Regulation and State Intervention in Non-Renewable Natural

Resources: the cases of the oil and gas sectors in Argentina�.

Enrique Kawamura

Universidad de San Andrés

April 1, 2014

Abstract

This paper develops a theoretical model based on stylized facts regarding regulation and perfor-

mance of the oil and gas sectors in Argentina after the fall of Convertibility. Regarding performance,

facts include an important drop in domestic production and investment, as well as a reversal in the

energy trade balance (from surplus to de�cit). Regulations include a direct government intervention

in upstream and downstream prices (below international ones), together with strong restrictions

to export. The model uses a self-enforcing equilibrium concept derived from Yared (2010) to endo-

geneize such policy variables such as upstream domestic prices and downstream energy prices. The

model can explain some of the regulation policies observed in the facts, including very low energy

prices and strictly positive subsidies.

1 Introduction

Beyond its well-known comparative advantage in agricultural resources, Argentina is also a relatively

important producer of some non-renewable resources, mainly, oil, gas and some metallic minerals (espe-

cially copper). Indeed, Argentina has been the third largest oil producer in South America in the early

1990s (after Venezuela and Brazil), although with a declining trend after 20001. Argentina became a net

�This paper is part of the IDB-funded research project "Fiscal Revenues and E¢ ciency in LAC Non-Renewable NaturalResource Sectors". I deeply thank the comments that Fernando Navajas gave to the di¤erent stages of this paper. I alsothank the coordinators, Andy Powell and Osmel Manzano, for their comments. The usual disclaimer applies.

1For example, by 1996 Argentina�s oil production represented a 16.4% of total oil production of South American coun-tries (and 27.2% when excluding Venezuela from the sample), while ten years later Argentina�s oil production representedonly 10.4% of total South-American oil production, according to the site of BP, http://www.bp.com.

1

exporter of crude-oil. Even though the net export of oil also decreased in the last decade, the country

maintains its position in this sector. Similarly, natural gas exploitation also evidenced an important

increase in Argentina. In 1990 Argentina�s natural gas production was 23:02 m3 billions, while in 2004

the production was 52:38 m3 billions. Argentina was also an important net exporter of gas until 2006.

The two non-renewable resources referred above belong to the group of those related to energy. This

feature may make these two sectors strong candidates to su¤er distorting interventions by the State

given that energy plays a key role in electoral competitions. For example, although Argentina was an

important net exporter of gas, beginning 2006 this feature reverted to negative foreign trade balances.

The main conjecture of why this happened is linked to a complex regulation system implemented by

the Administration who took power in 2003 in Argentina. Also, during the 2002 crisis the Argentine

government decided to introduce a tax on oil exports, a tax that still exists nowadays. Note that this

year coincides with the starting point of the increase in the international price of crude oil. These

two cases are just examples of the type of interaction between production and investment decisions by

the producers and the government as the major regulation authority. These interventions impact on

production and investment decisions. But, more importantly, such tax and regulation changes are likely

to be endogenous to the changes in the international and macroeconomic environment, changes that

impacts on the policy-maker incentives, which typically leads to time-inconsistency problems regarding

pre-committed tax or other regulations set ex-ante.

This paper presents a brief summary of major stylized facts about, not only production and in-

vestment in the oil and gas sectors in Argentina after the crisis, but also a description on changes in

regulation introduced by the National government during the period 2002-2012. In this period oil pro-

duction decreased about 25%, gas production decreased 15% between 2004 and 2012 and exploration

activity decreased until 2010. Resource (oil and gas) prices as well as energy prices were completely

controlled by the government. Both were kept way below their corresponding international prices. At

the same time, natural gas consumption heavily increased (about 48%). This combination of decline

in production, increase in domestic energy demand and domestic low prices implied an explosion in

the trade de�cit, especially in natural gas, whose physical volume reached values even higher than the

maximum trade surplus reached in 2004. As Navajas (2006) and other authors state, such regulation can

hardly be reconciled with standard theories regarding dynamic regulations or taxation in non-renewable

resources.

Given such facts, the paper proceeds to construct and develop a stochastic, intertemporal model of

a non-exhaustible resource with price regulations and quantity restrictions imposed by the state, whose

values arise endogenously in equilibrium. Embedded in a well-known model of non-renewable-resource

taxation, such as Deacon (1993), the framework considers explicit (incumbent) politicians who have

2

the capacity to set the values of such policies, but that could be removed from o¢ ce by some electoral

process. The equilibrium concept used for this purpose is an adaptation of the self-enforcing equilibrium

developed in the macro political economy papers by Yared (2010) and Acemoglu et al (2010a). This

concept internalizes self-enforcement problems faced by the regulators by requiring a non-deviation

condition by all relevant decision makers in equilibrium. Thus, the sequence of policy variables arising

in such equilibrium is by construction time consistent.

In the model, upstream �rms produce a non-renewable resource to be used as the only input for the

production of a �nal consumption good called energy, produced by a downstream �rm. These �rms face

two types of regulations: the prohibition to export as long as domestic demand is not satis�ed (which

occurs in the equilibrium analyzed in this paper) and prices completely set by the government. Energy

is one of the three goods consumed by households, the other two being a numeraire good and the third

an international commodity that is unrelated to energy uses. Households receive an endowment of labor

time (normalized to one) and a stochastic endowment of the international energy-unrelated commodity.

The government is one of many possible (identical) bureaucrats who consume from the unused �scal

revenues coming from the taxes on the energy-unrelated commodity paid by households. On the other

hand, the government also sets the price of energy received by the downstream �rm, and may potentially

subsidize the downstream �rm to cover losses. Also, part of the input necessary to produce energy may

have to be imported by the government in case that upstream �rms do not produce (extract) su¢ cient

quantities of the resource to cover the total demand for energy inputs.

The model also includes a political game between consumers and bureaucrats. The main assumption

is the existence of elections at the beginning of each period, where consumers may choose to re-elect

the incumbent government or else to replace it with another bureaucrat. The model assumes that

consumers enjoy some additional utility coming from the replacement of the incumbent (preference for

turnover). Also, any incumbent bureaucrat faces a utility cost of being expelled from power. Thus,

these two preference features shape the bureaucrat�s incentives to set low prices for energy to allow for

re-election to occur.

The main results of the theoretical model are as follows. The model predicts that domestic energy

prices are way below the international price of the resource as long as the cost of the incumbent govern-

ment to be expelled from power and the voters�preference for political turnover are both high enough

(together with a high enough lower bound in the realization of the endowment of the energy-unrelated

commodity). The intuition for this result is simple. Under the last condition tax revenues are high

to get enough consumption for the incumbent in case that his re-election take place. High utility cost

for the incumbent after loosing elections imply lower bene�ts from setting high energy prices (and low

subsidies) rather than low. The high value of preference for turnover make the incentives to reduce the

3

energy price even stronger, to ensure that voters have enough incentives to re-elect the incumbent. Note

that such conditions also ensure strictly positive subsidies paid to the energy producer (downstream

�rm), a fact that characterize the natural gas sector in Argentina especially after 2008.

Other results from the theoretical model include conditions under which resource imports occur in

equilibrium. Such conditions are those that imply a very low price-to-marginal-cost relation in the

upstream sector. In turn, conditions for such low domestic price are characterized by a �rst-order non-

linear Euler equation. Its (stochastic) terminal condition (given at the time of exhaustion of the resource)

implies that the domestic price in that terminal period is a constant fraction of the international price,

where this fraction re�ects the share of variable inputs in the resource-extraction technology. This Euler

equation may also be consistent with declining expected domestic prices, although such speci�c result

needs a much sharper quantitative characterization that is out of the scope of this paper.

It is important to stress that, to our knowledge, this is the �rst paper in presenting a dynamic,

endogenous, self-enforcing regulation model in the non-renewable resource sector. Indeed, the now

very extensive2 literature on taxation and regulation of non-renewable resources do not include such an

endogenous taxation analysis3. In particular, such papers do not take into account the endogeneity of

the risk of expropriation. However, for Latin American countries such as Argentina this type of risk

is clearly of a �rst-order importance, provided the regular instability of governments and, especially,

public policies in the region. The relevance of this risk has already been pointed out by Garnaut and

Clunies Ross (1975).

However, only much more recently there has been an increase in the analysis of formal models with

expropriation risk within the natural resource sector. An early example is the model by Gaudet, Laserre

and Long (1995). The basic version of this model assumes two periods and no commitment on behalf

of the government, and so the second-period royalty rate is (re)set after the �rst period�s tax rate has

been set. To my knowledge, this constitutes one of the few models assuming an explicit problem of time

inconsistency. Clearly, their model does not capture the problem of expropriation and other types of

risks associated to a not-fully-benevolent government. The model �lls this gap by assuming a politician�s

objective function that combines a rent-seeking behavior with a political-power-keeping behavior, the

2For comprehensive surveys on this literature, see the classic work by Heaps and Hellwell (1985) and the more recentby Lund (2010).

3The classical treatment based on Hotelling�s (1931) model is surveyed by section 5 in Heaps and Helliwell (1985).Within this tradition we �nd Manzano´s (2000) extension considering a "quality" dimension in the pro�t function appliedto the analysis of the Venezuelan market. For the case of Argentina, Vera (2000) presents a Hotelling model to analyzetwo types of royalties: a sales-based tax and a long-run sustainable revenue tax system. The extension to stochasticenvironments includes Leland (1978), Campbell and Lindner (1985), Ball and Bowers (1983), Lund (1992) and Zhang(1997). Postali (2007) applies an explicit option pricing technique to analyze the impact of taxation on the petroleumsector in Brazil.

4

second of which is more consistent with a benevolent uninformed government than the �rst type of

behavior.

Another relatd study is Bohn and Deacon (2000). These authors develop and estimates a theoretical

model of asset expropriation. When estimated with an international panel data they �nd that ownership

instability indeed reduces the speed of oil exploitation. This type of expropriation risk is also a major

subject in several chapters of the book by Hogan and Sturzenegger (2010)4. In the end, this expropriation

risk re�ects a time-inconsistency problem by a government that may want to extract more rents by

increasing tax rates over the original arrangement after the international price of the resource increases.

This time inconsistency issue is discussed in more detail when presenting the formal model presented

in this paper.

The rest of the paper is organized as follows. Section 2 presents some key facts about the oil and gas

sectors in Argentina after the abandonment of the currency board system in 2002. Section 3 presents the

model with the analysis. Section 4 presents the main results concerning the equilibrium characterization

of the model. Section 5 presents several policy implications coming from the equilibrium analysis of the

model. Finally 6 presents the conclusions with several suggestions for future research.

