\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 08, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2016/08\hfil Hamilton-Jacobi-Bellman equations]
{Analytic solutions for Hamilton-Jacobi-Bellman equations}
\author[A. Palestini \hfil EJDE-2016/08\hfilneg]
{Arsen Palestini}
\address{Arsen Palestini \newline
MEMOTEF, Sapienza University of Rome,
Via del Castro Laurenziano 9,
00161 Rome, Italy}
\email{Arsen.Palestini@uniroma1.it}
\dedicatory{Communicated by Ludmila S. Pulkina}
\thanks{Submitted August 14, 2016. Published January 10, 2017.}
\subjclass[2010]{35C05, 49L20, 91A23}
\keywords{Hamilton-Jacobi-Bellman equations; feedback equilibrium;
\hfill\break\indent differential game}
\begin{abstract}
Closed form solutions are found for a particular class of
Hamilton-Jacobi-Bellman equations emerging from a differential game
among firms competing over quantities in a simultaneous oligopoly
framework. After the derivation of the solutions, a microeconomic
example in a non-standard market is presented where feedback
equilibrium is calculated with the help of one of the previous formulas.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
In differential games and optimal control theory,
Hamilton-Jacobi-Bellman (HJB) equations have played a major role in the
previous decades. A wide range of models have been introduced and analyzed
whose solutions have been either determined or approximated with the help
of Dynamic Programming techniques, and especially feedback strategies are
considered a key solution concept.
Just to cite a few fundamental recent textbooks on the different aspects of
this issue: the main theoretical contribution is probably the volume by
Seierstad and Sydsaeter (\cite{Seierstad}, 1986), whereas the textbook
by Dockner et al.\ \cite{Dockner} is a major contribution including a wide
range of applications to a lot of economic models. J{\o}rgensen and
Zaccour \cite{Jorgensen2} focus on marketing models especially, furthermore
a very rich treatment on HJB equations is provided in Bardi and
Capuzzo Dolcetta \cite{Bardi}.
In recent decades, a variegated stream of literature has found a relevant
development on several aspects of HJB equations: a theoretical investigation
involving results on viscosity solutions can be found in Lions and
Souganidis \cite{Lions}, relevant properties of the value function in
infinite horizon optimal control problems are found out in Baumeister et al.\
\cite{Baumeister}, an iterative dynamic programming method for $2$-agent
games is introduced by Zhang et al.\ \cite{Zhang}, a study on differential
games with non-constant discounting is proposed by Marin-Solano and
Shevkoplyas \cite{Marinsolano}, necessary and sufficient conditions for
feedback equilibria in linear-quadratic games are established in \cite{Engwerda}.
As far as economic models are concerned, in the latest years the attention of
economic literature has been directed towards oligopoly models with
several different market effects: the survey carried out by J{\o}rgensen and
Zaccour \cite{Jorgensen1} provides an extensive outline of applications until 2007.
Other relevant contributions are due to Wirl \cite{Wirl}, who discusses the
properties of the optimal value function in a scenario with polluting emissions,
whereas Erickson \cite{Erickson} studies a dynamic advertising
framework. Prasad et al.\ \cite{Prasad} derive feedback equilibrium in an
advertising model where customers switch to competing brands, and HJB equations
are solved also by Colombo and Dawid \cite{Colombo4} in a scenario where
technological spillovers appear. On the other hand, Lambertini and
Mantovani \cite{Lambertini3} derive feedback strategies in a dynamic renewable
resource oligopoly under pre-emption and subject to voracity effects.
All these papers, and many others, feature solutions to the related HJB equations
in linear-quadratic structures and standard demand on the markets.
However, relatively little attention has been paid to industrial organization
models where the inverse demand function of the market has a non-standard form
such as the hyperbolic one. A preliminary microfoundation and some results in
such a setting can be found in Lambertini \cite{Lambertini1} (dynamic framework)
and Colombo \cite{Colombo1} (static framework), whereas in Lambertini and
Palestini \cite{Lambertini2}, the derivation and solution of the HJB equations
originating from this framework are presented and discussed.
