Lobbying over Exhaustible-Resource Extraction

CENTER FOR INTERNATIONAL ECONOMICS

Working Paper Series

Working Paper No. 2017-10

Lobbying over Exhaustible-Resource Extraction

Achim Voss and Mark Schopf

October 2017

Lobbying over Exhaustible-Resource Extraction∗

Achim Voss† Mark Schopf‡

October 10, 2017

Abstract

We characterize the resource-extraction path that is chosen by a government which

is influenced by a resource-supplier lobby group. The lobby group pays the govern-

ment in exchange for a deviation from welfare-maximization. We show how the de-

velopment of payments relates to the development of a conflict of interest between

profit-maximization and welfare-maximization. Due to stock-pollution damages, the

government prefers a lower long-run level of cumulative extraction than the lobby

group. Moreover, the resource suppliers’ aim of maximizing profit implies that the dis-

torted extraction may be too slow to maximize welfare, while flow-pollution damages

imply that it may be too fast.

Keywords: Environmental Policy; Exhaustible Resources; Political Economy; Lob-

bying; Time Consistency; Dynamic Programming

JEL Codes: D72; Q31; Q38; Q58

∗We would like to thank Edward B. Barbier, Hassan Benchekroun, Christopher Costello, Manuel Förster,Anke Gerber, B. Michael Gilroy, Jörg Lingens, Daniel Nachtigall, Hendrik Ritter, Marco Runkel, NathalieSchopf, Daniel Schultz, Gerhard Sorger, Gerard van der Meijden, Klaus Wälde and participants of the AURÖWorkshop in Kiel, the SURED in Ascona, the WCERE in Istanbul, the EEA-ESEM in Toulouse, the VfS Confer-ence in Münster, the Tinbergen Seminar in Amsterdam, and the RSERC in Berlin for helpful comments.†School of Economics and Social Sciences, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg,

Germany. Tel.: +49 40 42838 4529. E-mail: [email protected].‡Department of Economics, University of Hagen, Universitätsstraße 41, 58097 Hagen, Germany. Tel.:

+49 2331 987 2449. E-mail: [email protected].

1

1. Introduction

1 Introduction

It is common for environmental economists, policymakers and NGOs to assume that the

influence of natural-resource supplier interest groups distorts policy away from a “social

planner’s” ideal. For example, the Center for Responsive Politics, a watchdog NGO, sus-

pects that the American coal-mining industry uses payments to politicians to lobby against

environmental regulations (Rodriguez, 2015), World Bank representatives see “powerful

lobbies” as an obstacle to “the carbon price that theory recommends” (Fay and Hallegatte,

2015), and a lignite lobby has been accused of exerting influence on the German govern-

ment’s coalition agreement (Delfs, 2013). If we believe that lobbyists can influence policy,

we have to expect a distortion in favor of natural-resource suppliers almost by definition,

at least if there are no sufficiently strong counterforces. The aim of this article is to analyze

how the distortive influence of resource owners develops in the framework of a dynamic

model of resource extraction.

The literature in the tradition of the Grossman and Helpman (1994) common-agency

interest-group model assumes that interest groups offer conditional bribes to the govern-

ment – interpreted as contribution payments to politicians – in order to shift policy into their

preferred direction. This model has considerably enhanced our understanding of lobbies’

political influence. The political economy of exhaustible resources, however, raises some

specific questions, not least due to its inherently dynamic nature. How do policy and its

distortion develop over time? How does the resource owners’ interest in influencing policy

and, relatedly, how do contribution payments develop over time, as more and more of the

resource has been extracted? How do the government’s valuation of payments, the lobby’s

cost of paying them, and both parties’ bargaining power affect both policy and payments?

We characterize resource extraction in an equilibrium influenced by lobbying. As usual

in the literature, the government’s utility is a weighted sum of a utilitarian welfare function

and utility from contribution payments. In order to focus on the relation between the gov-

ernment and the resource-owner lobby, we assume that there are no other lobby groups.

Most interest-group models assume that lobby groups are first movers and offer contribu-

tion schedules that maximize their surplus, subject to the condition that the government is

indifferent about accepting them. In a dynamic setting, an active role for the government

seems more realistic. We therefore model policy and contribution payments as determined

via (Nash) bargaining. The solution is time-consistent, such that both the lobby and the

government always benefit from continued cooperation.

The first main contribution of this article is to provide a model of lobby influence on

resource extraction that simultaneously takes three important mechanisms on this market

into account: resource depletion, environmental pollution, and price elasticity of resource

demand. In our model, the government aims at maximizing welfare, net of environmental

damages. By contrast, the lobby group’s sole objective is intertemporal profit maximization.

It has an incentive to restrain supply in order to keep the price high. It turns out that

Achim Voss and Mark Schopf 2/54

1. Introduction

whether extraction in the lobbying equilibrium occurs faster or slower than in the social

optimum, depends crucially on the relative magnitudes of this incentive on the one hand,

and of environmental externalities on the other hand.

Because extraction costs are increasing in cumulative extraction and the marginal util-

ity of resource consumption is finite,1 welfare-maximizing and profit-maximizing extraction

would both decline to zero as cumulative extraction approaches a convergence level at which

further extraction would not pay off. However, due to stock-pollution and other environ-

mental damages, the profit-maximizing convergence level exceeds the welfare-maximizing

one. Thus, the lobby’s preferred extraction may be either too high or too low from the

social planner’s point of view as long as cumulative extraction is still small, but it definitely

exceeds the welfare-maximizing extraction in the long run.

Our model of the resource market is fairly general. For concreteness, the resource may

be thought of as a fuel like coal or oil. In the model, there is flow pollution, such as

soot or dust, and stock pollution, such as carbon dioxide or permanent landscape changes.

Our analysis suggests that, for example, empirical analyses of lobby influence on natural-

resource policy should take the amount of past extraction, the price elasticity of demand

and potential flow- and stock-pollution damages into account. This result also applies

to the welfare effects of monopolistic extraction. A conflict of interest between profit-

maximization and welfare-maximization may be absent temporarily, but in the long run,

profit-maximizing extraction will be too high.

We propose the basic model in relatively general functional forms. However, we analyt-

ically solve and discuss the lobbying equilibrium with linear-quadratic functions in order

to obtain clear-cut results. Using this analytical solution, our second contribution then is

to relate the conflict of interest and its development to the underlying economic parame-

ters. For instance, we show under which conditions the conflict of interest between welfare

maximization and profit maximization weakens or intensifies over time.

Thirdly, we characterize the contribution payments’ development and show how they

relate to the conflict of interest between the government and the resource lobby. The

present value of expected contributions is always positive, but the payments may decrease

or increase over time and even temporarily become negative. Moreover, in the long run,

contribution payments definitely approach zero as extraction costs increase towards a pro-

hibitively high level and the resource owners preferred extraction goes to zero. At an earlier

point in time, however, the cumulative extraction exceeds the socially optimal level. The

bargained extraction continues, but from that moment on, the government would always

switch to zero extraction if the bargaining would fail. Because the lobby no more has to

factor in that lobbying for higher current extraction deteriorates its future bargaining posi-

tion, its willingness to pay increases sharply. From a technical point of view, we thus show

how a non-negativity constraint affects dynamic bargaining.

1Cumulative extraction is the amount of the resource that has been extracted up to a given moment intime.


2. Relation to the Literature

In our model, the cooperation between the government and the lobby takes the form

of a bargaining in every period about that period’s extraction quantity and payment. The

assumption of repeated bargaining serves the purpose of guaranteeing time-consistency

and thus does not have to be understood literally; a time-consistent contract bargained at

the beginning of time would have to fix the same values for extraction and payments. Sim-

ilarly, the direct choice of extraction quantity is analytically convenient, but not decisive.

As a fourth contribution, we characterize the tax path that would implement such a path of

extraction. The allocation and distribution are equivalent to a direct choice of quantities

if the suppliers receive the tax revenues. Thus, even if the government can only influence

extraction indirectly, the model can be applied, and the tax still reflects both government

and resource-owner interests.

In our baseline model, we assume that failing to reach an agreement once makes it

impossible ever to cooperate again, which ensures tractability. However, in a first extensionwe also report results for a possibly more realistic setting in which the outside option

is to stop cooperation for one period, but the parties expect to cooperate again in the

future. We show that this recursive Nash bargaining solution of Sorger (2006) yields the

same bargained extraction path, and demonstrate under which conditions the contribution

payments coincide in both solution concepts.

Our political model assumes that the government is in power forever. This may best rep-

resent a country with weak institutions in which policy-makers are, for instance, autocrats

or bureaucrats with a large amount of discretion. In a second extension we also consider

the possibility that the government may be ousted with some probability and discuss the

special case of being in power for only one period in detail.

Our paper proceeds as follows. In the next section, we relate the paper to the literature.

In Section 3, we first introduce the model economy and the political agents and then

characterize the lobbying equilibrium with general functional forms. In Section 4, we apply

a linear-quadratic specification in order to characterize specific cases of this equilibrium

in which the lobbying distortion works in different directions or in which it develops in

different ways. In Section 5, we demonstrate how the bargained extraction path can be

implemented via taxes. In Section 6.1, we demonstrate how our model can alternatively

be solved with the recursive Nash bargaining solution. Section 6.2 treats the case of short-

lived governments. Section 7 concludes.

2 Relation to the Literature

Our article contributes to different strands of literature, namely lobbying, resource extrac-

tion, and dynamic bargaining. The government in our model aims to maximize welfare,

but it is willing to choose a different policy in exchange for lobby payments. This follows

the tradition of the Grossman and Helpman (1994, 2001) common-agency model, in which



several lobby groups offer policy-dependent payment schedules to the government. In our

model, there is only a resource-owner lobby, which allows to focus on the factors that shape

the bargaining outcome: the effects of stock and flow pollution, the price elasticity of re-

source demand, bargaining power and payment valuations. Instead of using a payment

schedule, we apply an asymmetric Nash bargaining solution to determine the policy and

the sharing of the surplus. This allows the government to have positive bargaining power

in the bilateral setting (cf. Grossman and Helpman, 2001, Section 7.5 and Goldberg and

Maggi, 1999).2

There are several applications of the Grossman and Helpman (1994, 2001) framework

to environmental policy. The first dynamic model of political influence on resource ex-

traction that is similar to ours is Barbier et al. (2005). They provide an insightful model

and empirical investigation of lobby pressure on governments in developing countries and

demonstrate that lobbying accelerates the speed of deforestation. The private agent in their

model sells the resource at the world-market price, so he would never have an incentive

to limit supply. By contrast, we take the price elasticity of resource demand and differ-

ent kinds of pollution into account, and show that their interaction implies that over the

course of extraction, the lobbying distortion may change its sign and temporarily disappear.

In contrast to Barbier et al. (2005), our model covers cases in which a higher lobby influ-

ence implies a slower resource extraction, namely if extraction reduces the resource price

more than it increases the marginal flow-pollution damage, and cumulative extraction is

low. Another model related to ours is that of Boyce (2010). He analyzes the political

influence of a fishery lobby. In his model, harvesters have a logarithmic utility function

of their resource extraction, such that a motive to restrain supply in order to maximize

profits is also absent. Additionally, we explicitly analyze the development of contribution

payments and the forces that determine their development. In a model of intertemporally

optimal deforestation, Andrés-Domenech et al. (2015) take demand reactions into account,

but they focus on parameter values that imply corner solutions. Finally, Schopf and Voss

(2016) use the Nash bargaining solution to analyze the influence of an environmental or-

ganization and an extraction firm from foreign countries on resource depletion. In this

application, the government and the organization prefer to preserve the resource, while

the firm’s profit-maximizing extraction declines over time. In contrast to the current paper,

the conflicts of interests are thus unambiguous. Instead, Schopf and Voss (2016) focus on

the development of the parties’ bargaining power. It may change over time as more of the

resource is extracted.

In our model of resource extraction, extraction is not limited by a physical stock, but

the resource is economically exhaustible, because extraction costs increase with cumulative

past extraction (cf. Livernois and Martin, 2001). This implies that there are no Hotelling

rents, but Ricardian rents due to increasing costs (Hartwick, 1982). Additionally, there may

2Note that in this literature, lobbying is not clearly distinguishable from bribery, so that our analysis canbe interpreted both ways. For a distinction, see Harstad and Svensson (2011).



be monopoly rents. There is a large literature analyzing how the governments of resource-

importing countries can capture the rents of foreign resource suppliers, either Hotelling

rents (see, e.g., Bergstrom, 1982, Daubanes and Grimaud, 2010 and Keutiben, 2014) or

monopoly rents or both (see, e.g., Wirl, 1994, 1995 and Rubio and Escriche, 2001). In our

model, resource suppliers are part of the same country as consumers, so that a welfare-

maximizing government has no particular interest in distributing rents away from them.

However, the government would like to avoid the monopolistic distortion of supply.

In our model, monopolistic resource extraction is slower than that of competitive, un-

regulated suppliers, which is a standard result in the literature (Solow, 1974; Krautkrae-

mer, 1998).3 A welfare-maximizing government might prefer even slower extraction, how-

ever, due to a second distortion, namely environmental damage. Accordingly, the insight

that monopolistic supply restriction can be a second-best solution for environmental exter-

nalities (Buchanan, 1969; Barnett, 1980) also applies to exhaustible resources (Sweeney,

1977). Conversely, governmental welfare maximization does not always mean slowing

down extraction, even if there are environmental externalities. It thus depends on market

parameters whether monopolistic extraction is too fast or too slow to maximize welfare.

Nevertheless, the accumulation of stock pollution implies that welfare maximization re-

quires curbing extraction from some point onwards (cf. Muzondo, 1993). This difference

between preferred extraction levels shapes the conflict of interest between the government

and the resource lobby in our model.

The last relevant strand of literature is that of dynamic cooperative games. Modeling

the bargaining between the lobby and the government in an intertemporal context requires

an assumption about the bargaining parties’ outside options and commitment. We assume

that there is no commitment, such that the bargaining solution has to be time-consistent.

If no agreement can be reached, the bargaining parties choose uncooperative strategies

forever, such that no further payments are made and the government enforces the welfare-

maximizing amount of extraction. This modeling assumption follows Petrosyan (1997) and

is used, for example, by Kaitala and Pohjola (1990), Fanokoa et al. (2011) and Jørgensen

et al. (2005). It may represent situations in which a bargaining failure destroys trust.

Additionally, it allows analytic solutions. For alternative approaches, see Boyce (2010),

who applies the truthful Markov perfect equilibrium of Bergemann and Välimäki (2003),

and Section 6.1 where we consider an application of the recursive Nash bargaining solution

of Sorger (2006).

3Situations where a monopolist chooses faster extraction are possible, but less common. For an overviewof the literature on monopolistic resource supply, see the list in Fischer and Laxminarayan (2005).


3. The Model

3 The Model

3.1 The Economy

Our aim is to characterize the extraction path of exhaustible resources under the influence

of a resource-owner lobby group. To prepare this, we first lay out the model economy and

characterize two benchmarks for the lobbying model: welfare maximization and monopo-

listic profit maximization. These are the polar cases that span the bargaining range of the

political model that we analyze subsequently.

We consider a partial-equilibrium model, such that the resource sector we consider is so

small that, for example, the interest rate is exogenous. With a resource extraction of qt ≥ 0in period t = 0, 1, 2, ..., let zt ≥ 0 denote cumulative extraction of all previous periods. Then

the equation of motion of z is

zt+1 = zt + qt. (3.1)

In the following, we drop t where no ambiguities arise.

Gross consumer surplus from consuming a quantity q is u(q). We assume u, and all

other functions defined in the following, to be at least twice continuously differentiable,

and the functions and their first and second derivatives to take finite values for all finite q.

