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# Irrelevance of Capital Structures with Complete Markets

# Irrelevance of Capital Structure with Complete Markets

```{contents} Contents

:depth: 2

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

- $\theta^i$ be the fraction of a 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 prospective next period random dividends $A k^\alpha e^\epsilon$.

Consumer $i$ also confronts 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 $c^i_1(\epsilon) = 0$ and solve for $-a^i(\epsilon)$

we obtain

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 arrange 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 defined in {eq}`debtlimit` depends on

Notice that $-\bar a^i(\epsilon;\theta^i)$ defined in {eq}`debtlimit` depends on

* his endowment $w_1^i(\epsilon)$ at time $1$ in state $\epsilon$

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

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

These constitute 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 his state-by-state debt limits and time $0$ and time $1$ budget constraints

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

$$

\begin{aligned}

$$

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

at time $0$, scaled Lagrange multiplier

at time $0$ and scaled Lagrange multiplier

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

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 $\epsilon$,

at time $1$ and state $\epsilon$,

then form the Lagrangian

$$

& + \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 \cr

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

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

\end{aligned}

$$

It consists of 4 functions that do the following things:

>

* `opt_k` computes the planner's optimal capital $K$

- First, create a grid for capital.

- Then for each value of capital stock in the grid, compute the left side of the planner's