2.1. Individuals, matching and firms

2.1.1. Preference representation

The per-period utility function of an individual is

(2.1) \begin{equation} U\left (C\right )-h\left (l\right )+u(q). \end{equation}

where $U(C)$ is the utility of consuming the general good $C$ , $h(l)$ is the disutility from supplying labor in the CM, and $u(q)$ is the utility of consuming the specific good in the DM. We assume that the functions $U$ and $u$ are continuously differentiable, strictly increasing, strictly concave, $U_{1},u_{1}\gt 0$ , $U_{11},u_{11}\lt 0$ , and the following boundary conditions hold: $u(0)=U_{1}(\infty )=u_{1}(\infty )=0$ , and $u_{1}(0)\lt \infty$ .Footnote ²⁰ Also, we assume that $h\left (l\right )=l$ . This simplifies the algebraic description of the CM decision problem and ensures that agents exiting the CM are identical.

2.1.3. Firms

Consider a firm of type $i\in I$ that takes the CM good’s relative price $p$ (in units of labor) as given. Following Menzio et al. (Reference Menzio, Shi and Sun2013), we define labor as the numéraire good. The firm hires labor on the spot market to produce the CM good and the DM good. One unit of labor is transformed into one unit of CM good linearly. In the DM, a firm takes the market tightness function $\theta$ as given, and chooses the measure of trading posts (viz., shops) $\text{d}N(x,q)$ to open in each submarket.Footnote ²¹ If $x$ is what a matched buyer is willing to pay for $q$ and $s(x,q)\,:\!=\,\rho \left (\theta \left (x,q\right )\right )$ , then $x\cdot s(x,q)$ is the firm’s expected revenue in submarket $(x,q)$ . To produce $q$ the firm must hire $c(q)$ units of labor. Hence $s(x,q)c(q)$ is its expected labor wage bill in submarket $(x,q)$ . We assume that $q\mapsto c(q)$ is a continuous convex function. The firm also pays a per-period fixed cost $k$ of creating the trading post in submarket $(x,q)$ .

The firm’s profit is:

(2.2) \begin{equation} \pi (p;\,k)=\max _{Y\in \mathbb{R}_{+}}\left \{ pY-Y\right \} +\max _{\text{d}N}\int _{\mathbb{R}_{+}^{2}}\left \{ s(x,q)\left [x-c(q)\right ]-k\right \} \text{d}N(x,q), \end{equation}

where $N$ is a positive measure on the Borel $\sigma$ -algebra $\mathcal{B}\left (\mathbb{R}_{+}^{2}\right )$ . The first term on the RHS is the firm’s value from operating in the CM. The second is its DM total expected value across all submarkets it chooses to operate in.

Note that the firms’ problem above (and also agents’ decision problems to be discussed below) do not explicitly depend on the aggregate distribution of agents. This is because of the nature of competitive search in the DM: Firms and buyers take matching probabilities as given when making their respective price posting and directed search decisions. The observed terms of trade posts and matching probabilities suffice to condition their decision processes. Moreover, CM preferences are quasilinear such that agents are identical at the end of the CM. We discuss this further in Section 2.4.1.

2.2. Money supply

At the start of each date $t$ , the total stock of money in the economy $M$ is known. We assume that $M$ grows at a constant rate $\tau$ :

(2.3) \begin{equation} \frac {M_{+1}}{M}=1+\tau, \end{equation}

We assume $\tau \gt \beta -1$ , where $\beta$ is the discount factor. New money stock $\tau M$ is injected lump sum to all agents at the end of date $t$ .

Since we define labor as the numéraire good, if we denote $\omega M$ as the current nominal wage rate, where $\omega$ is normalized nominal wage (i.e., nominal wage rate per units of $M$ ), then a dollar’s worth of money is equivalent to $1/\omega M$ units of labor. The variable $\omega$ will be endogenously determined in a monetary equilibrium.Footnote ²² If $M$ is the beginning-of-period aggregate stock of money in circulation, then $1/\omega =M\times 1/\omega M$ is the beginning-of-period real aggregate (per-capita) stock of money, measured in units of labor.

Denote (equilibrium) nominal wage growth as $\gamma (\tau )\equiv \omega _{+1}M_{+1}/(\omega M)$ . Later, for a stationary monetary equilibrium, we will require that equilibrium nominal wage grows at the same rate as money supply, i.e., $\gamma (\tau )\vert _{\left (\omega _{+1}=\omega \right )}=M_{+1}/M$ .

2.3. Individuals’ decisions

An individual is identified by her current money balance, $m$ (measured in units of labor). Given policy $\tau$ , her decisions also depend on the aggregate wage $\omega$ . Denote the relevant state vector as $\textbf{s}\,:\!=\,(m,\omega )$ .Footnote ²³ At the beginning of a period (ex ante), an individual decides either to work and consume in the CM or to be a buyer in the frictional DM.Footnote ²⁴ Next, we describe the decision problems of agents who are ex-post CM or DM buyers. We then describe an agent’s ex-ante decision problem of choosing which of CM or DM to go to.

2.3.1. Ex-post CM buyers

Suppose now we have an individual $\textbf{s}\,:\!=\,(m,\omega )$ who begins the current period in the CM. The individual takes policy, $\tau$ , and the sequence of aggregate prices, $\left (\omega, \omega _{+1},\ldots \right )$ , as given. Her value from optimally consuming $C$ , supplying labor $l$ , and accumulating end-of-period money balance $y$ , is

(2.4) \begin{align} W(\textbf{s})=\max _{\left (C,l,y\right )\in \mathbb{R}_{+}^{3}}\biggl \{ U(C)-h(l)+\beta \bar {V}\left (\textbf{s}_{+1}\right ):pC+y\leq m+l,\ \ m_{+1}=\frac {\omega y+\tau }{\omega _{+1}\left (1+\tau \right )}\biggr \}, \end{align}

where $\bar {V}:S\rightarrow \mathbb{R}$ is her continuation value function (see Section 2.3.3 on the following page). This continuation value function yields her next-period expected total payoff from state $m_{+1}$ . The continuation state for the individual, $m_{+1}$ , is derived as follows: At the end of the CM, the individual would have accumulated balance $y$ (measured in units of labor). In current units of nominal money, this is $\omega M\times y$ . At the beginning of next period, each individual gets a nominal transfer of new money $\tau M$ . In units of labor next period, the beginning-of-period balance would thus be $m_{+1}=\left (\omega My+\tau M\right )/\left (\omega _{+1}M_{+1}\right )$ . Replacing for $M/M_{+1}$ with the money supply process in (2.3) gives the expression for the individual’s continuation state $m_{+1}$ in (2.4).

