35. Models for Fiat Money

Jun He June 01, 2026

中文English 法定货币Fiat Money 搜寻货币Search Money 交换媒介Medium of Exchange Kiyotaki-Wright 纳什议价Nash Bargaining

35. 法定货币模型

35.1 设定

连续时间设定。
单位质量的连续统主体。
有测度 $m<1$ 的货币，内在无价值。
- 每个主体持有的货币要么 $0$、要么 $1$，故只有部分个体口袋里有一单位货币。
- 假设主体不能持有超过一单位货币，即已有一单位货币时不会为换货币而生产。
每个主体都能生产与消费：
- 专业化（specialization）：任何人都不能消费自己的产品，故人们需四处走动、遇见他人以购买消费品。
- 无物物交换（no bartering）：不存在这样一对个体——他们恰好为彼此生产所需之物，故无直接物物交换可能。
  - 单一巧合相遇（single coincidence meeting）：由上述假设，相遇的两人中只可能有一人想买另一人所产之物。设单一巧合相遇的概率为 $2\alpha\in(0,1)$。
- 消费 $q$ 单位所欲商品的效用 $u(q)$，$u(\cdot)$ 递增且凹，$u'(0)=\infty$。
- 生产 $q$ 单位商品的成本 $c(q)$，$c(\cdot)$ 递增且凸，$c'(0)=0$。
- 故存在唯一的一阶最优（社会福利最大化）生产水平 $q^{\star}$，满足 $u'(q^{\star})=c'(q^{\star})$。
连续贴现率 $\rho>0$。

35.2 问题

我们要找一个竞争均衡，其中只要"可能"就交易 $q$ 单位商品——"可能"指主体 $A$ 想买主体 $B$ 能生产之物，且 $A$ 恰好持有一单位货币、$B$ 恰好无货币。

35.2.1 贝尔曼方程

记无货币主体的价值函数为 $V_0$、有货币的为 $V_1$。两个贝尔曼方程：

$$ \rho V_0=\alpha m\big(-c(q)+V_1-V_0\big) \tag{35.1} $$

$$ \rho V_1=\alpha(1-m)\big(u(q)+V_0-V_1\big) \tag{35.2} $$

(35.1)：LHS 是无货币主体单位时间的价值变化，只在该主体卖出商品换得货币时发生 = 价值变化 $(-c(q)+V_1-V_0)$ × 单一巧合相遇某一方向（卖出）概率 $\alpha$ × 对手持有货币概率 $m$。
(35.2)：LHS 是有货币主体单位时间的价值变化，只在该主体买入商品花掉货币时发生 = 价值变化 $(u(q)+V_0-V_1)$ × 某一方向（买入）概率 $\alpha$ × 对手无货币概率 $1-m$。

35.2.2 交易数量的议价

在单一巧合相遇中：买方或卖方先向对方报价，对方可立即接受、或等 $\Delta$ 时间后给出还价，如此反复直到一方接受。买方报价以一单位货币购买 $q_b(\Delta)$ 单位商品；卖方报价以一单位货币出售 $q_s(\Delta)$ 单位商品。买方希望把 $q_b(\Delta)$ 设得尽量高，使卖方在接受报价与等待还价之间无差异：

$$ \underbrace{V_1-c(q_b(\Delta))}_{\text{seller accepts now}}=\underbrace{e^{-\rho\Delta}(V_1-c(q_s(\Delta)))}_{\text{seller waits to counter}} \tag{35.3} $$

卖方希望把 $q_s(\Delta)$ 设得尽量低，使买方无差异：

$$ \underbrace{V_0+u(q_s(\Delta))}_{\text{buyer accepts now}}=\underbrace{e^{-\rho\Delta}(V_0+u(q_b(\Delta)))}_{\text{buyer waits to counter}} \tag{35.4} $$

