Observationally Equivalent Financial Mechanisms in an OLG Model with Spatial Separation

This study introduces search frictions into a variant of overlapping generations environments, as in Smith (2002). In the model, this study compares financial intermediation to trading shares in decentralized secondary markets. The results show that under certain conditions, both financial mechanisms generate the same equilibrium outcome.


Introduction
Following Bryant (1980) and Diamond and Dybvig (1983), a body of literature that investigates the consequences of liquidity insurance with financial intermediaries has been persistently expanding. An important stream in such literature was provided by Smith (2002), who incorporated banks as described by Diamond and Dybvig (1983) into a monetary overlapping generations (OLG) model with spatial separation and showed the suboptimality of the Friedman rule (Friedman, 1969) 1 . In his model, there are two islands between which there is no communication (spatial friction). Furthermore, liquidity shocks are modeled by random relocation of agents. Then, a bank that is established by a coalition of agents provides deposit contracts, which play the role of liquidity insurance, to agents and relocated agents will withdraw their deposits before they move to the other island. 1 This type of financial intermediation has been blindly adopted in many previous studies, such as Schreft and Smith (2002), Haslag and Martin (2007), Matsuoka (2011), Ohtaki (2014), and Gupta and Makena (2020). However, even within the OLG model with spatial separation, such financial intermediation is not a unique financial mechanism. In contrast to previous studies, the aim of this study is to introduce another financial mechanism to the OLG model with spatial separation and compare its consequences to those of financial intermediation.
This study introduces the primary and secondary capital markets to the OLG model with spatial separation. Agents first enter primary and money markets, which are centralized, and invest their endowments in the production process and money. To simplify the argument, we assume that one share is issued for each unit of investment in the production process. Agents then buy or sell shares in the secondary market, which is assumed to be decentralized, rather than centralized. To construct decentralized secondary markets, agents must search for trading partners. In each decentralized secondary market of shares, a relocated agent will be a seller, and an agent who stays on the same island will be a buyer.
This study demonstrates the somewhat surprising result that, under certain conditions, equilibrium consequences under primary and secondary markets are the same as those in the presence of financial intermediation. This may imply the robustness of the results obtained in previous studies. In particular, this study shows the suboptimality of the Friedman rule.
Our findings contribute to the literature by comparing the direct and indirect finances. Jacklin (1987) has shown that, in a static economy as considered in Diamond and Dybvig (1983), direct and indirect finances attain the same allocation. On the other hand, in three-period OLG models, Bhattacharya and Padilla (1996) and Fulghieri and Rovelli (1998) argued that indirect finance is superior to direct finance in certain situations. The observational equivalence between direct and indirect finances in this study can be interpreted as a restatement of Jacklin's result in a two-period OLG model. The remainder of this paper is organized as follows. Section 2 introduces the model's ingredients. Section 3 considers financial intermediation as a benchmark. Section 4 considers an economy with primary and secondary markets. Section 5 provides the concluding remarks. Proofs of propositions are presented in the Appendix.

Ingredients of the Model
This section presents ingredients of the model. Time is indexed by t and runs discretely from minus infinity to plus infinity. Each period is divided into two stages; 1 and 2. Two island exist at distinct locations, and there is no communication between them. In stage 1 of each period, a single perishable commodity, called the consumption good, exists on each island. In addition, an intertemporal production technology exists, whereby 0 k ≥ units stored at stage 1 of date t At the beginning of each period, a new generation, consisting of a continuum of ex-ante identical agents with a unit mass, appears on each island and exists for three consecutive stages. Agents born in stage 1 of period t are young in period t, or their first and second stages, and old in period 1 t + or their third stage.
They aim to maximize their utility ( ) To simplify the argument, we assume that half of the agents become type α agents and the other half become type β agents, that is, At the end of each period, the type α agents on each island move to the other island, whereas the type β agents stay on the same island. It is assumed that type α agents, called movers, cannot receive the return of the storage investment, whereas type β agents, called nonmovers, can do so.
In the economy, a durable and intrinsically useless object referred to as money also exists. Money is issued by the central bank and its per capita stock at date t is denoted by M t . The per capita stock of money follows the equation , at the beginning of the period. To guarantee the nonnegativity of the net nominal interest rate, it is assumed that

Equilibrium with Financial Intermediation
As a benchmark, we first consider an equilibrium with financial intermediaries as in Smith (2002).

Timing of Trades
At the beginning of each period t, young agents cooperate with each other and establish a bank. They then deposit a certain amount of their after-tax/transfer 2 We can consider several backgrounds to our production technology; one called the storage technology, and one that is a linear production function  endowments into their bank. During stage 1 of the period, the bank and the old agents meet in a centralized spot market (of money). At this stage, the bank invests its deposits in storage technology and money. In stage 2 of the period, young agents learn their types, and in the second period, movers lose their connection to their banks. As a result, movers withdraw their deposits during this stage, and nonmovers withdraw their deposits during the third stage. We assume that the centralized spot market at stage 1 of each period t is competitive and denoted by P t the nominal price of the consumption good in the market. We also use q t to denote the per capita real money balance in each period t, that is, t t t q M P = . We define the inflation rate by

Definition of Equilibrium
We begin by considering the behavior of the banks. The bank established in period t is assumed to propose a "contract" to its agents. Here, a contract is a trip- which include a pair of per capita investments in storage technology and money. The first constraint for the bank is the restriction on deposits. Because each agent's after-tax/transfer endowment is The bank then invests in storage technology and money. Its portfolio constraint is given by where 1 0 t k + ≥ and 0 t t m P ≥ are the per capita amounts of the investments in storage technology and money, respectively. Because the returns of the investments must meet the total consumption, the following budget constraint should be considered: This inequality can be rewritten as which can be interpreted as an individual rationality (or participation) constraint for the bank. The final restriction is the liquidity constraint. Because movers lose their connection to their banks in the second period, they withdraw their money after they learn their types. Therefore, at the end of the first period, the bank must have sufficient liquidity to meet the needs of the movers: Finally, we assume that the bank adopts the welfare function , which is equal to each agent's expected lifetime utility function, as its objective function. A contract We can now define an equilibrium with financial intermediation as follows: Moreover, it is a monetary steady state (with financial intermediation) if it is independent of period t, that is, if there exists some ( ) for each period t.