2 Some facts about the oil and gas sectors in Argentina after

the Convertibility

This section presents what it is understood as major empirical facts characterizing the oil and gas sec-

tors in Argentina for the period following the abandonment of the currency board in 2002. From the

relationship between the government and producers, this period has been characterized by a steady

increase in regulations concerning prices and trade restrictions. This same period also shows an impres-

sive fall (after a recovery in both production and investment in exploration (in the latter case, at least

until 2010). Thus, it is apparent that those stronger regulations seemed to have a¤ected the incentives

to invest and to produce oil and especially natural gas. These and other facts are presented as follows.

1. Fall in production. In the period considered here, the decrease in domestic production of bothoil and natural gas (in this last case, since 2004) has been an increasing public concern. Figures

1 and 2 below document such fall for each of these two energy resources.

4Mainly: Tomz and Wright (2010), Rigobon (2010) and Schwartz and Trolle (2010). Also in this book, Wernerfelt andZeckhauser (2010) discuss several actions that �rms can take to avoid expropriation (that may include a slower speed ofproduction of the natural resource, as Bohn and Deacon (2000) predict). Also, Engel and Fisher (2010) discuss optimalauction desigm under the threat of expropriation.

5

Figure 1. Oil production in Argentina. 2002-2012. Source: IAPG

Figure 2. Natural gas production in Argentina (2002-2012). Source: IAPG.

Indeed, the cumulative decrease in oil production between 2002 and 2012 is above 25%. The

decrease in gas production between the 2004 peak and 2012 is more than 15%. As Navajas (2006)

and other authors state, such a decrease (together with a sharp increase in domestic gas use, see

�gure 5 below) explains a big portion of the energy problems faced in Argentina, especially after

2007.

2. Fall (and recovery?) in exploration investment. Those falls in production shown in fact 1can also be linked to a decrease in exploration activity. Figures 3 and 4 below illustrates this fact.

6

Figure 3. Number of exploratory wells in Argentina (2005-2012): oil. Source: IAPG.

Figure 4. Number of exploratory wells in Argentina: natural gas (2005-2012). Source: IAPG

These �gures strictly show that, between 2005 and 2010 (with some exceptions) the number of

wells dedicated to exploration of new reserves in oil and gas showed a tendency to decrease. This

seems to show a short run reversion in 2011 and 2012, possibly as an e¤ect of the restatization of

the major oil and gas company in Argentina, Yacimientos Petrolíferos Fiscales (YPF). However,

during all this period the quantity of wells are well below the technical level to increase production

signi�catively. Overall, beyond these short term �uctuations, the general level of this variable

suggests a strong lack of incentives to invest in the sector.

7

3. Reversion of international trade balance in gas: from surplus to de�cit. As a conse-quence of the decrease in domestic production, and also as a consequence of the increase in the

domestic demand for natural gas, then it was inevitable the reversion of the trade surplus, turning

soon into a de�cit.

Figure 5. Natural gas domestic consumption in Argentina (2000-2012). Source: Secretary of

Energy.

Almost moving in opposite directions, �gure 5 shows a steady increase (only interrupted by the

international �nancial crisis in 2008-2009) in natural gas consumption. The increase between

2002 and 2012 is in the order of 48%. As stated above, these �gures suggest that the immediate

consequence would be the reversion of the trade surplus in gas into a de�cit.

8

Figure 6. Trade balance in Argentina: natural gas (1999-2012). Source: IAPG.

The last �gure is a key aspect of the domestic natural gas sector in Argentina after the 2002 crisis.

Especially, after 2006, exports decreased to almost zero in 2011 and 2012 (they are not exactly zero

just because of the pre-committed sales to Chile). Imports, on the other hand, steadily increased

since 2008 until 2012. The trade de�cit in physical volumes is 43% higher than the peak of the

trade surplus in 2004.

4. Government intervention on producer prices (lower than international prices). Asearly as in February 2002 the Duhalde administration implemented by decree5 export taxes on

oil and gas (see fact 6 below). Several other policy decisions also introduced tight restrictions

regarding the capability of selling oil and gas abroad. In a sense, several of these policy decisions

implied a complete departure from the policy implemented by the Menem administration in the

ninenties, which included market deregulation and privatization of YPF6. These interventions have

implied a departure of domestic prices for oil and gas from international prices, at least until the

2008 crisis and later.

Figure 7. WTI monthly spot prices and volume-weighted average oil domestic prices for

Argentina (2002-2012). Sources: EIA and Secretary of Energy.

5Decree 310/02. This decree was later complemented by several resolutions of the Secretary of Energy (Resolutions337/04 and 394/07) introducing changes in the tax rate scale.

6For surveys on these 1990�s policies see Gadano (2006 and 2010), Roccaro and Fernandez (2005) and Scheimberg(2007). About the policy decisions after 2002 see Scheimberg (2011).

9

Figure 8. Natural gas prices: imports from Bolivia and average domestic producer prices.

Sources: Hidrocarburos Bolivia and Secretary of Energy.

Figure 7 shows that, after two years with domestic prices higher than the international WTI spot

price reference, then since 2004 the latter was systematically above the domestic one, except for

the international crisis years (which was clearly a transitory event). In the natural gas case the

di¤erence is even more clear, at least since 2007, the estimated import price from Bolivia7 is at

least 2.5 (typically, more than 3) times the volume-weighted average domestic price, as shown

in �gure 8. This price di¤erence, combined with a virtual prohibition to export gas (given by

the domestic-market priority rule introduced in 2004) seems to partially explain the decrease in

domestic production and the low level of investment exploration within the period in which YPF

was still private (mainly, until 2010, as seen above). If one accepts that the inclusion in economic

costs some measure of opportunity costs given by import prices, then such a regulation strongly

would lead to very high costs and low pro�ts, as Navajas (2006) and Scheimberg (2007 and 2011)

state, and unlike what other papers such as Kozulj (2005) states8. If one accepts that pro�tability

is key to understand incentives to invest then this may help explaining the lack of investment in

recent years. A recent paper by Barril and Navajas (2011) reinforces this view through a model-

based econometric test to study the determinants of the decline in production, using a panel-data

set at the area-level from 2004 through 2009. This exercise explains the rate of growth in natural

7The data in �gure 8 includes quarterly estimates from the site HidrocarburosBolivia.com between 2007 and 2009 andunit values of gas imports from o¢ cial data released by the Secretary of Energy. The lack of o¢ cial data of gas exportsfrom Bolivian sources force to provide such estimates (for a discussion, see Aguilar and Valdivia (2011)).

8Interestingly, this same author writes an article in 2012 arguing the necessity of adjusting tari¤s upwards, neglectingin part his diagnosis made in 2005. (See Kozulj (2012)).

10

gas production as a function of di¤erent variables, including the ratio of cumulative production to

resource size, exploration investment e¤ort, and some concession-renegotiation dummies. One of

their main results is that an increase of 1% in the ratio of cumulative production to resource size

implies a 20% decrease in the rate of growth of natural gas. The extension of the concession to the

company Pan American Energy in 2007 by the government of the Chubut province also shows a

signi�cant increase on the rate of growth of gas for that company. Ceteris paribus this regulation

implies a 39% increase in the growth rate. These and other results induce the authors to suggest

that such a decline can be attributed mostly to "depressed economic incentives" on intra marginal

reserves and depressing also incentives to invest.

5. Natural gas tari¤s (for �nal demand) much below the import prices (implying highlevels of subsidies). Since the 2002 crisis, the government kept tari¤s for �nal use in the naturalgas markets clearly below di¤erent measures of opportunity costs, mainly, those based on import

prices. Cont et al (2011) estimated such di¤erences, which are illustrated in the next �gure.

Figure 9. Natural gas: import prices from Bolivia and tari¤s for �nal use in the Cuty of Buenos

Aires and its Metropolitan Area. Source: Cont et al (2011).

In �gure 9 the blue line shows the upward evolution of the import price of natural gas from Bolivia.

The red and green lines show the evolution of tari¤s for �nal users in the city of Buenos Aires and

the metropolitan area (the Greater Buenos Aires area). For the 2008-2010 tari¤s the observation

corresponds to the upper scale of use (more than 1800 cubic meters per year). Even for such

intensive users of gas (some which correspond to the upper decile of the income distribution in

11

this area) the price paid is way below the cost of importing it. This is the origin of the increasing

value in the subsidies that the government had to pay to gas distributors as a compensation for

such price disparity9.

6. No major changes in taxation on the upstream of oil and gas. Regarding taxation onthis sector, upstream companies face two separate types of �scal costs. The �rst corresponds

to the regular taxes applied to all businesses. The main ones are corporate income taxes (with

a �at tax rate of 35%), value-added taxes (a �at rate of 21%) and a tax on major shareholder

assets. The �rst two are the most important regarding impact on �scal revenues. However,

until 2006 there has been no di¤erentiated treatment towards oil and gas companies regarding

their application. In that year the government released the Act 26154 with the intention of

providing investment and production incentives to the sector. Such incentives included a faster

accounting-based capital depreciation for computing the corporate income tax base and also faster

reimbursement of VAT retentions, among others. However, according to tax law experts, this Act

never was e¤ectively applied. The evidence above (especially in �gures 1-4) con�rms this absence

of any relevant reaction towards this particular tax policy measure. The second source of �scal cost

for companies comes from royalties. In the 1990�s the Menem�s Administration released Act 24145

(1992). According to this, the ownership of oil, gas and minerals passed from its original National

domain to the Provinces. The Constitution Reform in 1994 con�rmed this reform. This implied

that the Provinces were now the government levels receiving royalty revenues, not the National

government. However, the same legislation con�rmed the royalty rate that was in the old regime

(12% over output value at domestic prices) and since then such rate was not modi�ed. Thus, there

has been no relevant reform regarding the �scal cost imposed on upstream oil and gas companies

coming from royalty rate changes. As stated in fact 4, with the crisis following the abandonment

of the Convertibility plan in January 2002, the government in that year implemented measures to

increase its revenues. One of them was the introduction of a tax on oil exports. Initially the tax

rate was a �at 20% independent of the value of exports. However, in 2004 the government changes

this tax introducing a stepwise tax rate regime, up to a maximum of 45%, as a function of the

international price of a barrel. The increase in the rate was 3 percentage points per each $2 increase

in the price. In 2007 the government set a completely contingent tax rate that implied a price

cap of the barrel of $ 60.9. However, such export tax is currently (at least, since 2010) not very

relevant, given the low volume of oil exports (corresponding to oil types not used for the distilling

9Actually, Cont et al (2011) estimate the amount of subsidies that �nal users received as a consequence of such tari¤regulation. For example, they estimate that, in the 2008-2010, the upper decile of users received about US$ 102.5 millionsin implicit subsidies.

