Stochastic Model of Thin Market with an Indivisible Commodity

Since there are different trading mechanisms established in individual markets, and since these differences may not be neglected, each type of (thin) market requires a special treatment. Hence, we may divide the existing works into those describing auctions (cf. the bibliography by Klemperer, 1999), those devoted to labor market (cf, Roth and Xing, 1994, Crawford, 1991, or Coles and Smith, 1998, for instance) and those describing thin markets with securities (cf. Lo and MacKinlay, 1990, for instance).

the paper, the usage of this distribution is quite natural: similarly to various models of queuing systems, the Poisson distribution of the amounts is a consequence of a quite acceptable assumption that the appearance of the agents' offers (demands) during the time between the auctions is driven by the Poisson process. 3 The formula of the distribution of (P,Q) in this model has the form of a finite sum.
The "continuous" model is similar to the "finite participants" one. The difference lies in the assumption that, despite the "finite participants'' model, the jumps of the individual demand and supply curves may realize themselves in an infinite number of points of the real line and that the probability that they realize in a single point is zero. The formula of the joint distribution of (P,Q) has the form of a Lesbegue integral.
The paper is organized as follows -in Section 1, a general formula for the distribution of the price and volume is derived, in Section 2, the "finite participants" model is defined and the distribution of the price and the volume is derived, in Section 3, the same is done for the "continuous" model. Section 4 concludes the paper.

Stochastic Model of the Organized Auction
As it was already premised, we assume that the individual demand functions and the individual supply functions are random piecewise constant with integer values. Quite naturally, it is supposed that each demand function is left continuous with the zero limit in infinity, and that each supply function is right continuous with the zero limit in minus infinity. Moreover, we assume that the number of buyers with a non-zero demand function and the number of sellers with a non-zero supply function is finite almost sure. 4 Given these assumptions, each realization of the aggregate demand function D is piecewise constant left continuous with integer values and with the zero limit in infinity, and each realization of the aggregate supply function S is piecewise constant right continuous with integer values and with the zero limit in minus infinity.
As it was already mentioned, we assume that the market price P is determined as the average price maximizing the total traded volume Q, i.e.
It is easy to prove that, for each positive integer q, where Using the reformulation (3), the formula for the joint distribution of (P,Q) may easily be derived: For each positive integer q and each p R, where q is the probability distribution of the extended ) and 1 is the probability distribution of the extended random vector (X [1] , Y (1) ).

A Model with a Finite Number of Participants
Let us assume, until the end of the Section 2, that there are m buyers and n sellers participating in the market. Further, let us suppose that each individual demand function has at most one jump and that the magnitude of the jump is a Poisson random variable. Speaking more exactly, we assume that there exist x 1 , x 2 , …, x m and p 1 , p 2 , …, p m such that the i-th individual demand function is determined by the formula where I is the indicator function and D i is a Poisson random variable with an intensity p i . Analogously, we assume the existence of y 1 , y 2 , …, y n and q 1 , q 2 , …, q n such that the j-th individual supply function is determined by where S j is a Poisson random variable with the intensity q j . Finally, let us assume all the variables D i and S j to be mutually independent. Given our assumptions, the aggregate functions are Thanks to the fact that the distribution of the sum of independent Poisson random variables is Poisson with its the parameter equating the sum of the summed variables' parameters, for each -∞≤ x ∞ and S y y y q j for each -∞< y ∞ . Hence, we are getting that for each -∞≤ x ∞ and -∞< y ∞.
The following Theorem provides a little complicated but computable formula for the joint distribution of (P,Q): Proof. The formula (10) is an application of (4). The formula (11) is an application of (5). For a detailed proof, see Šmíd (2004), Section 1.
Remark. Note that the distribution of (P,Q) does not depend directly either on the number of agents or on the intensities of their arrival -it is uniquely determined only by δ(•) and σ(•).

A Model with a Continuous Expected Demand and Supply
In the present section, let us assume that there exist functions δ and σ such that Similarly to the "finite participants" model, it follows from the assumptions 3. and 5. that The joint distribution of the vector (P,Q) could be expressed by means of a Lebeque integral: Proof. The formula (13) is an application of (4). The formula (14) is an application of (5). For a detailed proof, cf. Šmíd (2004), Section 2.