Lobbying and Public Good Provision in a Federal Economy:
A Dynamic Approach
Bodhisattva Sengupta1
McGill University
Abstract: In this paper, we address the issue of dynamic lobbying within a federation
that consists of a central government and two regional governments. Regions lobby for
more central funds to produce a pure public good such as an improvement in environmental quality. Thus, bene…ts of lobbying are public, unlike what is traditionally assumed in
the public choice literature. Lobbyists capture a part of the central grant; the other part
is used to produce the public good. Allocation of central funds is dictated by two considerations: the level of lobbying as well as the e¤ ective use of central funds. Welfare to the
consumer equals utility from the public good, net of lobbying cost. Unlike the regional governments, the central government is non strategic. We investigate steady state behaviour
as well as welfare implications of di¤ erent lobbying protocols: e.g. open and closed loop
cooperative and non co-operative behaviour. We show that, for symmetric provinces, the
steady state lobbying level and stock of public good are greater than the case when lobbyists
are benevolent and cannot capture the rent. Second, introducing a rent-appropriating lobbyist may increase the welfare of the consumers compared to the case where the lobbyist is
‘benevolent’. Third, for a wide range of parameter values, non-cooperative lobbying results
in higher welfare for the consumer vis-à-vis cooperative lobbying protocols. Predictions of
Fershtman and Nitzan (1991, hereafter FN) emerge as limiting cases of our model.
JEL Classi…cation: H7; R1
Keywords: Lobbying, Public Good, Federalism
This version: 10 February, 2007
Please do not quote
1
Department of Economics, McGill University, 855 Sherbrook Street West, Montreal, Quebec, Canada.
E-Mail: bodhisattva.sengupta@mail.mcgill.ca. I am indebted to Ngo Van Long, Hassan Benchekroun,
Sanjay Banerji, Soham Baksi, Jennifer Hunt and participants of Montreal Natural Resource and Environmental Economics Workshop for helpful comments and criticisms. Usual disclaimer applies.
1
Introduction
Regional lobbying, in some form or other, is constitutionally valid in many federations,
despite being denounced by public choice theorists (Tullock 1967; Krueger 1974; Posner
1975). The main argument against lobbying is that the bene…ts of lobbying are private.
It leads to wastage of scarce resources. A direct policy implication is that in an ideal
world, lobbying must be discouraged. In reality, the lobbying power of provinces often
determines the pattern of centre-state transfers. Such a policy can be justi…ed, on a
normative basis, only if the bene…ts of federal grants are public in nature. At limit, the
public good produced within the region is a pure public good within the region. Examples
include environmental programme pursued by a certain province or provincial investment
in knowledge.2
The objective of the present paper is to examine the dynamic provision of a public
good in the presence of regional lobbying. The federation consists of a central government
and two provinces. There exists one lobbyist (who is the local politician,say) in each
province. Lobbying imposes a private cost on the residents of a region. A central grant
‡owing to a region is partly determined by regional lobbying. A fraction of the central
grant is converted into a public good via a …xed coe¢ cient technology. The rest might be
captured by the lobbyist for his/her own bene…t. Thus, the bene…t of lobbying is public.
Provinces di¤er in two dimensions: lobbying e¢ ciency (proxied by cost of lobbying) and
production e¢ ciency (ability to convert one unit of central grant into the stock of public
good). Lobbyists maximise the sum of the present values of the lifetime utility of the
province and the lifetime rent accruing to the lobbyist.
We consider di¤erent types of strategies and behavioural protocols that can be adopted
by the lobbyists. For example, lobbyists can undertake cooperative (maximising the sum
of utilities of both the provinces and of rents accruing to both the lobbyists ) or non
co-operative (maximising the sum of utility of his/her own province and personal rent)
protocols. Lobbyists can employ either open loop strategies, where they commit to a
time path of e¤orts, or closed loop strategies where individual e¤orts at any given instant
depends on the stock of public good at that moment (that is, Markov perfect, in the sense
of Maskin and Tirole, 2001).3 We compare and contrast our results with the scenario where
2
The EPA Superfund is an example of federally funded provincial environmental programme. The
Californian stem cell research programme, has it been partly funded by Federal government, would be a
good example for knowledge investment.
3
See appendix A1 for complete de…nitions. Economists are usually interested in properties of closed
loop, non cooperative solutions for two reasons. First, cooperation is hard to sustain. Second, unlike open
1
the lobbyist cannot ( or does not) capture the rent, and acts as a benevolent politician.
Speci…cally, we ask three questions. What are the level of lobbying and public good stock
at the steady state? What lobbying protocol is bene…cial for the provincial consumers? Are
the consumers better o¤ in the presence of a non-benevolent (i.e. rent-seeking) lobbyist?
We now turn to a discussion of our main results. First, at least in the symmetric
case, lobbying levels and the stock of public good increase compared to the case with the
benevolent lobbyist for all lobbying protocols. We also found conditions such that the
lobbying level is positive, i.e. the bene…t from the public good and the rent outweighs
the free rider problem. Second, using numerical analysis, we show that, at least for some
parameter values, welfare accruing to the consumer is higher for noncooperative lobbying
protocols than cooperative protocols. This result is in contrast with all standard models
of dynamic public good provision (e.g. Fershtman and Nitzan, 1991, hereafter FN). Third,
we show that non cooperative strategies with a rent capturing lobbyist can bestow higher
welfare to the consumer compared to a benevolent lobbyist. This provides us with a
normative yardstick against which to evaluate di¤erent lobbying protocols (e.g. as in List
and Mason, 2001). Fourth, we show that, using asymmetric provinces, higher e¢ ciency
in public good production might lead to perverse e¤ects on equalisation.
In a static setting, lobbying associated with public good is considered in Katz et
al. (1990). They seek to investigate the level of lobbying when two localities lobby
for a public good (such as removal of pollution). Removal of pollution is a pure public
good within a certain locality and a private good among the jurisdictions. Their basic
results and intuitions are very similar to the Tullock (1980) rent-seeking game in the sense
that the lobbying e¤orts are less than the utility (value) of the local public good. In
a recent paper, Cheikbossian (2006) considers the case where public goods within each
region have an interjurisdictionary spillover e¤ect. The aim of his paper is to identify
the conditions under which decentralised outcome (when regions lobby for and produce a
public good) provides higher surplus to the federation compared to the centralised outcome
(when central government itself produces a public good in each region). However, in our
opinion, these models, being static in nature, can not capture the dynamic reality of many
environmental public good.
Provision of public good in a dynamic setting has been …rst systematically studied by
FN. Extending the framework provided by Bergstrom et al. (1986) to a dynamic set up,
they compare two di¤erent types of strategies adopted by the players. The basic result is
loop equilibrium, a closed loop equilibrium is always subgame perfect, making it a desirable outcome for
this class of games.
2
a pessimistic one: compared to an open loop case, with Markov perfect non cooperative
strategies, each agent provides less e¤ort and thus generates a lower amount of public
good in the long run. Second, FN …nd that individual e¤orts are strategic substitutes: if
one agent provides less e¤ort, the opponent will provide more. It is also obvious that in
such a setting, a cooperative strategy (if possible) will yield the highest level of consumer
bene…t. However, the FN results are not robust to alternative speci…cations. Exploiting
the technique provided by Tsutsui and Mino (1990), Wirl (1996) showes that among the
alternative equilibriums (static, open loop, cooperative), linear4 Markov strategy provides
the worst outcome (i.e. lowest e¤ort and public good stock) vis-à-vis non linear ones.5
Thus the ‘pessimistic result’in FN depends on the assumption of linear strategies. Kessing
(2007) considers the discrete good case (i.e. a public good to be provided in the future
when the sum of contribution meets the cost) and, unlike FN, …nds that individual e¤orts
are strategic complements rather than substitutes.
Wirl (1994) considers lobbying in a dynamic set up. In order to explain why lobbying
expenditure is smaller than the prize itself, he shows that lobbying expenditure is strictly
less in the Markovian set up. In his analysis, lobbying is a constant sum game : thus
a bene…t to one group is a loss to the other. Other related studies include Tornel and
Lane (1999), Tornel and Velasco (1994) and Long and Sorger (2006). In these models,
competing interest groups (which can be interpreted as competing regions) lobby for and
grab productive resources (owned by central government) for private bene…t, thus a¤ecting
growth potentials of the economy.
To sum up, we abstract away from a world of pure private bene…ts of lobbying, as
in the public choice literature, since it may not be appropriate for many public goods.
On the other hand, we no longer believe that a lobbyist is necessarily benevolent, as
a straightforward reading of the standard models of dynamic provision of public good
suggests. Adopting a position somewhere in between, we show that noncooperative, closed
loop lobbying may provide more welfare to the consumer than other lobbying protocols.
And consumers are better o¤ with a rent seeking lobbyist compared to a benevolent
lobbyist under certain protocols.
The paper is divided into the following parts. In section 2, we set up the model. In
section 3, we discuss the open loop case. The e¤ects of closed loop or Markovian strategy
is discussed in the fourth section. Section 5 analyses the symmetric equilibrium. In section
4
5
That is, e¤orts are linear functions of the stock of public good.
