Dynamic Games in Environmental Economics PhD minicourse ...€¦ · Bård Harstad (UiO) Dynamic Environment 5 December 2017 5 / 59. 2-a. Motivation Countries may be able to commit

Dynamic Games in Environmental EconomicsPhD 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


2-a. Games with stocks - Stochastic games

From Mailath and Samuelsson (2006: 174-5):

games [where the stage game changes from period to period] arereferred to as dynamic games or, when stressing that the stage gamemay be a random function of the game’s history, stochastic games.The analysis of a dynamic game typically revolves around a set of gamestates that describe how the stage game variesEach state determines a stage gamethe 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"


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 stateand time, but not of other aspects of the historyThe strategy profile is a Markov (perfect) equilibrium if it is bothMarkov and a subgame-perfect equilibriumA strategy profile is a stationary Markov strategy if they are functionsof the state, but not of time or other aspects of the historyThe strategy profile is a stationary Markov (perfect) equilibrium if it isboth stationary Markov and a subgame-perfect equilibrium

Maskin and Tirole (2001, JET):

Markov strategies depend (only) on the coarsest partition of historiesthat are payoff relevantTwo histories h and h′ are payoff-irrelevant if, when other players’strategy satisfy σ−i (h) = σ−i (h′), then i cannot do strictly betterthan strategies σi (h) = σi (h′) that are not contingent on h vs h′.So, states/stocks that are not payoff-relevant should not matter.


2-a. Games with stocks - MPE - justifications

There are too many SPEs

hard to make predictionsmany 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


2-a. Motivation

Countries may be able to commit - to the extent that they are patientor 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?


2-a. Model: Timing


2-a. A Model

A model with n+ 1 stocks:

Vi ≡ ∑tui ,tδ

t

ui ,t ≡ Bi (gi ,t ,Ri ,t )− C (Gt )− k (ri ,t ) + e ∑j 6=irj ,t

Ri ,t = qRRi ,t−1 + ri ,t , j ∈ {1...n}\iGt = qGGt−1 +∑ gi ,t + θt , i ∈ {1, 2, ..., n}θt ∼ F

(0, σ2

)Continuation values,

Vi (Gt−1,R1,t−1, ...,Rn,t−1) ,Wi (qGGt−1 + θt ,R1,t , ...,Rn,t )


2-a. Two simplifications

Reducing the number of stocks to two. With perfect substitutes:

Bi (gi ,t + Ri ,t ) and yi ,t = gi ,t + Ri ,t , we can write:

ui ,t ≡ Bi (yi ,t )− C (Gt )− k (ri ,t ) + e ∑j 6=irj ,t

Gt = qGGt−1 +∑ yi ,t −∑iRi ,t + θt .

With heterogeneity only in bliss points: B (yi ,t − y i ), writeỹi ,t ≡ yi ,t − (y i − y) , and y ≡∑ y i/n, to get:

B (yi ,t − y i ) = B (ỹi ,t − y) ≡ B̃ (ỹi ,t ) andg̃i ,t ≡ ỹi ,t − Ri ,t = gi ,t + (y i − y) , so ∑ g̃i ,t = ∑ gi ,t ;Gt = qGGt−1 +∑ g̃i ,t + θt = qGGt−1 +∑ ỹi ,t − Rt + θt ;ui ,t = B̃ (ỹi ,t )− C (Gt )− k (ri ,t ) + e ∑

j 6=irj ,t .

Remove "tildes" and interpret yi ,t as consumption relative bliss.Bård Harstad (UiO) Dynamic Environment 5 December 2017 9 / 59

2-a. Reformulated model

Write continuation values as V (Gt−1,Rt−1) andW (qGGt−1 + θt ,Rt ).As a third simplification, k (ri ,t ) = kri ,t :

ui ,t ≡ B (yi ,t )− C (Gt )− kri ,t + e ∑j 6=irj ,t

yi ,t = gi ,t + Ri ,tRi ,t = qRRi ,t−1 + ri ,t , j ∈ {1...n}\iGt = qGGt−1 +∑ yi ,t − Rt + θt , i ∈ {1, 2, ..., n}Rt = ∑

iRi ,t

θt ∼ F(0, σ2

)K ≡ k − (n− 1) e

Example Q:

B (.) = −b2(y − yi ,t )2 , C (.) =

c2G 2t (Q)


2-a. Simplifications and implications

Lemma 0: Markov strategies depend only on Gt−1 andRt−1 ≡ ∑i Ri ,t−1So, same yi ,t even if Ri ,t differ!