Note

取 $\Delta\to0$ 得议价结果整理 (35.3)、(35.4) 得 (35.5)、(35.6)；由二者，$\Delta\to0$ 时 $\lim q_b(\Delta)=\lim q_s(\Delta)\equiv q$（双方的共同报价，即时议价无延迟）。把 (35.5) 与 (35.6) 的两边对应相除（(35.7)=(35.8)），$\Delta\to0$ 得 $$ > \frac{V_1-c(q)}{V_0+u(q)}=\frac{c'(q)}{u'(q)}\ \Rightarrow\ \frac{c'(q)}{V_1-c(q)}=\frac{u'(q)}{V_0+u(q)} \tag{35.9} > $$ 注意 $q$ 也是 $(V_0+u(q))(V_1-c(q))$ 的最大值点 (35.10)，f.o.c. $u'(q)(V_1-c(q))-c'(q)(V_0+u(q))=0$ 给出同样结果（纳什议价）。

Tip

注记 35.1 (35.9) 的议价结果有时称"棚内议价（in hut bargaining）"——双方坐进棚里议价，无外部选项作威胁点。亦可让每人同时与许多对手议价，则威胁点 = 不交易的价值，$q$ 解 $q\in\arg\max(V_0+u(q)-V_1)(V_1-c(q)-V_0)$；为简便我们用 (35.9)。

35.2.3 三个方程的系统

$$ \rho V_0=\alpha m\big(-c(q)+V_1-V_0\big) \tag{35.11} $$

$$ \rho V_1=\alpha(1-m)\big(u(q)+V_0-V_1\big) \tag{35.12} $$

$$ \frac{V_1-c(q)}{V_0+u(q)}=\frac{c'(q)}{u'(q)} \tag{35.13} $$

三个未知数 $V_0,V_1,q$。重写更显式：

Note

代入求解由 (35.11) 解 $V_0=\frac{-\alpha mc(q)}{\rho+\alpha m}+\frac{\alpha mV_1}{\rho+\alpha m}$，代入 (35.12) 得 $$ > V_1=\frac{\alpha(1-m)\big[(\rho+\alpha m)u(q)-\alpha mc(q)\big]}{\rho(\rho+\alpha)},\quad V_1-c(q)=\frac{(\rho+\alpha m)\big[\alpha(1-m)(u(q)-c(q))-\rho c(q)\big]}{\rho(\rho+\alpha)} \tag{35.14} > $$ $$ > V_0+u(q)=\frac{(\rho+\alpha(1-m))\big[\alpha m(u(q)-c(q))+\rho u(q)\big]}{\rho(\rho+\alpha)} \tag{35.15} > $$ 代入 (35.13) 得 $$ > \frac{(\rho+\alpha m)\big[\alpha(1-m)(u(q)-c(q))-\rho c(q)\big]}{(\rho+\alpha(1-m))\big[\alpha m(u(q)-c(q))+\rho u(q)\big]}=\frac{c'(q)}{u'(q)} \tag{35.16} > $$

由 (35.16)，LHS 与 RHS 都为正但小于 $1$（RHS 正因 $u',c'>0$；LHS 小于 $1$ 因分子小于分母）。故可得

$$ 0\le\frac{c'(q)}{u'(q)}<1\ \Leftrightarrow\ 0\le c'(q)

（因 $c'(q^{\star})=u'(q^{\star})$，$c'$ 递增、$u'$ 递减，$u'(0)=\infty$，$c'(0)=0$）。

35.2.4 解：均衡交易数量

一个显然解是非货币解 $V_1=V_0=q=0$，即完全不交易。这发生在人们对货币价值毫无信心时——无人接受货币，交易因而不可能。
(35.17) 给出的另一解是：只要交易可能，交易数量 $q$ 小于一阶最优 $q^{\star}$。直观上这是因为接受货币的一方不保证能立即用货币购买，故其对货币价值有所贴现，使货币价值对买卖双方不同。

Note

参考文献 - Kiyotaki and Wright. "On Money as a Medium of Exchange." Journal of Political Economy (1989). - Trejos and Wright. "Searching, Bargaining, Money, and Prices." Journal of Political Economy (1995).

