Does Inequality Lead to Greater Efficiency in the Use of Local Commons?
The Role of Strategic Investments in Capacity
Rimjhim M. Aggarwal* and Tulika A. Narayan**
Proposed running head (abbreviated title): Inequality and efficiency in commons
*Rimjhim M. Aggarwal, Dept. of Agricultural and Resource Economics, University of Maryland,
College Park, MD 20742. Email: aggarwal@arec.umd.edu .
**Tulika A. Narayan, Dept of Agricultural and Resource Economics, University of Maryland, College
Park, MD 20742. Email: tulika@arec.umd.edu.
1
Does Inequality Lead to Greater Efficiency in the Use of Local Commons?
The Role of Strategic Investments in Capacity
Abstract: This paper examines how inequality in access to credit affects the strategic behavior of
users of a common resource in their investment and extraction decisions. A dynamic two-stage game
is developed in which agents choose the level of sunk investment in capacity and the consequent
extraction path. The results show that agents invest in excess capacity, except when inequality is very
high. The relation between inequality and efficiency in resource extraction is found to be to be nonmonotonic, with the steady state resource stock being closest to the optimal level when either
inequality is very high or very low.
Keywords: Common Pool Resources, Inequality, Access to Credit, Strategic Investment,
Groundwater
2
I.
Introduction
In many common pool resource (CPR) contexts, a large investment in capacity is required prior to
extraction (such as in wells and pumping equipment for groundwater extraction and vessels and
tracking gear for fishing). Since the nature and magnitude of this investment often influences the
extraction options available, agents may strategically choose investment levels to affect the game in
extraction levels. While most of the literature on CPRs treats investments in capacity as exogenously
given,1 some recent papers have shown why it may be important to consider investments (jointly with
extraction) as strategic choice variables. For instance, Copeland (1990) examines the strategic choice
of investments that affects rival firm’s costs and benefits of extraction and identifies the conditions
under which the strategic effect may reinforce or diminish the free riding effect. Barham et..al. (1998)
examine the choice of investment as an entry deterrence strategy and use their model to explain the
structure and development of some resource extraction industries.
These papers on strategic investment assume that credit markets are competitive so that all the
users of the common resource face homogenous cost of credit for financing their investment and there
are no borrowing limits. This assumption may be particularly hard to justify in rural areas of less
developed countries (LDCs), where due to informational constraints, credit market imperfections are
pervasive. The extent of credit available through the formal credit market is generally rationed in most
LDCs and is generally closely determined by the amount of collateral (e.g. land or livestock) that can
be offered (Binswanger and Sillers, 1983). The residual demand for credit may be met, wholly or
partially, through the informal market where interest rates are much higher than in the formal market.2
This implies that although an entire community of users may have extraction rights on a resource, the
cost of credit (and hence the capacity to extract ), will differ across users depending on their ownership
of private (collateralizable) assets.
The purpose of the present paper is to examine how this heterogeneity (as measured by
difference in marginal cost of credit) affects the strategic behavior of agents in their investment and
extraction decisions, and to derive a relation between the extent of heterogeneity and steady state
3
resource stock and investment levels.3 In a recent paper, Baland and Platteau (1997) explored the
relation between heterogeneity in access to credit and efficiency in extraction from CPRs. They found
that the more unequal is the distribution of credit constraints, the more efficient is the appropriation
from CPRs in a non-cooperative setting.4
Baland and Plateau’s model is based on a static setting wherein strategic interaction in
extraction alone is considered. As argued before, in CPR contexts where a large prior investment in
capacity is required, agents may choose investments strategically and this may alter some of the results
derived from considering a static setting. In this paper, we develop a dynamic two-stage game in
which agents first choose the level of sunk investment in capacity and then the extraction path over the
infinite horizon. The extraction game and the investment game are linked in our model because
investment in sunk capacity functions as a commitment device to force exit of other agents from the
extraction game. This strategic role of investment has been examined extensively in the industrial
organization literature, however to the best of our knowledge, it has not been applied to study the
effect of heterogeneity in a dynamic CPR extraction game.5
Contrary to Baland and Plattaeau’s result, we find that once we incorporate the strategic effect
of investment in capacity, greater inequality does not necessarily lead to greater efficiency in use of
CPRs. The relation between inequality and efficiency in resource extraction is found to be nonmonotonic, with the steady state resource stock being closest to the socially optimal level when either
inequality is very high or very low.6 For moderate levels of inequality, we show that the resource
stock may in fact be lower than that under perfect equality. Further, we show that because of the
strategic role of investments in this setting, agents invest in excess capacity in general, except when
inequality is high.
Our model is motivated in large part by the case of groundwater extraction in India, although
as we explain later, it is applicable to a number of other CPR contexts. Property rights on groundwater
are generally poorly defined in most countries. In the specific case of India, landowners have the right
to drill wells on their own land and pump out as much water, as they desire. However, the fixed costs
4
of drilling a well and buying the pumping equipment are very high, particularly in semi-arid areas
where wells need to be very deep in order to intercept water-bearing fractures in the sub-strata. 7 The
depth of the well and the horsepower of the pumping equipment set de facto limits on how much water
each agent can withdraw in any given time period from the common aquifer. Thus, this is a case where
investments in capacity determine extraction choices in a fundamental way and agents are likely to
choose both investment and extraction levels strategically.
Although all landowners have rights over the groundwater underlying their plots, a majority of
small and marginal landowners in India have not invested in wells, while large farmers have invested
in multiple wells with very high pumping capacities (Shah, 1993; Aggarwal, 2000). Given the huge
subsidies on electricity supply, the marginal costs of pumping are very low and this has led to a rapid
decline in the water table in this region. In the competitive pumping race that has ensued, those who
have deeper wells have survived while those with shallower wells (generally the smaller landowners)
have been driven out over time (Bhatia, 1992). An important policy question being posed in this
context is regarding the effect that government’s policies on distribution of credit can have on
groundwater use, given the fact that direct regulation of groundwater is not administratively feasible in
the short run.
Similarly in the context of many fisheries, the choice of capacity (as measured by vessel size
and horsepower, net rigging and tracking gear) is critical in determining the type and size of catch.
Kurien (1992) in his case study of coastal fisheries in south India describes how with the expansion of
export markets in prawns in the mid 1960s, merchants from urban areas started to heavily invest in
vessels capable of deep-sea fishing. Traditional fishermen with relatively poor access to credit were
not able to take advantage of these opportunities and were slowly displaced as stocks depleted and
their traditional technologies became redundant. In contexts such as this, the dynamics of the resource
stock and the investments in capacity are likely to be strongly influenced by agents’ access to credit,
together with other bio-economic factors.
5
The rest of the paper is organized as follows. To fix ideas, we develop our model in the
context of groundwater. In Section II, we present the benchmark case of a single well owner within an
aquifer, who faces a competitive market for water. This case also defines the social optimum in our
setting. Then, in Section III, we extend this analysis to the case of two homogenous agents who
extract from the same groundwater aquifer. In Section IV, we introduce heterogeneity amongst these
agents in terms of the marginal cost of credit that they face. In Section V, we use the results from
Section IV to map a relation between inequality on the one hand and steady state stock level and
investment on the other hand. Finally, in Section VI, we conclude.
II. Sole Ownership
For completeness, we begin with the case of a single agent who has sole extraction rights to a
groundwater aquifer. Consider the following two-stage model. In the first stage, the agent chooses
how deep to drill the well and in the second stage chooses how much water to extract from it. The
depth of the well is an important determinant of the capacity of the well in the sense that whenever the
underlying water table falls below the base of the well, the well becomes dry. For example in figure 1,
water can be extracted from well 1 so long as the water table lies above line D, while the
corresponding level for well 2 is line E. We assume that for the simple aquifer we consider here, there
is a one to one relation between changes in the level of the water table over time and changes in the
underlying water stock.8 Thus given the depth of a well, one can define a lower bound on the water
stock given as X , such that whenever the water stock in the aquifer falls below X , the well becomes
dry. In other words, if w(t) denotes the water pumped at time t and X(t) is the level of water stock at
time t, then the capacity constraint condition can be written as
Capacity constraint condition :
w(t) >0 only if X(t)  X
Let I denote the investment made in drilling the well. This investment is directly related to the
depth of the well, which in turn, is inversely related to X . Thus the relation between I and X can be
written as
6
[1]
X ( I)
X =
where X ( I ) < 0 and X ( I )  0. The investment in the depth of well is regarded as a sunk
investment which has to be made once and for all, prior to extraction.9 The marginal cost of
investment, denoted by  , is assumed to be a constant that depends upon the rate of interest faced by
the agent in the credit market.
In the second stage, the agent chooses an extraction path w( X, t) that maximizes the present
value of net returns from extraction. We assume that agent has complete information about the water
stock at each instant in time and conditions his water extraction decision on both calendar time and the
current water stock. Following Gisser (1983), we assume that the cost of extracting water is an
increasing function of the extent of lift shown in figure 1 as AB. As discussed above, the extent of lift
at any time t, is inversely related to the water stock X(t). A cost function, widely used in the resource
economics literature, which captures these properties, is given as10
[2]
C (t ) 
cwt , X (t ) 
X (t )
where c is a constant. The well owner is assumed to be a price taker in the market for water, with the
price of water given by the constant p.
The agent’s optimization problem can be solved using backward induction by first solving the
second-stage problem conditional on the investment decision in the first-stage. The second-stage
optimization problem is given as

