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Royal Economic Society

Endogenous growth, convexity of damages and climate risk: how Nordhaus’ framework supports deep cuts in carbon emissions

Simon Dietz and Nicholas Stern London School of Economics

RES Manchester 2015

Gross underestimation of risk

• “There are very strong grounds for arguing

that [IAMs] grossly underestimate the risks of climate change”

• 1a. Underlying exogenous drivers of growth (in one-

good models)

• 1b. Damage functions that only work on annual
• utput
• 2. Quantitatively weak damage functions
• 3. Very limited distributions of risk

An illustration from Nordhaus’ DICE

• Production, without climate change
• ,
• Production with climate change
• 1
• The damage multiplier
• 1/1
• where π1 = 0 and π2 ≈ 0.003
• Temperature is a complex function of emissions,

climate sensitivity = 3degC

Examining the proposition

• In this paper we build on DICE by including
• 1a. Endogenous growth
• 1b. Climate damages to drivers of growth, i.e. capital

stocks or TFP

• 2. Strong convexity in the damage function
• 3. Large climate risk via climate sensitivity

parameter

Endogenous growth (i)

• A model of capital damages and learning by

investing (Arrow-Romer)

– Production

• 1
• – Capital accumulation

1

• 1
• – Partitioning of damages
• 1 1

1

Endogenous growth (ii)

• A model of endogenous TFP and damages to TFP

– Production

• 1

̅

• – Capital and TFP dynamics

1

• ̅ 1

1

̅ !

• "#

– Partitioning of damages

• 1 1

1

Convexity of damages

• Damage function suggested by Weitzman

(JPET, 2012) 1 1/1

$ %.'()

• Three scenarios:
• 1. Set π3 = 0, i.e. standard DICE (‘Quadratic’)
• 2. Set π3 so that D = 0.5 at T = 6 (‘Weitzman’)
• 3. Set π3 so that D = 0.5 at T = 4 (‘High’)

Climate sensitivity distribution

• 1

1 2 3 4 5 6 7 8

Source: own fit

• f IPCC AR5

WG1, SPM

Results: baseline growth

Results: optimal emissions cuts

2015 2055 2105 Standard 16% 27% 45% Quadratic damage 26% 59% 100% Weitzman damage 31% 74% 100% High damage 48% 100% 100% n.b. TFP model, random S

Results: optimal carbon prices

2015 2055 2105 Standard 44$/tC 106 237 Quadratic damage 110 435 1012 Weitzman damage 147 657 1012 High damage 329 1121 1012 n.b. TFP model, random S

Results: optimal atmospheric CO2

Conclusions

• Main aim was to explore assumptions

necessary and sufficient to sustain deep emissions cuts at the optimum

• Examine proposition that standard DICE

grossly underestimates climate risks to economy

• Shows deep cuts are optimal even if discount

rate is high

Supplementary slides

Results: baseline atmospheric CO2

Results: baseline temperature

Royal Economic Society

ROYAL ECONOMIC SOCIETY MANCHESTER, 2015 RICK VAN DER PLOEG OXFORD UNIVERSITY (WITH AART DE ZEEUW)

CLIMATE TIPPING AND ECONOMIC GROWTH

How to model catastrophes?

 Real possibility that a discontinuous change in damages or in

carbon cycle will take place. This change can be abrupt as with shifts in monsoonal systems, but loss of ice sheets resulting in higher sea levels have slow onsets and can take millennium or more to have its full effect (Greenland 7m and Western Antarctica 3m, say) and may already be occurring.

 9 big catastrophes are waiting to happen, not all at same time.  Collapse of the Atlantic thermohaline circulation is fairly

imminent and might occur at relatively low levels of global

• warming. This affects regions differently, but we capture this

with a negative TFP shock.

 We look at TFP calamity and also at K, P and climate sensitivity

• calamities. Expected time of calamity falls with global warming.

