Trade and Contract Enforcement - CiteSeerX

Trade and Contract Enforcement James E. Anderson Boston College and NBER

Leslie Young Chinese University of Hong Kong

February 3, 2000

Abstract We develop a simple model of trade with imperfect contract enforcement. Courts force the execution of defaulted contracts with parametric probability. In political economic equilibrium, the enforcement of contracts between agents of sovereign nations must be at a level mutually agreed between nations. The enforcement probability may be low, with the reluctant nation being associated with low levels of economic development in terms of model parameters. Low enforcement equilibrium can be a trap, as a better equilibrium may exist discretely far away. The model is replete with ‘paradoxical’ comparative statics, such as the paradox of immiserizing bargaining power. We define a tariff equivalent of imperfect enforcement which is useful in empirical investigations of trade and security.

Preliminary version. Comments welcome. This paper began while Anderson visited the Chinese University of Hong Kong, whose support and hospitality is gratefully acknowledged. Earlier versions were presented to the NBER Summer Institute, August, 1999 and the Econometric Society, January 2000. Helpful comments from participants are gratefully acknowledged, especially Jonathan Eaton. Correspondence to: James E. Anderson Dept. of Economics Boston College Chestnut Hill, MA 02467 USA [email protected]

Contractual insecurity appears to reduce international trade. The opinions of traders expressed in surveys suggest substantial insecurity and casual empiricism suggests its impact is significant. Formal inference from trade flow models shows that implicit transactions costs are large (McCallum, 1995; Roberts and Tybout, 1997). But it is difficult to assess the significance of contractual insecurity as opposed to other transactions costs or to realistically approach policy recommendations in the absence of a theory of contract insecurity. Imperfect contract enforcement is modeled in this paper as a parametric probability that courts will force the execution of a contract in default. Standard complete contract models, in contrast, provide no role for incremental enforcement effort --- actions are either contractable or not due to natural elements. 1 The model is used to suggest an explanation of the wide international variation in enforceability (World Economic Forum, 1997). Low enforcement presents something of a puzzle, since better enforcement is presumed to be good. Thus the World Bank has recently emphasized institutional quality to its clients. The incidence of poor institutions and the variety of institutional quality are explained by political economy in our model: traders on one side of the market will be harmed by better enforcement under plausible conditions. International trade contracts must be enforced across sovereign boundaries, hence enforcement is only as strong as both nations (and presumptively their traders) agree.2 Low enforcement can be a ‘trap’ as a better equilibrium may exist discretely far away. The correlation of poor institutions with poor countries is explained in terms of the model’s parameters. 3 Our companion paper (Anderson and Young, 1999) examines the positive and normative consequences of middlemen such as trading monopolies who provide an alternative to formal contract enforcement. The ambiguous benefit of better enforcement is due to market failure. Agents without executable contracts match randomly on a spot market. On the excess side of the market, some agents will return home without matching. This 1 2

Logically, it is impossible to contract directly on the performance of the court. Countries’ commitments to procedural ‘rules of the game’ must be self-enforcing.

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Trade and Contract Enforcement

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trading inefficiency is directly increased by better enforcement (fewer scarce side traders are in the spot market ex post) but indirectly reduced (more scarce side traders enter into trade ex ante). Excess side traders’ interest balances these two forces. Scarce side traders, in contrast, always support better enforcement. Paradoxical possibilities abound in the model. Traders on the excess side of the market may be subject to the paradox of scarcity: the larger the potential gains from trade, the less is the volume of trade. Poorer countries may have more intense need for imports (being more different from the exporters) but with bad institutions they may trade less. Excess side traders may also have the ‘paradox of immiserizing bargaining power’, losing from a parametric increase in their bargaining power. Imperfect enforcement impedes trade similarly to tariffs, so it is useful to compare and connect contract enforcement with tariffs. We set out a tariff equivalent of enforcement and show it is decreasing in enforcement effort. Thus we provide a rationale for empirical work on resistance to trade based on poor institutions (Anderson and Marcouiller, 1999). A sequel paper (Anderson, 2000), relates contract enforcement and trade taxation. As with a tariff, it is not obvious that better enforcement is always in a nation’s interests (or its politically active agents’ interests). As with a tariff, this is due to the effect of enforcement on international and domestic prices. Section 1 sketches the model. Section 2 develops the basic setup and qualifications. Despite the simplicity of the model, we claim its qualitative features are robust to most reasonable extensions. Section 3 sets out the tariff equivalent of imperfect enforcement. Section 4 presents the comparative static analysis with respect to enforcement. Section 5 considers the political economic equilibrium of enforcement. Section 6 concludes with a summary and some discussion of extensions. The first appendix reports simulation results and the second appendix provides proofs of propositions and reports comparative static analysis with respect to the bargaining power and the mean outside options of buyers and sellers. 3

Poor enforcement may also be explained by its cost. Fixed costs of good enforcement may be too large for the gains from trade to cover. We do not reject this cost side explanation, but abstract

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Sketch of the Model The simplest model of imperfect contract enforcement and trade

minimally must contain four elements. First, trade must incur ex ante costs which are sunk at the time of exchange. This provides a role for contracts in overcoming the holdup problem --- ex post bargaining over price which ignores sunk costs. Second, there must be costly search in the absence of an enforced contract. This is because with costless search, contracts are unnecessary. Each agent’s outside option in bargaining with any partner is to keep searching and obtain the most favorable price available to any agent from his side of the market, which in equilibrium is the standard competitive price. We assume search takes the form of random matching.4 Third, there must be a random component in agents’ outside options. This is because rational agents must sign contracts which will be broken when a random shock makes default preferable for some agent. In our model the outside option is shifted by a random component of trade cost which affects the ex post cost of returning home to transact. Fourth, imperfect enforcement requires that contract default is sometimes not remedied by the court.5 In an environment with these four elements, agents facing broken contracts will have an incentive to search rather than renegotiate. This is because default by an agent from the other side of the market reveals a more favorable than average outside option. Our setup thus generally produces a spot market along with a futures market. In order to focus on trading inefficiency the model takes as given the market prices which provide the ex ante outside options to buyers and sellers. This simplification is justified, we argue below, because it cleanly reveals mechanisms which must be at work in larger models in which home and foreign market prices are endogenous.

from it to concentrate on the hitherto neglected but important benefit side. 4 If in contrast we rule out search entirely, all broken contracts result in renegotiation. This seems a less interesting and relevant alternative to us because it cannot yield an explanation of trade as contracts become useless. See our companion paper where anarchic trade is based on random matching. The no-search alternative retains some of the flavor of our model in that improvements in enforcement have ambiguous effects on the traders, hence better enforcement is not always supported. 5 Failure to enforce is based on the realization of another independent random variable.