12

process to get gasoline and other type of fuels). In sum: this brief summary shows that taxation

legislation changes in oil and gas has not shown to have an important impact on investment and

extraction decisions. This was mainly due to government decisions implying almost null changes

in the �scal costs for those companies. Royalty rates did not change. Corporate income tax and

VAT rates remained also unchanged. Export taxes became important but also for a limited period

of years. As stated below, other regulations implied that, especially since 2010, were more relevant

to understand the production, investment and trade balance negative performances.

These �ve facts characterize key variables (regulated prices and tari¤s, production, energy consump-

tion and investment) of the oil and gas sectors in Argentina after the abandonment of the Convertibility.

For several authors who studied them in more detail some of those facts constitute a clear unsustain-

ability problem regarding the long run future of this sector. Also, for critics of the administration who

decided such policies these facts show some form of "irrationality", based on those sustainability issues.

The question is whether it is possible to at least qualitatively interpret or explain part of those facts.

The model presented in the next section is an attempt to do so.

3 A model of state intervention for Argentina in the gas (and

oil) markets

This section introduces an intertemporal theoretical model to understand the potential driving forces

behind the type of the state intervention observed in the gas sector and, less strongly, in the oil sector.

The model�s assumptions mainly re�ect institutionally relevant features of those two markets partially

collected in the last section.

3.1 The set-up

Consider an economy lasting for an in�nite number of periods dated as t = 0; 1; ::: In each period there

exists three types of agents:

3.1.1 Domestic consumers and downstream �rms.

Consumers care for leisure and two goods: energy and an internationally tradable numeraire good

denominated in dollars, called the consumption good from here on. There is a third good not consumed

by agents, an internationally tradable commodity whose units are also denominated in dollars, called

the exported commodity. Each consumer receives two endowments in each period t: labor time and a

13

random amount �t of the exported commodity. The endowment of time for each consumer is equal to

one unit per period t: The endowment of the commodity follows a stochastic process with a compact

support � ���; ���with � > 0: For simplicity, assume that the consumer can work as much as she wants

and produce the consumption good using a linear production with a stochastic productivity (equal to

the market wage) given by !t. Assume that the support of !t is a �nite selection of a compact set

� [!; �!] with ! > 0: The price of energy in dollars is �t. The model assumes that the consumer�s

preference is represented by

E0

( 1Xt=0

�t

"(zt)

1� 1"

1� 1"

+c�t l

1��t

�� (1� �)1��

#)(1)

where zt is the consumption of energy, lt is the consumption of leisure and ct the consumed quantity of

the consumption good. Here assume that " > 1 and 0 < � < 1: Consumers pay a tax on the endowment

of the exported commodity, whose rate is � t: Consumers cannot save nor borrow. Thus, the problem

of each consumer in each period t is to choose (zt; lt; ct) to maximize each term in (1) subject to the

following budget constraint:

�tzt + ct = (1� � t) �t + !t (1� lt) (2)

10.

Energy is traded in a regulated monopoly 11. The unique downstream �rm faces a production

function for zt given by

zt =

(yLt if zt � yLt

yLt + ymt if zt > yLt(3)

Here the variable yLt denotes the quantity of the input bought to the upstream �rm and ymt is the

quantity of the input imported from abroad. Given that the input for the downstream �rm is the same

10For reference, standard calculations show that the demand for energy is given by:

zdt =!"(1��)t

�"t

Thus, consumer�s welfare as a function of �; � and other exogenous variables is as follows

1

("� 1)

!(1��)t

�t

!"�1+ !�t +

�t (1� � t)!1��t

11Although historically the downstream market for gasoline and other oil-derived sources of energy presented somedegree of competition (see, e.g., Serebrisky, 2001), after 2003 the perception is that price is not the variable in which �rmscompete, given the strong degree of the intervention by the State to regulate the price.

14

resource as that produced by the upstream �rms, then its importing price is still pt. Thus the minimum

cost function is obviously linear:

CD (zt; pt) =

(�ptzt if zt � yLt

�ptyLt + pt

�zt � yLt

�if zt > yLt

(4)

This technology allows to infer that the government�s upstream-market regulations may a¤ect the down-

stream regulated energy monopoly12. The latter produces energy with the technology (3) to satisfy the

demand of consumers derived from the maximization of (1) subject to (2).

At the beginning of each period consumers decide (through an implicit voting mechanism) which

politician in power or replaces her for a di¤erent one (although with the same preferences). The binary

variable Gt (st) 2 f0; 1g captures this choice. Consumers derive a periodic utility �c > 0 from replacinga bureaucrat with a new one. This assumption comes directly from Yared (2010) and captures aversion

to political persistence.

3.1.2 The upstream �rms

In each period there is a large number of a unique type of (upstream) �rms. The basic set-up assumes

that each �rm is owned by risk-neutral shareholders. The �rm produces a natural, possibly exhaustible

resource, a second commodity whose main use is the production of energy (either within the country or

abroad). The technology of the upstream �rm comes is taken directly from Deacon (1993). Let xt be

the quantity of the variable input and Rt the stock of reserves available at the beginning of period t:

Then the following production function represents the upstream �rm technology:

�tF (xt; Rt) (5)

with

F (x;R) = x�R1�� (6)

where 0 < � < 1. That is, the production function is a constant-return-to-scale Cobb-Douglas. This

parametrization also corresponds to that in Deacon (1993) chosen for the quantitative exercise performed

in that paper. On the other hand, reserves in period t equals the cumulative reserve additions up to

12One possibly important issue is related to the way integration between downstream and upstream production ismodelled. This may be be important in the case of Argentina. However, the main point that the model answers doesnot seem to be strongly related to "transaction costs" that may arise in the relantionship between the upstream and thedownstream. To keep the model as simple as possible, and considering that such transaction costs problems may not be�rst order for this paper, the assumption is to consider two separate sectors for the upstream and the downstream.

15

that date less cumulative production. The term � denotes the total factor productivity for each �rm:

In this basic version �t is assumed to be exogenously stochastic. More precisely, each �t is assumed to

follow a �rst-order Markov process with support being a �nite subset of a compact set��; ���, with � > 0:

Reserves in period t equals the cumulative reserve additions up to that date less cumulative production.

Using Deacon�s notation, let Wt be the cumulative drilling and Yt be the cumulative output (extracted

resources). Thus

Rt = �(Wt)� Yt (7)

where

� (W ) = � � (1� exp (� W )) (8)

with � and strictly positive constants. The cumulative investment is in fact the addition of all

exploration investment decisions made up to any period t: Denote wt as the exploration investment

decision made at time t: Associated to this investment there is a drilling cost D (wt) expressed in

dollars. This D function is assumed to be equal to

D (wt) = wt (9)

This means that wt not only expresses a physical quantity of investment e¤ort but also a monetary

amount, given the linearity of the cost function D: Given this investment e¤ort, the law of motion for

Wt and Yt respectively are

Wt+1 = Wt + wt (10)

Yt+1 = Yt + �tx�tR

1��t (11)

The upstream �rm sells the quantity produced in potentially two markets. A �rst market is that where

the production of this good is delivered to the downstream �rm to produce energy. Thus, the variable

yLt introduced in subsection 3.1.1 denotes the upstream �rm production for domestic use. For these

sales, the upstream �rm receives a price equal to �pt: Below it will be assumed that the domestic price �ptare determined by the government. Thus, the �rm chooses yLt given the regulated price �pt: If allowed,

the upstream �rm would also sell part of the output abroad, at the international price pt: However, this

possibility depends on the regulation imposed by the government (see below) regarding a priority rule

for domestic sales. In each period the upstream �rm decides the input quantity xt and the investment

amount wt that maximizes the expected discounted pro�ts, taking the realization of the TFP shock �t;

the foreign price of the resource good pt; the regulated local price �pt; the domestic sale restriction �yt and

the price of the qt input good as given. The model assumes that the vector (pt; qt) follow a �rst-order

16

Markov process with �nite support being included in the compact [pmin; pmax] ��q; �q�; with pmin > 0;

q > 0. In principle, the latter is assumed to be independent of that governing the TFP shock �t and that

of the endowment shock !t: In subsequent extensions this independence assumption may be dropped

to examine its implication concerning the equilibrium.

The upstream �rm owner is assumed to be risk neutral with discount factor �; the same one as

that of consumers. Thus, the upstream �rm chooses a plan fxt; wtg1t=0 that maximizes the expecteddiscounted value of dollar pro�ts until depletion of the reserves. The exact expression of such pro�ts

is presented below after introducing the government policy decisions (since these decisions a¤ect the

explicit form of the pro�t function).

Finally, as a consequence of the interaction between the upstream �rms decisions and the government

decisions (to be presented below), as well as the exogenous variables, there exists an endogenous random

period T (whose at least one of the realizations may be in�nite) in which resource reserves are completely

exhausted. Thus, whenever T is �nite then in periods fT + jg1j=1j reserves are equal to 0, i.e., RT+j =0; and so the upstream �rm is shut down after reserves are exhausted.

3.1.3 Government decisions and upstream �rm pro�ts

This model assumes the existence of a (national-level) government run by one of many ex-ante identical

politicians or bureaucrats, whose preferences are de�ned over stochastic sequences of dollars consumed,

gt. They are represented by the following expected utility function:

E0

( 1Xt=0

�tgt

)(12)

where the discount factor � is the same as that of the upstream �rm owners and consumers.

The government sets the tax rate on potential foreign sales (which becomes e¤ective only when

exports are positive). Also, the government decides the price of energy �t and the price paid to the

upstream producers of the input good for domestic sales, �pt. The model assumes that the government

sets the restriction that upstream �rms�s local sales must satisfy the domestic demand from the energy

producer. There is of course an implicit assumption that the government has the power to close down

any upstream �rm that intends to violate such rule. Thus, upstream �rm can only export the remains

of production after servicing the local demand. If production is below the local demand, then all

production goes to the latter. On the other hand, given that the government directly sets the price

of energy, it subsidizes the energy producer when the latter incurs in losses. Let St denote the dollar

amount of subsidies provided by the government to the energy producer. Finally, the government taxes

17

a fraction � t 2 (0; 1) of each consumer�s wage income in terms of the numeraire good.The budget constraint relevant for the national government�s incumbent in period t is

gt = � t�t � St (13)

where St satis�es:

St = max

(pt!"(1��)t

�"t+ (�pt � pt) y

Lt �

!"(1��)t

�"�1t

; 0

)(14)

If the incumbent is �nally kicked out of power in a given period t, the former sets the policy variables so

as to obtain a maximum consumption in period t. These decisions include the con�scation of positive

pro�ts that the downstream �rm may get given the price �outt set by the exiting incumbent. In that

period the latter gets a utility loss of �N (1��)�

> 0 in addition to the loss of future rents (consumption).