In this article, I would like to extend such treatment by focussing on a class
of HJB equations which are strictly connected to oligopoly differential games
with hyperbolic inverse demand functions. Differently from recent papers
which analyze several formulations of linear-quadratic differential games
(i.e. \cite{Colombo1,Colombo2}), I take into account games whose structure
is based on polynomials having degree higher than $2$.
To solve the HJB equation, the approach I adopt is the same as in most
literature on differential games and optimal control applied to economics
(see, for example \cite{Dockner}): a guess for $V^*(\cdot)$ is chosen and
then the explicit formulation of its coefficients is established.
Here is a brief outline of the main results:
\begin{itemize}
\item A class of HJB equations having a polynomial term in one of the two
arguments of the unknown function is taken into account and solved in closed form.
\item An oligopoly differential game is introduced with a hyperbolic inverse
demand function. By deriving the HJB equation for this model, the structure we
obtain is the one that can be solved.
\item The application of the formula is exhibited and the Nash feedback
strategy is determined.
\end{itemize}
The remainder of this article is as follows. Section $2$ features the
main findings on the solution of a class of HJB equations with two different
choices of parameters. Section $3$ introduces an application to a $3$-firm
differential game where the inverse demand function of the market is hyperbolic.
The value function and the Nash feedback strategy are explicitly calculated.
Section 4 concludes and outlines some possible future improvements.
\section{Analytic solutions}
Consider $m \in \mathbb{Z}_+$ such that $m \geq 1$.
Let us introduce a family of HJB equations, having
$V(x,t) \in C^1((0,+\infty) \times [0,T])$ as their unknowns:
\begin{equation}
\frac{\partial V(x,t)}{\partial t}-\rho V(x,t)
= \sum_{l=0}^m \beta_l x^l +\alpha x^\gamma \frac{\partial V(x,t)}{\partial x},
\label{solvableHJB}
\end{equation}
where $\gamma \in \left\{0,1 \right\}$. Functions $V$ are defined on
$(x,t) \in (0,+\infty) \times [0,T]$, such that the boundary condition
$V(x,T)=0$ holds for all $x >0$.
To solve \eqref{solvableHJB}, we choose a suitable guess, turning out
to be a polynomial in $x$ having the same degree of the polynomial
which appears at the right-hand side of \eqref{solvableHJB}.
The two cases will be separated based on the value of $\gamma$.
The arguments of $V$ will often be omitted for simplicity during the
derivation of the solutions to \eqref{solvableHJB}.
\subsection{Solution for $\gamma=0$}
When $\gamma=0$, $\alpha$ is the only coefficient of the first-order partial
derivative of $V(\cdot)$ with respect to $x$, i.e.
\begin{equation}
\frac{\partial V}{\partial t}-\rho V
= \sum_{l=0}^m \beta_l x^l +\alpha \frac{\partial V}{\partial x}.
\label{solvableHJBgamma=0}
\end{equation}
To establish the solution to \eqref{solvableHJBgamma=0}, a preliminary
Lemma is helpful, to provide the solution formula for the linear dynamic
system whose unknowns are going to be the coefficients of the value function.