For a criticism of Tsutsui and Mino’s methodology as well as a discussion of some recent issues on
non-linear Markov strategies, see Rowat (2006).
3
6, we show some numerical results. Asymmetric provinces are discussed in section 7 and
section 8 concludes.
2
The Model
The central government distributes funds pi (i = 1; 2) to produce pure public good to two
regions. Given the fund, the production of public good in each region is
i pi
(
i
< 1).
There is no cost of provision of this good for the local government. However, the regions
lobby for the fund, and lobbying is costly. Lobbying expenditure of region i is Li . The
overall public good stock is E. This evolves through time by the following equation:
E_ =
i pi
mE
(2.1)
where m is the natural rate of ‘decay’of overall stock of public good. This is a fairly
standard representation of dynamic evolution of public good.6
For a real life example,
i pi
might be expenditure on pure public good such as puri…ca-
tion of air. Improved quality of air will bene…t the federation as a whole due to symmetric
spillover. Note that,
i
can be thought as an e¢ ciency parameter.
As we have mentioned before, the central government’s transfer to region i depends
on both Li and
i.
For simplicity, let us assume the linear transfer formula:
pi = Li +
The parameter
i:
(2.2)
is the relative weight between lobbying and e¢ ciency.7 Thus, the
central transfer formula consists of both discretionary and formulaic parts. The central
decision maker matches the lobbying e¤ort (discretionary), and provides an unconditional
block grant (based on e¢ ciency) to the regions.8 In our formulation, central government
is not a strategic player. This is also a somewhat standard simplifying assumption, e.g.
in Tornel and Lane (1999) as well as Wirl (1994).9
6
Our main results will follow if the spillover e¤ect is less than 1, i.e. if we are not dealing with a pure
public good.
7
A more general speci…cation is pi = (Li ; i ), with Li > 0; i > 0
8
In the above formulation, it seems that the central government does not face a budget constraint.
p
However, if we change the funding equation by pi = 2 Li + i , then this would re‡ect that central funds
have some alternative use: the marginal reward for lobbying is decreasing. Incorporating this does not
change our results signi…cantly.
9
A key feature of the …scal federalism literature (e.g. Koethenbuerger, 2006) is that central government
acts as a strategic agent. Assuming a naïve central governement, we abstract away from this issue.
4
The portion of the central fund which is not spent (‘lost in transit’) is appropriated by
the local politicians (lobbyist) as rent. The rent thus equals (1
i )pi .
We assume that the
lobbyist maximises his/her own rent plus the net utility of the population. In other words,
his/her behaviour is partly benevolent and partly sel…sh, and reasonably approximates the
behaviour of a local politician. We further assume that the regions derive identical bene…t
(U (E)) from the pure public good E. The lifetime utility function of a lobbyist is given
by:
Z1
e
t
(U (E) + (1
i )pi
Ci (Li )) dt
(2.3)
0
For the sake of tractability, we use the following assumptions of convenience:
E 2 . Here,
(a) The utility function is linear-quadratic: U (E) = E
2
1
positive number such that Emax = is the satiation level of the public good.
is a small
determines
the rate at which the marginal utility falls.
(b) The cost function is convex and has the following simple structure Ci (Li ) = 12 ci L2i ,
ci being a constant.
These two assumptions imply that we are dealing with a standard linear quadratic
game. Apart from tractability, another defence that has been put forward for such a game
is that it might represent "Good Taylor approximation of more general games" (Fudenberg
and Tirole, 1991).
Note that (2.3) can be decomposed into
Z1
e
t
E
0
2
E2
2
1
ci L2i dt + 4
2
i (1
i)
+ (1
i)
Z1
0
e
t
3
Li dt5
(2.3a)
The …rst term represents the consumer’s bene…t, while the second component (within
squared bracket) equals the total rent appropriated by the lobbyist. This representation
highlights the importance of inclusion of the cost term in lobbyists’utility. If the lobbyist
does not care for the cost, he/she would demand in…nite amount of fund.10
3
Open Loop Strategy
Open loop strategy requires that the players commit to a strategy from the beginning,
and the path is dependent on time. In other words, the strategy space is described by
10
If we put
i
= 1; we are back to the FN model (1991). Thus, it is a special case of our own model.
5
Li = f (t)8t . Such a strategy may work if both players make a binding agreement of
adhering to it.
3.1
Open Loop Cooperative Behaviour
If the lobbyists maximise their joint utility,11 the maximand becomes
Z1
e
t
E 2 + (1
2E
1
ci L2i dt
2
i )pi
0
Let
be the costate variable. Then the current value Hamiltonian for the problem
can be written as (plugging in the values for pi )
H 1 = 2E
E 2 + (1
1 )(L1
= 2E
E 2 + (1
1 )L1
Where, A2 = (
2
1
+
A3 = (1
+
1)
+ (1
1
1
c1 L21
c2 L22
2
2
+
( 1 (L1 + 1 ) + 2 (L2 + 2 ) mE)
1
1
c1 L21
c2 L22
2 )L2 + A3
2
2
+ ( 1 L1 + 2 L2 + A2 mE)
+ (1
2 ) (L2
+
2)
2)
2
1) 1
+ (1
2) 2
@2H 1
= 2ci < 0
The SOC is satis…ed since
@L2i
Let us assume an interior equilibrium (Li > 0).
The necessary conditions are
(1
i)
ci Li +
_ = ( + m)
E_ =
Proposition 1.
1 L1
+
2 L2
i
=0
(3.1.1)
2 (1
E)
(3.1.2)
+ A2
mE
(3.1.3)
The open loop co-operative behaviour results in the following steady
state values of E and Li
EsOc =
11
2A1 + (s1
+ s2 2 2 + A2 ) (m + )
m( + m) + 2 A1
1 1
In Mason and List (2001), the cooperative, or centralised outcome entails equal e¤orts for both regions.
6
LOc
is =
i
m(2 + si + msi ) 2 (A2 di
m( + m) + 2 A1
2
i
i
=
; si =
1
j j)
i
, di = si sj
ci
ci
i
The steady state is stable, in a saddlepoint sense. And, given certain restriction on
1
parameters, EsOc < :
Where,
, A1 =
i
Proof. See appendix 1.
1
1
is a measure of e¢ ciency of region i’s lobbyist ( the higher is , the
Note that,
ci
ci
lower is the e¤ort cost),and i is a measure of region i0 s e¢ ciency of use of the public fund.
So,
i,
the product of these two measures, is an overall measure of e¢ ciency of region i.
Comment 1: Lobbying may be zero if
or A2 is too high. From the neccesary
condition (3.1.3), A2 is the ‘autonomous’growth of E when lobbying is zero.
is the rate
at which MU of public good goes down. If these are high, the province may not altogether
engage in costly lobbying. Lobbying is highest when
= 0, i.e.when the centre allocates
only on the basis of lobbying. In that case, A2 = A3 = 0:
Oc . HowComment 2: Compared to the benevolent world12 , it seems that EsOc > EsB
Oc
ever, it is not necessarily true that LOc
is > LisB . Here the subscript B signi…es associated
variables in the model with benevolent lobbyists.
Comment 3: Unlike the benevolent case, the ratio of steady state lobbying levels is
no longer represented by ratio of relative e¢ ciency
We can also represent the open loop behaviour as a ‘closed loop solution’, where Li
is a function of E. To do this, we must solve the di¤erential equations in terms of E and
L explicitly, and make the necessary substitutions. The following proposition summarizes
our main …ndings with regard to the ‘closed loop’form.
Proposition 2.
The equilibrium open loop, cooperative lobbying can be represented
in the closed loop form as
Li = a0i
b0i E
where,
b0i =
i
2A1
2m +
p
8 A1 + (2m + )2 > 0
Oc
a0i = LOc
is + b0i Es > 0
12
See appendix 2.
7
Proof. See appendix 1.
Comment 4: The positive value of b0i parallels the F N result that e¤orts are strategic
substitutes( although, in case of cooperation, the word ‘strategic’is somewhat misleading).
If one region lobbies less, the other one will lobby more.
3.2
Open loop Non cooperative behaviour:
If the regions play a non co-operative game, let
i
be costate variable of region i:Then the
relevant Hamiltonian is
Hi = E
2
E 2 + (1
i )(Li
+
1
ci L2i +
2
i)
i ( 1 (L1
+
1)
+
2 (L2
+
2)
mE)
This is concave in control variable.
The necessary conditions are
(1
i)
ci Li +
_ i = ( + m)
E_ = A2 +
i
i i
(1
i Li
=0
(3.2.1)
E)
(3.2.2)
mE
(3.2.3)
In this case, we can summarise the steady state properties13 in the following theorem:
The steady state , stable values of public good stock and lobbying is
Proposition 3.
given by the following expressions
EsOnc =
LOnc
=
is
A1 + (s1
i
+ s2 2 2 + A2 ) (m + )
m( + m) + A1
1 1
m(1 + si (m + ))
A2
m( + m) + A1
di
j j
where the superscript nc implies no cooperation, O refers to open loop and the subscript
s stands for steady state.14
Proof. See appendix 1.