Moreover, we assume u′(0) > 0, u′′(q) < 0 and d2(qu′(q))dq2 < 0. Consumers take the market

price p of the resource as given and choose their consumption so as to maximize their net

surplus u(q) − p · q, implying p(q) = u′(q). In the following analysis, we use p both for the

price and for this stationary inverse demand function. For other variables and functions

defining them, we proceed in the same manner so as to economize on notation, and we

add clarifications where necessary. Similarly, we use the prime notation for the derivative

of a function of one variable if the function has no superscript.

Extraction costs c(q, z) ≥ 0 are strictly increasing and convex in current extraction:∂c(q,z)∂q

> 0, ∂2c(q,z)∂q2 ≥ 0. Cumulative extraction increases the marginal cost of current extrac-

tion, ∂2c(q,z)∂q∂z

> 0, and it increases the cost of extracting a positive quantity: ∂c(0,z)∂z

= 0 and∂c(q,z)∂z

> 0 for q > 0. Over time, extraction is limited because extraction costs increase in

past extraction (but not by a physical stock).

Finally, there are environmental damages x(q, z) ≥ 0, which are caused by current and

cumulative resource consumption and therefore referred to as flow pollution and stock

pollution, respectively. x(q, z) is assumed to be additively separable, as well as strictly

increasing and convex in each argument. Being identical to cumulative extraction, stock

pollution does not depreciate over time. Summarizing, the economy’s utilitarian welfare is

w(q, z) = u(q)− c(q, z)− x(q, z). (3.2)

A pair of sequences (qt+s)∞s=0 and (zt+s)∞s=0 starting in period t is feasible if it starts with


3. The Model

the given value zt and fulfills (3.1) for s = 0, 1, 2, .... Below, we assume that only feasible

sequences are considered. The agents in our model have an infinite planning horizon and

a discount rate r > 0, implying a discount factor δ = 1/(1 + r) ∈ (0, 1). Thus, the present

value of the discounted welfare stream evaluated in t, below just called intertemporal

welfare, can be written as

W ((zt+s)∞s=0) =∞∑s=0

δs · w(qt+s, zt+s). (3.3)

Environmental damages x(q, z) are external to resource owners, so that their profit is

π(p, q, z) = p · q − c(q, z) (3.4)

and their intertemporal profit evaluated in t is

Π ((zt+s)∞s=0) =∞∑s=0

δs · π(pt+s, qt+s, zt+s). (3.5)

We now discuss the welfare-maximizing and the profit-maximizing extraction paths.

First, consider a social planner’s choice of (zt+s)∞s=0 with the aim of maximizing

W ((zt+s)∞s=0) as defined in (3.3). The welfare-maximizing extraction as a function of cu-

mulative extraction, qw(z), is given by the following Bellman equation:

Ww(z) = maxq

[u(q)− c(q, z)− x(q, z) + δ ·Ww(z + q)] , (3.6)

where the superscript w denotes the planner’s optimal solution.4 Since this optimization

problem is stationary, the result is a state-dependent extraction function qw(z). As shown

in Appendix A, (3.6) implies the following Euler equation, which is the Hotelling rule

modified for stock-dependent cost effects, as well as flow- and stock-pollution damages:

u′(qwt )− ∂c(qwt , zt)∂q

− ∂x(qwt , zt)∂q

= δ ·[u′(qwt+1)− ∂c(qwt+1, z

wt+1)

∂q−∂x(qwt+1, z

wt+1)

∂q+ ∂c(qwt+1, z

wt+1)

∂z+ ∂x(qwt+1, z

wt+1)

∂z

], (3.7)

where qwt = qw(zt) and zwt+1 = zt + qwt . Thus, the current welfare created by marginal

resource extraction, which is its consumer benefit net of extraction cost and flow external-

ities, must equal the discounted welfare that could be gained from the resource if it were

extracted a period later, plus the additional extraction cost and environmental damages

due to the higher cumulative extraction.

Now suppose that a monopolist supplies the resource. Because the monopolist internal-

4We use this superscript because we need to define other values of intertemporal welfare later on. Forthe existence of the maximum in (3.6), see Lemma A.2 in Appendix A.


3. The Model

izes the price reaction p(q) = u′(q), (3.4) becomes

π(q, z) = p(q) · q − c(q, z), (3.8)

implying a stationary maximization problem. Using the superscript π for monopolistic

profit maximization, the Bellman equation and the Euler equation for this case are

Ππ(z) = maxq

[p(q) · q − c(q, z) + δ · Ππ(z + q)] , (3.9)

p(qπt ) + p′(qπt )qπt −∂c(qπt , zt)

∂q= δ ·

[p(qπt+1) + p′(qπt+1)qπt+1 −

∂c(qπt+1, zπt+1)

∂q+ ∂c(qπt+1, z

πt+1)

∂z

],

(3.10)

where qπt = qπ(zt) and zπt+1 = zt + qπt . The interpretation is similar to that of (3.7), except

that the monopolist does not care about environmental damages, but about his influence

on the price.

The welfare-maximizing extraction path (3.7) and the profit-maximizing extraction

path (3.10) coincide if along the whole path it holds that

p′(qt)qt + ∂x(qt, zt)∂q

= δ ·[p′(qt+1)qt+1 + ∂x(qt+1, zt+1)

∂q− ∂x(qt+1, zt+1)

∂z

]. (3.11)

This would be fulfilled if there were no stock pollution and a marginal increase of extraction

increased flow-pollution damage costs as much as it reduced revenues by reducing the

price. Finally, competitive, unregulated resource owners would neither internalize the

environmental damages nor their influence on the price. Thus, their Euler equation can be

derived as a special case either from (3.10) by substituting the price derivatives with zero,

or from (3.7) by dropping the environmental-damage derivatives.

Consider ∂w(0,z)∂q− ∂x(0,z)

∂z/r. This is the value of the marginal welfare of the first extracted

unit in the current period, net of the present value of its marginal future stock pollution

costs. Similarly, ∂π(0,z)∂q

is the marginal profit of the first extracted unit. In the following, we

refer to both as the first-unit gains for brevity. If cumulative extraction z exceeds zw, where

zw fulfills

∂w(0, zw)∂q

− 1r

∂x(0, zw)∂z

= u′(0)− ∂c(0, zw)∂q

− ∂x(0, zw)∂q

− 1r

∂x(0, zw)∂z

= 0, (3.12a)

then the first-unit welfare cannot be positive, and the non-negativity constraint is binding

for welfare-maximization, qw(z) = 0 (see Proposition A.1 in Appendix A). Likewise, if

cumulative extraction z exceeds zπ, where zπ fulfills

∂π(0, zπ)∂q

= p(0)− ∂c(0, zπ)∂q

= 0, (3.12b)

marginal extraction costs are so high that extraction is not profitable anymore, such that


3. The Model

qπ(z) = 0. The effects on future extraction costs are irrelevant for zw and zπ because∂c(0,z)∂z

= 0. Because environmental damages are an externality (and by p(0) = u′(0)), we

have zw < zπ.

For an easier characterization of the equilibrium extraction and the political equilib-

rium, we make two additional assumptions. Firstly, we follow Salant et al. (1983) in

assuming:

Assumption 1 For all q, z it holds that u′′(q)− ∂2c(q,z)∂q2 − ∂2x(q,z)

∂q2 + ∂2c(q,z)∂q∂z

< 0 and d2(qu′(q))dq2 −

∂2c(q,z)∂q2 + ∂2c(q,z)

∂q∂z< 0.

Then, we can characterize the extraction time as follows:

Lemma 1 Assumption 1 is sufficient for an infinite extraction time to be optimal. Thus, zi fori = w, π is reached asymptotically if the extraction function is qi(z).

Proof. See Salant et al. (1983).

We therefore refer to zw and zπ as convergence levels. Secondly, we make the following

assumption:

Assumption 2 The cost function is of the form c(q, z) = c1(q) + c2(z)q with c′′2(z) ≤ 0.

Then, we can characterize the development of extraction as follows:

Lemma 2 If z > zi, ∂qi(z)∂z

= 0 for i = w, π. If z < zi, ∂qi(z)∂z∈ (−1, 0), such that the respective

optimal extraction declines monotonically in z and, thus, over time.

Proof. See Appendix A.

Assumption 2 allows to characterize the political equilibrium in the next sections. It is

fulfilled by the cost function we assume for the illustrations in Section 4.

3.2 Political Agents

Having characterized extraction that is chosen optimally either from a social planner’s or

from a profit-maximizing resource owner’s point of view, we now turn to politically deter-

mined resource extraction. We consider a government and the sector of resource owners.

The government can choose the extraction quantity – for example via resource taxes or

extraction quotas. The government would prescribe the welfare-maximizing extraction if

nothing convinces it not to do so. In order to be allowed a more favorable extraction quan-

tity, the resource owners form a lobby organization that bargains with the government

about q and a contribution payment m. In line with the lobbying literature, m may be

interpreted as a campaign-contribution payment or any kind of bribe that goes directly to

the politicians’ instead of the state coffers. The bargained extraction quantity may again be


3. The Model

enforced via quotas; we therefore model a direct choice of q in the current section. Alter-

natively, we demonstrate how the lobby outcome can be implemented via resource taxes

in Section 5. Finally, the lobby can simply be a monopolistic resource owner trying to fend

off quantity regulation.

In each period, the government has the following utility function:

g(q,m, z) = w(q, z) + γm. (3.13)

Thus, the government cares about welfare w, but also derives utility γm from the lobby’s

payments. Let the present value of payments to the government discounted to period t, or

intertemporal payments, be denoted by

M ((mt+s)∞s=0) =∞∑s=0

δs ·mt+s. (3.14)

Intertemporal utility of the government is, along the lines of (3.3), the discounted sum of

the utility stream:5

G ((mt+s)∞s=0 , (zt+s)∞s=0) =

∞∑s=0

δs · g(qt+s,mt+s, zt+s) = W ((zt+s)∞s=0) + γM ((mt+s)∞s=0) .

(3.15)

The (collective) utility function of resource owners is

l(q,m, z) = π(q, z)− λm, (3.16)

which is the sector’s profit π net of the lobby’s cost of paying contributions λm. In π, the

price reaction is taken into account – see (3.8). The marginal-cost parameter λ may, for

example, reflect the coordination problems within the group. Intertemporal utility of the

lobby group is, along the lines of (3.5):

L((mt+s)∞s=0 , (zt+s)∞s=0) =

∞∑s=0

δs · l(qt+s,mt+s, zt+s) = Π ((zt+s)∞s=0)− λM ((mt+s)∞s=0) .

(3.17)

The form of the utility functions (3.13) and (3.16) is standard in the interest-group litera-

ture (cf. Grossman and Helpman, 1994, 2001). The fact that they are additively separable,

and linear in contribution-payment utility, allows the derivation of the lobbying equilibrium

in the following analysis.6

5While we assume that the government stays in power forever in order to concentrate on the developmentof the conflict of interest concerning resource extraction, we demonstrate how the bargaining result alters ifwe consider a short-lived government in Section 6.2.

6See Klein et al. (2008) for the complications that can arise when current choices affect future marginalutility from the control variables in settings without commitment.


3. The Model

In our dynamic setting, we have to assume how often bargaining takes place and what

happens in case of disagreement – which determines the parties’ outside options. Con-

cerning the bargaining frequency, we assume that in each period, the government and the

lobby bargain about payments and an extraction quantity for that period. Concerning the

outside options, we assume that if no agreement is reached, cooperation breaks down for-

ever (following Petrosyan, 1997). Then in all future periods, the government unilaterally

decides on the quantity, whereas the lobby does not pay contributions.

The outside-option assumption implies that the welfare-maximizing extraction path is

chosen forever after a disagreement. This allows to derive a closed-form solution. More-

over, it is always possible to determine a bargaining outcome that makes both parties better

off compared to their outside options. By the bargaining-frequency assumption, the solu-

tion will be time-consistent and does not require commitment (cf. Jørgensen and Zaccour,

2001). Thus, it will always be rational to anticipate continued cooperation in equilibrium.

However, the assumption of negotiation each period thus does not have to be understood

literally; to be time-consistent, a contract that is agreed on at the beginning of time has to

yield the same policy as such a periodwise negotiation.

Commitment exists in one sense, however: The fact that disagreement in the bargaining

process leads to a permanent termination of cooperation may be seen as a commitment

not to cooperate (Sorger, 2006). An alternative interpretation would be that the parties no

longer trust each other. Given that party contribution payments in exchange for a favor are

hardly enforceable in court, trust may be crucial.

Finally, the bargaining outcome is determined by the asymmetric Nash bargaining solu-

tion. Thus, the government plays an active role in the bargaining process, and its strength

is represented by the respective parameter in the Nash product. A take-it-or-leave-it offer

by the lobby, which is more typical in the literature, is a special case of this solution.7

3.3 Nash Bargaining Solution

In this section, we define the Nash bargaining solution. We first derive the outside options.

The government’s utility would be the maximized intertemporal welfare defined in (3.6),

and the lobby’s utility would equal the intertemporal profit (3.5) for q = qw(z). We can

define it recursively by

Πw(z) = π(qw(z), z) + δ · Πw(z + qw(z)). (3.18)

Next, we turn to the bargaining outcome. The values of the variables q and m on which the

bargaining parties agree are denoted by the superscript a. They will depend on cumulative

7 The take-it-or-leave-it offers – or contribution schedules – are compared with Nash bargaining in Voss andSchopf (2017). Given quasilinear utility functions, the resulting policies will be identical, but if there is onlyone organized sector as in the current model, payments and utility will only be identical if the government’sbargaining-weight parameter in the asymmetric Nash-bargaining function is zero.


3. The Model

extraction, thus, q = qa(z) and m = ma(z). As long as the intertemporal utility of both

bargaining parties is always higher with the bargained values than it would be in case of

disagreement, they can anticipate that qa(z) and ma(z) will also be chosen in the future.

We can therefore define the equilibrium intertemporal utility functions recursively as well:

Ga(z) = g(qa(z),ma(z), z) + δ ·Ga(z + qa(z)), (3.19a)

La(z) = l(qa(z),ma(z), z) + δ · La(z + qa(z)). (3.19b)

In the same way, W a(z) and Πa(z) define welfare and profit with bargained extraction, and

Ma(z) is the present value of payments. Finally, we define the Nash product and, thereby,

the bargained extraction and payment functions qa(z) and ma(z):

N(q,m, z) ≡ [g(q,m, z) + δ ·Ga(z + q)−Ww(z)]η · [l(q,m, z) + δ · La(z + q)− Πw(z)]1−η ,

(3.20a)

(qa(z),ma(z)) ∈ argmaxq,m

[N(q,m, z)|q ≥ 0] , (3.20b)

where η ∈ [0, 1] measures the bargaining power of the government relative to that of the

lobby group. We define µ ≡ γ/λ, the weighted objective function v(q, z) ≡ w(q, z) + µ ·π(q, z) and, using this, the weighted intertemporal objective function:

V ((mt+s)∞s=0 , (zt+s)∞s=0) =

∞∑s=0

δs · v(qt+s,mt+s, zt+s). (3.21)

We can then characterize qa(z) and ma(z) as follows:

Proposition 1 (Nash Bargaining Extraction Path and Payments) The solution of theNash bargaining (3.20) is given as follows. The extraction function qa(z) maximizes V (z); itthus solves the following weighted Bellman equation:

V a(z) = maxq

[v(q, z) + δ · V a(z + q)] . (3.22)

The intertemporal payments Ma(z) and the payments within the periods ma(z) then are:

Ma(z) = η

λ· [Πa(z)− Πw(z)] + 1− η

γ· [Ww(z)−W a(z)] , (3.23a)

ma(z) = Ma(z)− δ ·Ma(z + qa(z)). (3.23b)

Proof. See Appendix B.

Thus, q is chosen so as to maximize a weighted sum of intertemporal welfare and intertem-

poral profit. In this sense, the bargained extraction maximizes the joint product, and η


3. The Model

determines how it is shared. The weight µ depends on the government’s marginal utility γ

of receiving money relative to the lobby’s marginal cost λ of paying it.

