Dynamic Games in Environmental Economics PhD minicourse Part II: - - PowerPoint PPT Presentation

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

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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)

• f resource or pollution.

Note that the stock may or may not be "payoff relevant"

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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

• f 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 σi (h) = σi (h) that are not contingent on h vs h. So, states/stocks that are not payoff-relevant should not matter.

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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

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2-a. Motivation

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

• r 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?

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2-a. Model: Timing

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2-a. A Model

A model with n + 1 stocks: Vi ≡ ∑

t

ui,tδt ui,t ≡ Bi (gi,t, Ri,t) − C (Gt) − k (ri,t) + e ∑

j=i

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

• 0, σ2

Continuation values, Vi (Gt−1, R1,t−1, ..., Rn,t−1) ,Wi (qG Gt−1 + θt, R1,t, ..., Rn,t)

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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=i

rj,t Gt = qG Gt−1 + ∑ yi,t − ∑

i

Ri,t + θt. With heterogeneity only in bliss points: B (yi,t − yi), write

• yi,t

≡ yi,t − (yi − y) , and y ≡ ∑ yi/n, to get: B (yi,t − yi) = B ( yi,t − y) ≡ B ( yi,t) and

• gi,t

≡

• yi,t − Ri,t = gi,t + (yi − y) , so ∑

gi,t = ∑ gi,t; Gt = qG Gt−1 + ∑ gi,t + θt = qG Gt−1 + ∑ yi,t − Rt + θt; ui,t =

• B (

yi,t) − C (Gt) − k (ri,t) + e ∑

j=i

rj,t. Remove "tildes" and interpret yi,t as consumption relative bliss.

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2-a. Reformulated model

Write continuation values as V (Gt−1, Rt−1) and W (qG Gt−1 + θt, Rt). As a third simplification, k (ri,t) = kri,t: ui,t ≡ B (yi,t) − C (Gt) − kri,t + e ∑

j=i

rj,t yi,t = gi,t + Ri,t Ri,t = qRRi,t−1 + ri,t, j ∈ {1...n}\i Gt = qG Gt−1 + ∑ yi,t − Rt + θt, i ∈ {1, 2, ..., n} Rt = ∑

i

Ri,t θt ∼ F

• 0, σ2

K ≡ k − (n − 1) e Example Q: B (.) = −b 2 (y − yi,t)2 , C (.) = c 2G 2

t

(Q)

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2-a. Simplifications and implications

Lemma 0: Markov strategies depend only on Gt−1 and Rt−1 ≡ ∑i Ri,t−1 So, same yi,t even if Ri,t differ! Ri,t and Rt is a "public good" regardless of e

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2-b. Business as usual - L1

Lemma 1-BAU: V B

R = qRK/n.

Proof: At the investment-stage, i solves max

ri,t EW (qG Gt−1 + θt, qRRt−1 + ∑ i

ri,t) − kri,t ⇒ EWR (qG Gt−1 + θt, Rt) = k ⇒ Rt (Gt−1) , so V B (Gt−1, Rt−1) = W (qG Gt−1 + θt, R (Gt−1)) −K n [Rt (Gt−1) − qRRt−1] ⇒ V B

R = qRK/n.

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

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2-b. Business as usual - L2

Lemma 2-BAU: V B

G = −qG (1 − δqR) K/n

Proof: At the emission stage, B (yi,t) − C qG Gt−1 + θt + ∑ yi,t − Rt + δVG (G, R) = 0 (1) So yi,t = yt is a function of ξt + θt where ξt ≡ qG Gt−1 − Rt, and so is Gt. Inserted, the foc for Rt comes from: max

ri,t E

• B
• yB (ξ)
• − C
• G B (ξ)
• + δV
• G B (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) + δVG G (ξ)

+ δVR = k. (3)

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

In the symmetric equilibrium: V (G, R) = EB (y (ξ)) − EC (G (ξ)) −k n [qG Gt−1 − ξ − qRRt−1] +e (n − 1) n [qG Gt−1 − ξ − qRRt−1] +δV (Gt (ξ) , qG Gt−1 − ξ) Taking the derivative gives the lemma.