On the other hand, the expelled incumbent gets a total utility loss of �g

1��13:

The de�nition of subsidies in (14) merits a warning. According to empirical evidence (see, e.g., Cont

et al (2012)) the subsidies that gas companies receive are calculated as the di¤erence between the import

price and the domestic price of gas for �nal demand. In the notation of this paper, subsidies should be

de�ned as eSt � max(0; (pt � �t)

!"(1��)t

�"t� yLt

!)In this case, the upstream �rms would receive the price �t instead of �pt: This version of the model

considers a slightly more general assumption of a government potentially paying a price to the resource-

extracting upstream �rms di¤erent from the �nal-demand energy price, even though the marginal pro-

ductivity with respect to the resource in the downstream �rm is equal to one. As it will be clear below,

this seemingly more general assumption may lead to an empirically counterfactual result that prices

received by upstream �rms may not follow the same dynamics than then price of energy, which may

be read as not realistic. The task of checking how results below may change with this change in the

subsidies de�nition is left for future research.

Assume that

�

24(1� �)�

"��1�

q

# �1��

pmax + 1

35 � � : 0 < � < 1 (15)

13Note the government�s lack of access to credit markets or storage technologies. Even though bureaucrats in powerare all risk neutral, access to storage may change the features of the equilibrium regulations (see Cole and Kocherlakota(2001) for a risk-sharing problem with hidden income and hidden storage, albeit with risk averse agents). Possible futureresearch may explore how the addition of such technologies may change the results of this paper.

18

This condition will have an importance when considering equilibria with strictly positive subsidies.

The government regulations, together with assumptions on the exogenous stochastic variables and

parameters clari�ed below, will imply that, in equilibrium, yLt will be strictly less than zt in every period

t: In other words, upstream will just produce in equilibrium only to satisfy the local demand. Therefore,

in such an equilibrium

yLt = �tx�tR

1��t (16)

In such an equilibrium export taxes become irrelevant. All these conditions allow to write the (equilib-

rium) period- t pro�t for the upstream �rm as

�pt�tx�tR

1��t � qtxt � wt (17)

Given the objective of the upstream �rm, as long as exports are zero in the analyzed equilibrium, then

the government will a¤ect production and investment decisions by the upstream �rm only through the

regulated domestic prices f�ptg1t=0 :

3.2 The game

The interaction between producers and di¤erent government levels comes through a (repeated) game

whose main elements are taken from the sustainable taxation literature in macroeconomics,especially

the recent work by Yared (2010) and Acemoglu et al (2010)14. This timing is taken directly from Yared

(2010). The sequence of actions is as follows:

� At the beginning of period t; given observing history st; domestic consumers choose to re-elect theincumbent government (that is, Gt (st) = 1) or to choose a di¤erent one (that is, Gt (st) = 0).

� After the decision on the government, the incumbent politician chooses tax rates and regulationvariables.

� Markets open and consumer and upstream �rm owners make their respective decisions.

� Whenever is applicable, government replacement takes place at the end of the period.

Under this timing, the government faces the decision of investing in political reputation for period

t+1 on or just decide to extract as much surplus as possible provided that at the end of the period she14This concept is an extension to the well-known sustainable equilibrium concept in Chari and Kehoe (1990), Chang

(1998), Phelan and Stacchetti (2001) and Fernández-Villaverde and Tsyvinski (2004). In those papers the policy-makeris assumed to be benevolent, whereas in this model the policy-maker-regulator is a sel�sh (rent-seeking) politician. Thisadaptation allows to study "time-consistent taxation and regulation" within this particular framework.

19

will be removed from power. Hence, this timing implies that in every period t the bureaucrat in power

who would like to stay sets values for the policy variables that provides incentives to consumers to vote

again for her at the beginning of period t+1: At the same time, for this re-election to be rational for the

bureaucrat such policy must give her more utility value of remaining in power than of being expelled

from o¢ ce. These constraints are part of the equilibrium concept introduced below.

3.3 The equilibrium notion

Given the timing presented in the subsection above, and the absence of asymmetries of information in

the benchmark case, the basic equilibrium notion follows very closely that of Yared (2010) and Acemoglu

et. al. (2010b). In such an equilibrium the history of play at the beginning of period t includes the

realization of all stochastic variables up to period t; the history of play by the government and the

allocations chosen by �rms and consumers. Thus, this notion yields the following de�nition.

De�nition 1 A self-enforcing equilibrium is a plan of governments��X�; � �; �p�; ��

; a plan of

output, investment and input use by upstream �rms fy�; w�; x�g ; a plan for consumers fz�; c�; l�g andrandom period T � such that:

1. at every node each plan maximizes the continuation value of the corresponding agent;

2. at the random time T � reserves are equal to 0 from that period onwards.

As stated above, this de�nition borrows from that in Yared (2010) and Acemoglu et al (2010),

with the addition of the random period T when reserves are exhausted. It captures the fact that the

plans in the sustainable equilibrium do not present "time inconsistency" problems (given the constraints

in the model). In other words, given that in Emerging Markets countries such as Argentina policies

cannot be conceived as coming from a time consistent planner or government, but from (complex)

political processes, this model captures some dimensions of such process that are considered relevant

for the analysis of state intervention in the exhaustible-resources sector. It is a concept that allows

endogeneizing the values that di¤erent policy instruments take every period, explained by the political

process embedded in the game described in subsection 3.2.

4 Self-enforcing equilibrium characterization with no exports,

positive imports and positive subsidies.

This section characterizes the self-enforcing equilibrium when the upstream �rm does not export in any

given period t: As stated in the introduction, this type of equilibrium in fact re�ects mainly the features

20

of the natural gas and a possible future oil market in Argentina. Of course, the 0-export condition

is part of a self-enforcing equilibrium when several conditions over exogenous variables hold. Those

conditions are provided also in this section.

4.1 Upstream �rm�s optimal choice under no exports.

The equilibrium begins by the characterization of the optimal plan decided by each upstream �rm. The

following proposition characterizes the solution to this problem (all proofs are in Appendix B).

Proposition 2 Suppose a self-enforcing equilibrium with 0�exports. Thus the optimal policies by eachrepresentative �rm are given by the following equations: for period T 15

xT = [� � (1� exp (� WT ))� YT ]

���T �pTqT

� 11��

(18)

and for periods t < T

xt = [� � (1� exp (� Wt))� Yt]

��t [��pt � �t]

qt

� 11��

(19)

where

�t � �

(Et

�qt+1xt+1Rt+1

���� t+ 1 < T

�+ Et

"���T �pT(qT )

�

� 11������� t+ 1 = T

#)

= �

(Et

"��t+1 [��pt+1 � �t+1]

q�t+1

� 11������� t+ 1 < T

#+ Et

"���T �pT(qT )

�

� 11������� t+ 1 = T

#)(20)

The optimal level of investment wt in any self-enforcing equilibrium is given in the following expression.

wt = �Wt +1

ln� (1� �) �

(1� �)�+1

ln �t (21)

Finally, in period T the following condition must hold:2664YT � � �26641� � �ET�1

�h��T �pTq�T

i 11���

1� �

37753775"1� �

11��T

���pTqT

� �1��#� 0 (22)

15The upperscript � is dropped for simplicity.

21

Several comments are worth making regarding the upstream �rm optimal decision. Period T corre-

sponds to that in which reserves are exhausted. The exhaustion condition is equation (22). The latter

states that the accumulated production exceeds the accumulated investments. These are su¢ cient con-

ditions to ensure that there are no incentives to invest in this last period. In such period, the optimal

use of the input becomes static. Indeed, the value of its marginal product (valued at the domestic price

of the resource) equalizes the input price.

For periods before the exhaustion of the resource (t < T ) the demand for the input not only depends

on the current values of the TFP shock and prices (including the price of the resource set by the

government) but also on the variable �t;whose value depends on expectations of future values of those

same prices and TFP shocks (but not on present ones). Variable �t essentially embeds the expected

next-period marginal rate of substitution between the two inputs of the production function. Equation

(19) is obtained from a �rst-order condition that is a variant of the well-known Hotelling rule. It relates

the price-marginal-cost di¤erence of two consecutive periods. However, it explicitly takes into account

the fact that reserves could be exhausted in the next period (which is given by the second expression

within brackets in equation (20)). Given the stochastic nature of the model, such possibility cannot be

ruled out (and it should even have positive probability, at least for t not too small).

On the other hand, given the particular functional forms of this model (again, borrowed from Deacon

(1993)) the optimal investment decision depends on the accumulated investment spending Wt, as well

as on the expectation about future TFP shocks as well as future output and input prices. On the

one hand this feature is consistent with the view about the dependence of the investment in natural

resources to future "perceived macroeconomic risk" and future "perceived political risk", as discussed in

e.g., Navajas et al. (2005). Also, this dependence combined with a constant-return-to-scale technology,

a lack of commitment by the incumbent and the lack of access to public debt makes the decision on

export taxes independent on the investment behavior. The domestic price set in a given period t could

have only a¤ected the investment decision in t � 1; but such investment is already taken as given by

the incumbent. Had the incumbent partial commitment for, say, one period ahead, then the decision in

period t would also include one-period ahead tax rates, a¤ecting the investment decision in the current

period.

4.2 Self-enforcing equilibrium government policies.

The next steps is to present a characterization of the self-enforcing equilibrium. The following propo-

sition presents such characterization for the benchmark case of N = 1 and risk-neutral owners of the

upstream �rm.. The reasoning follows very closely that of Yared (2010). Thus, the main result is the

22

following:

Proposition 3 In any self-enforcing equilibrium with 0 exports and both positive imports and subsidiesthe policy decisions are characterized by the following properties:

1. For any period t :

(a) the self-enforcing equilibrium price of energy ��t follows the following law of motion:

�t�st�=

8>>>>>>>>>>>><>>>>>>>>>>>>:

�� (s) ��

"

"�1�V gv�s;vout(p;!)+�

1��

�!�(1��)

�pt if � (st�1) >

"pt�1h"�

pt��(st)

�1�+1i�

!t!t�1

�(1��)+"�1

"��pt�1

�t�1(st�1)�1�"+1

��!t�1!t

�(1��)+"�1

!pt

if � (st�1) 2"

"pt�1h"�

pt�(st)

�1�+1i�

!t!t�1

�(1��)+"�1

; "pt�1h"�

pt��(st)

�1�+1i�

!t!t�1

�(1��)+"�1

#� (s) �

�"

"�1�V gv (s;�v(s))!�(1��)

�pt if � (st�1) <

"pt�1h"�

pt�(st)

�1�+1i�

!t!t�1

�(1��)+"�1

(23)

where �V gv

�s; v

out(p;!)+�1��

�represents the marginal shadow value (for the government) of de-

creasing the consumer�s total expected discounted welfare when the latter is equal to the outside

option value (i.e., when consumers are indi¤erent between reelecting the incumbent and not

doing so), and also where �V gv (s; �v (s)) is the same marginal shadow value (for the govern-

ment) of decreasing the consumer0s total welfare when the incumbent is indi¤erent between

continuing in power and not doing so;

(b) for su¢ ciently high values of ! and both �g and �c then the energy producer receives strictly

positive subsidies with probability 1,

(c) for su¢ ciently high values of !, the tax rate on the endowment � �t is equal to 1:

2. For nodes st with t = T; then the self-enforcing-equilibrium domestic price �p�T paid to the upstream

�rms equals �pT :

3. For any period t < T; then the self-enforcing-equilibrium domestic price �p�t paid to the upstream

23

�rms satis�es:

ept � �ept= �Et

8><>:264(1� �)

24� 1�

t+1ept+1

qt+1

35�

1��

� 1

375�ept+1 + ept+1 � 2 (1� �)�

"�t+1ept+1q�t+1

# 11��

������� t+ 1 < T �

9>=>;(24)�� (1� �)�

21��Et

(�pT ��T �

q�T �

� 11������� t+ 1 = T �

)

where

ept � �pt � �tept � ��p�t � �t

and where �t follows (20).

