Essays on resource environmental economics

ESSAYS ON RESOURCE AND ENVIRONMENTAL ECONOMICS NEIL SEBASTIAN D’SOUZA NATIONAL UNIVERSITY OF SINGAPORE 2010 ESSAYS ON RESOURCE AND ENVIRONMENTAL ECONOMICS NEIL SEBASTIAN D’SOUZA MSc. (Distinction), BEng. (First Class) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2010 Acknowledgement I would like to thank Professor Aditya Goenka for his invaluable insight, patient guidance, and unstinting support. I have learnt so much from him. Most importantly, his open-mindedness and intellectual curiosity has instilled in me a deep appreciation for economics in its myriad forms. I think I am a better researcher for this. I would also like to thank my committee members, Professor Parimal Bag, Professor Tomoo Kikuchi, and Professor Julian Wright, who spent their valuable time to provide me with insightful feedback and support. I would like to thank all my teachers here at the National University of Singapore. I am grateful for all that they have taught me to get me to this point. I would like to specifically mention my batchmates, Himani, Nona, and Rica. I consider myself fortunate to have been in their company for the past four years. And finally, I would like to thank my family for their unwavering support, especially my wife Tanmayee. She patiently bore four years of my lifestyle as a student without complaint. Her calming influence and positive attitude were my constants for the past four years. I dedicate my thesis to her. i Contents Acknowledgement i Contents ii Summary iv List of Tables vi List of Figures vii The Games ‘Oil’igopolists Play 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 A description . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 An illustrative example . . . . . . . . . . . . . . . . . . . . . . 22 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4 ‘Oil’igopoly and heterogenous crude 35 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Does heterogeneity matter? . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 The model with heterogenous crude . . . . . . . . . . . . . . . . . . . 41 2.4 The theory of ‘oil’igopoly, heterogeneity, and policy relevance . . . . . 56 ii 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational Overemission of Carbon Dioxide 63 65 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 The Model Environment . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 The symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 The asymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.5 Policy implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Bibliography 98 Appendices 104 .1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 .2 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 .3 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 .4 Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 iii Summary The purpose of this thesis is to gain a better understanding of how strategic interactions amongst agents shape outcomes in some areas of resource and environmental economics. Chapter of the thesis takes another look at the theory of ‘oil’igopoly that predicts that countries holding larger proven reserves of crude oil tend to produce quantities that are larger in absolute size but smaller as a proportion of their reserves. However, the literature on the subject thus far has only focussed on the interactions amongst the oil producers in determining the equilibria attained. The influence of interactions between the oil producers and the non-oil producers on the price-output path(s) has not been formalized in the context of the theory of ‘oil’igopoly. We reconsider the results of the theory of ‘oil’igopoly wherein the