Boom Before Bust? The Industrial Organization of Groundwater

Transcription

1 Boom Before Bust? The Industrial Organization of Groundwater Ujjayant Chakravorty and E. Somanathan May 9, 203 Abstract Recent decades have seen an explosion in the drilling of wells and extraction of groundwater in India and China. According to many experts, the rise of groundwater extraction has led to increased food production and food security in these two populous nations. However, this trend has occurred at the same time as water tables have declined in many regions in India and China. We develop a model of the industrial organization of groundwater markets. We show that increasing scarcity of water may lead to a boom in "well-drilling," contrary to what one may expect. In our framework, resource scarcity reduces competition among water sellers, hence increases profits. Higher profits induce greater entry. Keywords: water scarcity, transport cost, market structure, development, Salop circle, spatial competition, Bertrand pricing We are grateful to Kelsey Jack, Gilbert Metcalf, Lorenzo Planas and Alain Ayong le Kama for helpful comments as well as workshop participants in Shanghai, Annecy and Clermont-Ferrand for valuable feedback. Professor of Economics, Tufts University. ujjayant.chakravorty@tufts.edu Professor, Economics and Planning Unit, Indian Statistical Institute, Delhi. som@isid.ac.in

2 Introduction China and India are the largest countries in the world, with a heavy dependence on groundwater for agriculture. The share of groundwater in Chinese irrigation has gone up from near zero in the 950s to about 20 percent early in this century, according to the Chinese Ministry of Water Resources (Wang, et al 2007). 2 Wang, et al note that groundwater is "critical" for the expansion of agriculture and the regional economy. The same story can be repeated for India and other countries in South Asia. For example, 7 million Indian farmers cultivate farms that are smaller than an acre (0.4 hectare). In the large Indian state of Uttar Pradesh, 60 percent of small farmers buy water from "water extraction devices" different types of tubewells. The area covered by pump irrigation in South Asia is estimated to be higher than under surface water irrigation (Shah, 2009). 3 There is plenty of evidence to suggest that the future growth in food production is intimately tied to the development of groundwater both in South and East Asia, which is home to billions of people. Unlike in developed nations such as the United States, the availability of pump irrigation has prevented the consolidation of agriculture in this region, and millions of small farmers are able to afford pump irrigation. Thus groundwater irrigation is intimately tied to the survival of small farms (size less than one acre or 0.4 hectares), especially in South Asia. However, there seems to be incontrovertible evidence that this large- scale extraction of groundwater has led to a steady fall in the water table in many India is the world s largest user of groundwater, accounting for about a quarter of world consumption. Groundwater accounts for about 60 percent of irrigated agriculture and 85 percent of drinking water supplies (World Bank 200). 2 In Northern China, about half of the irrigation water comes from groundwater, less in the Southern part of the country. Unlike surface water irrigation, the decentralized nature of groundwater use means that it is often under-reported in official statistics. 3 There is significant evidence to suggest that crop yields are higher with groundwater than in surface water irrigation and farmers prefer planting higher valued crops such as fruits and vegetables and orchards to groundwater. 2

3 parts of India and China. For example, a 2004 National Assessment in India found 29 percent of all groundwater blocks to be in the "critical, semi-critical or over-exploited categories" (World Bank, 200). 4 The goal of this paper is to develop an analytical model that explains this somewhat paradoxical phenomenon - an expansion of well drilling concomitant occurring at the same time as depletion of the water table, at least in some major irrigation-intensive regions of South and East Asia. More specifically, we seek to understand who buys and who sells groundwater and how government policy may affect the static and dynamic efficiency of water use. This is important not only because a large number of small farmers depend on groundwater for their water needs, but both small and large farmers participate as buyers and sellers in this market. 5 Typically, a pump could run for about 3,000 plus hours a year. Thus, government tax and subsidy policies will affect the decision to buy or sell water, and have important efficiency and welfare effects, that we study in this paper. Most of the economic literature on groundwater pumping has focused on the dynamics of depletion and market failures arising from common property access to water aquifers (e.g., Gisser, 983; Gaudet, Salant and Moreaux, 2003). There have been no studies on the industrial organization of groundwater. This is important because groundwater markets have specific characteristics that do not allow for standard economic principles to be applied. 4 The report also points out that the "potential social and economic consequences of continued weak or nonexistent groundwater management are serious, as aquifer depletion is concentrated in many of the most populated and economically productive areas" (World Bank 200). 5 A typical small farmer needs a pump for about hours a year. (Shah, 2009). However, once the investment is made, a pump can supply water to other farmers as well. 3

