First-Price auction symmetric equlibria with a general distribution

In this paper I obtain the mixed strategy symmetric equilibria of the first-price auction for any distribution. The equilibrium is unique. The solution turns out to be a combination of absolutely continuous distributions case and the discrete distributions case.

First-price auction symmetric equilibria with a general distribution

In this paper I obtain the mixed strategy symmetric equilibria of the first-price auction for any distribution. The equilibrium is unique. The solution turns out to be a combination of absolutely continuous distributions case and the discrete distributions case.

Abstract types and distributions in independent private value auctions

In this note, in an independent private values auction framework, I discuss the relationship between the set of types and the distribution of types. I show that any set of types, finite dimensional or not, can be extended to a larger set of types preserving incentive compatibility constraints, expected revenue and bidder’s expected utilities. Thus for example we may convexify a set of types making our model amenable to the large body of theory in economics and mathematics that relies on convexity assumptions. An interesting application of this extension procedure is to show that although revenue equivalence is not valid in general if the set of types is not convex these mechanism have underlying distinct allocation mechanism in the extension. Thus we recover in these situations the revenue equivalence.

Endogenous normative indices for the industry concentration

This paper presents new indices for measuring the industry concentration. The indices proposed (C n ) are of a normative type because they embody (endogenous) weights matching the market shares of the individual firms to their Marshallian welfare shares. These indices belong to an enlarged class of the Performance Gradient Indexes introduced by Dansby&Willig(I979). The definition of Cn for the consumers allows a new interpretation for the Hirschman-Herfindahl index (H), which can be viewed as a normative index according to particular values of the demand parameters. For homogeneous product industries, Cn equates H for every market distribution if (and only if) the market demand is linear. Whenever the inverse demand curve is convex (concave), H underestimates( overestimates) the industry concentration measured by the normative indexo For these industries, H overestimates (underestimates) the concentration changes caused by market transfers among small firms if the inverse demand curve is convex(concave) and underestimates( overestimates) it when such tranfers benefit a large firm, according to the convexity (or the concavity) of the demand curve. For heterogeneous product industries, an explicit normative index is obtained with a market demand derived from a quasi-linear utilility function. Under symmetric preferences among the goods, the index Cn is always greater than or equal the H-index. Under asymmetric assumptions, discrepancies between the firms' market distribution and the differentiationj substitution distributions among the goods, increase the concentration but make room for some horizontal mergers do reduce it. In particular, a mean preserving spread of the differentiation(substitution) increases(decreases) the concentration only if the smaller firms' goods become more(less) differentiated(substitute) w.r.t. the other goods. One important consequence of these results is that the consumers are benefitted when the smaller firms are producing weak substitute goods, and the larger firms produce strong substitute goods or face demand curves weakly sensitive to their own prices.