interactions between the oil and non-oil producers are explicitly modeled. We find that such a modeling approach not only allows us to capture the empirical regularity predicted by the theory of ‘oil’igopoly but is also able to generate a positive correlation between the price and quantity of oil traded, which is found in the data. Thus this modeling approach, which endogenizes the price of oil, forms a bridge between the theory of ‘oil’igopoly and the macroeconomy-oil price relationship literature pioneered by Hamilton (1983). We thus provide an alternative modeling approach that provides a deeper understanding of the drivers of the crude oil reservesproduction relationship. The theory of ‘oil’igopoly, however, abstracts away from the fact that crude oil is iv heterogeneous. In Chapter 2, we take into consideration the heterogeneity of crude oil and show how it affects the price-output relationship of crude oil production. We find that the preferences of oil-consuming nations influence the production decisions of the oil producers. In addition to matching the predictions of the theory of ‘oil’igopoly, we can show that oil-producing countries that have endowments of low grade crude tend to restrict their production compared with oil-producers that possess high grade crude. Considering the implications of the theory of ‘oil’igopoly in light of the heterogeneity of crude oil provides a deeper understanding of how energy policy in oil-consuming countries can affect the output decision of oil producers. In Chapter we consider the issue of catastrophic climate change. Fossil fuels when burnt for energy, release carbon dioxide into the atmosphere. Carbon dioxide, though not a pollutant in the classical sense, is the most important of the anthropogenic greenhouse gases. Despite strong and improving evidence of the role of excessive carbon dioxide emissions in hastening the arrival of climate-induced catastrophe, countries seem reluctant to reduce their dependence on fossil fuels. In this paper, we use the framework of a non-cooperative differential game to analyze why countries produce more carbon dioxide emissions than are deemed permissible given the current state of scientific understanding on the subject. We find that overemission of carbon dioxide, which results from the overconsumption of fossil fuels, is rational. v List of Tables 1.1 1.2 1.3 2.1 2.2 Panel-data estimation (log(Reserves/Production) to log(Reserves)) . . Panel-data estimation (log(Production) to log(Reserves)) . . . . . . . Panel-data estimation (log(Reserves/Production) to log(Reserves) and log(GWP)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Price-API Relationship . . . . . . . . . . . . . . . . . . . . . . . The Effect of Heterogenous Crude on the Reserves-Production Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 7 39 41 List of Figures 1.1 1.2 1.3 1.4 The The The The reserves–production relationship . . . . . . . . . . . . . . R/P–production relationship . . . . . . . . . . . . . . . . reaction functions of the oil and foodgrain producers . . . reaction functions of the bigger and smaller oil producers 3.1 3.2 Worldwide CO2 emissions . . . . . . . . . . . . . . . . . . . . . . . . 67 The growth rate of CO2 emissions (%) and the growth rate of GWP (%) 69 vii . . . . . . . . . . . . . . . . 