2.3.2. Ex-post DM buyers

Now we focus on an individual who has just decided to be a DM buyer. The buyer chooses which submarket (or trading post) $\left (x,q\right )$ to enter, taking the market tightness function $\left (x,q\right )\mapsto \theta \left (x,q\right )$ as given. The individual buyer, $\textbf{s}\,:\!=\,(m,\omega )$ , has initial value:Footnote ²⁵

(2.5) \begin{align} B(\textbf{s})&=\max _{x\in [0,m],q\in \mathbb{R}_{+}}\left \{ \beta \left [1-b\left (x,q\right )\right ]\left [\bar {V}\left (\frac {\omega m+\tau }{\omega _{+1}\left (1+\tau \right )},\omega _{+1}\right )\right ]\right .\nonumber \\ &\quad +\left .b\left (x,q\right )\left [u(q)+\beta \bar {V}\left (\frac {\omega \left (m-x\right )+\tau }{\omega _{+1}\left (1+\tau \right )},\omega _{+1}\right )\right ]\right \} . \end{align}

Consider the first two terms on the RHS of Equation (2.5): With probability $1-b\left (x,q\right )\boldsymbol{\,:\!=\,}1-\lambda \left (\theta \left (x,q\right )\right )$ the buyer fails to match with the trading post and must thus continue the next period with his initial money balance subject to inflationary transfer. With the complementary probability $b\left (x,q\right )\,:\!=\,\lambda \left (\theta \left (x,q\right )\right )$ he matches with a trading post $\left (x,q\right )$ , pays the seller $x$ in exchange for a flow payoff $u\left (q\right )$ , and then continues into the next period with his net balance, also subject to inflationary transfers.

2.3.3. Ex-ante

Given a money balance $z$ , the individual decides which markets to participate in, and her value becomes

(2.6) \begin{equation} \tilde {V}\left (z,\omega \right )=\max _{a\in \left \{ 0,1\right \} }\left \{ aW\left (z,\omega \right )+\left (1-a\right )B\left (z,\omega \right )\right \} . \end{equation}

As shown in Menzio et al. (Reference Menzio, Shi and Sun2013), the resulting value function $B$ in Equation (2.5) may not be strictly concave in $m$ .Footnote ²⁶ This is the case even if primitive functions are strictly concave. As a result, the value function $\tilde {V}$ may not be concave either.Footnote ²⁷ This implies that agents can be weakly better off by choosing a lottery over the pure participation outcomes. Suppose at the beginning of a period, an agent begins with money balance $m$ . If there is a non-empty subset $\left [z_{1},z_{2}\right ]$ containing $m$ such that any weighted average of the pure-action induced values $\tilde {V}(z_{1},\omega )$ and $\tilde {V}(z_{2},\omega )$ (weakly) dominates $\tilde {V}(m,\omega )$ , then the agent will optimally play a fair lottery $\left (\pi _{1},1-\pi _{1}\right )$ over the prizes $\left \{ z_{1},z_{2}\right \}$ . This yields the ex-ante value

(2.7) \begin{equation} \bar {V}\left (\textbf{s}\right )=\max _{\pi _{1}\in [0,1],z_{1},z_{2}}\left \{ \pi _{1}\tilde {V}(z_{1},\omega )+\left (1-\pi _{1}\right )\tilde {V}\left (z_{2},\omega \right ):\,\pi _{1}z_{1}+(1-\pi _{1})z_{2}=m\right \} . \end{equation}

2.4. Monetary equilibrium

In this paper, we restrict attention to the case of a monetary equilibrium. Hereinafter, whenever we refer to “monetary equilibrium,” or “equilibrium,” we mean a (Markovian) monetary equilibrium—one in which agent’s decision functions are time-invariant maps. In what follows, we first characterize the equilibrium strategy of firms (section 2.4.1), the equilibrium value and decision functions of agents in the CM (section 2.4.2) and in the DM (section 2.4.3), and then we describe the market clearing conditions in equilibrium (section 2.4.4). At the end of this section, we define formally the notion of a stationary monetary equilibrium (SME).

2.4.1. Equilibrium strategy of firms

A firm’s problem is static. We can characterize the equilibrium behavior of a firm given $p$ (in the CM). Free entry in the CM will render zero profits to firms in equilibrium, and thus, $p=1$ . Likewise, free entry and zero-profit in the DM with competitive search imply that

(2.8) \begin{equation} r(x,q)\,:\!=\,s\left (\theta (x,q)\right )\left [x-c(q)\right ]-k\leq 0,\qquad \text{and,}\qquad \theta (x,q)\geq 0, \end{equation}

where the weak inequalities would hold with complementary slackness: For a submarket $(x,q)$ such that $r(x,q)\lt 0$ , the firm optimally chooses not to post in the submarket. If $r(x,q)=0$ , then the firm is indifferent to creating different numbers of trading posts in submarket $(x,q)$ . We can also deduce that $r(x,q)\gt 0$ cannot be an equilibrium: If expected profit is positive, then this implies $\theta (x,q)=+\infty$ , and thus $s\left (\theta (x,q)\right )=0$ which yields a contradiction to the case.Footnote ²⁸ We will restrict attention to an equilibrium where Equation (2.8) also holds for submarkets not visited by any buyer.Footnote ²⁹