4 A Sellers B Figure : Sellers on a Salop circle with unit circumference. 2 The Model We consider a situation where a continuum of identical farms are distributed uniformly along a Salop circle of unit circumference. For analytical convenience, we ignore intra-farm distances and assume all farms to be points in space. The demand for water at any farm is assumed to be a non-increasing function q(p) where p is the price of water. Suppose there are n wells spaced evenly on the circle (see Figure 2). Let the average cost of pumping a unit of water to the surface be a constant c > 0. Let the cost of transporting a unit of water over distance x on the surface be t(x) > 0 with t (x) > 0. Later, we will discuss how more general pumping and transport costs may affect our results. For now, we will suppose that t(x) is linear so that t(x) = tx. So the marginal cost of pumping groundwater to a farm at a distance x > 0 is given by c + tx. The presence of transport costs introduces differentiation among buyers because it costs more to deliver water to buyers located further away. To simplify our model, we abstain from dealing with a monopoly, and assume that the monopoly price is "high enough." The presence of monopoly 4

5 sellers in our model will not affect our results, as we will show later. However, for now, we focus on getting the main result in a setting where sellers of water participate in a purely oligopolistic market. Assumption The monopoly price of water at the wellhead is at least c+ t. n The above assumption may hold if the cost of pumping and transport costs of water are sufficiently low. Proposition Under Assumption, the price of water paid by a buyer located at distance x from the nearest well is decreasing in x. Proof : The nearest seller s marginal cost of delivering water is c + tx < c + t ( n x), the marginal cost of the second nearest seller, (except for a buyer equidistant from two sellers). Therefore, the nearest seller can capture the entire market at x by selling at just under the second-nearest seller s marginal cost. This will be profit-maximizing for the nearest seller provided the monopoly price at x is at least c + t ( x). However, this price is less n than or equal to c + t which is less than or equal to the monopoly price at n x = 0 (Assumption ). Since the monopoly price is increasing in the marginal cost, and the marginal cost is increasing in x, this means that the monopoly price at x is greater than c + t ( n x). Hence, ( ) p(x) = c + t n x, () which is decreasing in x. Even though it costs more to transport water to farms further away, the more distant farms pay a lower price than nearby farms. The reason is Bertrand competition between a seller and his immediate neighbor. The price is simply equal to the nearest neighbor s marginal cost of water in order to cut that neighbor out of the market. Although the market is completely 5

6 segmented, with each farm buying water exclusively from the nearest well, the market is a series of Bertrand duopolies, not a series of spatial monopolies. 6 Figure 2 shows the layout of the sellers for any given n. Each seller will split the market with his two neighbors on either side. On each side, he will supply to buyers located from zero to seller is given by. Thus the total profit to any water π(n) = 2 [p(x) (c + tx)] q(p(x))dx. 0 Substituting for p(x) from (), we see that π(n) = 2t [ 2x]q(p(x))dx. (2) 0 n Note that the mark-up is independent of the cost of pumping water to the surface (as long as c is not so high that Assumption fails). This is because the Bertrand competitors cost of pumping water to the surface is the same and the mark-up depends on the difference in marginal costs of the two sellers. We note a further consequence of. By, with c and t fixed, a decrease in n raises the price at every farm. Since, by, this price is less than the monopoly price at every farm, this must increase the profit at every farm. Since it also expands a seller s market (see (2)), it must result in an increase in every seller s profit. We record this as Remark π(n) is decreasing in n for fixed c and t. 6 This result - in which the more distant buyers pay a lower price, has been observed in generic models of spatial competition, see for e.g., Norman and Thisse (996) in which firms price discriminate among uniformly located buyers with inelastic demand. 6