Ri ,t and Rt is a "public good" regardless of e


2-b. Business as usual - L1

Lemma 1-BAU: V BR = qRK/n.Proof: At the investment-stage, i solves

maxri ,tEW (qGGt−1 + θt , qRRt−1 +∑

iri ,t )− kri ,t ⇒

EWR (qGGt−1 + θt ,Rt ) = k ⇒ Rt (Gt−1) , soV B (Gt−1,Rt−1) = W (qGGt−1 + θt ,R (Gt−1))

−Kn[Rt (Gt−1)− qRRt−1] ⇒

V BR = qRK/n.

Note: Since VR is a constant, VGR = 0, and VG does not depend onR.‖


2-b. Business as usual - L2

Lemma 2-BAU: V BG = −qG (1− δqR )K/nProof: At the emission stage,

B ′ (yi ,t )− C ′(qGGt−1 + θt +∑ yi ,t − Rt

)+ δVG (G ,R) = 0 (1)

So yi ,t = yt is a function of ξt + θt where ξt ≡ qGGt−1 − Rt , and sois Gt . Inserted, the foc for Rt comes from:

maxri ,tE[B(yB (ξ)

)− C

(GB (ξ)

)+ δV

(GB (ξ) ,R

)]− kri ,t (2)

which gives the foc, determining ξt = ξB as a constant:

−E[B ′ (y (ξ)) y ′ (ξ)− C ′ (G (ξ))G ′ (ξ) + δVGG ′ (ξ)

]+ δVR = k.

(3)


2-b. Business as usual - L2 - proof continued

In the symmetric equilibrium:

V (G ,R) = EB (y (ξ))− EC (G (ξ))

−kn[qGGt−1 − ξ − qRRt−1]

+e (n− 1)

n[qGGt−1 − ξ − qRRt−1]

+δV (Gt (ξ) , qGGt−1 − ξ)

Taking the derivative gives the lemma.‖


2-b. Business as usual - Observations

Since VG is a constant, (1) can be differentiated:

y ′∗ =dy ∗i ,tdθt

= −dy ∗i ,tdRt

=−C ′′

nC ′′ − B ′′ , and (4)

G ∗′t =dG ∗tdθt

= −dG∗t

dRt= 1+ ny ′∗ =

−B ′′nC ′′ − B ′′ .

The foc for ri ,t (3) becomes

−E[B ′ (.) y ′ − C ′ (.)

(ny ′ + 1

)+ δVG

(ny ′ + 1

)]+ δVR = k.

Combined with B ′ (.) = C ′ − δVG , and (4), we get:−E

[(C ′ − δVG

)y ′ − C ′ (.)

(ny ′ + 1

)+ δVG

(ny ′ + 1

)]+ δVR = k

E[(C ′ − δVG

) ( C ′′ − B ′′nC ′′ − B ′′

)]= k − δVR (5)

E[B ′ (.)

(C ′′ − B ′′nC ′′ − B ′′

)]= k − δVR . (6)


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-ordercondition of (2) is:

EB ′′ (yi )(dyidR

)2+ B ′ (yi )

d2yi(dR)2

− C ′′ (G )[ndyidR− 1]2

−[C ′ (G )− δVG

] [nd2yi(dR)2

]≤ 0

When we substitute with (4) and differentiate it, we can get:

EB ′′C ′′ (C ′′ − B ′′)(nC ′′ − B ′′)2

− B ′ (n− 1)[(C ′′)2 B ′′′ − (B ′′)2 C ′′′

(B ′′ − nC ′′)3

]≤ 0.


2-b. Business as usual - Emissions

Proposition

The equilibrium consumption yBi is independent of Ri andsuboptimally large, given R. From (1):

B ′ (yi ,t ) = C ′(qGGt−1 + θt +∑ yi ,t − Rt

)+ qG (1− δqR )K/n

Country i pollutes less but j 6= i pollutes more if Ri is larger, fixingRj∀j 6= i . From (4):

∂gnoi /∂Ri = −C ′′(n− 1)− B ′′nC ′′ − B ′′ < 0,

∂gnoj /∂Ri =C ′′

nC ′′ − B ′′ > 0 ∀j 6= i ,

dG ∗tdRt

=B ′′

nC ′′ − B ′′ < 0.