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Traders in our model must incur trade costs prior to exchange.6 We think of the cost as primarily reflecting information acquisition and we assume traders are heterogeneous in their ability to become informed about foreign trade.7 The sunk cost also includes shipment and design costs common to all traders. In the event of a failure to trade, traders must incur reverse shipment and design costs to return to their domestic markets. Crucially, trade costs are subject to idiosyncratic shocks discovered only after committing to trade, hence the traders’ outside options are affected by the cost shock.8 Agents without enforced contracts match once from the set of opposite side traders and if matched they bargain over the exchange price in the spot market.9 Each agent’s outside option in the bargaining is his net price at home, the home market price net of the trade cost shock. The bargained price is a convex combination of the outside options. In the forward market, agents have an incentive to contract on price prior to committing their trade costs, since the bargained price cannot reflect sunk trade costs. We assume costless search in the forward market. With perfect contract enforcement, trade will be efficient: all agents whose gain from trade exceeds their expected trade cost will enter trade and all trade will be executed. Contracts are incomplete because it is impossible to contract on the performance of the enforcement system.10 Imperfect contract enforcement is 6

Sunk trade costs are well documented empirically (Roberts and Tybout, 1997; Bernard and Wagner, 1998). 7 A more realistic but inessentially complicated story is that trading firms hire a heterogeneous supply of labor, so that the rents accrue to high ability workers. 8 Assuming idiosyncratic shocks implies that risk diversification is possible, hence justifying risk neutrality. The cost shock plausibly includes common elements (such as exchange rate movements) but idiosyncratic elements are highly plausible. With risk neutral traders, only the reverse shipment aspect of the cost shock plays any role in the model, and here the need to quickly arrange redesign and shipment home seems likely to meet idiosyncratic cost. 9 This simplification of a more general costly search model is extremely convenient. The model developed here is related to the search literature founded on Diamond (1981, 1982). With search friction there is a ‘thick-market externality’ which resembles our transactions externality (but is opposite in sign). Holdup occurs naturally with search frictions (Acemoglu and Shimer, 1999). We initiate an imperfect contract in this setting. Our random matching setup mirrors the treatment by Casella and Rauch (1998) of informational inefficiency in a trade model. 10 In contrast, the complete contracts literature analyzes optimal (and implicitly perfectly enforceable) contracts subject to limits on what specific attributes of a transaction can be contracted. See McLaren (1998) for interesting work analyzing trade and contract limitations.

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parameterized as the probability that a contract in default will be enforced, taken to reflect the rules and procedures governing enforcement in the event of default. Agents choose to default when they receive favorable outside options. The probability of execution is equal to the probability of no default plus the product of the probability of default and the probability of enforcement, assuming the random processes of enforcement and trade costs are independent. Equilibrium entry into trade occurs when the marginal traders are able to cover expected trade cost by the expected gain from trade, the expected price less the home market price. The expected price facing traders prior to their decision to enter trade is a compound of the contract price and the spot market price, with the latter being a compound of the expected bargained price and the outside option for those who fail to match. Our model generally reaches an equilibrium where ex ante supply and demand are not equal. Since there are gains from trade, transactions are inefficient and costless improvements in enforcement would be socially efficient. When enforcement is endogenously chosen, enforcement decisions occur logically prior to contracting or trading. Forward-looking traders on each side of the market seek the institutions of contract enforcement which maximize their expected economic returns. In the context of international trade, sovereign nations will agree to the strongest enforcement (institutions) which the traders from both sides of the market will support. We show below that traders from the scarce side of the market always gain from improved enforcement. Their position is identical to traders facing a trade barrier. On the excess side of the market, however, traders’ interests in better enforcement are positively linked to the probability of a match, which in turn depends on the balance of two factors. First, a rise in the enforcement probability reduces the number of partners from the scarce side who will be seeking a match on the spot market, tending to lower the probability of a match. But second, better enforcement improves the price for traders from the scarce side of the market, induces more participation and hence tends to raise the probability of a match. The incomplete contract approach can be simultaneously criticized as hand-waving and defended as realistic (Tirole, 1999). We use it for tractability and discuss robustness of the

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Conditions are provided in which political economic equilibrium emerges in a low enforcement trap which condemns the agents to lower welfare than is possible. The model yields low enforcement equilibrium when arameter values associated with low stages of economic development are present. In contrast to ‘big push’ models of low income traps (e.g. Murphy, Shleifer and Vishny, 1989), the poverty trap arises in the absence of fixed costs or coordination failures.

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Equilibrium with Imperfect Contract Enforcement We develop a model of equilibrium exchange in contracts on the forward

market and in executed trade on the spot market. Each trader transacts one unit of the good in exchange for money, with each exchange accounting for an infinitesimal share of the (spot or forward) markets. Here and the next two sections, contract enforcement is exogenous. All agents are risk neutral and differ in the deterministic component of their trade cost, reflecting differential abilities to become informed about foreign markets. They commit to trade if the expected cost is weakly less than the gain expected from trade, the expected trade price less the expected home market net price (outside option). The expected price a trader receives is built up from the component prices which emerge from spot market bargaining and from contract execution at contract prices determined in the forward market. Decision makers solve backwards from the spot market to the default decision and its outcome to the forward market to the decision to enter trade. In equilibrium the subjective probability of matching used to determine expected payoffs is equal to the objective probability resulting from the numbers of agents who enter on each side of the market and either default or honor their contracts. Contracting in the forward market occurs first in time. In this market all traders can costlessly search, and all participants expect to at least break even on a contract. Next, traders sink their trade costs. Then they discover the random component of trade cost which affects their outside options. At this point the traders consider whether to default on the contract. Default is chosen only by excess side traders under our simplifying assumption that restricts the support of model to generalization below.

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the distribution of scarce side trade cost shocks. Default by the excess side occurs when the shock falls below a critical value µ* determined below. Default occurs with probability 1 − φ(µ* ); φ ∈(0,1). The partner of a defaulter appeals to the court11 , which enforces a portion of contracts in default θ; θ ∈[0,1]. The probability of execution of a contract is: (2.1)

β = φ + θ(1− φ) . Agents whose partners default know their partners have better outside

options than the average spot market trader, so if the court does not enforce the contract they are better off entering the spot market and making a random match. On the spot market, all matches result in Nash bargaining over the price, incorporating the random shocks to outside options. Finally exchange occurs: some with executed contracts at the contract price, some with matches at the bargained prices and some with unmatched traders going home to trade at their outside options. We set out the decision tree of the excess side traders in Figure 1. The scarce side traders face the same stages of decision, their tree is less complicated (they always sign contracts and hey never default because they never receive a sufficiently favorable shock). In Figure 1, branches from a point represent decisions by the trader while branches from a horizontal line represent externally imposed outcomes. Contracting traders can formally decide not to trade, but there is a penalty which makes this a dominated decision. The conditional expected prices are listed at the bottom, p C for an executed contract, p * for a bargained price on the spot market between randomly matched traders, and b or c for return to the domestic market without matching. The match probability is π .

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We might naturally think of the court as being in the defaulter’s country. Only a court in the defaulter’s country is able to seize assets of the defaulter to compel execution. But even a court in the plaintiff’s country is bound by international conventions in its treatment of foreign citizens who might happen to have assets which can be seized. We present a reduced form ‘court’, but note that a structural model of international commercial law must utimately explain conventions by mutual self-interest of the kind we feature.