From (2.8), we can deduce that

(2.9) \begin{equation} s(x,q)\equiv \mu \circ b\left (x,q\right )=\begin{cases} \frac {k}{x-c(q)} & \iff x-c(q)\gt k\\ 1 & \iff x-c(q)\leq k \end{cases}. \end{equation}

Observe that the firm’s probability of matching with a buyer, $s(x,q)\,:\!=\,\rho \left (\theta \left (x,q\right )\right )$ depends only on the posted terms of trade $(x,q)$ . Likewise, the buyer’s probability of matching with a firm is $b(x,q)\,:\!=\,\lambda \left (\theta \left (x,q\right )\right )$ , for a given the matching technology $\mu :[0,1]\rightarrow [0,1]$ . Thus, in any submarket with positive measure of buyers, the market tightness, $\theta (x,q)\equiv b(x,q)/s(x,q)$ , is necessarily and sufficiently determined by free entry into the submarket. Moreover, the terms of trade of a submarket $(x,q)$ is sufficient to identify the submarket. This implies that firms’ and agents’ optimal decision processes do not depend on the equilibrium distribution of agents. They only depend on the distribution through the aggregate statistic $\omega$ as a result of inflation. The equilibrium will be (partially) block recursive.

In equilibrium, there is a relation between $q$ and $(x,b)$ . That is, in any equilibrium, each active trading post will produce and trade the quantity:

(2.10) \begin{equation} q=Q(x,b)\equiv c^{-1}\left [x-\frac {k}{\mu (b)}\right ], \end{equation}

given payment $x$ and its matching probability $s=\mu (b)$ . This relation allows us to perform a change of variables, and re-write the buyers’ problems below in terms choices over $(x,b)$ , instead of over $(x,q)$ .

2.4.3. Equilibrium DM buyer

Observe that since $\bar {V}(\cdot, \omega ),W(\cdot, \omega )\in \mathcal{V}[0,\bar {m}]$ (i.e., are continuous, increasing and concave), then by (A.1), $\bar {V}\left (\cdot, \omega \right )\in \mathcal{V}[0,\bar {m}]$ . In an equilibrium, the DM buyer’s problem in (2.5) can be re-written as

(2.13) \begin{align} B(\textbf{s})=\max _{x\in [0,m],b\in [0,1]}\biggl \{\beta (1-b)\left [\bar {V}\left (\frac {\omega m+\tau }{\omega _{+1}\left (1+\omega \right )},\omega _{+1}\right )\right ]\nonumber \\ +b\biggl [u\left (Q(x,b)\right )+\beta \bar {V}\left (\frac {\omega \left (m-x\right )+\tau }{\omega _{+1}\left (1+\tau \right )},\omega _{+1}\right )\biggr ]\biggr \}. \end{align}

It appears as if the buyer is choosing his matching probability $b$ along with payment $x$ . This is comes from a change of variables utilizing the equilibrium relation (2.10) between quantity $q$ and terms of trade $(x,b)$ . From this we can begin to see that there will be a trade-off to the buyer, in terms of an extensive margin (i.e., trading opportunity $b$ ), and, an intensive margin (i.e., trading quantity given payment $x$ ).

The operator defined by (2.13) does not preserve concavity: The objective function in (2.13) is not jointly concave in the decisions $\left (x,b\right )$ and state $m$ , since it is bilinear in the function $b$ and the value function $\bar {V}$ , or the flow payoff function $u$ . However, using lattice programming arguments, we can still show that the resulting DM buyers’ optimal choice functions for $(x,b)$ , denoted by $\left (x^{\star },b^{\star }\right )$ , are monotone, continuous, and have unique selections.

Proposition 2 (DM value and decision functions)

For a given price sequence $\left \{ \omega, \omega _{+1},\ldots \right \}$ , the following properties hold.

1. 1. For any $\bar {V}(\cdot, \omega _{+1})\in \mathcal{V}\left [0,\bar {m}\right ]$ , the DM buyer’s value function is increasing and continuous in money balances, $B\left (\cdot, \omega \right )\in \mathcal{C}\left [0,\bar {m}\right ]$ .

2. 2. For any $m\leq k$ , DM buyers’ optimal decisions are $b^{\star }\left (m,\omega \right )=x^{\star }\left (m,\omega \right )=q^{\star }\left (m,\omega \right )=0$ , and $B\left (m,\omega \right )=\beta \bar {V}\left [\phi (m,\omega ),\omega _{+1}\right ]$ , where $\phi (m,\omega )\,:\!=\,(\omega m+\tau )/\left [\omega _{+1}(1+\tau )\right ]$ .

3. 3. At any $\left (m,\omega \right )$ , where $m\in \left [k,\bar {m}\right ]$ and the buyer matching probability is positive $b^{\star }\left (m,\omega \right )\gt 0$ :

1. (a) The optimal selections $\left (x^{\star },b^{\star },q^{\star }\right )(m,\omega )$ and $\phi ^{\star }(m,\omega )\,:\!=\,\phi \left [m-x^{\star }\left (m,\omega \right ),\omega \right ]$ , are unique, continuous, and increasing in $m$ .

2. (b) The buyer’s marginal valuation of money $B_{1}(m,\omega )$ exists if and only if the marginal ex-ante value of money $\bar {V}_{1}\left [\phi (m,\omega ),\omega \right ]$ exists.

3. (c) $B(m,\omega )$ is strictly increasing in $m$ .