[3]
Max
w t , X ( t )

 pwt , X (t )  


0
cw(t , X (t ))   t
 e dt
X (t ) 
[A] X  r  wt , X (t ) 
[B] 0  wt , X (t )  w
[C] wt , X (t ) = 0
for
X(t) < X (I )
[D] X o  X (0)
7
where  is the discount rate and r is the natural recharge rate of water. Equation [A] governs the
stock transition over time. Constraint [B] implies that at each instantaneous point in time there is an
upper bound, w , on the amount of water that can be extracted.11 To allow for the possibility of
complete exhaustion, we assume that w >r. Constraint [C] ensures that there cannot be any extraction
whenever the stock of water falls below X (I ) , and [D] gives the initial value of the stock.
Since the maximand in the above problem is linear in the control variable wt , X (t ) , the
equilibrium is a bang-bang solution. The optimal extraction path is given by the following most rapid
approach path (MRAP)
[4]
w

w S (t , X (t ))  r
0

if X (t )  Max{ X S , X ( I )}
if X (t )  Max{ X S , X ( I )}
if X (t )  Max{ X S , X ( I )}
where XS is the steady state stock level given as
[5]
 c   2 c 2  4  cpr
X 
2 p
S
The optimal solution represented in (4) and (5) is independent of the initial state vector and
depends only on calendar time and the current state vector so it is a feedback equilibrium.12
Given this solution to the second-stage problem, in the first-stage the sole owner chooses the
level of sunk investment in the depth of the well such that the marginal costs of investment equal the
discounted marginal benefits from extraction. Let us assume that the initial stock level is greater than
XS. Let IS be the level of investment that corresponds to XS in [1]. From (4) it is clear that along the
optimal path, the agent does not extract any water whenever the stock level falls below XS. Thus the
marginal benefits from investing beyond IS are zero. However, for any level of investment I< IS, the
total benefit from investing is given by



cw  δt
cr  δt
 pw 
 e dt    pr 
 e dt
X(t) 
X (I) 

s( X (I))
s( X (I))
[6]
B( I ) 