35. Models for Fiat Money

35.1 Set-up

Consider a continuous time set-up.
There is a continuum of agents with unit mass.
There is measure $m<1$ of money available, which is intrinsically worthless.
- Money holding by each agent is either 0 or 1. So, only part of the individuals have one unit of money in their pocket.
- Assume that agent cannot hold more than one unit of money, i.e. when they already have one unit of money in pocket, they won't produce in exchange for money.
Each agent can produce and consume.
- Specialization: there is specialization among agents, i.e. anyone cannot consume his own product. So, people need to go around and meet other guys to buy their consumption good.
- No bartering: there is no such pair of individuals that they produce desired good for each other. So, no direct bartering is possible.
  - Single coincidence meeting: by the assumption above, it is only possible for one of the two meeting individuals want to buy the good produced by the other. Suppose the probability of single coincidence meeting is $2\alpha\in(0,1)$.
- Utility of consuming $q$ unit of desired goods is $u(q)$ where $u(\cdot)$ is increasing and concave, and $u'(0)=\infty$.
- Cost of producing $q$ unit of goods is $c(q)$ where $c(\cdot)$ is increasing and convex, and $c'(0)=0$.
- So, there exists a unique first best (social welfare maximizing) production level $q^{\star}$ such that $u'(q^{\star})=c'(q^{\star})$.
The continuous-discount rate is $\rho>0$.

35.2 The problem

We want to find a competitive equilibrium where $q$ units of goods are traded whenever "possible", where "possible" means that agent $A$ wants to buy goods that agent $B$ can produce, and agent $A$ happens to have one unit of money while agent $B$ happens to have no money.

35.2.1 The Bellman equation

Denote the value function for agents without money by $V_0$, and denote the value function for agents with money by $V_1$. Then, we have the following two Bellman equations:

$$ \rho V_0=\alpha m\big(-c(q)+V_1-V_0\big) \tag{35.1} $$

$$ \rho V_1=\alpha(1-m)\big(u(q)+V_0-V_1\big) \tag{35.2} $$

For (35.1), the LHS is the change in value per unit of time for agent without money, which only happens when agent sells good and gets money, which is calculated by the change in value $(-c(q)+V_1-V_0)$ multiplied by the probability of one direction (selling) single coincidence meeting happens ($\alpha$, because $2\alpha$ is for "I want to buy your good" or "you want to buy my good", which are symmetric) multiplied by the probability of the counter party having money ($m$).
For (35.2), the LHS is the change in value per unit of time for agent with money, which only happens when agent buys good and spends money, which is calculated by the change in value $(u(q)+V_0-V_1)$ multiplied by the probability of one direction (buying) single coincidence meeting happens ($\alpha$) multiplied by the probability of the counter party having no money ($1-m$).

35.2.2 Trade quantity bargaining

Suppose that in the single coincidence meeting: either buyer or seller first makes an offer to the other side, and the other side can take the offer immediately or wait for $\Delta$ amount of time and then give back a counter offer, then this process goes repeatedly until one side takes the offer. The buyer makes the offer of purchasing $q_b(\Delta)$ unit of goods (produced by the seller) with one unit of money; the seller makes the offer of selling $q_s(\Delta)$ unit of goods (produced by the seller) for one unit of money. The buyer hopes to set $q_b(\Delta)$ as high as possible such that the seller is indifferent between accepting the offer and waiting to give counter offer, i.e.

$$ \underbrace{V_1-c(q_b(\Delta))}_{\text{seller accepts now}}=\underbrace{e^{-\rho\Delta}(V_1-c(q_s(\Delta)))}_{\text{seller waits to counter}} \tag{35.3} $$

The seller hopes to set $q_s(\Delta)$ as low as possible such that the buyer is indifferent between accepting the offer and waiting to give counter offer, i.e.

$$ \underbrace{V_0+u(q_s(\Delta))}_{\text{buyer accepts now}}=\underbrace{e^{-\rho\Delta}(V_0+u(q_b(\Delta)))}_{\text{buyer waits to counter}} \tag{35.4} $$

Note

Taking $\Delta\to0$ to get the bargaining result Rearrange (35.3) and (35.4) to get (35.5) and (35.6); from these, as $\Delta\to0$ we have $\lim q_b(\Delta)=\lim q_s(\Delta)\equiv q$ (the common offer by both sides, i.e. instant bargaining without delay). Dividing the corresponding sides of (35.5) by (35.6) ((35.7)=(35.8)), and taking $\Delta\to0$, $$ > \frac{V_1-c(q)}{V_0+u(q)}=\frac{c'(q)}{u'(q)}\ \Rightarrow\ \frac{c'(q)}{V_1-c(q)}=\frac{u'(q)}{V_0+u(q)} \tag{35.9} > $$ Note that this $q$ is also a maximizer of $(V_0+u(q))(V_1-c(q))$ (35.10), whose f.o.c. $u'(q)(V_1-c(q))-c'(q)(V_0+u(q))=0$ gives the same result (Nash bargaining).