7 t c n t c E C c D A M( x ) B Figure 2: Price and marginal cost paths for seller A. M is location of buyer midpoint between adjacent sellers A seller s profit must be as large as the fixed cost of digging the well and installing the pump, which we aggregate and denote by F > 0. 7 With free entry, the equilibrium well density n e must satisfy π(n e ) = F. (3) Is this equilibrium socially optimal? Since the transport cost t is unlikely to be amenable to policy intervention, we take it as given and ask whether taxes or subsidies on the unit pumping cost c and fixed cost of a well F may improve social welfare. Proposition 2 In Bertrand equilibrium with free entry, if q (p) is sufficiently small for all p, then the equilibrium well density and aggregate water use are 7 This may include the cost of obtaining a permit for the tubewell and an electricity connection, in areas where the farm is connected to the grid. 7

8 2 3 (a) small n Fig. 3. Saw-tooth price path with peak ks where (b) large n Figure 3: Saw-tooth price path with peaks where sellers are located greater than the well density and water use that a social planner who can tax and subsidize F would choose. Proof : The total net social surplus, assuming Bertrand pricing, is given by n [S(n) F ] = n [π(n) + C(n) F ] (4) where S(n) is the social surplus from the farms served by one seller when well density is n, and C(n) is the consumer surplus from one seller s farms. The social planner can choose n by adding a tax to the right-hand side of the free entry condition (3). The planner chooses n to maximize the net social surplus (4). Hence, the socially optimal well density n o satisfies the first-order condition S(n) + n o S (n o ) = F. 8

9 or π(n o ) = F [n o S (n o ) + C(n o )]. (5) Since, by Remark, π(n) is decreasing, it is enough to show that π(n e ) < π(n o ). Since π(n e ) = F, we need to show that the term in brackets on the right-hand side of (5) is negative. Let s(p(n, x)) denote the social surplus accruing from the farm at distance x from a seller. Now ( S (n) = d ) 2 s(p(n, x))dx dn 0 = s(p(n, ( ) )) n (6) s s(p(n, x))dx. (7) n The first term above is the fall in total social surplus accruing to farms in a seller s market owing to the shrinkage of that market when n increases. The second term is the increase in social surplus in a seller s market owing to the fall in the price, and consequent expansion in water consumed at all farms in that seller s market. Now s ds s(p(n, x)) = n dp p n = q (p) ( t n 2tx (8) ) ( ) t. (9) The third factor above follows from () while the first two are explained in the file "Figure for Prop 2.pdf". By assumption, q (p) is bounded by a small constant, so the second term in (7) can be ignored. So ns (n) = s ( ) n = k ( ) n where k(x) is the consumer surplus of the farm at x. At x =, price equals marginal cost, so social surplus equals consumer surplus. Now k(x) < k ( ) n 2 9

10 for all x [ ) ( 0, because p(x) > p ). So C(n) = 2 k(x)dx < ( ) k dx = k ( ) n. Hence So ns (n) < C(n). C(n) + ns (n) < 0. Therefore, π(n o ) > F. Since π(n) is decreasing in n (because the price at every farm is decreasing in n, and the measure of farms that constitute a seller s market is decreasing in n), and π(n e ) = F by free entry, therefore n e > n o. Since the price of water at every farm is decreasing in n, therefore water use in equilibrium is socially excessive. 3 Depletion Consider a Bertrand equilibrium. Suppose the available water supply falls as a result of last period s extraction. Note that in the short run (keeping n fixed), at x =, p(x) must rise. For if it did not, then by the no-arbitrage condition, prices would remain unchanged at every x. But then water demand would exceed water supply. It follows (by the no-arbitrage condition) that the new short-run equilibrium involves a rise in the price at every x by a uniform amount that is just sufficient for the resulting decrease in demand to match the shortfall in supply. Figure 3 shows the rise in price as a result of water scarcity in the basin. Prices must rise until aggregate demand equals supply. Now this must increase profit, because the unit cost of delivering water 0

