Contingent Markets

It's a Wonderful Life

Market Interdependence

Building & Loan → Housing Market:: George's institution gives loans to regular working families (like Mr. Martini the immigrant) who traditional banks would reject. Each loan creates a new homeowner in Bailey Park instead of a slum renter.
Housing → Labor Market:: When families own homes in Bailey Park, they're not paying high rents. They have stability, their kids can go to school longer (like Harry going to college), and workers can take better jobs rather than whatever low-wage positions are available.
Labor Market → Retail/Main Street:: With better jobs and no crushing rents, people have disposable income. Gower can run his pharmacy, Martini can open his bar, Ernie can run his taxi business. The town supports diverse small businesses.
Retail/Main Street → Building & Loan:: Those successful small businesses and stable working families put their savings back in the Building & Loan. Creating the funds for more loans.

It's a Miserable Life

Market Interdependence

Potter's Bank → Housing Market:: Without the Building & Loan, working families cannot get loans from Potter's bank. Potter rejects them, forcing everyone to rent from him in Potter's Field slums at extortionate prices.
Housing → Labor Market:: Families trapped paying Potter's high rents cannot save or invest in education. Children leave school early for work. Workers must accept whatever Potter offers, creating a captive labor force with suppressed wages.
Labor Market → Retail/Main Street:: With minimal disposable income after paying Potter's rents, families cannot support local businesses. Main Street becomes Pottersville - pawn shops, bars, and predatory lenders. Small business diversity disappears.
Retail/Main Street → Potter's Bank:: Without competitive businesses or savings alternatives, all wealth flows to Potter. His bank monopolizes deposits and lending. The economy stagnates in extractive equilibrium with Potter controlling all capital flows.

A Pure Trade Economy Example

Market interdependence grows complex quickly, so we focus on pure trade economies --- economies without production.

Consider an economy with two agents (Adam and Eve), one consumption good (apples), and two periods (today and tomorrow). Each agent owns an apple tree that yields a fixed harvest in each period; we call this the agent's endowment.

In a pure-trade economy, harvesting is effortless --- no inputs are required. Adam and Eve simply decide how many apples to buy or sell from their endowments.

A Pure Trade Economy Example

(continuation)

There are two markets in the economy. In the first market, one can buy or sell apples for delivery today. In the second market, one can buy or sell apples for delivery tomorrow. All trading happens today. No trades (only deliveries) happen tomorrow.

An agent maximizes the discounted utility of consumption given the budget constraint:

$$\max_{\substack{\phantom{1}\\\mathrm{s.t.}\\ \phantom{1}\\p_0 \cdot c_0 + p_1 \cdot c_1 \le p_0 \cdot e_0 + p_1 \cdot e_1}} u(c_0) + \delta \cdot u(c_1)$$

where $c_0$ is the amount of apples consumed today, $c_1$ is the amount of apples consumed tomorrow, $e_0$ is the endowment of apples today, and $e_1$ is the endowment of apples tomorrow.

The Economy

The Environment In Which Our Story Takes Place

These are the building blocks of our plot:

Time

Time is discrete and the horizon is finite, $t=0,1,\ldots,T$.

At period $t$, consumers learn which state $s_t$ is realized. Consumers have symmetric information — they share a common filtration $\mathcal{F}=\left(\mathcal{F}_t\right)_{t\in T}$.

States

A state is a pair $s_t=(t,I_t)$ where $t$ is the time period, $I_t$ is an information set in the partition $\mathcal{F}_t$.

At each state $s_t$, there is a single composite good. Let $c_{s_t}$ denote the quantity consumed.

Consumers' Preferences

There are $J$ consumers and each consumer $j$ has a utility function $u^j$ for the consumption good and maximizes the expected discounted utility $U^j = \sum_{t=0}^T \delta^t \, \mathbb{E}[u^j(c_{s_t})]$ for $j=1,\ldots,J$.

Click here to see how to set up consumers' discounted expected utility given the time horizon and the information structure (filtration).

Contingent Goods

Definition

A contingent good is a contract to deliver one unit of the consumption good at some specified date only if some pre-specified state occurs.

Two contingent goods are distinct whenever they differ in the date or event that triggers the delivery --- even if the underlying physical good is identical. Their prices need not be equal. For instance, a barrel of oil delivered on December 31st, 2027 is a different contingent good than one delivered on July 2nd, 2029.

Prices

Let $p_{s_t}$ denote the market price of the contingent good (contract) that delivers one unit of consumption good when state $s_t$ occurs in period $t$ and delivers nothing otherwise.

Agents buy and sell contingent good contracts exclusively in period zero; $p_{s_t}$ is paid upfront. The spot price $\pi_{s_t}$, by contrast, is paid in period $t$ when state $s_t$ occurs.

Complete Markets

When every contingent good can be traded in period zero, we say markets are complete. If some contingent good cannot be traded — a missing market — we say markets are incomplete.

Endowments

Definition

An individual's endowment, $\mathbf{e}_j=(e^j_{s_t})$, is the basket of (contingent) goods that individual $j$ possesses prior to any trade or consumption takes place.