Along the lines of (3.7) and (3.10), the bargaining yields an Euler equation that is a

weighted sum of the planner’s and the monopolist’s Euler equations:

p(qat )−∂c(qat , zt)

∂q− ∂x(qat , zt)

∂q+ µ ·

[p(qat ) + p′(qat )qat −

∂c(qat , zt)∂q

]

= δ ·

p(qat+1)− ∂c(qat+1, zat+1)

∂q−∂x(qat+1, z

at+1)

∂q+ ∂c(qat+1, z

at+1)

∂z+ ∂x(qat+1, z

at+1)

∂z

+ µ ·[p(qat+1) + p′(qat+1)qat+1 −

∂c(qat+1, zat+1)

∂q+ ∂c(qat+1, z

at+1)

∂z

], (3.24)

where qat = qa(zt) and zat+1 = zt + qat . The higher the government values the contributions

relative to the lobby’s cost of paying them, the more the lobby determines the path. In

line with (3.12), we also note that there is a convergence level za, such that qa(z) = 0 for

z ≥ za. It is defined by

∂w(0, za)∂q

+ 1r

∂w(0, za)∂z

+ µ · ∂π(0, za)∂q

= 0. (3.25)

Thus, za is between zπ and zw, and the weighting parameter µ determines how close it is

to which of these levels. Extraction continues after the welfare-maximizing convergence

level has been reached, and the payment must compensate the government for continuing

extraction.

While the bargaining power of the government η does not influence the extraction, it

does influence the payments. Their present value as defined in Proposition 1 takes away

all of the resource owners’ additional profit if the government has all the bargaining power

(η = 1), and only compensates the government for not maximizing welfare in the opposite

case (η = 0). In any case, the present value of payments is positive.

In a full-commitment situation, the lobby could pay Ma(z0) at the beginning of time.

Without the possibility to commit, the present value of payments has to beMa(z) as defined

in Proposition 1 in each period. Substituting (3.23a) into (3.23b) and simplifying, the

payments within the periods can be determined as:

ma(zt) = η

λ· [π(qat , zt)− π(qwt , zt)] + 1− η

γ· [w(qwt , zt)− w(qat , zt)]

+ δ ·η

λ·[Πw(zat+1)− Πw(zwt+1)

]+ 1− η

γ·[Ww(zwt+1)−Ww(zat+1)

]. (3.26)

First consider z ≥ zw, such that qw(z) = 0. Then the terms in the second line cancel out.

Thus, ma(z) is positive for the same reason that makes Ma(z) positive – the lobby obtains

higher profit if cooperation is successful, but welfare is reduced. For z < zw, however,

the payments also take into account how the future outside options change due to current


3. The Model

cooperation. These outside options are intertemporal welfare and profit along the welfare-

maximizing extraction path. The government’s part of (3.26) is always positive because it

reflects the welfare loss from deviating for one period from the welfare-maximizing extrac-

tion path. The lobby’s part, however, can temporarily be negative; deviating one period

from the government’s preferred extraction path to the bargained one does not necessarily

increase intertemporal profits.

Given that cumulative extraction increases over time, the development of payments can

be understood using the derivative of ma(z) with respect to z. While they generally depend

on the specific functions, as we will see in the next section, we can see that there are two

threshold levels of z at which this development changes qualitatively. The first time this

happens when the government’s convergence level zw is anticipated to be crossed in the

next period. This is the case when z = zw, where

zw ≡ zw − qa(zw). (3.27)

The second time is when zw is actually crossed. We characterize these potential changes as

follows:

Proposition 2 (Development of Payments for z = zw and z = zw.) Suppose that η = 0or ∂qw(zw)

∂z= 0. Then ∂ma(z)

∂zis continuous for all z ∈ [0, za). Now suppose that η > 0 and

∂qw(zw)∂z

< 0. Then, ∂ma(z)∂z

jumps upwards at z = zw, and it jumps downwards at z = zw.

Proof. See Appendix B.

In general, as z increases from one period to the next, all components that determine

ma(z) change over time. However, for Πw(z) it holds that

∂Πw(zt)∂z

=[∂π(qwt , zt)

∂q+ δ

∂Πw(zt + qwt )∂z

]∂qwt∂z

+ ∂π(qwt , zt)∂z

+ δ∂Πw(zt + qwt )

∂z, (3.28)

that is, the envelope theorem is not applicable because extraction is not chosen in a profit-

maximizing way. Πw(z) declines in z near the government’s convergence level, because

then ∂π(qwt ,zt)∂q

> 0 by qwt < qπt , and ∂qwt∂z∈ (−1, 0). However, Πw(z) becomes zero at z = zw

and remains so afterwards. Moreover, ∂qw(z)∂z

can jump discontinuously from a negative

value to zero at this point as the government’s optimization becomes constrained. Then,∂Πw(z)∂z

jumps discontinuously at z = zw.

This discontinuity in ∂Πw(z)∂z

leads to discontinuities in ∂ma(z)∂z

at z = zw and z = zw.

The derivative first jumps at z = zw, when the lobby’s future outside option in case of

current cooperation, Πw(zat+1), is not reduced by the increase in z anymore. Accordingly,

its willingness to pay to ensure cooperation increases sharply. By contrast, the lobby’s

current outside option, π(qwt , zt) + δΠw(zwt+1), stops deteriorating by the increase in z when

z = zw. Accordingly, its willingness to pay to avoid non-cooperation decreases sharply.

While the dynamic version of the Nash bargaining solution laid out in the current section


4. Explicit Example

Table 1: Explicit functions.

Functions Explicit forms

u(q) =(ρq − ρqq

2 q)· q

p(q) = u′(q) = ρq − ρqqqc(q, z) =

(κqzz + κq + κqq

2 q)· q

x(q, z) = χzz + χqq + χqq2 q

2

can be applied quite generally, we now turn to a linear-quadratic specification. This allows

to characterize into which direction the lobby influence distorts resource extraction, based

on the properties of the welfare and profit function and on the amount of cumulative

extraction.

4 Explicit Example

4.1 The Lobbying Equilibrium

We now derive the extraction paths and payments for a linear-quadratic specification. The

assumed explicit forms of the functions are summarized in Table 1. Collecting terms, we

have

w(q, z) = (βw − κqzz) · q − αw2 q2 +

(q

r− z

)· χz, (4.1a)

π(q, z) = (βπ − κqzz) · q − απ2 q2, (4.1b)

v(q, z) = (βa − κqzz) · q − αa2 q2 +

(q

r− z

)· χz, (4.1c)

where

αw ≡ κqq + ρqq + χqq, απ ≡ κqq + 2ρqq, αa ≡αw + µαπ

1 + µ, (4.2a)

βw ≡ ρq − κq − χq −χzr, βπ ≡ ρq − κq, βa ≡

βw + µβπ1 + µ

. (4.2b)

We assume all coefficients to be positive if not stated otherwise, and we assume ρq >

κq + χq + χz/r, implying βπ > βa > βw > 0. Before explaining the effects at work in the

utility functions (4.1), let us for the moment take them as given and consider their dynamic

optimization as in (3.6), (3.9) and (3.22). To do so, we note that Assumption 1 implies

αi > κqz for i = w, π, a, (4.3)


4. Explicit Example

and we define

ψi ≡2

αi +√α2i + 4

rκqz (αi − κqz)

≤ 1αi

for i = w, π, a. (4.4)

The following proposition summarizes the welfare-maximizing, profit-maximizing and the

bargained resource extraction paths:

Proposition 3 (Explicit Example: Extraction Paths) Welfare-maximizing, profit-maximizing and the bargained extraction are functions of the amount of cumulative resourceextraction z:

qi(z) =

ψi · (βi − κqzz) if z ≤ zi,

0 if z > zi,for i = w, π, a, (4.5)

where zi = βi/κqz. Cumulative extraction levels along paths starting in t develop as follows:

zit+s =

zi − (1− ψiκqz)s · (zi − zt) if zt ≤ zi,

zt if zt > zi,for i = w, π, a, (4.6)

where s = 0, 1, 2, ... and 0 < 1− ψiκqz ≤ 1.

Proof. See Appendix C.

An optimal extraction path is characterized by two properties. The first is the convergence

level, that is, the level of cumulative extraction that would imply zero extraction. In the

previous section, we have seen that it is defined by zero first-unit gains. By (4.1), these

first-unit gains are

∂w(0, z)∂q

− χzr

= βw − κqzz,∂π(0, z)∂q

= βπ − κqzz,∂v(0, z)∂q

− χzr

= βa − κqzz,

respectively. They are zero for zi ≡ βi/κqz. Because the first-unit profit does not include

flow-pollution damage costs (χq) or the future stock-pollution damage costs (χz/r), we

have βπ > βa > βw and zπ > za > zw. Thus, when z = zw, it would be welfare-maximizing

to stop extracting, but the bargained extraction would be positive.

The second property is the amount of extraction given any level of cumulative extraction

below the convergence level. Because optimal extraction then is positive, we always have

zt+s > zt, which by (4.5) implies that qi(z) decreases over time. However, by (4.6) zi is

only reached asymptotically.

The amount of extraction given βi−κqzz is determined by the respective ψi term. Since

ψi does not change the total amount of cumulative extraction in the long run, but only

how rapidly this amount is approached, ψi represents the speed of convergence. A larger ψi


4. Explicit Example

implies that qi(z) is higher for a given z, but as z then increases, qi(z) also declines more

rapidly.

What then determines the size of ψi? This term summarizes the decrease in marginal

gains due to effects both within the current period and in the future. It is reduced by κqzbecause higher extraction today increases future extraction costs, and a higher r increases

ψi because stronger discounting makes effects in the future less relevant. In (4.1), we can

see that αw and απ are the (absolute) slopes of the marginal welfare and profit functions,

respectively. If they are high, marginal welfare and profit strongly decrease in q, which

speaks for postponing resource extraction in order to smooth out marginal welfare and

marginal profit over time. Marginal consumer surplus is part of marginal welfare, but it

also reflects the resource price. Thus, ρqq counts double for marginal profit, namely for the

price and for its change in reaction to higher supply. On the other hand, environmental

damages are not part of profit. Thus, a monopolist’s extraction may be too slow from a

social planner’s point of view due to a market-power effect, or too fast due to a marginalflow-pollution effect. The former exceeds the latter if ρqq > χqq, which implies απ > αa > αw.

A higher lobby influence due to a higher µ increases the weighted average of the bar-

gaining parties’ first-unit gains βa − κqzz and, thus, the bargained convergence level za.

On the other hand, it increases the speed of convergence if and only if αw > απ. The

bargained extraction path then implies intertemporal welfare and profit resulting from the

Nash bargaining:

Proposition 4 (Explicit Example: Nash Bargaining Welfare and Profit) Intertemporalwelfare and profit resulting from the Nash bargaining are

W a(z) = Aw|a · qa(z)2 −Bw|a · qa(z)− χzz

1− δ , (4.7a)

Πa(z) = Aπ|a · qa(z)2 −Bπ|a · qa(z) ≥ 0, (4.7b)

where

Ai|a ≡1ψa− αi

2

1− δ (1− ψaκqz)2 for i = w, π, (4.8a)

Bi|a ≡βa − βi

1− δ (1− ψaκqz)for i = w, π. (4.8b)

Proof. See Appendix C. Substituting ψa into Aπ|a, we see that the numerator must exceed

αa− 12απ = 1

2κqq + χqq + µ1+µ (ρqq − χqq) > 0. Thus, Aπ|a ≥ 0 and Bπ|a < 0, so that Πa(z) > 0

for qa > 0.

From Section 3.3, we know that disagreement implies q = qw(z) and m = 0 forever. In case

of disagreement in the bargaining, intertemporal welfare and profit thus are:

Proposition 5 (Explicit Example: Nash Bargaining Disagreement Welfare and Profit)


4. Explicit Example

In case of disagreement in the bargaining, intertemporal welfare and profit are

Ww(z) = Aw|w · qw(z)2 − χzz

1− δ , (4.9a)

Πw(z) = Aπ|w · qw(z)2 −Bπ|w · qw(z) ≥ 0, (4.9b)

where

Ai|w ≡1ψw− αi

2

1− δ (1− ψwκqz)2 > 0, for i = w, π, (4.10a)

Bπ|w ≡βw − βπ

1− δ (1− ψwκqz)< 0. (4.10b)

Proof. See Appendix C. Substituting ψa into Aπ|w, we see that the numerator must exceed

αw − 12απ = 1

2κqq + χqq > 0. Thus, Aπ|w > 0 and Bπ|w < 0, so that Πw(z) > 0 for qw > 0.

Finally, we can determine the equilibrium payments:

Proposition 6 (Explicit Example: Nash Bargaining Payments) The present value of pay-ments is

Ma(z) = η

λ·Aπ|a · qa(z)2 −Bπ|a · qa(z)−

[Aπ|w · qw(z)2 −Bπ|w · qw(z)

]+ 1− η

γ·Aw|w · qw(z)2 −

[Aw|a · qa(z)2 −Bw|a · qa(z)

]. (4.11)

The payments in any period can be determined as

ma(z) = η

λ·[(βπ − κqzz) ∆a,w(z)− απ

2 ∆2a,w(z)

]− 1− η

γ·[(βw − κqzz) ∆a,w(z)− αw

2 ∆2a,w(z)

]+ δ ·

η

λ·[Aπ|w ·∆2w+

a,w (z)−Bπ|w ·∆w+a,w(z)

]− 1− η

γ· Aw|w ·∆2w+

a,w (z), (4.12)

where

∆a,w(z) ≡ qa(z)− qw(z), ∆2a,w(z) ≡ qa(z)2 − qw(z)2, (4.13a)

∆w+a,w(z) ≡ qw(z + qa(z))− qw(z + qw(z)), ∆2w+

a,w (z) ≡ qw(z + qa(z))2 − qw(z + qw(z))2.

(4.13b)

Proof. Knowing W a(z) and Πa(z) from Proposition 4, and Ww(z) and Πw(z) from Proposi-

tion 5, we can determine the payments using Proposition 1.

Intertemporal utility of the government and the lobby, Ga(z) and La(z), are then given by

the weighted sums (3.15) and (3.17). The development of ma(z) depends on the develop-

ment of the difference between the bargained extraction qa(z) and the welfare-maximizing


4. Explicit Example

extraction qw(z) in its different shapes as shown in the Proposition.8 In detail, this develop-

ment thus depends on the parameters that determine qa(z) and qw(z), i.e., on ψi and βi for

i = a, w (see Proposition 3). Therefore, we will now turn to discuss the different possible

cases.

4.2 The Lobbying Distortion and its Development

We have seen that the bargained extraction path is a compromise between the welfare-

maximizing and the profit-maximizing one. Compared to welfare-maximization, this com-

promise implies a lobbying distortion. We characterize this distortion and its development

as the difference between the bargained and the welfare-maximizing extraction quantity

(that would, for any z, be the starting point of the extraction path if the two parties could

not agree). Using Proposition 3, the lobbying distortion is

∆a,w(z) ≡ qa(z)− qw(z) =

(ψa − ψw) (βa − κqzz) + ψw (βa − βw) if z < zw,

ψa (βa − κqzz) if z ∈ [zw, za) ,

0 if z ≥ za.

(4.14)

In what follows, we also demonstrate how this lobbying distortion and its development

determine the payments within the periods for different levels of cumulative extraction.

The third line of (4.14) simply reflects that for z ≥ za, the parties would not agree on

positive extraction. Because the government would also prefer zero extraction, ∆a,w(z) = 0,

and the payments are zero as well (see Proposition 6). As discussed in Sections 3.1 and

3.3, in equilibrium, za will not be reached in finite time, such that this case would only be

relevant if z were already at this level in the beginning of the planning horizon.