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2-b. Business as usual - Observations

Since VG is a constant, (1) can be differentiated: y ∗ = dy ∗

i,t

dθt = −dy ∗

i,t

dRt = −C nC − B , and (4) G ∗

t

= dG ∗

t

dθt = −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)

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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: EB (yi) dyi dR 2 + B (yi) d2yi (dR)2 − C (G)

• ndyi

dR − 1 2 −

• C (G) − δVG
• n d2yi

(dR)2

• ≤ 0

When we substitute with (4) and differentiate it, we can get: EBC (C − B) (nC − B)2 − B (n − 1)

• (C )2 B − (B)2 C

(B − nC )3

• ≤ 0.

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2-b. Business as usual - Emissions

Proposition

The equilibrium consumption y B

i

is independent of Ri and suboptimally large, given R. From (1): B (yi,t) = C qG Gt−1 + θt + ∑ yi,t − Rt + qG (1 − δqR) K/n Country i pollutes less but j = i pollutes more if Ri is larger, fixing Rj∀j = i. From (4): ∂gno

i

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

j

/∂Ri = C nC − B > 0 ∀j = i, dG ∗

t

dRt = B nC − B < 0.

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2-b. Business as usual - Investments

Proposition

From the proof of Lemma 2-B, Rt = qG Gt−1 − ξB − qRRt−1, so: ∂rno

i

/∂R− = −qR/n, ∂rno

i

/∂G− = qG /n. If one country pollutes more, every country invests more in the next period If one country invests more, every other country invests less in the next period. There is a unique symmetric MPE.

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2-b. Business as usual - Transition

Proposition

If θt is large, every country pollutes less at t and t+1, invests more at t+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 = 1 qR ∂rt+2 ∂θt ,(7) where ρ ≡ C nC − B ∈ (0, 1) .

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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) = max

Rt EθW (qG Gt−1 + θt, Rt) − K

n (Rt − qRRt−1) ⇒ V ∗

R

= qRK/n. Note: Since VR is a constant, VGR = 0, and VG does not depend on R.

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2-b. First Best - L2

Lemma 2-FB: V ∗

G = −qG (1 − δqR) K/n

Proof: At the emission stage, B (yi,t) − nC qG Gt−1 + θt + ∑ yi,t − Rt + nδVG (G, R) = 0 (8) So yi,t = yt is a function of ξt + θt where ξt ≡ qG Gt−1 − Rt, and so is Gt. Inserted, the foc for Rt comes from: max

Rt Eθ [B (y ∗ (ξt + θt)) − C (G ∗ (ξt + θt)) + δV (G ∗ (ξt + θt) , R)

−K n (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)

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2-b. First Best - L2 - proof continued

With symmetric investments: V (G, R) = E [B (y ∗ (ξ∗ + θt)) − C (G ∗ (ξ∗ + θt))] −K n [qG Gt−1 − ξ∗ − qRRt−1] +δV (G ∗ (ξ∗) , qG Gt−1 − ξ∗) Taking the derivative gives the lemma.

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2-b. First Best - Observations

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

i,t

dθt ≡ −dy ∗

i,t

dRt = −C nC − B/n, and (10) G ∗

t

= dG ∗

t

dθt = −dG ∗

t

dRt = 1 + ny ∗ = −B/n nC − B/n. The foc for ri,t (9) becomes EθB (y ∗ (ξ + θt)) −y ∗ + δVR +

• C (G ∗ (ξ + θt)) − δVG

1 + ny ∗ = K n .

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2-b. First Best - Observations continued

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

• C − δVG

−ny ∗ +

• C − δVG

1 + ny ∗ + δVR

• =

K n Eθ

• C − δVG

+ δVR

• =

K n EθC = K/n − δVR + δVG = K n (1 − δqG ) (1 − δqR) . (11) And with B (.) = n (C − δVG ), we get: EB (.) = K (1 − δqG ) . (12)

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

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

t (qG Gt−1 − Rt + θt) > y ∗ t (qG Gt−1 − Rt + θt) .