An individual's income is the value of his/her endowment basket.

In a pure trade economy, the supply of each contingent good is simply the aggregate endowment of that good — there is no production. Supply is perfectly inelastic.

Complete Markets Equilibrium

Assume there are $J$ consumers and $K$ contingent goods.

Definition: Market Equilibrium

An equilibrium is a list of prices (the price of each contingent good) $\mathbf{p}=(p_{1},p_{2},\ldots,p_{K})$ and a basket of goods $\mathbf{x}_j=(x^j_{1},x^j_{2},\ldots,x^j_{K})$ for each consumer $j$ so that:

Given the prices $\mathbf{p}$, consumer $j$ chooses $\mathbf{x}_j$ to maximize his/her discounted expected utility, $U^j$, subject to the budget constraint, $\mathbf{p}\cdot \mathbf{x}_j \leq \mathbf{p}\cdot \mathbf{e}_j$.
Given the consumer decisions $\mathbf{x} = (\mathbf{x}_j)$, the market of each contingent good $k$ clears (supply equals demand), $\sum_{j=1}^{J} e^j_{k}=\sum_{j=1}^{J} x^j_{k}$.

To solve the consumer's maximization problem in (1), we use Recipe #1 and the fact that the consumer's income is the value of her endowment, $i=\mathbf{p}\cdot\mathbf{e}$.

Notice that we have $K$ markets and $K-1$ (relative) prices to find. So we solve the system of market-clearing equations, demand of good $k$ = supply of good $k$, in $K-1$ markets of our choosing.

Finding a market equilibrium

A dynamic example without uncertainty

We consider an economy with two periods, no uncertainty, and two consumers, Adam and Eve, whose endowments are $\mathbf{e}_A=(100,4)$ and $\mathbf{e}_E=(9,25)$, and utilities

$$U^A(c_0,c_1)=\sqrt{c_0}+\delta\sqrt{c_1} \qquad\text{and}\qquad U^E(c_0,c_1)=\sqrt{c_0}+\delta\sqrt{c_1}.$$

For simplicity, set $p_0=1$.

Answer: Why can we do this normalization?

Answer

Divide both sides of the budget constraint by $p_0\gt 0$:

$$\begin{align*} p_0c_0+p_1c_1&\le p_0e_0+p_1e_1\\ \iff\; c_0+p\,c_1&\le e_0+p\,e_1,\qquad\text{where } p=\frac{p_1}{p_0}. \end{align*}$$

The budget set, and so each consumer's demand, depends only on the relative price $p$: multiplying $p_0$ and $p_1$ by the same positive number changes nothing. Only relative prices are determined in equilibrium, so we are free to measure prices in units of good 0 by setting $p_0=1$; then $p_1=p$.

Individual demands.

Each consumer solves $\max\sqrt{c_0}+\delta\sqrt{c_1}$ s.t. $c_0+p_1 c_1=i$. The FOCs give $\frac{1}{2\sqrt{c_0}}=\lambda$ and $\frac{\delta}{2\sqrt{c_1}}=\lambda p_1$, so $\sqrt{c_0}=\frac{1}{2\lambda}$ and $\sqrt{c_1}=\frac{\delta}{2\lambda p_1}$. So, $\sqrt{c_1}=\frac{\delta}{p_1}\,\sqrt{c_0}$ and $c_1=\frac{\delta^2}{p_1^2}\,c_0$. Substituting $c_1=\frac{\delta^2}{p_1^2}\,c_0$ into the budget constraint and solving for $c_0$ yields,

$$c_0^*(p_1,i)=\frac{p_1\, i}{p_1+\delta^2}.$$

And substituting back the optimal $c_0$ into $c_1=\frac{\delta^2}{p_1^2}\,c_0$ we obtain

$$c_1^*(p_1,i)=\frac{\delta^2\, i}{p_1(p_1+\delta^2)}.$$

Since the income of Adam is $i^A=100+4p_1$, Adam's demand for current consumption is $c_0^A(p_1)=\dfrac{p_1}{p_1+\delta^2}(100+4p_1)$.

Since the income of Eve is $i^E=9+25p_1$, Eve's demand for current consumption is $c_0^E(p_1)=\dfrac{p_1}{p_1+\delta^2}(9+25p_1)$.

Market equilibrium price.

The market demand for good 0 is

$$D_0(p_1)=\frac{p_1}{p_1+\delta^2}(109+29p_1).$$

Equating with total supply $S_0=100+9=109$:

$$\begin{align*} \frac{p_1}{p_1+\delta^2}(109+29p_1)&=109 \;\Rightarrow\; 109p_1+29p_1^2=109p_1+109\delta^2\\ \Rightarrow\; 29p_1^2&=109\delta^2 \;\Rightarrow\; p_1^*=\delta\sqrt{\frac{109}{29}} = 1.93872\,\delta. \end{align*}$$

Market equilibrium quantities.

To obtain the actual equilibrium quantities, we plug the market equilibrium price $p_1^* = 1.93872\,\delta$ back into Adam and Eve's respective individual demands.