0
8
where s( X ( I )) denotes the time at which the stock attains the lower bound X . To ease notation, let
S(I)= s( X ( I )) . Differentiating [6] with respect to I, gives the marginal benefit from investing at any
level I< IS as
[7]


c
cr
B ( I )  S ( I )(w  r)( p 
) e   S (I )  
 X ' ( I ) e  S ( I )
2
X
(
I
)

 δ X (I)
It can be shown that B’(I) is strictly positive and downward sloping for I< IS. 13
Figure 2 shows the marginal benefit and cost curves of investment. The agent chooses the
level of investment that equates the marginal cost of investment (  ) with the marginal benefit of
investment. In figure 2, if    S , then the agent invests IS and drives the stock to the steady state
level, XS. On the other hand, if  >  S , then the optimal choice of investment is less than IS and given
by the intersection of the marginal benefit and cost curves in figure 2. In this case, the optimal steady
state stock level is less than XS. Under the assumption that the well owner is a price taker in the
market for water, this solution also defines the social optimum in this setting.
III.
Homogenous Agents
In this section we consider the case of two agents (i =1, 2) who extract from a common groundwater
aquifer and are homogenous in all respects. As opposed to the case of sole-ownership, in a two-person
case, the two-stage model is much more complex because of strategic behavior. In contrast to the
static formulations of the game in the literature, our dynamic 2 person formulation allows agents to
make decisions at more than one point of time depending on the information that is being revealed
during the play of the game. For ease in exposition, we have divided this section in two parts. In the
first part, we present the Markov perfect equilibrium solution for the case usually modeled in the
groundwater literature where only the extraction decision (and not the capacity choice decision) is
taken into account (see for instance, Provencher and Burt, 1993; Gisser, 1983). Such a setting is
useful in situations where capacity can be quickly adjusted to any changes in extraction needs and/or
9
the costs of setting up capacity are negligible and so capacity does not represent a rigid constraint. In
the second part of this section we relax this assumption and present the two-stage model with capacity
and extraction choice.
III.1 Extraction choice with no capacity constraints
The optimization problem for agent 1 is given as
[8]


cw t , X (t )   t
e dt
Max   pw1 t , X (t )   1
w1
X (t )

0
s.t [A] X  r  w1 t , X (t )   w2 t , X (t ) 
[B] 0  w1 t , X (t )  w
[C] w1 (t , X (t )) = 0 for
X(t) < X (I )
[D] X o  X (0)
The unique Markov perfect equilibrium trajectory is symmetric for both agents and is given as
[9]
w

w N t , X (t )   r / 2
0

if X (t )  X N
if X (t )  X N
if X (t )  X N
where
[10]
X
N
 c   2 c 2  2 pcr

2 p
On comparing equations [5] and [10] it is clear that XN < XS. This is the standard result of
over-exploitation when agents do not fully internalize the externalities generated in the use of the
commons. The gross payoffs from extraction in this case are given as
s( X N )
[11]

N


0


cw   t
cr   t
 pr
 pw 
e   

e
X (t ) 
2 2X N 

s( X N ) 
10
where s(XN) is the time at which the stock reaches the steady state level given as XN in (10).
III.2 Two stage game with capacity and extraction choice
Now let us consider the case where agents choose their investment in capacity in the first stage of the
game. In second stage of the game, both agents observe each other’s capacity level and choose their
extraction path. As we show below, agents may now choose investment levels strategically in order to
force exit or deter entry of the other agent. In order to keep the analysis fairly general, we do not make
any a priori assumptions regarding which agent makes the investment choice first.14 Instead, we
explore all possible options regarding the sequence of moves and compare the agents’ payoffs under
these different possibilities. As we show below, in the homogenous agents case, the sequence of
moves does not matter in any fundamental way. In the heterogeneous agents case, the sequence of
moves plays an important role and thus is discussed in greater detail. The equilibrium concept we use
in this multi-stage game is that of sub-game perfection under the assumption of complete information.
Given this setting, the set J of strategies available to each agent can be categorized as: J = {D,
A, E} where D stands for drive out, A for accommodate and E for exit. For agent 1 to drive out agent
2, two conditions must be satisfied. First, in stage 1 of the game, agent 1’s investment should be
greater than that of agent 2, i.e. I1 > I2. Second, in stage 2, agent 1 must drive down the stock to a
level beyond X (I2) in finite time. By definition, adoption of the drive out option by any agent implies
forced exit for the other agent and together these imply that there exists a time period td, such that for
all t > td, there is only one agent in the game. As opposed to this, the strategy to accommodate implies
that there are two agents in the game for all t. For accommodation to work, both players must invest at
the same level. Finally, note that each agent always has the option of exiting out of the game in finite
time, irrespective of what the other agent does.
Let us begin by analyzing the case when agent 2 makes the investment choice first (the case
when agent 1 moves first is symmetric). Recall that XN was found to be the Nash steady state stock
level in the absence of capacity constraints. Let IN denote the level of investment in capacity that
11
corresponds to XN from [1]. Note that (IN, IN) cannot be the equilibrium investment strategy in the
presence of capacity constraints. This is because each agent by investing a small amount,  , above IN
can drive out the other agent and get larger profits. To examine when such a drive-out strategy would
be chosen, we lay out the extraction paths and the associated payoffs when agent 1 pursues the drive
out strategy after having observed agent 2 invest at a certain level given as I2. We then compare this
drive out payoff to payoff from the two other strategies i.e. exit and accommodation.
If I2 < IS (where IS is the equilibrium steady state investment level under sole-ownership), then
agent 1 would choose to invest IS and drive down the stock to XS. This case is similar to the sole
ownership case. However, agent 2 can foresee this possibility and would not invest less than IS if he
expects agent 1 to invest IS. Thus (IS , I2 < IS) cannot be an equilibrium investment strategy. Now
consider the payoffs from the different strategies when agent 2 invests I2 > IS. Given I2 > IS, agent 1’s
extraction path in the second stage, under the drive out strategy (D) is given as
w