Possible Tipping Points Duration before effect is fully realized (in years) Additional Warming by 2100 0.5-1.5 C 1.5- 3.0C 3-5 C Reorganization of Atlantic Meridional Overturning Circulation about 100 0-18% 6-39% 18- 67% Greenland Ice Sheet collapse at least 300 8-39% 33- 73% 67- 96% West Antarctic Ice Sheet collapse at least 300 5-41% 10- 63% 33- 88% Dieback of Amazon rainforest about 50 2-46% 14- 84% 41- 94% Strengthening of El Niño-Southern Oscillation about 100 1-13% 6-32% 19- 49% Dieback of boreal forests about 50 13-43% 20- 81% 34- 91% Shift in Indian Summer Monsoon about 1 Not formally assessed Release of methane from melting permafrost Less than 100 Not formally assessed.

Probabilities of Various Tipping Points from Expert Elicitation

Previous work

 Gollier (2012): Markov 2-regime switching model &

exogenous risk of big drop in GDP growth  much higher SCC.

 Threat of doomsday scenario: Bommier et al. (2013).  Regime shifts with uncertain arrival of catastrophe:

 Partial equilibrium: Tsur & Zemel (1996), Karp & Tsur (2011),

Naevdal (2006), Polasky, de Zeeuw & Wagener (2011).

 General equilibrium: Lemoine and Traeger (2014) use Ramsey

model to understand effect of release of permafrost as instantaneous doubling of ECS and of learning and multiple

• catastrophes. Cai, Judd and Lontzeck (2015) similar and focus
• n shock to damage function and numerical challenge.

Messages

 Chance of catastrophe can lead to much higher SCC

without an extremely low discount rate provided hazard rises sharply with temperature. The motive is to avert risk.

 There is also a social benefit of capital (SBC) which gives a

rationale for precautionary capital accumulation and being better prepared.

 Calibrate a global IAM with Ramsey growth with both

catastrophic and marginal climate damages.

 Show role of convexity of the hazard function.  Show effect of more intergenerational inequality aversion

and thus more risk aversion on SCC and SBC: i.e., on carbon tax and capital subsidy.

Climate disaster and Ramsey growth

 Concave time-separable utility function.  Concave and CRTS production function.  Factors of production: capital K, labour, fossil fuel

and renewables. All factors are imperfect substitutes.

 Fossil fuel E is abundant at cost d .  Supply of renewable R is infinitely elastic at cost c.  Extremely simple carbon cycle: nothing stays up

permanently in the atmosphere, constant decay rate.

 Hazard of catastrophic drop in TFP is H(P) and is

modelled with Poisson process with H  0

Climate disaster and Ramsey growth

, ,

max E ( ( )) subject to ( ) ( ( ), ( ), ( )) ( ) ( ) ( ) ( ), 0, (0) , ( ) ( ) ( ), 0, (0)

t C E R

e U C t dt K t AF K t E t R t dE t cR t C t K t t K K P t E t P t t P



  

 

                    



, ( ) , , ( ) (1 ) , , 1, Pr[ ] 1 exp ( ( )) , 0.

t

P A t A t T A t A t T T t H P s ds t                         



Backward induction

 For time being, damages only result from calamities.  Solve post-catastrophe problem as standard Ramsey

problem to give post-calamity value function:

 Solve before-catastrophe problem from the HJB:

( , ).

A

V K 



 



, ,

'( ) ( , ),

( , ) Max ( ) ( , ) ( , , ) ( , )( ) with optimality conditions ( , , ) , ( , ) / ( , ) 0, ( ( , ) ( , ) ( , ) , ) .

