The parameter sweeping described in the previous section is useful for illustrating the main predictions of the model. The caveats discussed above around multiplicity chal- lenge the validity of conclusions drawn from comparative exercises based merely on exploring changes in equilibrium outcomes after variations in the fundamental param- eters of the game. The references to the equilibrium computations in this section are then illustrative only and serve the purpose of supporting a qualitative description of the mechanics of the game 27.
The game presented here is such that the architecture of buyer-seller relations and the prices underlying these are a result of the surplus-sharing rules, the heterogeneity in the matching quality and the existence of linking costs. Below I summarise in seven statements, the main ways in which these three sets of parameters can affect the ob- served networks and prices, to turn, in the next chapter to the discussion on structural estimation.
27For these illustrations, consider the same simple setting of two identical buyers with a unit demand
over an indivisible product that can be supplied by two sellers that are constrained to producing for, at most, one or other buyer in each period. The key parameters we are interested in are the bargaining power of the second buyer,b2, (given that that of the first buyer was fixed at 0.5 for the whole of the
sweeping exercise), the cost of linking with a new buyer,chigh, and the quality of the matches of the
second buyer with either seller,ρ21 andρ22. Once more, recall that the there is no cost of breaking a
link and that the cost of re-linking with anold supplier is fixed for the whole exercise. Similarly,ρ11
andρ12are set so the quality of the match between buyer one and seller one is higher than that between
First, a buyer with no bargaining power (take-it-or-leave it offers from sellers) will not link when costs of forming a link are sufficiently high. This is a straightforward statement. When one buyer has no bargaining power, the probability of observing only the other buyer trading increases with the costs of forming a new link. As in oursweeping
exercise, fixing the bargaining parameter of buyer 1,b1= 0.5, note that wheneverb2 = 0,
buyer 2 doesn’t trade, in the presence of non-zero costs of linking. When the costs of forming a new link are the same as those of maintaining a link, buyer 2, even with no bargaining power might still form links and the ergodic distribution over states show networks in which only buyer 1 trades with probability 0.46. As the cost of forming links increases, networks with buyer 1 as the only buyer in the market arise with a steady state probability of above 0.85 (withcabove a certain threshold, this probability is 0.99). Remember that the no-linking alternative for the buyer was set to a value unappealingly low. Then, under this set up, it can be seen that when the costs of forming a new link are non-zero, if a buyer is trading, then the profit-sharing rule cannot be one in which the seller captures all the surplus in a relation. The bargaining parameters to be estimated with our data, then, need to be consistent with this observation.
Second and related to the previous point, when both buyers have at-least-some bar- gaining power, networks with one buyer only occur with probability close to zero. The out-of-the-market option for the buyer was set low enough so that trade always domi- nates the option of not-linking. So long as each buyer can extract at least some of the surplus produced in trade, they will both be active. This can be under networks that are coordinated, so that there is no clumping of both buyers choosing the same seller or with miss-coordination, happening when both buyers choose the same seller. Note however that buyers’ linking choices occur simultaneously so no purposeful coordination is possible. Recall as well that the presence of private shocks prevent buyers from mak- ing certain conjectures over rivals’ choices. Clumping occurs with significant non-zero probability only when buyer 2 has no bargaining power. Above that, irrespective of the parametrisation of the heterogeneity in the matching quality, both buyers trade and they do so by linking with different sellers. In our exercise, this implies observing a network where the only two links are g11= 1 and g22= 1 or one in which these areg12= 1 and
g21= 1.
Third, in equilibrium, buyers choose different suppliers. In the game proposed here, sellers are constrained to supplying one unit of the product only. Whenever two buyers link in negotiation with the same seller, one of them (precisely the one that offers lower gains from trade to the seller) will not trade in that period, paying -if any- the associated cost of having linked and obtaining the outside value. This induces the buyer moving away from that particular seller, onto the second best option. The other buyer, who offers the largest gains to the seller, will trade. However, given that larger negotiation
sub-graphs for the seller improve her bargaining position, this will drive the price up, making states of clumping less valuable for the remaining buyer, even when she succeeds to trade. This result is important for the structural estimation, as the links we observe in the data will be the result of equilibrium behaviour that does not involve unrealised links (which we, obviously, cannot observe).