w  r
0

D
1
[12]
if
X (t )  X ( I 2   )
if
X (t )  X ( I 2   )
if
X (t )  X ( I 2   )
For ease in presentation, it would be helpful to distinguish between two kinds of payoffs, the gross
discounted payoffs from extraction (not accounting for investment costs), denoted by  and the net
discounted payoffs denoted by  =  - investment costs. The gross payoff to agent 1 from extracting
along this path can be written as

[13]
D
1

S ( I 2  )

0

cw t
cr
( pw 
)e dt   ( pr 
)e  t dt
X (t )
X
(
I


)
2
S ( I 2  )
As one would expect, this drive out payoff received by agent 1 is a decreasing function of I2 (proof in
appendix A1).
By definition, under the above drive out strategy, agent 2 is forced to exit and his extraction
path is given as
12
if X (t )  X ( I 2 )
w
w2E  
0
[14]
if X (t )  X ( I 2 )
The gross payoff to agent 2 (payoff for agent 1 is symmetric) from extracting along this path can be
written as
[15]
 2E ( I 2 ) 
S (I2 )

0

cw   t
 pw 
e dt
X (t ) 

Under complete information, agent 2 knows that he would be driven out in the second stage.
Therefore, given  , he chooses investment optimally in the first stage. For  >0, the optimal net exit
payoff is given as
S (I E )
[16]
 ( ) 
E

0

cw    t
 pw 
e dt   I E ,
X (t ) 

S (I ) 

cw   t
E
e dt   I 
   pw 
I

arg
max
where,
X (t ) 
 0 

The above net exit payoff has a special significance in this setting. Each agent can always guarantee
for himself this minimum payoff by exiting out of the extraction game in finite time, irrespective of
the actions of the other agent. Thus  E ( ) represents the reservation payoff in this setting.
Definition 1: A strategy (Ii, wi) for agent i ( i=1, 2) is individually rational if the net payoff to agent i
from this strategy is at least as large as his net exit payoff.
As opposed to driving out agent 2, agent 1 can also accommodate him. If both agents invest
IN and accommodate each other then the gross payoffs are given by 
is an unstable equilibrium since 
N
N
in [11]. As argued earlier, this
<  1D (I2 = IN) and so agent 1 prefers to drive out agent 2 by
investing IN +  if the latter invests IN. Agent 2 can foresee this and would never invest IN. Thus
accommodation is not possible at investment level of IN. However, since  1D (I2) is a decreasing
function of I2 there exists an INC > IN, such that for I2 = INC
13
[17]
 1D ( I 2  I NC )   1N
That is, if agent 2 invests INC then agent 1’s gross payoffs from driving him out are the same as his
gross payoffs from accommodating him.
Suppose agent 2 invests INC, now let us compare the net payoffs under the different strategies
available to agent 1. The net payoff under the exit strategy is given by equation [16]. Under the
strategy of accommodating by investing INC, the net payoffs are given as
[18]
 1A` ( I NC )  
N
 I NC  .
The net payoffs to agent 1 from driving out agent 2 are given as
[19]
 1D ( I NC )   1D  ( I NC   )
Using [17], it is clear that the strategy of accommodation yields strictly higher net payoffs to agent 1
than the strategy of drive out, if agent 2 invests INC. The case is symmetric when agent 1 moves first
and invests INC . Thus, irrespective of which agent moves first (or if they both move simultaneously),
the following proposition follows
Proposition 1: In the homogenous agents case with capacity constraint, irrespective of which agent
moves first, if it is individually rational for both agents to accommodate each other at INC then
a) (INC, INC) are the equilibrium investment levels.
b)
XN is the steady state stock level.
Proof: In the appendix (A2).
Recall that XN was shown to be the steady state stock level in the case without capacity
constraints also. However, the difference in the case with capacity constraints is that strategic
behavior leads both agents to invest in excess capacity, since INC>IN.
14
IV
Heterogeneous Agents
In this section we assume that agents are homogenous in all respects except in terms of the cost of
credit they face. 15 Let the cost of credit be denoted by  1 and  2, respectively, for the two agents.
We define inequality as 1  2 . In order to focus on the pure effect of inequality, we need to
compare states that are not Pareto ranked.16 Therefore, we consider (1, 2)  P,