4. (d) the optimal policy functions $b^{\star }$ and $x^{\star }$ , respectively, satisfy the first-order conditions

   (2.14) \begin{align} &u\circ Q\left [x^{\star }(m,\omega ),b^{\star }(m,\omega )\right ]+b^{\star }(m,\omega )\left (u\circ Q\right )_{2}\left [x^{\star }(m,\omega ),b^{\star }(m,\omega )\right ] \nonumber \\ &\quad =\beta \left [\bar {V}\left (\phi \left (m,\omega \right ),\omega _{+1}\right )-\bar {V}\left (\phi ^{\star }\left (m,\omega \right ),\omega _{+1}\right )\right ], \end{align}
   and,
   (2.15) \begin{equation} \left (u\circ Q\right )_{1}\left [x^{\star }(m,\omega ),b^{\star }(m,\omega )\right ]=\frac {\beta }{1+\tau }\left (\frac {\omega }{\omega _{+1}}\right )\bar {V}_{1}\left [\phi ^{\star }(m,\omega ),\omega _{+1}\right ]. \end{equation}

Proposition 2 states the properties of a DM agent’s value and policy functions. It extends the results from Menzio et al. (Reference Menzio, Shi and Sun2013) to the setting with non-zero inflation. The proof is given in the appendix, we summarize the idea of the proof: Part 1 of the proposition states that $B\left (\cdot, \omega \right )\in \mathcal{C}\left [0,\bar {m}\right ]$ is increasing and continuous in $m$ . This observation uses standard results from optimization and is contained in Lemma 1 of the appendix. Part 2 is proven in Lemma 2 in the appendix, and its insight here is simple: If buyers do not carry enough money to at least pay for a trading post’s fixed cost, no firm will want to set up that post in equilibrium, and so the buyers get nothing. Part 3(a) is proven in Lemma 3 using the fact that a log-transform of a DM buyer’s objective function is jointly concave in the choice variables $\left (x,b\right )$ , and is continuous in $m$ (fixing the aggregate state $\omega$ ). It nevertheless satisfies an increasing difference—and therefore, supermodularity—property. Thus, by lattice programming arguments, we can show that the DM buyer’s optimal policies are increasing in $m$ . Lemmata 4 and 5 in the appendix, together establish the following properties (Parts 3(b) and 3(c) of Proposition 2): Whenever a buyer has a chance of matching, her value function is differentiable. As a result, we can also characterize her best response in terms of a matching probability (extensive margin) and spending level (intensive margin) via Euler equations—see Part 3(d) in Proposition 2—and this is proven in Lemma 6.

2.4.4. Market clearing

In equilibrium, the total production of CM good equals its demand, $Y=C$ . Given equilibrium policy functions, $x^{\star }$ and $b^{\star }$ , the equilibrium distribution of money $G$ , and wage $\omega$ , Equation (2.10) pins down market clearing conditions for each submarket in the set of equilibrium submarkets. Money demanded must also equal money supplied:

(2.16) \begin{equation} \frac {1}{\omega }=\int m\text{d}G(m;\,\omega )\gt 0. \end{equation}

Since $M$ is the beginning of period aggregate stock of money in circulation, the LHS of (2.16), $1/\omega =M\times 1/\omega M$ , is the beginning of period real aggregate stock of money, measured in units of labor. The RHS of (2.16) is beginning of period aggregate demand, or holdings, of real money balances measured in the same unit.

2.5. Existence of a SME with a unique distribution

For the rest of the paper, we focus on a stationary monetary equilibrium (SME), which comprises the characterizations from Section 2.4, where prices are constant over time: $\omega =\omega _{+1}$ .

Definition 1. A stationary monetary equilibrium (SME), given an exogenous monetary policy $\tau$ , is: (i.) a list of value functions $\textbf{s}\mapsto (W,B,\bar {V})(\textbf{s})$ , satisfying the Bellman equation, jointly in (2.4), (2.5), (2.6) and (2.7); (ii.) a list of corresponding decision rules $\textbf{s}\mapsto (l^{\star },y^{\star },b^{\star },x^{\star },q^{\star },z^{\star },\pi ^{\star })(\textbf{s})$ supporting the value functions; (iii.) a market tightness function $\textbf{s}\mapsto \mu \circ b^{\star }(\textbf{s})$ given a matching technology $\mu$ , satisfying firms’ profit maximizing strategy (2.9) and (2.10) at all active trading posts; (iv.) a wage rate function $\textbf{s}\mapsto \omega (\textbf{s})$ satisfying the money stock adding up condition (2.16); and (v.) an ergodic distribution of real money balances $G(\textbf{s})$ satisfying an equilibrium law of motion

(2.17) \begin{equation} G(E)=T(G)(E)\,:\!=\,\int P(\textbf{s},E)\text{d}G(\textbf{s}),\qquad \forall E\in \mathcal{B}\left (S\right ), \end{equation}

where is the Borel $\sigma$ -algebra generated by open subsets of the product state space $S$ , and, $\textbf{s}\mapsto P({\textbf {s}},\cdot )$ is a Markov kernel induced by agents’ best responses $(l^{\star },x^{\star },q^{\star },z^{\star },\pi ^{\star })$ and equilibrium matching $\mu \circ b^{\star }$ .

In Online Appendix D we show that a composite Bellman function for each agent satisfies Banach’s fixed point theorem. Then, from Propositions 1 and 2, we know that agents’ decision functions are monotone and continuous. This implies that for a fixed $\omega$ , the equilibrium Markov operator on a current distribution of agents $G$ is a monotone map and satisfies measurability conditions. This implies a monotone mixing property as a result of the equilibrium self-map (2.17) on the space of distributions $G$ . These allow us to conclude that there is a unique fixed point (in a weak-convergence sense). Finally, we also show that there is at least one fixed point in the space of $\omega$ satisfying the SME conditions by utilizing the intermediate value theorem.Footnote ³⁰