Characterization
This subsection characterizes the monetary equilibrium with financial intermediation. We first verify that, at any monetary equilibrium, it must hold that 1 0 t i + ≥ for each t. In fact, the combination of Equations (2) and (3)  In addition, note that the bank established in period t wishes to keep the amount of money holding as small as possible if 1 0 t i + > . In this case, the liquidity constraint (4) plays an important role. This prevents banks from setting the amount of money to 0. In fact, the liquidity constraint must hold with equality because the movers' consumption at a monetary equilibrium is positive due to the boundary condition imposed on u. On the other hand, when 1 0 t i + = , investments in storage technology and money are completely substitutable because their rates of return become equal to each other. As a result, there might be mul-3 Note that ( )( ) As a corollary of this proposition, we can say that a monetary steady state, denoted by ( ) We use this characterization to obtain the observational equivalence between equilibrium outcomes under different financial mechanisms.

Equilibrium with Primary and Secondary Markets
In contrast to previous works such as Smith (2002) and Haslag and Martin (2007), this section introduces the primary and secondary markets of capital into the model. Our secondary markets are decentralized, and agents must, therefore, search for trading partners 4 .

Timing of Trades
In stage 1 of each period t, young and old agents meet in a centralized spot market. At this stage, young agents invest their after-tax/transfer endowment in storage technology and money. We assume that one share is issued for each unit of storage investment. Therefore, investment in storage technology may be interpreted as issuing new shares in the primary market. In stage 2 of that period, young agents match in pairs. This matching is random, but without loss of generality, a type α agent is assumed to meet a type β agent. In each pairwise meeting (decentralized secondary market), each agent's portfolio is common knowledge and the surplus from the trade is split by generalized Nash bargain-4 The idea of embedding search into an OLG model was introduced by Zhu (2008) ing. Each agent consumes returns from money or storage investments in the third stage. We assume that the centralized spot market of each period t is competitive, and we use P t to denote the nominal price of the consumption good in the market. We also denote by q t the per capita real money balance in the centralized market in each period t, that is, t t t q M P = . We define the inflation rate

Definition of Equilibrium
Because agents are ex ante identical, we consider a symmetric situation with respect to young agents' choices in their first stage. At stage 1 of each period t, young agents invest their after-tax/transfer endowments in the storage technology and money. This portfolio constraint is described by The outcome of a pairwise meeting between type α and type β agents is assumed to be a maximizer of the following generalized Nash bargaining problem: where ( ] 0,1 θ ∈ is the bargaining power of the type α agent. Here, Equation (7) provides the set of possible pairs ( )  We can now define an equilibrium with search as follows: Definition 2 Moreover, it is a monetary steady state (with search) if it is independent of period t, that is, there exists some ( )

Observationally Equivalence
To characterize a monetary equilibrium with search, we first assume that agent α's bargaining power is equal to one, that is, 1 θ = . The generalized Nash bargaining described by Equations (6) where the last inequality is the participation constraint for nonmovers. One can immediately investigate that a solution to this problem is as follows: is also the solution to the generalized Nash bargaining problem, provided that θ is sufficiently close to 1.
We can now characterize a monetary steady state in this economy.
Note that Equation (16) implies that, in a monetary steady state, movers' consumption is less than or equal to that of nonmovers. This follows from the strict concavity of u and the fact that ( ) ( )( ) ( ) ( ) . This guarantees the solution to the Nash bargaining problem with 1 θ = . In addition, note that, without any doubt, the characterization in the last proposition is equivalent to that obtained in the previous section. Therefore, we can conclude that the financial mechanisms considered in Sections 3 and 4 are observationally equivalent.

Properties of Monetary Steady State
We close this section by discussing several properties of monetary steady states (with search). The following properties are well known in the existing literature on the OLG model with spatial separation and financial intermediation. However, in contrast to previous studies, nearly all of which assumed a constant relative risk aversion with an index less than or equal to one, we show the same properties under a more general class of utility functions. The first proposition guarantees the existence and uniqueness of a monetary steady state. Finally, we examine the optimality of the Friedman rule. We define the equilibrium welfare given the gross rate of growth of the money stock, 1

Concluding Remarks
The OLG model with spatial separation, developed by Champ, Smith, and Williamson (1996) and Smith (2002), is a very tractable model for describing situations in which agents face liquidity shortages. Although previous works assumed that agents remedy liquidity shortages via liquidity insurance provided by financial intermediaries such as banks, this study considers that agents who face a liquidity shortage seek trading partners and liquidate their share of returns of the intertemporal production technology. By comparing these two financial mechanisms, this study shows that such indirect and direct finances yield the same equilibrium outcomes under appropriate conditions. Our observational equivalence is based on several technical requirements. First, the sizes of agents who face and do not face a liquidity shortage must be the same. If they are not, some agents lose opportunities for liquidation, and the economy will be worse off than in this study. Second, the bargaining power of agents who face a liquidity shortage must be sufficiently close to unity. If the bargaining power of agents who face a liquidity shortage is sufficiently close to zero, for example, Equation (15)