H^t_t & = \check A_{22}^{t-1}(\check A_{21} + \check A_{22} H^0_0 ) \end{aligned}

$$

An optimal decision rule for the Stackelberg's choice of $u_t$ is

An optimal decision rule for the Stackelberg leader's choice of $u_t$ is

$$

u_t = - F \check y_t \equiv - \begin{bmatrix} F_z & F_x \cr \end{bmatrix}

### Comments and Interpretations

After all, at the end of the day, it will turn out that because we set

$\check z_0 = z_0$, it will be true that $z_t = \check z_t$

Because we set $\check z_0 = z_0$, it will turn out that $z_t = \check z_t$

for all $t \geq 0$.

Then why did we distinguish $\check z_t$ from $z_t$?

q_{1t} \end{bmatrix} + \begin{bmatrix} 0 \cr 1 \end{bmatrix} x_t

```

This specification assures that from the point of the view of a firm 1,

This specification assures that from the point of the view of firm 1,

$q_{2t}$ is an exogenous process.

Here

x_t = - \tilde F X_t

$$

and it's state evolves according to

and its state evolves according to

$$

\tilde X_{t+1} = (\tilde A - \tilde B \tilde F) X_t

we recover

$$

x_0 = - \tilde F \tilde X_0

x_0 = - \tilde F \tilde X_0 ,

$$

which will verify that we have properly set up a recursive

### Time Consistency of Follower's Plan

Since the follower can solve its problem using dynamic programming its

problem is recursive in what for it are the **natural state variables**,

The follower can solve its problem using dynamic programming because its

problem is recursive in **natural state variables**,

namely

$$

## Computing Stackelberg Plan

Here is our code to compute a Stackelberg plan via a linear-quadratic

dynamic program as outlined above

Here is our code to compute a Stackelberg plan via the linear-quadratic

dynamic program describe above

```{code-cell} python3

# Parameters

## Time Series for Price and Quantities

The following code plots the price and quantities

The following code plots the price and quantities produced by the Stackelberg leader and follower.

```{code-cell} python3

q_leader = yt[1, :-1]

### Explanation of Alignment

If we inspect the coefficients in the decision rule $- \tilde F$,

we can spot the reason that the follower chooses to set $x_t =

If we inspect coefficients in the decision rule $- \tilde F$,

we should be able to spot why the follower chooses to set $x_t =

\tilde x_t$ when it sets $x_t = - \tilde F X_t$ in

the recursive formulation of the follower problem.