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р ┘How sellers' capacities affect equilibrium in finite markets
Ruslan Shavshin, Marina Sandomirskaia (HSE University)
Keywords: directed search, finite market, friction, market inefficiency, market concentration
Directed search theory develops a deep understanding for endogenous matching of buyers and sellers on finite markets, optimal pricing schemes, and, as a result, the market inefficiency (Wright et al. 2021). In the benchmark models, sellers generally have one unit of homogeneous product, and the main attention is paid to the mechanics of the friction process, the resulting failure of effective matching, and the redistribution of bargaining power between the two market sides determined by their sizes.
In this paper we assert that one important dimension of the finite market, that is the capacity of sellers, has been poorly accounted in previous studies (Burdett, Shi, and Wright 2001), (Lester 2010), (Geromichalos 2012), and we fill this drawback. Considering sellers with more than one product at each seller increases the availability of a product for buyers and relaxes the friction. This is a bit more applicable setting since sellers, for instance, operating on marketplace, often have some reserve of a given product, which capacity depends on the size of the firm. Moreover, even if the total number of products in the market is large, this doesn t necessarily mean that the cheapest seller can satisfy the whole market. We determine the exact condition on the sellers capacities under which the market demonstrates the friction: if at least two sellers can serve all buyers, then the friction disappears and the market is indistinguishable from that without any capacity constraints.
In this paper we consider the finite market with ns sellers and nb buyers which are not necessarily grow to infinity. We start with the symmetric sellers with equal capacities 1 d" h < nb. Sellers simultaneously post prices and then buyers independently send their requests for the purchase. The probability of being served is more complicated than in the case of a unique product per seller, as it depends both on the capacity and the actual demand. We derive an analytical solution of buyers game and, consequently, the equilibrium price in sellers game. This is a direct extension of formula (45) from (Wright et al. 2021). This generalized formula provides immediate implications for understanding the toughness of competition on the market. It allows to calculate the matching probability and the utilities of seller as a function of h. For small h d" 2 it was previously shown that more capable firms get larger utility, but this tendency changes with the growth of the number of products at every seller. Analyzing the comparative statics we deduce that, because of the intensification of competition, with the grows of availability, resulting price decreases, and even though the number of deals increase, this doesn t compensate loses. This means that the utility of the firm is a bell-shaped function of h. Thus, there exists an optimal seller capacity in the symmetric setting, and this capacity is not so large, such that some deficit is optimal for sellers. Also this framework contributes into our understanding the influence of market concentration on the competition.
We also present a well-formulated maximization problem for sellers in the case of their non-equal capacities. We incorporate big and small sellers with respect to their capacities and conclude that large firms enjoy extra market power and propose higher prices than small. This is due to the buyers get higher chance to buy the product at a large seller, and thus small sellers have to compete more aggressively in order to sell a little number of products, while large sellers may sell less often per unit of product, but with greater marginality.
To sum up, this study enriches the setting of classic finite market into the capacity accounting and makes the decision of sellers and classic relationship price matching efficiency firm size more clear. It clarifies the nature of buyers competitive behavior which in turn affects the seller expectation of the demand, and highlights the influence of product distribution across sellers.
References
Kenneth Burdett, Shouyong Shi, and Randall Wright. Pricing and matchingwith frictions . In: Journal of Political Economy 109.5 (2001), pp. 1060 1085.
Athanasios Geromichalos. Directed search and optimal production . In: Journalof Economic Theory 147.6 (2012), pp. 2303 2331
Benjamin Lester. Directed search with multi-vacancy firms . In: Journal of Eco-nomic Theory 145.6 (2010), pp. 2108 2132.
Randall Wright et al. Directed Search and Competitive Search Equilibrium: AGuided Tour . In: Journal of Economic Literature 59.1 (Mar. 2021), pp. 90 148
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