11 t c n c D A M( x ) B Figure 4: Bertrand Equilibrium with Scarce Water. Price rises uniformly to equate Aggregate Demand and Supply to any location has not changed, but prices have gone up. Hence, in the long run, there must be entry sufficient to bring profits back down to F. However, the average price will remain high enough to just exhaust the available water supply. We see that a decline in water availability has led to a boom in well density. Suppose there is further decline in water supply in the subsequent period. Perhaps, in this last period, the only available water is the natural recharge. Now if the water supply has declined enough, then the process described above will lead to a large increase in prices, so large that Bertrand pricing disappears because prices are constrained by the monopoly price. Note that, if we consider a hypothetical continuous decline in the water supply W, the monopoly price would first be reached at the wellhead since the marginal cost of extraction is lowest there. Further increases in W would result in a

12 monopoly price that increases in x upto some level of x and then declines with slope t along the Bertrand price line to. For this range of W, since prices continue to increase at some farms, profits increase and so entry increases. This case is shown in Figure 3. The boom continues. Eventually, for a low enough W, Bertrand pricing must be completely eliminated with monopoly pricing prevailing all the way till. Further declines in W cannot result in further price increases. Instead, sellers will stop selling to the least profitable customers, which are the ones located farthest from them. Profits will decline and there will be exit. Not all farms will get positive water supply. The boom turns into a bust. this result can be stated as follows. Proposition 3 If pumping costs remain constant but aggregate water available shrinks with use, firm profits increase, leading to increased entry followed by an instantaneous collapse of the industry. In Fig.6, we plot equilibrium firm entry as a function of available stock of water. If the stock of water is higher than W then the number of firms in the market is constant. Any new entrant will be unable to cover the fixed cost of entry F. However when the stock of water falls below additional entry wi W firms must reduce the total quantity of water they sell, so that prices must rise at each location (while unit costs of water delivery at each location remain unchanged), to equate aggregate demand to supply. However, firm profits must go up because now prices are closer to monopoly prices, even though quantity sold has gone down at each location. This induces more entry, which in turn increases aggregate water withdrawals, further raising prices. This must continue until monopoly prices are reached first at the wellhead, then in either direction. The number of firms reaches a maximum when water prices are monopolistic at each location. Further entry must increase prices and reduce profits, resulting in a gap in water delivery between any two tubewells. Finally all firms must exit when there is not enough water to 2

13 t c n P m c D A M( x ) B Figure 5: Increased Water Scarcity leads to Monopoly Pricing starting at the Well-head. Monopoly Prices rise with Distance cover a sufficiently large service area under monopoly pricing. This level of the stock of water is denoted in the figure by W 0. Below this level, no firm will supply water. Note that we have assumed that c remains constant as W falls. More realistically, there may first be a rise in c as the water table falls. Consider first the case when depletion implies a fall in the water table, but no change in the aggregate amount of water that is available. Then depletion leads to a rise in the pumping cost c, which drives up prices at each location. Quantity supplied at each location falls, leading to lower profits. This will result in a gradual exit from the industry. Thus in the case in which depletion results in a pure increase in the cost of pumping, we do not get a boom in well drilling. Well density will fall. However, if water demand is very inelastic, this effect may be small. When the bottom of the aquifer is reached, then W will start to fall and the process described above may set in. 3

14 n max n * W 0 W Figure 6: Firm Entry as a function of the Stock of Water However in reality, over extraction may manifest itself both through a reduction in the amount of available water W and an increase in the cost of extraction. The unit pumping cost is likely to be linear in the depth of the water table, since the potential energy expended will be the gravitational force times the depth. If the aquifer is rectangular in shape, then the volume of the water will also be linear in the depth of the water table. It is not clear how aquifer depletion will impact the pumping cost and the stock of water. If for example, the stock of water declines faster than the change in the depth of the water table, such as in a triangular aquifer, we may see a boom in well drilling followed by a bust. However, if the pumping cost changed faster than the stock of water, the boom will not exist and there will be a steady exit of firms from the industry as costs rise. The cost of pumping is directly a function of energy prices. Most tubewells at least in South Asia, use electricity where it is available, and diesel 4