1   2

2
where P = (1 ,  2 ) : 1  0,  2  0,

 0

Without loss of generality, agent 2 can be assumed to be the agent with the higher cost.
Let us begin by examining how the net payoffs from the different strategies discussed in the
previous section vary with  i (i = 1, 2). In the previous section we noted that for any given level of
 i, if one agent invests INC then for the other agent the net payoff from accommodation exceeds that
from drive out. The following lemma establishes how the net payoff from accommodation at INC
compares to the net exit payoff as 
i
varies
Lemma 1: There exists a  0> 0, such that
a) If i < 
b)
0
,
the net payoff from accommodation at INC is individually rational
If i >  0, , the net payoff from accommodation at INC is not individually rational
Proof: in the appendix (A3). Intuitively, Lemma 1 follows from the following two facts. First, at 
= 0, the net accommodation payoff to agent i is higher than the net exit payoff (see appendix A3).
Second, as  i increases, the net payoffs from both strategies decrease, with the rate of decrease for
accommodation strategy being higher than that for the exit strategy. This implies that there exists a
certain critical level of marginal cost of investment, which we call  0, beyond which it becomes
optimal for agent i to exit out of the game.
In the remaining part of the paper, we will assume that    0. This implies that when
agents are homogenous (perfect equality), accommodation at INC, is individually rational for both
15
i
agents and proposition 1 holds. Now consider what happens as inequality increases, keeping the mean
level of costs at the same level. For the low cost agent (agent 1), accommodation at INC continues to
be individually rational. However, for agent 2 it is likely that if inequality becomes fairly high (and
thus his marginal cost of credit becomes sufficiently high), it may become optimal to exit. Based on
this logic, the following two propositions outline the relation between inequality and steady state stock
and investments levels.
Proposition 2: If agents are heterogenous such that  1<  2 <  0, then they accommodate each other
at INC, irrespective of which agent moves first. The equilibrium investment levels are (INC, INC) and the
steady state stock level is XN.
Proof: From Lemma 1 we know that when  1 and  2 are both strictly less than  0, then the
accommodation strategy at INC is individually rational for both players. Therefore, the proof follows
directly from proposition 1.
For the case when agents are homogenous or when inequality is small, the sequence of moves
does not matter (proposition 1 and 2). However, when inequality becomes large such that  2 >  0,
the sequence of moves becomes important. As we show in the proof of proposition 3 below, the agent
with higher marginal costs (agent 2) is indifferent between moving first or second because he
ultimately exits in either case. However, the agent with lower marginal costs (agent 1), strictly prefers
to move second. Intuitively, this happens because for agent 1, the investment level required to drive
out agent 2 depends on whether he can observe agent 2’s investment level before he makes his own
investment decision. If agent 1 moves second, he can observe agent 2’s investment at I 2E , and just
needs to invest  above I 2E in order to drive out agent 2. However, if agent 1 moves first, he has to
make sure that agent 2 does not invest more than him and drive him out. Thus he has to invest at a
level such that agent 2 is indifferent between driving him out and exiting the game. Call this
16
investment level I 2 . Since I 2 > I 2E , this implies that agent 1 has to invest in excess capacity, when he
moves first, in order to rule out the threat of being driven out by agent 2. 17
Proposition 3: For large mean preserving deviations in marginal costs, such that  1 <  0 but  2 >
 0, it is no longer individually rational for both players to accommodate each other. Agent 1 exits
out of the game and thus the equilibrium investment levels are given as [Max (IS, I 2m + ), I 2E ] and
the steady state stock level is given by Min[XS, X (I 2E   ) ], where
I 2m
E