4. Subsidies are strictly positive in every state st such that

[� � (1� exp (� Wt))� Yt]

��t [��pt � �t]

�

q�t

� 11��

< !"(1��)t �t

�st��"

where �t (st) is given by (23).

5. The self-enforcing-equilibrium exhaustion period T � is characterized by the minimum T such that"TXj=1

(�1)j�1�1� � (1� �) �

(1� �)��T�j

�"�jl=1

��T�l

�[��pT�l � �T�l]

qT�l

��� 11��#

+

� �ET�1

�h�2�T pTq�T

i 11���

1� �� 1

3775"1� �

11��T

��2pTqT

� �1��#

(25)

� 0

In this equilibrium, the incumbent at the national government level never leaves o¢ ce.

24

The last proposition presents the main results regarding key aspects of the self-enforcing-equilibrium

prices set by the incumbent bureaucrat. A major aspect is that a su¢ cient condition for positive

subsidies is a su¢ ciently high values of the vector (�g; �c) : This means that when the threat against the

incumbent bureaucrat of punishing him had the bureaucrat deviate from the equilibrium policy (which

includes subsidies) then the latter are part of the equilibrium. The interpretation is not too di¢ cult

to see. High values of such parameters induce the incumbent to maintain energy prices low enough in

any state s: Of course, the motivation to keep these prices low is to induce consumers to re-elect the

incumbent next period. Such incentives are stronger the higher are both �c and �g:

Regarding the evolution of the self-enforcing equilibrium energy prices �t, a remarkable feature

is the existence of state-contingent lower and upper bounds � (s) and �� (s) ; which is the analog of

the bounds-on-the-income-tax result in proposition 3 (equation (40) in Yared (2010). Clearly, such

bounds come from the incentive-compatibility constraints, one for the government (stating that the

government prefers to follow the self-enforcing equilibrium strategies rather than deviating from it, with

the automatic consequences of not being re-elected), and the other for the consumers (stating that each

consumer prefers to re-elect the incumbent rather than electing a new bureaucrat). The appendix shows

the recursive versions of such constraints. Interestingly, the second expression in (23) states that, as

long as the last period energy price is within the current period bounds, then the current-period price

is increasing in the last-period price. This shows some (weak) tendency towards an increase in the price

of energy, although this is subject to the condition of past prices within the bounds. This feature may

be read as partially consistent with the evidence observed for Argentina after 2011.

The result of the tax rate on the exportable endowment equal to 1 is partially a consequence of

both the preferences assumed for consumers (the voters) and the fact that consumers do not value the

exportable commodity. Given the consumers�s preferences, a marginal increase in the tax rate decreases

the consumers� utility by an amount which, �rst, is proportional to the endowment � and, second,

whose absolute value is decreasing in the labor productivity !. On the other hand, given the linearity

of bureaucrats�s preferences, the period-t utility is increasingly linear. In this case, a marginal increase

in the tax rate which is also proportional to the endowment �: Thus, for the bureaucrat, a marginal

increase in the tax rate on the one hand increases both his utility but on the other hand makes the

incentive constraint for the consumers tighter (since it represents a decrease in their utility). However,

with the assumption of ! being high enough the second e¤ect is milder than the �rst one, implying

that the net e¤ect of increasing � is always positive, this providing the �nal result of full taxation of

the exportable commodity (given that energy prices are low enough, and so compensating for this full

taxation outcome).

25

Another feature of this equilibrium is the independence of the domestic price of the natural resource

(produced by the upstream �rm) from the price of energy (produced by the downstream �rm). The

reasons for this independence can be found in two key aspects of this model: on the one hand, the

linearity of the Bernoulli utility function for the bureaucrats, and, on the other hand, the linearity in

the technology of energy production. These two features then bring the bene�t (for the modeler) of

allowing for a relatively simple characterization of the equilibrium law of motion of these prices, but at

the cost of predicting an independence between those two prices that may be at odds with the empirical

evidence in Argentina.

5 Interpreting facts with the model results and some policy

implications.

Some of the facts presented in section 2 can be read using the results of proposition 3. Part (1.b) presents

a su¢ cient conditions for fact 5 (energy price much lower than the import price) with probability one.

Note that these su¢ cient conditions include a su¢ ciently high value of the endowment of the minimum

realization of the commodity shock. The rationale for this result is as follows. High enough values of both

�g and �c induce the bureaucrat to set very low energy prices to keep incentives for re-election, implying

subsidies. For this to be a self-enforcing equilibrium the bureaucrat should get enough revenue from

other sources. That revenue undoubtedly comes from commodity taxes. Thus, one way of interpreting

fact 5 is from this part of proposition 3 is that regulations would keep energy prices low enough following

electoral incentives, being that "sustained" by tax revenues from other commodity (i.e., tax on cereals

exports such as soybean).

Part 3 of proposition 3 characterizes the domestic producer price dynamics before reserves exhaus-

tion. Unfortunately, equation (24) does not look simple enough to state under which conditions �p�t <

pt: For the depletion equilibrium period T �, however, it is true that the domestic price is strictly less

than the international price given that � < 1: For earlier periods, however, the Euler equation that

determines the dynamics of the domestic producer price �p�t does not lead to such inequality in a sharp

way. Nevertheless, a su¢ cient condition for such inequality to hold in periods before T � is that the term

inside the �rst conditional expectation is strictly negative, which, at the same time, could occur when,

on expectation, ep�t+1 is strictly greater than ept+12and less than or equal to �ept+1: There are clearly less

demanding conditions for �p�t < pt to hold, but the last discussion at least gives a rough idea that such

inequality depends on the expectations of future domestic prices. Given that the stochastic terminal

condition for �p�T � is known, it is possible that an iteration of the expectation on the right hand side

26

of equation (24) would lead to the de�nitive answer. However, the problem is that such expression is

highly non-linear, and thus such conditions can only be studied by means of numerical simulations.

Part 4 of proposition 3 characterizes the states under which resource imports occur in equilibrium,

which is what has occurred with natural gas in Argentina since 2008, according to fact 3. Given the

characterization of energy prices, a su¢ cient condition for such inequality to hold is a domestic price

much lower than the input price qt; i.e., with a low domestic-price-to-marginal-cost relationship. Part of

this is documented in fact 4, but given lack of reliable data on unit production cost the whole inequality

cannot be put under test properly. However, fact 4 suggests that part of the explanation of energy

imports in Argentina is indeed the low incentives to produce, given the low domestic producer price,

which is also what fact 1 yields.

Regarding policy implications, as stated in the section above, a major lesson from proposition (3

is that political incentives may be the cause for strictly positive subsidies. Another lesson is that the

parameters in�uencing the energy price do not a¤ect the resource domestic price as long as subsidies

are positive. This means that, from the perspective of the resource production and investment, political

considerations do not seem to a¤ect the natural resource sector decisions, at least at the margin. On

the other hand, the equilibrium law of motion for the natural resource domestic price is the direct con-

sequence of a "rent-seeking" bureaucrat decision. Such law of motion partially determines the behavior

of the hitting time T: Although it is not possible to get precise sharper analytical characterization re-

sults on T it is clear that the bureaucrat decision (assuming that his preferences depend on his own

consumption) yields directly this equilibrium hitting time.

What kind of policy (or institutional) implications do these result imply? A direct answer would

require the full characterization of the Pareto-e¢ cient allocation, which is out of the scope of this

paper. However, a general intuition emerges directly from the model itself. The latter assumes that

policy makers are sel�sh agents who consume units of the numeraire goods. In making decisions, the

bureaucrat only takes into account political incentives coming from the consumers, who are in charge of

electing the bureaucrat in charge of the government in every period. Bureaucrats have no explicit interest

in the "long-run sustainability" of upstream production (extraction) of the non-renewable resource.

The intuition then is that a more benevolent government would weight the bene�ts of a possibly low

domestic price paid to the upstream �rms for the resources (and low expected future prices embedded in

the variable �t) with the cost of such policy to the upstream �rm in the long run, especially concerning

the distribution of the equilibrium stopping time T �:

Paradoxically, the main characterization result states that, if the objective were to reduce the amount

of subsidies paid by the government, either the parameter �c or �g should not be "high enough". Thus,

reducing the value of these parameters would relax the government�s incentives to keep the price of

27

energy very low. However, as long as equilibrium subsidies paid by the government are still positive,

then the latter policy would not a¤ect the domestic price of the resource paid to upstream �rms. The

latter only depends on exogenous variables and parameters. This also applies clearly to the exhaustion

time T �: Thus, in such equilibrium no policy can change the dynamics of the domestic price of the

resource. Again, the only possible reform to analyze is a discrete change of the values of �g and/or

�c; so that in equilibrium subsidies may become null. However, in that type of equilibria, the domestic

price may be indeterminate. This issue is left for future research.

6 Concluding remarks

This paper develops a theoretical intertemporal model of to give one possible explanation for several

stylized facts about production, investment and especially, government regulation of the Argentine oil

and gas sectors. The assumptions of the model partially re�ect some of the distinctive features of some

of the sectors described above. In terms of its results, the theoretical model, although very stylized in

several dimensions, can qualitatively replicate several features of such regulations. The model predicts

downstream prices below the import prices (as observed in the data) as long as the threat of exclusion

from power for the regulator is strong enough. The model predicts that, with low enough resource

regulated prices relative to the variable input price then the economy imports resources from abroad.

This prediction is clearly consistent with the observed natural gas imports in Argentina after 2008.

The model also predicts a forward looking behavior for the domestic-producer price of the resource.

This behavior is at least consistent with a decrease in investment coming from a perception of lower

future (expected) prices of the resource. That is, this model would explain the decline in exploration

investment seen in the data as the result of a expected future decline in domestically regulated prices.