2-b. Business as usual - Investments

Proposition

From the proof of Lemma 2-B, Rt = qGGt−1 − ξB − qRRt−1, so:

∂rnoi /∂R− = −qR/n,∂rnoi /∂G− = qG/n.

If one country pollutes more, every country invests more in the nextperiod

If one country invests more, every other country invests less in thenext period.

There is a unique symmetric MPE.


2-b. Business as usual - Transition

PropositionIf θt is large, every country pollutes less at t and t+1, invests more att+1, and less at t+2. Steady state is reached after two periods. From(4):

∂gt∂θt

= −ρ and ∂gt+1∂θt

= −qG (1− ρ) = −∂rt+1∂θt

=1qR

∂rt+2∂θt

,(7)

where

ρ ≡ C′′

nC ′′ − B ′′ ∈ (0, 1) .


2-b. First Best - L1

Lemma 1-FB: V ∗R = qRK/n.Proof: At the investment-stage, i should solve

V ∗ (Gt−1,Rt−1) = maxRtEθW (qGGt−1 + θt ,Rt )−

Kn(Rt − qRRt−1)

⇒V ∗R = qRK/n.

Note: Since VR is a constant, VGR = 0, and VG does not depend onR.‖


2-b. First Best - L2

Lemma 2-FB: V ∗G = −qG (1− δqR )K/nProof: At the emission stage,

B ′ (yi ,t )− nC ′(qGGt−1 + θt +∑ yi ,t − Rt

)+ nδVG (G ,R) = 0 (8)

So yi ,t = yt is a function of ξt + θt where ξt ≡ qGGt−1 − Rt , and sois Gt . Inserted, the foc for Rt comes from:

maxRtEθ [B (y

∗ (ξt + θt ))− C (G ∗ (ξt + θt )) + δV (G ∗ (ξt + θt ) ,R)]

−Kn(Rt − qRRt−1)

which gives the foc for ri ,t , determining ξt = ξ∗ as a constant.

K/n =

EθB′ (y ∗ (ξ + θt ))

(−y ∗′

)−(C ′ (G ∗ (ξ + θt ))− δVG

) (−G ∗′

)+ δVR(9)


2-b. First Best - L2 - proof continued

With symmetric investments:

V (G ,R) = E [B (y ∗ (ξ∗ + θt ))− C (G ∗ (ξ∗ + θt ))]

−Kn[qGGt−1 − ξ∗ − qRRt−1]

+δV (G ∗ (ξ∗) , qGGt−1 − ξ∗)

Taking the derivative gives the lemma.‖


2-b. First Best - Observations

Since VG is a constant, foc for yi ,t (8) can be differentiated to get:

y ′∗ =dy ∗i ,tdθt≡ −

dy ∗i ,tdRt

=−C ′′

nC ′′ − B ′′/n , and (10)

G ∗′t =dG ∗tdθt

= −dG∗t

dRt= 1+ ny ′∗ =

−B ′′/nnC ′′ − B ′′/n .

The foc for ri ,t (9) becomes

EθB′ (y ∗ (ξ + θt ))

(−y ∗′

)+ δVR

+(C ′ (G ∗ (ξ + θt ))− δVG

) (1+ ny ′∗

)=Kn.


2-b. First Best - Observations continued

Combined with B ′ (.) = n (C ′ − δVG ), we get:

Eθ[(C ′ − δVG

) (−ny ∗′

)+(C ′ − δVG

) (1+ ny ′∗

)+ δVR

]=

Kn

Eθ[(C ′ − δVG

)+ δVR

]=

Kn

EθC′ = K/n− δVR + δVG =

Kn(1− δqG ) (1− δqR ) . (11)

And with B ′ (.) = n (C ′ − δVG ), we get:

EB ′ (.) = K (1− δqG ) . (12)


2-b. Comparing Business as usual vs. the First Best

By comparing (8) with (1), we have

yBt (qGGt−1 − Rt + θt ) > y ∗t (qGGt−1 − Rt + θt ) . (13)If either σ2 = 0 or Q, then we can compare (5)-(6) with (11)-(12):

EyBt(qGGt−1 − RBt + θt

)< Ey ∗t (qGGt−1 − R∗t + θt ) ,

EGB > EG ∗ ⇒RBt < R

∗t .