3.1. A novel computational scheme

Recall that the directed search problem makes the value function $\tilde {V}$ $\left (\cdot, \omega \right )$ non-concave. Since there may exist lotteries that make agents better off than playing pure strategies over participating in DM or CM, we have to devise a means for finding these lotteries that convexify the graph of the function $\tilde {V}\left (\cdot, \omega \right )$ . A common way to do this is to discretize the function’s original domain of $\left [0,\bar {m}\right ]$ . Then, around each finite element of the domain, one must check if there is a linear segment that approximately convexifies $graph\left [\tilde {V}\left (\cdot, \omega \right )\right ]$ . This is prone to compounded errors, especially if the grid is coarse.Footnote ³¹ This approximation scheme works fine when we only have a lottery where the lower bound in the domain $\left [0,\bar {m}\right ]$ is included, i.e., a lottery on a set like $\left \{ 0,z_{2}\right \}$ , where $z_{2}\lt \bar {m}$ , as is the case of Sun and Zhou (Reference Sun and Zhou2018).Footnote ³² However, it becomes less accurate when lotteries may exist on upper segments of the function, i.e., lotteries on sets like $\left \{ z'_{1},z'_{2}\right \}$ , where $0\lt z'_{1}\lt z'_{2}\lt \bar {m}$ , but we have no prior knowledge of what the lower bound $z'_{1}$ is. When there is inflation, multiple lottery sets may arise, which will be discussed further.

Our computational contribution exploits the insight that $\tilde {V}\left (\cdot, \omega \right )$ has a bounded and convex domain, thus there exists a smallest convex set that contains its graph. This set is indeed $\text{graph}\left [\bar {V}\left (\cdot, \omega \right )\right ]$ , where $\bar {V}\left (\cdot, \omega \right )$ was defined in (2.7). The rest of the work then can be done by using a standard and robust convex-hull algorithm to back out a finite set of coordinates representing the convex hull, i.e., $\text{graph}\left [\bar {V}\left (\cdot, \omega \right )\right ]$ . Given these points, we approximate the function $\bar {V}\left (\cdot, \omega \right )$ by interpolation on a chosen continuous basis function. We use the family of shape-preserving, linear B-splines. With this algorithm, we can very quickly and directly determine the entire set of possible lotteries that exists with an arbitrarily high precision, for any given non-convex/concave function $\tilde {V}\left (\cdot, \omega \right )$ . For more details and the full algorithm for computing a SME, please see our Online Appendix E.

3.2. Statistical calibration

The CM and DM preference functions are, respectively,

\begin{equation*} U\left (C\right )-h(l)=\frac {C^{1-\sigma _{CM}}}{1-\sigma _{CM}}-l,\ \ \text{and,}\ \ u\left (q\right )=\bar {U}_{DM}\left [\ln {{\left (q+\underline {q}\right )}-{\ln {{\left (\underline {q}\right )}}}}\right ], \end{equation*}

where $\underline {q}=1\times 10^{-8}$ . Following Menzio et al. (Reference Menzio, Shi and Sun2013), the matching function specifies that a trading post’s matching probability as a function of a buyer’s matching probability is $\mu (b)=1-b$ .Footnote ³³ Note that there is no parameter required for the DM production technology, i.e., we had assumed that $c\left (q\right )=q$ .

In Table 1, we list the parameters of the model. The values of parameters $\tau$ and $\beta$ are externally pinned down, while the remaining parameters $\left (\sigma _{CM},k,\bar {U}_{DM}\right )$ are jointly calibrated to match an empirical money demand curve (including its shift and elasticity) and labor hours statistics. We will now provide an intuitive explanation of our identification and calibration strategy.

Table 1.

Benchmark estimates

^a Mean nominal interest and inflation rates in the data are annual.

^b The auxiliary statistics (data) are from a spline function fitted to the data on annual observations of the (3-month T-bill) nominal interest rate ( $i$ ) and Lucas–Nicolini New-M1-to-GDP ratio ( $M/PY$ ).

3.2.2. Money demand: identification and internal calibrations

In the model, we can map the taste parameter $\sigma _{CM}$ from the observed aggregate money demand relationship. The risk aversion parameter $\sigma _{CM}$ affects money demand through the individual money demand condition (see Equation (B.2) in our Online Appendix for detail), its related envelope condition embedded in marginal continuation value function $\bar {V}_{1}$ , and through aggregation in the overall SME. From Theorem 1, $\bar {V}_{1}$ depends on probable ex-post DM or CM outcomes. Hence, ex-ante $\bar {V}_{1}$ depends on DM and CM preferences. That is, the CRRA parameter $\sigma _{CM}$ influences equilibrium money demand.Footnote ³⁴

Likewise, from Equations (2.6), (2.10) and (2.13), the ex-post market participation problem depends on cost parameter $k$ . In turn, this influences ex-post participation value function $\tilde {V}$ . This feeds into $\bar {V}_{1}$ through the ex-ante lottery problem in Equation (2.7) and the optimal money demand condition (see Equation (B.2) in our Online Appendix for detail).

Since we focus on the long-run equilibrium, we calibrate the pair $\left (\sigma _{CM},k\right )$ to minimize the distance between the model-implied aggregate money demand relationship and an auxiliary (spline) money-demand model. The auxiliary model is fitted to long-run data from 1915 to 2007.Footnote ³⁵ Our approach of using long-run data is similar in spirit to Lucas (Reference Lucas2000).

Figure 2 depicts the model’s aggregate money demand curve (solid black line), with a three-month Treasury-bill measure of the nominal interest rate ( $i)$ and the Lucas and Nicolini (Reference Lucas and Nicolini2015) “New” M1-to-GDP ratio ( $M1/PY$ ) on the horizontal and vertical axes, respectively. The long-run data is shown as scatter points with various shapes: circles for pre-WWII observations, squares for post-WWII and pre-Great-Recession observations. The dashed line is the auxiliary, empirical money demand curve used as our target for indirectly estimating the model’s money demand (solid curve). The scatter plots indicate that the empirical money demand has shifted in several regimes in the historical data (see also, Ireland Reference Ireland2009). Following Lucas (Reference Lucas2000), we can consider our approach as specifying a model-implied money demand curve that is a “halfway-house” between these different historical episodes. Indeed, from Figure 2, we can see that the solid curve (model) lies in between the various sub-samples and is close to the empirical (auxiliary) money demand curve.