Regarding the policy lessons, this model suggests that, to change government incentives regarding

price and quantities regulations in energy-related-non-renewable sectors, some political reforms regard-

ing electoral incentives are needed. In particular, how the electoral system favors either political turnover

or political stability is not neutral relative to government incentives to keep prices below international

ones. As discussed in section 5, changes in �g or �c may a¤ect such incentives in subtle ways. It is

important to keep producing further research to get a more precise answer about which type of electoral

reforms may improve government regulations to induce a reversion in the decline of both investment

and production.

As discussed in several parts of this paper, there is a lot of room for other extensions and improve-

ments of this basic model. First, as stated in subsection 3.1.3, the type of subsidies considered on this

version of the model may be di¤erent from that employed by the Argentine government in the gas sec-

28

tor after 2004. Also, although the model can qualitatively predict the equilibrium behavior of di¤erent

policy variables, however, it does not provide precise quantitative results to compare then with the

quantitative facts provided in section 2. In particular, how the equilibrium dynamics for resource and

energy prices, as well as its impact on production and investment and the average duration of reserves,

vary with the values of parameters �g and �c; may help energy regulation designers to understand not

only what type of changes in political constraints to implement but also more quantitative answers

regarding such constraints. This is left for future research.

One major assumption in this model is the long-lived politician who in equilibrium never leaves power

and so sets taxes and regulations. However, as di¤erent authors (such as Spiller and Tommasi, 2003),

the political dynamics in Latin America is a much more complex phenomenon, and such complexity

may have di¤erent implications regarding the stability of di¤erent political actors. Spiller and Tommasi

(2003) stress how bad incentives coming from the way that the political systems work may imply that

politicians act as if they only care about the short run. Indeed, Manzano and Monaldi (2008) stress

this point applied to the oil sector. As stated below, di¤erent assumptions on preferences or di¤erent

political types can be introduced to the benchmark model to deal with more complex political dynamics.

However, such assumption change usually is not free: more complex political dynamics may threaten

the possibility of getting clean results out of the modi�ed model.

Another aspect is the exogeneity of the type of concession contracts assumed in this model. In

reality, concession contracts are (endogenous) outcomes of auctions. The model could then be extended

to introduce an analysis of an optimal auction mechanism, in the spirit of the analysis by Engel and

Fisher (2010)16, that then yields di¤erent regulations and taxation systems depending on the type of

contract arising from such mechanism. Particularly important is the paper by Stroebel and van Benthem

(2012). The latter develops a dynamic contracting problem taken from Thomas and Worrall (1994) and

applied to non-renewable resources. The theoretical results regarding the contract design is tested

using a WoodMackenzie dataset of hydrocarbon contracts, showing that the theoretical predictions are

consistent with this evidence. Such an exercise may be embedded in the model here to get simultaneously

results concerning regulations and contract features.

Such extension endogeneizing contracts seems important to understand recent news from the oil and

gas sectors in Argentina about political declarations threatening private companies to unilaterally �nish

concession contracts. Such a threat is clearly absent in the model. Therefore, it would be an interesting

extension of this model to introduce an assumption that the government can "kick the company out" of

16However, the original Engel and Fisher paper only deals with a mechanism design problem within model of oilproduction much more stylized than in this paper. Thus, the possibility of extending the analysis by Engel and Fisher(2010) or even using part of the standard auction theory literature in this model may lead to a potential complexity thatmay not allow for analytical solutions.

29

the concession. However, for this mechanism to become an interesting one the model there are certain

assumptions that need special consideration. For example, such an extension may need an assumption

on what would be the alternative if a company is expelled from the industry. The assumption on this

alternative is not obvious for the case of Argentina. This consideration and others are all left for future

research.

Other relevant issues have been left out of the analysis. One of them is the relationship between

corporate governance, regulation design and political incentives. This issue is basically motivated by

the recent decision from the Argentine government to re-nationalize YPF in 2012. This fact induces

to think that governance problems may start interacting in subtle ways with the policy and regulation

decisions. In the benchmark model the owners of the upstream �rms are risk neutral and unique for

each �rm. An extension would entail multiplier owners for the upstream �rms, even with risk averse

preferences instead of risk neutral. But then a possible problem (already present in the traditional

GEI literature) is the lack of de�nition of the objective of the �rm. With multiple risk averse owners

and incomplete �nancial markets there is usually disagreement about the relevant discount factor. The

static GEI literature solves this problem introducing voting mechanisms inside the �rm. For such a

dynamic model, an adaptation of such voting is required. A possible idea is to use an adaptation of the

self enforcing voting model by Maggi and Morelli (2006), meaning that the voting decision in each stage

must be part of a perfect equilibrium notion. This of course will clearly complicate the equilibrium

characterization, but it could be useful to analyze such an interaction.

Another venue for future research is the consideration of asymmetric information between the reg-

ulator and the �rms. Taking ideas from the well-known model by Osmundsen (1998), who, assuming

also full commitment17, considers the optimal contract under asymmetric information, as well as the

tax systems that implement such contract. The extension of the model in this paper would consider

also an extension of Osmundsen�s analysis from a two-period model to the in�nite horizon case18.

Finally, a more recent issue opened in the Argentine gas and oil sector with the recent discovery

of shale gas and shale oil reserves in the Vaca Muerta area, in the province of Neuquen. This new

and unconventional source of gas seems to play a key role in the future regarding regulations in the

gas sector. It is then a natural next step to extend the model to include this type of discoveries of

new technologies. How the availability of these new technologies a¤ect the incentives by the incumbent

17As Osmundsen (1998) points out (an argument borrowed from La¤ont and Tirole, 1988) the combination of both noncommitment and asymmetric information makes the characterization of the optimal contract less clear. This is the mainreason of why Osmunden (1998) assumes perfect commitment.18The recent paper by Ales et al (2012) adapts Yared (2010) to an asymmetric information environment to explain

political turnover. This extension seems very useful to adapt it to a non-renewable resource endogenous regulation (andtaxation) problem like the one in this paper.

30

national-government politician to regulate the gas market in certain ways is an almost mandatory

exercise to provide precise policy recommendations for the future gas sector.

7 Appendix: proofs

7.1 Proof of Proposition 2

Suppose �rst the upstream �rm�s problem in period T: The latter can be written as follows:

V uT (sT ; RT ; �pT ) = max

(xT )2R+

��pT �Tx

�TR

1��T � qTxT

where the choice of wT is set to 0 here (later on we check that this is indeed optimal under the condition (22)).

The �rst order condition with respect to xT yields directly the input demand (18). Thus the upstream �rm�s

value in period T is

V uT (sT ; RT ; �pT ) =

���

qT

���T �pT

� 11��

(1� �)RT (26)

For periods t < T the upstream �rm�s Bellman equation is as follows:

V ut (st; Rt; �pt) (27)

= max(xt;wt)2R2+

��pt�tx

�tR

1��t � qtxt � wt

+�Et�V ut+1

�st+1;� � (1� exp (� (Wt + wt)))� Yt � �tx

�tR

1��t ; �pt+1

��� t+ 1 < T�

+�Et

"���

qT

���T �pT

� 11��

(1� �)�� � (1� exp (� (Wt + wt)))� Yt � �tx

�tR

1��t

������ t+ 1 = T

#)

The �rst-order conditions are as follows:

��tx��1t R1��t (28)(

�pt � �Et

�@V u

t+1

@Rt+1(st+1; Rt+1; �pt+1)

���� t+ 1 < T

�� (1� �)�

�1���Et

"��T �pTq�T

� 11������� t+ 1 = T

#)= qt

31

(Et

�@V u

t+1

@Rt+1(st+1; Rt+1; �pt+1)

���� t+ 1 < T

�+ Et

"���

qT

���T �pT

� 11��

(1� �)

����� t+ 1 = T

#)(29)

�� exp (� (Wt + wt))

= 1

The envelope condition is as follows

@V ut

@Rt(st; Rt; �pt)

= (1� �) �tx�tR

��t(

�pt � �Et

�@V u

t+1

@Rt+1(st+1; Rt+1; �pt+1)

���� t+ 1 < T

�� (1� �)�

�1���Et

"��T �pTq�T

� 11������� t+ 1 = T

#)

Replacing here the equation (28) the latter becomes

@V ut

@Rt(st; Rt; �pt)

=(1� �) �tx

�tR

��t qt

��tx��1t R1��t

=

�1� �

�

�qtxtRt

Thus replacing the latter in the �rst order condition (28) then

��tx��1t R1��t(

�pt � �Et

��1� �

�

�qt+1xt+1Rt+1

���� t+ 1 < T

�� (1� �)�

�1���Et

"��T �pTq�T

� 11������� t+ 1 = T

#)= qt

Solving for xt yields the demand for input (19). Replacing also in the �rst order condition 29 we have that(Et

��1� �

�

�qt+1xt+1Rt+1

���� t+ 1 < T

�+ Et

"���

qT

���T �pT

� 11��

(1� �)

����� t+ 1 = T

#)�� exp (� (Wt + wt))

= 1

Solving for wt yields the optimal investment plan (21). Note that in period T it must happen that the

accumulated output is not less (in fact, equal to) reserves. This condition is actually what de�nes T: Thus T

32

is the minimum t such that

� �

26641� � �ET�1�h

��T �pTq�T

i 11���

1� �

3775 � YT +

2664� �26641� � �ET�1

�h��T �pTq�T

i 11���

1� �

3775� YT

3775 � 11��T

���pTqT

� �1��

which is clearly equivalent to inequality (22). Under this condition there is clearly no incentive to invest so wuT= 0:

Finally, note that it is possible to get

�t = �

(Et

"��t+1 [��pt+1 � �t+1]

q�t+1

� 11������� t+ 1 < T

#+ Et

"���T �pT(qT )

�

� 11������� t+ 1 = T

#)

as stated in the proposition. So, solving forward:

�t

= �Et

"���T �pT(qT )

�

� 11������� t+ 1 = T

#

+�

�Et

���

11��t+1 q

� �1��

t+1 [��pt+1 � ��t+2]1

1��

����� t+ 1 < T

��with

�t+2 � Et

h��t+2q

��t+2 [��pt+2 � �t+2]

� 11����� t+ 2 < T

ig

+Et+1

"���T �pT(qT )

�

� 11������� t+ 1 = T

#

The �nal expression comes from induction.

7.2 Proof of Proposition 3

Step 1 Using the traditional primal approach let us write the consumer and bureaucrat�s welfare as function

of allocations and exogenous prices and shocks. From (18) and (19) we solve for �pt as a function of xt

and other variables:

�pT =qT��T

�xTRT

�1��(30)

33

�pt =1

�

qt�t

�xtRt

�1��+ �t

!(31)

where �t depends on future values �pt (which the bureaucrat takes as exogenous in t given the lack-of-

commitment assumption).