As usual, we can represent the open loop strategies into a closed loop form in the
following proposition :
13
14
Comments analogous to 1,2, 3 apply here as well.
Given some restrictions on the parameters, there is no “over-accumulation” of steady state stock.
8
Proposition 4.
The closed loop form of open loop ,non-cooperative lobbying can be
represented as
Li = a1i
b1i E
where,
b1i =
i
2m +
2A1
p
4 A1 + (2m + )2 > 0
Onc
a1i = LOnc
>0
is + b1i Es
Proof. See appendix 1.
Now we can compare the cooperative and non cooperative strategies. The steady
state behaviours are reported in the following proposition:
Proposition 5.
For the open loop case, if
is small enough (to prevent overprovi-
sion of public good), then cooperative strategy induces more public good as well as higher
lobbying expenditure at the steady state.
Proof. See appendix 1.
Onc
Comment 5: The necessary and su¢ cient conditions for EsOc > EsOnc and LOc
is > Lis
turn out to be that of preventing overaccumulation of public good.
Comment 6: The proposition parallels the fact that in noncooperative case, agents
try to free ride and thus provide less e¤ort. Therefore, the noncooperative stock of public
good and e¤orts (here, lobbying is analogous to e¤ort) will be less than the cooperative
case
The following lemmas compare the closed loop forms of co-operative and non cooperative lobbying.
Lemma 1
b0i > b1i : as E goes up, the marginal reduction in lobbying is higher in
cooperaive case than in non cooperative case.
This follows immediately from comparing
b1i =
i
2A1
2m +
p
4 A1 + (2m + )2
2m +
p
8 A1 + (2m + )2
and
b0i =
i
2A1
9
The above inequality implies that, given a decrease in opponent’s lobbying, a province
will lobby for more in case of cooperative lobbying. Therefore the degree of strategic
substitutability goes down.
Lemma 2
a0i > a1i : when E is small (i.e. almost 0), the e¤ ort is higher in
cooperative case than in non co-operative case.
Onc and E Oc >
This is a corollary to proposition 5 and the above lemmas : LOc
s
is > Lis
m
Oc
Oc
Onc
Onc
Onc
> . Thus, Lis + b0 Es > Lis + b1 Es
Es as long as
A2 + A1 s1 d1 2 2
As before, if we write LOc
= a0 b0 E and LOnc
= a1 b1 E , then solving this
i
i
a0 a1
and
set of linear equations, we can see that the unique solution is at EA =
b0 b 1
1
LA =
(a1 b0 a0 b1 ) : Since a0 > a1 and b0 > b1 , there exists a positive EA such
b0 b1
that these equations are satis…ed. However, its not certain whether LA is positive or not.
The denominator is always positive. The numerator can be written as
b0 LOnc
is
Onc
b1 LOc
is + b0 b1 (Es
EsOc ), which is negative if b0 LOnc
is
b1 LOc
< 0. It is
is
not possible to compare these two analytically.
These two possibilities are summarised in the following two graphs. On the horizontal
axis, we have the state variable, i.e. stock of public good, and the vertical axis represents
lobbying.
L
LOc
LOnc
E
Figure 1: LA > 0; open loop strategies
and
L
LOc
LOnc
E
Figure 2: LA < 0; open loop lobbying
10
The discussion can be summarised in the following proposition:
For open loop case, if E is small, cooperative lobbying dominates
Proposition 6.
non cooperative e¤ ort. On the other hand, when E is large, it may or may not dominate
the latter. However, given an increase in E, the marginal decrease in lobbying e¤ ort is
higher in cooperative case.
4
Closed loop Strategies
Stationary closed loop strategies employ Li = Fi (E) , i.e. the control variable is a function
of the state variable . Each lobbyist understands that lobbying creates more public good,
which is related with utility. Therefore, it is reasonable to base costly lobbying e¤ort on
the current value of stocks. As before, we begin with cooperative strategy.
4.1
Closed Loop Cooperative Strategy
For co-operative solution the utility is the sum of two regions . The relevant HJB equation
is,
V (E) = max 2E
Li
E 2 + (1
i )(Li
+
i)
1
ci L2i + V 0 (E)(
2
i pi
mE)
Where V (E) is the value function.
Maximisation of the LHS implies
Li =
Where
i
=
i (si
+ V 0 (E))
(4.1.1)
i
ci
Substituting this back into the RHS, the HJB equation reduces to
V (E) = 2E
1
1
E 2 + A1 + A3 + (V 0 (E))2 A1 + V 0 (E)(A1 + A2 )
2
2
mEV 0 (E)
Given the linear quadratic structure of the game, we posit that the value function is
c
quadratic. Let V (E) = a + bE + E 2 be the trial solution
2
Then
V 0 (E) = b + cE
and the lobbying e¤ort is
Li =
i (si
+ b + cE)
11
(4.1.1a)
The cooperative structure implies that we are solving a dynamic optimisation problem,
and there is no di¤erence between open and closed loop linear outcomes. The di¤erence,
if any, occurs when the lobbyists employ non-linear strategy.
4.2
Closed Loop Non Cooperative Strategy
For non cooperative strategy, each region maximises its own bene…t. Consequently, each
of the regions has its own value function. Let the function be denoted as Wi (E) : The
HJB equation of lobbyist is give by,
Wi (E) = max E
2
Li
E 2 + (1
i )(Li
+
1
ci L2i + Wi0 (E)(
2
i)
i pi
mE)
Again, we postulate a quadratic value function ( to ensure linear strategies)
fi 2
E
2
Wi = di + ki E +
Proposition 7. The coe¢ cients of the value function are determined by the following
equations:
fi =
ki =
1
2
2m +
i i
A2 fi + fi
di =
2fj
i
fi
i i
+ kj fi
m+
i
+ ki
+ki sj
i
+ si
j j
q
4
j j
j j
i i
+ 2m +
+ fi si
i i
+ fi sj
+
2fi
i i
fj
j j
i
ki
i i
si
i i
2
i
+ ki2
i i
2fj
j j
ci fi 2i
2
j j
ci fi si
+ ki kj j j + ki si i
1 2 2 1 2
ki ci si 2i
k ci i
ci s
2 i
2 i
2
i
+1
i
2
i
The negative value of fi is chosen to ensure stability. The steady state value of E is
given by
EsCN c =
1
+
2
m
A1 + A2 + 1 1 k1 +
f1 1 1 f2 2 2
2 2 k2
Proof. See appendix 1.
The lobbying e¤ort is given by Li =
i (si
+ ki + fi E): For stability, we need that
fi < 0:
Unfortunately, it is not possible to solve the equations explicitly. In the next section,
we impose some restrictions on the parameter space in order to obtain analytical solutions.
12
5
Symmetric Equilibria
5.1
Steady State Public Good Stocks and Lobbying
We restrict ourselves to the set of symmetric solution: both regions have similar e¢ ciency
and cost parameters.
Let
= x; ci = x, such that
i
i
=
We have,
A1 = 2
2
A2 = 2
x
2 2
x
Open Loop Cooperative Outcome:
LOc
1s =
m (m + ) + 4x
EsOc =
4x
2
2
m
(m + )
(1
x
+ (m + ) 2 (1
x ) + 2x2
m (m + ) + 4 x
4x2
x )+2
2
2
2
There are two sources of increase in relative e¢ ciency: either through increase in
(= x) or through decrease in c(= x):
Open Loop Noncoopearive Outcome:
LOnc
1s =
m (m + ) + 2 x
EsOnc =
2x
2
2
m
1
(m + ) (1
x
+ (m + ) 2 (1
x )+1
x ) + 2x2
m (m + ) + 2 x
2x2
2
2
2
Oc ;E Onc > E Onc ; LOnc > LOnc ; LOc > LOc as long as
It is clear that15 EsOc > EBs
s
1s
1s
Bs
1Bs
1Bs
= x < 1.
Comment 7: The subscript B refers to the benevolent lobbyist case. Thus, at least
for the symmetric outcome, the lobbying and public good stock will be greater than the
world of benevolent lobbyist.16
Closed Loop Noncoperative Outcome:
Needless to say, we shall have fi = f; ki = k and di = d
Proposition 8.
The symmetric solution implies the following value function
W (E) = d + kE +
15
16
See appendix 2.
See proposition 1, comment 2.
13
f 2
E
2
where,
1
f=
6x
2
k=
(2m + )
q
2f
2
2f x
m+
2
12x
+ (2m + )2
+ 2f x2 2 + 1
3f x 2
Proof. See appendix 1.
