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Page 1: Legally Binding Environmental Agreements - uio.no · Bård Harstad (UiO) Legally Binding April 2016 5 / 29. 2-b. Model: Assumptions Standard economic assumptions: Each country act

Legally Binding Environmental Agreements

Bård Harstad

UiO

April 2016

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Motivation

Most environmental problems are dynamic in nature

Pollution cumulates over time

Technological solutions are also relevant

Takes time to develop

Countries/district often act independently

Must study the game between them

Agreements may be legally binding

But agreements might be made on some aspects

...but not everything of interest

How should such an incomplete contract look like?

What will it be in equilibrium?

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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] 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"

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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 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

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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

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2-b. Model: Assumptions

Standard economic assumptions:

Each country act 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

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2-b. Model: Timing

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2-b. Model: Timing

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2-b. Model: Timing

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2-b. A Model

A model with n+ 1 stocks:

Ui ≡ ∑tui ,tδ

t

ui ,t ≡ B (gi ,t + Ri ,t )− C (Gt )− kri ,tRi ,t = qRRi ,t−1 + ri ,t , j ∈ {1...n}\iGt = qGGt−1 +∑

igi ,t + θt , i ∈ {1, 2, ..., n}

θt ∼ F(0, σ2

)Continuation values,

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

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2-b. Simplifications

Simplifying the model:

ui ,t ≡ B (yi ,t )− C (Gt )− kri ,tyi ,t = gi ,t + Ri ,tGt = qGGt−1 +∑ yi ,t − Rt + θt , i ∈ {1, 2, ..., n}Rt = ∑

iRi ,t

θt ∼ F(0, σ2

)Write continuation values as U(Gt−1,Rt−1) andW (qGGt−1 + θt ,Rt ).

Example Q:

B (.) = −b2(y − yi ,t )2 , C (.) =

c2G 2t (Q)

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2-b. Simplifications

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" even with no spillovers

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2-c. Business as usual - L1

Lemma 1-BU: UBR = qRk/n.Proof: At the investment-stage, i solves

maxri ,tEW (qGGt−1 + θt , qRRt−1 +∑

iri ,t )− kri ,t ⇒

EWR (Gt−1,Rt ) = k ⇒ Rt (Gt−1) , so

UB (Gt−1,R (Gt−1)) = W (Gt−1,R (Gt−1))

−kn[Rt (Gt−1)− qRRt−1] ⇒

UBR = qRk/n.

Note: Since UR is a constant, UGR = 0, and UG does not depend onR.‖

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2-c. Business as usual - L2

Lemma 2-BU: UBG = −qG (1− δqR ) k/nProof: At the emission stage,

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

)+ δUG (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)

)+ δU

(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) + δUGG

′ (ξ)]+ δUR = k.

(3)

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

In the symmetric equilibrium:

U (G ,R) = EB (y (ξ))− EC (G (ξ))

−kn[qGGt−1 − ξ − qRRt−1]

+δU (Gt (ξ) , qGGt−1 − ξ)

Taking the derivative gives the lemma.‖

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2-c. First Best

The first best will have the same UG and UR

(i) The first-best emission levels are functions of the technology stocks,R ≡ {R1, ...,Rn}:

g ∗i (R) = y i − Ri −cn (ny + qGG− + θ − R) + δqG (1− δqR ) k

b+ cn2. (4)

(ii) The symmetric first-best investments are:

r ∗i = y −qRnR− +

qGnG− − (1− δqR )

(1− δqGcn2

+1b

)k.

(iii) Combined, the first-best pollution stock is:

G ∗ = ∑i∈N

g ∗i (R∗) + qGG− =

(1− δqG ) (1− δqR )cn

k +b

b+ cn2θ. (5)

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2-c. BAU compared to FB

With business as usual, countries pollute too much and invest too little:

rbaui = y − qRnR− +

qGnG−

−[(b+ cn)2

cb (b+ c) n

(1− δqR

n

)k − (1− δqR )

δqGcn2

k

]< r ∗i ,

gbaui

(Rbau

)= y i − Ri −

c (ny + qGG− + θ − R) + δqG (1− δqR ) k/nb+ cn

> g ∗i(Rbau

)> g ∗i (R

∗) . (6)

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2-e. Short-Term Agreements

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2-c. Short-Term Agreements

With short-term agreements, countries pollute the optimal amount, giventhe stocks, but investments are suboptimally low:

r sti = r ∗i −(b+ cn2

bcn

)kn(n− 1) < r ∗i ,

g sti(Rst)= g ∗i

(Rst)> g ∗i (R

∗) .

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2-c. Short-Term Agreements

Compared to business as usual, short-term agreements lead to:(i) lower pollution,

Eg st(r st)= Egbau

(rbau

)− n− 1n (b+ c)

(1− δqR

n

)k;

(ii) lower investments,

r sti = rbaui − (n− 1)2

n (b+ c)

(1− δqR

n

)k;

(iii) lower utilities if the period is short, i.e., if(kn

)2(n− 1)2 > (1− δqR )

2 +(b+ c) (bcσ)2

(b+ cn2) (b+ cn)2. (7)

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

PropositionThere is a unique symmetric MPE.

The slopes of the continuation value, UG and UR , 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

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2-e. 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.

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2-f. Long-Term Agreements

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2-f. 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-f. Long-Term Agreements - Investments

Proposition

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.

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

Proof:When the level g coi is already committed to, the first-order condition fori’s investment is:

0 = B ′ (g coi + Ri ) + δUR − k ⇒ (8)

y coi = B ′−1 (k − δUR ) , Rcoi = B ′−1 (k − δUR )− g coi , (9)

r coi = B ′−1 (k − δUR )− g coi − qRRi ,−. (10)

If UcoR 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 (10), everyone understands thatnegotiating the gi s is equivalent to negotiating the ri s.

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

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 for allcountries.This implies thatRt = ∑i

(B ′−1 (k − δUR )− g coi

)= nB ′−1 (k − δUR )−EGt + qGGt−1.

Thus, we can write the value function as:

U (G−,R−) = maxEGi

B(B ′−1 (k − δUR )

)− EC (EGt + θt )

−kn

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

)+EδU

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

).

Note that the Envelope theorem gives UR− = qRk/n.

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

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

EC ′ (G ) = k/n− δUR + δUG = (1− δqG ) (1− δqR ) k/n. (11)

This is the same pollution level as in the first best (??).Since investments are suboptimally low, the pollution level aresuboptimally small ex post.Combining (11) with (8),

B ′ (yi ) /n− EC ′ (G ) + δUG = (k − δUR ) /n− k/n+ δUR

=δqRkn

(1− 1

n

).

gi ,t = Eg ∗i(Rlt)− δqRk

n

(n− 1b+ cn2

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2-f. 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 ) + δUG = 1

n (k − k) +δqR kn

(1− 1

n

)> 0.

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

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