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Figure 1. Decision Tree of Excess Side Traders

Forward Market Contract at pc

No contract

Trade Commitment Trade at cost t

Trade at cost t

No trade Realization of cost µ Default decision

Default if µ 0. The reduced form solution to the critical value of shock which determines

default is obtained by substituting from (2.5) into (2.3) and solving for: (2.6)

µ*b = v(µ *b ,θ ) ≡ − µ D (µ*b )

[1− φ(µ*b )](1− θ) > 0. φ(µ *b ) + θ[1− φ(µ*b )]

The existence of a critical value is identical to the issue of the existence of a forward market. In the Appendix we prove: Proposition 1: A unique forward market equilibrium exists. Note that the critical value and hence µ D and β are functions only of the distribution parameters and of the probability of enforcement. The contract price in excess demand equilibrium is given by p C = p b − µ D (πω + 1 − π )(1− β) / β . A similar series of steps yields the contract price in excess supply equilibrium as p s + µ D [π (1− ω) + 1 − π ](1− β) / β . Now we build up the expected price facing the scarce side. Scarce side traders on the spot market face a mix of random draws and defaulters. The probability of a scarce side trader meeting a defaulter in the spot market is equal to the probability that an excess side trader will match. For example, in excess demand equilibrium, the probability that a seller matches with a defaulting buyer is equal to the probability that a buyer in the spot market matches with a seller. This is because the number of sellers who search is equal to the number of successful defaulting buyers while the total number of buyers in the spot market

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is equal to the number of successful defaulters plus the number of buyers without contracts. The expected bargained prices conditional on default are thus p * + πωµ D for sellers on the scarce side p * − π (1− ω )µ D for buyers on the scarce side.

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The unconditional expected prices for scarce side traders are (2.7)

p˜ s = βpC + (1− β )[p* + πωµ D ] in excess demand equilibrium p˜ b = βpC + (1− β)[ p* + π (1− ω)µ D ] in excess supply equilibrium.

Here we adopt the notational convention that a tilde denotes a scarce side trader. Note that for the scarce side sellers, the contract price is superior to their expected spot price, hence they always sign: p C − (p * + πωµ D ) = (1− π )(b − p* ) − µ D [(πω + 1− π )(1− β )/ β + πω ] > 0. The unconditional expected prices are the basis of the entry decision. As functions of the match probability π and the exogenous variables they are:17 in excess demand equilibrium p b = πp* + (1− π )b (2.8)

p˜ s = βpb +(1- β )p* − µ D (1− β)(1− π ); in excess supply equilibrium p˜ b = βps + (1− β) p* + µ D (1− β)(1− π ) p s = πp* + (1− π )c.

On the excess side we substitute in the solution values for contract price and simplify. The intuition for the excess side prices is straightforward. On the scarce side, the expected price is a compound of the contract price and the expected scarce side price conditional on default, the latter including the expected effect of bargaining with a population containing some defaulters. The contract price, as we have seen, is equal to the uncovered expected spot price for the excess side plus the option value of default. That portion of the option value of default which reflects its value in bargaining cancels the less favorable price the scarce side trader must receive or pay. The remainder of the effect of the default value is the value of favorable costs on the outside option for those who do not match, ± µ D (1− β )(1− π ) depending on which side is in excess. Hence we have (2.8).

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Considerable (and harmless) simplification of the comparative static analysis below is achieved by imposing a discrete distribution for trade costs such that the solution values µ*b , µ D are invariant to θ .18 Concretely, we assume for what follows that trade costs have a two point distribution. Then β varies positively with θ and it is simpler to treat parametric changes in β as the enforcement policy variable. 2.2 The Equilibrium Match Probabiliy Now consider the determination of the equilibrium match probability. The active traders are those whose expected costs fall weakly short of their expected return from entering the trading zone. Traders differ in the size of their expected trade costs due to differing ability. Formally, we order the traders from lowest cost to highest to form the deterministic trade cost schedules t i (q), i = s,b . Let p denote a certainty equivalent price. Demand and supply schedules are defined as: s[p − c] = {q | t s (q) ≤ p − c} d[b − p] = {q | t b (q) ≤ b − p}. In this setup, the equilibrium will result in either excess demand with unmatched buyers or excess supply with unmatched sellers unless by chance s[p* − c] = d[b − p* ]. In rational expectations equilibrium the ex ante subjective probability of a match for the excess side and of a match with a defaulter for the scarce side must be equal to the same ex post objective probability. The objective probability of making a random match facing a buyer in excess demand equilibrium is given by the value of π which satisfies: (2.9)

π=

(1− β )s[ p˜ s − c] . d[b − pb ] − βs[ p˜ s − c]

The numerator on the right hand side is equal to the number of sellers who are active in the spot market, those whose contracts are broken. The denominator is 17

Excess side traders earn the uncovered spot price whether or not they hold contracts due to the indifference condition. 18 For the continuous case or for large discrete changes in the enforcement probability, the more elaborate structure of (9.1) is required to properly evaluate comparative static derivatives.

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equal to the number of buyers who are active in the spot market, the total number of buyers who committed to trade less the number whose contracts are executed. Proposition 2 If d[b − p* ] > s[ p* − c], a unique excess demand trading equilibrium exists in the spot market. Proof: see the Appendix. The equilibrium requires π = f (π ,⋅) where f (π ,⋅) denotes the right hand side of (2.9). The intuition of the proof is that we show is that fπ < 0 hence f (π ) must cross the 450 line in the space of {π , f (π )}. Equation (2.9) is solved for the equilibrium probability of a match (in the interval where π < 1), Π(β,ω,b,c) = {π | f (π ,β,ω,b,c) − π = 0}. Then the unique solution for Π is substituted back into the price equations to yield the certainty equivalent prices as functions of the parameters of the model. Figure 2 illustrates excess demand equilibrium in the spot market model. Recall that p b is endogenous as a function of OS/OD. Equilibrium is found where the ex post objective probability agrees with the ex ante subjective probability used in the construction of p b : π=

(1− β )OS / OD . 1− βOS / OD

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Figure 2. Spot Market Equilibrium price

b

s[p-c]

pb

s p˜

p*

d[b-p] c

O

S

D

quantity

The economics of this model are simple, but its qualitative properties should obtain in several more realistic scenarios. (1) We assume a homogeneous product market, but the model extends to a stylized differentiated products market where ex ante, all varieties are equally valuable. On both sides of the market, firms must incur costs of design to enter the international market. The random component of these design costs plausibly affects the outside option

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each side faces ex post in redesign to use the home market. This extension is empirically significant because Rauch and Trindade (1999) report that ethnic Chinese networks are more important trade stimulators for differentiated goods than for homogeneous goods. (2) Standard general equilibrium models of production and exchange add only inessential complexity. The outside options remain constant if goods production is Ricardian. Trader’s costs use real resources in a diminishing returns process. A significant complexity comes in with the political economy of non-Ricardian economies, as the interests of traders (rent due to ability or fixed factors) can differ from those of other domestic agents. Our conclusions remain in convex economies if traders only are politically organized. Nonconvex economies add only familiar elements. The results of the model may not be robust to alternative risk bearing arrangements and alternative contractual failure structures. The risk of contract failure may not be diversifiable; if it is not, the risk attitudes of traders must be built into their behavior from the beginning. Risk aversion should lower trade relative to the risk neutral assumption of this paper, but the effect of enforcement improvement on trade and traders’ interests remain to be investigated. Alternative models of contract imperfection may produce different results. Nevertheless, our main message seems likely to hold up: weak enforcement will be preferred by the scarce side nation’s traders, while the desirability of better enforcement for the excess side is dependent on the parameters.