\begin{lemma}\label{solutionofthe systemforgamma=0}
The solution to the dynamic system
\begin{equation}
\begin{pmatrix}
\dot{y}_0(t) \\
\dot{y}_1(t) \\
\dots \\
\dot{y}_{m-1}(t) \\
\dot{y}_m(t)
\end{pmatrix}
=
\begin{pmatrix}
\rho & \alpha & 0 & \cdots & \cdots & 0 \\
0 & \rho & 2 \alpha & 0 & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & \cdots & 0 & \rho & m \alpha \\
0 & 0 & \cdots & 0 & 0 & \rho
\end{pmatrix}
\begin{pmatrix}
y_0(t) \\
y_1(t) \\
\dots \\
y_{m-1}(t) \\
y_m(t)
\end{pmatrix}
+\begin{pmatrix}
\beta_0 \\
\beta_1 \\
\dots \\
\beta_{m-1} \\
\beta_m
\end{pmatrix},
\label{dynamicprototype}
\end{equation}
endowed with final conditions $y_j(T)=0$, for all $j=0,1,\ldots, m$, and
for all $\alpha, \beta_0, \beta_1, \ldots, \beta_m \in \mathbb{R}$, is given by
\begin{equation}
y^*_j(t)=\sum_{k=0}^m \gamma_{j,k} (t-T)^k e^{\rho(t-T)}+C_j,
\label{generalsolution}
\end{equation}
where:
\begin{itemize}
\item if $j \in \left\{0,1, \ldots, m-2\right\}$, then
$$
C_j=-\gamma_{j0}=-\frac{1}{\rho}
\Big(\beta_j+\sum_{k=1}^{m-j} (-1)^k \big(\frac{\alpha}{\rho}\big)^k
\big(\Pi_{l=j+1}^{j+k}l \big) \beta_{j-k} \Big),
$$
and the following recurrence relation among coefficients holds:
\begin{gather*}
\gamma_{j,k}=\frac{(j+1)\alpha \gamma_{j+1,k+1}}{k} \quad \text{for } k=1,
\ldots, m-j, \\
\gamma_{j,k}=0 \quad \text{ for } k=m-j, \ldots, m;
\end{gather*}
\item if $j=m-1$, then
\begin{gather*}
C_{m-1}=-\gamma_{m-1,0}=-\frac{1}{\rho}
\Big(\beta_{m-1}-\frac{m \alpha}{\rho}\beta_m \Big),\\
\gamma_{m-1,1}=\frac{m \alpha \beta_m}{\rho}, \quad
\gamma_{m-1,k}=0 \text{for } k=2, \ldots, m;
\end{gather*}
\item if $j=m$, then $C_m=-\beta_m/\rho$, $\gamma_{m,0}=\beta_m/\rho$,
$\gamma_{m,k}=0$ for $k=1, \ldots, m$.
\end{itemize}
\end{lemma}
\begin{proof}
Since the matrix in \eqref{dynamicprototype} has $\lambda=\rho$ as its
unique eigenvalue, having algebraic multiplicity $m+1$ and geometric
multiplicity $1$, all the solutions to the dynamic system can be expressed
in the form
\begin{equation*}
y^*_j(t)=\sum_{k=0}^m \gamma_{j,k} (t-T)^k e^{\rho(t-T)}+C_j,
\end{equation*}
where, by imposing the final condition $y^*_j(T)=0$, we obtain that
$C_j=-\gamma_{j0}$ for all $j=0, 1, \ldots, m$.
The system can be solved by starting from the last equation, which is
solvable by separation of variables, and then proceeding upwards
by successive substitutions. It is easy to work out the initial two solutions:
\begin{gather*}
y^*_m(t)=\frac{\beta_m}{\rho} \left(e^{\rho (t-T)}-1 \right),\\
y^*_{m-1}(t)=\frac{1}{\rho}\Big(\beta_{m-1}-\frac{m \alpha \beta_m}{\rho}\Big)
\left(e^{\rho (t-T)}-1 \right)+\frac{m \alpha \beta_m}{\rho}(t-T) e^{\rho (t-T)}.
\end{gather*}
Subsequently, solving upwards and employing substitutions, we can determine the
iteration for all the solutions.
The general Cauchy problem
\begin{gather*}
\dot{y}_j(t)=\rho y_j(t)+(j+1)y^*_{j+1}(t)+\beta_j \\
y_j(T)=0
\end{gather*}
has the unique solution
\begin{equation}
y^*_j(t)=\frac{\beta_j}{\rho}\big(e^{\rho(t-T)}-1 \big)
+(j+1) \alpha \Big(\int_T^t y^*_{j+1}(s) e^{-\rho s}ds \Big) e^{\rho t}
\label{jthsolutionforgamma=0}
\end{equation}
for $j=0, \ldots, m-1$. To determine the recurrence relations between coefficients,
consider the general formulation \eqref{generalsolution} of the $j$-th solution.