@f
@f
It can be seen that
< 0 and
> 0; where A = 2m +
@
@A
The symmetric SS stock of public good is given by
EsCN c =
2
2x
2
+ 2kx 2 + 2x2
m 2f x 2
As before, if we posit the lobbying e¤ort as LCnc = a2
b2 =
1
6x
f=
q
(2m + )
and
a2 =
12x
2
2
(5.1.1)
b2 E, then
+ (2m + )2
(s + k)
For the ‘closed loop form’of open loop lobbying protocols,
q
p
1
i
2
b1 =
4 A1 + (2m + ) =
8x 2 + (2m + )2
2m +
2m +
2A1
4x
q
p
1
i
b0 =
2m +
2m +
8 A1 + (2m + )2 =
16x 2 + (2m + )2
2A1
4x
The following lemma compares the slopes of closed loop forms non cooperative lobbying.
Lemma 3 For symmetric case, b1
b2 : That is, the marginal reduction in lobbying
for closed loop strategy is lower than open loop counterpart.
Proof. As x ! 0, b2 !
2m +
Now,
b1
b2 ,
or,
b2 + b1
i.e.
1
6x
and b1 !
2m +
( applying L’Hôpital’s rule) so b1 = b2
0
(2m + )
q
12x
1
4x
2
+ (2m + )2
q
2m +
8x
14
2
+ (2m + )2
0
when x 6= 0, this implies
p
)2 A
12x 2 + A2
3 A
p
8x
2
+ A2
If x = " > 0 and very small, then the di¤erence
0 j where, A = 2m +
0 and both expressions becomes
equal.
Di¤erentiating the expression w.r.t. x;
8 2
12 2
+3 p
2 p
2 12x 2 + A2
2 8x 2 + A2
2
p
p
12 2
12
+p
> 0; since 8x 2 + A2 < 12x 2 + A2
= p
12x 2 + A2
8x 2 + A2
Since this is an increasing function, the di¤erence is strictly positive as x increases .
In other words, b1 > b2 when x > 0:
Thus, given an decrease in Lj (and therefore,E), the increase in lobbying e¤ort by
region i is least in non cooperative closed loop case: the degree of strategic substitutability
further goes down.
15
6
Numerical Analysis
We continue our analysis of symmetric equilibria. Unless otherwise mentioned, we willl
take
= 1;
= m = :5; and
= :01: Sensitivity analysis shows that our results are robust.
6.1
Steady State Lobbying and Public Good
In this section, our concern will be to look at the steady state values of stock and lobbying
levels under di¤erent lobbying protocols. Speci…cally, three possibilities arise.
6.1.1
Constant, both c and
increases proportionally
The basic message is the following: keeping
constant, when both alpha and cost in-
creases, lobbying e¤ort goes down and steady state stock increases. Usually, the cooperative strategy entails highest stock of public good as well as highest lobbying. This is
consistent with standard public good result: in a centralised environment, the e¤orts as
well as public good stock would be highest.
We have plotted x(= c) on the x-axis. It can be seen that EsOnc and EsCnc almost
overlap. In reality, the former is slightly greater throughout.
5
4.5
4
3.5
3
2.5
2
EOC
1.5
1
EONC
ECNC
x
Figure 3: SS Stock for
= :1
We redraw the same variables for a higher value of :
16
9.
7
9.
1
8.
5
7.
9
7.
3
6.
7
6.
1
5.
5
4.
9
4.
3
3.
7
3.
1
2.
5
1.
9
1.
3
0.
7
0.
1
0.5
0
12
10
8
EOC
6
EONC
4
ECNC
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Figure 4: SS stock for
Therefore, as
=1
increases, the steady state stock of public good will increase for a given
and c:
The following pair of diagrams depict SS lobbying for a given
and increasing x. As
and c increase, cost of lobbying
goes up and incentive for lobbying goes down without any e¤ ect on relative e¢ ciency.
This discourages lobbying.
12
10
8
LOC
LCNC
6
LONC
4
2
9.7
8.9
8.1
7.3
6.5
5.7
4.9
4.1
3.3
2.5
1.7
0.9
0.1
0
Figure 5: SS lobby with
= :1
It is somewhat hard to distinguish, but cooperative lobbying (LOC) is greater than
both closed loop noncooperative (LCN C) and open loop cooperative (LON C) lobbying
levels. LON C is slightly greater than LCN C: e¤ort in open loop is higher than the closed
loop case.
And
17
12
10
8
LOC
LONC
LCNC
6
4
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 6: SS lobby with
0.9
=1
Therefore, with increasing , lobbying will be higher for a given x:
Result 1: Keeping
constant, as
and c increase, SS lobbying falls and SS output
goes up.
6.1.2
Constant c, increase in e¢ ciency through increase in
Now we want to analyse the speci…c case when the increase in
in : As
comes through increase
goes up, the relative e¢ ciency of the province increases, but also the share to
the lobbyist goes down. So it is not clear whether the lobbyist will have enough incentive
to lobby more.
It can be seen that co-operative lobbbying increases, but noncooperative lobbying
actually decreases as
: Here, we have taken c = x = 1
in the horizontal axis.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
LOC
LONC
0.96
0.89
0.82
0.75
0.68
0.61
0.54
0.4
0.47
0.33
0.26
0.19
0.12
LCNC
0.05
and plot
increases through increase in
Beta
Figure 7: SS Lobby with
18
= :5
However, when
decreases, the trend is somewhat altered. With increased , lobbying
goes up for all lobbying protocols.
3
2.5
2
LOC
1.5
LONC
1
LCNC
0.5
0.95
0.89
0.83
0.77
0.71
0.65
0.59
0.53
0.47
0.41
0.35
0.29
0.23
0.17
0.11
0.05
0
Beta
Figure 8: SS lobby with
= :1
The change in lobbying behaviour comes somewhere between
= :3 and
= :4. Thus,
with non cooperative lobbying, decrease in share of rent actually reduces lobbying e¤ort
is ‘high enough’.
In both cases though, SS stock of public good increases
12
10
EOC
6
EONC
4
ECNC
2
0
0.
4
0.
47
0.
54
0.
61
0.
68
0.
75
0.
82
0.
89
0.
96
Stock
8
0.
05
0.
12
0.
19
0.
26
0.
33
if
Beta
Figure 9: SS stock with
19
= :5
18
16
14
SS Stock
12
EOC
10
8
EONC
6
ECNC
4
2
0.95
0.89
0.83
0.77
0.71
0.65
0.59
0.53
0.47
0.41
0.35
0.29
0.23
0.17
0.11
0.05
0
Beta
Figure 10: SS Stock with
It can be seen that lower
Result 2: As
= :1
fosters higher lobbying as well as higher public good stock.
increases through increasing
, two opposite e¤ ects act on lobbying.
E¢ ciency of lobbying goes up, but the rent of the lobbyist goes down. For cooperative
lobbying, the …rst e¤ ect dominates the latter. For non cooperative lobbying, the second
e¤ ect is dominant when agents are relatively impatient. On the other hand, public good
stock increases, even for non cooperative lobbying. Thus increase in
outweighs decrease
in L.
6.1.3
Constant
, increase in e¢ ciency through decrease in c
Both lobbying and steady state stock increases with decreasing c, as revealed by the
following diagrams. For this example,
= :8;
= m = :5;
= 1: We have plotted
the horizontal axis.17
14
12
10
LOC
8
LONC
6
LCNC
4
2
0.
16
2
0. 56
17
4
0. 12
18
7
0. 45
20
2
0. 98
22
1
0. 33
24
3
0. 32
27
0
0. 17
30
3
0. 67
34
6
0. 66
40
3
0. 83
48
3
0. 57
60
2
0. 56
79
9
1. 22
18
64
2. 3
30
15
0
Beta
Figure 11: SS lobby with decreasing c
17
Compare with Fig. A2.2
20
along
For output
45
40
35
30
EOC
25
EONC
20
ECNC
15
10
5
0.
13
3
0.
14
3
0.
15
4
0.
16
7
0.
18
2
0.
20
0
0.
22
2
0.
25
0
0.
28
6
0.
33
3
0.
40
0
0.
50
0
0.
66
7
1.
00
0
2.
00
0
0
Beta
Figure 12: SS stock as c decreases
When c decreases, the e¢ ciency of lobbying increases without any change in lobbying
incentive. This will increase lobbying, ceteris paribus, and also increase steady state stock.
Result 3: As c goes down,
increases without any e¤ ect on lobbying incentive. This
leads to higher lobbying and higher public good stock.
21
6.2
Welfare analysis: The Need for a Lobbyist
In this section, we perform the welfare18 analysis in a symmetric world. We look at a
speci…c example: when increase in e¢ ciency comes through increase in . Let us recall that
increase in
will increase output, but will decrease lobbying for non co-operative outcomes.
Therefore, the e¤ect on welfare of the consumer (utility net of cost) becomes ambiguous
in nature. For the following set of diagrams, we have taken E0 = 0;
and c = 2; unless otherwise mentioned.
6.2.1
= m = :5;
= 1;
is plotted in the horizontal axis.
Components of Welfare with Rent dissipation: Di¤erent Lobbying Protocols
Total welfare( that of lobbyist and population) is depicted in the following …gure:
7
6
5
WOC
4
WON
3
WCN
2
1
0.