31 33 Chapter The Games ‘Oil’igopolists Play 1.1 Introduction The reserves-production ratio is an important metric to understand the market for non-renewable energy sources such as crude oil and natural gas. If taken at face value, the reserves-production ratio gives us the number of years of supply that are remaining given the reserves that are in place today. As noted by Flower (1978) and Mitchell (1996), the reserves-production ratio needs to be paid attention to by policy analysts as they provide signals regarding the ability of an oil producer to vary production. Economists have been mindful of the reserves-production ratio. The theory of ‘oil’igopoly developed by Loury (1986) and extended by Polasky (1992) gives a simple yet elegant prediction, that countries holding larger proven reserves of oil tend to produce quantities that are larger in absolute size but smaller as a proportion of their reserves. In other words, the reserves-production ratio is increasing in the reserves that a country or company possess. Polasky (1992) found support for this reserves-production relationship when using data for oil-producing countries and companies. Pickering (2008) also found support for this reserves-production relationship is said to be submodular in (z, θ) if it is the case that for all x = (z, θ) and x = (z , θ ) in Z × Θ we have f (x) + f (x ) ≥ f (x ∨ x ) + f (x ∧ x ) A function f : Z × Θ → is said to satisfy decreasing differences in (z, θ) if for all pairs (z, θ) and (z , θ ) in Z × Θ, it is the case that z ≤ z and θ ≤ θ implies f (z , θ) − f (z, θ) ≥ f (z , θ ) − f (z, θ ) n Let Z and Θ be subsets of and l respectively. A function f : Z × Θ → is said to be supermodular in (z, θ) if it is the case that for all x = (z, θ) and x = (z , θ ) in S × Θ we have f (x) + f (x ) ≤ f (x ∨ x ) + f (x ∧ x ) A function f : Z × Θ → is said to satisfy increasing differences in (z, θ) if for all pairs (z, θ) and (z , θ ) in Z × Θ, it is the case that z ≤ z and θ ≤ θ implies f (z , θ) − f (z, θ) ≥ f (z , θ ) − f (z, θ ) Theorem 23 A function f : Z × Θ → decreasing differences on Z 106 is submodular if and only if f has Proof. Let Z ⊂ m . For z ∈ Z, we will denote by (z−ij , zi , zj ) the vector z, but with zi and zj replaced by zi and zj .Suppose, without loss in generality, that zi ≥ zi and zj ≥ zi . Suppose that z = (z1 , ., zm ) and z = (z1 , ., zi , zj , ., zm ). Let w = (z1 , ., zi , zj , ., zm ) and w = (z1 , ., zi , zj , ., zm ).Thus we have, w∧w =z w∨w =z Since f is submodular on z we have f (z; θ) + f (z ; θ) ≥ f (z ∨ z ; θ) + f (z ∧ z ; θ) Rearranging and using the notation for w and w we get, f (z1 , ., zi , zj , ., zm ; θ) + f (z1 , ., zi , zj , ., zm ; θ) ≥ f (z1 , ., zi , zj , ., zm ; θ) + f (z1 , ., zi , zj , ., zm ; θ) To see that decreasing differences on Z implies submodularity on Z, consider any z and z in Z. We are required to show that f (z; θ) + f (z ; θ) ≥ f (z ∨ z ; θ) + f (z ∧ z ; θ) 107 If z ≥ z or z ≤ z , this inequality trivially holds. So suppose that z and z are not comparable under ≤. For notational convenience, arrange the coordinates of z and z so that z ∨ z = (z1 , ., zk , zk+1 , ., zm ) z ∧ z = (z1 , ., zk , zk+1 , ., zm ) As z and z are not comparable under ≤, we must have < k < m. Now for ≤ i ≤ j ≤ m, define z i,j = (z1 , ., zi , zi+1 , ., zj , zj+1 , ., zm ) Then we have, z 0,k = z ∧ z z k,m = z ∨ z z 0,m = z z k,k = z . Since f has decreasing differences on Z, it must be that for all ≤ i < k ≤ j < m 108 f (z i+1,j+1 ; θ) − f (z i,j+1 ; θ) ≤ f (z i+1,j ; θ) + f (z i,j ; θ) Thus we have for k ≤ j < m i+1,j+1 f (z k,j+1 ; θ) − f (z 0,j+1 ; θ) = Σk−1 ; θ) − f (z i,j+1 ; θ)] i=0 [f (z i+1,j ≤ Σk−1 ; θ) + f (z i,j ; θ) i=0 f (z = f (z k,j ; θ) − f (z 0,j ; θ). As this inequality holds for all j satisfying k ≤ j < m. it follows that the lefthand side is takes its highest value when j = m − 1, while the right-hand side takes its lowest value when j = k. Therefore, f (z k,m ; θ) − f (z 0,m ; θ) ≤ f (z k,k ; θ) + f (z 0,k ; θ) Which is exactly, f (z; θ) + f (z ; θ) ≥ f (z ∨ z ; θ) + f (z ∧ z ; θ) Since z and z were chosen arbitrarily, we have shown that f is submodular on Z. 109 Theorem 24 Let Z be an open sublattice of f : Z ×Θ → m for a given θ. A C function is submodular on Z for a given θ if and only if for all z ∈ Z, we have ∂ f (z; θ) ≤0 zi zj (for all i, j = 1, ., m, i = j). Proof. Let f be a C function on Z ⊂ m for a given θ ∈ Θ. By theorem 2, f is submodular on Z for a given θ ∈ Θ if and only if for all z ∈ Z, for all distinct i and j, and for all > and δ > 0, we have, f (z−ij , zi + , zj + δ; θ) − f (z−ij , zi + , zj ; θ) ≤ f (z−ij , zi , zj + δ; θ) + f (z−ij , zi , zj ; θ) Dividing both sides the positive quantity δ and letting δ → 0, we see that f is submodular on Z if and only if for all z ∈ Z, for all distinct i and j, and for all > 0, ∂f (z−ij , zi + , zj ; θ) ∂f (z−ij , zi , zj ; θ) ≤ ∂zj ∂zj Subtracting the right-hand side from the left-hand side, dividing by the positive quantity , and letting → 0, f is seen to be submodular on Z for a given value of θ if and only if for all z ∈ Z, and for all distinct i and j, we have ∂ f (z; θ) ≤0 zi zj (for all i, j = 1, ., m, i = j). 110 Theorem 25 Let Z = S × T . Let S be a compact sublattice on sublattice of + n−1 and let Θ be a sublattice on k + +, let T be a and f : Z × Θ → + be a continuous function on Z = S × T for each fixed t ∈ T and θ ∈ Θ. Suppose that f is submodular on Z for each fixed θ ∈ Θ. Let the correspondence ρ from T to S be defined as ρ∗ = argmaxf (s, t)|x ∈ S 1. For each t ∈ T , ρ(t; θ) is a nonempty, compact poset of + and admits a greatest element. 2. ρ∗ (t ; θ) ≥ ρ∗ (t; θ) for all t ≤ t Proof. Since f is continuous on S for all t ∈ T and S is compact, ρ∗ (t; θ) is nonempty for all t ∈ T . Fix t and consider a sequence sn in ρ∗ (t; θ) converging to an s ∈ S. Then, for any s ∈ S, we have f ((sn , t), θ) ≥ f ((s, t), θ). By continuity of f ((•, t), θ) we have f ((s, t), θ) ≥ f ((s , t), θ). This implies that s ∈ ρ∗ (t; θ). Thus, ρ∗ (t; θ) is closed and compact as S is compact. Furthermore, the meet and the join of any two elements in ρ∗ (t; θ) is a sublattice that admits a greatest element. This completes the proof of part 1. Now consider t and t such that t ≥ t . Let s ∈ ρ∗ (t; θ) and s ∈ ρ∗ (t ; θ). By optimality we have f ((s, t); θ) − f ((s , t); θ) ≥ f ((s, t ); θ) + f ((s , t ); θ) 111 Since f is submodular in Z = S × T , we know from Theorem that it would display decreasing differences in X. Suppose that s ≥ s , then f ((s, t); θ) − f ((s , t); θ) ≤ f ((s, t ); θ) + f ((s , t ); θ) From these two equations we have a contradiction.This means that s ≤ s . Therefore, we have ρ∗ (t ; θ) ≥ ρ∗ (t; θ) for all t ≤ t. The following is the statement of Kukuskin’s theorem (1994) Theorem 26 Suppose N is a finite set and for each i ∈ N there is a compact set Xi of reals and an upper hemi-continuous correspondence φi : Si → Xi , where Si = j∈N/(i) Xj allowing a decreasing single-valued selection. Then there exists a point x0 ∈ X = xi∈N Xi such that x0i ∈ Φi ( .2 j=i x0j ) for all i ∈ N . Appendix B Oil Producers We can write down the maximization problem faced by the oil producer as follows: i i maxU1 (f1C ) + δU2 (f2C ) subject to oi1T + oi2T ≤ O1i 112 and oi1T ≥ 0, oi2T ≥ The Lagrangian function is i i Z = U1 (f1C ) + δU2 (f2C ) + λ(O1i − oi1T − oi2T ) and the Kuhn-Tucker conditions are Zoi1T = U1 − λ ≤ 0, oi1T ≥ 0, oi1T Zoi1T = Zoi2T = δU2 − λ ≤ 0, oi2T ≥ 0, oi2T Zoi2T = Zλ = O1i − oi1T − oi2T ≥ 0, λ ≥ 0, λZλ = If we assume that oi1T and oi2T are both 0, then the utility will be zero. Hence, we must consider the case where at least one of them is non-zero. Suppose that oi1T = and oi2T > 0. This implies that λ = δU2 > (Since we assume that utility function to be increasing). This in turn implies that the constraint binds, that is, O1i = oi1T − oi2T . The same argument can be made if we assume that oi1T > and oi2T = 0. 