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The Tariff Equivalent of Imperfect Enforcement

Since imperfect enforcement restricts trade volume, it has a tariff equivalent in volume terms. Formalizing this idea is useful for empirical work. On the scarce side of the market imperfect enforcement is equivalent to a tariff in all respects; a useful framework for a future WTO agenda which may include negotiations designed to improve contractual efficiency. The actual trade volume exchanged is that on the scarce side of the market. In the case of excess demand equilibrium the volume exchanged is s[βpb + (1− β )p* ]. The ‘tariff equivalent’ of the imperfect enforcement is obtained

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by first solving the reduced form probability of a match γ ( p*, β,b,c) and then defining the hypothetical buyers’ price implied by the actual volume: pt ( p* , β,b,c) = {p:d[b − p] = s[βpb + (1− β)p * − µ D (βπω + 1− π ) − c]} where p b ( p* , β,b,c) = Π( p* ,β, b,c)p* + [1−Π(⋅)]b. Then the ad valorem tariff equivalent of imperfect enforcement is (3.1)

τ = T( p* ,β ,b,c) = −1+ p t ( p* ,β, b,c) / p˜ S ( p* ,β ,b,c) .

The analysis is very similar to the tariff equivalent of a quota, and may be illustrated by erecting a vertical line in Figure 2 at the volume . The ad valorem tariff equivalent should intuitively fall with rises in β, better enforcement lowers the tariff equivalent. Formal proof that this is so is deferred to Section 4, where the effect of β on buyers’ and sellers’ prices is also analyzed. For excess supply equilibrium, a very similar structure yields a tariff equivalent; the details are left to the reader. The tariff equivalent setup can be used in empirical work such as that reported by Anderson and Marcouiller (1999). The tariff equivalent of imperfect enforcement defined here is volume equivalent and welfare equivalent for the scarce side of the market. The equilibrium of the model requires that the ex ante subjective probability of a match is equal to the ex post objective probability of a match. On other dimensions is not equivalent to a tariff and in particular it yields no revenue. Thus, while enforcement and tariff policies are related, they are not close substitutes.

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Comparative Statics of Enforcement The key building block to understanding the interests of the agents in

enforcement is the effect of the enforcement parameter on the prices and the volume of trade. The analysis begins with the effect of the enforcement level β on the equilibrium match probability. Let a function subscripted by a variable label denote partial differentiation of the function with respect to the variable. Differentiating the RHS of (2.9) with respect to β using (9.1) ∂Π / ∂β = fβ / (1− fπ ). Since 1 − fπ > 0 , the probability response to enforcement ∂Π / ∂β is signed by:

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 ε s ∂˜p s  π f β =  −(1− π ) + (1− β + βπ) s where ˜p ∂β  1 − β  ∂p˜ s = p b − p* + (1− π )µ D = (1− π )(b − p * + µ D ) > 0 ∂β ε s ≡ p˜ ss' / s.

The negative first term under the bracket of (4.1) represents the direct effect of a rise in β on the probability of a match in the ex post market: fewer sellers will be available as more contracts are binding. But the positive second term implies that indirectly, the rise in β raises the ratio of sellers to buyers by raising the sellers’ expected price, thereby increasing the probability of an ex post match. With a sufficiently large elasticity of supply, the indirect effect dominates and π is increasing in β . The traders’ interests depend on the certainty equivalent prices they face. As for the buyers in excess demand equilibrium, differentiating (2.8): (4.2)

dp b ∂Π = −(b − p*) . dβ ∂β

The buyers’ certainty equivalent price falls and the buyers gain whenever the probability of a match is increasing in enforcement, which requires a sufficiently large elasticity of supply. For the sellers, their interests vary with the effect of enforcement on their expected price, obtained by differentiating (2.8), d˜p s ∂˜p s ∂˜p s ∂Π = + where dβ ∂β ∂π ∂β (4.3)

∂p˜ s = (1− π )(b + µ D − p* ) > 0 ∂β ∂p˜ s = − β(b − p* ) + (1− β )µ D < 0. ∂π

The sellers’ expected price can be shown to rise with β ; see the Appendix. Thus traders on the scarce side of the market always prefer better enforcement while the traders on the excess side of the market may or may not prefer better enforcement. The excess side traders gain as better enforcement mitigates the negative externality they inflict on each other by entering a

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crowded market. Both sides always gain from improved enforcement if the elasticity of supply is sufficiently great. Similar manipulations for excess supply equilibrium yield the same qualitative conclusions: the probability of a match is increasing in enforcement effort and the welfare of traders on both sides of the market is improving in enforcement effort provided the elasticity of demand is sufficiently large. With a small elasticity, traders on the scarce buyers side support a rise in enforcement while traders on the excess sellers side oppose it. Summarizing results: Proposition 3: (a)with imperfect contract enforcement, for sufficiently large elasticities of supply and demand, all traders prefer better enforcement from any given level of effort. (b) Traders on the scarce side of the market always prefer better enforcement. (c) With elasticities sufficiently small, traders on the excess side of the market oppose better enforcement. Proof: See the Appendix. The intuition of Proposition 3 is simple: there is a negative transactions externality on the excess side of the market due to marginal traders entering a crowded market not internalizing the negative effect they have on the probability of a deal being reached by other excess side traders. The direct effect of better enforcement worsens that externality because at a constant volume of trade from the scarce side it removes potential scarce side partners from the spot market. Indirectly, better enforcement increases the number of traders from the scarce side. Better enforcement is favored on the excess side if the supply response from the scarce side is sufficiently large to overcome the negative direct effect. The tariff equivalent of imperfect enforcement defined in Section 2 is decreasing in β , as intuitively it should be. This follows for excess demand equilibrium from the sign of (9.3) after differentiating (3.1) with respect to the enforcement probability:

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1 ∂τ ∂ log p t ∂log p˜ s = − 1 + τ ∂β ∂β ∂β  εs  ∂log p˜ s = −  b + 1 < 0.  ε˜  ∂β The elasticity of demand is marked with a ~ to denote evaluation at the quantity associated with the quantity supplied. It is useful to reflect on the properties of the model as perfect enforcement approaches. As β → 1, the probability of an ex post match falls to zero. Then all traders will avoid the market unless they are matched in a contract ex ante. This contract equilibrium will be the usual competitive equilibrium with the price determined by d[pC ] = s[pC ].

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The Political Economic Equilibrium of Enforcement

The key element distinguishing international trade from domestic trade is that contract enforcement must be in the interest of traders from both countries simultaneously, since sovereign nations cannot be compelled to enforce a contract not in their citizens’ interest. We assume that the enforcement choice is made prior to any contracts being signed or any trade taking place. Side payments are assumed to be impossible. Since the scarce side of the market always favors better enforcement while the excess side has an ambiguous interest, a political economic equilibrium occurs where the interest of traders on the excess side is locally optimized. Multiple political economic equilibria may occur. Section 5.1 develops the formal analysis and Section 5.2 draws out the implications, relating low enforcement to underlying parameters and to countries’ stage of economic development. 5.1 Optimal Enforcement Consider first excess demand equilibrium. The interior optimum condition for buyers requires ∂Π / ∂β = 0 at a point where ∂ 2 Π / ∂β 2 > 0. This requires f β = 0. Evaluation of f β to characterize the optimum is in general a complex matter. There may or may not exist an interior optimum and there may be