05
0.
08
0.
11
0.
14
0.
17
0.
2
0.
23
0.
26
0.
29
0.
32
0.
35
0.
38
0.
41
0.
44
0.
47
0.
5
0
B et a
Figure 13: Total welfare as
increases
We can see that the cooperative welfare (W OC) is higher than non cooperative open
loop (W ON ) or closed loop(W CN ) welfare and all of them are increasing in :The total
welfare for non cooperative outcomes almost coincide.
As mentioned before, total welfare can be decomposed into two parts: lobbyists’welfare
(or rent) and consumers’welfare (utility net of cost). In the following …gures, we depict
the decompositions.
18
The relevant formulae have been documented in appendix 3.
22
1.4
1.2
1
WROC
0.8
WRON
0.6
WRCN
0.4
0.2
0.
05
0.
09
0.
13
0.
17
0.
21
0.
25
0.
29
0.
33
0.
37
0.
41
0.
45
0.
49
0
Beta
Figure 14: Lobbyists lifetime rent as
As
goes up, and
increases
goes to 1, the lifetime welfare of the lobbyists go to zero
(mimicing the F N world). The lobbyist would get higher return (W ROC) from cooperative lobbying.than the noncooperative open loop (W RON ) or closed loop(W CON )
protocols.
Again, welfare from non cooperative strategies almost coincide, although
W RON > W RCN .
Now, regarding the consumer’s welfare,19
7
6
5
4
WCOC
3
WCON
2
WCCN
1
0.
05
0.
08
0.
11
0.
14
0.
17
0.
2
0.
23
0.
26
0.
29
0.
32
0.
35
0.
38
0.
41
0.
44
0.
47
0.
5
0
-1
Beta
Figure 15: Consumers’welfare as
increases
It can be seen that for a range of , welfare accruing to the consumer is higher for non
cooperative lobbying ( and closed loop strategy,W CCN gives higher welfare than open
loop, W CON ) than cooperative lobbying (W COC). Intitial welfare is negative, because
utility from the public good is less than the cost of lobbying. For cooperative case, lobbying
19
We are depicting UC , in terms of appendix 3.
23
is su¢ ciently high for relatively lower values of
compared to other protocols. This makes
it undesirable from consumers’point of view.The range of
where the consumer prefers
non cooperative, closed loop lobbying (here, it is approximately 0 :35) is higher for higher
values of
and m. In other words, it goes down when people are more patient and/or the
depreciation rate is low.
6.2.2
Need for a Not-so-benevolent (NB) Lobbyist
Closed loop non-cooperative lobbying: In a benevolent world there is no rent for
lobbyist. In that case, the consumer gets total welfare: this is maximised with cooperative
lobbying protocol. However, for a certain range of , non cooperative closed loop protocol
with NB lobbyist will provide highest bene…t to consumer. This can be demonstrated in
the following diagram:
WCCN-WCCB
0.1
0.
05
0.
09
0.
13
0.
17
0.
21
0.
25
0.
29
0.
33
0.
37
0.
41
0.
45
0.
49
0
-0.1
-0.2
-0.3
-0.4
-0.5
Beta
Figure 16: Di¤erence in welfare levels
Here, W CCB measures the cooperative welfare in a benevolent world. We have the
result that at least, for certain values of
;noncooperative, closed loop welfare with NB
lobbyist will be higher than all lobbying protocols within benevolent world. This justi…es
the use of a NB lobbyist. We have used W CCN because it is hard to sustain cooperative
outcome as well as any open loop strategy (which requires a power to commit ).20 However,
this is not true if
= m = :1: Thus, as people gets more impatient or the rate of dissipation
goes up, there is a rational for having a politicians who are not entirely benevolent and
use closed loop, non cooperative strategies.
Open loop protocols: Let us assume that agents can commit to a time path of e¤orts.
We can compare non cooperative protocols. For
20
See fn. 3
24
= m = :5;
WCON-WCB
0.2
0.1
0.
05
0.
09
0.
13
0.
17
0.
21
0.
25
0.
29
0.
33
0.
37
0.
41
0.
45
0.
49
0
WCON-WCB
-0.1
-0.2
-0.3
-0.4
Beta
Figure 17: Di¤erence in Welfare:
= m = :5
Thus, if we have open loop lobbying, for a wide range of
(approximately, 0:2
0:5);
NB non-cooperative lobbying (fosters W CON )will dominate corresponding strategy in
benevolent world (generates W CB). Note that, as
On the other hand, as
goes to 1; the di¤erence dies down.
and m goes down, then NB non cooperative strategy com-
pletely dominates the other counterpart.
WCON-WCB
5
4
3
2
1
Beta
Figure 18: Di¤erence in Welfare:
Here, as before, as
0.
5
0.
05
0.
08
0.
11
0.
14
0.
17
0.
2
0.
23
0.
26
0.
29
0.
32
0.
35
0.
38
0.
41
0.
44
0.
47
0
-1
= m = :1
goes to 1, the di¤erence ceases to be. For all other values of , a
politician who is only partly benevolent, will bene…t the consumers.
7
Extension: Asymmetric Provinces
In this section, we investigate asymmetric provinces: i.e. assume that the provinces di¤er
by
i.
A potential implication for such asymmetry is welfare di¤ erential (or inequality )
among the provinces, an issue which can not be dealt in analysis of symmetric equilibrium.
Let us recall that, the regional welfare to consumers is de…ned by
25
Wic =
Z1
e
t
1
ci L2i dt
2
U (E)
0
Since both regions reap identical bene…t from the (pure) public good, it is clear that the
region where the lobbying cost is higher will experience lower welfare. To investigate the
e¤ect of asymmetry on lobbying cost, we consider the case of open loop non cooperative
lobbying. The oher lobbying protocols will have similar results.
The lobbying e¤ort of each region is given by
Li (t) = Lsi + b1i (E(t)
Es )
Here,
i
b1i =
2m +
2A1
p
4 A1 + (2m + )2 =
ik
<0
For non cooperaive, open loop case, the steady state lobbying e¤orts are given by
following expressions
Ls1 =
m(1 + s1 (m + ))
(A2 d1
m( + m) + A1
2 2)
1
Ls2 =
m(1 + s2 (m + ))
(A2 d2
m( + m) + A1
1 1)
2
Then, the di¤erential in lobbying is given by
L1 (t)
L2 (t) = (Ls1 Ls2 ) + (
| {z } |
2 )k(E(t)
1
A
Let E0 = 0: Then E(t)
Es ) k(E(t)
There can be two sources of changing
Es )
i:
Es )
}
{z
B
0
change in
i
and change in ci . We take up
comes from variation in
i
. As
each of them, one at a time.
7.1
Variation Through :
Assume that all variation in
i
goes up, each province
has higher productivity, but the lobbyist has less motivation for lobby. The result can be
summarised in the following proposition:
Proposition 9 : If
i
>
j
(i.e. the region i is relatively more e¢ cient in production
of the public good) then Li (t) > Lj (t) if
i
26
is not too high. In other words, the more
e¢ cient region lobbies more vigorously, contributes more to public good (
given the central rule pi = Li +
i ),
i pi
>
j pj ,
cL2i
=
2
bears comparatively higher lobbying cost
and, as a result, enjoys less lifetime welfare than the other region.
Proof. See appendix 1
Variation through c
7.2
Now assume that the variations come from c1 6= c2 , but
1
=
2
= .
We have a useful proposition:
Proposition 10
The province which is more cost e¢ cient pays higher lobbying
cost and enjoys lower welfare, when all di¤ erences in relative e¢ ciency
i
=
ci
is
explained through variations in cost.
Proof. See appendix 1.
Discussion:
If one province bears higher lobbying cost than the other , we can say that there is
a transfer of welfare, through the pure public good channel, from that province to the
other. Normally, the concept of equalisation within a province implies that the direction
of transfer of welfare occurs from the e¢ cient province to the ine¢ cient province. Here,
if relative e¢ ciency is based on cost di¤erences, this is the case. The cost e¢ cient region
lobbies vigorously, adds more stock to the public good, but achieves smaller lifetime welfare
than the ine¢ cient province. On the other hand, if relative e¢ ciency is based on productive
e¢ ciency ( i ), then results are ambiguous. If the productive e¢ ciency parameter in region
i ( i ) is high, then the steady state lobbying by region i goes down ( due to lower incentive
for lobbying) compared to the other region. This implies, at least in the vicinity of steady
state, welfare is transfered from the ine¢ cient province to the e¢ cient province.
8
Conclusion
In this paper, we have explored the dynamic provision of a pure public good under various
protocols of lobbying and a non benevolent lobbyist. Our basic results are summarised
below.
First, individual lobbying e¤orts are strategic substitutes. The degree of strategic
substitutability is highest under the cooperative case. It falls as we move to the open loop
noncooperative case. Finally, it is lowest for the closed loop non cooperative case.