We turn to the case z ∈ [zw, za), which is represented by the second line of (4.14).

Then, qw(z) = 0, such that ∆a,w(z) = qa(z) is positive. After cumulative extraction has ex-

tended beyond the welfare-maximizing convergence level, the resource owners’ influence

still implies positive extraction. The payments in these periods reflect this distortion. By

Proposition 6, ma(z) for z ∈ [zw, za) is

ma(z) = η

λ·[(βπ − κqzz) qa(z)− απ

2 qa(z)2]

︸︷︷︸>0

−1− ηγ·[(βw − κqzz) qa(z)− αw

2 qa(z)2]

︸︷︷︸<0

.

In every period, the payments at least compensate the government for the welfare loss in

that period and the additional stock-pollution damages in the future, due to the choice of

qa > 0 instead of qw = 0. On the other hand, the payments are at most as high as the

8The dependence on the different ∆ versions in (4.13) is used in order to conserve the structure of ma(z)with the familiar parameters. However, it is shown in Appendix D that ma(z) can also be expressed asdepending only on ∆a,w(z).


4. Explicit Example

resource owners’ additional profit. Thus, payments are definitely positive, and increase

with η.

The development of payments can be understood by bearing in mind that z always in-

creases, so that differentiating ma(z) helps to understand the qualitative behavior between

one period and another:

∂ma(z)∂z

= η

λ·[−κqzqa(z)︸︷︷︸

<0

+ [βπ − κqzz − απqa(z)]︸︷︷︸>0

∂qa(z)∂z︸︷︷︸<0

]

− 1− ηγ·[−κqzqa(z)︸︷︷︸

<0

+ [βw − κqzz − αwqa(z)]︸︷︷︸<0

∂qa(z)∂z︸︷︷︸<0

].

Firstly, a higher z implies higher costs of extracting qa. This directly reduces the resource

owners’ profits, so that they are less willing to pay for implementing a positive qa instead

of qw = 0, and it increases the welfare loss this would entail, so that the government would

demand a higher compensation. The former effect speaks in favor of declining, the latter in

favor of increasing payments over time. Secondly, a higher z reduces qa, which reduces both

profits and the welfare loss from cooperation. This indirect effect of z speaks in favor of

declining payments over time. Thus, payments definitely decline over time if the bargaining

power of the government is high enough for profits to determine the compensation.9 By

contrast, if the lobby receives all the gains of cooperation (η = 0), so that the welfare

loss determines the compensation, the effects work into opposing directions. In that case,

payments are always declining in z if the indirect, positive effect of z on welfare outweighs

the direct, negative one. Else, if the direct effect outweighs the indirect one and η is

small, payments increase for zw and z values not much higher, but for some z reach a

maximum and then converge towards zero.10 Since additional marginal-cost increases

are irrelevant if no extraction takes place, the direct effect vanishes with the equilibrium

extraction quantities. Note that a higher η both implies higher payments and makes it less

likely that payments increase. Thus, they can only increase over time (for z ∈ [zw, za)) if

they are small in the first place.

We now to the first line of (4.14). This case is relevant in the beginning of the planning

horizon if it then holds that z < zw. Now, implementing the lobbying equilibrium means

choosing qa(z) instead of some positive qw(z). The payment formula thus becomes more

complex:

ma(z) = η

λ·[(βπ − κqzz) ∆a,w(z)− απ

2 ∆2a,w(z)

]− 1− η

γ·[(βw − κqzz) ∆a,w(z)− αw

2 ∆2a,w(z)

]+ δ ·

η

λ·[Aπ|w ·∆2w+

a,w (z)−Bπ|w ·∆w+a,w(z)

]− 1− η

γ· Aw|w ·∆2w+

a,w (z). (4.12)

9We demonstrate in Proposition D.1 in Appendix D that η (1 + µ) ≥ 1/2 is sufficient.10For η = 0, payments initially increase over time if αw < απ, see Proposition D.1 in Appendix D.


4. Explicit Example

Table 2: Lobbying-equilibrium cases.

z < zw

Case Relation ∆a,w(0) ∆′a,w(z)I ψw < ψa > 0 < 0II ψw = ψa > 0 = 0III ψaβ/βw > ψw > ψa ≥ 0 > 0IV ψw > ψaβ/βw > ψa < 0 > 0

For z ≥ zw, ∆a,w(z) > 0 and ∆′a,w(z) < 0.For z → za, ∆a,w(z) = 0 and ∆′a,w(z) = 0.

The first line reflects the same effects as discussed above for a binding constraint, z ∈[zw, za). But now ∆a,w(z) may, as we discuss in detail below, be positive or negative and

change its sign over time. If, for example, ∆a,w(z) < 0, then there is a welfare loss in

the current period due to cooperation, for which the government has to be compensated.

The second line reflects the change in the future outside options due to current coopera-

tion. The future intertemporal welfare and profit along the welfare-maximizing path are

declining in z. Cooperation thus impairs the future outside options if ∆a,w(z) > 0, and

improves them if ∆a,w(z) < 0. Depending on whether the lobby’s part or the government’s

part in the second line dominates, and depending on the sign of ∆a,w(z), this change in

future intertemporal utilities can lead to higher or lower payments. If ∆a,w(z) > 0 (and

η > 0), payments may even temporarily turn negative when the reduction of the lobby’s

future outside option exceeds its profit gain in the current period (see Proposition D.3 in

Appendix D). Furthermore, the development of payments is much more complex than for

z ≥ zw and also depends on the sign and the development of the lobbying distortion.

For the discussion of this development, it is useful to note that, by (4.2) and (4.4):

ψw R ψa R ψπ ⇔ αw Q αa Q απ ⇔ χqq Q ρqq.

This leads to four distinct cases in which the lobbying distortion and, thus, the contribution

payments develop in a qualitatively different manner, namely χqq > ρqq ⇔ ψw < ψa (Case

I), χqq = ρqq ⇔ ψw = ψa (Case II), and for χqq < ρqq first ψwβa/βw > ψw > ψa (Case III) and

finally ψw > ψaβa/βw > ψa (Case IV). In what follows, we discuss the lobbying distortion

and its development in each of the Cases I–IV, which are summarized in Table 2. Since

the influences in the payment formula are quite involved, we discuss them in a relatively

informal manner, referring to Appendix D for formal derivations. We illustrate the cases

using diagrams for specific parameter values, namely r = 1/15, ρq = 100, κqz = 1/10,

κq = 0, κqq = 0, χz = 2, and χq = 0 so that βπ = 100 and βw = 70. Initially, cumulative

extraction is zero, z0 = 0. We assume λ = γ = 1, so that the lobby’s policy weight is

µ = 1 and βa = 85, and the bargaining power is symmetric, η = 1/2. The relation of the

remaining economic parameters, ρqq and χqq, constitutes the four cases.


4. Explicit Example

Figure 1: Extraction path and cumulative extraction path for χqq > ρqq (Case I).

Case I: χqq > ρqq.

In Case I, the marginal flow-pollution effect outweighs the market-power effect, χqq >

ρqq, so that ψw < ψa < ψπ. Because βa > βw, ∆a,w(z) is definitely positive: Due to all kinds

of pollution, the government would reduce extraction in case of disagreement.

Figure 1 shows the paths of extraction (left-hand-side figure) and cumulative extrac-

tion (right-hand-side figure) for χqq = 5, ρqq = 2. The dashed gray curves are the profit-

maximizing paths and the dotted gray curves are the welfare-maximizing paths, if they

were followed from t = 0 on. The cumulative extraction of the monopolist, zπ, would

always exceed that of the social planner, zw. Moreover, by χqq > ρqq, the monopolist does

not smooth extraction as much over time as the social planner. Thus, qπt quickly decreases,

and is below qwt from some moment onwards. The equilibrium extraction path that the

bargaining parties agree on, qa(z), is a compromise between these extremes, shown as the

black curve in Figure 1. The cumulative extraction converges to z, which is a weighted

average between zπ and zw.

From the point of view of this lobbying equilibrium, qπt and qwt are only hypothetical

reference paths once qa(z) has been chosen for a while. By contrast, the dash-dotted black

curve represents the extraction qw(z) that would be chosen in the corresponding period

after disagreement, given that z up to that period has been determined by the bargained

extraction. Each point along that curve represents extraction in the first period of deviation

from the lobbying equilibrium to the welfare-maximizing path, so that each point is the

beginning of an extraction path converging to q = 0, while cumulative extraction would

converge to zw from then on. This only changes when z ≥ zw; then the threat would be

to choose q = 0 immediately and forever. Each term of the right-hand side of (4.14) is

positive, so that ∆a,w(z) > 0 for z < za; the government would always switch to lower

extraction. At the same time, the size of this change declines in z, both in the periods in

which the non-negativity constraint is binding for the government and in those in which


4. Explicit Example

Figure 2: Contribution payment path for χqq > ρqq (Case I).

it is not; ∆′a,w(z) < 0 for z < za. This can also be seen in Figure 1: the vertical difference

between the solid black curve and the dash-dotted curve is always declining.

Figure 2 shows the development of contribution payments. Because ∆a,w(z) > 0, choos-

ing qa instead of qw in the current period implies a higher z in the future. This worsens the

lobby’s future outside option. However, the profit gain in the current period dominates, so

that the payments are always positive (see Proposition D.2 in Appendix D).

In Case I, the decline of the lobbying distortion over time leads to declining payments

for z < zw (see Proposition D.2). For z ≥ zw, we discussed above that payments could

increase initially, but definitely vanish in the long run. With symmetric bargaining power,

which we assumed for the figure, payments are always declining in z for z ≥ zw (see

Proposition D.1). Since

∂qw(zw)∂z

= −ψwκqz < 0

and η > 0, we know from Proposition 2 that ∂ma(z)∂z

jumps upwards at z = zw, and that

it jumps downwards at z = zw. This can be seen in the figure, where payments increase

between zw and zw. For z ≥ zw, an increase in z does not deteriorate the lobby’s futureoutside option in case of current cooperation anymore. The increase in z deteriorates its

current outside option until z = zw, when the outside option becomes zero. Thus, the

lobby’s willingness to pay for cooperation increases at z = zw, and its willingness to pay to

avoid non-cooperation declines at z = zw.

Case II: χqq = ρqq.

The developments of extraction and cumulative extraction for the second case are de-

picted in Figure 3. In this knife-edge case, the marginal flow-pollution effect and the

market-power effect cancel out: χqq = ρqq (= 2 in the figure) and thus ψw = ψa = ψπ. If


4. Explicit Example

Figure 3: Extraction paths and cumulative extraction paths for χqq = ρqq (Case II).

Figure 4: Contribution payment path for χqq = ρqq (Case II).

we additionally had χq = χz = 0, the bargained, the profit-maximizing and the welfare-

maximizing extraction would coincide. Accordingly, the difference between qa(z) and qw(z)is driven solely by the difference in first-unit gains or, equivalently, the difference between

the convergence levels, as long as the government and the lobby group want positive ex-

traction. Thus, for z < zw, (4.14) simplifies to

∆a,w(z) = ψa (βa − βw) ,

such that ∆′a,w(z) = 0; each period, qa(z) and qw(z) decrease by the same amount. Only

when the non-negativity constraint becomes binding for the government, this cannot con-

tinue; qw(z) is then and remains zero, while qa(z) continues to decline.

Figure 4 shows the development of contribution payments. They remain at a positive,


4. Explicit Example

Figure 5: Extraction paths and cumulative extraction paths for χqq < ρqq and ψwβw < ψaβa(Case III).

constant level as long as qw(z + qa) > 0. Once z ≥ zw, the payments display the same

discontinuity and long-run behavior as in Case I.

Cases III and IV: χqq < ρqq.

Now suppose that χqq < ρqq, so that ψw > ψa > ψπ. ∆a,w(z) is then ambiguous, because

βa > βw. However, we have

∆′a,w(z) = − (ψa − ψw)κqz > 0.

Thus, if ∆a,w(z) is positive for z = 0, then it will remain so as z increases. We have

∆a,w(0) ≥ 0 ⇔ ψaβa − ψwβw ≥ 0.

This defines Case III. The market-power effect is stronger than the marginal flow-pollution

effect – as q is increased, the decrease in marginal revenue, and even the decrease in the

joint marginal gains, is stronger than that of marginal welfare. However, the additional

pollution effects in βw outweigh this. The welfare-maximizing path would then imply

extraction below qa, which in turn is below qπ. Thus, the government would still reduce

extraction in case of disagreement. In Figure 5 (where χqq = 2, ρqq = 5), this can be

seen the same way as in the previous cases. ∆a,w(z) > 0 and ∆′a,w(z) > 0 imply that the

preferred extraction quantities diverge over time. Accordingly, and in contrast to Cases I

and II, payments increase – see Figure 7a. Given that the bargaining parties can anticipate

high payments in the future, payments may even be negative, and they can be declining in

z for small values of z; in Proposition D.3, it is shown under which conditions this is the

case. Once qw(z+ qa) = 0, the development of payments is similar to that in Cases I and II.


4. Explicit Example

Figure 6: Extraction paths and cumulative extraction paths for χqq < ρqq and ψwβw > ψaβa(Case IV).

If, on the other hand, ψwβw > ψaβa, we have Case IV. The government’s preferred

extraction exceeds the bargained extraction for small z, in particular for z = 0: ∆a,w(0) < 0.

But as time goes by, qw(z) again decreases more than qa(z). Thus, it becomes lower than

the bargained extraction for large z, in particular for z = zw < za. Put another way, in total,

the lobby group would want to extract more than the government, but a profit-maximizing

extraction path would imply a stronger postponing of extraction. Substituting ∆a,w(z) = 0in (4.14) shows that there is a switching-level of z, defined by

z = za − ψwψw − ψa

(za − zw)(

= zw − ψaψw − ψa

(za − zw)). (4.15)

Therefore, the lobbying distortion – the absolute value of ∆a,w(z) – first declines until qw

and qa coincide; afterwards, the two diverge again as ∆a,w(z) increases, until the non-

negativity constraint on the welfare-maximizing extraction becomes binding; see the left-

hand-side of Figure 6 (where χqq = 0, ρqq = 5).

Figure 7b shows the development of contribution payments. The curve first slopes

downwards. Payments are zero in the period when ∆a,w(z) = 0. Afterwards, they definitely

turn negative as ∆a,w(z) becomes positive (see Proposition D.3). In the figure, the curve

slopes upwards and the payments become positive before qw(z) starts to be constrained.

In general, it is possible that the payments are still negative when z approaches zw (see

Proposition D.3), but once it has crossed zw, the behavior resembles that of the other

cases, and the payments are definitely positive again.

Discussion.

In the long run, the government would always want to extract less of the resource than

the resource owners. In Cases I and II, the government’s preference to smooth extraction


4. Explicit Example

(a) ψwβw < ψaβa (b) ψwβw > ψaβa

In Figure 7b, contribution payments are negative around t = 45.

Figure 7: Contribution payment paths for χqq < ρqq (Cases III and IV).

over time to avoid high flow-pollution damages is also stronger than the resource owners’

preference for such a smoothing out; the consumers’ willingness to pay is relatively fixed.

Because the resource owners prefer to extract a relatively high share of what they want to

extract in total, the reduction in their preferred extraction from one period to the next is

stronger than that of the government’s. Thus, the bargained extraction declines faster than

the welfare-maximizing extraction, the disagreement about optimal extraction is reduced,

and the lobbying distortion and, thus, the payments decline over time. These cases possibly

fit coal. Its emissions – like soot – cause high flow-pollution damages, and the existence of

a world market reduces the importance of market power.

In Case III, the resource owners have a stronger preference for smoothing extraction

over time than the government, because the market-power effect outweighs the marginal

flow-pollution effect. This means that the government would always extract a high share

of its totally desired extraction in the current period. Conversely, the welfare-maximizing

extraction reacts stronger to the increase in z than the bargained extraction does, such

that the difference between them increases. This case may represent natural gas, which is

relatively clean, but where market power is relevant due to pipeline restrictions and a lack

of LNG terminals (Lise and Hobbs, 2009).