Figure 2.

Lucas and Nicolini (Reference Lucas and Nicolini2015) money demand annual data (1915–2007), model (green-dashed line) and auxiliary regression model target (red-dashed line). The red dot refers to the sample average for nominal interest.

3.3. Calibrated SME: equilibrium functions

In Figure 3, we plot the SME value functions $\left (\tilde {V},\bar {V}\right )$ in the benchmark economy. In the benchmark economy, our algorithm finds two lottery segments. We know that the graph of $W\left (\cdot, \omega \right )$ is that of an affine function. This is because the CM utility function is quasilinear, so that $W$ is linear in $m$ . The non-convex/concave value function for DM buyers is $B\left (\cdot, \omega \right )$ . We do not plot these in the figure since the upper envelope of these two graphs give us $\tilde {V}\left (\cdot, \omega \right )$ , the thick solid green line shown in the figure. (Note that due to relative scales, the lower segment or the affine part of $\tilde {V}\left (\cdot, \omega \right )$ attributable to $W\left (\cdot, \omega \right )$ may appear “flat” in the figure but it is actually an increasing affine graph.) Denote $\text{conv}\left \{ \cdot \right \}$ as the convex-hull set operator. The solid magenta graph is the graph of $\bar {V}\left (\cdot, \omega \right )$ obtained through our convex-hull approximation scheme, once we have located all the intersecting coordinates between the set $\text{graph}\left [\tilde {V}\left (\cdot, \omega \right )\right ]$ and the upper envelope of the set $\text{conv}\left \{ \text{graph}\left [\tilde {V}\left (\cdot, \omega \right )\right ],\left (0,0\right ),\left (\bar {m},0\right )\right \}$ .

Figure 3.

Value functions for benchmark economy.

Sustaining the equilibrium value functions are the policy functions $\left (l^{\star },b^{\star },x^{\star },q^{\star }\right )$ , and the lottery policies $\left (\pi _{1},1-\pi _{1}\right )$ and $\left (\pi '_{1},1-\pi '_{1}\right )$ over the prize supports $\left (z_{1},z_{2}\right )$ and $\left (z'_{1},z'_{2}\right )$ , where $\pi _{1}\left (m,\omega \right )=\left (z_{2}-m\right )/\left (z_{2}-z_{1}\right )$ and $\pi '_{1}\left (m,\omega \right )=\left (z'_{2}-m\right )/\left (z'_{2}-z'_{1}\right )$ .

The other policy functions can be seen in Figure 4. Consider the panel depicting the graph of the CM labor supply function. As shown in Proposition 1, labor supply is affine and decreasing in money balance. There are three shaded patches in the Figure’s panels. The darker (and narrowest) patch corresponds to the region where $m\in [0,k)$ . In this region, an agent will never match nor trade in the DM. The orange patches (one of which overlaps the dark-red patch) are the regions of the agent’s state space in which a lottery may be played—i.e., $\left [z_{1},z_{2}\right ]$ and $\left [z'_{1},z'_{2}\right ]$ . What matters for each agent in the SME is then the loci of these policy functions outside of the orange patch, but including the points on its boundary. These will be consistent with the equilibrium’s ergodic state space of agents. As shown in Proposition 2, the policy functions $\left (b^{\star },x^{\star },q^{\star }\right )$ are monotone in $m$ in the relevant subspace where an agent can exist at any point in time. The relevant ergodic subspace of $\left [0,\bar {m}\right ]$ in equilibrium is given by $\left \{ z_{1},\left [z_{2},z'_{1}\right ],\left [z'_{2},\bar {m}\right ]\right \} =\left \{ 0,\left [0.52,0.54\right ],\left [0.98..,\bar {m}\right ]\right \}$ in the benchmark economy in Figure 3 or Figure 4.

Figure 4.

Markov policy functions in the benchmark economy.

Given the information about our benchmark SME’s active or relevant agent state space and the corresponding policy functions, we can simulate an agent’s outcomes and also compute the equilibrium distribution of real money holdings.Footnote ³⁷ To do so, one may begin from any initial agent named $\left (m,\omega \right )$ and apply the decision rules computed earlier, as in Figure 4. Details of the algorithms for simulating the SME outcomes can be found in our Online Appendix F. We now proceed to discuss the equilibrium trade-offs faced by agents (i.e., the model mechanism) in the next section.

4.1. CM labor supply (money demand) and inflation

First, we consider the effect of inflation on the intensive margin of decision in the CM. This is summarized by the CM agents’ labor supply (which also implies money demand) response to inflation:

Figure 5.

Labor supply falls with inflation.

Notes: For reference, the circled-blue (squared-red) marker corresponds to the response of an agent with an average money balance under SME $(\tau =0)$ (SME $(\tau =10)$ ). Those who work turn out to be the agents who have zero initial money balance. The inset figure zooms in to a subset of the graphs to emphasize the relative positions of the averages.

The solid-blue (dashed-red) graph in Figure 5 depicts the SME labor supply as a function of $m$ when $\tau =0\%$ ( $\tau =10\%$ ) per annum. For illustration, the circled-blue and squared-red markers, respectively, correspond to the labor supply responses of agents with an average money balance under SME $(\tau =0)$ and SME $(\tau =10)$ . In both cases, the average is zero since the agents who work in the CM are those with zero initial money balances. The figure demonstrates that their labor supplies—and also their money demands—fall with inflation. That is, with higher inflation (tax) agents will tend to carry less money balances over time.

4.2. DM competitive-search extensive-intensive margins and inflation

Next we report the effects of inflation on the DM responses, where the extensive margin arises.

Figure 6.

DM-buyers’ matching probability and its elasticity with respect to money holdings shift down with higher inflation.

