Money, urbanization, and product diversification: a money-search approach

I consider an extension of the second-generation money-search model (Trejos and Wright, 1995) to study the relationship between varieties of products and chances of trade in the matching market. In the analysis, money is costly to store (e.g., fiat money). In the second-generation model, money is indivisible but goods are divisible. Agents are classified by their abilities of producing goods and their tastes for consumptions. For example, an agent who loves product k buys the products from a seller who can produce good k. Agents live infinitely long and agents are not allowed to have more than one unit of money. The matching process is a Poisson-process random-matching.

In the extended model, in advance to the start of entire matching process, agents make investments in own abilities of making varieties. For example, a I(n) dollars investments gives the agent skills of producing n + 1 varieties (one variety is a gift). In the analysis, we assume I(n) is positive and strictly increasing and convex in n; hence, dI/dn > 0 and d²I/dn² > 0. The investment decisions are affected by the distribution of agents’ tastes, the money stock, and the chance of meeting another agent (see below for the formal arguments).

Let a_i be the probability of single-coincidence for agent i. Now suppose agents i and j are paired. Then the probability of double-coincidence is given by a_ia_j. Similarly, the probability of no coincidence is given by (1 − a_i)(1 − a_j). Therefore, the probability of single-coincidence trade is given by 1 − a_ia_j − (1 − a_i)(1 − a_j) = a_i + a_j − 2a_ia_j. Taking the taste for consumption is as given, the probability of single-coincidence is determined by the ability of the seller how widely create the varieties, hence, n.

Let λ₀ > 0 be the arrival rate (interpreted as a form of urbanization). Without loss of generality, we assume λ₀ ≤ 1 by setting the length of one period to be sufficiently short. Let μ be the money stock (e.g., population share of buyers). Then the probability of monetary trade per period is λ = λ₀μ(1 − μ)(a_i − a_j − a_ia_j). The equilibrium of investment decision is dependent on the investment decisions of other agents (e.g., λ affects the value function and it is affected by the probability of single-coincidence of the other agent); thus, the equilibrium is Nash equilibrium. In this research, the equilibrium is analyzed numerically using standard CRRA utility and quadratic cost functions such as u(q) = q^1−ρ/(1 − ρ) and c(q) = q², respectively (in the numerical analysis, the rate of risk aversion is ρ = 0.15). Then we find an equilibrium that determines the range of varieties endogenously. In addition, we also find an increase of per-capita real-money holding increases the variety if μ < θ, where 0 ≤ θ ≤ 1 is buyer’s bargaining (in general, as we will see, the equilibrium is socially irrational if μ > θ and μ = θ implies there is no stable equilibrium).