(13) If either σ2 = 0 or Q, then we can compare (5)-(6) with (11)-(12): EyB

t

• qG Gt−1 − RB

t + θt

• <

Ey ∗

t (qG Gt−1 − R∗ t + θt) ,

EG B > EG ∗ ⇒ RB

t

< R∗

t .

Transition rule (7) continues to hold if ρ is replaced by the larger ρ∗ = C / (nC − B/n). So, BAU-transition is too slow and too much relying on investments. But yB reacts less to θ than does y ∗, while G B reacts more to θ than does G ∗. If θ < 0 very small, then it is possible that G B < G ∗. If θ > 0 very large, then it is possible that yB > y ∗. Note that the equilibrium play is independent of future regime.

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2-b. First Best - Q

With (Q), (12) becomes: Eb (y − yi,t) = K (1 − δqG ) ⇔ Eyi,t = y − K b (1 − δqG ) With (10), we can write: yi,t = Eyi,t − cnθt cn2 + b = y − K b (1 − δqG ) − cnθt cn2 + b. Similarly, we get: G ∗

t

= K cn (1 − δqG ) (1 − δqR) + bθt cn2 + b g ∗

i,t

= K cn2 (1 − δqG ) (1 − δqR) − qG Gt−1 − cnθt cn2 + b Ri,t = y − K b (1 − δqG ) − K cn2 (1 − δqG ) (1 − δqR) + qG Gt−1.

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2-b. Business as usual - Q

From (6), Q gives: Eb (y − yi,t) = nc + b c + b (k − δVR) . With (4), we get: yi,t = Eyt − cθt nc + b = y − 1 b nc + b c + b (k − δVR) − cθt nc + b. Furthermore, EcG = nc + b c + b (k − δVR) + δVG

∑ gi,t = 1

c nc + b c + b (k − δVR) + 1 c δVG − ncθt nc + b − qG Gt−1 Rt = ny − b + cn cb nc + b c + b (k − δVR) − 1 c δVG + qG Gt−1.

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

Proposition

1

In each case, VG and VR are the same. Thus, when analyzing this period, 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 pollute more and invest less next period.

4

If one pollutes less, everyone invests less and pollutes more next period.

5

Countries pollute too much and invest too little in BAU compared to FB.

6

The dynamic common pool problem (with strategic substitutes) is worse than its static counterpart.

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2-c. Short-Term Agreements

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2-c. Short-Term Agreements: Initial Observations

Negotiating the gi,ts is equivalent to negotiating the yi,ts at the negotiation/emission stage. At the negotiation stage (i.e., the emission stage), the countries are identical w.r.t. yi,t regardless of differences in Ri,ts. We should thus expect a symmetric equilibrium when it comes to yi,t = gi,t + Ri,t So, the more a country has invested, the smaller is the negotiated quota 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?

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

Lemma 1 and 2 hold with similar proofs as before: V ST

G

= −qG (1 − δqR) K/n and V st

R = 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) and G (ξ) are given by (10). This gives: EC (G) = k + δUG − δUR = (1 − δqG ) (1 − δqR) K/n + (K + en) ( 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.

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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, and compared to (12) yS < y ∗. Compared to (5) we have G S < G B, but compared to (11) we have G S > G ∗. Hence, RS

t < R∗ t .

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2-c. Short-Term Agreements: Q

Since Eb (.) = n (k − δVR) EcG = k − δVR + δVG and Eb (y − yi,t) = n (k − δVR) yi,t = y − 1 bn (k − δVR) − cθt nc + b/n Gt = k c − 1 c δVR + 1 c δVG + bθt/n nc + b/n

∑ gi,t

= k c − 1 c δVR + 1 c δVG + bθt/n nc + b/n − qG Gt−1 − θt Rt = ny − b + cn2 bc (k − δVR) − 1 c δVG + qG Gt−1.