By integrating, we have
\begin{align*}
&y^*_j(t) \\
&=\frac{\beta_j}{\rho}\big(e^{\rho(t-T)}-1 \big)
+(j+1) \alpha e^{\rho t} \int_T^t
\Big(\sum_{k=0}^m \gamma_{j+1,k} (s-T)^k e^{-\rho T}+C_{j+1} e^{-\rho s} \Big) ds \\
&=\frac{\beta_j}{\rho}\big(e^{\rho(t-T)}-1 \big)
+(j+1) \alpha \Big[\sum_{k=0}^m \gamma_{j+1,k} \frac{(t-T)^{k+1}}{k+1}e^{\rho (t-T)}
+ \frac{C_{j+1}}{\rho}
\big(1- e^{\rho (t-T)} \big)\Big] \\
& =-\frac{1}{\rho}(\beta_j-(j+1) \alpha C_{j+1})(1- e^{\rho(t-T)})+(j+1)
\alpha \sum_{k=0}^m \gamma_{j+1,k}
\frac{(t-T)^{k+1}}{k+1} e^{\rho(t-T)},
\end{align*}
which gives the recurrence relations among coefficients:
$$
C_j=-\frac{1}{\rho}(\beta_j-(j+1) \alpha C_{j+1}), \quad
\gamma_{j,0}=-C_j, \quad
\gamma_{j,k}=\frac{(j+1) \alpha \gamma_{j+1,k+1}}{k}.
$$
\end{proof}
\begin{theorem} \label{thm2}
The function $V^*(x,t)=\sum_{l=0}^m A^*_l(t)x^l$ with
\begin{gather*}
A^*_0(t)=\sum_{k=0}^m \gamma_{0,k} (t-T)^k e^{\rho(t-T)}+C_0 \\
A^*_1(t)=\sum_{k=0}^m \gamma_{1,k} (t-T)^k e^{\rho(t-T)}+C_1 \\
\dots \\
A^*_{m-1}(t)=\frac{m \alpha \beta_m}{\rho}(t-T)
e^{\rho(t-T)}+\frac{1}{\rho}\big[\frac{m \alpha \beta_m}{\rho}-\beta_{m-1}\big]
\left(1-e^{\rho (t-T)} \right) \\
A^*_m(t)=\frac{\beta_m (e^{\rho (t-T)}-1)}{\rho}
\end{gather*}
is a solution of \eqref{solvableHJBgamma=0}.
\end{theorem}
\begin{proof}
Consider the guess $V(x,t)=\sum_{l=0}^m A_i(t) x^l$ and substitute it
into \eqref{solvableHJBgamma=0} to obtain
\begin{equation*}
\sum_{l=0}^m \dot{A}_l(t) x^l- \rho \sum_{l=0}^m A_l(t) x^l
=\sum_{l=0}^m \beta_l x^l +\alpha \Big(\sum_{l=1}^m l x^{l-1} A_l(t) \Big),
\end{equation*}
after eliminating the first term of the sum on the right-hand side.
Collecting terms with all the powers of $x$ in both sides leads to the
dynamic system
\begin{equation}
\begin{gathered}
\dot{A}_0(t)=\rho A_0(t)+\alpha A_1(t)+\beta_0 \\
\dot{A}_1(t)=\rho A_1(t)+2 \alpha A_2(t)+\beta_1 \\
\dot{A}_2(t)=\rho A_2(t)+3 \alpha A_3(t)+\beta_2 \\
\dots \\
\dot{A}_{m-1}(t)=\rho A_{m-1}(t)+m \alpha A_m(t)+\beta_{m-1} \\
\dot{A}_m(t)=\rho A_m(t)+\beta_m\,,
\end{gathered}
\label{SystemAjdotgamma=0}
\end{equation}
which should be endowed with the following set of final conditions
satisfying the boundary condition: $A_j(T)=0$, for $j=0,1,\ldots, m$.
By Lemma \ref{solutionofthe systemforgamma=0}, the solution of
\eqref{SystemAjdotgamma=0}
amounts to
\begin{equation}
\begin{gathered}
A^*_0(t)=\sum_{k=0}^m \gamma_{0,k} (t-T)^k e^{\rho(t-T)}+C_0 \\
\dots \\
A^*_{m-1}(t)=\frac{m \alpha \beta_m}{\rho}(t-T) e^{\rho(t-T)}
+\frac{1}{\rho}\big[\frac{m \alpha \beta_m}{\rho}-\beta_{m-1}\big]
\left(1-e^{\rho (t-T)} \right) \\
A^*_m(t)=\frac{\beta_m (e^{\rho (t-T)}-1)}{\rho},
\end{gathered} \label{systemsolutiongamma=0}
\end{equation}
where coefficients $C_j$ and $\gamma_{j,k}$ are defined as in
Lemma \ref{solutionofthe systemforgamma=0}.