B B K

B C E R B B K P B B E R A B P K

U C V K P

V K P U C V K P AF K E R dE cR C K V K P E P AF K E R d V K P V H P V K V K K P AF K E P R c        



                   

Precautionary saving and curbing risk of calamity

 The Euler equation has a precautionary return  or

social benefit of capital (SBC):

 The SCC is:

1/

( , , , ) with ( , ) ( ) 1 ( ) 1 0. '( )

B K A B K A

C Y K d c A C V K C H P H P U C C



                                      

 

 

 

 

 

 

 

 

 

 

 

' ( ) ( ') ( ') ( ') ' ' ( ) ( ') '

( ) ( ) ( ) exp ' ( ) ( ) ( ) exp / ' ( ) .

B A s B t s t

t B A B t

V V H P s r s s H P s ds U C s H P s ds

s s t ds H P s V s V s ds U C t

    

 

 

       

      

 

 

Interpretation

 ‘Doomsday’ scenario has VA = 0, so the discount rate

is increased  frantic consumption and less

• investment. Mr. Bean!

 But if world goes on after disaster, precaution is

• needed. Since consumption will fall after disaster,

SBC > 0 and the discount rate is reduced. This calls for precautionary capital accumulation (if necessary internalized via a capital subsidy)

 The SBC is bigger if the hazard and size of the

disaster are bigger. And if intergenerational inequality aversion (CRIIA) is bigger.

Illustrative calibration of hazard function

 Use H(826) = 0.025 and H(1252) = 0.067  So doubling carbon stock (rise in temperature with 3 degrees) brings

forward expected time of calamity from 40 to 15 years.

After-disaster, naïve and before-disaster steady states

After disaster Naive solution Constant hazard h = 0.25 Linear hazard Quadratic hazard EIS = 0.8 Capital stock (T $)

276 392 472 530 486 436

Consumption (T $)

41.3 58.6 59.4 59.6 59.2 58.9

Fossil fuel use (GtC/year)

7.3 10.4 11.0 9.7 7.7 7.7

Renewable use (million GBTU/year)

8.2 11.7 12.4 12.7 12.2 11.8

Carbon stock (GtC)

1218 1731 1838 1623 1281 1279

Precautionary return (%/year)

0.76 1.24 0.99 0.57

SCC ($/tCO2)

22.4 56.9 51.0

Precautionary capital can be negative if hazard function is very convex

 Steady-state pre-disaster K is bigger than naive K iff:  This is always so if hazard constant and SCC zero.  With convex enough hazard function effects of SCC can

• utweigh effect of SBC, so inequality need not hold.

 With quartic SCC is very high and SBC very low. The high

carbon tax averts disaster so much that there is less need for precautionary capital accumulation. Put differently, precautionary capital is bad as it induces more fossil fuel use, more global warming and a relatively big increase in hazard of climate disaster. So avoid Green Paradox.

1 * *

.

B B

d d

 

     



                

Role of intergenerational inequality aversion

 Much debate is about discount rate but CRIIA = 1/

is at least as important.

 Higher  or lower CRRA and CRIIA has two effects:

 Lower CRRA, so lower SBC, less precautionary saving and thus

less fossil fuel demand and emissions. Need lower carbon tax

 Lower CRIIA so more prepared to sacrifice consumption and

have a higher carbon tax.

 With  = 0.8 and linear hazard first effect

dominates: lower SCB and lower SCC so less capital before disaster and less sacrificing of consumption.

Gradual damages A(Temp) and the SCC

 Before-disaster SCC has in general 3 components:

     

 

     

 

( ( ') conventional Pigouvian social cost of carbon ( ( ') 'raising the stakes' effec

( )

' ( ) ( ) ' ( ) ' ( ) ( ) ( ) , , ( ) ' ( )

s t s t

H P s ds t H P s ds A P t

t

A P s F s U C s e ds U C t H P s V K s P s e ds U C t

   



 

       



       

 

 

   



 

t ( ( ') 'risk averting' effect

, 0 .