 I 2 when agent 2 moves first

2

 I when agent 1 moves first
Proof: In appendix (A4).
V.
Inequality, Investment and Steady State Stock Levels
In this section we use the results from the previous section to map a relationship between inequality on
the one hand, and investment and steady state stock levels on the other hand. In figures 4a and 4b, 
2,
is shown along the horizontal axis. The origin represents the point of perfect equality at which  2 =
1 =    0. As one moves to the right along this axis,  2 increases while  1 decreases, preserving
the mean at  . Thus a movement to the right along this axis represents increasing levels of inequality
with the same mean level of marginal costs. The aggregate investment levels and the steady state
stock levels are shown along the vertical axis in figures 4a and 4b, respectively.
In a small neighborhood around the origin where  2   0, it is individually rational for each
player to invest INC and accommodate the other player (proposition 2). So the aggregate investment
levels and the steady state stock levels are the same as in the case of perfect equality. We label this as
the range of low inequality in figure 4. As 
2
increases further such that  2 >  0, it is no longer
individually rational for player 2 to invest INC and stay in the game indefinitely (lemma 1). It follows
from proposition 3 that in a small neighborhood to the right of  2   0 , agent 2 invests I 2E and is
17
driven out by agent 1 who invests I 2m +  and drives down the stock to X ( I 2E   ) . Thus the
steady state stock level and the investment levels are lower in this neighborhood than under perfect
equality.
As inequality increases further, aggregate investment falls monotonically. However, the
relationship between steady state stock level and inequality is somewhat more complex. Note that the
steady state stock level falls sharply at  2 =  0 and thereafter increases as inequality increases. For
the case where  2 >  0, we can distinguish between the following ranges for increasing values of 
2
(see figure 4).
1) Moderate inequality: where aggregate investment is ( I 2m + I 2E +  ) < 2INC and the steady state
stock level is X ( I 2E   ) , where X ( I 2E   ) <XN < XS.
2) High Inequality: where aggregate investment is ( I 2m + I 2E +  ) < 2INC and the steady state stock is
X ( I 2E   ) , where XN < X ( I 2E   ) < XS.
3) Very High Inequality: where aggregate investment is ( I 2E +IS) < 2INC and the steady state stock is
XS.
Thus, we find that when we relax the assumption of exogenous investment in capacity and
allow for strategic choice of investment in a dynamic setting, the relationship between inequality and
steady state stock level is non-monotonic. In particular, starting from perfect equality, as inequality
increases, there is a range (which we refer to as the range of moderate inequality) where steady state
stock level is lower than the level under perfect equality. On the other hand, in the low and high
ranges of inequality, the steady stock level is at least as large as that under perfect equality. In the
very high inequality range, steady state stock is at the first best level.
In the above analysis we have assumed that there are no borrowing limits and that agents can
borrow whatever they want at a constant cost which differs amongst agents. Another form of
inequality that is common in developing countries is quantity rationing in the amount of credit given to
18
agents. This is typically true in the context of formal credit markets where the rate of interest is set
artificially low by the government and there is excess demand for credit. Available credit is rationed,
with borrowing limits that are positively related to ownership of collaterizable assets such as land. It
is interesting to observe that our results on the non-monotonic relationship between inequality and
steady state stock hold in this case too.
To see this, assume that the cost of credit is same for both agents and is set at a low level
(below 
0
) . Let I iMAX denote the maximum investment agent i can make given his exogenously
given borrowing limit. We define inequality in this case as I 1MAX  I 2MAX . When inequality is zero
so that I iMAX is the same for both the agents and is such that I iMAX  I NC , proposition 1 still holds.
As inequality in distribution of credit amount increases, we increase I 1MAX and decrease I 2MAX
preserving the mean at the same level as that under perfect equality. Note that as long as I iMAX  I NC
for both agents, proposition 2 again holds and we are in the region of low inequality in figure 4.
When level of inequality increases further so that I 2MAX  I NC (and by assumption of mean
preserving spread, I 1MAX > I NC ), the steady state stock would be X ( I 2MAX   ) which is greater than
the steady state stock level under perfect equality. As inequality increases further, the steady state
stock level rises, as before. For I 2MAX < I 2S , the steady state stock level is the same as that under sole
ownership.
VI.
Summary and Conclusions
Previous work on common pool resources has generally focused on strategic interaction amongst
agents in extraction choices alone. The papers that have examined the strategic effects of investment,
on the other hand, have assumed credit markets to be fully competitive so that all agents face uniform
cost of credit for financing their investment. This assumption may be difficult to justify, in the context
of CPR use in rural areas of LDCs where credit market imperfections are pervasive. In this paper we
19
develop a two-stage model where agents (homogenous in all respects except their cost of credit)
choose the level of sunk investment in capacity and subsequently, the extraction path over the infinite
horizon. Sunk investments serve as a commitment device in this model to force exit. Since the cost of
credit influences these investment choices, heterogeneity amongst agents in terms of their access to
credit affects both capacity and extraction choices.
Using this model we find that because of the strategic role of investments in this setting,
agents invest in excess capacity in general. This excess capacity result is similar to that found in entry
deterrence models (Tirole, 1988). Furthermore we show that with increase in inequality, aggregate
investment falls monotonically. In other words, aggregate investment approaches the optimal level as
inequality increases. By contrast, the relation between inequality and efficiency in resource extraction
is non-monotonic. The steady state resource stock is found to be closest to the socially optimal level
when either inequality is very high or very low. For moderate levels of inequality, we show that the
resource stock may in fact be lower than that under perfect equality. This result is in sharp contrast to
the monotonicity result found in models that do not consider the strategic role of investment in
capacity.
In many CPR contexts, such as that of groundwater in semi-arid India, direct regulation of
extraction rates is regarded as infeasible in the short run. An important indirect policy tool here is the
administrative distribution of important inputs, such as credit, used in groundwater extraction. Some
previous papers based on static settings (such as that by Baland and Platteau, 1997), have argued that
the more unequal is the distribution of credit, the higher is the efficiency in use of commons. The
contribution of our paper lies in showing that the relation between inequality in cost of credit and
efficiency in resource use may not necessarily be monotonic. In fact, we show that moderate levels of
inequality may lower resource stocks even below that under equality.
How large is this range of moderate inequality is an open empirical question, the answer to
which is likely to be very context specific.18 Given the diversity in institutional background, resource
dynamics and market conditions facing agents in different CPR contexts it is not possible (nor
20
desirable) to derive any general results on the relation between inequality and efficiency in CPR
extraction. The purpose of this paper has been to examine the role of strategic investments in capacity
and the plausibility of a non-monotonic relation in a context where such investments are important. In
some other CPR contexts such as that of fisheries, for instance, investment in capacity (in the form of
boats and nets) may not be sunk to the same degree as investment in well capacity. Unlike the case of
well investment, which tends to be very lumpy and irreversible, investment in fishing capacity may be
revised more frequently. Thus the commitment value of such investment and its effectiveness in
forcing exit may also be lower.19 In the context of fisheries, our assumption of extraction costs being
linear in the amount extracted, may also be more difficult to justify. However, once this assumption of
linearity of extraction costs is dropped, the second stage game becomes much more complex to
analyze.20
Our basic model can be extended in several other directions. An important extension would
be to examine whether our results change qualitatively when there are more than two agents. Previous
papers that have modeled only extraction choice and not capacity choice (such as Baland and Platteau,
1997; Dayton-Johnson and Bardhan, 1996) found the results to be qualitatively similar in a multiple
agent setting. Finally, in the CPR literature there has also been a lot of interest in examining efficiency
in extraction from a CPR when users have some market power (Cornes et. al,. 1986). The strategic
effects of investment are likely to be much more complex in such a setting where agents have to
consider the impact of their investments on the product market as well.
21
Appendix
A1) Proof:  1D is a decreasing function of I2 , for I2>IS.
 =
s( X (I2 ))

