Dynamic Games in Environmental Economics PhD minicourse Part II: Stochastic Games and Contracts Bård Harstad UiO 5 December 2017 Bård Harstad (UiO) Dynamic Environment 5 December 2017 1 / 59
Content of the Day 2 a. Games with stocks - Stochastic games b. Markov-perfect equilibria as "business as usual": A dynamic common-pool problem c. Short-term agreements and Hold-up problems d. Optimal long-term contracts e. Duration f. Renegotiation Design Bård Harstad (UiO) Dynamic Environment 5 December 2017 2 / 59
2-a. Games with stocks - Stochastic games From Mailath and Samuelsson (2006: 174-5): games [where the stage game changes from period to period] are referred to as dynamic games or, when stressing that the stage game may be a random function of the game’s history, stochastic games . The analysis of a dynamic game typically revolves around a set of game states that describe how the stage game varies Each state determines a stage game the appropriate formulation of the set of states is not always obvious In resource/environmental economics, the typical state is the stock(s) of resource or pollution. Note that the stock may or may not be "payoff relevant" Bård Harstad (UiO) Dynamic Environment 5 December 2017 3 / 59
2-a. Games with stocks - Markov perfect equilibria Mailath and Samuelsson (2006: Ch 5): A strategy profile is a Markov strategy if they are functions of the state and time, but not of other aspects of the history The strategy profile is a Markov (perfect) equilibrium if it is both Markov and a subgame-perfect equilibrium A strategy profile is a stationary Markov strategy if they are functions of the state, but not of time or other aspects of the history The strategy profile is a stationary Markov (perfect) equilibrium if it is both stationary Markov and a subgame-perfect equilibrium Maskin and Tirole (2001, JET): Markov strategies depend (only) on the coarsest partition of histories that are payoff relevant Two histories h and h � are payoff-irrelevant if, when other players’ strategy satisfy σ − i ( h ) = σ − i ( h � ) , then i cannot do strictly better than strategies σ i ( h ) = σ i ( h � ) that are not contingent on h vs h � . So, states/stocks that are not payoff-relevant should not matter. Bård Harstad (UiO) Dynamic Environment 5 December 2017 4 / 59
2-a. Games with stocks - MPE - justifications There are too many SPEs hard to make predictions many SPEs are are not renegotiation proof MPE is "simplest form of behavior that is consistent with rationality" (Maskin and Tirole, 2001) Experimentally support in complex games (Battaglini et al 2014) Robust to, for example, finite time Meaningful to study (incomplete) contracts Bård Harstad (UiO) Dynamic Environment 5 December 2017 5 / 59
2-a. Motivation Countries may be able to commit - to the extent that they are patient or ratify treaties by (writing) national laws. There may also be costs of noncompliance not easily modelled. Agreements may also be legally binding or sanctioned But: agreements might be made on some aspects ...but not on everything of interest What is the consequence of such incomplete contract ? What is the optimal/equilibrium contract? Bård Harstad (UiO) Dynamic Environment 5 December 2017 6 / 59
2-a. Model: Timing Bård Harstad (UiO) Dynamic Environment 5 December 2017 7 / 59
2-a. A Model A model with n + 1 stocks: ≡ ∑ u i , t δ t V i t B i ( g i , t , R i , t ) − C ( G t ) − k ( r i , t ) + e ∑ u i , t ≡ r j , t j � = i R i , t = q R R i , t − 1 + r i , t , j ∈ { 1 ... n }\ i q G G t − 1 + ∑ g i , t + θ t , i ∈ { 1 , 2 , ..., n } G t = � 0 , σ 2 � θ t ∼ F Continuation values, V i ( G t − 1 , R 1 , t − 1 , ..., R n , t − 1 ) , W i ( q G G t − 1 + θ t , R 1 , t , ..., R n , t ) Bård Harstad (UiO) Dynamic Environment 5 December 2017 8 / 59
