Financial Markets

Now we assume there are no contingent markets in period zero. The consumption good(s) available for consumption in a given time period or state, must be purchased at the spot (i.e. the time period/state in which the good is available for consumption and not beforehand).

If a consumer wants to transfer income across time periods or states, the consumer must buy or sell financial assets.

Financial Assets

We assume that there are $K$ financial assets indexed by $k=1,\ldots,K$.

A financial asset $k$ is represented by a list of gross-return rates $\left(r_k^{s_t}\right)$, where $s_t$ is a state that occurs in period $t$ and $r_k^{s_t}$ is the gross rate of return (i.e., the price of the asset in state $s_t$ at time $t$ divided by the price of the asset when the asset was issued). We set the convention that the gross return of the asset is $-1$ at the state in which the asset is issued.

N.B. This financial structure is exogenous but the rate of returns of the assets will be endogenously determined in equilibrium.

Below we have the returns of an asset for a two-period economy in which there is uncertainty regarding the future (state $=1$ occurs with probability $2/3$ and state $=2$ occurs with probability $1/3$). The asset return is two in state 1 and zero in state 2:

The Economy

With the exception of the absence of contingent good markets, the existence of spot good markets and financial markets — the structure of the economy: number of time periods, number of consumers, their preferences and endowments, the information structure, etc. is identical as in the Contingent Markets Economy.

But now consumer $i$ besides choosing the consumption basket $\mathbf{c}_i=(c_i^{s_t})$ must also choose a portfolio $\mathbf{z}_i=(z_i^k)$, where $z_i^k$ is the dollar amount that consumer $i$ invests in financial asset $k$.

$z_i^k \gt 0 \;\Rightarrow\;$ the consumer buys financial asset $k$
$z_i^k \lt 0 \;\Rightarrow\;$ the consumer sells financial asset $k$

Contingent Markets vs Financial Markets

A Two Period Example With Uncertainty

We consider an economy with:

One consumption good, $c$.
Two periods, $t=0,1$.
Two states (are learned) in period $t=1$: $s=1,2$.
Probability of state 1 is $p$.
We learn/discover which state happened, only in period 1.
In period 0, there is uncertainty regarding the state of the economy in period 1.

We have 3 contingent goods:

Consumption today $c_0$.
Consumption tomorrow if state 1 occurs, $c_1$.
Consumption tomorrow if state 2 occurs, $c_2$.

Prices are:

The period zero ($t=0$, today) prices for the contingent goods are $p_0$, $p_1$ and $p_2$.
The respective spot prices are $\pi_0$, $\pi_1$ and $\pi_2$.

For a financial asset $a$, we write:

$z_a$: for the dollar amount invested in the asset (positive or negative).
$r_a^1$: for the rate of return of the asset if state 1 happens.
$r_a^2$: for the rate of return if state 2 happens.

Contingent Markets vs Financial Markets

A Two Period Example With Uncertainty (continuation)

We assume two financial assets, $a$ and $b$ and also that the consumer utility is given by,

$$U(c_0,c_1,c_2)=u(c_0)+\delta\cdot \left( p\cdot u(c_1)+(1-p)\cdot u(c_2)\right)$$

and her endowment is $\mathbf{e}=(e_0,e_1,e_2)$.

Contingent Markets:

One budget (at $t=0$):

$$\begin{align*}p_0\cdot c_0+p_1\cdot c_1+p_2\cdot c_2\le\\ \le p_0\cdot e_0+p_1\cdot e_1+p_2\cdot e_2\end{align*}$$

Choice variables are: $c_0,c_1$ and $c_2$

Financial Markets:

Three spot budgets:

$$\pi_0\cdot c_0+z_a+z_b\le \pi_0\cdot e_0$$ $$\pi_1\cdot c_1\le \pi_1\cdot e_1+z_a\cdot r_a^1+z_b\cdot r_b^1$$ $$\pi_2\cdot c_2\le \pi_2\cdot e_2+z_a\cdot r_a^2+z_b\cdot r_b^2$$

Choice variables are: $c_0,c_1,c_2,z_a$ and $z_b$.

Equilibrium in the Financial Markets Economy

Definition: A financial markets economy is in equilibrium if:

Given financial assets returns and goods' spot prices, each consumer maximizes his/her utility.
Prices are such that markets for goods clear (same as in the contingent goods economy), supply must equal demand.
Financial asset prices (equivalently, returns) are such that each financial asset market clears. That is, financial markets are in zero net-supply.

To understand the zero net-supply condition, consider the economy with only two agents, Anna and Bob. If Anna buys 3 dollars worth of asset $k$, $z_k^A=3$ then Bob must sell 3 dollars worth of asset $k$, $z_k^B=-3$. In general if we have $N$ consumers in the economy for any asset $k$ we must have

$$\sum_{i=1}^N z_k^i=0.$$

Finding a Financial Markets Equilibrium

Adam and Eve's Example

We revisit the Adam and Eve example from Contingent Markets. Same economy: endowments $\mathbf{e}_A=(100,4)$ and $\mathbf{e}_E=(9,25)$, and utilities

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

But now there are no contingent markets. Instead, a single bond with gross return $r$ is available. Each consumer chooses how much $z$ to invest in the bond:

$$c_0 = e_0 - z, \qquad c_1 = e_1 + r\cdot z.$$

Individual demand for the bond.