113 Suppose that λ = 0. Then, we have O1i > oi1T + oi2T , U1 ≤ 0, and δU2 ≤ 0. In this case, if oi1T = and oi2T > 0, then it must be that U1 < 0, and δU2 = 0. Since we consider utility to be an increasing function, this cannot be true. The same argument holds if we consider oi1T > and oi2T = Thus, it must be that in equilibrium O1i = oi1T + oi2T . Foodgrain Producers We can write down the maximization problem faced by the foodgrain producer as follows: maxU1 (oi1C ) + δU2 (oi2C ) subject to i ≤ ftC ≤ fti The Lagrangian function is i i )+ ) + λ2 (f2i − f2C Z = U1 (oi1C ) + δU2 (oi2C ) + λ1 (f1i − f1C and the Kuhn-Tucker conditions are i i i i Zf1C = U1 − λ1 ≤ 0, f1C ≥ 0, f1C Zf1C =0 114 i ≥ 0, λ1 ≥ 0, λ1 Zλ1 = Zλ1 = f1i − f1C i i i i Zf2C = δU2 − λ2 ≤ 0, f2C ≥ 0, f2C Zf2C =0 i Zλ2 = f2i − f2C ≥ 0, λ2 ≥ 0, λ2 Zλ2 = i i If we assume that f1C or f2C are 0, then the utility will be zero in that period. i i Hence, it must be that ftC > 0. Consider the case where f1C > 0. This implies that λ1 = U1 > (as we assume that the utility function is increasing). This i = f1i . We can provide a similar argument for the case where enures that f1C i i = fti . > 0. Thus, it must be that in equilibrium ftC f2C .3 Appendix C We will first show that problem (3.2) has a solution di (t) = d∗i (where d∗i is constant). We will first show that if the solution d∗i exists, then it is admissible and that there are functions φ∗ , F ∗ , and λ∗ that satisfy the necessary conditions. Then we will show that the solution d∗i is optimal. Consider the constant solution d∗i . Then, the solution to the differential equation F˙ (t) = φ( n i=1 ∗ d(t), G)((1 − F (t))) is F ∗ (t) = Ae−φ t + 1. When t = 0, F (0) = 0. ∗ Thus A = −1. Therefore, F ∗ (t) = −e−φ t + 1. 115 Hi = e−δt [(1 − F (t))(U (di (t)) + V (ci (t)) − D(ci (t)) +F (t)(1 − γi )(U (d¯i ) + V (ci (t)) − D(ci (t)) −φ(di (t) + dj (t), G)((1 − F (t)))Ki ] j=i dj (t), G)(1 − F (t)) +λ(t)φ(di (t) + j=i (44) From the first order conditions we get U (c∗i (t)) = Di (c∗ (t)) (45) (where, c∗i (t) is the optimal choice of clean energy utilized). λ(t) = −e−δt U (di (t)) − Ki φ (di (t) + j=i dj (t), G) . φ (di (t) + j=i dj (t), G) (46) ˙ ¯ + V (ci (t)) − D(ci (t)) λ(t) = e−δt [−(U (di (t)) + V (ci (t)) − D(ci (t)) + γi (U (d) dj (t), G))] − (φ(di (t) + +Ki (φ(di (t) + j=i dj (t), G))λ(t). j=i Rearranging the differential equation (47) and solving we get 116 (47) ¯ + γi (V (ci (t)) − D(ci (t))) λ(t) = eφt [B + (U (di (t)) − U (d) +Ki (φ(di (t) + dj (t), G)) j=i −(δ+φ)t e (−(δ + φ(di (t) + j=i dj (t), G)) ] (48) Using the transversality condition and F ∗ (t) we get B = 0. Therefore we have ¯ + γi (V (ci (t)) − D(ci (t))) + K(φ(di (t) + λ∗ (t) = eφt [(U (di (t)) − U (d) dj (t), G)) j=i −(δ+φ)t e (−(δ + φ(di (t) + j=i dj (t), G)) ] (49) From equations (46) and (49) we get U (d∗i ) φ∗ = (U (d∗i ) − U (d¯i ) + (V (ci (t)) − D(ci (t))) + Kδ ∗ (δ + φ ) (50) The admissibility of the solutions can be checked by substituting them in the first order conditions. We now show the optimality of the constant solution d∗i . Let ∞ e−δt [(1 − F ∗ )(U (d∗i ) + V (c∗ ) − D(c∗ )) V∗ = +F ∗ (1 − γ)(U (d¯i ) + V (c∗i ) − D(c∗i ) − φ(Σd∗i , G)((1 − F ∗ )K]dt (51) 117 and ∞ V e−δt [(1 − F )(U (di (t)) + V (c) − D(c)) = +F (1 − γ)(U (d¯i ) + V (ci ) − D(ci ) − φ(Σdi , G)((1 − F )K]dt (52) Now ∞ ∗ V −V e−δt [(1 − F ∗ )(U (d∗i ) + V (c∗i ) − D(c∗i )) = +F ∗ (1 − γ)(U (d¯i ) + V (c∗i ) − D(c∗i ) − φ(Σd∗i , G)((1 − F ∗ )K] ∞ e−δt [(1 − F )(U (di ) + V (c∗i ) − D(c∗i )) − +F (1 − γ)(U (d¯i ) + V (c∗i ) − D(c∗i ) − φ(Σdi , G)((1 − F )K]dt(53) ∞ V∗−V e−δt [(1 − F ∗ )U (d∗i ) = +(1 − F )U (di ) + (F − F ∗ )(1 − γ)(V (c∗i ) − D(c∗i ) +F ∗ U (d¯i ) − F U (d¯i ) −φ(Σd∗i , G)((1 − F ∗ )K + φ(Σdi , G)((1 − F )K] Adding and subtracting F U (d∗i ) to the above expression and rearranging, we get 118 ∞ V∗−V e−δt [(1 − F )(U (d∗i ) − U (di )) = +(F − F ∗ )U (d∗i ) − (F − F ∗ )U (d¯i ) +(F − F ∗ )(1 − γ)(V (c∗i ) − D(c∗i ) −φ(Σd∗i , G)((1 − F ∗ )K + φ(Σdi , G)((1 − F )K]dt Adding and subtracting (1 − F )(φ(Σd∗i ) − φ(Σdi )) to the above expression and rearranging, we get ∞ ∗ V −V e−δt [(1 − F )(U (d∗i ) − U (d)) = −K(φ(Σd∗i − φ(Σdi )) + (F − F ∗ )(1 − γ)(U (d∗i ) − U (d¯i ) +(V (c∗i ) − D(c∗i ) − φ(Σd∗i , G)K]dt (54) Since U (·) is concave and φ(·) is convex we get ∞ V∗−V e−δt [(1 − F )(U (d∗i ) − K(φ (Σd∗i ) ≥ +(F − F ∗ )(1 − γ)(U (d∗i ) − U (d¯i ) + (V (c∗i ) − D(c∗i ) −φ(Σd∗i , G)K]dt (55) From equations (46), (47), and (55) we get 119 ∞ V∗−V ˙ e−δt [−(1 − F )λ∗ φ (Σd∗i )(d∗i − di ) + (F − F ∗ )λ− ≥ (F − F ∗ )λ∗ φ(Σd∗i )]dt ≥ (since .4 ∞ (F − F ∗ )λ˙ = − ∞ (1 − F )λφ(Σdi )dt + ∞ λφ(Σd∗i )(1 − F ∗ )dt) Appendix D In the literature on differential games, one usually refers to the concepts of weak and strong time consistency. The difference between these two properties can be outlined as follows: Weak time consistency Consider a game played over t = [0, ∞) and examine the trajectories of the state variables, denoted by x(t). The equilibrium is weakly time consistent if its truncated part in the time interval t = [T, ∞), with T ∈ (0, ∞), represents an equilibrium also for any subgame starting from t = T , and from the vector of initial conditions xT = x(T ). Strong time consistency Consider a game played over t = [0, ∞). The equilibrium is strongly time consistent, if its truncated part is an equilibrium for the subgame, independently of the conditions regarding state variables at time T , x(T ). Strong time consistency requires the ability on the part of each player to account for the rivals behaviour at any point in time, i.e., it is, in general, an attribute of closed-loop equilibria, and corresponds to subgame perfectness. Weak time consistency is a milder requirement and does not ensure, in general, that the resulting Nash equilibrium be subgame perfect.13 13 For a more detailed analysis of these issues, see Dockner et al. (2000), pp. 98-107); see also Basar and Olsder (1995, ch. 6). 120 The game considered by us in this paper belongs to the class of games that are linear in the state variable. Fershtman (1987) (pp. 225-229) has shown that in such a game if the open-loop Nash equilibrium exists then it is also the feedback equilibrium and so is strongly time consistent. 121 [...]... the oil-macroeconomy literature pioneered by Hamilton (1985), In section 1.2 we test the robustness of the reserves-output relationship alluded to by the theory of ‘oil’igopoly Section 1.3 presents a description of the model as well as an illustrative example Section 1.4 consists of the conclusion 1.2 Empirical analysis Given that oil is a nonrenewable resource, one would expect a relationship between... oil from consuming nations also needs to be accounted for when modeling the oil markets Oil is used by most producing countries as a means of obtaining consumption goods Fluctuations in the price of goods and services consumed by the producing nations would most certainly have an impact on their extraction decisions In addition, the theory was developed using the Nash equilibrium, open loop concept rather... focusses on the reserves-production relationship, provides valuable insight into the production behavior of oil-producing countries and firms However, the