```{math}

:label: sstack1

q_{1t} + (1+\beta + c_1) q_{1t+1} - \beta q_{1t+2} = c_0 - c_2 q_{2t+1}

- q_{1t} + (1+\beta + c_1) q_{1t+1} - \beta q_{1t+2} = c_0 - c_2 q_{2t+1}

```

Equation {eq}`sstack1` is a second-order difference equation in the sequence

**Remarks:** We have formulated the Stackelberg problem in a space of

sequences.

The max-min problem associated with Lagrangian

The max-min problem associated with firm 2's Lagrangian

{eq}`sstack4` is unpleasant because the time $t$

component of firm $1$'s payoff function depends on the entire

future of its choices of $\{q_{1t+j}\}_{j=0}^\infty$.

component of firm $2$'s payoff function depends on the entire

future of its choices of $\{q_{2t+j}\}_{j=0}^\infty$.

This renders a direct attack on the problem cumbersome.

imply that for $t \geq 1$

$$

x_t = \sum_{j=1}^t H_j^t \check z_{t-j}

\check x_t = \sum_{j=1}^t H_j^t \check z_{t-j}

$$

where

- the continuation value $- y_t P y_t$ earned by a continuation

Stackelberg leader who inherits state $y_t$ at $t$

- the value of a **reborn Stackelberg leader** who inherits state

$z_t$ at $t$ and sets $x_t = - P_{22}^{-1} P_{21}$

$z_t$ at $t$ and is free to set $x_t = - P_{22}^{-1} P_{21}$

The difference between these two values is a tell-tale sign of the time

inconsistency of the Stackelberg plan