Transition rule (7) continues to hold if ρ is replaced by the largerρ∗ = C ′′/ (nC ′′ − B ′′/n). So, BAU-transition is too slow and toomuch relying on investments.But yB reacts less to θ than does y ∗, while GB reacts more to θ thandoes G ∗.If θ < 0 very small, then it is possible that GB < G ∗.If θ > 0 very large, then it is possible that yB > y ∗.Note that the equilibrium play is independent of future regime.Bård Harstad (UiO) Dynamic Environment 5 December 2017 25 / 59

2-b. First Best - Q

With (Q), (12) becomes:

Eb (y − yi ,t ) = K (1− δqG )⇔ Eyi ,t = y −Kb(1− δqG )

With (10), we can write:

yi ,t = Eyi ,t −cnθtcn2 + b

= y − Kb(1− δqG )−

cnθtcn2 + b

.

Similarly, we get:

G ∗t =Kcn(1− δqG ) (1− δqR ) +

bθtcn2 + b

g ∗i ,t =Kcn2

(1− δqG ) (1− δqR )− qGGt−1 −cnθtcn2 + b

Ri ,t = y −Kb(1− δqG )−

Kcn2

(1− δqG ) (1− δqR ) + qGGt−1.


2-b. Business as usual - Q

From (6), Q gives:

Eb (y − yi ,t ) =nc + bc + b

(k − δVR ) .

With (4), we get:

yi ,t = Eyt −cθtnc + b

= y − 1bnc + bc + b

(k − δVR )−cθtnc + b

.

Furthermore,

EcG =nc + bc + b

(k − δVR ) + δVG

∑ gi ,t =1cnc + bc + b

(k − δVR ) +1c

δVG −ncθtnc + b

− qGGt−1

Rt = ny −b+ cncb

nc + bc + b

(k − δVR )−1c

δVG + qGGt−1.


2-b. Business as Usual and the first best: Lessons

Proposition1 In each case, VG and VR are the same. Thus, when analyzing thisperiod, it is irrelevant whether in the next period there is BAU or FB.

2 There is a unique symmetric MPE.3 If one invests more, everyone consumes more and the others pollutemore and invest less next period.

4 If one pollutes less, everyone invests less and pollutes more nextperiod.

5 Countries pollute too much and invest too little in BAU compared toFB.

6 The dynamic common pool problem (with strategic substitutes) isworse than its static counterpart.


2-c. Short-Term Agreements


2-c. Short-Term Agreements: Initial Observations

Negotiating the gi ,ts is equivalent to negotiating the yi ,ts at thenegotiation/emission stage.

At the negotiation stage (i.e., the emission stage), the countries areidentical w.r.t. yi ,t regardless of differences in Ri ,ts.

We should thus expect a symmetric equilibrium when it comes toyi ,t = gi ,t + Ri ,tSo, the more a country has invested, the smaller is the negotiatedquota

Since the countries are symmetric wrt yi ,t , the yi ,t’s are first best,given Rt .

But what are the equilibrium (noncooperatively set) investment levels?


2-c. Short-Term Agreements: Solving the Model

Lemma 1 and 2 hold with similar proofs as before:V STG = −qG (1− δqR )K/n and V stR = qRK/n.The quotas are first best, and thus given by (8). Differentiating (8)gives, as we know, (10). The foc wrt r is as in (3), where y ′ (ξ) andG ′ (ξ) are given by (10). This gives:

EC ′ (G ) = k + δUG − δUR = (1− δqG ) (1− δqR )K/n+ (K + en) (1− 1/n)EB ′ (yi ) = n (n− δqR )K + n2 (n− 1) e

The foc for ri ,t becomes

−E[B ′ (.) y ′ − C ′ (.)

(ny ′ + 1

)+ δVG

(ny ′ + 1

)]+ δVR = k.


2-c. Short-Term Agreements: Solving the Model - cont.

Combined with B ′ (.) = n (C ′ − δVG ), we get:

−E[n(C ′ − δVG

)y ′ − C ′ (.)