\end{proof}
\subsection{Solution for $\gamma=1$}
In this case, \eqref{solvableHJB} takes the form
\begin{equation}
\frac{\partial V}{\partial t}-\rho V= \sum_{l=0}^m \beta_l x^l +\alpha x \frac{\partial V}{\partial x},
\label{solvableHJBgamma=1}
\end{equation}
The next theorem intends to exhibit the solution strategy.
\begin{theorem} \label{thm3}
The function $V^*(x,t)=\sum_{l=0}^m A^*_l(t)x^l$ with
\begin{gather*}
A^*_0(t)=\frac{\beta_0 \left(e^{\rho (t-T)}-1 \right)}{\rho} \\
A^*_1(t)=\frac{\beta_1 \left(e^{(\rho+\alpha) (t-T)}-1 \right)}{\rho+\alpha} \\
A^*_2(t)=\frac{\beta_2 \left(e^{(\rho+2\alpha) (t-T)}-1 \right)}{\rho+2\alpha} \\
\dots \\
A^*_m(t)=\frac{\beta_m \left(e^{(\rho+m \alpha) (t-T)}-1 \right)}{\rho+m \alpha},
\end{gather*}
is a solution of \eqref{solvableHJBgamma=1}.
\end{theorem}
\begin{proof}
Call $V(x,t)=A_0(t)+A_1(t)x+ \cdots +A_m(t) x^m$, where
$A_j(t) \in C^1([0,T])$, for all $j=0, \ldots, m$. By replacing it
in \eqref{solvableHJB}, we have
\begin{equation*}
\sum_{l=0}^m \dot{A}_l(t) x^l- \rho \sum_{l=0}^m A_l(t) x^l
=\sum_{l=0}^m \beta_l x^l +\alpha \Big(\sum_{l=0}^m l x^l A_l(t) \Big).
\end{equation*}
Collecting terms with powers of $x$ yields an $m+1$-equations dynamic system
\begin{equation}
\begin{gathered}
\dot{A}_0(t)=\rho A_0(t)+\beta_0 \\
\dot{A}_1(t)=(\rho+\alpha) A_1(t)+\beta_1 \\
\dot{A}_2(t)=(\rho+2 \alpha) A_2(t)+\beta_2 \\
\dots \\
\dot{A}_m(t)=(\rho+m \alpha) A_m(t)+\beta_m
\end{gathered},
\label{SystemAjdotgamma=1}
\end{equation}
subject to the set of final conditions satisfying the boundary data: $A_j(T)=0$,
for $j=0,1,\ldots, m$.
System \eqref{SystemAjdotgamma=1} can be easily solved by separation of
variables in each ODE. Plugging the solutions into the expression of $V(x,t)$,
we achieve the solution to \eqref{solvableHJBgamma=1}:
\begin{equation}
V^*(x,t)=\frac{\beta_0 \left(e^{\rho (t-T)}-1 \right)}{\rho}+\sum_{l=1}^m \frac{\beta_l
\left(e^{(\rho+l \alpha) (t-T)}-1 \right)x^l}{\rho+l \alpha}.
\label{solutiongamma=1}
\end{equation}
\end{proof}
\section{A microeconomic application}
Consider $N$ firms engaging in a Cournot competition, producing homogeneous
goods and bearing a cost for developing R\&D in their own sectors.
This typical setup describes an oligopolistic game evolving over time,
where players aim to maximize their own payoff
(For a rich overview of such models, see \cite{Dockner}.).