' ( ) ( ) ( ) ' ( )

s t

B A

H P s ds t

V V t T

H P s s s e ds U C t

 



   

  







Catastrophic and marginal climate damages

Naïve solution 20% shock in TFP 10% shock in TFP after shock linear quadratic after shock linear quadratic Capital stock (T $) 378 271 492 465 323 431 421 Consumption (T $) 57.1 40.8 58.3 58.2 48.7 57.8 57.8 Carbon stock (GtC) 1502 1107 1287 1161 1303 1425 1320

Temperature (degrees Celsius)

4.00 2.68 3.33 2.88 3.38 3.77 3.44 Precautionary return (%/year) 1.10 0.90 0.57 0.49 SCC ($/GtCO2) 15.4 11.0 54.8 71.2 13.2 29.8 41.5 marginal 15.4 11.0 4.3 5.7 13.2 3.8 4.7 risk averting 35.0 51.9 12.4 24.2 raising stakes 15.4 13.7 13.7 12.5

Effect of climate sensitivity on damages

Carbon and capital catastrophes

Naïve solution CS jumps from 3 to 4 20% drop in P 20% drop in K After calamity Before calamity Capital stock (T $) 379 372 382 381 433 Consumption (T $) 57.1 56.3 57.3 57.1 57.6 Carbon stock (GtC) 1503 1374 1400 1490 1534 Temperature (degrees Celsius) 4.00 4.82 3.69 3.96 4.09 Precautionary return (%/year) 0.05 0.03 0.57 SCC ($/GtCO2) 15.5 26.7 26.5 16.9 18.5 Marginal 15.5 26.7 4.1 3.8 3.8 risk averting 2.2 1.4 2.5 raising stakes 20.2 11.7 12.2

Conclusions

 Small risks of climate disasters may lead to a much

bigger SCC even with usual discount rates. Rationale is to avoid risk.

 Also need for precautionary capital accumulation.  Need estimates of current risks of catastrophe and

how these increase with temperature.

 Recoverable shocks such as P or K calamities are

less problematic.

 Catastrophic changes in system dynamics

unleashing positive feedback may be much more dangerous than TFP calamities.

Extension: North-South perspective

 Carbon taxes and capital stocks  Non-coop bias in carbon tax, not in precautionary

return on capital

Regime shifts

Other extensions

 Adaptation capital (sea walls, storm surge barriers) increases with

global warming: trade-off with productive capital.

 Positive feedback in the carbon cycle changes carbon cycle dynamocs

(e.g., Greenland or West Antarctica ice sheet collapse).

 Multiple tipping points with different hazard functions and lags (Cai,

Judd, Lontzek). ‘Strange’ cost-benefit analysis (Pindyck).

 Learning about probabilities of tipping points, but also about whether

they exist all (cf. ‘email-problem’). How to respond to a tipping point which may never materialize?

 Second-best issues: Green Paradox can lead to ‘runaway’ global

warming if the system is tipped due to more rapid depletion of oil, gas and coal in face of a future tightening of climate policy (Ralph Winter, JEEM, 2014).

Pure rate of time preference,  0.014 Elasticity of intertemporal substitution,  0.5 (and 0.8) Share of capital in value added,  0.3 Share of fossil fuel (oil, gas, coal) in value added,   0.0626 Share of fossil fuel in total energy,  0.9614 Share of energy in value added,  0.0651 Share of labour in value added, 1     0.6349 Depreciation rate of manmade capital,  0.05 Initial level of GDP, Y0 63 trillion US $ Initial capital stock, K0 200 trillion US $ Initial fossil fuel use, E0 468.3 million G BTU = 8.3 GtC Initial renewable use, R0 9.4 million G BTU Total factor productivity, A 11.9762 Cost of fossil fuel, d 9 US $/million BTU = 504 US $/tC Cost of renewable, c 18 US $/million BTU Initial stock of carbon, P0 826 GtC = 388 ppm by vol. CO2 Pre-industrial carbon stock 596.4 GtC = 280 ppm by vol. CO2 Fraction of carbon that stays up in atmosphere,  0.5 Eventual climate shock,  0.2 (and 0.1) Equilibrium climate sensitivity 3 (and 4)