Fourth, when faced with more than one potential buyer, the link that prevails with a given seller is the one that maximises the gains from trade for the seller which de- pends, via negotiated prices, on the relative bargaining powers and the matching quali- ties. Other things equal28 buyers with lower bargaining power are preferred and better matches are preferred as well. Note however, that the effect of the quality of the match on the seller’s choice is mediated by the bargaining parameter itself and the higher the bargaining power of the buyer, the lower the effect of the quality of the match on the price and, therefore, on the seller’s profits.
Fifth, the presence of heterogeneity in the matching qualities induces sorting but not necessarily the most efficient outcome. This is immediately related to the shape of the period profits. As the quality of the matching ρij enters additively in the unit
profits of the buyer, it is clear that, other things equal, buyers would in principle be inclined to linking with their best match. However, when the best match for both buyers corresponds to the same supplier, in equilibrium, the outcome network corresponds to that which exhibits higher gains from trade from the seller’s perspective, with the “discarded” buyer moving away to the second best alternative. This does not always imply that in the presence of matching-level heterogeneity the fully efficient network arises. To see this, consider a case in which ρ21 > ρ22 > ρ11 > ρ12 and ρ11+ρ22 >
ρ21+ρ21. In such a setting, both buyers prefer, other things equal, seller one. However,
the gains from trade for that seller are higher with buyer 2. This can force buyer 1 to link with his second best alternative, seller 2. Although this outcome maximises seller 1’s and buyer 2’s profits, it leads to a network that does not produce the industry efficient outcome. The decentralised linking protocol proposed here leads to the prevalence of the network that maximises industry-wide gains from trade, within capacity constraints and in the presence of matching heterogeneity in the ρordering of this example, only if
ρ11+ρ22≤ρ21+ρ12.
Sixth, higher costs of linking generate stronger persistence in pre-existing links and drive prices up. This as well constitutes a natural result from the structure of the game. The effect of the cost of forming a new link operates in two ways. As it negatively affects the period profits for the buyer, irrespective of the result in the bargaining stage, other things equal, higher linking costs lower the value of opening new links for a buyer. With
it, the future value of moving away from the current supplier is also low, making the relative gains from the current relation higher and driving the price up, for a fixed set of bargaining parameters. This can then mimic settings in which the buyer allocates systematically orders to its existing supplier(s).
Seventh, the network structure affects both the prices, via outside options, and the discrete choice decision in a way that is similar to entry / exit problems. In our simple setting, buyers are allowed to choose one link only, linking decisions are non-coordinated and other buyers’ choices affect the profitability of a given link directly. This feature resembles the strategic aspect of standard entry / exit games in which firms decide whether to enter a market (start a relationship with a seller) with profits being dependent on whether she finds herself a monopolist in the market after entry (she is the only party negotiating with the seller) or whether she is competing with other firms (she bargains with a seller who is sustaining multiple negotiations at the same time). On top of this
direct effect, the choices of rivals situated in other areas of the graph play a strategic role as well. To see this, consider a buyer facing two different possible suppliers,s1ands2 and
assume that the buyer we are interested in, buyer 1, negotiates and trades withs1 and
another buyer, buyer 2, trades withs2, under set costs of linking, qualities of the matches
and other exogenous primitives. Upon successful trade, each buyer needs to make a new choice of suppliers for the next period. Buyer 2 will be perceived as more likely to re-link withs2than any other supplier. From buyer 1’s perspective, then, the probability of any
state of the world arising with buyer 2 linking with any seller different froms2 is lower.
This affects the continuation value for all the choices buyer 1 could potentially make in the current period and the equilibrium prices associated to these. To continue with the analogy to entry / exit games, the problem presented here could be interpreted as one in which each seller represents a market and each buyer is a firm who needs to decide what market to enter. Each market can hold one monopolist only, whose cost (price to be paid for the garment) depends on (the identity of) other incumbents or entrants both via the competition in the current period and the probability distribution over states that can be reached in the future. A problem that is similar to the one described here is that studied byAguirregabiria and Ho(2010), who find this network effect as the underlying mechanism supporting entry deterrence. However, for the purpose of their model and its estimation, they propose an independence assumption across markets (sellers in analogy to our setting) that simplifies the game substantially.