(ny ′ + 1

)+ δVG

(ny ′ + 1

)]+ δVR = k.

EC ′ = k − δVR + δVG ,EB ′ (.) = n (k − δVR )

Suppose σ = 0 or Q. Compared to (6), we have: yS < yB , andcompared to (12) yS < y ∗. Compared to (5) we have GS < GB , butcompared to (11) we have GS > G ∗. Hence, RSt < R

∗t .


2-c. Short-Term Agreements: Q

SinceEb (.) = n (k − δVR )

EcG = k − δVR + δVGand

Eb (y − yi ,t ) = n (k − δVR )

yi ,t = y −1bn (k − δVR )−

cθtnc + b/n

Gt =kc− 1c

δVR +1c

δVG +bθt/nnc + b/n

∑ gi ,t =kc− 1c

δVR +1c

δVG +bθt/nnc + b/n

− qGGt−1 − θt

Rt = ny −b+ cn2

bc(k − δVR )−

1c

δVG + qGGt−1.


2-c. Short-Term Agreements vs. BAU - Lessons

PropositionThere is a unique symmetric MPE.

The slopes of the continuation value, VG and VR , are as in BAU andFB.

Emission levels are ex post optimal (given Rt), but investments aresmaller than at the FB, and thus consumption is also smaller.

Under Q, we can also see that compared to BAU, ST leads to lessemissions but also less investments:

Eg st(r st)= Egbau

(rbau

)− n− 1n (b+ c)

(e (n− 1) +

(1− δqR

n

)K);

r sti = rbaui −

(n− 1)2

n (b+ c)

(e (n− 1) +

(1− δqR

n

)K).


2-c. Short-Term Agreements vs. BAU - Lessons continued

PropositionIf investments are important, countries can thus be worse off with STthan in BAU. Under Q:

uST < uBAU ⇔ (14)(e +

Kn

)2(n− 1)2 > (1− δqR )2 +

(b+ c) (bcσ)2

(b+ cn2) (b+ cn)2.

This condition is more likely to hold for large e and n, and small σ.


2-c. Short-Term Agreements vs. BAU - Lessons continued

Short term agreements can be harmful

Intuition:

Countries invest less in fear of being "held up" in future negotiationsCountries invest less when the problem is expected to be solved in anycaseIf investments are important, this makes the countries worse off ex ante(before the investment stage)

Agreements can be harmful ex ante - because they reduce incentivesto invest.

When the investments are sunk, then an agreement is always betterthan no agreement.

If e is large, investments are already suboptimal so short-termagreements are then likely to be harmful.

If σ is large, then y respond too little under BAU, and ST is morereassuring.


2-d. Long-Term Agreements

The timing is reversed: gi ,t is negotiated first, then i invests.

With this timing, there is no hold-up problem in this period

But still underinvestments - particularly if duration short


2-d. Long-Term Agreements - Analysis (i)

Proof:

When the level g coi is already committed to, the first-order conditionfor i’s investment is:

k = B ′ (g coi + Ri ) + δVR ⇒ (15)y coi = B

′−1 (k − δVR ) , Rcoi = B ′−1 (k − δVR )− g coi , (16)r coi = B

′−1 (k − δVR )− g coi − qRRi ,−. (17)

If V coR is constant (confirmed below), the second-order condition isB ′′ ≤ 0, which holds by assumption.If the negotiations fail, the default outcome is the non-cooperativeoutcome, giving everyone the same utility.

Since the ri s follow from the gi s in (17), everyone understands thatnegotiating the gi s is equivalent to negotiating the ri s.


2-d. Long-Term Agreements - Analysis (ii)

Since all countries have identical preferences w.r.t. the ri s (and theirdefault utility is the same) the ri s are going to be equal for every i .

Symmetry requires that ri , and thus ζ ≡ gi + qRRi ,−, is the same forall countries.

This implies that Rt = ∑i(B ′−1 (k − δVR )− g coi

)=

nB ′−1 (k − δVR )−EGt + qGGt−1. Thus, we can write the valuefunction as:

V (G−,R−) = maxEGi

B(B ′−1 (k − δVR )

)− EC (EGt + θt )

−Kn

(nB ′−1 (k − δVR )− EGt + qGGt−1 − qRRi ,−

)+EδV

(EGt + θt , nB ′−1 (k − δVR )− EGt + qGGt−1

).

