This simplification of BCG’s setup helps us by

{cite}`Modigliani_Miller_1958`- creating a benchmark economy to compare with outcomes in BCG’s

`- creating a benchmark economy to compare with outcomes in BCG’s

incomplete markets economy

- creating a good guess for initial values of some equilibrium objects

to be computed in BCG’s incomplete markets economy via an iterative

algorithm

- illustrating classic complete markets outcomes that include

- indeterminacy of consumers’ portfolio choices

- indeterminacy of firms' financial structures that underlies a

Modigliani-Miller theorem

Modigliani-Miller theorem {cite}`Modigliani_Miller_1958

- introducing `Big K, little k` issues in a simple context that will

recur in the BCG incomplete markets environment

The economy lasts for two periods, $t=0, 1$.

There are unit measure continua of two types of consumers named $i=1,2$.

There are two types of consumers named $i=1,2$.

A scalar random variable $\epsilon$ with probability density $g(\epsilon)$ affects both

### Objects appearing in a competitive equilibrium

We let

Let

- $a^i(\epsilon)$ be consumer $i$’s purchases of claims

- $a^i(\epsilon)$ be consumer $i$ ’s purchases of claims

on time $1$ consumption in state $\epsilon$

- $q(\epsilon)$ be a pricing kernel for one-period Arrow

securities

- $\theta_0^i \geq 0$ be consumer $i$’s intial share of

the firm, $\sum_i \theta_0^i =1$

- $\theta^i$ be the fraction of the firm’s shares purchased by

consumer $i$ at time $t=0$

- $- \bar a^i(\epsilon; \theta^i)$ be debt limits constraining consumer $i$ ' s issues of claims

on time $1$ consumption in state $\epsilon$

- $V$ be the value of the representative firm

- $\tilde V$ be the value of equity issued by the representative

firm

$q(\epsilon)$, a given value of a firm $V$ that has chosen capital stock $k$, a price of

equity $\tilde V$, and given prospective random dividends next period $A k^\alpha e^\epsilon$.

Consumer $i$ also faces a state-by-state borrowing limit that restricts quantities of Arrow securities that he can issue.

If we evaluate consumer $i$'s time $1$ budget constraint at zero consumption $c^i(\epsilon) = 0$ and solve for $-a^i(\epsilon)$

we obtain

$$

- \bar a^i(\epsilon;\theta^i) = w_1^i(\epsilon) +\theta^i A k^\alpha e^\epsilon

$$

The quantity $- \bar a^i(\epsilon;\theta^i)$ is the maximum amount that it is feasible for consumer $i$ to repay to

his Arrow security creditors at time $1$ in state $\epsilon$.

To implement trading with one-period Arrow securities, we must impose on agent $i$ the state-by-state debt limits

$$

-a^i(\epsilon) \leq - \bar a^i(\epsilon;\theta^i)

$$

Notice that consumer $i$'s borrowing limit depends on

* his endowment at time $1$ in state $\epsilon$

* his share $\theta^i$ of a representive firm's dividends

These are the two sources of **collateral** that back the consumer's issues of Arrow securities that pay off in state $\epsilon$

Consumer $i$ chooses a scalar $c_0^i$ and a function

$c_1^i(\epsilon)$ to maximize

$$

u(c_0^i) + \beta \int u(c_1^i(\epsilon)) g (\epsilon) d \epsilon

$$

subject to time $0$ and time $1$ budget constraints

subject to his state-by-state debt limits and the time $0$ and time $1$ budget constraints

$$

\begin{aligned}

c_0^i & \leq w_0^i +\theta_0^i V - \int q(\epsilon) a^i(\epsilon) d \epsilon - \theta^i \tilde V \cr

c_1^i & \leq w_1^i(\epsilon) +\theta^i A k^\alpha e^\epsilon + a^i(\epsilon)

c_1^i(\epsilon) & \leq w_1^i(\epsilon) +\theta^i A k^\alpha e^\epsilon + a^i(\epsilon)

\end{aligned}

$$

Attach Lagrange multiplier $\lambda_0^i$ to the budget constraint

at time $0$ and scaled Lagrange multiplier

at time $0$, scaled Lagrange multiplier

$\beta \lambda_1^i(\epsilon) g(\epsilon)$ to the budget constraint

at time $1$ and state $\epsilon$ and form the Lagrangian

at time $1$ and state $\epsilon$, and scaled Lagrange multiplier

$\beta \phi_1^i(\epsilon) g(\epsilon)$ to the debt limit at time $1$ and state $\epsilon$,

then form the Lagrangian

$$

\begin{aligned}

L^i & = u(c_0^i) + \beta \int u(c^i_1(\epsilon)) g(\epsilon) d \epsilon \cr

& + \lambda_0^i [ w_0^i + \theta_0^i - \int q(\epsilon) a^i(\epsilon) d \epsilon -

\theta^i \tilde V - c_0^i ] \cr

& + \beta \int \lambda_1^i(\epsilon) [ w_1^i(\epsilon) + \theta^i A k^\alpha e^\epsilon

+ a^i(\epsilon) c_1^i(\epsilon) g(\epsilon)] d \epsilon

+ a^i(\epsilon) c_1^i(\epsilon) ] g(\epsilon) d \epsilon \cr

& + \beta \int \phi_1^i(\epsilon) [ - \bar a^i(\epsilon; \theta^i) - a^i(\epsilon) ] g(\epsilon) d \epsilon

\end{aligned}

$$

First-order necessary conditions for an optimum with respect to

$c_0^i, c_1^i(\epsilon)$, and $a^i(\epsilon)$ are

Off corners, first-order necessary conditions for an optimum with respect to

$c_0^i, c_1^i(\epsilon)$ and $a^i(\epsilon)$ are

$$

\begin{aligned}

q(\epsilon) = \beta \left( \frac{u'(c_1^i(\epsilon))}{u'(c_0^i)} \right) g(\epsilon)

```

To deduce a restriction on the consumer’s choice of $\theta^i$ we

To deduce a restriction on equilibrium prices, we

solve the period $1$ budget constraint to express

$a^i(\epsilon)$ as