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several optima. To illustrate the interesting possibility of multiple interior optima, consider a special case. Suppose the traders from the supply side have a step cost function, with any volume up to s 0 available with cost t 0 while with cost t 1 > t 0 , any volume is available. For regions where ε S is infinite, the buyers’ certainty equivalent price is decreasing in β while at s 0 where the elasticity is equal to zero, the certainty equivalent price is increasing in β. In these circumstances, there are two political economic interior equilibria, at s 0 with the associated interior value of β and at perfect enforcement. By putting more steps into the step function, the number of equilibria rises, with multiple interior equilibria. The special case of the step function for trade costs is more general than it might first appear. Infrastructure capacities lie behind the trade costs borne by individual traders. It is reasonable to assume that at larger volumes, eventually it is necessary to shift to a larger capacity, with costs to be carried by the traders. (Of course there can be regions of decreasing cost as well --- these are suppressed here for simplicity.) The motivation in terms of individual heterogeneity is simply one convenient case. Even interpreted as a distribution of heterogeneous costs, the distribution of trade costs might frequently assume the ‘lazy-S’ cumulative density shape which characterizes the normal distribution as well as a number of other commonly used distributions. In the lower and upper reaches of the lazy-S, elasticities of supply are very high, which mimics the ‘treads’ of the step function. Further insight on the effect of trade cost elasticities qt' (q) / t on the sign of ∂Π / ∂β is provided by our simulation results reported in the Appendix. They show that with constant elasticity trade cost functions, buyers’ welfare is monotone decreasing in enforcement with unitary trade cost elasticity on both sides of the market and monotone increasing in enforcement with supply trade cost elasticity equal to 3 and demand trade cost elasticity equal to 1. Now consider the case of excess supply equilibrium. The interior optimum for sellers requires dP / dβ = 0 where d 2 P / dβ 2 < 0. From (9.6) this requires

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23

gβ = 0 .

Again the step function case yields multiple optima. Trading costs which step upward on the foreigners’ demand side will create regions of infinite and zero demand elasticity. Supply price is increasing in β in the first region and decreasing in β in the second region. Optima are found at the volumes associated with the steps and at perfect enforcement. 5.2 Implications: Enforcement and Development Casual empiricism suggests that low enforcement is associated with low levels of economic development. If enforcement levels are endogenously determined reflecting the interests of traders, then low development is associated with traders who oppose improvements in enforcement.19 We now attempt to associate the determinants of a preference for low security with low levels of economic development. A necessary condition for opposition to better enforcement is belonging to the excess side of the market. The excess state is associated with high bargaining power, with relatively low trade costs and with several possible configurations of reservation prices and elasticities of demand and supply. A sufficient condition for opposition on the excess side is a low elasticity of supply of traders from the scarce side of the market. We develop the sources of being on the excess side in turn. Consider first systematic asymmetries in bargaining power. Thus suppose that low development is associated with low enforcement levels due to high bargaining power. At first blush, the idea may seem implausible because clearly the bargaining power of developed countries exceeds that of developing countries taking each as nations. However, the bargaining power relevant in the model is that between typical agents from developing vs. developed countries. Here it may be plausible to assume that developing country agents have more power than their counterparts from developed countries. Casual empiricism 19

Notice that the low income trap we analyze is based on trade costs, but does not involve the fixed costs that are the focus of the big push literature. However, enforcement may well require

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24

suggests the validity of the assumption, as does an evolutionary argument. Because the developed country’s domestic institutions are powerful, its people are able to prosper despite their weakness in individual bargaining situations. In contrast, agents from underdeveloped countries must be powerful in order to prosper because their domestic institutions are weak; agents in such countries must defend themselves in many domestic situations where law enforcement is weak. In the next subsection we confirm that in our model higher bargaining power improves payoffs under conditions similar to those which suffice for better enforcement to worsen payoffs. Moreover, even without systematic differences in bargaining power inherent in the populations; behind the scenes, selection mechanisms at different stages of development may plausibly determine the asymmetric bargaining powers of those agents who enter international trade.20 The state of excess demand requires d[b − p* ] − s[p* − c] = d[(1− ω )(b − c)] − s[ω(b − c)] > 0 . The condition on the left hand side is decreasing in ω since − d' −s' < 0. This means greater bargaining power for the buyers increases the likelihood of meeting the excess demand condition. Similar results hold for the excess supply condition, so high bargaining power generally makes it more likely that the traders having it will be on the excess side of the market. 21 setup costs which must be jointly paid. This big push type coordination problem reinforces the low income trap. 20 Even if both countries have the same underlying distribution of bargaining power among agents, commercial survival in domestic interactions in a country with weak institutions is likely to select tough bargainers while commercial survival in a country with strong institutions may select weak bargainers. Reinforcing this selection mechanism, in a long run social evolutionary equilibrium the country with weak institutions is likely to have a population distribution of bargaining strengths which is systematically tougher. Bargaining strength asymmetries are linked to time preference asymmetries in Rubinstein’s (1982) alternating offers model of bargaining, and casual empiricism suggests far more impatience among agents from developed countries than among agents from developing countries. (The causal flow from the strength of institutions to the strength of bargaining is a rich topic for future research, going far beyond the scope of this paper. The equilibrium links between formal and informal sectors modeled by Marcouiller and Young, 1995, are a useful backdrop for this future research.) 21

A more complex question is to sign the effect of the bargaining power on the optimal amount of enforcement for the excess side of the market. The enforcement optimal enforcement levels are implied by (4.1) or (4.2). The implicit relationship between the bargaining power and the optimal enforcement level is too complex for

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The model also indicates that relatively low trade costs may be associated with belonging to the excess side of the market. If we assume that the trade costs are labor intensive, then low wage countries would have low trade costs compared to high wage countries. This may be plausible but awaits more development of a theory and empirics of trade costs. It may be plausible to believe that for most products, the sunk trade costs are greater for sellers than for buyers. Then markets with poor enforcement would be tend to be in excess demand. (Our treatment has given excess demand priority for this reason.) The worst contractual insecurity problems would be found with imports into developing countries, confirmed by casual empiricism. Turning to the reservation prices, the excess demand condition is increasing in b − c if (1− ω )d' −ωs' > 0 . For equal bargaining power, when demand is more elastic than supply, rises in the buyers’ reservation price or decreases in the sellers’ reservation price will make excess demand more likely. With demand less elastic than supply, rises in b − c make excess demand less likely. Finally, a sufficient condition for traders to oppose better enforcement is belonging to the excess side of the market combined with an elasticity of trade cost from the scarce side which is sufficiently low. Thin markets are empirically associated with low levels of development and may well be associated with low elasticities of supply of traders from the developed countries. 5.3 Development and Bargaining Power Enforcement is a policy variable while bargaining power is a parameter. If we think of bargaining power as reflecting social norms, then we may speculate that over time the norms will evolve toward locally efficient values. In the context of our model, we can then examine the effect on agents’ payoffs of changes in their bargaining power. The direct effect is trivial; agents always gain from improved bargaining power for given match probability. However, the match probability changes with bargaining power, and this change affects even

analytic methods. Simulations not reported in the Appendix show that the optimal enforcement chosen by the excess side of the market is generally increasing in the bargaining power of that side of the market.