Second, if the lobbyists are non-benevolent, the levels of lobbying as well as the stock
27
of public good will be higher in a symmetric steady state as compared to a world with
benevolent lobbyists who do not capture any rent. This holds true for both co-operative
and non cooperative lobbying protocols.
Third, the e¤ect of increase in relative e¢ ciency on lobbying is ambiguous. When cost
decreases, lobbying goes down. On the other hand, if production e¢ ciency increases, lobbying under non cooperative protocols goes down if the depreciation rate and/or discount
factor is too high. With similar parameter values, this e¤ect is absent under benevolent
lobbying.
We …nd two results on consumer welfare. These lead to the policy prescription that,
when the rate of depreciation and/or discount factor is high, it is better to follow the noncooperative, closed loop protocol (that is, the protocol having the most desirable economic
properties) and a non-benevolent lobbyist. This suggestion is based on two observations:
First, in a non benevolent world, non cooperative, closed loop lobbying yields higher
welfare than all other protocols for a speci…c range of parameter values.
Second, non cooperative, closed loop lobbying with a non benevolent lobbyist may foster
higher welfare compared to all other protocols in a benevolent world.
The latter result highlights the importance of having a non benevolent lobbyist.21 This
is further bolstered by the following result: welfare under the open loop, non cooperative
protocol with a non benevolent lobbyist may be signi…cantly higher than open loop, non
cooperative protocol in a benevolent world.
These results are further reinforced if we allow for asymetric provinces: higher e¢ ciency
may lead to perverse e¤ects on equalisation prospects.
Brie‡y, we mention some further issues which can be analysed using this framework.
First, we have assumed that the central government is not a strategic agent. Relaxation
of this assumption implies a hierarchical structure with leader-follower aspects, leading to
a Stackelberg equilibrium. Second, introduction of re-election probability of the lobbyists
that grab too much of resources will endogenise the choice of provincial e¢ ciency. Thus
there remain legitimate issues for future research.
21
New York Times (02/07/2006) reports that small towns in US are actually hiring professional lobbyists
for a fee to get federal funds for projects such as construction, reparation of bridges and sewer plants, which
have a public good aspect. Our model is applicable to these situations.
28
Appendix 1
De…nitions
Problem (A1.P)
Let i 2 f1; 2; :::N g denote an agent. Let ui (c(t); x(t); t) denote the instantaneous
utility function of the agent i at time t; where c =< ci >ni=1 is the set of control variables
and x =< ci >ni=1 the set of state variables. The objective of each agent is to choose ci in
order to maximise the lifetime utility
max
ci
ZT
ui (c(t); x(t); t)e
it
dt
0
such that the transition equation
x(t)
_
= f (x(t); c(t); t)
and some boundary conditions are satis…ed.
i
is the rate of discount.
De…nition D1: Markovian( or closed loop) Nash Equilibrium:
The n-tuple ( i )ni=1 of functions is called a Markovian Nash equilibrium if , for all
i 2 f1; 2; :::ng an optimal control path of the problem (P) exists and given by
ci (t) =
i (x(t); t).
Such strategies
i
is called a Markovian or closed loop strategy.
De…nition D2: Open loop Nash Equilibrium:
The n-tuple ( i )ni=1 of functions is called a open loop Nash equilibrium if , for all
i 2 f1; 2; :::ng an optimal control path of the problem (P) exists and given by
ci (t) =
i (t).
Such strategies are called open loop strategies.
Theorem A1.1
For in…nite horizon autonomous problems, the Markovian strategy collapses to stationary Markovian strategy, i.e.
i (x(t); t)
=
i (x(t))
Proof. See Dockener et al (2000)
We have used these de…nitions and properties to formulate our problem.
Proof of Certain Propositions
Proof of Proposition 1:
Steady State Values of Open loop Cooperative Strategy
The necessary conditions are
29
(1
i)
ci Li +
_ = ( + m)
E_ =
1 L1
+
2 (1
2 L2
Thus
Li
=
i
E)
2
1
+ (
Dividing the …rst equation by ci and let
Li = (si + )
=0
i
i
=
2
2)
+
ci
mE
, we get
i
si
i
2
And L2 =
Thus,
E_ = 1 L1 +
L1 +
2 s2
2 s1
1
2
2
L1
2 s1
+
2 s2
+ A2
mE
1
Using (A) and (B)
L_ 1 = 1 _ = 1 (( + m)
=
( + m)
1
2 (1
L1
E))
2 (1
s1
E)
1
= ( + m) (L1
s1
1)
2
1 (1
E)
_ L_ 1 = (0; 0)
From these the SS values can be calculated, when E;
2A1 + (s1 1 1 + s2 2 2 + A2 ) (m + )
EsOc =
m( + m) + 2 A1
m(2
+
s
2 (A2 d1 j j )
1 + ms1 )
LOc
1s = 1
m( + m) + 2 A1
2
1
2
2
2+ 2
and A2 =
1
2
c1
c2
As
0 for stability check, the1coe¢ cient matrix is
+m
2 1
B
C
@ 1 1+ 2 2
A
m
Here, A1 =
+
1
The trace is
> 0 and the determinant is
m
2
1 1
2
2 2
the system is stable in a saddlepoint sense.
p
1
The Eigenvalues are 21
8 A1 + (2m + )2
2
And,
2A1 + (s1
1
+ s2 2 2 + A2 ) (m + )
<
m( + m) + 2 A1
(2A1 + (s1 1 1 + s2 2 2 + A2 ) (m + )) m( + m)
1 1
,
30
2 A1 < 0
m2 < 0. Thus
(m + ) (A2 + s1
1 1
+ s2
2 2)
< m(m + )
,
m
m
>
)
A2 + s1 1 1 + s2 2 2
A2 + 1 + 2 A1
If this condition is maintained, we can prevent ‘over-accumulation’of public good.
<
Proof of Proposition 2
Closed Loop Representation of Open Loop Cooperative Strategy:
The homogeneous part of the equations can be written as
1 1+ 2 2
E_ =
L1 mE
1
and
L_ 1 = ( + m)L1 + 2
1
E
After substitution, we have the following equation in E
1 1+
•
E
E_ (m( + m) + 2 1 C) E = 0, where C =
2 2
1
Solving this, we get
E(t) = C1 exp(r1 t) + C2 exp(r0 t)
Here,
ri =
1
2
1
2
p
8 A1 + (2m + )2 (i = 1; 0)
The complete solution for E is thus
E = C1 exp(r1 t) + C0 exp(r0 t) + EsOc
When t = 0; C1 + C2 = E0
EsOc
Along the steady path, when t ! 1, E = EsOc ) C1 = 0
Thus C2 = E0
) E = Es + (E0
EsOc
EsOc ) exp(r0 t)
Substituting back in the homogeneous equation of L1 ;
L_ 1 = ( + m)L1 + 2 1 f(E0 Es ) exp(r0 t)g
Solving this, we get the homogeneous solution.
2 1
L1 = C3 et(m+ )
E Oc E0 exp(r0 t)
r0
m s
and the complete solution
2 1
L1 = C3 et(m+ )
E Oc E0 exp(r0 t) + L1s
r0
m s
When t = 0;
2 1
C3
E Oc E0 = L1 (0) LOc
1s
r0
m s
When t ! 1; then, along the stable path, L1 = LOc
1s : Thus C3 = 0
The stable path for the i th province is thus characterised by
2 i
Oc
LOc
E Oc E0 exp(r0 t)
i = Lis
r0
m s
31
= LOc
is +
2 i
+m
r0
(EsOc
E)
2
pi
(EsOc E)
+
+m
8 A1 + (2m + )2
2 i
p
(EsOc E)
= LOc
is +
1
1
2
m + 2 + 2 8 A1 + (2m + )
4
p i
= LOc
(EsOc E)
is +
2m + + 8 A1 + (2m + )2
p
i
2m +
8 A1 + (2m + )2 (E EsOc )
= LOc
is +
2A1
p
@LOc
i
Thus
= i 2m +
8 A1 + (2m + )2 < 0
@E
2A1
= LOc
is +
1
2
1
2
The intercept is
LOc
is
i
p
2m +
2A1
Proof of Proposition 3:
8 A1 + (2m + )2 EsOc > 0
Steady State Values of Open Loop, Non Cooperative Strategy
The necessary conditions are
ci Li +
i
_ i = ( + m)
E_ =
Using the fact that Li =
i (si
+
i ),
(1
i
i Li
=0
i i
E)
mE + A2
we can write the system of di¤erential equation
as
L_ i = ( + m)Li +
and
E_ =
i Li
i
E
i (1
+ si + si m)
mE + A2
The steady state of the system reveals that
At SS, the system of equations are
( + m)(L1
s1
1)
+
1
E
1
= 0
( + m)(L2
s2
2)
+
2
E
2
= 0
1 L1
+
2 L2
32
mE + A2 = 0
Solving this,one gets
LOnc
=
is
i
m(1 + si (m + ))
A2
m( + m) + A1
di
j j
and
+ s2 2 2 + A2 ) (m + )
m( + m) + A1
where the superscript nc implies no cooperation,O refers to closed loop and the subEsOnc =
A1 + (s1
1 1
script s stands for steady state.