theory has been developed by only taking into consideration the interactions amongst the oil-producing countries For an internationally traded commodity like oil, it is not just the supply behavior of oil-producing countries that warrants consideration,... contemporaneously known about and economically viable given prevailing market conditions Before we embark on the econometric exercise, we plot the data to get a feel for the relationship between the production and reserves of crude oil We consider two plots, both of which are cross-sectional plots (for 2007) for the 38 countries under consideration Figure 1.1 plots the production (in tmb) versus the reserves... the reserves-production relationship has to take into account the state of the global economy as its effects are significant Thus, a theoretical model that explicitly takes into consideration the interactions between the oil and non-oil producing countries would be needed to reflect this empirical regularity This is what we shall consider in the next section 6 We use real GWP in billions of 2005 dollars... maximization problem of the foodgrain producer j: 14 maxoj Σ2 δ t−1 Ut (oj ) t=1 tC (1.15) tC subject to j qt oj = pt ftT tC j 0 ≤ ftC ≤ ftj and j pt Σj ftT = qt Σi oi tT (where the first constraint is the period budget constraint, the second is the resource constraint, and the third is the market clearing condition) Given the foodgrain producer’s optimization problem, we can define a value function for... maximization problem can be converted into a one-shot problem because each agent is going to consume all of their stock in the final period This allows us to fold the game back via backward induction and solve the resulting problem Proposition 2 The production decisions of the oil and foodgrain producers are strategic complements Proof Consider the oil producers problem as described by equation (1.7)... coordination failure between the decisions of the oil producers and the consuming countries Such an interpretation is in keeping with Killian’s (2009) analysis of the relationship between oil and the macroeconomy Proposition 3 The production decisions of the oil producers are strategic substitutes Proof Consider the oil producers problem as described by equation (1.7) 17 j (Σj (f1T ))(oi ) 1T Σi (oi... intertemporal maximization problem of the oil producer i: i i maxftC Σ2 δ t−1 U (ftC ) t=1 11 (1.7) subject to i qt oi = pt ftC tT i 0 ≤ oi ≤ Ot t i oi + oi ≤ O1 1 2 and j pt Σj ftT = qt Σi oi tT (where the first constraint is the individual budget constraint, the second represents the intratemporal resource constraint, the third represents the intertemporal resource constraint, and the fourth constraint gives... f1 in period 1.) We can combine equations (1.1) and (1.2) to get the amount of foodgrains that the oil producer i can consume in period 1 as: i f1C = j (Σj (f1T ))(oi ) 1T Σi (oi ) 1T (1.3) Equation (1.3) tells us that not only does the amount of foodgrains consumed by an oil producer i depend on her own decision of how much oil to trade, but also on the decision of other oil and foodgrain producers . ESSAYS ON RESOURCE AND ENVIRONMENTAL ECONOMICS NEIL SEBASTIAN D’SOUZA NATIONAL UNIVERSITY OF SINGAPORE 2010 ESSAYS ON RESOURCE AND ENVIRONMENTAL ECONOMICS NEIL SEBASTIAN. example. Section 1.4 consists of the conclusion. 1.2 Empirical analysis Given that oil is a nonrenewable resource, one would expect a relationship between the reserves and production for an oil-producing. obtaining consumption goods. Fluctuations in the price of goods and services consumed by the producing nations would most certainly have an impact on their extraction decisions. In addition, the theory