Notes: For reference, the circled-blue (squared-red) marker corresponds to the response of an agent with an average money balance under SME $(\tau =0)$ (SME $(\tau =10)$ ).

In both main panels of Figure 6, the solid-blue graphs correspond to SME $(\tau =0)$ and the dashed-red ones are for SME $(\tau =10)$ . As in the previous figure, the circled-blue and squared-red markers (with inset figure) correspond to the matching-probability response of agents with an average money balance conditional on being in the DM, under SME $(\tau =0)$ and SME $(\tau =10)$ , respectively. The uniform shift down in the matching probability function, in Figure 6 (left panel), is the extensive margin response to inflation: All else equal, higher inflation induces agents in equilibrium to face lower matching rates with DM trading posts. This margin, or force, also corresponds to a shift down in agents’ $m$ -wealth elasticity of matching probabilities in Figure 6 (right panel). That is, these agents would be optimally less $m$ -wealth sensitive in their desired matching rates.

Now consider the intensive margin of competitive search, as illustrated in Figure 7:

Figure 7.

DM-buyers’ payments schedule $x$ and inflation.

Notes: For reference, the circled-blue (squared-red) marker corresponds to the response of an agent with an average money balance under SME $(\tau =0)$ (SME $(\tau =10)$ ).

However, from the previous observation, inflation also lowers money holdings for agents entering each DM. This turns out to be particularly the case for the average DM agents. The average DM agent ends up with an outcome of matching at a lower probability, offering less payment, and this corresponds to a less elastic matching probability with respect to money balance.

Consider the firms’ side of the DM. In Figure 8 we compute the pricing function implied from the equilibrium behavioral functions for DM total payments in submarkets ( $x$ ) and traded goods ( $q$ )—i.e., $p\,:\!=\,x/q$ . Higher inflation causes the function $p$ to uniformly shift up.

Figure 8.

DM (submarkets) pricing function ( $p$ ) and inflation.

4.3. DM speed of trading and CM liquidity-management participation

The last discussion is also connected to how fast agents expect to spend their monies in the DM. This will be tied to how often they re-enter the CM to manage their liquidity positions. To see this, we illustrate the implied expected transactions per dollar, $b\circ x/m$ , under each inflationary equilibrium, SME $(\tau =0)$ and SME $(\tau =10)$ .

Although $b$ and $x$ uniformly fall with inflation (see the previous observation), each agent optimally would have a per-dollar expected expenditure response function $b\circ x/m$ that may rise or fall with inflation. This is shown in Figure 9 (left panel). However, on average they expect to be “trading faster” and offloading their money balance each time they expect to trade in the DM. Moreover, Figure 9 (right panel) shows that high- $m$ agents are also less sensitive in their speed of trading with respect to $m$ than low- $m$ agents.

Figure 9.

DM-buyers’ (implied) expected transactions per dollar, $b\circ x/m$ , and inflation.

Notes: For reference, the circled-blue (squared-red) marker corresponds to the response or elasticity of an agent with an average money balance under SME $(\tau =0)$ (SME $(\tau =10)$ ).

The equilibrium best responses above suggest to us a few key insights about agents at different money positions: At a given inflation rate, the higher money-balance (“rich”) agents are less elastic with respect to money balance in terms of their matching probabilities and velocities of spending in the DM. With higher inflation, although average outcomes of DM buyer matching rates and payments become lower, the average outcome of DM speed of transactions become higher. (This finding is further reinforced by the observation that the top-10% of DM agents trade faster relative to the bottom-10% as inflation rise. See Figure 11.)

4.4. Intensive-extensive margin: distributional effects of inflation

The previous discussion centers on equilibrium decision or allocation functions, as well as the responses of the average money-balance agents, conditional on them being in the CM or DM. We now consider agents across the entire distribution of money holding. In addition, instead of illustrating comparative equilibria using an equilibrium with 0% inflation and one with 10% inflation per annum, we now consider a set of economies ordered by different inflation at (just above) the Friedman rule to 10% per annum.

We now use the calibrated model to demonstrate how the intensive-versus-extensive-margin tension resolves, in the face of higher inflation. On the horizontal axes of the following figures, we increase the (quarterly) steady-state inflation rate, $\tau$ , within the set $\left (\beta -1,0.025\right ]$ . On the vertical axes, we measure relevant statistics for each corresponding economy under policy $\tau$ .

We begin with the distribution of agents in the DM (or the DM-conditional distribution of agents) since fundamental source of the trade-off arises in the DM. We shall see that the pattern induced by inflation on the distributions of outcomes here will also emerge in the aggregate distribution of agents’ money holdings.

In Figure 10 (top left and right panels), the dashed and circled lines denote the bottom- and the top-ten percent of outcomes, of the respective SME( $\tau$ ) distributions (conditional on agents in the DM), at each inflation rate. The “rich” face a slower decline in their total payments ( $x^{\star }$ ) for DM goods relative to the “poor,” with respect to higher inflation (Figure 10 (top-left panel)). Also, while matching probabilities of all buyers fall with inflation, the “rich” experience a slower decline in these probabilities relative to the “poor” (see Figure 10 (bottom panel)). Therefore, there is an increased dispersion in total payments and trading probabilities.

Figure 10.

Buyer matching probabilities and quantities—Mean (top panels, solid), 90% (solid-dotted) and 10% (dashed) percentiles of DM-conditional distribution of agents.

Another way to see the increased dominance of the extensive or trading-opportunity margin as inflation rises is as follows. Consider Figure 11. Since the probability of not getting matched, $1-b^{\star }\left (m\right )$ , increases with inflation for all agents in the DM, this exacerbates the cost of holding money for DM buyers who are unmatched, especially those holding higher money balances. We observe that across the distribution of agents, matching probabilities $b^{\star }\left (m\right )$ decrease with inflation. However, what matters for DM agents is how quickly they can dispose of a given amount of liquidity they carry into each DM round to exchange for DM goods. Above, we introduced a useful summary statistic: the (average) payment in the DM across buyers, $bx$ / $m$ . This ratio increases with inflation. This means that within the DM-conditional distribution, the dispersions in matching rates, payments, and transaction velocities increase with inflation, with the “rich” agents facing a less steep decline in these outcomes than the “poor” agents. This symptom is the consequence of what we outlined above regarding the dominance of the DM extensive margin as inflation rises.