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the scarce side of the market because it affects the contract price. The “paradox of immiserizing bargaining power” is possible, whereby agents lose from increased bargaining power. When the paradox obtains, behavior may evolve slowly toward less costly norms. The second appendix shows that a rise in the bargaining power of sellers affects the payoffs of buyers and sellers in excess demand equilibrium by: dp b ∂Π = π (b − c) − (b − p* ) dω ∂ω dp˜ s = (1− β + βπ )(b − c) dω −[β (b − p* ) − (1− β)µ D ] The appendix shows that ∂Π > 0. ∂ω

∂Π . ∂ω

For both buyers’ and sellers’ prices, the positive direct effect of a rise in the seller’s bargaining power22 is offset by the indirect effect of bargaining power which reduces both prices by raising the match probability. The appendix shows that for a high elasticity of supply, excess side buyers gain from a rise in the scarce side sellers’ bargaining power. In contrast, scarce side traders must gain from a rise in their bargaining power regardless of elasticities. Similar results follow for the case of excess supply. The conditions when excess side agents would prefer lower bargaining power are similar to the conditions when excess side agents would prefer better enforcement. Low enforcement trap equilibria may thus be associated with globally ‘bad’ norms which fail to evolve. 5.4 The Paradox of Scarcity Low levels of economic development may be associated with high prices of imports even in the absence of formal barriers to trade, high b in terms of our model. Behind this are the deep parameters of technology and endowments which the theory of international trade emphasizes along with the plausible idea that b diverges the more the country differs from the developed countries. 22

For sellers, the direct effect is ambiguous but ordinarily should be positive.

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With imperfect contract enforcement in excess demand equilibrium, a high ‘autarky’ price b can imply less trade --- the paradox of scarcity. When the paradox operates, the greater the gains from trade the smaller is the volume of trade. The intuition from standard models is that the higher is b , the larger is the volume of trade, all else equal. In terms of the present model, too, a rise in the buyers’ mean outside option will intuitively raise the bargained price and hence the constraining supply of exports. With imperfect enforcement, however, the second appendix shows that with high elasticities of demand and supply, both buyers’ and sellers’ prices may paradoxically fall. The sellers’ expected price and thus the volume of trade falls when the elasticity of demand is sufficiently large. Comparative statics with respect to sellers’ outside options also reveal paradoxical possibilities for both sides of the market. Here, it is logical to think of low levels of development associated with excess supply equilibria for exports. Then the paradox of scarcity is that the lower the outside option c , the less is trade, all else equal.

6

Conclusion We have developed a simple model of imperfect contract enforcement.

The model implies the simultaneous existence of forward and spot markets for homogeneous products. We use the model to analyze the enforcement policies of countries on either side of the market. We offer conditions under which weak enforcement is the political economic equilibrium outcome. Paradoxical comparative static results abound. Our companion paper (Anderson and Young, 1999) considers middlemen institutions which perform some of the same role as imperfect enforcement but which introduce their own distortions. A sequel paper (Anderson, 2000) initiates consideration of trade policy with imperfect enforcement. The political economy in this paper is very simple: only the traders on each side earn rents, and enforcement is set at the strongest level which is supported by both groups. Two more complex political economy structures suggest themselves, one international and one domestic. On the domestic front, a general equilibrium analysis may be useful. Within a single country, say Home,

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the competing enforcement interests of Home importers and Home exporters (and Home producers of import-competing and export goods) can be traded off in a fully articulated interest group competition model of the type articulated by Grossman and Helpman (1995). On the international front, it may be useful to consider negotiations over trade policy and enforcement both sequentially and simultaneously. Do commitments on enforcement make liberal trade negotiations easier or more difficult? Another worthwhile extension is to incorporate possibilities of heterogeneous agents with respect to enforcement technologies. Ethnic ties have been emphasized as overcoming informational and enforcement imperfections in empirical work by Rauch and Trindade (1999). It should be interesting to explore the effects of official enforcement and trade policy in a model of private enforcement.

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References

Acemoglu, Daron and Robert Shimer (1999), “Holdup and Efficiency with Search Friction”, International Economic Review, 40, 827-849. Anderson, James E. and Douglas Marcouiller (1999), “Trade, Insecurity and Home Bias: An Empirical Investigation”, NBER Working Paper No. 7000. Anderson, James E. and Leslie Young (1999), “Trade and the Rule of Law”. Anderson, James E. (2000), “Tariffs and Contract Enforcement”. Bernard, Andrew B. and Joachim Wagner (1998), “Export Entry and Exit by German Firms”, NBER Working Paper No. 6538. Casella, Alessandra and James Rauch (1998), “Overcoming Informational Barriers to International Resource Allocation: Prices and Group Ties”, NBER Working Paper No. 6628. Diamond, Peter A. (1981), “Mobility Costs, Frictional Unemployment and Efficiency”, Journal of Political Economy, 89, 798-812. Diamond, Peter A. (1982), “Aggregate Demand Management in Search Equilibrium”, Journal of Political Economy, 90, 881-94. Grossman, Gene and Elhanan Helpman (1995), “Protection for Sale”, American Economic Review, 84, 833-850. Marcouiller, Douglas and Leslie Young (1995), “The Black Hole of Graft: the Predatory State and the Informal Economy”, American Economic Review 85, 630-646. McCallum, John, (1995), “National Borders Matter: Canada-US Regional Trade Patterns”, American Economic Review, 85, 615-623. McLaren, John (1999), “Supplier Relations and the Market Context: A Theory of Handshakes”, Journal of International Economics, 48, 121-38. Kevin Murphy, Andrei Shleifer, and Robert Vishny (1989), “Industrialization and the Big Push,” Journal of Political Economy 97, 1003-1026. Rauch, James R. and Vitor Trindade (1999), “Ethnic Chinese Networks in International Trade”, NBER Working Paper No. 7189. Roberts, Mark and James Tybout (1997), “The Decision to Export in Colombia: An Empirical Model of Entry with Sunk Costs”, American Economic Review, 87, 545-564.

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Rubinstein, Ariel (1982), “Perfect Equilibrium in a Bargaining Model”, Econometrica, 50, 97-109. Tirole, Jean (1999), “Incomplete Contracts: Where Do We Stand?”, Econometrica, 67, 741-82. World Economic Forum (1997), The Global Competitiveness Report 1997 (Geneva: World Economic Forum).

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Appendix 1: Simulation of Changes in Enforcement

The simulation analysis is based on a model in which the distributions of trade costs result in constant trade cost elasticities. This results in demand and supply functions which have constant elasticity with respect to the gain from trade: d[b − pb ] = δ(b − pb )ε s[ps − c] = σ( ps − c)η . Welfare for buyers is equal to the buyers’ surplus, equal to the rent to low trade cost individuals in our model. δ 1+ε S= (b − p b ) ε +1 . 1+ε Linearity is implied by the cases of ε = 1 and η = 1. We report results only for the case of excess demand. The benchmark parameters are: b = 2 c =1 δ = 10 σ = 1 ω = 0.5 µ D = −0.1 ε =1 η = 1 and η = 3. These parameter values guarantee excess demand in equilibrium. Using these parameter values and the definitions of the certainty equivalent prices and of the match probability in the excess demand case, we can simulate to solve for the equilibrium match probability and the buyers’ welfare. A feature of the location parameters chosen is that the buyer’s have a lot of market power: in the linear case at the efficient market equilibrium the elasticity of demand is equal to 21 while the seller’s supply elasticity is equal to 2.1. Thus all the imperfectly secure equilibria we show below are superior to the secure equilibrium for the buyer. The results of the charts illustrate the behavior of the match probability and of the buyer’s welfare as the enforcement probability changes. We report the buyer’s welfare as the percentage gain over the buyer’s welfare in the efficient equilibrium defined by market clearance. For the linear supply case η = 1, the results of the charts below show that the match probability is decreasing in the enforcement effort and thus welfare of buyers in excess demand equilibrium is decreasing in enforcement. For η = 3 , welfare and the match probability are increasing in the enforcement probability up to an interior maximum reached at 64.2%. For the constant elasticity case, these results show that the effect of improved enforcement reverses at a fairly low value of the elasticity of trade costs.