Each consumer solves $\max_{z}\;\sqrt{e_0 - z}+\delta\sqrt{e_1 + rz}$. The FOC gives

$$\frac{-1}{2\sqrt{e_0 - z}}+\frac{\delta\, r}{2\sqrt{e_1 + rz}}=0 \quad\Rightarrow\quad \delta\, r\,\sqrt{e_0 - z}=\sqrt{e_1 + rz}.$$

Squaring and solving for $z$:

$$\delta^2 r^2(e_0 - z)=e_1 + rz \quad\Rightarrow\quad z^*(r)=\frac{\delta^2 r^2\, e_0 - e_1}{r(1+\delta^2 r)}.$$

Adam's demand: $z_A(r)=\dfrac{100\,\delta^2 r^2 - 4}{r(1+\delta^2 r)}$.

Eve's demand: $z_E(r)=\dfrac{9\,\delta^2 r^2 - 25}{r(1+\delta^2 r)}$.

Equilibrium bond return.

The bond is in zero net supply, so $z_A + z_E = 0$:

$$\frac{100\,\delta^2 r^2 - 4 + 9\,\delta^2 r^2 - 25}{r(1+\delta^2 r)}=0 \quad\Rightarrow\quad 109\,\delta^2 r^2 = 29.$$

Solving:

$$r^* = \frac{1}{\delta}\sqrt{\frac{29}{109}} = \frac{0.51581}{\delta}.$$

Comparison with contingent markets.

In the contingent markets equilibrium, the price of tomorrow's consumption was $p_1^*=\delta\sqrt{109/29}$. Notice that

$$\frac{1}{r^*}=\delta\sqrt{\frac{109}{29}}=p_1^*.$$

The bond return is the reciprocal of the contingent price. With the bond, consumers can replicate the same trades as in the contingent market — the financial markets equilibrium allocation coincides with the contingent markets allocation.

Finding a Financial Markets Equilibrium

Adam and Eve's Example: Summary

Equilibrium bond return: $r^*=\dfrac{0.51581}{\delta}$

	Adam	Eve	Economy
$e_0$	$100$	$9$	$109$
$e_1$	$4$	$25$	$29$
$z^*$	$\dfrac{84.966\,\delta}{1.939 + \delta}$	$\dfrac{-84.966\,\delta}{1.939 + \delta}$	$0$
$c_0^*$	$\dfrac{193.872 + 15.035\,\delta}{1.939 + \delta}$	$\dfrac{17.449 + 93.966\,\delta}{1.939 + \delta}$	$109$
$c_1^*$	$\dfrac{51.581 + 4\,\delta}{1.939 + \delta}$	$\dfrac{4.642 + 25\,\delta}{1.939 + \delta}$	$29$

Finding the Equilibrium

Back To The Two Period With Uncertainty Example

We can normalize spot prices to be equal to one then using the spot budget constraints to eliminate consumption to express the utility of consumer $i$ as a function of the asset portfolio and endowments:

$$\begin{align*}&U_i(z_a^i,z_b^i)=u(e_0^i - z_a^i - z_b^i)\\ &+\;\delta\cdot\bigl(p\cdot u(e_1^i + z_a^i r_a^1 + z_b^i r_b^1)+(1-p)\cdot u(e_2^i + z_a^i r_a^2 + z_b^i r_b^2)\bigr)\end{align*}$$

Now we can apply Recipe #0 to the unconstrained maximization problem:

$$\max\limits_{z_a^i,\, z_b^i}\; U_i(z_a^i, z_b^i)$$

The FOCs are:

$$\begin{align*}\diff{U_i}{z_a^i}&=-u'(c_0^i)+\delta\bigl(p\cdot r_a^1\cdot u'(c_1^i)+(1-p)\cdot r_a^2\cdot u'(c_2^i)\bigr)=0\\[6pt] \diff{U_i}{z_b^i}&=-u'(c_0^i)+\delta\bigl(p\cdot r_b^1\cdot u'(c_1^i)+(1-p)\cdot r_b^2\cdot u'(c_2^i)\bigr)=0\end{align*}$$

N.B. We use $c_0^i, c_1^i, c_2^i$ here just to save space. Remember that $c_0^i = e_0^i - z_a^i - z_b^i$,
$c_1^i = e_1^i + z_a^i r_a^1 + z_b^i r_b^1$, and $c_2^i = e_2^i + z_a^i r_a^2 + z_b^i r_b^2$.

Solving the FOCs, we obtain the demands for the financial assets $z_a^{i}(\mathbf{r}_a, \mathbf{r}_b)$ and $z_b^{i}(\mathbf{r}_a, \mathbf{r}_b)$.