D
1
0


 δt

cw  δt
cr
 pr 
 e dt
 pw 
 e dt 
X(t) 
X (I 2  ) 

s( X (I2 )) 


 1D  X ( I 2   ) δs( X(I2 )) 

 s( X ( I 2   ))
c
cr


e

(w  r) p 

2
I 2
X(I 2  )   X ( I 2   )
I 2
δ X (I 2  ) 


In the two-person case,
X (t )
 r  2w
t

s( X ( I ))
1

 X (I )
2w  r

 X ( I 2   ) s ( X ( I 2  )) 
 1D
cr
c
 w  r 


e

p

Therefore,
=



2

I 2
X ( I 2   ) 
I 2
  X ( I 2   )  2 w  r 
From [1],
 X (I 2 )
 D
 0 therefore 1 <0 if the following condition holds
I 2
I 2

cr
c
 w  r 


 p 
X ( I 2   ) 
 X ( I 2   )  2w  r 
2
Since
wr
< 1, the above condition would hold if
2w  r


cr
c
  X ( I 2   )  X S  I 2    I S
  p 
X (I 2   ) 
 X (I 2   ) 
2
For the case being considered, I2 > IS and so the above condition is satisfied.
A2) Proof of proposition 1: Suppose agent 2 moves first. To check if (INC, INC) are the
equilibrium investment levels consider what happens if there is a one step unilateral deviation by
agent 1 to INC + (where  > 0), in order to drive out agent 2. As discussed above, the drive out
strategy gives strictly less net payoffs than the accommodation strategy thus this deviation is not
profitable for agent 1. Now consider a unilateral deviation by agent 1 to INC -. Since INC -. <
INC, agent 1 cannot drive out agent 2. Agent 2, however, would now find it optimal to drive out
agent 1. Since the net exit payoff is lower than the payoffs from accommodation at INC, this
deviation is also not profitable for agent 1. Under complete information, agent 2 can foresee that
agent 1 would invest INC and accommodate if he invests INC. Following the same logic as for
agent 1, he would not find it optimal to deviate from INC. Thus (INC, INC) are the equilibrium
investment levels . By symmetry, the equilibrium would be the same if agent 1 moved first.
A3) Proof of Lemma 1: In the neighborhood of i = 0, the net payoff from accommodation
strategy is higher than that under the exit strategy for agent i. The derivative of the net payoff
function from accommodation strategy (at investment level of INC ) is -INC (from [18]). On the
other hand, the derivative of the net exit payoff function is - I iE (from [16]). INC is not a function
of  i (see [17]), but I iE is a decreasing function of  (from [16]). This implies that starting from
a neighborhood of  i =0 where the net accommodation payoff is higher than the net exit payoff,
as  i increases the net accommodation payoff falls at a constant rate while the net exit payoff
falls at a decreasing rate (figure 3). This is sufficient to prove the existence of 0 in lemma 1.
A4) Proof of proposition 3: Consider the equilibrium under the two possible sequences of moves:
Case I: Agent 2 moves first. Since the option of accommodation by investing at INC is no longer
individually rational for agent 2, he can either drive out agent 1 or exit out of the game himself.
Since agent 1 faces a lower cost of investment and invests after observing agent 2, drive out by
agent 2 is not a feasible option here. Agent 2’s only option is to exit and thus he chooses to invest
I 2E to maximize his net exit payoff. Given this, agent 1 chooses to invest [max (IS, I 2E +)].
Case II:– Agent 1 moves first. If agent 1 invests I 2E +, then agent 2 having observed agent 1’s
investment, will invest a little more and drive him out. To avoid being driven out, agent 1 will
invest at a level where agent 2’s net payoff from driving out agent 1 equals his net payoff from
exit. Denote this level of investment by I 2 . Thus agent 1 will invest max (IS, I 2 +) whereas
agent 2 will invest I 2E .
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A
B
D
E
Figure 1: Schematic diagram of groundwater aquifer
Irrigation surface
A
Marginal
Benefit
Marginal
Cost
B
S
O
IS
Investment
Figure 2: Marginal benefit and costs of investment under sole-ownership
Net
Payoff
Net exit payoff
0
O
Figure 3: Net payoff from exit and accommodation strategies.
2
Net payoff from accommodation
Investment
2INC
2IN
IS
Low
Inequality
Moderate
Inequality
2 
High
Inequality
Very High
Inequality
2
(a)
Steady state
Stock
XS
XN
Low
Inequality
2 
Moderate
Inequality
High
Inequality
Very High
Inequality
2
0
(b)
Figure 4: Effect of Inequality on Investment and Steady State Stock
NOTES
1. A few important examples include, Gordon (1954), Clark (1976), Dasgupta and Heal (1979), Levhari
and Mirman (1980), Gisser (1983), Provencher and Burt (1993).
2. Studies from all over the developing world point to these interest rate differentials. For instance,
Aleem’s survey of Chambar region of Pakistan found interest rates to vary between 18% to 200% per
annum in the informal sector, compared to formal sector rate of around 12% (cited in Ray, 1998).
3
As explained above, differences in marginal costs of credit arise largely because of asset inequality.
However, as an anonymous referee has pointed out, the treatment of these types of inequality is quite
different. First, note that an increase in asset inequality does not always imply an increase in inequality in
costs of credit. Thus for example, a fall in the poor agent’s asset may not necessarily increase the usurious
interest rates charged to him. Second, asset inequality can affect CPR use through several other channels
also, such as through affecting participation in other factor markets or affecting the availability of exit
options (Dayton-Johnson and Bardhan, 1996). The purpose of the present paper is to focus solely on the
effect that occurs through the role of strategic investments when agents have differential cost of credit.
4
They assume a concave effort function and show that if one moves from a situation where none of the
agents is credit constrained to a situation where some are credit constrained, then the constrained agents are
likely to reduce their extraction levels. As a response, the unconstrained agents are likely to increase their
level of extraction. However, because of the concavity assumption, it follows that the increase less than
compensates for the decrease. Thus total extraction is likely to fall.