Note that the Envelope theorem gives VR− = qRK/n.


2-d. Long-Term Agreements - Analysis (iii)

With this, we can also take the derivative wrt Gt−1 to getVG = −qG (1− δqR )Kn.With this, the foc wrt EGt is:

EC ′ (G ) = K/n− δVR + δVG = (1− δqG ) (1− δqR )K/n. (18)

This is the same pollution level as in the first best.

Since investments are suboptimally low if δqR > 0 or e > 0 , thepollution level is suboptimally small ex post.

Combining (18) with (15),

B ′ (yi ) /n− EC ′ (G ) + δVG = (k − δVR ) /n−K/n+ δVR

=1n(k −K ) + δqRK

n

(1− 1

n

).


2-d. Long-Term Agreements - Lessons

PropositionThere is a unique MPE.

The value function is linear with Lemma 1 and 2 as before.

The smaller is the quota g ti , the larger is ri ,t : ∂ri ,t/∂gi ,t = −1 whileri ,t : ∂ri ,t/∂gj ,t = 0.Conditional on g ti , every country invests too little if either δqR > 0 ore > 0.

The negotiated gi + qRRi ,− is the same for every country.

EC ′(GT)=EC ′ (G ∗), even if investments are too low.

Thus, quotas are too low ex post:B ′ (yi ) /n−EC ′ (G ) + δVG = 1n (k −K ) +

δqRKn

(1− 1n

)> 0.

If there is no uncertainty, gi ,t (R) < g ∗i ,t (R) if either δqR > 0 ore > 0.


2—e. Long-Term Agreements: Multiple Periods


2—e. Long-Term Agreements - Investments

Given quotas {gi ,t , ...gi ,T }, i invests as to ensure:

B ′(g lti + Ri ,t

)= k − kδqR , t < T , (19)

B ′(g lti + Ri ,t

)= k −KδqR/n, t = T .

Investments are first-best for t < T iff e = 0, but they are smaller att = T if δqR > 0.

The smaller is the quota gi ,t , the larger is ri ,t : ∂ri ,t/∂gi ,t = −1 whileri ,t : ∂ri ,t/∂gj ,t = 0.The quotas should be smaller than what is "ex post optimal"(particularly if duration short)


2—e. Long-Term Agreements - Quotas (i)

At the start of t = 1, countries negotiate emission levels for everyperiod t ∈ {1, ...,T}.In equilibrium, all countries enjoy the same default utilities.Just as before, they will therefore negotiate the quotas such that theequilibrium investment will be the same for all is.Using (19) for each period t < T , this implies:

Rt = ∑i

(B ′−1 (k (1− δqR ))− gi ,t

)= nB ′−1 (k (1− δqR ))− E (Gt − qGGt−1)⇒

ri ,t = (Rt − qRRt−1) /n, ∀t= B ′−1 (k (1− δqR ))− E (Gt − qGGt−1) /n (20)−qRB ′−1 (k (1− δqR )) + qRE (Gt−1 − qGGt−2) /n,

t ∈ {2, ..,T − 1} .


2—e. Long-Term Agreements - Quotas (ii)

At the start of the first period, i’s continuation value can be writtenas:

V = max{EGt}t

1− δT−11− δ B

(B ′−1 (k (1− δqR ))

)(21)

−T

∑t=1

δt−1[Kri ,t + C

(EGt +

t

∑t ′=1

θt ′qt−t ′G

)]+δT−1B

(B ′−1 (k − δqRK/n)

)+δTV

(EGT +

T

∑t ′=1

θt ′qT−t ′G ,RT

),

where ri ,t is given by (20) and RT is given by (19).

Lemma (1) VR = qRK/n. (2) VG = −qGK (1− δqR ) /n.Proof. Both parts follow from (21) by applying the envelope theorem.


2—e. Long-Term Agreements - Quotas (iii)

The first-order condition of (21) w.r.t. any EGt , t ∈ {1, ..,T}, gives:

EC ′ = (1− δqR ) (1− δqG )K/n

The second-order condition, −C ′′ < 0, always holds.Since investments are suboptimally low, the emissions are too smallconditional on equilibrium technology stocks.