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

Proposition

There is a unique symmetric MPE. The slopes of the continuation value, VG and VR, are as in BAU and FB. Emission levels are ex post optimal (given Rt), but investments are smaller than at the FB, and thus consumption is also smaller. Under Q, we can also see that compared to BAU, ST leads to less emissions but also less investments: Egst rst = Egbau rbau − n − 1 n (b + c)

• e (n − 1) +
• 1 − δqR

n

• K
• rst

i

= rbau

i

− (n − 1)2 n (b + c)

• e (n − 1) +
• 1 − δqR

n

• K
• .

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

Proposition

If investments are important, countries can thus be worse off with ST than in BAU. Under Q: uST < uBAU ⇔ (14)

• e + K

n 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.

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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 negotiations Countries invest less when the problem is expected to be solved in any case If investments are important, this makes the countries worse off ex ante (before the investment stage)

Agreements can be harmful ex ante - because they reduce incentives to invest. When the investments are sunk, then an agreement is always better than no agreement. If e is large, investments are already suboptimal so short-term agreements are then likely to be harmful. If σ is large, then y respond too little under BAU, and ST is more reassuring.

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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

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

Proof: When the level gco

i

is already committed to, the first-order condition for i’s investment is: k = B (gco

i

+ Ri) + δVR ⇒ (15) yco

i

= B−1 (k − δVR) , Rco

i

= B−1 (k − δVR) − gco

i ,

(16) rco

i

= B−1 (k − δVR) − gco

i

− qRRi,−. (17) If V co

R

is constant (confirmed below), the second-order condition is B ≤ 0, which holds by assumption. If the negotiations fail, the default outcome is the non-cooperative

• utcome, giving everyone the same utility.

Since the ris follow from the gis in (17), everyone understands that negotiating the gis is equivalent to negotiating the ris.

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

Since all countries have identical preferences w.r.t. the ris (and their default utility is the same) the ris are going to be equal for every i. Symmetry requires that ri, and thus ζ ≡ gi + qRRi,−, is the same for all countries. This implies that Rt = ∑i

• B−1 (k − δVR) − gco

i

= nB−1 (k − δVR) −EGt + qG Gt−1. Thus, we can write the value function as: V (G−, R−) = max

EGi B

• B−1 (k − δVR)

− EC (EGt + θt) −K n

• nB−1 (k − δVR) − EGt + qG Gt−1 − qRRi,−
• +EδV
• EGt + θt, nB−1 (k − δVR) − EGt + qG Gt−1
• .

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

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

With this, we can also take the derivative wrt Gt−1 to get VG = −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 , the pollution level is suboptimally small ex post. Combining (18) with (15), B (yi) /n − EC (G) + δVG = (k − δVR) /n − K/n + δVR = 1 n (k − K) + δqRK n

• 1 − 1

n

• .

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2-d. Long-Term Agreements - Lessons

Proposition

There is a unique MPE. The value function is linear with Lemma 1 and 2 as before. The smaller is the quota g t

i , the larger is ri,t: ∂ri,t/∂gi,t = −1 while

ri,t: ∂ri,t/∂gj,t = 0. Conditional on gt

i , every country invests too little if either δqR > 0 or

e > 0. The negotiated gi + qRRi,− is the same for every country. EC G T =EC (G ∗), even if investments are too low. Thus, quotas are too low ex post: B (yi) /n−EC (G) + δVG = 1

n (k − K) + δqR K n

• 1 − 1

n

> 0. If there is no uncertainty, gi,t (R) < g ∗

i,t (R) if either δqR > 0 or

e > 0.