Finally, in Case IV resource owners would initially lobby for lower extraction to increase

the resource price and, thus, revenues. However, in the long run they want to extract more

because they do not care about stock pollution damages. Thus, the conflict of interest

between welfare maximization and profit maximization turns around at some point and

the lobbying interests from then on resemble those in Case III. This may possibly apply to

crude oil, where earlier on, the market power of suppliers was a dominant topic in research


5. Resource Taxes

and politics, but nowadays, environmental effects seem to be much more relevant.

5 Resource Taxes

The lobbying-equilibrium policy has been derived as a direct choice of extraction quantities.

It is also possible to implement the extraction path via resource taxes. Assume that resource

suppliers are so small that they take the price path including the resource tax as given, and

they cannot coordinate their extraction quantities. Only through their lobby organization’s

influence on policy can they internalize the effect of supply on the price. Then, along the

lines of (3.10), the Euler equation of a resource supplier is

p(qt)− τt −∂c(qt, zt)

∂q= δ ·

[p(qt+1)− τt+1 −

∂c(qt+1, zt+1)∂q

+ ∂c(qt+1, zt+1)∂z

], (5.1)

where τt is the resource tax of the current period.

The tax can be used to implement the extraction path bargained between the lobby and

the government, (3.24). Comparing the two Euler equations, it must hold that

τat − δ · τat+1 = 11 + µ

·∂x(qat , zt)

∂q− µ · ∂p(q

at )

∂qqat

− δ ·[∂x(qat+1, z

at+1)

∂q− µ ·

∂p(qat+1)∂q

qat+1 −∂x(qat+1, z

at+1)

∂z

]. (5.2)

Because the extraction path qat+s is known, it is straightforward to derive the tax path. For

the explicit example, the tax path is given as follows:

Proposition 7 (Explicit Example: Tax Path) The tax path(τat+s

)∞s=0

that implements the

extraction path(qat+s

)∞s=0

by price-taking resource suppliers is defined by

τat+s = βπ − βa + [αa − (απ − ρqq)] qat+s (5.3)

for s = 0, 1, 2, .... Equivalently, as a state-dependent policy rule we have

τa(z) = βπ − βa + [αa − (απ − ρqq)] qa(z). (5.4)

Proof. Using the explicit functions from Table 1 in (5.2) yields:

τat − δ · τat+1 = 11 + µ

·χq + (χqq + µρqq) qat − δ ·

[χq + (χqq + µρqq) qat+1 − χz

]. (5.5)

We can substitute qat and qat+1 from Proposition 3, which yields a difference equation for τt.

Solving it and choosing a starting value τ0 that leads to a non-explosive path yields (5.3).

The tax path consists of two parts. The first, βπ − βa, corrects for the different convergence

levels. The resource tax converges to this part in the long run, where it must just keep


6. Extensions

firms from extracting. The second part is proportional to

αa − (απ − ρqq) = χqq + µρqq1 + µ

. (5.6)

Here, απ − ρqq is the slope of a competitive resource supplier’s marginal profit function,

such that αa − (απ − ρqq) ensures that within a period, marginal profit decreases as fast as

the joint marginal gains of the government and the lobby. If the lobby’s weight µ is very

high, (5.6) goes to ρqq and βa goes to βπ, so that the resource suppliers are induced to

act like a monopolist. If µ is zero, (5.6) is χqq and βa is βw, implying a purely Pigouvian

taxation.

Finally, note that implementing the lobbying equilibrium by resource taxation requires

the tax receipts to be distributed to the suppliers as a lump-sum payment. While the point

in time at which this occurs is irrelevant in principle, in line with our lobbying model we

would expect the tax receipts to be paid back in each respective period.

6 Extensions

6.1 Recursive Nash Bargaining Solution

The Nash bargaining solution discussed in Section 3.3 presupposes that the government

and the lobby never return to cooperation if negotiations fail. The present value of pay-

ments reflects the additional profit and the lost welfare due to the cooperation, compared

to permanent welfare maximization. In equilibrium, there are always gains from coop-

erating and thus cooperation is time-consistent. Nevertheless, it might not be consistent

to assume that, once negotiations failed, it will never take place again. Arguably, such a

behavior would require commitment to non-cooperation.

To cope with this issue, Sorger (2006) proposes the recursive Nash bargaining solution.

It assumes that if bargaining failed, the government and the lobby would choose non-

cooperative strategies, but only for that period. The bargaining parties rationally expect

themselves to cooperate again a period later because there will again be gains from co-

operating. In contrast to the model in Section 3.3, the strategies in case of disagreement

must take into account how bargaining positions are changed a period later. Because the

parties never commit themselves to future behavior, the recursive Nash bargaining solution

is time-consistent.

In the recursive Nash bargaining solution, we denote the agreement outcome by the su-

perscript ar and the disagreement outcome by d. We thus need to define strategies for both

bargaining parties’ choice variables, m and q, for the cases of agreement and disagreement.

Payments have no direct intertemporal effect, implying md(z) = 0. Moreover, the problem

is still stationary, and (3.22) in Proposition 1, according to which the bargained extraction

is independent of the outside options and maximizes a weighted sum of intertemporal wel-


6. Extensions

fare and intertemporal profit, still applies. Thus, the bargained extraction does not depend

on the non-cooperative solution and is the same as in our earlier model: qar(z) = qa(z).Thus, the lobby’s and the government’s intertemporal utilities in case of disagreement

consist of profit or welfare in the period of disagreement, respectively, plus the resulting

future intertemporal utility along the lobbying-equilibrium extraction path:

Gd(z) = w(qd(z), z) + δ ·[W a(z + qd(z)) + γMar(z + qd(z))

], (6.1a)

Ld(z) = π(qd(z), z) + δ ·[Πa(z + qd(z))− λMar(z + qd(z))

], (6.1b)

where W a(z) and Πa(z) are identical to the values in the Nash bargaining solution of

Section 3.3. We can derive the intertemporal payment function as we derived Ma(z) in

Proposition 1:

Mar(z) = η

λ·[Πa(z)− Ld(z)

]+ 1− η

γ·[Gd(z)−W a(z)

]. (6.2)

Define zd∞t+s = zt+qd(zt)+qd(zt+qd(zt))+ .... For s = 1, this is the amount of cumulative ex-

traction that comes about in case of disagreement, while for s > 1, it is the result of perma-nent disagreement starting in period t – that is, if the bargaining parties would not return to

the bargaining table a period after disagreeing, but the government would always choose

the disagreement extraction. Moreover, define qdt = qd(zt) and the values of intertemporal

profit and welfare implied by permanent disagreement as Πd∞(zt) = ∑∞s=0 δ

sπ(qdt+s, zd∞t+s)and W d∞(zt) = ∑∞

s=0 δsw(qdt+s, zd∞t+s). Then, repeatedly inserting (6.1) into (6.2) yields:

Mar(zt) = η

λ·[Πa(zt)− Πd∞(zt)

]+ 1− η

γ·[W d∞(zt)−W a(zt)

]− lim

s→∞δs ·

[η

λ· Lar(zd∞t+s)−

1− ηγ·Gar(zd∞t+s)

]

= η

λ·[Πa(zt)− Πd∞(zt)

]+ 1− η

γ·[W d∞(zt)−W a(zt)

]. (6.3)

Inserting (6.3) in (6.1) and rearranging yields:

Gd(z) = W d∞(z) + η · δ ·µ ·[Πa(z + qd(z))− Πd∞(z + qd(z))

]+W a(z + qd(z))−W d∞(z + qd(z))

, (6.4a)

Ld(z) = Πd∞(z) + 1− ηµ· δ ·

µ ·[Πa(z + qd(z))− Πd∞(z + qd(z))

]+W a(z + qd(z))−W d∞(z + qd(z))

, (6.4b)

where the curly-bracketed terms represent the future gains of cooperation in case of dis-

agreement in the current period. The government chooses qd(z) so as to maximize (6.4a).

If the lobby gets all the future gains of cooperation (η = 0), the government chooses

qd(z) = qw(z), which would maximize intertemporal welfare if this extraction path were


6. Extensions

continued in the next period. A period later, it will then have to be compensated for the

difference to maximal welfare. Else, if the government gets some of the future gains of

cooperation (η > 0), it has a mixed motivation of maximizing welfare and impairing the

lobby’s future outside option. Then, qd(z) exceeds (is lower than) qw(z) if the gains of

cooperation increase (decline) with the future cumulative extraction.

Along the lines of (3.12), the non-negativity constraint on the government’s disagree-

ment extraction qd would be binding if z ≥ zd, with the convergence level zd defined by

∂w(0, zd)∂q

+ δ · ∂Wd∞(0, zd)∂z

= −η · δ ·µ ·[∂Πa(zd)∂z

− ∂Πd∞(0, zd)∂z

]+ ∂W a(zd)

∂z− ∂W d∞(0, zd)

∂z

. (6.5)

If the future gains of cooperation increase with the future cumulative extraction, zd exceeds

zw, and vice versa. zd can exceed za if the future gains of cooperation increase with the

future cumulative extraction. Then, qd is always positive along the lobbying-equilibrium

extraction path.11

Comparing intertemporal utilities between the recursive Nash bargaining solution and

our earlier model, we note that bargained intertemporal welfare and profit are the same,

so that the present values of payments make the difference. From (6.3) and Proposition 1,

Mar(z)−Ma(z) = η

λ·[Πw(z)− Πd∞(z)

]+ 1− η

γ·[W d∞(z)−Ww(z)

]. (6.6)

SinceWw(z) is maximized intertemporal welfare, the welfare difference cannot be positive.

However, the profit difference can be positive, depending on whether the lobby prefers

qd(z) or qw(z). Only if profits are lower in case of permanent disagreement than in case of

welfare maximization, the difference in payments can be positive.12 If qd(z) = qw(z), the

payments and thus the intertemporal utilities coincide. This is definitely the case if η = 0,

see (6.4a), or if z ≥ zd and z ≥ zw, because then extraction in case of disagreement is

zero in both solution concepts. The outside option then is zero extraction and payments

forever in our earlier model, and postponing extraction and payments for one period in the

recursive Nash bargaining solution.

Equivalent to (3.26), we can also derive the payments within the periods:

mar(zt) = η

λ·[π(qat , zt)− π(qdt , zt)

]+ 1− η

γ·[w(qdt , zt)− w(qat , zt)

]+ δ ·

η

λ·[Πd∞(zat+1)− Πd∞(zd∞t+1)

]+ 1− η

γ·[W d∞(zd∞t+1)−W d∞(zat+1)

]. (6.7)

If the non-cooperative convergence level is lower than the lobbying-equilibrium conver-

11For the explicit example, see Lemma E.3 in Appendix E.12For the explicit example, see Lemma E.4 in Appendix E.


6. Extensions

gence level, zd < za, the payment function mar(z) has two kink points.13 The first is at

z = zd ≡ zd − qa, when the future disagreement extraction becomes zero in case of agree-

ment in the current period, and the second at z = zd, when the current extraction in case of

disagreement becomes zero. By contrast, if the non-cooperative convergence level exceeds

the lobbying-equilibrium convergence level, zd ≥ za, mar(z) develops smoothly.

We return to the explicit example of Section 4. The bargained extraction, qa(z), is the

same as in our earlier model and thus stated in Proposition 3, and intertemporal welfare

and profit, W a(z) and Πa(z), are stated in Proposition 4.

The recursive Nash bargaining solution is derived in Appendix E. Resource extraction

in case of disagreement, qd(z), is linear in cumulative extraction, but we can only implicitly

define the speed of convergence, ψd ≤ ψw, which we do in Lemma E.2.

We define

βd ≡ βa −βa − βw

1 + η(1+µ)δ(ψd−ψa)κqz1−δ(1−ψaκqz)

. (6.8)

Then resource extraction in case of disagreement is given as follows:

Proposition 8 (Explicit Example: Recursive Nash Bargaining Disagreement Extrac-

tion) The extraction in case of disagreement is a function of the amount of cumulative re-source extraction z:

qd(z) =

ψd · (βd − κqzz) if z ≤ zd,

0 if z > zd,(6.9)

where zd = βd/κqz.

Proof. The function is derived in Appendix E, in particular in Lemma E.1 and E.2. The

relation between zd, zw and za is summarized in Lemma E.3.

This yields intertemporal welfare and profit in case of permanent disagreement:

Proposition 9 (Explicit Example: Recursive Nash Bargaining Permanent-

Disagreement Welfare and Profit) In case of permanent disagreement in the bargaining,intertemporal welfare and profit are

W d∞(z) = Aw|d · qd(z)2 −Bw|d · qd(z)− χzz

1− δ , (6.10a)

Πd∞(z) = Aπ|d · qd(z)2 −Bπ|d · qd(z) ≥ 0, (6.10b)

13Given that ∂qd(zd)/∂z < 0 (cf. Proposition 2), η > 0 is not necessary for the discontinuity if zd 6= zw.


6. Extensions

where

Ai|d ≡1ψd− αi

2

1− δ (1− ψdκqz)2 > 0 for i = w, π, (6.11a)

Bi|d ≡βd − βi

1− δ (1− ψdκqz)for i = w, π, a. (6.11b)

Proof. The functions are derived in Appendix E, in particular in Lemma E.1 and E.2. Ai|d >

0 follows from ψd ≤ ψw together with (4.10a).

Intertemporal utility of the government and the lobby in case of non-cooperation for one

period, Gd(z) and Ld(z), are then given by (6.4a) and (6.4b). Finally, we can determine

the equilibrium payments:

Proposition 10 (Explicit Example: Recursive Nash Bargaining Payments) The presentvalue of payments is

Mar(z) = η

λ·Aπ|a · qa(z)2 −Bπ|a · qa(z)−

[Aπ|d · qd(z)2 −Bπ|d · qd(z)

]+ 1− η

γ·Aw|d · qd(z)2 −Bw|d · qd(z)−

[Aw|a · qa(z)2 −Bπ|w · qa(z)

]. (6.12)

The payments in any single period can be determined as

mar(z) = η

λ·[(βπ − κqzz) ∆a,d(z)− απ

2 ∆2a,d(z)

]− 1− η

γ·[(βw − κqzz) ∆a,d(z)− αw

2 ∆2a,d(z)

]+ δ ·

η

λ·[Aπ|d ·∆2d+

a,d (z)−Bπ|d ·∆d+a,d(z)

]− 1− η

γ·[Aw|d ·∆2d+

a,d (z)−Bw|d ·∆d+a,d(z)

],

(6.13)

where

∆a,d(z) ≡ qa(z)− qd(z), ∆2a,d(z) ≡ qa(z)2 − qd(z)2, (6.14a)

∆d+a,d(z) ≡ qd(z + qa(z))− qd(z + qd(z)), ∆2d+

a,d (z) ≡ qd(z + qa(z))2 − qd(z + qd(z))2.

(6.14b)

Proof. KnowingW a(z) and Πa(z) from Proposition 4 andW d(z) and Πd(z) from Proposition

9, we can determine the payments using (6.3) and (6.7).

Intertemporal utility of the government and the lobby resulting from the recursive Nash

bargaining, Gar(z) and Lar(z), are then given by (3.15) and (3.17).