The relevant coe¢ cient
0
1 matrix is
+m
0
1
B
C
B 0
C
+
m
2 A, trace: m + 2 > 0
@
m
1
2
Determinant: m 1 1 m 2 2
1
Trace: m + 2 > 0
1
2
Eigenvalues : m + ; 12
p
1
2 2
m3
m
2
2m2 < 0
4 A1 + (2m + )2
Thus the system is stable, in a saddlepoint sense.
and, to prevent over accumulation,
A1 + (s1 1 1 + s2 2 2 + A2 ) (m + )
1
<
m( + m) + A1
,
(m + ) (A2 + s1
1 1
+ s2
2 2)
< m(m + ):
,
m
A2 + s1 1 1 + s2 2
Proof of Proposition 4:
<
2
Closed Loop Form of Open Loop Non-Cooperative Strategy
The stable path of E is
E(t) = EsOnc + (E0
EsOnc ) exp(r3 t)
Where, r3 is the negative eigenvalue of the coe¢ cient matrix and is given by
p
1
4 A1 + (2m + )2
r3 = 21
2
The non cooperative, ‘closed loop form’of open loop lobbying would be
LOnc
= LOnc
i
is
=
LOnc
is
= LOnc
is +
E
i
i
EsOnc
E
EsOnc
2m +
The slope is
@LOnc
i
= i 2m +
@E
2A1
+m
r3
p i
m + 12 + 21 4 A1 + (2m + )2
p
4 A1 + (2m + )2
E EsOnc
2A1
p
4 A1 + (2m + )2 < 0
33
and the intercept
LOnc
jE=0 = LOnc
i
is
p
4 A1 + (2m + )2
2m +
i
EsOnc
2A1
Proof of Proposition 5:
Comparison of Steady States: Open Loop Cooperative and Non-Cooperative
Strategies:
2A1 + (s1
EsOc =
,
(2A1 + (s1
1 1
+ s2 2 2 + A2 ) (m + )
A1 + (s1 1 1 + s2 2 2 + A2 ) (m + )
> EsOnc =
m( + m) + 2 A1
m( + m) + A1
1 1
+ s2
2 2
+ A2 ) (m + )) (m( + m) + A1 )
(A1 + (s1
,
m A1 m A 1 A2
1 1
+ s2
A1 A2 m A1 s1
1 1
A1 m (m + ) > A1 (m + ) (A2 + s1
1 1
2 2
+ A2 ) (m + )) (m( + m) + 2 A1 ) > 0
m A1 s2
2 2
A1 s1
1 1
A1 s2
2 2+
m2 A1 > 0
,
,
+ s2
m
>
A2 + s1 1 1 + s2 2 2
Also, for region 1,
m(2 + s1 + ms1 ) 2 (A2 d1
LOc
1s =
m( + m) + 2 A1
,
(m(2 + s1 + ms1 )
2 (A2
d1
2 2)
2 2 )) (m(
> LOnc
1s =
,
m3 + m2 > m (m + ) (A2 + A1 s1
m>
(A2 + A1 s1
d1
m(1 + s1 + ms1 )
(A2 d1
m( + m) + A1
+ m) + A1 )
(m( + m) + 2 A1 ) (m(1 + s1 + ms1 )
,
2 2)
d1
2 2)
1 1
+ s2
(A2
d1
2 2 ))
>0
2 2)
or
m
>
A2 + A1 s1 d1 2 2
The denominator is equal to A2 + s1
2 2(
since d1 = s1
s2 ). This is the
same condition of preventing over accumulation.
Proof of Proposition 7:
Value Function and Steady State Output for Closed Loop Non Cooperative
Strategy:
The HJB equation is
34
2 2)
Wi (E) = max E
2
Li
E 2 + (1
i )(Li
+
1
ci L2i + Wi0 (E)(
2
i)
i pi
The FOC yields
Li =
i (si
+ Wi0 (E))
fi 2
E
2
Let Wi (E) = di + ki E +
Then Wi0 = ki + fi E
HJB equation is
fi 2
E =
di + ki E +
2
i + A2 ki + ki i + si
+ki si
+ ki sj
i i
+E(A2 fi
2
i
j j
mki + fi
+kj fi
j j
ki
i
fi
i
+ fi si
i i
2
+E fi fj j j mfi 12
Equate coe¢ cients of E 2
fi
= f1 f2 j j mfi
2
Equate coe¢ cients of E
A2 fi
mei + fi
+kj fi
j j
fi
i
+ fi si
+
si i i + ki kj
i i
ki2 i i ki ci si 2i
i i
+ 2ki fi
+ fi sj
+
j j
fi2 i i
+ 12 fi2
i i
i i
+ 2ei fi
i i
+ fi sj
+ ki fj
ki ci fi 2i
1
2 2
2 ci fi i
1
2
i i
i i
+ ki fj
1
2 2
2 ci si i +
j j
2
i
ci fi si
+ 1)
j j
ki ci fi 2i
j j
j j
2
1 2
2 ki ci i
ci fi si
2
i
+ 1 = ki
Equate constants:
di =
i
+ A2 ki + ki
+ki si
i i
i
+ ki sj
+ si
ki
i
2
i
j j
+
si i i + ki kj
i i
2
ki i i ki ci si 2i
j j
2
1 2
2 ki ci i
Thus the equations under considerations are
q
1
2
fi =
2fj j j
4 i i + 2m +
2m +
2fj j j
2 i i
Similarly,
A2 fi + fi i fi i i + kj fi j j + fi si i i + fi sj j j ci fi si
ki =
m+
2fi i i fj j j + ci fi 2i
1
di = ( i + ki i + si i ki i i si i i + ki kj j j + ki si i i
+ki sj
2
i
j j
The lobbying is Li =
Now,
E_ = 1 p1 +
=
1 L1
=
1 1 (s1
=
1 1 s1
+
+
2 p2
2 L2
+ ki2
i (1
ki ci si
i i
2
i
2
1 2
2 ki ci i
1
2 2
2 ci si i
2
i
+1
1
2 2
2 ci si i )
+ ki + fi E)
mE
+ A2
mE
+ k1 + f1 E) +
2 2 s2
+ A2 +
This will be stable if f1
1 1
2 2 (s2
+ k2 + f2 E) + A2
1 1 k1
+
+ f2
2 2
2 2 k2
+ E (f1
m<0
35
mE
1 1
+ f2
2 2
m)
mE)
For stability, we assume the negative values of fi
and the SS value is given by
1 1 s1
EsCN c =
1 1 s1
+
2 2 s2
=
1
+
+
m
2 2 s2
f1
+
1 1 k1
f2
1 1
+
2 2 k2
2 2
A1
2
Proof of Proposition 8:
Symmetric Solution for Closed Loop Non Cooperative Value Function
for symmetric solution, f1 = f2 = f (say). Thus,
p
f = x1 2 m + 12
x 2 f 12 4m + 4m2 + 2 + 4x
2
8mx
2
f
4x
2
f + 4x2
4 2
f
Solving this, we get
q
1
f=
(2m
+
)
(2m + )2 + 12x 2
6x 2
One of the roots of f is negative, the other one is positive. We choose the negative
root, i.e.
q
1
(2m + )
6x 2
For symmetric case,
f=
12x
2
+ (2m + )2
ki = k Thus
k=
m+
1
2f x
2
f
2
fx
+ fx
2
k + f (1
Solving this,we get
1
2f
2f x 2 + 2f x2
k=
m+
3f x 2
Similarly, the value of d is obtained.
2
x ) + 2f x2
2
+1
+1
Proof of Proposition 9:
E¤ect of Increasing
i
on Provincial Lobbying E¤ort:
The di¤erential in lobbying is given by
L1 (t)
L2 (t) = (Ls1 Ls2 ) + (
| {z } |
2 )k(E(t)
1
{z
A
Let E0 = 0: Then E(t)
Let
2,
Es )
while c1 = c2 = c: This implies
0
0: Thus, for non coopc
erative solution, region 1 will lobby more, and incur higher lobbyisng costs = L1 (t)2
2
than region 2 if A > 0.
1
>
Es ) k(E(t)
B
Using the de…nitions, we have
m(1 + s1 (m + ))
(A2 d1
Ls1 = 1
m( + m) + A1
2 2)
36
1
>
2
and B
Es )
}
+ cm2 cm2 1 + 22 c 31 c 1 22
c2 m + c 21 + c 22 + c2 m2
m(1 + s2 (m + ))
(A2 d2 1 1 )
Ls2 = 2
m( + m) + A1
2
cm + cm 2 cm 2
cm2 2 + 21 c 32 c 21 2
1 2 + cm
=
c2 m + c 21 + c 22 + c2 m2
Note that, the denominators are equal. The di¤erence, if any, will come from the
=
cm + cm
cm
1
1
1 2
numerators. Thus,
Ls1
Ls2 > 0
,
cm + cm
cm
1
cm + cm
,
(
1) ( 1
2
1 2
cm
2
+
1
2)
2
+c (
+ cm2
+ cm2
1 2
2
1
1)
2
cm2
cm2
2
+ mc (
2
2
+
2
2
+
1
3
1
c
+
2
1
2
1 ) (m
c
3
2
c
+
2
1 2
c 21 2
>0
1) > 0
Now,
1
(
>
1
Let
2
+
M
!