Tip

Remark 35.1 The bargaining result $q$ in (35.9) is sometimes called "in hut bargaining", which means that when two parties sit down to bargain with each other, they don't have outside option to think about and to be used as threat point to the counter party. We can alternatively form the bargaining as everyone can simultaneously bargain with many counter parties so the threat point is the value of not trading, and thus $q$ solves $q\in\arg\max(V_0+u(q)-V_1)(V_1-c(q)-V_0)$. But for simplicity and for a reasonable result, we proceed by (35.9).

35.2.3 The system of three equations

$$ \rho V_0=\alpha m\big(-c(q)+V_1-V_0\big) \tag{35.11} $$

$$ \rho V_1=\alpha(1-m)\big(u(q)+V_0-V_1\big) \tag{35.12} $$

$$ \frac{V_1-c(q)}{V_0+u(q)}=\frac{c'(q)}{u'(q)} \tag{35.13} $$

with three unknowns $V_0,V_1,q$. We can rearrange to have a more explicit expression:

Note

Substitute and solve From (35.11), $V_0=\frac{-\alpha mc(q)}{\rho+\alpha m}+\frac{\alpha mV_1}{\rho+\alpha m}$, substitute into (35.12): $$ > V_1=\frac{\alpha(1-m)\big[(\rho+\alpha m)u(q)-\alpha mc(q)\big]}{\rho(\rho+\alpha)},\quad V_1-c(q)=\frac{(\rho+\alpha m)\big[\alpha(1-m)(u(q)-c(q))-\rho c(q)\big]}{\rho(\rho+\alpha)} \tag{35.14} > $$ $$ > V_0+u(q)=\frac{(\rho+\alpha(1-m))\big[\alpha m(u(q)-c(q))+\rho u(q)\big]}{\rho(\rho+\alpha)} \tag{35.15} > $$ Plug (35.14) and (35.15) into (35.13): $$ > \frac{(\rho+\alpha m)\big[\alpha(1-m)(u(q)-c(q))-\rho c(q)\big]}{(\rho+\alpha(1-m))\big[\alpha m(u(q)-c(q))+\rho u(q)\big]}=\frac{c'(q)}{u'(q)} \tag{35.16} > $$

From (35.16), both the LHS and RHS are always positive but less than 1 (RHS is positive because of the assumptions on $u(\cdot)$ and $c(\cdot)$; LHS is less than 1 because the numerator is smaller than the denominator). So, we can conclude that

$$ 0\le\frac{c'(q)}{u'(q)}<1\ \Leftrightarrow\ 0\le c'(q)

where the last equivalence comes from $c'(q^{\star})=u'(q^{\star})$ and our assumptions that $u(\cdot)$ is increasing and concave, $u'(0)=\infty$, $c(\cdot)$ is increasing and convex, and $c'(0)=0$.

35.2.4 The solution: equilibrium trading quantity

One obvious solution from (35.17) is the non-monetary solution such that $V_1=V_0=q=0$, which means that no trade happens at all. This happens when people altogether have no confidence in the value of money at all, so none of them would accept the money, and thus this trade becomes impossible.
The other solution coming out of (35.17) is that the traded quantity $q$ whenever trading is possible is less than the first best $q^{\star}$. This equilibrium has strictly positive trading, but the level of trading is less than the first best intuitively due to the fact that the party accepting money is not guaranteed for sure to be able to immediately use that money for purchase, so the party accepting money has to discount the value of money, which makes the value of money different to the buying and selling parties.

Note

References - Kiyotaki and Wright. "On Money as a Medium of Exchange." Journal of Political Economy (1989). - Trejos and Wright. "Searching, Bargaining, Money, and Prices." Journal of Political Economy (1995).