Welfare, Match Probability and Enforcement Trade cost elasticity of supply=3 0.8000 0.7000 0.6000 0.5000 welfare

0.4000 0.3000 0.2000

match probability

0.1000 0.0000 0.0

0.2

0.4

0.6

0.8

1.0

Enforcement probability

Welfare, Match Probability and Enforcement linear trade costs case 2.5000

2.0000

1.5000

welfare

1.0000

0.5000 match probability 0.0000 0.0

0.2

0.4

0.6

Enforcement Probability

0.8

1.0

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Appendix 2: Properties of Equilibrium

9.1 Proofs of Propositions Proof of Proposition 1: We may graph the fixed point problem (2.6) in the space of µ*b , v(µ*b ,θ) . We note that  (1− φ)(1− θ)  D φ' v µ (µ b ,θ ) = − µ ' −µ D − µ L while v(µ H ,θ ) = 0 < µ H . Then the graph of v must cross the 45 degree line somewhere on the interval [µ L , µ H ]. || We have the intuitive comparative statics: dµ *b / dθ = vθ / (1− v µ ) < 0 (9.1)

dµ D / dθ = (φ' /φ)(µ*b − µ D )dµ*b / dθ < 0 dβ / dθ = (1− φ) + (1− θ )φ' dµ *b / dθ .

Proposition 2 states that spot market equilibrium exists and is unique. From the text we have the objective probability of a match in excess demand equilibrium as: (1− β )s[ p˜ s − c] f (π ,β ,ω,b,c) = where d[b − pb ] − βs[ p˜ s − c] p˜ s = βpb + (1− β)p * − µ D (1− β )(1− π ) pb = πp* + (1− π )b p* = ωb + (1− ω)c. Equilibrium is found where f (π ) − π = 0; π ∈[0,1]. We show that f (π ) crosses the 45 degree line on the unit interval. We also show that fπ < 0 , so the crossing point is unique. Let ε i for i = s,b refer to the elasticity of supply for sellers and of demand for buyers where the latter is defined to be positive.23 Then:

23

That is, level and

ε s ≡ ps' /s and ε b = pd'/ d where in each case the price is set at the appropriate s' and d' are positive.

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Trade and Contract Enforcement fπ =

(9.2)

33

π ∂log(s / d) (1− β + βπ ) where 1−β ∂π

∂ log(s / d)  ε s ∂˜p s ε b ∂pb  = s + b  ∂π  p˜ ∂π p ∂π  ∂p˜ s = −β(b − p* ) + µ D (1− β ) < 0 ∂π ∂pb = −(b − p* ) < 0, hence ∂π fπ < 0. Proof of Proposition 2 (Existence and Uniqueness of Spot Market Equilibrium): Excess demand equilibrium guarantees that d[b − p* ] > s[ p* − c] for β < 1. This implies that f (1,⋅) =

(1− β )s[ p* − c]/ d[b − p* ] < 1. We know 1 − βs[p* − c]/ d[b − p * ]

s[⋅]/ d[⋅] is falling in π by (9.2). As s[⋅]/ d[⋅] reaches 1, f (π ,⋅) = 1. Since f (π ,⋅) is continuous, f (π ) − π = 0 has a solution in the unit interval. Since fπ < 0 , the solution is unique. Similar steps ensure the existence of a unique excess supply trading equilibrium. Whether the unique equilibrium is in excess supply or excess demand is determined by whether s[p* − c] > ( 0. ∂β  1− β p b ∂π  ∂π 1− β

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The buyers and sellers price responses in excess supply equilibrium are obtained as follows. The equilibrium is characterized by: p s = πp* + (1− π )c p˜ b = βpC + (1− β)[ p* − π (1− ω )µ D ] p C = p s + µ D [π (1− ω) + 1− π ](1− β )/ β (9.4)

µ D = E[µ s | µ s < µ * ] g(π ,⋅) ≡

(1− β )d[b − p˜ b ] s[p s − c] − βd[b − ˜pb ]

P(β ,ω,b,c,µ D ) = [π | π = g(π ,⋅)]. The partial derivatives of the prices with respect to β and π are:

(9.5)

∂p˜ b = β( p* − c) − µ D (1− β) > 0 ∂π ∂p˜ b = −(1− π )[ p* − (c − µ D )] < 0 ∂β ∂p s = p* − c > 0. ∂π

The match probability derivative corresponding to (4.1) is: g ∂P = β where ∂β 1 − gπ (9.6)

gπ =

π ∂log(d / s) (1− β + βπ ) 1−β ∂π

gβ =

π  ∂log(d / s)  −(1− π ) + (1− β + βπ )   1−β  ∂β 

where ∂ log(d / s)  ε b ∂˜pb ε s ∂p s  = − b + s  0. ∂β  ˜p ∂β  Expected prices respond to changes in enforcement according to: (9.7)

dp s ∂P = ( p* − c) and dβ ∂β

34

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(9.8)

d˜pb ∂˜p b ∂˜p b dπ = + dβ ∂β ∂π dβ d˜p b ∂˜p b  π ε s ∂ps  π(1− π ) ∂˜pb (1− gπ ) = 1 + (1− β + βπ ) − < 0. dβ ∂β  1− β p s ∂π  1− β ∂π

35

.

Using (9.5) and (9.6) in (9.8), the buyers’ price is falling in β regardless of the magnitude of the derivative of π with respect to β . From (9.7), for a sufficiently large demand elasticity, the sellers’ price is rising in β . Otherwise, it is falling in β .|| 9.2 More Comparative Statics We use the model to consider the effect of changes in the bargaining power and outside options on welfare. We first must analyze their effect on the match probability. We show that the direct effect of the parameter changes on buyer’s and seller’s expected price is opposite in sign to the indirect effect through the match probability. Paradoxical results abound, such as the “paradox of immiserizing bargaining power”: either side of the market may be made worse off by an increase in its bargaining power. We highlight the most important possibilities. 9.2.1 Bargaining Power, Outside Options and the Match Probability The bargaining power and trade cost distribution parameters of the model have comparative static effects on π which we explore in this section. The analysis is run by the comparative static derivative of π in (2.9) with respect to ω , the sellers’ bargaining power, and with respect to the outside options b and c . As for increases in the bargaining power of sellers in excess demand equilibrium, ∂Π / ∂ω is signed by24 ∂Π 1− β + βπ ∂log(s / d) =π ∂ω (1− fπ )(1− β ) ∂ω (9.9) . ∂ log(s / d)  ε s ∂p˜ s ε b ∂pb  = s + b  > 0. ∂ω  ˜p ∂ω p ∂ω  Here, 1 − fπ is positive by (9.2). We have ∂p˜ s / ∂ω = (1− β + βπ)(b − c) > 0 and ∂p b / ∂ω = π (b − c) > 0 , hence the match probability is increasing in the bargaining power of sellers in excess demand. A symmetric treatment of the excess supply case shows that increases in the bargaining power of buyers will raise the match probability. Thus increases in the bargaining power of the scarce side of the market raise the match probability. The outside option parameters affect the match probability in excess demand equilibrium as follows. The rate of change in the match probability with respect to the buyers’ outside option is:

24

Note that

∂p˜ s / ∂ω = (1− β + βω)(b − c) − µ D βπ > 0 and ∂p b / ∂ω = π (b − c) > 0 .