In a simplified version of this scenario, each player chooses a strategy,
denoted by a control variable,
to maximize an objective function which is the integral of the discounted
flows of her profits. The notation to be employed is standard for
industrial organization models. I am going to borrow it mainly
from \cite{Lambertini1}. It is exposed in the following list together
with some hypotheses
\begin{itemize}
\item $u_i(t) \in U_i \subseteq \mathbb{R}_+$ is the strategic variable for the
$i$-th player, representing output level, and $u=(u_1, \ldots, u_N)$
is a vector of strategies. Each control set $U_i$ may be either bounded,
such as $[0, \overline{u}]$, or unbounded, such as $[0, +\infty)$,
in compliance with the inverse demand structure of the market;
\item $p(u(t))$ is the inverse demand function of the market,
decreasing in the sum of all outputs;
\item $c u_i(t)$ is the linear production cost borne by the $i$-th player,
where $c>0$ is the marginal cost parameter;
\item $\pi_i(u(t),t)=(p(u(t))-c)u_i(t)$ is the profit gained by the $i$-th
firm at time $t$;
\item the horizon of the competition is finite, i.e. the game evolves over
a compact time interval $[0,T]$;
\item the discount factor of profits is the same for all players:
$e^{-\rho t}$, where $\rho>0$ is the force of interest on the market,
also considered as a measure of how much players discount their future profits;
\item $k_i(t)$ is player $i$'s state variable, describing physical capital or
capacity, which accumulates over time in compliance with a given dynamics
$G(k_i(t))$ to allow continuous production.
Consider the most general case for the state set, i.e.
$k_i(t) \in K_i \subseteq \mathbb{R}_+$, meaning that, depending on the
cost structure, the state set may be either bounded or unbounded.
The initial conditions of such accumulation process are $k_i$, for
$i=1, \ldots, N$, i.e. $k_i$ are the capacity levels at instant $t$, where
$t \in [0, T)$ is the initial instant of the game;
\item the $i$-th firm bears a further cost induced by accumulation of its
own physical capital. The cost function $C_i(k_i)$ is a non-negative function
of the $i$-th physical capital. Generally, $C_i(k_i)$ is either a linear or
a convex function;
\item there is no scrap value or salvage value at time $T$
(This requirement is equivalent to considering a no prize game.);
\item the game is played simultaneously;
\item players are symmetric, meaning that their productive characteristics
make the oligopoly symmetric. In an oligopolistic competition, symmetry
among firms can be described in several ways: same initial capital endowment,
same number of workers earning the same wages, same output having the same
production costs, and so on. Consequently, they cannot be distinguished and
for this reason we can search for a symmetric solution of the game.
Asymmetric scenarios are more complex, and such frameworks may be
investigated in future research.
\end{itemize}
Firm $i$ solves the following optimization program:
\begin{equation}
\max_{u_i \geq 0} \int_0^T e^{-\delta t}
[\pi_i(u_1(t), \ldots, u_N(t))-C_i(k_i(t))] dt,
\label{standardproblem}
\end{equation}
subject to
\begin{equation}
\begin{gathered}
\dot{k}_i(t)=G(k_i(t))-u_i(t), \\
k_i(0)=k_{i0}.
\end{gathered}
\label{standardCauchy}
\end{equation}
Usually, the search for the feedback (or Markov-perfect) equilibrium is
pursued by solving the related system of HJB equations,
having the optimal value functions
$V_i(k_i,t)$ as its unknowns.
I am going to confine my attention to the explicit solution to the related HJB
equations, without taking into account the issue of sufficient conditions.
Such a topic is widely discussed in many important contributions such as
the textbooks by Bertsekas \cite{Bertsekas1,Bertsekas2},
and the volumes treating dynamic programming with applications to economics
and management science, i.e. Kamien and Schwartz (\cite{Kamien}
is the most recent edition) and Dockner et al.\ \cite{Dockner}.
Basically, under simple regularity assumptions which are verified in most
solvable models, the existence of a solution to the HJB equations corresponds
to the existence of a feedback solution to the differential game.
Call $k_i$ the $i$-th level of capacity at time $t$, where $t \in [0, T)$
is the initial time instant of the game. Hence, the $2$ arguments of $V_i$
are $k_i$, i.e. the initial level of capital, and $t$, i.e. the initial time.
Hence, $k=(k_1, \ldots, k_N)$ is the vector of initial data, and the $i$-th player's
optimal value function is
\begin{equation}
V_i(k_i,t)=\int_{t}^T e^{-\delta s} \left[\pi_i(u(s), s)-C_i(k_i) \right]ds.
