equivalent (as % of Actual C) Technology Level, W Technology Level, W Provide a handwriting diagram.

23 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 23 Dynamic Property: If > 1, there is a common growth path along the barrier. If < 1, there is a strong initial value dependence.... This is because of the direction of tech growth. - If > 1, typically an economy crosses the boarder from non-conversion region to conversion region. Every time W goes up, the economy shifts to the left, and hence it moves along the barrier curve. - If < 1, typically an economy crosses the barrier curve from conversion region to non-conversion region. Since W is increasing on average, once it enters into nonconversion region, it tends to stay in the same region forever. Hence, although < 1 seems to be economically more plausible, it generates a bit strange model behavior. Provide 2 handwriting diagrams here.

24 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 24 5 Comments on Empirical Work 5.1 Parameters Common to This Literature (discount rate): Typically, easy to estimate. Should be close to the risk-free rate. 0 (trend growth rate of W ) and 0 (volatility of W ): Typically, they are very important. But, in most cases, they are hard to estimate. For example, in Leroux et al. (2009), (one of) exogenous shock process (GBM) is the "economic value of biodiversity". However, in our framework, 0 and 0 are the trend growth rate and volatility of output (or consumption), which is much easier to estimate. 5.2 Parameters Specic to Our Model (marginal cost of conversion): Should be small and should have only small effects (maybe).

25 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 25 (elasticity of substitution btw R and C): Since this is so important, we recommend not to choose one value for. Rather, it is better to show the results for several values of. (relative importance of R in u): Note that also absorbs the difference in measurement unit. Unfortunately, for, it seems to need some eldwork/questionnaire, or the like. Such a kind of research is often called environments pricing. However, our model provides some hints. 1 myopic: ln P = ln 1 optimal: ln P = ln 2 (ln R (ln R ln C) ln C) (on barrier) Assuming is given, the only variable that is difcult to observe is the shadow price of reserved environment P, since C is simply consumption and R is the % of reserved land in one country/area. Hence, having these numbers we can recover. Note importantly that (i) if P is the price of the service ow of R, myopic model should

26 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 26 be used, while (ii) if P is the price of the reserved land R as stock, optimal model should be used. Hence, the meaning of P in, say, a questionnaire should be clear in this difference. Also, if an economy is inside of the non-conversion region, the shadow price should be higher than that on the barrier (hence, it is not really applicable if < 1). Finally, you may be tempted to use the following results. 1 myopic: ln W = ln 1 optimal: ln W = ln 2 (ln R (ln R ln C) ln C) (on barrier) However, as mentioned earlier, since adjusts the difference in measurement units. Since usually tech growth is measured in anonymous unit, it seems to be hard to use these expressions.

27 General Equilibrium Approachto Evaluate Real Option Value ofreserved Environments 27 6 Conclusion We conducted the real option exercise for reserved environments in a very simple general equilibrium framework. Our general equilibrium framework has two major merits: (i) It reveals the mechanism of the change in environment value. (ii) It helps to identify some key parameters. In our model, we nd (elasticity of substitution btw C and R) plays an important role. Literally, means how environments are irreplaceable. If environments are really irreplaceable (inelastic), then the demand for R increases as people becomes richer. The value function is at under our functional and parametric assumptions. Hence, (a) If the option value is ignored, the mistake can be huge. However, (b) It also implies that the mistake is not really painful.