Royal Economic Society

Why finance ministers favor carbon taxes, even if they do not take climate change into account

Ottmar Edenhofer, Max Franks, Kai Lessmann 01.04.2015

MOTIVATION MODEL SETUP RESULTS

The climate problem at a glance

Resources and reserves to remain underground:

• 80 % coal
• 40 % gas
• 40 % oil

The Globalisation Paradox: A Trilemma

The Globalisation Paradox: A Trilemma

The Globalisation Paradox: A Trilemma

Resource rents as solution

Research questions

• Most economists agree on carbon pricing to address the

climate externalty, many prefer taxes.

Research questions

• Most economists agree on carbon pricing to address the

climate externalty, many prefer taxes.

• What is the role of a carbon tax under the assumption that no

climate externality exists?

Research questions

• Most economists agree on carbon pricing to address the

climate externalty, many prefer taxes.

• What is the role of a carbon tax under the assumption that no

climate externality exists?

• Can carbon taxes finance infrastructure more efficiently than

capital taxes when input factors are mobile?

Research questions

• Most economists agree on carbon pricing to address the

climate externalty, many prefer taxes.

• What is the role of a carbon tax under the assumption that no

climate externality exists?

• Can carbon taxes finance infrastructure more efficiently than

capital taxes when input factors are mobile?

• What are the supply side dynamics when resource importing

countries tax carbon?

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom.

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom. Carbon tax captures part of the Hotelling rent.

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom. Carbon tax captures part of the Hotelling rent.

• 2. No green paradox:

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom. Carbon tax captures part of the Hotelling rent.

• 2. No green paradox:

Demand side fully determines extraction.

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom. Carbon tax captures part of the Hotelling rent.

• 2. No green paradox:

Demand side fully determines extraction. Carbon taxes postpone extraction,

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom. Carbon tax captures part of the Hotelling rent.

• 2. No green paradox:

Demand side fully determines extraction. Carbon taxes postpone extraction, and reduce cumulative emissions.

Results

• 1. In Nash equilibrium, carbon tax more efficient than capital tax.

Both taxes subject to race to the bottom. Carbon tax captures part of the Hotelling rent.

• 2. No green paradox:

Demand side fully determines extraction. Carbon taxes postpone extraction, and reduce cumulative emissions.

• 3. Both results are robust under different strategic settings:

(Non-)cooperative importers, (non-)strategic exporter.

MOTIVATION MODEL SETUP RESULTS

Household: max

• t=0

U(Ct/Lt) (1 + ρ)t , Ct(1 + τC,t) = wtLt + rtKt − It + ΠF

Firm: max

Household: max

• t=0

U(Ct/Lt) (1 + ρ)t , Ct(1 + τC,t) = wtLt + rtKt − It + ΠF

Firm: max

= ⇒ FK = r(1 + τK), FR = p + τR, FL = w(1 + τL) Household: max

• t=0

U(Ct/Lt) (1 + ρ)t , Ct(1 + τC,t) = wtLt + rtKt − It + ΠF

Government: max

• t=0

Lt U(C/L) (1 + ρ)t , ζ ∈ {K, R, C, L} I G + Taxtransfer = rτKK + τRR + τCC + wτLL Gt+1 = Gt(1 − δ) + I G

Firm: max

= ⇒ FK = r(1 + τK), FR = p + τR, FL = w(1 + τL) Household: max

• t=0

U(Ct/Lt) (1 + ρ)t , Ct(1 + τC,t) = wtLt + rtKt − It + ΠF

Resource exporter: Resource market: max

• t=0

ptRt − ct t

Rsupply =

• j

Rdemand

p = pj ∀j

Resource exporter: Resource market: Capital market: max

• t=0

ptRt − ct t

Rsupply =

• j

Rdemand

p = pj ∀j

• j

K supply

=

• j

K demand

r = rj ∀j

Nash equilibrium, two sub-games,

Nash equilibrium, two sub-games,

Nash equilibrium, two sub-games,

Nash equilibrium, two sub-games,

Nash equilibrium, two sub-games, solved for

non-cooperative behavior max

Wi, given τ j

• r

Nash equilibrium, two sub-games, solved for

non-cooperative behavior max

Wi, given τ j

• r

cooperative behavior of governments max

W1 + W2

MOTIVATION MODEL SETUP RESULTS

MOTIVATION MODEL SETUP RESULTS

• Numerical solution due to high complexity (dual game structure,

intertemporal optimization, two international markets, etc.)