Step 2 Both the bureaucrat�s consumption and the consumer�s welfare can be written as follows:

gt = � t�t � St

where

St =

8><>: max

�0; (pt � �t)�

�"t !

(1��)" +

�1�

�qt�t

�xtRt

�1��+ �t

�� pt

��tx

�tR

1��t

�; t � T

maxn0; (pt � �t)�

�"t !

(1��)"t

o; t > T

and

U ct =1

("� 1)

!(1��)t

�t

!"�1+ !�t +

�t (1� � t)

!1��t

Step 3 Given the expressions obtained in step 2, we can start writing the Bellman equations characterizing

the bureaucrat�s optimal program. Step 3 starts from writing the Bellman equation for periods t > T:

We start by writing the value of deviating from the self-enforcing equilibrium path for both the exiting

bureaucrat and the consumer. Dropping time subindices the former is

V g;out (s) = max�;�

��� + (� � p)��"!(1��)"

�Clearly � o = 1: The �rst order condition with respect to � is

(1� ")��"�2 + "p��"�1 = 0

This yields the standard Amoroso-Robinson�s formula:

�out (p) =p

1� 1"

Note that this expression implies that when deviating from the equilibrium path the government extracts

34

all pro�ts from the downstream �rm. Thus the respective deviating values are:

V g;out (bs) = � +

�p

"� 1

��"p

"� 1

��"!(1��)" = � +

�"� 1p

�"�1!(1��)"

""

where bs � (p; �; !)and so, since after deviation from the incumbent the consumers expect that the new incoming bureaucrat

would also deviate from the equilibrium policy, thus obtaining

vout (p; !) =

1("�1)

�!(1��)

p

�"�1"

��"�1+ !�

1� �

Now we turn to the on-equilibrium values. Assuming positive subsidies in equilibrium (a condition that will

be checked below) the problem is:

V g (s; v (s)) = max�;�;fv(s0)s02Sg

(�� + (� � p)��"!(1��)" + �

Xs02S

V g (s0; v (s0))Q (s; s0)

)(32)

subject to

v (s) =1

("� 1)

�!(1��)

�

�"�1+ !� +

� (1� �)

!1��+ �

Xs02S

v0 (s0)Q (s; s0) (33)

and

v0(s0) � vout (p0; !0) +

�c

1� �; 8s0 2 S (34)

V g (s0; v0 (s0)) � V g;out (bs0)� �g

1� �; 8s0 2 S (35)

35

The Lagrangian associated to the problem is:

Lt>T = �� + (� � p)��"!(1��)" + �Xs02S

V g (s0; v0 (s0))Q (s; s0)

+�

"1

("� 1)

�!(1��)

�

�"�1+ !� +

� (1� �)

!1��+ �

Xs02S

v0(s0)Q (s; s0)� v (s)

#+Xs02S

�Q (s; s0) � (s0)hv0(s0)� vout (p0; !0)� �c

i+Xs02S

�Q (s; s0) (s0)

�V g (s0; v0 (s0))� V g;out (bs0) + �g

1� �

�

The �rst order conditions are

�

�1� �

!1��

�8><>:< 0 ) � � = 0

= 0 ) � � 2 [0; 1]> 0 ) � � = 1

(36)

��"�"p��1 � ("� 1)

�!(1��)" � �

�!(1��)

�"�1��" = 0 (37)

� (s0) (1 + (s0))� � (s0) = � (38)

where, by envelope-theorem arguments:

�� = @V g

@v; �� (s0) = @V g (s0; v0 (s0))

@v (s0)

together with the constraints.

The second �rst-order condition yields

"p =��!�(1��) + ("� 1)

��

Thus � (�; s) is

� (�; s) =

�"

�!�(1��) + "� 1

�p (39)

Note that � (�; s) < p if and only if

" < �!�(1��) + "� 1

36

equivalent to

� > !1��

which implies that � � = 1: The rest of the argument follows that in the proof of lemma 3 and proposition

3 of Yared (2010). First we need to show that v is contained in a compact set [vs; �vs] for s 2 S: First,

note that v cannot go to �1 since � > 0; ! > 0; � > 0; � 2 [0; 1] : Thus, for every s 2 S; v has a

lower bound, called vs: Also, since the government would never choose � below �, de�ned as the value

of � that satis�es:

���" � (p� �)!(1��)" = 0

then�!(1��)

�

�"�1has an upper bound, and so has v: Call �vs the upper bound for v at s 2 S: To show

that the interval is also closed, let the sequence fvjg1j=1 with vj being in the set of feasible v0s (whichwas shown to be bounded), such that vj ! v0. We need to show that v0 is also in the set. Clearly, if

vj is in the set, there exist optimal choices f� (vj) ; � (vj)g1j=1. Each element of this sequence satis�esall the constraints, including the weak inequalities. Thus, the limit of f� (vj) ; � (vj)g1j=1 satis�es theincentive compatibility constraints. Also, for each j; the vector (� (vj) ; � (vj)) is contained in a compact

subset of R2: Thus, by the Dominated Convergence Theorem the limit of f� (vj) ; � (vj)g1j=1 allows theconsumer to achieve v0: Thus, v0 must be in the set. Also, to prove that V g is strictly concave in v;

take two di¤erent values for v in the set [vs; �vs] ; say, v0 and v": Associated to them there are optimal

plans (� (v0) ; � (v0)) and (� (v") ; � (v")) : Consider the transformation r (v) � [� (v)]1�" : Thus, for

both values of v we have

v � 1

("� 1)!(1��)("�1)r (v) + !� +

� (1� � (v))

!1��+ �

Xs02S

v0 (s0)Q (s; s0)

Thus the right hand side is clearly linear in (r; �) : Thus, the convex combination � [r (v0) ; � (v0)] +

(1� �) [r (v") ; � (v")] clearly satis�es this equality for v� � �v0 + (1� �) v": Now, the bureaucrat�s

consumption can be rewritten as follows

�� +�r (v)� pr (v)

�""�1

�!(1��)"

This function is linear in � but strictly concave in r: Thus, � [r (v0) ; � (v0)] + (1� �) [r (v") ; � (v")]

allows the bureaucrat to achieve a strictly higher consumption and hence utility. This shows that V g is

strictly concave in v:

The last argument allows to follow the argument given in the proof of proposition 3 in Yared.

37

De�ne

� (s) � �V gv

�s;vout (p; !) + �c

1� �

�;

�� (s) � �V gv (s; �v (s)) ; �v (s) such that V

g (s; �v (s)) = V g;out (bs)� �g

1� �

Suppose �rst that � (st�1) < � (s) : The incentive constraint (34) and concavity of V g implies that

� (st) � � (s) : Thus, from (38) we have that � (st) > 0; implying that � (st) = � (s) : Suppose also

that � (st�1) > �� (s) : Then, the incentive constraint (35) and the monotonicity of V g implies that � (st)

� �� (s) : Thus, from (38) we have that (st) > 0 and so � (st) = �� (s) : Therefore, as in Yared (2010)

this argument shows that

��st�=

8><>:� (s) if � (st�1) < � (s)

� (st�1) if � (st�1) 2�� (s) ; �� (s)

��� (s) if � (st�1) > �� (s)

(40)

This expression allows to get the expression for the equilibrium value of �t (st) ; t > T: In particular the

energy price in period t; history st follows:

�t�st�=

8>>>>>>>>>>>><>>>>>>>>>>>>:

�� (s) ��

"

"�1�V gv�s;vout(p;!)+�

1��

�!�(1��)

�p if � (st�1) > "pt�1h

"�

pt��(st)

�1�+1i�

!t!t�1

�(1��)+"�1

"��pt�1

�t�1(st�1)�1�"+1

��!t�1!t

�(1��)+"�1

!pt

if � (st�1) 2"

"pt�1h"�

pt�(st)

�1�+1i�

!t!t�1

�(1��)+"�1

; "pt�1h"�

pt��(st)

�1�+1i�

!t!t�1

�(1��)+"�1

#� (s) �

�"

"�1�V gv (s;�v(s))!�(1��)

�p if � (st�1) < "pt�1h

"�

pt�(st)

�1�+1i�

!t!t�1

�(1��)+"�1

which is the law of motion that appears in the proposition statement. Note that "

"�1�V gv�s;vout(p;!)+�

1��

�!�(1��)

<

1 if and only if

!(1��) < �V gv

�s;vout (p; !) + �

1� �

�This inequality is clearly ensured when both �c and �g are large enough, since V g

v

�s; v

out(p;!)+�1��

�is

strictly decreasing in its second argument. This ensures that for any st then �t (st) < pt, implying

strictly positive subsidies for every t: Also, the latter implies that � (s) > !1��; implying that �!t = 1

in this equilibrium.

38

Step 4 Let any node such that t = T . Assuming strictly positive subsidies, in this case then we can write the

government�s Bellman equation in the following form:

V gT (sT ; RT ; vT ) = max

xT ;�T ;�T ;fvT+1(s0)s02Sg

�pT �Tx

�TR

1��T � qT

�xT + �T �T + (�T � pT )�

�"T !

(1��)"T

+�Xs02S

V g (s0; vT+1 (s0))Q (s; s0)

)

subject to the same constraints (33), (34) and (35), as in step 3 (together with non-negativity constraints

on xT ). In an interior solution, the choices of �T ; �T and�vT+1 (s

0)s02Ssatisfy the same �rst-order

conditions as in step 3. Thus, the expressions obtained in step 3 also hold for t = T for such variables.

Regarding the choice of xT then its FOC is

�pT �Tx��1T R1��T =

qT�) x�T =

��2pT �TqT

� 11��

RT

Thus replacing the latter on the right hand side of the Bellman equation the �rst two terms simply yield:

�pT �Tq�T

� 11��

RT (1� �)�2�1��

Thus the equilibrium domestic price of the commodity at time of complete exhaustion is equal to

�p�T = �pT

as stated in the proposition. A su¢ cient condition for strictly positive subsidies in period T is:

�1� ��!�(1��)

� ���!�(1��) + "� 1

�"�1�!"(1��)"�" > (1� �) (pmax)

"

��2pmaxq

� �1��

�1

1���

where �� � maxs �� (s) ; � � mins � (s) : Again, when both �c and �g are large enough, and when ! islarge, then such inequality holds.

39

Note also that the Bellman equation can be also

V gT (sT ; RT ; vT )

=

�pT �Tq�T

� 11��

RT�2�1�� (1� �) +

+ max�T ;�T ;fvT+1(s0)s02Sg

(�T �T + (�T � pT )�

�"T !

(1��)"T + �

Xs02S

V g (s0; vT+1 (s0))Q (s; s0)

)

Thus, let

V g (sT ; vT ) = max�T ;�T ;fvT+1(s0)s02Sg

(�T �T + (�T � pT )�

�"T !

