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: .
**Tulika A. Narayan, Dept of Agricultural and Resource Economics, University of Maryland, College Park, MD 20742. Email: .
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 non-monotonic, 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
- 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 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 non-monotonic, 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 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.
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 , such that whenever the water stock in the aquifer falls below , 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)
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 . Thus the relation between I and can be written as
[1] = ( I )
where ( I ) < 0 and ( 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 as[10]
[2]
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]
[A]
[B]
[C] = 0 for X(t) <
[D]
where is the discount rate and 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,, on the amount of water that can be extracted.[11] To allow for the possibility of complete exhaustion, we assume that >r. Constraint [C] ensures that there cannot be any extraction whenever the stock of water falls below , and [D] gives the initial value of the stock.
Since the maximand in the above problem is linear in the control variable, the equilibrium is a bang-bang solution. The optimal extraction path is given by the following most rapid approach path (MRAP)
[4]
where XS is the steady state stock level given as
[5]
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
[6]
where denotes the time at which the stock attains the lower bound. To ease notation, let S(I)= . Differentiating [6] with respect to I, gives the marginal benefit from investing at any level I< IS as
[7]
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 , then the agent invests IS and drives the stock to the steady state level, XS. On the other hand, if >, 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 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]
s.t [A]
[B]
[C]= 0 for X(t) <
[D]
The unique Markov perfect equilibrium trajectory is symmetric for both agents and is given as
[9]
where
[10]
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
[11]
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.