• Calibration: Two symmetric countries to avoid that results are driven by

asymmetries.

• Flexibility of modelling framework also allows for calibration to setups

with specific regions (e.g. USA, EU, Australia, and OPEC).

Single instrument portfolio

Single instrument portfolio

Mixed portfolio

Timing and volume effects

Timing and volume effects

No green paradox: Demand for infrastructure fully determines supply side dynamics

The optimal financing of infrastructure with a carbon tax from an importing government’s perspective implies

< rt − δ. Thus, extraction is postponed (see, e.g., Edenhofer and Kalkuhl, 2011).

Assumptions about strategic behavior of exporter

• Portfolios, which include the carbon tax τR yield higher NPV
• f consumption in importing countries.
• This finding is independent of whether the exporter may

interact strategically or not.

Summary of results

• 1. Carbon tax more efficient than capital tax.

Summary of results

• 1. Carbon tax more efficient than capital tax.

asymmetry between capital and carbon as tax base,

Summary of results

• 1. Carbon tax more efficient than capital tax.

asymmetry between capital and carbon as tax base,

• nly the resource stock gives rise to rent.

Summary of results

• 1. Carbon tax more efficient than capital tax.

asymmetry between capital and carbon as tax base,

• nly the resource stock gives rise to rent.
• 2. Carbon tax delays extraction, reduces cumulative emissions.

Timing of infrastructure demand fully determines supply side dynamics.

Summary of results

• 1. Carbon tax more efficient than capital tax.

asymmetry between capital and carbon as tax base,

• nly the resource stock gives rise to rent.
• 2. Carbon tax delays extraction, reduces cumulative emissions.

Timing of infrastructure demand fully determines supply side dynamics.

• 3. Results are robust under different sorts of strategic behavior:

Cooperating importers, strategic exporter.

Policy conclusions

• Carbon pricing can help to mitigate the race to the bottom.

Policy conclusions

• Carbon pricing can help to mitigate the race to the bottom.
• The supply side dynamics of carbon pricing matter, but pose

no environmental problem.

Policy conclusions

• Carbon pricing can help to mitigate the race to the bottom.
• The supply side dynamics of carbon pricing matter, but pose

no environmental problem.

• Rethink role of environmental policy:

Not only environmental ministers should favor carbon pricing, but also finance ministers.

Backup slides

There is far more carbon in the ground than emitted in any basline scenario

The scarcity rent of CO2 emissions

• Fossil fuel rents decrease with the

ambition of climate policy.

The scarcity rent of CO2 emissions

• Fossil fuel rents decrease with the

ambition of climate policy.

• If the optimal CO2 price is

implemented globally, this loss is

• vercompensated by the carbon

rent.

The scarcity rent of CO2 emissions

• Fossil fuel rents decrease with the

ambition of climate policy.

• If the optimal CO2 price is

implemented globally, this loss is

• vercompensated by the carbon

rent.

• The revenues of the carbon tax or

auctioning of emission permits can be used to finance tax reductions, infrastructure investments, or debt reduction.

Volume effects under behavioral assumptions

The resource rent

Welfare evaluation

Welfare evaluation

No problem with time inconsistency

No problem with time inconsistency

If taxing carbon is so good, why do we not see more

• f it in reality?
• 1. In the past: ignorance on the part of policy-makers. Today not true

anymore in many places.

