## Overview

This notebook formulates and computes a plan that a **Stackelberg

This lecture formulates and computes a plan that a **Stackelberg

leader** uses to manipulate forward-looking decisions of a **Stackelberg

follower** that depend on continuation sequences of decisions made once

and for all by the Stackelberg leader at time $0$.

$q_{10}, q_{20}$ are given numbers that serve as initial

conditions at time $0$.

By incurring a cost of change

By incurring a cost equal to

$$

\gamma v_{it}^2

\gamma v_{it}^2, \quad \gamma > 0 ,

$$

where $\gamma > 0$, firm $i$ can change its output according

firm $i$ can change its output according

to

$$

where

$c_0 = \frac{\beta a_0}{2 \gamma}, c_1 = \frac{\beta a_1}{\gamma}, c_2 = \frac{\beta a_1}{2 \gamma}$.

This equation can in turn be rearranged to become the second-order

difference equation

This equation can in turn be rearranged to become

```{math}

:label: sstack1

Operating on both sides of equation {eq}`sstack2` with

$\beta^{-1}$ times this inverse operator gives the follower's

decision rule for setting $q_{1t+1}$ in the

**feedback-feedforward** form.

**feedback-feedforward** form

```{math}

:label: sstack3

subject to initial conditions for $q_{1t}, q_{2t}$ at $t=0$.

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

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

sequences.

The max-min problem associated with Lagrangian

This renders a direct attack on the problem cumbersome.

Therefore, below, we will formulate the Stackelberg leader's problem

Therefore, below we will formulate the Stackelberg leader's problem

recursively.

We'll put our little duopoly model into a broader class of models with

the same conceptual structure.

the same structure.

## Stackelberg Problem

Let $u_t$ be a vector of decisions chosen by the Stackelberg leader

at $t$.

The $z_t$ vector is inherited physically from the past.

The $z_t$ vector is inherited from the past.

But $x_t$ is a decision made by the Stackelberg follower at time

$t$ that is the follower's best response to the choice of an

The two subproblems are designed

- to respect the protocol in which the follower chooses

- to respect the timing protocol in which the follower chooses

$\vec q_1$ after seeing $\vec q_2$ chosen by the leader

- to make the leader choose $\vec q_2$ while respecting that

$\vec q_1$ will be the follower's best response to

$\vec q_2$

- to represent the leader's problem recursively by artfully choosing

the state variables confronting and the control variables available

the leader's state variables and the control variables available

to the leader

**Subproblem 1**

We'll compute the present value earned by the Stackelberg leader.

We'll compute it two ways (they give identical answers -- just a check

on coding and thinking)

We'll compute it two ways and get the same answer.

In addition to being a useful check on the accuracy of our coding, computing things in these two ways helps us think about the structure of the problem.

```{code-cell} python3

v_leader_forward = np.sum(βs * π_leader)