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(9.10)

Trade and Contract Enforcement ∂Π 1− β + βπ ∂log(s / d) =π ∂b (1− fπ )(1− β ) ∂b

∂ log(s / d)  ε s εb  =  s [(1− β)ω + β(πω + 1− π )] + b [πω + 1 − π ] > 0. ∂b p  ˜p 

36

.

The effect of an increase in the reservation price of sellers on the probability of a match is given by: ∂Π 1 − β + βπ  ε s εb  (9.11) =π [(1− β + βπ)(1− ω)] +  s b [π (1− ω )] > 0 . ∂c (1− fπ )(1− β)  ˜p p  These effects come because rises in either outside option raise both buyers’ price and sellers’ price at constant match probability, thereby raising the ratio of supply to demand. For excess supply equilibrium, the value of the probability of a match satisfies: (1− β )d[b − p˜ b ]/ s[ pS ] π= ≡ g(π, β,ω, b,c) where 1 − βd[⋅]/ s[⋅] p s = πp* + (1− π )c p˜ b = βpC + (1− β)p * − (1− β )π (1− ω)µ D p C = p s + µ D [π (1− ω) + 1− π ]/ β . Drawing on the parallels with the excess demand case, the effects of changes in the reservation prices on the match probability in the excess supply case are given by their effect on the magnitude of excess supply at constant match probability. This is signed by: ∂P 1− β + βπ ∂log(d / s) =π ∂b (1− gπ )(1− β ) ∂b = −π (9.12)

1− β + βπ {[βπω + 1 − β]ε b / p˜ b + πωε s / p s} < 0 (1− gπ )(1− β)

∂P 1− β + βπ ∂log(d / s) =π ∂c (1− gπ )(1− β ) ∂c

1− β + βπ {βπ (1− ω )ε b / p˜ b + (1− πω)ε s / ps } < 0. (1− gπ )(1− β) Thus increases in outside options from either side of the market reduce the match probability in excess supply equilibrium. The intuition is that by increasing price (at constant match probability) the rise in outside option lowers the ratio of demand to supply. = −π

9.2.2 Welfare, Bargaining Power and Outside Options Bargaining power and outside options affect buyer’s and seller’s welfare directly by their impact on the bargained price and indirectly through their impact on the match probability and thus the certainty equivalent buyer’s and seller’s prices. For the case of excess demand we have

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dp b ∂Π = π (b − c) − (b − p* ) dω ∂ω dp˜ s (9.13) = (1− β + βπ )(b − c) dω ∂Π . ∂ω For both buyers’ and sellers’ prices, the positive direct effect of a rise in the seller’s bargaining power is offset by the indirect effect of bargaining power which reduces both prices by raising the match probability. For high elasticities, buyers can gain and sellers can lose from a rise in the sellers’ bargaining power. This is the “paradox of immiserizing bargaining power”. Similar results follow for the case of excess supply. To uncover more insight into the paradox, consider the case of excess demand with an increase in the seller’s bargaining power. Substituting from (9.9) into (9.13) and multiplying through by D = (1− β)(1− fπ ) , we obtain the price response to the seller’s bargaining power as: dp b ∂p b D= (1− β) dω ∂ω ε s  ∂p b ∂˜p s ∂˜p s ∂pb  +π(1− β + βπ) s  −  where p˜  ∂π ∂ω ∂π ∂ω  +[(1− β )µ D − β (b − p* )]

∂pb = π (b-c) > 0 ∂ω ∂p˜ s (9.14) = −β(b − p* ) + (1− β )µ D < 0 ∂π ∂pb = −(b − p* ) < 0 ∂π ∂p˜ s = (1− β + βπ)(b − c) > 0 ∂ω D = (1− β )(1− fπ ) > 0. The bracket term in the second line of the expression for dp b / dω is negative, which may be seen by substituting the price derivatives in and simplifying. The indirect effect on p b of a rise in the bargaining power of sellers is thus negative. The direct effect is positive. When the elasticity of supply is sufficiently large, the indirect effect dominates and buyers in excess demand equilibrium benefit from a fall in their bargaining power. As to the seller’s welfare, dp˜ s ∂p˜ s D = (1− β) dω ∂ω (9.15) ε b  ∂pb ∂˜p s ∂˜p s ∂pb  + + π (1− β + βπ) b  − + . p  ∂π ∂ω ∂π ∂ω  The bracket term in the second line can be shown to be positive using (9.14). Thus scarce side sellers always gain from a rise in their bargaining power.

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For changes in outside options in the case of excess demand we have: dp b ∂Π = [1− π + πω] − (b − p* ) db ∂b s dp˜ ∂Π = [β(1− π + πω) + (1− β )ω ] − (b − p* )β db ∂b (9.16) b dp ∂Π = [π (1− ω )] − (b − p* ) dc ∂c s d˜p ∂Π = [(1− β + βπ )(1− ω)] − (b − p* )β . dc ∂c An increase in the outside option from either side of the market must increase both buyers and sellers’ prices directly, the square bracket terms, but reduces them indirectly due to the price decreasing effect of the resulting rise in the match probability. This opens the door to paradoxical results. Consider the case of a rise in the buyers’ mean outside option. In excess demand equilibrium this would seem sure to improve the sellers’ welfare. Instead, the improvement in the match probability may so improve the expected price of non-contracting buyers that the contract price facing sellers falls. To explore this possibility in more detail, substitute from (9.11) into (9.16) and multiply both sides by D = (1− β)(1− fπ ) to form: dp˜ s ∂p˜ s D = (1− β) db ∂b (9.17) . ε b  ∂˜p s ∂pb ∂˜p s ∂pb  +π (1− β + βπ ) b  − . p  ∂π ∂b ∂b ∂π  Using the partial derivatives implicitly given in (9.16), the bracket term in the second line can be shown to be negative. Then the sellers’ price is reduced by a rise in buyers’ outside options if the elasticity of demand is sufficiently large. Thus, paradoxically, sellers can be made worse off by when their partners’ outside option worsens. This lowers the constraining trade volume so we have the paradox of scarcity: the greater is the gain from trade the less is the volume of trade. On the excess (buyers’) side of the market we have: dp b ∂pb D = (1− β ) db ∂b (9.18) ε s  ∂˜p s ∂pb ∂˜p s ∂pb  +π(1− β + βπ) s  − . p˜  ∂b ∂π ∂π ∂b  The bracket term in the second line is negative, so the buyers’ expected price is paradoxically lowered by a rise (worsening) of his outside option when the supply response is sufficiently large. Interestingly, (9.17)-(9.18) can both be negative (paradoxical price response on both sides), they can both be positive or they can differ in sign. Related results hold for the excess supply case. Based on the methods of this appendix, the reader can generate other paradoxical results.