• 2. Practical problems, caused e.g. by spacially differentiated taxes, complex

trading rules for non-uniformly mixes pollutants, etc...

• 3. Institutional problems:

Cost-effectiveness ranked lower in regulators list of multiple policy

• bjectives.

Ethical implications: Tax debases notion of environmental quality (Kelman, 1981); emission permits as ’right to pollute’.

• 4. Resistance from those with vested interest in preservation of existing

system. ’... all of the main parties involved [have] reasons to favor [command-and-control policies]: firms, environmental advocacy groups,

• rganized labor legislators and bureaucrats’ (Stavins, 1998, p.72).

Source: Hanley et al. (2007)

Why might public spending be too low? How can additional revenues from climate policy enhance welfare?

• 1. Weak institutions (non-OECD).
• 2. Existing allocation of public funds inefficient. New revenues

from climate policy free to allocate.

• 3. Myopia towards projects with long term benefits. Climate

policy might supply both funds and political momentum to implement such projects.

• 4. If in contrast projects with long term benefits were realized,

there might be a lack of fiscal tools to finance high up-front costs, e.g. political debt-limit.

Source: Siegmeier et al. (2015)

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

• Governments engage in Nash game using policy instruments:

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

• Governments engage in Nash game using policy instruments:
• Repeat...

for each player j

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

• Governments engage in Nash game using policy instruments:
• Repeat...

for each player j

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

• Governments engage in Nash game using policy instruments:
• Repeat...

for each player j

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

• Governments engage in Nash game using policy instruments:
• Repeat...

for each player j

Model setup - solution algorithm

• Households, firms and the resource owner are Stackelberg

followers of governments.

• Governments engage in Nash game using policy instruments:
• Repeat...

for each player j

• ...until policy instruments converge.

Numerical Model: Details

CES production function

CES production function F(K, G, R, L) = (α1Rs1 + (1 − α1)Z s1)

Z(K, G, L) = (α2X s2 + (1 − α2)Ls2)

X(K, G) = (α3K s3 + (1 − α3)G s3)

CiES social welfare function W =

• t

Lt (Ct/Lt)1−η 1 − η 1 (1 + ρ)t Parameter values σ1 α1 σ2 α2 σ3 α3 η ρ 0.5 0.1 0.7 0.42 1.1 0.65 1.1 0.03 , si = σi−1

σi

Intertemporal optimization: Household

max

• t=0

U(C/L) (1 + ρ)t , s.t. C(1 + τC) = wL + rK s + ΠF + Taxtransfer − I It = K s

taking ΠF

as given. Use discrete Maximum Principle with Hamiltonian: HHH

= U(Ct/Lt) + λt

• (1 + (rt − δ)) K s

... + ΠF

• FOCs and TC:

Lη−1

/C η

λt−1(1 + ρ) = λt (1 + rt(1 + τC,t) − δ) , 0 = (IT − (1 − δ)K s

Intertemporal optimization: Resource exporter

max

• t=0

(pt − ct − τRO,t)Rt + Ψt t

, ct(St) = rt

• 1 + χ2

χ1 ((S0 − St)/S0)χ3

• subject to
• t

Rt ≤ S0 where Rt = St − St+1, S0 is given, and Ψt = τRO,tRt is taken as given. Hamiltonian: HRO

= (pt − ct − τRO,t) Rt + λR(St − Rt) + Ψt, FOCs and TC: λR

λR

χ1S0 S0 − St S0 χ3−1 , λR

Intertemporal optimization: Government

max

τ

W =

T

• t=0

Lt U(Ct/Lt) (1 + ρ)t subject to I G + Taxtransfer = rτKK + τRR + τCC + wτLL Gt+1 = Gt + I G

t − δGt

and

• the international market clearing conditions,
• the maximization problems of households, firms, and the

resource exporter,

• their respective FOCs and TCs

Some parameter values

References (general)

• f Capital Supply, 2012, American Economic Review

Royal Economic Society