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2—e. Long-Term Agreements: Multiple Periods

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2—e. Long-Term Agreements - Investments

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

i + Ri,t

• =

k − kδqR, t < T, (19) B glt

i + Ri,t

• =

k − KδqR/n, t = T. Investments are first-best for t < T iff e = 0, but they are smaller at t = T if δqR > 0. The smaller is the quota gi,t, the larger is ri,t: ∂ri,t/∂gi,t = −1 while ri,t: ∂ri,t/∂gj,t = 0. The quotas should be smaller than what is "ex post optimal" (particularly if duration short)

Bård Harstad (UiO) Dynamic Environment 5 December 2017 43 / 59

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

At the start of t = 1, countries negotiate emission levels for every period t ∈ {1, ..., T}. In equilibrium, all countries enjoy the same default utilities. Just as before, they will therefore negotiate the quotas such that the equilibrium 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 − qG Gt−1) ⇒ ri,t = (Rt − qRRt−1) /n, ∀t = B−1 (k (1 − δqR)) − E (Gt − qG Gt−1) /n (20) −qRB−1 (k (1 − δqR)) + qRE (Gt−1 − qG Gt−2) /n, t ∈ {2, .., T − 1} .

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

At the start of the first period, i’s continuation value can be written as: V = max

{EGt}t

1 − δT −1 1 − δ B

• B−1 (k (1 − δqR))
• (21)

−

T

∑

t=1

δt−1

• Kri,t + C
• EGt +

t

∑

t=1

θtqt−t

G

• +δT −1B
• B−1 (k − δqRK/n)
• +δT V
• EGT +

T

∑

t=1

θtqT −t

G

, RT

• ,

where ri,t is given by (20) and RT is given by (19). Lemma (1) VR = qRK/n. (2) VG = −qG K (1 − δqR) /n.

• Proof. Both parts follow from (21) by applying the envelope theorem.

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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 small conditional on equilibrium technology stocks. Using (19), B − EC (G) n + nδVG = (1 − δqR) (k − K) B glt

i + Ri

• − EB (g ∗

i + Ri)

= (1 − δqR) (n − 1) e In Example Q, this becomes simply glt

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

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2—e. Long-Term Agreements - Lessons

Proposition

Investments are suboptimally low at t < T if e > 0, and at t = T if either 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 post

• ptimal if e > 0 in every period, and in the last period if either e > 0
• r δqR > 0.

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

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

The optimal length T balances the cost of underinvestment when T is short and the cost of the uncertain θs is increasing in T. In period T, countries invest suboptimally, not only because of the domestic hold-up problem, but also because of the international one. When all countries invest less, ui declines. This cost of the hold-up problem in period T, relative to any period t < T, can be written as: H = −b 2 (yi,t − yi)2 − b 2 (yi,T − yi)2 − K (ri,t − ri,T ) (1 − δqR) = −b 2 k − δqRK b 2 + b 2 k − z b 2 − K k − z b − k − δqRK b

• =

δqR b

• e + K

n e

• 1 − δqR

2

• + δqRK

2n

• (n − 1)2 .

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

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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, Ec 2 (Gt)2 = Ec 2

• EGt +

t

∑

t=1

θtqt−t

G

2 = c 2 (EGt)2 + c 2σ2 1 − q2t

G

1 − q2

G

• .

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

c 2σ2δt−1 1 − q2t

G

1 − q2

G

• =

cσ2 2 (1 − q2

G ) T

∑

t=1

δt−1 1 − q2t

G

• =

cσ2 2 (1 − q2

G )

• 1 − δT

1 − δ − q2

G

• 1 − δT q2T

G

1 − δq2

G

• .

(22)

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

If all future agreements last T periods, then the optimal T for this agreement is: min

T L(T) +

• δT −1H + δT L
• T

∞

∑

τ=0

δτ

T

• ⇒

0 = L(T) + δT ln δ

• H/δ + L
• T
• /
• 1 − δ
• T

= L(T) + δT ln δ

• H/δ + L
• T
• /
• 1 − δ
• T

= −δT ln δ   

cσ2/2 1−q2

G

• 1

1−δ − q2T +2

G

(1+ln q2

G / ln δ)

1−δq2

G

• −

H/δ+L( T) 1−δ

T

   , (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, and there is a unique T minimizing the loss.