15 otherwise, for running their tubewells. Farmers pay a flat fee for electricity use, which is not correlated with the quantity they consume. Thus in regions where electricity is used, we may observe the boom and bust cycle with depletion of groundwater. On the other hand, in regions where electricity is not available, and farmers use diesel, they pay per unit costs, even if they are subsidized. The costs of pumping should rise with increased depletion and we should observe a reduction in tubewell pumping over time. 3. Zero Transport Costs Let us now consider the case in which the unit transport cost t is zero. This may happen if evapotranspiration losses are low, for instance, or the cost of piping the water is low, such as in parts of Gujarat where portable plastic pipes can be installed at relatively low cost to move water between farms. In this case, the price charged to each consumer must be constant since there is no spatial heterogeneity due to space. However, there can not be any Bertrand equilibrium because then price must be at marginal cost, which implies zero profits and firms will be unable to cover their fixed cost F. The area serviced by firms will be limited by the capacity of the pump. If a new entrant comes in, prices will fall to c under Bertrand competition. So the entrant will not be able to recover his fixed cost. Entry will, therefore, be deterred. We will thus get a series of local monopolies. Less water will be used at each location - which is good from the point of view of depletion. But there will be no boom before the bust. The choice of capacity can be made endogenous. Assume that for an fixed investment of F, we can get a pump that services an area of radius R. Then each tubewell owner will maximize the expression 2π m R(F ) F where π m is the monopoly profit at any location x and the function R(F ) 5

16 gives the radius of the pump as a function of fixed costs. The solution to this problem is given by 2π m R (F ) =. If unit transport cost t > 0, and there is Bertrand pricing, then our last result (we should state it formally) says that a decline in the water supply will be accompanied by an increase in prices, and in the stage when c is increasing, a fall in well density, followed by an increase in well density when W falls faster than c (the boom), followed by a bust when there is a transition to monopoly pricing and some farms fail to get water. On the other hand, if t = 0, then there is no Bertrand regime. It is always local monopoly, and there will be no boom phase before the bust. Empirically, the interesting question is whether t = 0 and there is a capacity constraint that determines the size of the local monopoly, or whether t > 0 and we have Bertrand duopoly. In the latter case, the boom-and-bust phenomenon is predicted and in the former, only bust. Also, of course, we only get the price decreasing with distance only with Bertrand pricing. 3.2 Policies to address depletion Taxes on the pumping cost c or on the fixed cost F will reduce depletion. A tax on the pumping cost c will raise the price line vertically by the amount of the tax which in turn will lead to a reduction in water use at each location. However, beyond a certain level of tax, there will be monopoly pricing beginning at the well-head. A tax will reduce aggregate water use and by reducing profits, will lead to a fewer number of wells. Similarly, a tax on F will lead to reduced entry and higher pries and lower water use. 4 Concluding Remarks This paper addresses the issue of groundwater depletion, which is increasingly becoming a serious problem in the developing world. The dramatic 6

17 increase in groundwater extraction is responsible for the steady increase in grain production in the most populous countries in the world, especially India and China. At the same time, because of subsidies in the price of electricity and diesel, a large number of small farmers have continued to engage in farming by drilling tubewells, with significant impact on the water table in many regions. We develop a model that shows that at least under some conditions, depletion of the water stock may actually reduce competition and increase profits, leading to a boom in tubewell drilling and an inevitable bust when the water supply is near exhaustion. These conditions are likely to arise when energy costs are heavily subsidized, so that the fall in the water table is not matched by a concomitant rise in pumping costs. However, if energy costs rise as the water table subsides, it may lead to reduced profits and a slow exit from the industry. From a policy point of view, our framework, although highly stylized and simple, provides a range of predictions, that can be tested empirically. Low fixed costs in well drilling and low energy costs of pumping will lead to a higher well density. Higher costs of water transport and losses in conveyance will increase well density. However, in the limit when the cost of transportation goes to zero, we obtain a series of spatial monopolies which serve an area constrained by their pump capacity. In regions with significant recharge of water (e.g, from rainfall), there may be enough water each period to thwart the anti-competitive effects of scarcity - hence one may see a lower tubewell density. The reverse may hold for arid and water scarce regions. Future research can focus on a more explicit dynamic formulation of scarcity and its effects on pricing. We have restricted ourselves to developing a static model of competition between tubewells, which generates interesting insights on the effect of water depletion. However, we have abstained from explicit consideration of discount rates and other parameters that may impact dynamic extraction. 7