2)
+c
2
1
2
1
+
<0,
2
2
+ mc (m +
1) < 0
be the mean of the two e¢ ciency parameters:
+
2
1
M
2
Then we can write
1
=
M
+ " where " > 0
and
2
1
2
1
+
2
2
=(
M
=
+
2
M
"
=2
M
+ ")2 + (
")2 = 2
M
2
M
+ 2"2
The condition becomes
2
M
+c 2
2
M
So, if " > 0 but not too great, then Ls1
+ 2"2 <
mc(1
m)
Ls2 > 0. But the sign reverses when " is greater
than a certain threshold. Note that, if either
or m are too high (such that the sum > 1),
then the sign will be reversed for all ".
Note that, if " is su¢ ciently high, then
about the path of L1 (t)
1
L2 (t):
37
>
2
! Ls1 < Ls2 , but nothing can be said
Proof of Proposition 10:
E¤ect of Higher Cost on Provincial Lobbying E¤ort:
We have,
m(1 + s1 (m + ))
(A2 d1 2 2 )
Ls1 = 1
m( + m) + A1
c2 m + m
m + m2 m2
2 3
=
=
m c1 c2 + 2 c1 + 2 c2 + m2 c1 c2
m(1 + s2 (m + ))
(A2 d2 1 1 )
Ls2 = 2
m( + m) + A1
c1 m + m
m + m2 m2
2 3
=
=
m c1 c2 + 2 c1 + 2 c2 + m2 c1 c2
It is clear that if c1 < c2 , then Ls1 > Ls2
Li (t) = Lsi +
i k(E(t)
c2
(c1 ; c2 )
c1
(c1 ; c2 )
Es )
Thus, cost to region i is
ci
T Ci (t) = (Li (t))2
2
ci 2 2
ci
k (E(t)
= (Lsi )2 +
2
2 i
ci
1 2 2
= (Lsi )2 +
k (E(t)
2
2 ci
Since
Ls1
c2
=
s
L2
c1
1
s 2
c2
c1 (c2 )2
2 c1 (L1 )
=
2 = c
2
1
s
c2 (c1 )
1
2 c2 (L2 )
Es )2 + ci Lsi i k(E(t)
Es )2 + Lsi k(E(t)
Es )
Es )
(putting in
If c2 > c1 , then 12 c1 (Ls1 )2 > 21 c2 (Ls2 )2
Maintaining the same assumption,
1 2 2
1 2 2
k (Es E(t))2 >
k (Es E(t))2
2 c1
2 c2
and
Ls1 k(E(t)
Es ) > Ls2 k(E(t)
Es ), since c2 > c1 ) Ls1 > Ls2
Thus, (c2 > c1 ) ) (T C1 (t) > T C2 (t))
38
i
=
ci
)
Appendix 2
Lobbying Behaviour and Public Good Stock in the ‘Benevolent Lobbyist’
Environment
The original FN paper does not consider lobbying. Their problem (using our notation),
is the following:
Z1
e
t
E2
2
E
1 2
ci x dt
2 i
0
s:t:E_ =
xi
mE
xi represents individual contributions (e¤orts) for provision of public good.
We have re-interpreted the model in terms of federal lobbying. We replace individuals
by regions, e¤orts by lobbying, and add the assumption of a federal government that cares
for lobbying as well as e¢ ciency in production. The problem of a lobbyist in modi…ed
model becomes,
Z1
e
t
E2
2
E
1
ci L2i dt
2
0
s:t:E_ =
i pi
mE
The cooperative lobbying scenario yields the following pair of stable steady state lobbying (for the …rst region) and public good stock:
Oc
EBs
=
(m + )A2 + 2A1
m( + m) + 2 A1
LOc
1Bs = 2
1
(A2.1)
m
A2
m( + m) + 2 A1
The subscript B indicates the relevant environment.
For the open loop non co-operative scenario,
Onc
EBs
=
LOnc
1Bs =
(m + )A2 + A1
m( + m) + A1
1
m
A2
m( + m) + A1
39
(A2.2)
It is clear that in both protocols, the ratio of steady state lobbying equals ‘relative
e¢ ciency’, i.e.
Lk1Bs
=
Lk2Bs
Here, k denotes di¤erent protocols. As
i
1
2
goes up, region i will lobby more . This par-
ticular result parallels Dixit’s(1987) assertion that a player who has a strategic incentive,
will overexert.
Given relevant condition to prevent over accumulation
<
m
, it can be shown
A2
Oc > E Onc and LOc > LOnc
that EBs
Bs
1Bs
1Bs
For a symmetric solution, with
i
= x; ci = x and
= :01;we can write
Cooperative SS:
2
+ 2x2 2 (m + )
Oc = (m + )A2 + 2A1 = 4x
EBs
m( + m) + 2 A1
m (m + ) + 0:04x 2
2m 0:04x2 2
m
A2
LOc
=
2
=
1
1B
m( + m) + 2 A1
m (m + ) + 0:04x 2
Non co-operative SS:
2
+ 2 2 x2 (m + )
Onc = mA2 + A2 + 1 1 + 2 2 = 2x
EBs
m + 1 1 + 2 2 + m2
m (m + ) + 0:02x 2
m 0:02x2 2
m
A2
=
LOnc
=
1
1Bs
m( + m) + A1
m (m + ) + 0:02x 2
The following …gures plot the steady lobbying when increases through two di¤erent
channels. We have kept the same parameter values as …gure 7 and …gure 11, respectively.
Here, we have analysed only the open loop cases ( since closed loop NC behaviour follows
the open loop counterpart). LON B : non cooperative lobbying, LOCB : cooperative
lobbying.
2
1.8
1.2
1
0.8
LONB
LOCB
0.6
0.4
0.
4
0.
47
0.
54
0.
61
0.
68
0.
75
0.
82
0.
89
0.
96
0.2
0
0.
05
0.
12
0.
19
0.
26
0.
33
Lobbying
1.6
1.4
Beta
Figure A2.1: Increase in lobbying as
40
increases
Thus, as beta increases through increase in
the …gure A.2.1,
= m = :5;
, both types of lobbying increases. For
= 1; and c = 1:
30
Lobbying
25
20
LONB
15
LOCB
10
5
4.71
1.14
0.65
0.45
0.35
0.28
0.24
0.21
0.18
0.16
0.15
0.13
0
Beta
Figure A2.2: Increase in lobbying as c decreases
Here, lobbying increases with decreasing c, same as the model with rent dissipation.
For the …gure A2.2 , we have
= m = :5;
= 1;
= :8. initial c = 6:
Results can be further generalised if we consider another version of (2.3a), i.e.
Z1
e
t
E
2
E2
1
ci L2i dt +
2
0
Here,
2
4
i (1
i)
+ (1
i)
Z1
0
e
t
3
Li dt5
0 is a weight between lobbyists’ rent and consumers’ welfare.Throughout
the main discussion, we have assumed
= 1, while
benevolent lobbyist.
41
= 0 correspondes to the world of a
Appendix 3
Welfare Analysis
The welfare of the lobbyist as well as the regions are maximised for co-operative strategy : this is trivially true. However, in real world, it is not possible to enforce cooperative
strategy. Let us recall that the welfare of a province could be decomposed as
Z1
e
t
2
1
ci L2i dt + 4
2
E2
2
E
0
i (1
i)
+ (1
i)
Z1
e
0
t
3
Li dt5
The …rst term gives the lifetime utility of the population, while the second term represents the rent accruing to the lobbyist. .
For all protocols, public good stock and lobbying evolve according to
E = Es + (E0
Es ) exp(rt)
and
L = Ls + b (E0
Es ) exp(rt)
where, r is the negative eigenvalue, E0 is the initial stock, Es is the …nal stock. The
subscripts of L are explained in the same way.
Let E0 Es =
Z1
Z1
t
e Edt = e
0
t (E
s
+ exp(rt)) dt =
Es
+
r
0
E 2 = Es2 + 2 exp(2rt) + 2Es exp(rt)
Z1
2
E2
2Es
e t E 2 dt = s +
+
2r
r
0
L2 = L2s + b2 2 exp(2rt) + 2Ls b exp(rt)
Z1
L2
b2 2
2Ls b
+
e t L2 dt = s +
2
2r
r
0
And,
Z1
b
Ls
e t Ldt =
+
r
0
Thus, the consumers’welfare is
2
Es
Es2
2Es
c L2s
b2 2
2Ls b
UC =
+
+
+
+
+
r
2
2r
r
2 2
2r
r
and the lobbyists’s welfare is
(1
) Ls
b
UR =
+
+
r
For all examples, we have taken E0 = 0, so that = Es
42
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