(1��)"T + �

Xs02S

V g (s0; vT+1 (s0))Q (s; s0)

)(41)

Then

V gT (sT ; RT ; vT ) =

�pT �Tq�T

� 11��

RT�2�1�� (1� �) + V g (sT ; vT )

Note that the equality (41) is the same Bellman equation as in periods t > T:

Step 5 Suppose now any node st such that t < T:Thus, the Bellman�s equation can be written in this form:

V gt (st; Yt; Rt; vt)

= maxxt;� t;�t;fvt+1(s0)s02Sg

("pt �

1

�

qt�t

�xtRt

�1��+ �t

!#�tx

�tR

1��t + � t�t + (�t � pt)�

�"t !

(1��)"t

+�Xs02ScT

V gt+1

�s0; vt+1 (s

0) ;� � (1� exp (� (Wt + wt)))� Yt � �tx�tR

1��t

�Q (st; s

0)

+� (1� �)�2�1��

Xs02ST

"�pT �Tq�T

� 11�� �

� � (1� exp (� (Wt + wt)))� Yt � �tx�tR

1��t

�#Q (st; s

0)

+�Xs02ST

V g (s0T ; vT (s0))Q (st; s

0)

)

where the maximization is subject to the same constraints (33), (34) and (35). Also, recall that the value

of the Lagrange multiplier �T (s0) in state s0 may depend on vT (s0). In fact, only �� and � depends on

�v; since in the rest of the range for �T the latter is equal to �t; which is independent of vT : Thus, the

40

�rst-order conditions with respect to xt yields:

� (1� �)�2�1��

Xs02ST

�pT �Tq�T

� 11��

Q (st; s0) + �

Xs02ScT

@V gt+1

@Yt+1Q (st; s

0)

=xt�

��pt � �t

xt� qt

��tx�tR

1��t

�while the FOC with respect to vt+1 (s0) (for s0 2 ScT ) is

�t+1 (s0)�1 + t+1 (s

0)�� �t+1 (s

0) = �t (st)

and the FOC with respect to vt+1 (s0) (for s0 2 ST ) is

�T (s0) (1 + T (s

0)) + �Q (st; s0) �T (s

0) = �t (st)

By the Envelope condition:

@V gt

@Yt= (1� �) �tx

�tR

1��t

8<: 2��qt�t(xt)

1��R��2t

��

hpt � 1

�(�t)

iRt

9=;+�(1� �) �tx

�tR

��t � 1

�24� (1� �)�2�1��

Xs02ST

�pT �Tq�T

� 11��

Q (st; s0) + �

Xs02ScT

@V gt+1

@Yt+1Q (st; s

0)

35Using the �rst-order condition with respect to xt :

@V gt

@Yt= (1� �) �tx

�tR

1��t

8<: 2��qt�t(xt)

1��R��2t

��

hpt � 1

�(�t)

iRt

9=;+ xt�

��pt � �t

xt� qt

��tx�tR

1��t

�

=2

�(1� �) qt

xtRt+

�1� (1� �) �t

x�tR�t

��pt �

�t�

�� qtx

1��t

�2�tR1��t

41

Thus the �rst-order condition with respect to xt is

pt ��t�� qtx

1��t

�2�tR1��t

= � (1� �)�2�1��

XsT2ST

�pT �Tq�T

� 11��

Q (st; s0)

+�X

st+12ScT

�2

�(1� �) qt+1

xt+1Rt+1

+

�1� (1� �) �t+1

x�t+1R�t+1

��pt+1 �

�t+1�

�

� qt+1x1��t+1

�2�t+1R1��t+1

#Q (st; s

0)

Recalling that

�pt =1

�

qt�t

�xtRt

�1��+ �t

!Then

xtRt=

��tqt

� 11��

[��pt � �t]1

1��

Thus, the �rst-order condition with respect to xt can be rewritten as

pt ��t�� 1

�

��pt �

�t�

�= � (1� �)�

2�1��

XsT2ST

�pT �Tq�T

� 11��

Q (st; s0)

+�X

st+12ScT

2

�(1� �)

��t+1 [��pt+1 � �t+1]

q�t+1

� 11��

Q (st; s0)

+�X

st+12ScT

8><>:2641� (1� �)

24� 1�

t+1 [��pt+1 � �t+1]qt+1

35�

1��375�pt+1 � �t+1

�

�

� 1�

��pt+1 �

�t+1�

��Q (st; s

0)

where �t follows (20). De�ning

ept � �pt � �tept � ��pt � �t

42

then the latter expression is

ept � ept�

= � (1� �)�1+�1��Et

(�pT �Tq�T

� 11������� t+ 1 = T

)

+�X

st+12ScT

Et

8><>:2 (1� �)

"�t+1ept+1q�t+1

# 11��

+

2641� (1� �)

24� 1�

t+1ept+1

qt+1

35�

1��375 ept+1 � ept+1

�

������� t+ 1 < T

9>=>;The expression just obtained corresponds to (24) in the statement of the proposition.

Step 6 To get the expression (25) is based on the following progression in determining accumulated natural

resource output:

Y1 = R0�1

1��0

�[��p0 � �0]

q0

� �1��

Y2 =

"�

�1� � (1� �) �

(1� �)��1

��R0�

11��0

�[��p0 � �0]

q0

� �1��#�

11��1

�[��p1 � �1]

q1

� �1��

= �

�1� � (1� �) �

(1� �)��1

��

11��1

�[��p1 � �1]

q1

� �1��

�R0�1

1��0 �

11��1

�[��p0 � �0]

q0

� �1���[��p1 � �1]

q1

� �1��

Y3 =

(�

�1� � (1� �) �

(1� �)��2

�� �

�1� � (1� �) �

(1� �)��1

��

11��1

�[��p1 � �1]

q1

� �1��

�R0�1

1��0 �

11��1

�[��p0 � �0]

q0

� �1��)�

[��p1 � �1]q1

� �1��

�1

1��2

�[��p2 � �2]

q2

� �1��

= �

�1� � (1� �) �

(1� �)��2

��

11��2

�[��p2 � �2]

q2

� �1��

���1� � (1� �) �

(1� �)��1

��

11��1 �

11��2

�[��p1 � �1]

q1

� �1���[��p2 � �2]

q2

� �1��

+R0�1

1��0 �

11��1 �

11��2

�[��p0 � �0]

q0

� �1���[��p1 � �1]

q1

� �1���[��p2 � �2]

q2

� �1��

43

In general

Yt = �

�1� � (1� �) �

(1� �)��t�1

��

11��t�1

�[��pt�1 � �t�1]

qt�1

� �1��

���1� � (1� �) �

(1� �)��t�2

��

11��t�2 �

11��t�1

�[��pt�2 � �t�2]

qt�2

� �1���[��pt�1 � �t�1]

qt�1

� �1��

+�

�1� � (1� �) �

(1� �)��t�3

��

11��t�3 �

11��t�2 �

11��t�1

�[��pt�3 � �t�3]

qt�3

� �1��

�[��pt�2 � �t�2]

qt�2

� �1���[��pt�1 � �t�1]

qt�1

� �1��

�:::+ (�1)t�1R0�1

1��0 :::�

11��t�1

�[��p0 � �0]

q0

� �1��

:::

�[��pt�1 � �t�1]

qt�1

� �1��

So

Yt =tXj=1

(�1)j�1 ��1� � (1� �) �

(1� �)��t�j

�"�jl=1

��t�l

�[��pt�l � �t�l]

qt�l

��� 11��#

Thus, replacing the latter in the expression (22) yields the desired result.

Step 7 The �nal step corresponds to check the conditions for which subsidies are strictly positive for t < T .

Subsidies in these nodes are strictly positive if and only if

��t!

�(1��)t + "� 1

�"�1"�"!

"(1��)t p1�"t

�1� �t!

�(1��)t

�> (pt � �pt)Rt

24� 1�

teptqt

35�

1��

where �pt follows (24) and �t follows (20). Given that Rt � �; and also

ept � �pt � �tept � ��pt � �t

44

ept= �ept + �

Xst+12ScT

Et

264264(1� �)

24� 1�

t+1ept+1

qt+1

35�

1��

� 1

375�ept+1 + ept+1�2 (1� �)�

"�t+1ept+1q�t+1

# 11�������� t+ 1 < T

35�� (1� �)�

21��Et

(�pT �Tq�T

� 11������� t+ 1 = T

)

� �ept + Xst+12ScT

�Et

8><>:(1� �)�

24� 1�

t+1ept+1

qt+1

35�

1�� ept+1 + ept+1������� t+ 1 < T

9>=>;� �ept + X

st+12ScT

�Et

8><>:max��ept+1� �

1��;ept+1�

264(1� �)�

24� 1�

t+1

qt+1

35�

1�� ept+1 + 1375������� t+ 1 < T

9>=>;Note that

max

��ept+1� �1��

;ept+1� � ep 11��t+1

conditional on ept+1 � 1. Thus, when imposing a transversality condition on ept and after some algebrathen

ept � �ept + Xst+12ScT

�Et

8><>:ep1

1��t+1

264(1� �)�

24� 1�

t+1

qt+1

35�

1�� ept+1 + 1375������� ept+1 � 1; t+ 1 < T

9>=>;

45

� �ept+

Xst+12ScT

�Et

26640B@�ept+1 + X

st+22ScT

�Et+1

8><>:ep1

1��t+2

264(1� �)�

24� 1�

t+2

qt+2

35�

1�� ept+2 + 1375������� ept+2 � 1; t+ 2 < T

9>=>;1CA

11��

264(1� �)�

24� 1�

t+1

qt+1

35�

1�� ept+1 + 1375������� ept+1 � 1; t+ 1 < T

375� �ept + X

st+12ScT

�Et

264(1� �)�

24� 1�

t+1

qt+1

35�

1�� ept+1 + 1375 (�ept+1) 1

1��

+1Xj=1

�1+jEt

264(1� �)�

24� 1�

t+1

qt+1

35�

1�� ept+1 + 1375

8><>:�jl=1264(1� �)�

24� 1�

t+1+l

qt+1+l

35�

1�� ept+1+l + 13759>=>; (�ept+j) 1

1��

Since the support of pt, �t and qt; and since

�

24(1� �)�

"��1�

q

# �1��

pmax + 1

35 = � : 0 < � < 1

then the right hand side is is bounded by �pmax plus a constant multiplied by11�� : This implies that

epthas a �xed upper bound. Call � this upper bound. Then as long as

�1� ��!�(1��)

� ���!�(1��) + "� 1

�"�1�!"(1��)"�" > pmax�

"��1��

q

# �1��

then subsidies are strictly positive for every node with t < T: Again, a su¢ cient condition for this is to

make both �g and �c large enough.

�

46

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51