5
The most widely studied form of heterogeneity in investment game literature arises from temporal
symmetry (incumbent and entrant) which may give one firm (the incumbent) an absolute cost advantage
because of accumulation of capital or access to superior technology ( see for instance, Stackelberg, 1934;
Spence, 1977; Dixit, 1980; Fudenberg and Tirole, 1986) . Fudenberg and Tirole (1985) consider a setting
where firms have different access to the capital market but their purpose is to study predatory pricing
behavior rather than investment behavior. We have not come across any paper in this literature that
examines the impact of increase in inequality on investment behavior.
6
A recent working paper by Dayton-Johnson and Bardhan (1996) also suggests that the relation between
heterogeneity in asset endowments and efficiency in resource use may be non-monotonic. However, the
structure of their model is very different. In particular, they have a two period setting in which the capacity
level of agents is uniquely determined by their exogenously given asset endowments. Thus, agents choose
only the effort level in the first period (in the second period, it is always true that agents would apply
maximum effort).
7
According to the climatic classification of India, the semi-arid region includes the whole of Punjab and
parts of Haryana, Uttar Pradesh, Madhya Pradesh, Rajasthan, Gujrat, Maharashtra, Andhra Pradesh,
Karnataka and Tamilnadu (for a map see http://www.mapsofindia.com/maps/india/climaticregions.htm).
8
This assumption is fairly common in both theoretical and empirical literature on groundwater extraction
(see for instance, Knapp and Olson, 1995). In the specific case of a bathtub aquifer (Gisser, 1983) this
relation is given as X = (AS) H , where AS is area times storavity of the aquifer.
9
The horsepower of the pumping equipment may also be an important determinant of capacity. The
assumption that investment in capacity is made only at one time may be somewhat restrictive but can be at
justified given the observation that such investments tend to be quite lumpy. Empirical papers on well
investment have found that expansions in well capacity occur only once in every 5-10 years (Pender and
Asokan, 1993; Aggarwal, 2000). The implications of dropping this assumption are discussed in note 18.
10
See for instance, the survey of different resource problems in Clark (1985) and Conrad and Clark (1991).
11
In some resource contexts, it may be more reasonable to assume that the investment choice in first period
affects w (in addition to
X ). For instance, in the case of fisheries investment in fishing gear may also
affect w . However, to keep the analysis simple we assume w to be independent of I. Our calculations
(under reasonable assumptions regarding parameter values) suggest that our central results on the relation
between inequality and resource efficiency are not qualitatively affected by this assumption.
12
See also Conrad and Clark (1991) for a similar optimization problem.
13
Proof avaiable with the authros.
14
The results in entry deterrence have been found to be quite sensitive to the assumptions made regarding
the timing of the moves (Tirole, 1988 provides a survey).
15
We assume here that both agents can get all the credit they demand at this cost. Towards the end of
section V, we discuss what happens if instead there are differential borrowing limits.
16
Clearly if
 1 increases without an accompanying change in  2, this would imply a reduction in
efficiency of resource extraction.
17
In real life contexts, it is hard to say a priori which particular sequence of moves is more plausible. In
some CPR contexts, the agent with higher marginal costs (generally the poorer agent) may not be able to
afford to wait and thus may move first. In other contexts, it may be true that the agent with lower marginal
costs plays a leadership role and moves first although it is less advantageous for him.
18
Results from empirical studies on the relation between inequality and efficiency in use of CPRs are
mixed (Baland and Platteau, 1997 provide a survey). However, almost all these studies have tested for only
a linear relationship between inequality and resource efficiency. An important exception is a recent analysis
by Bardhan (2000) of determinants of cooperation in surface irrigation systems in South India. Bardhan
also found a sort of U shaped relation between inequality in landholding and existence of water related
conflicts.
19
Instead of considering investment decision to be made once and for all we could have considered
investment that is irreversible (hence cannot be revised downward, except for depreciation) but could be
revised upwards. This would have complicated the analysis quite a lot. Some papers in the investment
game literature have analyzed such a setting and these papers could throw some light on what we might
observe in our case. Thus, for instance, Fudenberg and Tirole (1983, 1986) consider two ways to model
such a situation. The first one is a “precommitment” or open loop equilibrium where agents commit
themselves to the entire time path of investment. This equilibrium is found to be exactly the same as if both
firms built their entire capital stocks at the start. The second option is to look for a Markov perfect
equilibrium. The capital stock becomes an additional state variable here, which complicates the analysis.
Fudenberg and Tirole (1983) find the existence of multiple of equilibria in this setting, some of which
involve excess capacity and others do not.
20
An alternate procedure to simplify the analysis might be to consider a general reduced form payoff
function that has the desired properties instead of a differential game for the second stage.
Such a
procedure, however, would make it difficult to relate our results to the steady state level of the resource.
We are grateful to the co-editor for drawing our attention to this alternative procedure.