Using (19),

B ′ − EC ′ (G ) n+ nδVG = (1− δqR ) (k −K )B ′(g lti + Ri

)− EB ′ (g ∗i + Ri ) = (1− δqR ) (n− 1) e

In Example Q, this becomes simply

g lti (R) = g∗i (R)− (1− δqR ) (n− 1) e/b.


2—e. Long-Term Agreements - Lessons

PropositionInvestments are suboptimally low at t < T if e > 0, and at t = T ifeither e > 0 or δqR > 0.

The agreement becomes tougher to satisfy towards the end.

Investments are larger if quotas are smaller.

Therefore, the optimal quotas are smaller than what is ex postoptimal if e > 0 in every period, and in the last period if either e > 0or δqR > 0.

The larger is e, the smaller are the optimal quotas compared to whatis ex post optimal.


2—e. The Optimal Duration: The cost of the hold-up

The optimal length T balances the cost of underinvestment when Tis short and the cost of the uncertain θ′s is increasing in T .

In period T , countries invest suboptimally, not only because of thedomestic hold-up problem, but also because of the international one.When all countries invest less, ui declines. This cost of the hold-upproblem in period T , relative to any period t < T , can be written as:

H = −b2(yi ,t − y i )2 −

b2(yi ,T − y i )2 −K (ri ,t − ri ,T ) (1− δqR )

= −b2

(k − δqRK

b

)2+b2

(k − zb

)2−K

(k − zb− k − δqRK

b

)(1− δqR )

=δqRb

(e +

Kn

) [e(1− δqR

2

)+

δqRK2n

](n− 1)2 .

Note that H increases in e, n, qR , and K , but decreases in b.


2—e. The Optimal Duration: The cost of uncertainty

The cost of a longer-term agreement is associated with θ.Although EC ′ and thus EGt , are the same for all periods,

Ec2(Gt )

2 = Ec2

(EGt +

t

∑t ′=1

θt ′qt−t ′G

)2=

c2(EGt )

2 +c2

σ2(1− q2tG1− q2G

).

The last term is the loss associated with the uncertainty.For the T periods, the total present discounted value of this loss is:

L(T ) =T

∑t=1

c2

σ2δt−1(1− q2tG1− q2G

)=

cσ2

2 (1− q2G )T

∑t=1

δt−1(1− q2tG

)=

cσ2

2 (1− q2G )

[1− δT1− δ − q

2G

(1− δT q2TG1− δq2G

)]. (22)


2—e. The Optimal Duration: The trade-off

If all future agreements last T̂ periods, then the optimal T for thisagreement is:

minTL(T ) +

(δT−1H + δT L

(T̂))( ∞

∑τ=0

δτT̂′

)⇒

0 = L′(T ) + δT ln δ(H/δ+ L

(T̂))

/(1− δT̂ ′

)= L′(T ) + δT ln δ

(H/δ+ L

(T̂))

/(1− δT̂ ′

)= −δT ln δ

cσ2/21−q2G(

11−δ −

q2T+2G (1+ln q2G / ln δ)1−δq2G

)−H/δ+L(T̂ )

1−δT̂ ′

, (23)assuming that some T satisfies (23).

Since(−δT ln δ

)> 0 and the term in the brackets increases in T ,

the loss decreases in T for small T but increases for large T , andthere is a unique T minimizing the loss.Bård Harstad (UiO) Dynamic Environment 5 December 2017 50 / 59

2—e. The Optimal Duration: The optimal T

Substituting for T̂ = T and (22) in (23) gives:

Hδ=

cσ2q2G2 (1− q2G ) (1− δq2G )

(1− δT q2TG1− δT

− q2TG

(1+

ln(q2G)

ln δ

)),

(24)where the r.h.s. increases in T . T = ∞ is optimal if the left-handside of (24) is larger than the right-hand side even when T → ∞:

cσ2q2G2 (1− q2G ) (1− δq2G )

≤ Hδ. (25)

If e and n are large but b is small, then H is large and (25) is morelikely to hold.

If (25) does not hold, the T satisfying (24) is larger. If c or σ2 arelarger, (25) is less likely to hold and, if it does not, (24) requires T todecrease.


2—e. The Optimal Duration: Lessons

Proposition1 There is a hold-up problem (with under-investments) for every finiteagreement.