Next we impose the market clearing conditions

$$\sum_{i=1}^N z_a^i = 0 \qquad\text{and}\qquad \sum_{i=1}^N z_b^i = 0$$

and solve for $\mathbf{r}_a$ and $\mathbf{r}_b$.

The Two Period With Uncertainty Example

A Numeric Illustration

Let's assume that:

$u(c)=\ln(c)$, $p=1/2$ and $\delta=0.9$
Asset $a$ is a bond: $r_a^1 = r_a^2 = r \gt 0$
Asset $b$ pays only in state 1: $r_b^1 = s \gt r$ and $r_b^2 = 0$

The corresponding FOCs are:

$$\begin{align*}\diff{U_i}{z_a^i}&=-\frac{1}{e_0^i - z_a^i - z_b^i}+\frac{0.45\, r}{e_1^i + z_a^i r + z_b^i s}+\frac{0.45\, r}{e_2^i + z_a^i r}=0\\[6pt] \diff{U_i}{z_b^i}&=-\frac{1}{e_0^i - z_a^i - z_b^i}+\frac{0.45\, s}{e_1^i + z_a^i r + z_b^i s}=0\end{align*}$$

Solving for the demands:

$$\begin{align*}z_a^i &= -\frac{9\, e_1^i\, r + 29\, e_2^i(r-s) + 9\, e_0^i\, r\, s}{38\, r(r-s)}\\[8pt] z_b^i &= \frac{9\, e_0^i\, r(2r-s)\,s + e_1^i\, r(-20r+29s) + e_2^i(20r^2 - 11rs - 9s^2)}{38\, r(r-s)\,s}\end{align*}$$

Market clearing: letting $S_0=\sum_i e_0^i$, $S_1=\sum_i e_1^i$, $S_2=\sum_i e_2^i$,

$$\begin{align*}&\sum_i z_a^i = -\frac{9\,S_1\, r + 29\,S_2(r-s) + 9\,S_0\, r\, s}{38\, r(r-s)} = 0\\[8pt] &\sum_i z_b^i = \frac{9\,S_0\, r(2r-s)\,s + S_1\, r(-20r+29s) + S_2(20r^2 - 11rs - 9s^2)}{38\, r(r-s)\,s} = 0\end{align*}$$

Notice that we can ignore the denominators when solving for $r$ and $s$. We obtain:

$$r = \frac{20\,S_1\,S_2}{9\,S_0(S_1+S_2)} \qquad\text{and}\qquad s = \frac{20\,S_1}{9\,S_0}$$

The same illustration as above but keeping $p$ and $\delta$ as parameters.

$u(c)=\ln(c)$, $p\in(0,1)$, $\delta\in(0,1)$
Asset $a$ is a bond: $r_a^1 = r_a^2 = r \gt 0$
Asset $b$ pays only in state 1: $r_b^1 = s \gt r$ and $r_b^2 = 0$

$$\begin{align*}\diff{U_i}{z_a^i}&=-\frac{1}{e_0^i - z_a^i - z_b^i}+\frac{\delta p\, r}{e_1^i + z_a^i r + z_b^i s}+\frac{\delta(1-p)\, r}{e_2^i + z_a^i r}=0\\[6pt] \diff{U_i}{z_b^i}&=-\frac{1}{e_0^i - z_a^i - z_b^i}+\frac{\delta p\, s}{e_1^i + z_a^i r + z_b^i s}=0\end{align*}$$

$$\begin{align*}z_a^i &= \frac{(p-1)\,\delta\,r(e_1^i + e_0^i\,s) - e_2^i(r-s)(1+p\,\delta)}{r(r-s)(1+\delta)}\\[8pt] z_b^i &= \frac{\delta\,e_0^i\,r\,s(r-p\,s) + e_2^i(r-s)(r+p\,s\,\delta) + e_1^i\,r\bigl(-r+s(1+\delta-p\,\delta)\bigr)}{r(r-s)\,s\,(1+\delta)}\end{align*}$$

Market clearing: letting $S_0=\sum_i e_0^i$, $S_1=\sum_i e_1^i$, $S_2=\sum_i e_2^i$,

$$\begin{align*}&\sum_i z_a^i = \frac{(p-1)\,\delta\,r(S_1 + S_0\,s) - S_2(r-s)(1+p\,\delta)}{r(r-s)(1+\delta)} = 0\\[8pt] &\sum_i z_b^i = \frac{\delta\,S_0\,r\,s(r-p\,s) + S_2(r-s)(r+p\,s\,\delta) + S_1\,r\bigl(-r+s(1+\delta-p\,\delta)\bigr)}{r(r-s)\,s\,(1+\delta)} = 0\end{align*}$$

Ignoring the denominators and solving for $r$ and $s$:

$$r = \frac{S_1\,S_2}{\delta\,S_0\bigl((1-p)\,S_1 + p\,S_2\bigr)} \qquad\text{and}\qquad s = \frac{S_1}{p\,\delta\,S_0}$$

Financial Markets