## Overview

This lecture can be viewed as an application of the {doc}`quantecon lecture <lqcontrol>`.

This lecture can be viewed as an application of this {doc}`quantecon lecture <lqcontrol>` about linear quadratic control

theory.

It formulates a discounted dynamic program for a firm that

chooses a production schedule to balance

In the tradition of a classic book by Holt, Modigliani, Muth, and

Simon {cite}`Holt_Modigliani_Muth_Simon`, we simplify the

firm’s problem by formulating it as a linear quadratic discounted

dynamic programming problem of the type studied in this {doc}`quantecon <lqcontrol>`.

dynamic programming problem of the type studied in this {doc}`quantecon lecture <lqcontrol>`.

Because its costs of production are increasing and quadratic in

production, the firm wants to smooth production across time provided

production, the firm holds inventories as a buffer stock in order to smooth production across time, provided

that holding inventories is not too costly.

But the firm also prefers to sell out of existing inventories, a

But the firm also wants to make its sales out of existing inventories, a

preference that we represent by a cost that is quadratic in the

difference between sales in a period and the firm’s beginning of period

inventories.

We compute examples designed to indicate how the firm optimally chooses

to smooth production and manage inventories while keeping inventories

We compute examples designed to indicate how the firm optimally

smooths production while keeping inventories

close to sales.

To introduce components of the model, let

- $d(I_t, S_t) = d_1 I_t + d_2 (S_t - I_t)^2$, where

$d_1>0, d_2 >0$, be a cost-of-holding-inventories function,

consisting of two components:

- a cost $d_1 t$ of carrying inventories, and

- a cost $d_1 I_t$ of carrying inventories, and

- a cost $d_2 (S_t - I_t)^2$ of having inventories deviate

from sales

- $p_t = a_0 - a_1 S_t + v_t$ be an inverse demand function for a

be the present value of the firm’s profits at

time $0$

- $I_{t+1} = I_t + Q_t - S_t$ be the law of motion of inventories

- $z_{t+1} = A_{22} z_t + C_2 \epsilon_{t+1}$ be the law

- $z_{t+1} = A_{22} z_t + C_2 \epsilon_{t+1}$ be a law

of motion for an exogenous state vector $z_t$ that contains

time $t$ information useful for predicting the demand shock

$v_t$

We can express the firm’s profit as a function of states and controls as

$$

\pi_t = - (x_t' R x_t + u_t' Q u_t + 2 u_t' H x_t )

\pi_t = - (x_t' R x_t + u_t' Q u_t + 2 u_t' N x_t )

$$

To form the matrices $R, Q, H$, we note that the firm’s profits at

To form the matrices $R, Q, N$ in an LQ dynamic programming problem, we note that the firm’s profits at

time $t$ function can be expressed

$$

\begin{equation}

\begin{split}

\pi_{t} =&p_{t}S_{t}-c\left(Q_{t}\right)-d\left(I_{t},S_{t}\right) \\

=&\left(a_{0}-a_{1}S_{t}+v_{t}\right)S_{t}-c_{1}Q_{t}-c_{2}Q_{t}^{2}-d_{1}I_{t}-d_{2}\left(S_{t}-I_{t}\right)^{2} \\

=&a_{0}S_{t}-a_{1}S_{t}^{2}+Gz_{t}S_{t}-c_{1}Q_{t}-c_{2}Q_{t}^{2}-d_{1}I_{t}-d_{2}S_{t}^{2}-d_{2}I_{t}^{2}+2d_{2}S_{t}I_{t} \\

=&-\left(\underset{x_{t}^{\prime}Rx_{t}}{\underbrace{d_{1}I_{t}+d_{2}I_{t}^{2}}}\underset{u_{t}^{\prime}Qu_{t}}{\underbrace{+a_{1}S_{t}^{2}+d_{2}S_{t}^{2}+c_{2}Q_{t}^{2}}}\underset{2u_{t}^{\prime}Hx_{t}}{\underbrace{-a_{0}S_{t}-Gz_{t}S_{t}+c_{1}Q_{t}-2d_{2}S_{t}I_{t}}}\right) \\

=&-\left(\underset{x_{t}^{\prime}Rx_{t}}{\underbrace{d_{1}I_{t}+d_{2}I_{t}^{2}}}\underset{u_{t}^{\prime}Qu_{t}}{\underbrace{+a_{1}S_{t}^{2}+d_{2}S_{t}^{2}+c_{2}Q_{t}^{2}}}

\underset{2u_{t}^{\prime}N x_{t}}{\underbrace{-a_{0}S_{t}-Gz_{t}S_{t}+c_{1}Q_{t}-2d_{2}S_{t}I_{t}}}\right) \\

=&-\left(\left[\begin{array}{cc}

I_{t} & z_{t}^{\prime}\end{array}\right]\underset{\equiv R}{\underbrace{\left[\begin{array}{cc}

d_{2} & \frac{d_{1}}{2}S_{c}\\

I_{t}\\

z_{t}

\end{array}\right]\right)

\end{split}

\end{equation}

$$

where $S_{c}=\left[1,0\right]$.

**Remark on notation:** The notation for cross product term in the

QuantEcon library is $N$ instead of $H$.

QuantEcon library is $N$.

The firms’ optimum decision rule takes the form

x_{t+1} = (A - BF ) x_t + C \epsilon_{t+1}

$$

The firm chooses a decision rule for $u_t$ that maximizes

$$

E_0 \sum_{t=0}^\infty \beta^t \pi_t

$$

subject to a given $x_0$.

This is a stochastic discounted LQ dynamic program.

Here is code for computing an optimal decision rule and for analyzing

its consequences.

Notice that the above code sets parameters at the following default

values

- discount factor β=0.96,

- discount factor $\beta=0.96$,

- inverse demand function: $a0=10, a1=1$

- cost of production $c1=1, c2=1$

- costs of holding inventories $d1=1, d2=1$

order to shed light on the role that inventories play by comparing outcomes

with those under our two other versions of the problem.

The bottom right panel displays an production path for the original

The bottom right panel displays a production path for the original

problem that we are interested in (the blue line) as well with an

optimal production path for the model in which inventories are not

useful (the green path) and also for the model in which, although