2 It is optimal to reduce/delay this cost by increasing the duration ofthe agreement.

3 On the other hand, the uncertainty aggregates over time.4 The trade-off implies that the optimal duration is longer if e is largewhile σ2 is small, for example.

5 Thus, with weak intellectual property rights, the climate treaty shouldlast longer.


2—f. Renegotiation and Updating of Agreements

For long-term agreements, gi

2—f. Long-Term Agreements with Renegotiation (i)

Consider (first) one-period long-term agreements

With no renegotiation, i.e., in the default outcome (f.ex., if therenegotiations fail), i’s interim utility is:

W dei = B(gdei + Ri

)− C

(qGG− +∑

N

gdej + θt

)+ δV .

If the countries renegotiate with side payments, then i can expect1/n of the renegotiation surplus.Thus, i’s expected utility at the start of the period is:

E

(W dei +

1n ∑j

(W rej −W dej

)− kri

), (26)

where W rej is j’s utility after renegotiation.


2—f. Long-Term Agreements with Renegotiation (ii)

Maximizing (26) w.r.t. ri gives the first-order condition:

k =[B ′(gdei + Ri

)+ δVR

] (1− 1

n

)(27)

+1nE∂

(∑N

W rej

)/∂R − ∑

j∈N\i

δVRn.

Note that VR drops out. The term E∑N W rej can be written as:

E max{g rei }i

∑N

B(g rej + Rj

)− C

(qGG− +∑

N

g rej + θt

)+ δV .

Since the right-hand side is concave in Ri , the second-order conditionof (27) holds. Furthermore, ∂

(∑N W rej

)/∂R is not a function of

gdei . Thus, if gdei increases, the right-hand side of (27) decreases and,

to restore equality, Ri must decrease.


2—f. Long-Term Agreements with Renegotiation (iii)

Thus, ri decreases in gdei .

First-best investments require that E∂ ∑N W rej /∂Ri ,t = K .Inserting that into (27), we get:

k =[B ′(gdei + Ri

)+ δVR

] (1− 1

n

)+1nK − ∑

j∈N\i

δVRn⇒

k −K/n = B ′(gdei + Ri

)(1− 1

n

)⇒

B ′(gdei + Ri

)= K + en.

Compared to the ex post optimal quotas, we have

B ′(gdei + Ri

)− EB ′ (g ∗i + Ri ) = en+ δqRK .


2—f. Renegotiation - Lessons

Proposition1 The emission quotas are always renegotiated to be ex post optimal2 Investments are larger if default quotas are smaller3 If quotas are suffi ciently small, investments are first best4 Compared to the ex post optimal quotas, the initial quotas should besmaller if e or δqR are positive/large.

5 Countries promise tough cuts in the future, but renegotiate themsubsequently

6 This procedure implements the first best.


Note Assumptions

Standard economic assumptions:

Each country acts as one player/individual (I abstract from "domestic"politics)Each country maximizes a well-specified objective function

Additional assumptions:

Symmetry: All countries are quite similarTime of investments and emissions alternatePollution cumulate over time and across countriesEverything is observableShort-term emissions are contractible


Important Contributions

R&D and Environmental Agreements

Aldy and Stavins (2005, 2007), Barrett (2005), Karp and Zhao (2009),Hong and Karp (2010), Golombek and Hoel (2005, 2006), Hoel and deZeeuw (2009)But: Contracts complete or absent, no dynamics

Dynamic (Differential) Games

Dockner, Jorgensen, Long and Sorger (2000), Friedman (1974), Ploegand de Zeeuw (1992), Fehrstman and Nitzan (1991), Sorger (1998),Dutta and Sundaram (1993), Dutta and Radner (2004, 2006, 2009)But: Neither R&D nor contracts (and multiple MPEs)

Contracts, Hold-up and Renegotiation Design

Hart and Moore (1988), Harris and Holmstrom (1987), Gatsios andKarp (1992)Aghion, Dewatripont and Rey (1994), Guriev and Kvasov (2002), Edlinand Hermalin (1996), Che and Hausch (1999)But: Typically 2x2 games


Motivation

Dynamic Games in Environmental Economics PhD minicourse ...€¦ · Bård Harstad (UiO) Dynamic Environment 5 December 2017 5 / 59. 2-a. Motivation Countries may be able to commit

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