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2—e. The Optimal Duration: The optimal T

Substituting for T = T and (22) in (23) gives: H δ = cσ2q2

G

2 (1 − q2

G ) (1 − δq2 G )

• 1 − δT q2T

G

1 − δT − q2T

G

• 1 + ln
• q2

G

• ln δ
• ,

(24) where the r.h.s. increases in T. T = ∞ is optimal if the left-hand side of (24) is larger than the right-hand side even when T → ∞: cσ2q2

G

2 (1 − q2

G ) (1 − δq2 G ) ≤ H

δ . (25) If e and n are large but b is small, then H is large and (25) is more likely to hold. If (25) does not hold, the T satisfying (24) is larger. If c or σ2 are larger, (25) is less likely to hold and, if it does not, (24) requires T to decrease.

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2—e. The Optimal Duration: Lessons

Proposition

1

There is a hold-up problem (with under-investments) for every finite agreement.

2

It is optimal to reduce/delay this cost by increasing the duration of the 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 large while σ2 is small, for example.

5

Thus, with weak intellectual property rights, the climate treaty should last longer.

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2—f. Renegotiation and Updating of Agreements

For long-term agreements, gi <Eg ∗

i to encourage R&D

This is clearly not "renegotiation proof" With renegotiation, gi <Eg ∗

i cannot hold ex post

Harder to encourage R&D?

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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 the renegotiations fail), i’s interim utility is: W de

i

= B

• gde

i

+ Ri

• − C
• qG G− + ∑

N

gde

j

+ θt

• + δV .

If the countries renegotiate with side payments, then i can expect 1/n of the renegotiation surplus. Thus, i’s expected utility at the start of the period is: E

• W de

i

+ 1 n ∑

j

• W re

j

− W de

j

• − kri
• ,

(26) where W re

j

is j’s utility after renegotiation.

Bård Harstad (UiO) Dynamic Environment 5 December 2017 54 / 59

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

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

• B

gde

i

+ Ri

• + δVR

1 − 1 n

• (27)

+1 nE∂

• ∑

N

W re

j

• /∂R − ∑

j∈N\i

δVR n . Note that VR drops out. The term E∑N W re

j

can be written as: E max

{g re

i }i ∑

N

B

• gre

j + Rj

− C

• qG G− + ∑

N

gre

j + θt

• + δV .

Since the right-hand side is concave in Ri, the second-order condition

• f (27) holds. Furthermore, ∂
• ∑N W re

j

• /∂R is not a function of

gde

i . Thus, if gde i

increases, the right-hand side of (27) decreases and, to restore equality, Ri must decrease.

Bård Harstad (UiO) Dynamic Environment 5 December 2017 55 / 59

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

Thus, ri decreases in gde

i .

First-best investments require that E∂ ∑N W re

j /∂Ri,t = K.

Inserting that into (27), we get: k =

• B

gde

i

+ Ri

• + δVR

1 − 1 n

• + 1

nK − ∑

j∈N\i

δVR n ⇒ k − K/n = B gde

i

+ Ri 1 − 1 n

• ⇒

B gde

i

+ Ri

• = K + en.

Compared to the ex post optimal quotas, we have B gde

i

+ Ri

• − EB (g ∗

i + Ri) = en + δqRK.

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2—f. Renegotiation - Lessons

Proposition

1

The emission quotas are always renegotiated to be ex post optimal

2

Investments are larger if default quotas are smaller

3

If quotas are sufficiently small, investments are first best

4

Compared to the ex post optimal quotas, the initial quotas should be smaller if e or δqR are positive/large.

5

Countries promise tough cuts in the future, but renegotiate them subsequently

6

This procedure implements the first best.

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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 similar Time of investments and emissions alternate Pollution cumulate over time and across countries Everything is observable Short-term emissions are contractible

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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 de Zeeuw (2009) But: Contracts complete or absent, no dynamics

Dynamic (Differential) Games

Dockner, Jorgensen, Long and Sorger (2000), Friedman (1974), Ploeg and 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 and Karp (1992) Aghion, Dewatripont and Rey (1994), Guriev and Kvasov (2002), Edlin and Hermalin (1996), Che and Hausch (1999) But: Typically 2x2 games

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