The second-generation money-search model (Trejos and Wright [
At the beginning of the entire matching process, money is randomly distributed among agents at probability represented by μ ∈ (0, 1), and it is transferred among them through monetary transactions. Both in barter and monetary transactions, bilateral bargaining determines the quantity of trade (e.g. Nash bargaining solution). The seller in the monetary trade must not be a money holder, as the model does not allow agents to carry more than two units of money. If there is no coincidence, there is no trade. The formal mathematical model is developed as follows.
Let qd be the quantity of consumption and qs be the quantity of the sale (production) of a particular product type. Given that an agent consumes a product that he likes, we let the utility function be uʹ(qd) for all agents, where uʹ(qd) > 0 and uʺ(qd) < 0. In addition, we let the cost function be c(qs) for all agents for all product types, where cʹ(qs) > 0 and cʺ(qs) ≥ 0. Furthermore, we assume an Inada condition: uʹ(0) 
cʹ(0) and uʹ(∞) 
cʹ(∞). The coincidence of wants is supposed to be stochastic, and its probability is given by α ∈ (0,1). The probability of double coincidence of wants is α2 and that of no coincidence is (1 − α)2, so that the probability of single coincidence is calculated as
1 − α2 − (1 − α)2 − α(1 − α) = α(1 − α) (1)
For a seller, once he is paired, the probability of monetary trade is then α(1 − α)μ. Similarly, for a buyer, it is α(1 − α)(1 − μ).
Let V0(t) and V1(t) be the value functions of the nonmoney holder and money holder at period t, respectively, and τ > 0 be the length of each period; hence, the periodical discount rate is approximated by τβ for sufficiently small τ. Let λ0 > 0 be the arrival rate at each moment in the Poisson process. Without loss of generality, we can make τ sufficiently small for λ0 < 1 (e.g., agents are paired once per period at most). Let p0 = α(1 − α)μλ0 be the probability for a buyer meeting a seller to make a monetary transaction in the matching process and p1 = α(1 − α)(1 − μ)λ0 be that of a seller meeting a buyer to make a monetary transaction. Let p2 = α2λ0 be the probability of barter trade between both the buyer and seller. In this specification, τpm represents the probability of each event m ∈ {0, 1, 2} per period. Let 
be the time derivative of Vm(t) so that
Then, for τ → 0, the Bellman equation for the seller satisfies
where 
is the net utility from barter trade and 
is the instantaneous social welfare maximizer (Appendix 1). Similarly, the Bellman equation for buyer satisfies
where γ is the constant utility flow to store a unit of money for a period; hence γ < 0 implies that money is costly to store (fiat money) and vice versa for γ > 0 (commodity money).
In the equilibrium, we have qd = qs = q, since demand and supply must be equal, where q is given as a Nash bargaining solution (Appendix 2) that satisfies V1(t) − V0(t) = (1 − θ)u(q) + θc(q) (5)
where θ ∈ [0, 1] represents the bargaining power of the buyer in the Nash product. The two Bellman equations are then solved as
In Equation (6), F(q) is given by
where λ = α(1 − α)λ0 is the arrival rate of a single coincidence.
Remark 1. F(q) is 
-shaped in q iff μ < θ and λ > (θ − μ)−1(1 − θ)β; ∩-shaped in q iff μ > θ and λ > (θ − μ)−1θβ; and otherwise monotonically increasing in q.
Proof. By differentiating F(q), we find that
Since utility and cost functions are monotonically increasing in q, we can say that F(q) is ∪-shaped if and only if F ʹ(0) < 0 and F ʹ(∞) > 0; it is ∩-shaped if and only if F ʹ(0) > 0 and F ʹ(∞) < 0, monotonically increasing if and only if F ʹ(0) > 0 and F ʹ(∞) > 0, and monotonically decreasing if and only if F ʹ(0) < 0 and F ʹ(∞) < 0. Since uʹ(0) ≫ cʹ(0) and uʹ(∞) ≪ cʹ(∞), the sign of F ʹ(q) at q = 0 is identical to the coefficient of uʹ(0) and at q → ∞ to that of cʹ(∞) in Equation (8). By rearranging the terms of (1 − θ)β – λ(θ − μ) ⋛ 0 for q = 0 and θβ + λ(θ − μ) ⋛ 0 for q → ∞, we obtain the conditions for each shape of F(q) as stated in this remark. We then find that the two inequalities do not satisfy the condition for the monotonically decreasing case.
Q.E.D.
The classification of F(q) stated in Remark 1 is depicted in
Remark 2. If θ = 1, as a dictatorial buyers model, F(q) cannot be ∩-shaped.
Since (1 − θ)uʹ(q) + θcʹ(q) > 0, 
⋛ 0 is equivalent to F ʹ(0) ⋛ 0; hence, the equilibria are given by F(q) = 0, as depicted in
to get
and 
as demand and supply of the individual i. Let us assume equal bargaiing power between the two agents. Since there is no transfer of money in barter trade, its Nash bargaining solution is given by maximizing
and
. Substituting the conditions to maximize with respect to 
and 
and rearranging the terms provide the first order condition as
is socially optimum, it satisfies the first order condition, as 
and
. If it is not a social optimum, it cannot satisfy the first order condition consistently with the equal bargaining power assumption. Therefore, the quantity of trade in barter trade is the social optimum.
be bargaining power of the buyer (measured in an increasing manner). Then, the Nash bargaining solution for monetary trade is given by