\label{ithvaluefunction}
\end{equation}
The $i$-th HJB equation reads
\begin{equation}
\begin{aligned}
\frac{\partial V_i(k,t)}{\partial t}-\rho V_i(k,t)
=\max_{u_i \geq 0} \Big\{& \pi_i(u(t), t)-C_i(k_i)
+\frac{\partial V_i(k,t)}{\partial k_i}(G(k_i)-u_i) \\
& +\sum_{j \neq i} \frac{\partial V(k_i,t)}{\partial k_j}(G(k_j)-u_j)\Big\},
\end{aligned}\label{ithHJB}
\end{equation}
endowed with the transversality condition $V_i(k_i,T)=0$, representing the
vanishing of \eqref{ithvaluefunction} at the final instant of the game.
The concept of Nash feedback equilibrium of a game deserves to be briefly
recalled: an $N$-tuple $(u_1^*(k,t), \ldots, u_N^*(k,t))$ is a feedback
Nash equilibrium if for all $j=1, \ldots, N$, $u_j^*(k,t)$ is a maximizer
of $V_j$ when all the remaining players play strategy $u_l^*(k,t)$,
for all $l \neq j$.
An approach which is commonly adopted in such problems involves the determination
of symmetric solutions, i.e. such as $u^*_1=u^*_2= \cdots=u^*_N$, which requires
suitable symmetry assumptions
and basically transforms a differential game into an optimal control problem
with a single agent. What follows is an Example showing the derivation of
a feedback equilibrium in a problem where the inverse demand function of
the market is hyperbolic.
\begin{example} \label{examp4} \rm
Sticking to the above notation, consider a market in which $3$ firms compete
over quantity and where the inverse demand function is hyperbolic,
i.e.
\begin{equation}
p(u_1, u_2 , u_3)=\frac{A}{\sum_{j=1}^3 u_j},
\label{inversehyperbolicdemand}
\end{equation}
where $A>0$ is the market reservation price.
See \cite{Lambertini1} and \cite{Lambertini2} for a
theoretical explanation.
The dynamic constraints are the kinematic equations:
$$
\dot{k}_i(t)=\alpha k_i(t)-u_i(t),
$$
where $\alpha>0$ indicates the growth rate of the physical capital.
The cost induced by the development of $k_i$ for the $i$-th firm is
$C_i(k_i)=\frac{k_i^2}{5}+\frac{k_i^4}{10}$, which is convex for $k_i \geq 0$.
Given such data, the PDE \eqref{ithHJB} becomes
\begin{align*}
&\frac{\partial V_i(k,t)}{\partial t}-\rho V_i(k,t) \\
&=\max_{u_i \geq 0} \Big\{\Big(\frac{A}{\sum_{j=1}^3 u_j}-c \Big)
u_i-\frac{k_i^2}{5}-\frac{k_i^4}{10}
+\frac{\partial V_i(k_i,t)}{\partial k_i}(\alpha k_i-u_i) \\
&\quad + \sum_{j \neq i} \frac{\partial V_i(k_i,t)}{\partial k_j}
(\alpha k_j-u_j)\Big\}.
\end{align*}
As can be simply verified, the expression to be maximized on the right-hand
side is concave in variables $u_i$, meaning that the existence of maximizers
is ensured. Maximizing the expression on the right-hand side yields
(whenever possible, arguments are omitted to lighten the notation):
\begin{equation*}
-c+A \frac{\sum_{j \neq i} u_j}{(u_1+u_2+u_3)^2}
-\frac{\partial V_i}{\partial k_i}=0,
\end{equation*}
leading to the $3$-equation system
\begin{gather*}
-c+A \frac{u_2+u_3}{(u_1+u_2+u_3)^2}-\frac{\partial V_1}{\partial k_1}=0, \\
-c+A \frac{u_1+u_3}{(u_1+u_2+u_3)^2}-\frac{\partial V_2}{\partial k_2}=0, \\
-c+A \frac{u_1+u_2}{(u_1+u_2+u_3)^2}-\frac{\partial V_3}{\partial k_3}=0.