Figure 11.

Top-left: on average, agents expect to have a higher per-dollar spending rate in the DM, i.e., to “trade faster.” Top-right: inflation tends to make the rich trade faster than the poor—the dispersion (90/10 ratio) in expected DM spending per-dollar rises with inflation. Bottom: agents trade faster on average in DM and return to CM quicker.

At the distributional level, we again see the intensive-versus-extensive-margins tension through Figure 11 (top-right panel): as inflation rises, their average spending per dollar rises faster relative to the “poor.” A consequence of this is that agents would also go to the CM more often to manage their liquidity, as shown in Figure 11 (bottom panel). We summarize what we have thus far:

In summary, the extensive margin effect creates more dispersion in the matching, payment and speed-of-trading outcomes in the DM. This tends to work against the redistributive or compression effect of inflation. For low inflation, the latter dominates and for higher inflation, the former takes over. Figure 12 shows the effect of this tension on the DM-conditional distribution of money, across inflation regimes. We summarize this in the following observation:

Figure 12.

Inflation and DM-conditional money distributions’ inequality statistic (ratio of 90-th to 10th percentile).

4.5. Overall money and pricing distributions and inflation

We next show that the non-monotone inequality effect of inflation in the DM gets inherited in the overall (DM-and-CM) distribution of money and prices.

Figure 13 reports an alternative Gini measure for money holdings inequality for the overall distribution.Footnote ³⁹ The green square marker in Figure 13 (bottom panel) denotes a reference SME at zero inflation, or at $\tau =0$ . The red diamond marker is at an SME with annual inflation of 10%.

Figure 13.

Inflation and the Gini coefficient for the overall money distribution.

Finally, a feature of competitive search in this Menzio et al. (Reference Menzio, Shi and Sun2013)-type model is that there is potentially an equilibrium-determined dispersion in goods’ terms-of-trade or pricing outcomes. This provides further motivation to study the effect of inflation in this Menzio et al. (Reference Menzio, Shi and Sun2013) type of monetary, heterogeneous-agent model where agents are free to choose their participation in markets, which gives rise to endogenous trading opportunities. As illustrated in Figure 14, both the average price and price dispersion increase with each inflationary equilibrium.

Figure 14.

Inflation and price dispersion. Mean (solid), 90% (solid-dotted) and 10% (dashed) percentiles of prices.

In summary, the extensive margin of search is an additional conduit for inflation to impact on the cross section of money holdings by affecting their heterogeneous matching probabilities. Unlike the results in earlier heterogeneous-agent monetary models (see, e.g., Imrohoroğlu and Prescott Reference Imrohoroğlu and Prescott1991b; Erosa and Ventura Reference Erosa and Ventura2002), inflation may not necessarily be a redistributive tax that reduces (money) wealth inequality. With a trade-off between inflation tax on the intensive margin of allocations and inflation incentivizing agents to trade faster on the extensive margin, we get non-monotone distributional consequences—a U-shaped inflation-money-inequality relationship.

5.1. Cross-model welfare cost comparisons

In Table 2, we compare our model’s welfare cost of inflation with some well-known studies in the literature. In representative-agent models such as Lucas (Reference Lucas2000) and Lagos and Wright (Reference Lagos and Wright2005) (which has additional bargaining frictions), the comparative-steady-state welfare cost of inflation can be quite high. However, this welfare cost tends to be lower when one revisits the question in a heterogeneous-agent version of the models. It is well known that the redistributive margin of inflation tax is always present in heterogeneous-agent models. This margin tends to reduce the inefficiency of holding money in the presence of inflation (see, e.g., Camera and Chien Reference Camera and Chien2014; Kocherlakota Reference Kocherlakota2005; Erosa and Ventura Reference Erosa and Ventura2002). This is also the case in random-matching versions of such models (see, e.g., Chiu and Molico Reference Chiu and Molico2010; Molico Reference Molico2006). In contrast, in heterogeneous-agent models such as Imrohoroğlu and Prescott (Reference Imrohoroğlu and Prescott1991b), with more free parameters to govern frictions, one could obtain a welfare cost of inflation as high as 0.9 percent per annum in CEV terms.

Table 2.

Welfare cost (CEV) from 0% to 10% (p.a.) inflation economy

^a Annualized CEV cost (relative to zero-inflation economy).

^b CIA: Cash-in-advance model.

^c RM: Random matching model.

^d HA: Heterogeneous agent model.

^e RA: Representative agent model.

^f TIOLI: Take-it-or-leave-it bargaining.

Figure 16.

Transition from zero- to ten-percent-inflation SME. Left: aggregate money. Right: real wage rate.

We also calculate the (mean) welfare cost of inflation between a zero-inflation and a ten-percent-inflation SME, taking into account the effects of transitional dynamics. Figure 16 shows the transition of the aggregate state variable in terms of total money holdings (left panel) and its inverse statistic which is the model’s real wage rate, $\omega$ (right panel). The vertical axes are measure in percentage deviations from the respective outcomes in the new or terminal SME. The economy is assumed to be in the initial SME( $\tau$ ) where money supply growth rate is $\tau _{-1}=0$ percent. At date $t=-1$ , money supply growth rate jumps to $\tau _{-1}^{\prime }=\tau ^{\prime }=10$ percent per annum. The economy reacts in date $t=0$ and takes some time to transit to the new SME( $\tau ^{\prime }$ ). We use a standard shooting algorithm to compute the transition. Total welfare cost of inflation, along the transition is 1.31 percent of the initial SME’s consumption, as summarized in Table 2 for our benchmark economy.