Because we can only implicitly define ψd, we now illustrate the differences between

our model approaches for specific parameter values. First, the disagreement extraction

in the recursive Nash bargaining solution is lower than in our earlier model for Case I

if the future gains of cooperation are not too important – that is, if η (1 + µ) < 1/δ (see

Lemma E.3). Payments then decline over time in both approaches before the respective


6. Extensions

non-cooperative convergence level is reached. For the special case of η (1 + µ) = 1, the

present value of payments is definitely higher in the recursive Nash bargaining solution

than in our earlier model in Case I (see Lemma E.4).14 Then, qd < qw leads to lower

profits in case of disagreement in the recursive Nash bargaining solution than in our earlier

model, and this loss outweighs the respective welfare loss, so that Mar > Ma. For Case II,

we find that qd = qw, so that mar = ma, and for Cases III and IV, we find that Mar < Ma

for the special case of η (1 + µ) = 1 and βw = βπ (see Lemma E.4). Finally, the payment

functionsmar(z) andma(z) are quadratic functions of the difference between the bargained

extraction and the extraction in case of disagreement (compare (D.1) in Appendix D to

(E.9) in Appendix E).

Summing up, we find that the assumptions about behavior in case of disagreement lead

to the same policy. However, it depends on the economic parameters whether the flexibility

to return to the bargaining table benefits the government or the lobby. Thus, depending

on the market conditions, the government would either have an interest in committing to

welfare maximization in case of disagreement or to remain flexible.

6.2 Short-lived Governments

Suppose that the mixed-motivation government of the previous sections is in power in the

current period, but that it will be ousted and succeeded by a purely welfare-maximizing

government (or social planner) with probability 1 − σ after each period. Once it has lost

power, it cannot return to office. We denote the outcome of the Nash bargaining between

the lobby and the short-lived government by the superscript as. Then, the government’s

and the lobby’s expected intertemporal utilities are

Gas(z) = g(qas(z),mas(z), z) + δ · [σ ·Gas(z + qas(z)) + (1− σ) ·Ww(z + qas(z))] , (6.15a)

Las(z) = l(qas(z),mas(z), z) + δ · [σ · Las(z + qas(z)) + (1− σ) · Πw(z + qas(z))] . (6.15b)

Furthermore, suppose that the outside option is a permanent termination of cooperation

as in Section 3.3. Then, extraction in case of disagreement is qw(z) and the respective

intertemporal utilities are given by (3.6) and (3.18). For example, the choice may be to

implement better institutions in case of disagreement. Along the lines of Proposition 1, the

bargaining maximizes

w(q, z) + µ · π(q, z) + δ · σ · [W as(z + q) + µ · Πas(z + q)]

+ (1− σ) · [Ww(z + q) + µ · Πw(z + q)], (6.16)

whereW as(z) = w(qas(z), z)+δ·σ ·W as(z + qas(z)) + (1− σ) ·Ww(z + qas(z)) and Πas(z)is defined equivalently. The expected intertemporal payments Mas(z) = mas(z) + δσ ·

14In Section 4.2, we use µ = 1 and η = 1/2, so that η (1 + µ) = 1.


6. Extensions

Mas(z + q) share the expected surplus along the lines of Proposition 1:

Mas(z) = η

λ· π(qas(z), z) + δ · [σ · Πas(z + qas(z)) + (1− σ) · Πw(z + qas(z))]− Πw(z)

− 1− ηγ· w(qas(z), z) + δ · [σ ·W as(z + qas(z)) + (1− σ) ·Ww(z + qas(z))]−Ww(z).

(6.17)

The payment in a given period thus is mas(z) = Mas(z)− δσ ·Mas(z + qas(z)).For σ = 1, we are in our earlier model from Section 3.3. For σ < 1, the government and

the lobby take into account that the bargaining ends with probability 1−σ after each period.

σ affects the equilibrium of this model in two ways. Firstly, σ directly increases the future

probability of continuing the lobbying equilibrium, which makes current continuation more

valuable, and secondly, it changes the jointly optimal bargained extraction, qas 6= qa. While

the model with explicit functions can be solved implicitly for σ ∈ (0, 1) (similar to that in

Section 6.1), we focus on the extreme case of σ = 0, so that the government and the lobby

cooperate for only one period. For this case, qas and mas can be derived explicitly. We

discuss the solution for qas and its difference to qa below; for the explicit formula of mas,

see Appendix F.

We define

ψas ≡ ψa + (ψa − ψw)2 (1− ψaαa)2ψw (1− ψaαa) + ψ2

aαa, (6.18a)

βas ≡ βw + (βa − βw) (2− ψwαw)ψ2w (1− ψaαa) + ψ2

a

, (6.18b)

where ψas R ψw ⇔ ψa R ψw, ψas > ψa ⇔ ψa 6= ψw, βas > βw and βas R βa ⇔ ψa R ψw.

Then resource extraction in the period of cooperation is given as follows:

Proposition 11 (Explicit Example: Short-Lived Government Extraction) For σ = 0, thebargained extraction is a function of the amount of cumulative resource extraction z:

qas(z) =

ψas · (βas − κqzz) if z ≤ zw,

1αa· (βa − κqzz) if z ∈ (zw, zas] ,

0 if z > zas.

(6.19)

where zas = βas/κqz. For z ≤ zw, qas > qa > qw if ψa > ψw, and qas = qa > qw if ψa = ψw.If ψa < ψw, the relation between qas, qa and qw depends on z, with switching levels of z alongthe lines of (4.15) for ψaβa < ψwβw. For z ∈ (zw, zas], qas > qa ≥ qw = 0.

Proof. The proof follows along the lines of the proof of Proposition 3. The relation between

qas, qa and qw follows from (4.5) and (6.19) together with (4.2b), (4.4) and (6.18).

If the non-negativity constraint on qw is binding, but that on qas is not, qas is chosen so as

to maximize w(q, z) + µ · π(q, z). Then qas exceeds qa, because there is only one period of


7. Conclusions

extraction. Furthermore, the profit gain and welfare loss and, thus, the payments in the

period of cooperation are higher for σ = 0 than for σ = 1 (see Proposition F.1).

If the non-negativity constraint on qw is not binding, we have to distinguish between

three cases, ψa > ψw, ψa = ψw and ψa < ψw. If ψa > ψw, qas exceeds qa. In every future

period, extraction will be below qa, so that there is a strong incentive to extract more in

the period of cooperation. Then, the profit gain and welfare loss are again higher for σ = 0than for σ = 1, and this effect outweighs the respective changes in the future outside

options. Thus, the payments in the period of cooperation are higher for σ = 0 than for

σ = 1 (see Proposition F.1).

For ψa = ψw, the q and m values coincide. In every future period, qw is constantly lower

than qa, so that higher current extraction does not affect the future lobbying distortion.

Thus, in the period of cooperation the government and the lobby behave as if they would

cooperate forever.

If ψa < ψw, the bargained speed of convergence, ψas, and the bargained first unit gains,

βas, are compromise values between ψa and ψw as well as between βa and βw. We find that

the lobby then is willing to pay more for σ = 0 than for σ = 1: mas − ma is increasing

in η (see Proposition F.1). Whether the government also demands more depends on the

parameter values as well as on the amount of cumulative extraction.

To summarize, while a higher probability of remaining in power clearly implies a longer

deviation from welfare maximization, the distortive influence of the lobby can definitely

be higher in the periods of cooperation if the probability of remaining in power is lower.

7 Conclusions

In this article, we analyze resource extraction determined by the bargaining of a govern-

ment and a lobby group. Equilibrium extraction is a compromise path between welfare

maximization and profit maximization. The influence of the lobby increases with the gov-

ernment’s preference for contribution payments and decreases with the lobby’s cost of

collecting them. Depending on how strongly resource extraction increases marginal flow-

pollution damages and decreases the resource price, extraction is either too fast or too

slow, compared to welfare maximization. Again depending on these market parameters,

the distance of the bargained extraction to the welfare-maximizing one may increase or

decrease over time. Only when cumulative extraction has become so high that it would

be welfare-maximizing to completely stop extraction, will the lobbying distortion decline

over time in any case – in the very long run, as extraction costs increase ever further, the

lobby’s preferred extraction quantity also approaches zero. Extraction continues for too

long, however, causing inefficiently high pollution.

We derive an analytical solution of our model with linear-quadratic functions that we

also use to analyze the lobby’s contribution payments to the government. Along with


7. Conclusions

the conflict of interest, they vanish in the very long run, but prior to that, contribution

payments decline if the marginal flow-pollution effect outweighs the market-power effect,

remain constant if the effects offset each other, and increase at least temporarily if the

market-power effect dominates. The contribution payments may turn negative temporar-

ily if there is an intertemporal profit loss from cooperating for one period – that is, if

intertemporal profits would be higher if welfare maximization started in the current period

than it would be if welfare maximization started later. This is the case if the increased ex-

traction strongly reduces the (welfare-maximizing) extraction that the government would

choose in case of disagreement in the future, which in turn worsens the lobby’s future

bargaining position. Nonetheless, the bargained extraction is always a deviation from in-

tertemporal welfare maximization, such that the present value of anticipated payments to

the government has to be positive.

In general, lobbying for higher current extraction implies that the economy will in-

herit higher cumulative extraction in the next period. The government would then choose

even lower extraction if it should switch to welfare maximization, implying lower profits,

compared to a situation in which welfare maximization would start right away. The lobby

always has to factor this in in the bargaining process. When cumulative extraction becomes

so high that welfare maximization implies zero extraction, however, this factor vanishes,

which implies a strong increase in the lobby’s willingness to pay.

Additionally, we demonstrate how the bargained extraction path can be implemented

via resource taxes. In our linear-quadratic application, they consist of a constant part, cor-

recting for the difference in preferred convergence levels, and a part that is proportional to

the lobbying-equilibrium extraction quantities, correcting for the marginal flow-pollution

effect and the market-power effect.

Most of our results are derived under the assumption that ending cooperation once

would end it forever. However, we also demonstrate that the bargained extraction is the

same if the threat is, instead, not to cooperate for the current period only. The contribution

payments, however, are in general different. They are only identical if the outside option is

identical, which is the case particularly if the government’s bargaining power is zero. Then,

the best it can do when deviating for one period from the bargained extraction path is to

behave as if it maximized intertemporal welfare, because it will be compensated exactly

for this welfare loss from the next period onwards. Furthermore, contribution payments

coincide in both settings for a given level of cumulative extraction if the respective dis-

agreement extraction quantities are zero. This is the case once both convergence levels are

reached.

Additionally, we analyze the effect of a shorter government tenure, focusing on the

case in which the government is definitely ousted after one period. While the deviation

from welfare-maximization will naturally be smaller compared to an indefinite lobbying

equilibrium, the distortion in the period in which the government is in power may be higher

than it would be in a permanent lobbying equilibrium. Since governmental institutions are


A. Intertemporal Maximization

in a better bargaining position if they can guarantee continued cooperation, it may be an

interesting topic for future research to analyze whether governments create permanent

non-elected bureaucracies in order to collect a higher bargaining rent.

The analysis in this article contributes to our understanding of the political economy

of resource extraction. Firstly, it is important to bear in mind that the conflict of interest

between welfare maximization and profit maximization can operate in different directions,

such that a lobby would either try to convince the government to increase or decrease

extraction, and it can change over time. Secondly, if we can observe bribes or “contribution

payments” that are very low, this does not necessarily indicate a low lobby influence. In

a dynamic relationship, what the bargaining parties expect for the future is decisive, as

well as how current cooperation changes future bargaining positions. Finally, if money is

exchanged for a favorable policy as it is in our setting, the bargained policy does not depend

on the behavior in case of disagreement, as long as both parties expect their cooperation

to continue.

Appendix

A Intertemporal Maximization

In this appendix, we first discuss the intertemporal maximization of Section 3.1 in more

detail. To prepare, we summarize some useful properties of w(q, z) and π(q, z) for further

reference:

Lemma A.1 (Properties of w(q, z) and π(q, z)) For a given value of z, w(q, z) = u(q) −c(q, z)− x(q, z) as defined in (3.2) has a unique myopic maximizer qw(z) = argmaxq w(q, z)for which qw(z) ≥ 0. Similarly, π(q, z) = p(q)q − c(q, z) as defined in (3.8) has a uniquemyopic maximizer qπ(z) ≥ 0.

Proof. By strict concavity of u(q) and convexity of c(q, z) + x(q, z) in q, w(q, z) is strictly

concave in q. Then, qw(z) = argmaxq w(q, z) is implicitly defined by the complementary-

slackness conditions ∂w(qw(z),z)∂q

≤ 0, qw(z) ≥ 0 and qw(z)∂w(qw(z),z)∂q

= 0. qw(z) follows along

the same lines because p(q)q = u′(q)q is strictly concave by assumption.

Even though q ≥ 0, Lemma A.1 implies that there are effective bounds both to the q values

and to intertemporal welfare and profit along extraction paths that maximize the respective

intertemporal values:

Lemma A.2 (Effective bounds of W((zt+s)∞s=0

)and Π

((zt+s)∞s=0

)) For a welfare-

maximizing extraction path, the sum W ((zt+s)∞s=0) defined in (3.3) is bounded, and for aprofit-maximizing extraction path, the sum Π ((zt+s)∞s=0) defined in (3.5) is bounded.



Proof. A welfare-maximizing extraction path starting with zt never includes qt+s > qw(zt+s)as defined in Lemma A.1 for any s, because qw(zt+s) maximizes w(q, zt+s) and adding to

future cumulative extraction cannot increase welfare. We can thus assume q ∈ [0, qw(z)] for

welfare-maximization. Since w is monotonously increasing in q up to w(qw(z), z), it follows

that w(q, z) ∈ [w(0, z), w(qw(z), z)]. By the Envelope theorem,

dw(qw(z), z)dz = ∂w(qw(z), z)

∂z= −∂c(q

w(z), z)∂z

− ∂x(qw(z), z)∂z

≤ 0, (A.1)

so that along a feasible path, w(qw(z), z) is non-increasing and intertemporal welfare is

bounded above. Moreover, since it is possible to choose qt+s = 0 ∀ s such that zt+s =zt ∀ s, maximized W ((zt+s)∞s=0) is bounded below by w(0, zt)/(1− δ). Along the same lines,

Π ((zt+s)∞s=0) is bounded.

Since, therefore, the maximum in the Bellman equation (3.6) is well-defined (cf. Sydsæter

et al., 2008, Sec. 12.3), we can apply the usual steps for describing the optimal path of the

control variable. We denote optimal extraction given z by qw(z) or, for brevity, just by qw.

The first-order condition is

∂w(qw, z)∂q

+ δ · ∂Ww(z + qw)∂z

= 0. (A.2)

Differentiating the Bellman equation and substituting (A.2) yields the Envelope Condition:

∂Ww(z)∂z

= ∂w(qw, z)∂z

+ δ · ∂Ww(z + qw)∂z

= ∂w(qw, z)∂z

− ∂w(qw, z)∂q

. (A.3)

Evaluating (A.3) for z = zt+1 = zt + qw(zt) and q = qwt+1 = qw(zt+1) and substituting into

the first-order condition (A.2) for z = zt and q = qwt = qw(zt) yields the Euler equation:

∂w(qwt , zt)∂q

= δ ·[∂w(qwt+1, zt+1)

∂q−∂w(qwt+1, zt+1)

∂z

], (A.4)

which is written more explicitly in (3.7). Proceeding along the same lines for profit-

maximization yields (3.10).

Finally, we define the level of cumulative extraction for which intertemporal maximiza-

tion would imply the non-negativity constraint on q to be binding:

Proposition A.1 (Binding non-negativity constraint) Suppose that cumulative extractionis at a level zw such that

∂w(0, zw)∂q

+ 1r

∂w(0, zw)∂z

= 0, (A.5)

or at a higher level z ≥ zw. Then and only then, the non-negativity constraint q ≥ 0 is binding



for welfare-maximization, qw(z) = 0, and this remains so in the future. Likewise,

∂π(0, zπ)∂q

= 0 (A.6)

defines the levels of cumulative extraction z ≥ zπ for which the non-negativity constraint isbinding for profit-maximization.