\end{gather*}
Summing the equations yields
$$
u_1+u_2+u_3=\frac{2A}{3c+\frac{\partial V_1}{\partial k_1}
+\frac{\partial V_2}{\partial k_2}+\frac{\partial V_3}{\partial k_3}}.
$$
We impose symmetry in order to find out the symmetric solution, so we call
$u:=u_1=u_2=u_3$, $k:=k_1=k_2=k_3$, and $V:=V_1=V_2=V_3$.
Consequently, we achieve a unique relation leading to the optimal output $u^*$:
\begin{equation*}
u^*=\frac{2A}{9 (c+\frac{\partial V}{\partial k})}.
\end{equation*}
Before proceeding, we should note that the search for a symmetric solution
transforms the differential game into an optimal control problem, having only
$V(k,t)$ as its unknown.
However, if we assumed symmetry a priori, the quantity to be maximized would
have become linear in the unique strategic variable $u$, and consequently would
have admitted no maximum points.
When such a maximization problem does not admit stationary points, we have to
discuss the behaviour at the boundary of the strategy space.
On the other hand, the cross derivatives $\frac{\partial V_i}{\partial k_j}$
are not meaningful any longer, hence they should be removed from the unique
HJB equation. By replacing $u^*$ into the unique HJB we achieve the following:
\begin{gather*}
\frac{\partial V}{\partial t}-\rho V
=\frac{A}{3}- \frac{2A c}{9 \big(c+\frac{\partial V}{\partial k}\big)}
-\frac{k_i^2}{5}-\frac{k_i^4}{10}+\alpha k \frac{\partial V}{\partial k}
-\frac{2A}{9 \big(c+\frac{\partial V}{\partial k}\big)}
\frac{\partial V}{\partial k} \\
\Longleftrightarrow
\frac{\partial V}{\partial t}-\rho V=\frac{A}{3}- \frac{2A}{9 \big(c+\frac{\partial V}{\partial k}\big)} \big(c+\frac{\partial V}{\partial k}\big)
-\frac{k_i^2}{5}-\frac{k_i^4}{10}+\alpha k \frac{\partial V}{\partial k} \\
\Longleftrightarrow \frac{\partial V(k,t)}{\partial t}-\rho V(k,t)
=\frac{A}{9}- \frac{k_i^2}{5}-\frac{k_i^4}{10}
+\alpha k \frac{\partial V(k,t)}{\partial k}.
\end{gather*}
which is a PDE belonging to the class of \eqref{solvableHJBgamma=1},
with parameters $\beta_0=\frac{A}{9}$, $\beta_2=-\frac{1}{5}$,
$\beta_4=-\frac{1}{10}$, $\beta_j=0$ for all
$j \in \mathbb{Z}_+\setminus \{0,2,4\}$. We can directly apply formula
\eqref{solutiongamma=1} to achieve the optimal value function
\begin{equation*}
V^*(k,t)=\frac{A \left(e^{\rho (t-T)}-1 \right)}{9\rho}
-\frac{\left(e^{(\rho+2 \alpha)(t-T)}-1 \right) k^2}{5(\rho+2\alpha)}-
\frac{\left(e^{(\rho+4 \alpha)(t-T)}-1 \right) k^4}{10(\rho+4 \alpha)},
\end{equation*}
and finally, by substitution, the optimal feedback strategy:
\begin{equation*}
u^*(k,t)=\frac{2A}{9 \Big(c-\frac{2 \left(e^{(\rho+2\alpha)(t-T)}-1 \right)k}
{5(\rho+2 \alpha)}-\frac{2 \left(e^{(\rho+4\alpha)(t-T)}-1 \right)k^3}
{5(\rho+4\alpha)}\Big)}.
\end{equation*}
\end{example}
\subsection*{Concluding Remarks}
Analytic solutions have been worked out for a class of HJB equations arising
from a differential game of oligopoly among firms engaging in competition over
outputs in a market subject to a hyperbolic inverse demand function.
Possible future developments of the present work are either the derivation of
the solutions to \eqref{solvableHJB} if parameter $\gamma$ is different from
$0$ and $1$ or a discussion on the same problem played over an infinite time
horizon. Furthermore, hyperbolic demand structures are an issue which deserves
to be further developed in general, also in settings which are
far from differential games scenarios.
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