2-a. Two simplifications Reducing the number of stocks to two. With perfect substitutes: B i ( g i , t + R i , t ) and y i , t = g i , t + R i , t , we can write: B i ( y i , t ) − C ( G t ) − k ( r i , t ) + e ∑ u i , t ≡ r j , t j � = i q G G t − 1 + ∑ y i , t − ∑ G t = R i , t + θ t . i With heterogeneity only in bliss points: B ( y i , t − y i ) , write y i , t − ( y i − y ) , and y ≡ ∑ y i / n , to get: � y i , t ≡ y i , t − y ) ≡ � B ( y i , t − y i ) = B ( � B ( � y i , t ) and y i , t − R i , t = g i , t + ( y i − y ) , so ∑ � g i , t = ∑ g i , t ; � � g i , t ≡ q G G t − 1 + ∑ � g i , t + θ t = q G G t − 1 + ∑ � G t = y i , t − R t + θ t ; y i , t ) − C ( G t ) − k ( r i , t ) + e ∑ � B ( � u i , t = r j , t . j � = i Remove "tildes" and interpret y i , t as consumption relative bliss. Bård Harstad (UiO) Dynamic Environment 5 December 2017 9 / 59
2-a. Reformulated model Write continuation values as V ( G t − 1 , R t − 1 ) and W ( q G G t − 1 + θ t , R t ) . As a third simplification, k ( r i , t ) = kr i , t : B ( y i , t ) − C ( G t ) − kr i , t + e ∑ ≡ u i , t r j , t j � = i = g i , t + R i , t y i , t R i , t = q R R i , t − 1 + r i , t , j ∈ { 1 ... n }\ i q G G t − 1 + ∑ y i , t − R t + θ t , i ∈ { 1 , 2 , ..., n } G t = = ∑ R t R i , t i � 0 , σ 2 � θ t ∼ F K ≡ k − ( n − 1 ) e Example Q: B ( . ) = − b 2 ( y − y i , t ) 2 , C ( . ) = c 2 G 2 (Q) t Bård Harstad (UiO) Dynamic Environment 5 December 2017 10 / 59
2-a. Simplifications and implications Lemma 0: Markov strategies depend only on G t − 1 and R t − 1 ≡ ∑ i R i , t − 1 So, same y i , t even if R i , t differ! R i , t and R t is a "public good" regardless of e Bård Harstad (UiO) Dynamic Environment 5 December 2017 11 / 59
2-b. Business as usual - L1 Lemma 1-BAU: V B R = q R K / n . Proof: At the investment-stage, i solves r i , t E W ( q G G t − 1 + θ t , q R R t − 1 + ∑ max r i , t ) − kr i , t ⇒ i E W R ( q G G t − 1 + θ t , R t ) = k ⇒ R t ( G t − 1 ) , so V B ( G t − 1 , R t − 1 ) = W ( q G G t − 1 + θ t , R ( G t − 1 )) − K n [ R t ( G t − 1 ) − q R R t − 1 ] ⇒ V B R = q R K / n . Note: Since V R is a constant, V GR = 0, and V G does not depend on R . � Bård Harstad (UiO) Dynamic Environment 5 December 2017 12 / 59
2-b. Business as usual - L2 Lemma 2-BAU: V B G = − q G ( 1 − δ q R ) K / n Proof: At the emission stage, B � ( y i , t ) − C � � � + δ V G ( G , R ) = 0 q G G t − 1 + θ t + ∑ y i , t − R t (1) So y i , t = y t is a function of ξ t + θ t where ξ t ≡ q G G t − 1 − R t , and so is G t . Inserted, the foc for R t comes from: � � � � � � �� y B ( ξ ) G B ( ξ ) G B ( ξ ) , R max r i , t E B − C + δ V − kr i , t (2) which gives the foc, determining ξ t = ξ B as a constant: � � + δ V R = k . B � ( y ( ξ )) y � ( ξ ) − C � ( G ( ξ )) G � ( ξ ) + δ V G G � ( ξ ) − E (3) Bård Harstad (UiO) Dynamic Environment 5 December 2017 13 / 59
2-b. Business as usual - L2 - proof continued In the symmetric equilibrium: V ( G , R ) = E B ( y ( ξ )) − E C ( G ( ξ )) − k n [ q G G t − 1 − ξ − q R R t − 1 ] + e ( n − 1 ) [ q G G t − 1 − ξ − q R R t − 1 ] n + δ V ( G t ( ξ ) , q G G t − 1 − ξ ) Taking the derivative gives the lemma. � Bård Harstad (UiO) Dynamic Environment 5 December 2017 14 / 59
2-b. Business as usual - Observations Since V G is a constant, (1) can be differentiated: dy ∗ = − dy ∗ − C �� y �∗ i , t i , t = = nC �� − B �� , and (4) d θ t dR t dG ∗ = − dG ∗ − B �� = 1 + ny �∗ = G ∗� t t = nC �� − B �� . t d θ t dR t The foc for r i , t (3) becomes � � � + δ V G � �� + δ V R = k . B � ( . ) y � − C � ( . ) ny � + 1 ny � + 1 − E Combined with B � ( . ) = C � − δ V G , and (4), we get: �� � � � + δ V G � �� + δ V R = k C � − δ V G y � − C � ( . ) ny � + 1 ny � + 1 − E �� � � C �� − B �� �� C � − δ V G = k − δ V R E (5) nC �� − B �� � � C �� − B �� �� B � ( . ) E = k − δ V R . (6) nC �� − B �� Bård Harstad (UiO) Dynamic Environment 5 December 2017 15 / 59
2-b. Business as usual - 2nd order conditions At the emission stage, the 2.O.C. is trivially satisfied. It is also at the investment stage if Q. Otherwise, the second-order condition of (2) is: � dy i � 2 � � 2 + B � ( y i ) d 2 y i ndy i E B �� ( y i ) ( dR ) 2 − C �� ( G ) dR − 1 dR � � � � n d 2 y i C � ( G ) − δ V G − ≤ 0 ( dR ) 2 When we substitute with (4) and differentiate it, we can get: � � ( C �� ) 2 B ��� − ( B �� ) 2 C ��� E B �� C �� ( C �� − B �� ) − B � ( n − 1 ) ≤ 0 . ( nC �� − B �� ) 2 ( B �� − nC �� ) 3 Bård Harstad (UiO) Dynamic Environment 5 December 2017 16 / 59
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