Finally, note that up to here we have imposed symmetry in three relevant aspects of the buyers’ characteristics: we have assumed that they charge the same price in their domestic markets ri =r, that they place orders of the same size qi =q and that they
larger buyers will win the competition for a seller when many-to-one situations arise. Although higherri’s will also increase the surplus the seller can extract, this is only via
the gains from trade for he buyer in the bargaining process. This implies that, capacity constraints aside and heterogeneity assumed away, a supplier will prefer a large buyer with low end-market prices than a smaller buyer with high prices. The cost of the inputs enter linearly in the sellers profit function as well, having still a greater impact in the stability of a given link than that ofri.
All of these29 are observed in the data, together with the active links (who trades with whom) and the prices at which trade takes place. The game proposed here models these two as results of the equilibrium behaviour consistent with values of three (sets of) parameters: the cost of linking, the bargaining powers of the buyers and the shape of the matching-specific heterogeneity component. The following chapter proposes a first exploration to the structural estimation of these using observed matches and prices.
29
The Structural Approach
4.1
Introduction
The previous chapter described a dynamic game of incomplete information in which buyers choose a supplier for a product, from a list of available sellers in a market. The game gives, as its main outcome, a configuration of a buyer - seller network and a set of contracts associated to it. These were the result of a process of competition between buyers for suppliers of heterogeneous qualities and a sunk cost of starting a relation. This chapter discusses and assesses the applicability of the econometric approach devel- oped in Lee and Fong(2013) for estimating the structural parameters of the game. In terms of its structure, their estimation algorithm resembles that of Bajari et al.(2007) with two stages performing a forward simulation routine to compute value functions and an iterative procedure that evaluates candidate parameters to compare the results pro- duced by these with conditional choice probabilities recovered non-parametrically from the data, as in Hotz and Miller(1993). There are two aspects in which Lee and Fong’s algorithm explicitly differs from that ofBajari, Benkard, and Levin. First, a fixed point problem in prices-to-values is nested in the computation of value functions, looping from computed values to prices and back to values iteratively until convergence to guarantee internal consistency of these two. This is similar to what was done in Chapter 3 when realising an example of a small game and computing its equilibrium(a). Second, the estimation of the structural parameters is done via the minimisation of a distance score that compares conditional choice probabilities that need to be computed for each pa- rameter candidate. In the exercise I perform in this chapter using simulated data, this proves very costly in terms of computer times and the advantages of the method over alternatives that rely on computing fully the equilibrium of the game for each candidate are dubious.
There are a number of specificities of the data and the problem at hand that makeLee and Fong’s algorithm not immediately applicable. This has led to the introduction of extensions and alternative steps in the algorithm in three dimensions: (i) the way in which conditional choice probabilities are obtained from the data; (ii) the use of data on prices; (iii) the construction of the score to minimise. The extensions related to (i) and (iii) follow suggestions inBajari et al.(2007) andHotz et al.(1994), while those in (ii) respond to the availability of additional information in our data. These extensions lead to sixteen different ways of applying the algorithm to my setting.
After framing the techniques used here in the empirical IO literature, the goal of this chapter is to present the algorithm and its alternatives and discuss the results of applying them to estimating the parameters of the simple game presented in Chapter 3 using simulated data. The main aim of this exercise is to shed light on the suitability of the methods to my setting and in no way constitutes an attempt of evaluating the merits of the econometrics underlyingLee and Fong’s algorithm in broader terms. What I present here pursues a different goal and does not display the rigour necessary to propose claims that could exceed the limits of this first exploration.
The next section,4.2, discusses the econometric approach, with references to the recent literature on the estimation of dynamic games of incomplete information. Then, in4.3, I present the structure of the chosen estimation algorithm, following the developments in Lee and Fong (2013). Section 4.4, describes the operational assumptions needed to implement the econometric approach to estimate the proposed game using our data. Finally, section 4.5 presents the results from a small Monte Carlo exercise that studies the performance of the algorithm and its variations and discusses the main difficulties encountered when applying it to our setting.