Proof. Suppose zero extraction were chosen in all future periods. Intertemporal welfare

evaluated in the current period then is w(qt, zt) + w(0, zt + qt)/r. Zero extraction in the

current period is optimal if

∂w(0, zt)∂q

+ 1r

∂w(0, zt)∂z

≤ 0. (A.7)

By ∂c(0,z)∂z

= 0 and by additive separability of x(q, z), we have ∂2w(0,z)∂q∂z

= 0 and ∂2w(0,z)∂z2 =

−∂2x(0,z)∂z2 ≤ 0, such that the left-hand side is weakly declining in z; thus, (A.7) must hold

for z ≥ zw. Because this applies in any future period, positive extraction cannot become

optimal again later on. The derivation of zπ follows along the same lines. Reverting the

arguments, the optimality of qw(z) > 0 or qπ(z) > 0, respectively, follows.

Proof of Lemma 2. For z > zw, ∂qwt∂z

= 0 follows from Proposition A.1. For z < zw, differ-

entiating (3.7) with respect to z and rearranging yields:

∂qwt+1∂z

=Γt ∂q

wt

∂z+ Ωtδ

(1 + ∂qwt

∂z

)Γt+1δ

(1 + ∂qwt

∂z

) , (A.8)

where

Γt = −∂2w(qwt , zt)∂q2 + ∂2w(qwt , zt)

∂q∂z, (A.9a)

Ωt = −∂2w(qwt , zt)∂q∂z

+ δ

[∂2w(qwt+1, z

wt+1)

∂q∂z−∂2w(qwt+1, z

wt+1)

∂z2

]. (A.9b)

Assumption 1 implies Γt > 0 and Assumption 2 implies Ωt > 0. By (A.8), ∂qwt∂z

/∈ (−1, 0)then implies ∂qwt+1

∂z> 0. ∂qwt+1

∂z> 0 in turn implies ∂qwt+2

∂z> 0, so that ∂qwt

∂z/∈ (−1, 0) implies

∂qwt+s∂z

> 0 for all s = 1, 2, 3, .... Since ∂qwt+s∂z≤ 0 must hold for some s = 1, 2, 3, ... to ensure

that lims→∞ zwt+s = zw, ∂qwt

∂z∈ (−1, 0) must hold. Because the derivatives of w(qt, zt) are

continuous (see Section 3.1), ∂qwt∂z

is continuous by (A.8). The derivation of ∂qπt∂z

follows

along the same lines.


B. Proofs of Section 3.3

B Proofs of Section 3.3

Proof of Proposition 1. We now derive the Nash bargaining solution, as presented in

Proposition 1. For a solution of the Nash bargaining with q > 0, (3.20) yields the first-

order conditions:

η

∂w(qa(z),z)∂q

+ δ · ∂Ga(z+qa(z))∂z

Ga(z)−Ww(z) + (1− η)∂π(qa(z),z)

∂q+ δ · ∂L

a(z+qa(z))∂z

La(z)− Πw(z) = 0, (B.1a)

ηγ

Ga(z)−Ww(z) − (1− η) λ

La(z)− Πw(z) = 0. (B.1b)

Rearranging (B.1b) and substituting this into (B.1a), we get

∂w(qa(z), z)∂q

+ δ · ∂Ga(z + qa(z))

∂z+ µ

[∂π(qa(z), z)

∂q+ δ · ∂L

a(z + qa(z))∂z

]= 0, (B.2)

where the payments in the intertemporal utility terms cancel out, so that the agreement

policy can be written as in (3.22). For the existence of the maximum, the arguments in

Lemma A.2 can be applied. Rearranging (B.1b) and collecting the payment terms yields

(3.23a). (3.23b) then follows from rearranging (3.14). This completes Proposition 1.

Proof of Proposition 2. In ∂Πw(zt)∂z

from (3.28), the derivatives of π(qt, zt) are continuous

(see Section 3.1) and ∂qwt∂z

is continuous for z 6= zw (see Proof of Lemma 2 in Appendix A).

Thus, ∂Πw(zt)∂z

is continuous for z 6= zw. Next, note that, since qw(z) = 0 and π(qw(z), z) = 0for z > zw,

limz→zw+

∂Πw(z)∂z

= 0, (B.3a)

and that, since ∂π(0,zw)∂q

> 0 and ∂π(0,z)∂z

= 0 (see Section 3.1),

limz→zw−

∂Πw(z)∂z

= limz→zw−

[∂π(qw(z), z)

∂q

∂qw(z)∂z

+ δ∂Πw(z + qw(z))

∂z

[1 + ∂qw(z)

∂z

]]

= ∂π(0, zw)∂q

∂qw(zw)∂z

+ δ∂Πw(zw)

∂z

[1 + ∂qw(zw)

∂z

]

⇒ ∂Πw(zw)∂z

=∂π(0,zw)

∂q∂qw(zw)

∂z

1− δ[1 + ∂qw(zw)

∂z

] ≤ 0. (B.3b)

Differentiating (3.26) and rearranging yields:

∂ma(zt)∂z

= η

λ

[∂π(qat , zt)

∂q

∂qat∂z

+ ∂π(qat , zt)∂z

+ δ∂Πw(zt + qat )

∂z

(1 + ∂qat

∂z

)− ∂Πw(zt)

∂z

]

− 1− ηγ

[∂w(qat , zt)

∂q

∂qat∂z

+ ∂w(qat , zt)∂z

+ δ∂Ww(zt + qat )

∂z

(1 + ∂qat

∂z

)− ∂Ww(zt)

∂z

].

(B.4)


C. Proofs of Section 4.1

Because the derivatives of π(qt, zt) and w(qt, zt) are continuous, ∂Πw(z)∂z

is continuous for

z 6= zw, ∂Ww(z)∂z

is continuous and ∂qat∂z

is continuous for z 6= za. Thus, ∂ma(zt)∂z

is continuous

for z 6= zw, zw, za. For z = zw and z = zw, which are relevant for Proposition 2, there may

be discontinuities. Applying (B.3) yields:

limz→zw+

∂ma(z)∂z

− limz→zw−

∂ma(z)∂z

= −ηλ·

∂π(0,zw)∂q

∂qw(zw)∂z

1− δ[1 + ∂qw(zw)

∂z

] · δ [1 + ∂qa(zw)∂z

]≥ 0, (B.5a)

limz→zw+

∂ma(z)∂z

− limz→zw−

∂ma(z)∂z

= η

λ·

∂π(0,zw)∂q

∂qw(zw)∂z

1− δ[1 + ∂qw(zw)

∂z

] ≤ 0, (B.5b)

where Lemma 2 implies ∂qa(zw)∂z

∈ (−1, 0). (B.5a) and (B.5b) are equal to zero if η = 0 or if∂qw(zw)

∂z= 0. Else, if η > 0 and ∂qw(zw)

∂z< 0, then (B.5a) is positive and (B.5b) is negative.

C Proofs of Section 4.1

We guess that there exist constants Y i0 and Y i

1 such that the following state-dependent

extraction functions solve the maximizations (3.6), (3.9) and (3.22) for the functions (4.1):

qi(z) =

Yi

0 + Y i1 z if Y i

0 + Y i1 z ≥ 0,

0 if Y i0 + Y i

1 z < 0,for i = w, π, a. (C.1)

Thus, we expect the quadratic utility functions to lead to extraction functions that are linear

in the state z as long as positive extraction is optimal. For stability of the dynamic system

(cf. Gandolfo, 1996, Chapter 3), we conjecture that

0 < 1 + Y i1 ≤ 1 for i = w, π, a (C.2)

must hold. To solve for the coefficients, we first use them to state the intertemporal utility

functions (3.6), (3.9) and (3.22) in an explicit form.

Lemma C.1 (Intertemporal Utility Functions) Suppose that qi(z) from (C.1) is the opti-mal extraction function for i = w, π, a. Then,

Ww(z) =βw + Y w0

Y w1κqz

1− δ (1 + Y w1 )q

w(z)−αw2 + 1

Y w1κqz

1− δ (1 + Y w1 )2 q

w(z)2 − χzz

1− δ , (C.3a)

Ππ(z) =βπ + Y π0

Y π1κqz

1− δ (1 + Y π1 )q

π(z)−απ2 + 1

Y π1κqz

1− δ (1 + Y π1 )2 q

π(z)2, (C.3b)

V a(z) =βa + Y a0

Y a1κqz

1− δ (1 + Y a1 )q

a(z)−αa2 + 1

Y a1κqz

1− δ (1 + Y a1 )2 q

a(z)2 − χzz

1− δ . (C.3c)


C. Proofs of Section 4.1

Proof. First suppose that Y i0 + Y i

1 z < 0, implying qi(z) = 0 forever because Y i1 ≤ 0 by (C.2)

and z cannot decrease. (4.1a), (4.1b) and (4.1c) become w(0, z) = −χzz, π(0, z) = 0 and

v(0, z) = −χzz, respectively. Then, Ww(z) = − χzz1−δ , Ππ(z) = 0 and V a(z) = − χzz

1−δ , which

is consistent with (C.3). Let us turn to the unconstrained case. Substituting (C.1) into the

equation of motion (3.1) yields zit+1 = zit + Y i0 + Y i

1 zit. Hence,

zit+s = zit +s−1∑ν=0

qit+ν = zit + sY i0 + Y i

1

s−1∑ν=0

zit+ν . (C.4)

After some substitutions and rearrangements, we get

zit+s =(1 + Y i

1

)szit + Y i

0

s−1∑ν=0

(1 + Y i

1

)ν=(1 + Y i

1

)s (Y i0Y i

1+ zit

)− Y i

0Y i

1. (C.5)

Substituting into (C.1) yields:

qit+s =(1 + Y i

1

)s (Y i

0 + Y i1 zt). (C.6)

Using (4.1a) for i = w, we then have (after some rearrangements)

w(qwt+s, zwt+s) =(βw + Y w

0Y w

1κqz

)(1 + Y w

1 )s (Y w0 + Y w

1 zwt )

−(αw2 + 1

Y w1κqz

)[(1 + Y w

1 )s (Y w0 + Y w

1 zwt )]2

+[Y w

0Y w

1+(

1r− 1Y w

1

)(1 + Y w

1 )s (Y w0 + Y w

1 zwt )]χz. (C.7)

Using this in (3.3) then yields (C.3a). (C.3b) and (C.3c) can be obtained in the same man-

ner using (4.1b) for i = π or (4.1c) for i = a instead of (4.1a) for i = w, and substituting

the results into (3.5) or (3.21), respectively.

Using these intertemporal utility functions, we can explicitly derive the respective coeffi-

cients:

Lemma C.2 (Benchmark Extractions) In the extraction functions (C.1), the coefficientsare Y i

0 = ψi · βi and Y i1 = −ψi · κqz, where ψi is defined in (4.4).

Proof. By (3.6), qw(z) is the q that maximizes w(q, z) + δ ·Ww (z + q). Substituting (4.1a)

and (C.3a), we can derive a first-order condition for q, into which we again substitute

qw(z) = Y w0 + Y w

1 z:

βw − κqzz − αw (Y w0 + Y w

1 z)

+ δ ·[Y w

1 βw + Y w0 κqz

1− δ (1 + Y w1 ) −

Y w1 αw + 2κqz

1− δ (1 + Y w1 )2 [Y w

0 + Y w1 [Y w

0 + (1 + Y w1 ) z]]

]= 0. (C.8)


D. Proofs of Section 4.2

This condition must also hold for z = z + qw(z) = z + Y w0 + Y w

1 z. This generates two

equations in two unknowns. These contain quadratic terms, but only the stated coefficients

fulfill (C.2). qπ(z) and qa(z) are derived in the same way.

Using the coefficients from Lemma C.2 in (C.5) and (C.6) yields (4.5) and (4.6) in Propo-

sition 3. Using them in the formulas from Lemma C.1 for i = a proves Proposition 4, and

using them in the formulas from Lemma C.1 for i = w proves Proposition 5. We complete

the proof of Proposition 3 by proving the inequalities 0 < 1− ψiκqz ≤ 1:

Lemma C.3 (Parameter Restrictions) Using the coefficients from Lemma C.2, (C.2) be-comes 0 < 1 − ψiκqz ≤ 1 for i = w, π, a, for which αi > κqz as assumed in (4.3) is necessaryand sufficient.

Proof. Substituting (4.4) into 0 < 1− ψiκqz and rearranging yields

2κqz < αi +√α2i + 4

rκqz (αi − κqz) for i = w, π, a, (C.9)

for which αi > κqz is necessary and sufficient. By (4.4), this implies ψi > 0, such that

1− ψiκqz ≤ 1.

D Proofs of Section 4.2

Substituting qw and qa from Proposition 3 into the payments formula of Proposition 6 and

simplifying yields:

ma(z) = 1γ

Θ1 · [qa(z)− qw(z)] + Θ2 · [qa(z)− qw(z)]2 if z < zw,

(βa − βw) qa(z) + Θ3 · qa(z)2 + ηµBπ|wqw(z)−Θ4 · qw(z)2 if z ∈ [zw, zw) ,

(βa − βw) qa(z) + Θ3 · qa(z)2 if z ∈ [zw, za) ,(D.1)

where

Θ1 ≡ −ηµBπ|w ·δ (1− δ)

1− δ (1− ψwκqz)2 ·ψw (ψa − ψw)κ2

qz

1− δ (1− ψaκqz)R 0 ⇔ ψa R ψw,

(D.2a)

Θ2 ≡1− δ

1− δ (1− ψwκqz)2

[η (1 + µ)

(1ψa− αa

2

)+ 1ψw− αw

2

]> 0, (D.2b)

Θ3 ≡ η (1 + µ)(

1ψa− αa

2

)− 1ψa

+ αw2 , (D.2c)

Θ4 ≡1

1− δ (1− ψwκqz)2

[η (1 + µ)

(1ψw− αa

2

)− 1ψw

+ αw2

]. (D.2d)


D. Proofs of Section 4.2

Writing the coefficients in this form allows a comfortable characterization of the develop-

ment of payments for different parameter relations. We start with z ≥ zw, so that qw(z) = 0and qw(z + qa) = 0.

Proposition D.1 (Contribution Payments for z ≥ zw) For z ∈ [zw, za), we have ma(z) >0. For z → za, ma(z) asymptotically converges towards zero. For this convergence to bemonotone,

η (1 + µ) ≥ 12

1−αw − αa

21ψa− αa

2

(D.3)

is necessary and sufficient. For this inequality to hold, η(1+µ) ≥ 1/2 or αw ≥ απ are sufficient.αw ≥ απ is necessary if η = 0. If the inequality does not hold, ma(z) is increasing for z = zw

and declining for large z, in particular for z ≥ (za + zw) /2.

Proof. Using Θ3 > −1/ψa:

ma(z) > 1γ

[(βa − βw) qa(z)− 1

ψaqa(z)2

]= 1γ

(κqzz − βw)︸︷︷︸>0

qa(z) > 0, (D.4)

except asymptotically where qa(z)→ 0; then, ma(z)→ 0. Furthermore,

∂ma(z)∂z

<1γ

[(βa − βw)− 2

ψaqa(z)

](−ψaκqz) = 1

γ(2κqzz − βa − βw)︸︷︷︸R0⇔ zR(za+zw)/2

(−ψaκqz)︸︷︷︸<0

, (D.5)

so that contribution payments at the latest start declining when z = (za + zw) /2. They

are always declining in z if ∂ma(zw)∂z

≤ 0. Evaluating ∂ma(zw)∂z

≤ 0 yields (D.3). Along

the lines of the proof of Proposition 5, αw > αa2 so that η(1 + µ) ≥ 1/2 is sufficient for

the (weak) inequality in (D.3) to hold. Finally, the bracketed term in (D.3) is negative if

αw ≥ 1/ψa ≥ αa, so that αw ≥ απ is sufficient for (D.3), and it is necessary for (D.3) if

η = 0.