Notice that in equilibrium Adam sells good 0 since $e_0^A \gt x_0^A$ and buys good 1 since $x_1^A \gt e_1^A$, while Eve does the opposite.

Finding a market equilibrium

An Example: Summary

$\delta$

Equilibrium price: $p_1^*=1.93872\,\delta={}$

	Adam	Eve	Economy
$e_0$	$100$	$9$	$109$
$e_1$	$4$	$25$	$29$
$i$
$c_0^*$			$109$
$c_1^*$			$29$

Finding a market equilibrium

A static example with uncertainty

We consider an economy with one period, two-state uncertainty, and two consumers, Adam and Eve, whose endowments are $\mathbf{e}_A=(100,4)$ and $\mathbf{e}_E=(9,25)$. State 1 happens with probability $q$ and state 2 happens with probability $1-q$, so the expected utilities are

$$\begin{align*} U^A(c_1,c_2)&=q\sqrt{c_1}+(1-q)\sqrt{c_2}\quad\text{and}\\ U^E(c_1,c_2)&=q\sqrt{c_1}+(1-q)\sqrt{c_2}. \end{align*}$$

For simplicity, set $p_1=1$.

Answer: Why can we do this normalization?

Answer

Divide both sides of the budget constraint by $p_1\gt 0$:

$$\begin{align*} p_1c_1+p_2c_2&\le p_1e_1+p_2e_2\\ \iff\; c_1+p\,c_2&\le e_1+p\,e_2,\qquad\text{where } p=\frac{p_2}{p_1}. \end{align*}$$

The budget set, and so each consumer's demand, depends only on the relative price $p$: multiplying $p_1$ and $p_2$ by the same positive number changes nothing. Only relative prices are determined in equilibrium, so we are free to measure prices in units of good 1 by setting $p_1=1$; then $p_2=p$.

Individual demands.

Each consumer solves $\max q\sqrt{c_1}+(1-q)\sqrt{c_2}$ s.t. $c_1+p_2 c_2=i$. The FOCs give $\frac{q}{2\sqrt{c_1}}=\lambda$ and $\frac{1-q}{2\sqrt{c_2}}=\lambda p_2$, so $\sqrt{c_1}=\frac{q}{2\lambda}$ and $\sqrt{c_2}=\frac{1-q}{2\lambda p_2}$. So, $\sqrt{c_2}=\frac{1-q}{q\,p_2}\,\sqrt{c_1}$ and $c_2=\frac{(1-q)^2}{q^2p_2^2}\,c_1$. Substituting $c_2=\frac{(1-q)^2}{q^2p_2^2}\,c_1$ into the budget constraint and solving for $c_1$ yields,

$$c_1^*(p_2,i)=\frac{q^2p_2\, i}{q^2p_2+(1-q)^2}.$$

And substituting back the optimal $c_1$ into $c_2=\frac{(1-q)^2}{q^2p_2^2}\,c_1$ we obtain

$$c_2^*(p_2,i)=\frac{(1-q)^2\, i}{p_2\left(q^2p_2+(1-q)^2\right)}.$$

Since the income of Adam is $i^A=100+4p_2$, Adam's demand for consumption in state 1 is $c_1^A(p_2)=\dfrac{q^2p_2}{q^2p_2+(1-q)^2}(100+4p_2)$.

Since the income of Eve is $i^E=9+25p_2$, Eve's demand for consumption in state 1 is $c_1^E(p_2)=\dfrac{q^2p_2}{q^2p_2+(1-q)^2}(9+25p_2)$.

Market equilibrium price.

The market demand for good 1 is

$$D_1(p_2)=\frac{q^2p_2}{q^2p_2+(1-q)^2}(109+29p_2).$$

Equating with total supply $S_1=100+9=109$:

$$\begin{align*} \frac{q^2p_2}{q^2p_2+(1-q)^2}(109+29p_2)&=109\\ \Rightarrow\; 109q^2p_2+29q^2p_2^2&=109q^2p_2+109(1-q)^2\\ \Rightarrow\; 29q^2p_2^2&=109(1-q)^2\\ \Rightarrow\; p_2^*&=\frac{1-q}{q}\sqrt{\frac{109}{29}} = 1.93872\,\frac{1-q}{q}. \end{align*}$$

Market equilibrium quantities.

To obtain the actual equilibrium quantities, we plug the market equilibrium price $p_2^* = 1.93872\,\frac{1-q}{q}$ back into Adam and Eve's respective individual demands.

Notice that in equilibrium Adam sells good 1 since $e_1^A \gt x_1^A$ and buys good 2 since $x_2^A \gt e_2^A$, while Eve does the opposite.

Finding a market equilibrium

An Example: Summary

$q$

Equilibrium price: $p_2^*=1.93872\,\frac{1-q}{q}={}$

	Adam	Eve	Economy
$e_1$	$100$	$9$	$109$
$e_2$	$4$	$25$	$29$
$i$
$c_1^*$			$109$
$c_2^*$			$29$