We continue with z < zw, so that qw(z) ≥ 0 and qw(z + qa) ≥ 0 are not binding. We have

to distinguish between the four lobbying-equilibrium cases. We start with Cases I and II in

which ψw ≤ ψa.

Proposition D.2 (Contribution Payments for z < zw and ψa ≥ ψw) Suppose thatψa > ψw (Case I) or ψa = ψw (Case II) and that z ∈ [0, zw). Then ma(z) > 0. Moreover,∂ma(z)∂z

< 0 in Case I and ∂ma(z)∂z

= 0 in Case II.

Proof. In both cases, qa(z) − qw(z) > 0; see (4.14). Additionally, ψa > ψw implies Θ1 > 0for Case I and ψa = ψw implies Θ1 = 0 for Case II. Thus, all (remaining) parts of (D.1) are

positive. Moreover,

∂ma(z)∂z

= 1γ

[Θ1 + Θ2 · 2∆a,w(z)

](ψw − ψa)κqz. (D.6)


E. Proofs of Section 6.1

In both cases, all parts of the square-bracketed term are positive. ψa > ψw in Case I then

implies ∂ma(z)∂z

< 0, and ψa = ψw in Case II implies ∂ma(z)∂z

= 0.

Finally, we characterize the development of ma(z) in Cases III and IV, in which ψa < ψw:

Proposition D.3 (Contribution Payments for z < zw and ψa < ψw) Define

z1 = zw − ψa1− ψwκqzψw − ψa

(za − zw) = ψwβw − ψaβa(ψw − ψa)κqz

, (D.7a)

z2 = z1 −Θ1

2Θ2

1− ψaκqzψw − ψa

za − zw

βa − βw= z1 −

Θ1

2Θ2 (ψw − ψa)κqz, (D.7b)

z3 = z2 −Θ1

2Θ2

1− ψaκqzψw − ψa

za − zw

βa − βw= z2 −

Θ1

2Θ2 (ψw − ψa)κqz, (D.7c)

where z1 < z2 < z3 by Θ1 < 0. We have ma(z) < 0 for z ∈ (z1, z3), m(z) = 0 for z ∈ z1, z3and ma(z) > 0 otherwise. Moreover, sign(∂m

a(z)∂z

) = sign(z − z2).Suppose that ψwβw ≤ ψaβa (Case III). Then z1 ≤ 0. This excludes cases in which ma(z) is

first positive and declining. Depending on the parameter values, both z2 and z3 may be belowzero or exceed zw. Thus, ma(z) can be positive, negative, or first negative and then positivefor z ∈ [0, zw]. If ma(z) > 0 for z ∈ [0, zw], ma(z) is also increasing for z ∈ [0, zw]. If ma(z)is negative or first negative and then positive, ma(z) may be increasing for z ∈ [0, zw], or firstdeclining and then increasing.

Now suppose that ψwβw > ψaβa (Case IV). Then 0 < z1 < zw. Thus, ma(z) is positive anddeclining for z ∈ [0, z1]. ma(z) may then take its minimum at a negative value at z2 ∈ (z1, z

w)or still be declining at zw. In the former case, ma(z) may become positive at z3 ∈ (z1, z

w) orstill be negative at zw.

Proof. In both cases, ψa < ψw implies Θ1 < 0. In Case III, ψwβw ≤ ψaβa so that z1 ≤ 0 and

in Case IV, ψwβw > ψaβa so that 0 < z1 < zw. We can write ma(z) and ∂ma(z)∂z

as functions of

z:

ma(z) = 1γ·Θ2 · (ψaκqz − ψwκqz)2 · (z1 − z) · (z3 − z) , (D.8a)

∂ma(z)∂z

= 1γ·Θ2 · (ψaκqz − ψwκqz)2 · (z2 − z) · (−2) . (D.8b)

Those levels of cumulative extraction for which ma(z) and ∂ma(z)∂z

change their signs directly

follow from (D.8).

E Proofs of Section 6.1

In this section, we demonstrate how to provide the solutions for the linear-quadratic ver-

sion of the recursive Nash bargaining solution in Section 6.1. We guess that the extraction



function in case of disagreement has the same linear form as (C.1) from the benchmark

cases:

qd(z) =

Yd

0 + Y d1 z if Y d

0 + Y d1 z ≥ 0,

0 if Y d0 + Y d

1 z < 0,(E.1)

where the stability condition is

0 < 1 + Y d1 ≤ 1. (E.2)

We can now determine the intertemporal welfare and profit in case of permanent disagree-

ment.

Lemma E.1 (Recursive Nash Bargaining Permanent-Disagreement Welfare and Profit)

Suppose that qd(z) from (E.1) is the disagreement extraction function. Then,

W d∞(z) =βw + Y d0

Y d1κqz

1− δ(1 + Y d

1

)qd(z)−αw2 + 1

Y d1κqz

1− δ(1 + Y d

1

)2 qd(z)2 − χzz

1− δ , (E.3a)

Πd∞(z) =βπ + Y d0

Y d1κqz

1− δ(1 + Y d

1

)qd(z)−απ2 + 1

Y d1κqz

1− δ(1 + Y d

1

)2 qd(z)2. (E.3b)

Proof. The proof follows along the lines of the proof of Lemma C.1.

We can use these intertemporal-utility functions to determine the unknown coefficients:

Lemma E.2 (Recursive Nash Bargaining Disagreement Extraction) In the disagreementextraction function (E.1), the coefficients are Y d

0 = ψd ·βd and Y d1 = −ψd ·κqz, where ψd ≤ ψw

is implicitly defined by

η (1 + µ)(

1κqz− ψd

)(ψa − ψd)2 − αw − κqz

αa − κqz

(1κqz− ψa

)(ψw

1− ψwαw+ ψd

)(ψw − ψd) = 0

(E.4)

and βd is defined in (6.8). This implies qd(z) = ψd · (βd − κqzz) for positive extraction. Replac-ing the unknown coefficients in Lemma E.1 and simplifying, we obtain Proposition 9.

Proof. By (6.4a), qd(z) is the q that maximizes

W d∞(z) + η · δ ·µ ·[Πa(z + qd(z))− Πd∞(z + qd(z))

]+W a(z + qd(z))−W d∞(z + qd(z))

.

Substituting W a(z) and Πa(z) from Proposition 4 and W d∞(z) and Πd∞(z) from Lemma

E.1, we can derive a first-order condition for q, into which we substitute qd(z) = Y d0 + Y d

1 z



and qa(z) from Proposition 3:

βw − κqzz − αw(Y d

0 + Y d1 z)

+ δ ·

Y d1 βw + Y d

0 κqz

1− δ(1 + Y d

1

) − αwYd

1 + 2κqz1− δ

(1 + Y d

1

)2

[Y d

0 + Y d1

[Y d

0 +(1 + Y d

1

)z]]

+ η (1 + µ) δ ·

(ψaαa − 2)κqz1− δ (1− ψaκqz)2ψa

[βa − κqz

[Y d

0 +(1 + Y d

1

)z]]

+ αaYd

1 + 2κqz1− δ

(1 + Y d

1

)2

[Y d

0 + Y d1

[Y d

0 +(1 + Y d

1

)z]]− Y d

1 βa + Y d0 κqz

1− δ(1 + Y d

1

) = 0.

(E.5)

This condition must also hold for z = z + qa(z) = z + ψa (βa − κqzz). This generates two

equations in two unknowns. Subtracting the first from the second and rearranging yields:

βa − κqzz1− δ (1− ψaκqz)

· δ (1− δ)1− δ

(1 + Y d

1

)2 ·

η (1 + µ)(1 + Y d

1

) (ψaκqz + Y d

1

)2

− αw − κqzαa − κqz

(1− ψaκqz)(

ψwκqz1− ψwαw

− Y d1

)(ψwκqz + Y d

1

) = 0. (E.6)

In line with the previous sections, we define Y d1 ≡ −ψd · κqz as stated in Lemma E.2.

Together with (E.6), this yields (E.4). Substituting Y d1 = −ψd · κqz in (E.2) yields 0 <

1 − ψdκqz ≤ 1. Thus, the first term in (E.4) is positive, so that the second term must also

be non-negative for the equality to hold. This yields ψd ≤ ψw. To derive Y d0 ≡ ψd · βd, we

solve (E.5) for Y d0 . This equation depends on

(Y d

1

)4. The curly-bracketed term in (E.6)

defines(Y d

1

)3, so that we can derive

(Y d

1

)4by multiplying with Y d

1 . Substituting(Y d

1

)4and

afterwards(Y d

1

)3into the equation for Y d

0 yields, together with Y d1 = −ψd · κqz, (6.8).

(6.8) then implies the following convergence-levels relation:

Lemma E.3 (Relation between zd, zw and za) The relation between zd, zw and za is givenas follows:

za > zd > zw if ψd > ψa, (E.7a)

za > zd = zw if ψd = ψa, (E.7b)

za > zw > zd if ψd < ψa and 1 + η (1 + µ) δ (ψd − ψa)κqz1− δ (1− ψaκqz)

> 0, (E.7c)

zd > za > zw if ψd < ψa and 1 + η (1 + µ) δ (ψd − ψa)κqz1− δ (1− ψaκqz)

< 0. (E.7d)

The future gains of cooperation are increasing in z if (E.7a) or (E.7d) hold, they stay constantif (E.7b) holds and they are declining in z if (E.7c) holds. η (1 + µ) < 1/δ is sufficient for



the last inequality in (E.7c) to hold and η (1 + µ) > 1/δ is necessary for the last inequality in(E.7d) to hold.

Proof. (E.7) directly follows from (6.8) together with zi = βi/κqz for i = w, a, d. From

the discussion of (6.5), we know that zd R zw is equivalent to increasing, constant and

declining future gains of cooperation. This and (E.7) prove the second to last sentence in

the proposition. For (E.7c) and (E.7d), note that

1 + η (1 + µ) δ (ψd − ψa)κqz1− δ (1− ψaκqz)

≷ 0 ⇔ ψd ≷η (1 + µ)− [1− δ (1− ψaκqz)] / (δψaκqz)

η (1 + µ) ψa.

The numerator on the right side of the equivalence increases with ψaκqz and goes to

η (1 + µ)− 1/δ for ψaκqz → 1. This proves the last sentence in the proposition.

Furthermore, we can determine the difference in payments for η (1 + µ) = 1 before the

non-cooperative convergence levels are reached.

Lemma E.4 (Difference in Payments for η (1 + µ) = 1 and z < zw, zd) Supposeη (1 + µ) = 1. Then for z < zw, zd, Mar(z) −Ma(z) is zero if ψw = ψa and it is positive ifψw < ψa. For βw = βa, the difference is negative if ψw > ψa.

Proof. Substituting (4.9a), (4.9b) and (6.10) into (6.6) for η (1 + µ) = 1 and rearranging

yields:

αw2 −

αa2

1− δ (1− ψwκqz)2 · [ψw (βd − κqzz)]2 −αw2 −

αa2

1− δ (1− ψdκqz)2 · [ψd (βd − κqzz)]2

+ βa − βd1− δ (1− ψwκqz)

· ψw (βd − κqzz)− βa − βd1− δ (1− ψdκqz)

· ψd (βd − κqzz)

+ βw − βd1− δ (1− ψwκqz)

· ψw[βa − βw + αw − αa

2/ψw − αw

(βw + βd

2 − κqzz)]

. (E.8)

For ψw = ψa, the first line is zero because then αw = αa, the second line is zero because

then ψd = ψw by (E.7b), and the third line is zero because then βd = βw by (E.7b). For

ψw < ψa, the third line is positive because then βw > βd by (E.7c) and 0 ≤ αw−αa2/ψw−αw ≤ 1.

Furthermore, for any X > 0, X2

1−δ(1−Xκqz)2 and X1−δ(1−Xκqz) are increasing in X. Thus, the

first line of (E.8) is positive because ψw < ψa implies αw > αa and ψw > ψd, and the second

line of (E.8) is positive because ψw > ψd. For ψw > ψa and βw = βa, the first line of (E.8)

is negative because ψw > ψa implies αw < αa and ψw > ψd, the second line is zero because

βw = βa, and the third line is zero because βw = βa implies βd = βw by (6.8).


F. Proofs of Section 6.2

Substituting qa and qd from Propositions 3 and 8 into the payments formula of Proposition

10 and simplifying yields:

mar(z) = 1γ

Θ2 ·

[qa(z)− qd(z)

]2if z < zd,

(βa − βw) qa(z) + Θ3 · qa(z)2 + η (1 + µ)Ba|dqd(z)− Θ4 · qd(z)2 if z ∈

[zd, zd

),

(βa − βw) qa(z) + Θ3 · qa(z)2 if z ∈[zd, za

),

(E.9)

where

Θ2 ≡1− δ

1− δ (1− ψdκqz)2

[η (1 + µ)

(1ψa− αa

2

)+ 1ψw− αw

2

]

− 1− δ1− δ (1− ψdκqz)2

(1ψw− αw

2 + 1ψa

)ψa (ψw − ψd)ψw (ψa − ψd)

(> Θ2 if ψd > ψa) , (E.10a)

Θ4 ≡1

1− δ (1− ψdκqz)2

[η (1 + µ)

(1ψd− αa

2

)− 1ψd

+ αw2

](E.10b)

and where Θ3 is defined in (D.2c). The inequality in (E.10a) follows from ψd ≤ ψw.

F Proofs of Section 6.2

Similar to (D.1), we can then derive the payments in the period of cooperation:

mas(z) = 1γ

1−δ

1−δ(1−ψwκqz)2

[η (1 + µ) 2ψw(1−ψaαa)+ψ2

aαa2ψ2a

+ 1ψw− αw

2

]· [qas(z)− qw(z)]2 if z ≤ zw,

(βa − βw) · qas(z) +[η (1 + µ) αa

2 + αw2 − αa

]· qas(z)2 if z ∈ (zw, zas] .

(F.1)

Finally, we can state the difference in payments:

Proposition F.1 (Explicit Example: Difference in Payments between σ = 0 and σ = 1)

For z ≤ zw, mas −ma is positive if ψa > ψw, and it is zero if ψa = ψw. If ψa 6= ψw, mas −ma

is increasing in η. For z ∈ (zw, zas], mas −ma is positive.

Proof. Substituting (D.1) and (F.1) into mas(z)−ma(z) and rearranging yields

mas(z)−ma(z) =

1γ

η (1 + µ) δ2(1−δ)(1−ψaκqz)2(ψa−ψw)4κ2qz

2[1−δ(1−ψaκqz)][1−δ(1−ψwκqz)2][1−δ+2δ(1−ψaκqz)ψwκqz+δψ2aκ

2qz]ψa

·[

[1−δ(1−ψwκqz)(1−ψaκqz)](ψaβa−ψwβw)−(ψa−ψw)ψaκqzβa[1−δ(1−ψwκqz)](1−ψaκqz)(ψa−ψw) − κqzz

]2+ 1−δ

1−δ(1−ψwκqz)2

(1ψw− αw

2

)·

[qas(z)− qw(z)]2 − [qa(z)− qw(z)]2

if z ≤ zw,

(1− ψaαa) (κqzz − βw) · qas(z) +[η (1 + µ) αa(1−ψaαa)2

2 + αw(1−ψ2aα

2a)

2

]· qas(z)2 if z ∈ (zw, zas] .

(F.2)


References

If ψa > ψw, the second and the third line of (F.2) are positive and, by Proposition 11,

qas > qa > qw, which implies that the forth line of (F.2) is positive, such that mas > ma. If

ψa = ψw, the second line of (F.2) is zero and, by Proposition 11, qas = qa, which implies

that the forth line of (F.2) is zero, such that mas = ma. The derivative of mas − ma with

respect to η is positive for z ≤ zw and ψa 6= ψw, and it is positive for z ∈ [zw, zas), which

proves the second sentence in the proposition. The last line of (F.2) is positive, such that

mas > ma for z ∈ [zw, zas).

References

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