

International youth conference "Geometry & Control"
April 18, 2014 13:10, Moscow, Steklov Mathematical Institute of RAS






Optimal R&D Policy in a Model Based on Exhaustible Resources
Konstantin Besov^{} ^{} Steklov Mathematical Institute, Russian Academy of
Sciences, Moscow, Russia

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Abstract:
We study an infinitehorizon optimal control problem arising from an
endogenous growth model in which both production and research require an
exhaustible resource. The model is a development of the earlier considered
problem [2] (going back to Jones [4, 5]) of optimal
extraction and use of a finite stock of some resource:
\begin{gather}
\tag{1}
Y(t)=A(t)^\kappa L^Y(t)^\alpha R_1(t)^{1\alpha}\qquadwhere\quad \alpha\in(0,1)\quadand\quad\kappa>0,
\tag{2}
\dot A(t)=A(t)^\theta L^A(t)^\eta R_2(t)^{\beta}\qquad where\quad \eta\in(0,1],\quad\beta\in[0,1\eta],
\quadand\quad\theta\in (0,1].
\end{gather}
Here $Y(t)$ is the output produced at time $t$ and $A(t)$ is the current
knowledge stock. The resource is divided between production ($R_1(t)$) and
research ($R_2(t)$). The total amount of the extracted resource cannot
exceed the initial supply $S_0>0$ of the resource:
$$
\tag{3}
\int_{0}^{\infty} [R_1(t)+R_2(t)] dt\leq S_0.
$$
The population (total labor supply) is fixed at a certain level $L>0$. Part
of the labor $L^Y(t)$ is employed in production, while the other part
$L^A(t)\in[0,L)$ is allocated to research:
$$
\tag{4}
L^A(t)+L^Y(t)\equiv L.
$$
We consider a discounted logarithmic utility function of the output as a
measure of welfare:
$$
\tag{5}
J_0(A(\cdot),L^A(\cdot),R_1(\cdot))=\int_0^\infty e^{\rho t}\ln Y(t) dt\to \max,
$$
where $\rho>0$ is a subjective discount rate.
As our study has shown [2], in the nonexceptional case of
$(1\beta)\theta<1$, the labor and resource allocated (optimally) to
research gradually decrease and ultimately vanish. Accordingly, the
expansion of the knowledge stock is limited and stops or virtually stops at
some moment and the output depletes to zero in the long run. However,
experience suggests that there may occur jumps (transitions) from one
technological trajectory to another. So we develop the above model further
in order to take account of the possibility of such a jump. Namely, we
assume that the moment of a jump $T$ is a random variable such that
$$
\tag{6}
P(T<t+\Delta t\mid t\le T)=\nu L^A(t)\Delta t+o(\Delta t),\qquad
o(\Delta t)/\Delta t\to 0\quadas \Delta t\to 0.
$$
Then the probability density function for the random variable $T$ is
$$
\nu L^A(t)e^{\nu\mathfrak L(t)},\qquadwhere\qquad \mathfrak L(t)=\int_0^tL^A(s) ds
\quadfor 0\le t<\infty,
$$
provided that $\mathfrak L(\infty)=\int_0^\infty L^A(s) ds=\infty$. If
$\mathfrak L(\infty)<\infty$, then there is a positive probability that the
jump will not occur at all, i.e. $p(T=\infty)=e^{\nu\mathfrak
L(\infty)}>0$.
It is important to note that the process described by relations
(1)–(4) is now of finite duration with probability
$1e^{\nu\mathfrak L(\infty)}$. Hence the integral in (5) must be
taken only over the interval $[0,T]$ rather than over $[0,\infty)$.
However, some estimate of the knowledge stock accumulated by the moment
$t=T$ should also be taken into account because the accumulated knowledge
$A(T)$ augments the productivity of the production means and hence
increases the welfare on the remaining time interval $[T,\infty)$. Thus, we
come to the following functional measuring the welfare:
$$
J_T(A(\cdot),L^A(\cdot),R_1(\cdot))=\int_0^T e^{\rho t} \ln Y(t) dt+e^{\rho T}V(A(T)).
$$
To determine the value of the accumulated knowledge $A(T)$, we consider an
auxiliary simple optimization problem on the interval $[T,\infty)$, which
yields
$$
V(A(T))=C+\frac\kappa\rho\ln A(T),
$$
where $C=C(\rho,\kappa,L)$ is a constant.
In this situation it is natural to aim at maximizing the expectation of the
utility functional $J_T(A(\cdot),L^A(\cdot),R_1(\cdot))$ considered as a
function of the random variable $T$. After some transformations, we reduce
the problem to an equivalent infinitehorizon optimal control problem.
Using standard results, we show the existence of an optimal control in the
resulting problem. Then we apply the recent version of the Pontryagin
Maximum Principle [1] (see also [3]) and analyze the solutions
of the arising Hamiltonian system of the PMP. In particular, it is
interesting to compare the behavior of optimal controls in this problem
with that in problem (1)–(5).
Materials:
slides.pdf (125.8 Kb),
abstract.pdf (88.7 Kb)
Language: English
References

K. O. Besov, “On necessary optimality conditions for infinitehorizon economic growth problems with locally unbounded instantaneous utility function,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 284, 56–88 (2014) [Proc. Steklov Inst. Math. 284, 50–80 (2014)].

S. Aseev, K. Besov, and S. Kaniovski, “The problem of optimal endogenous growth with exhaustible resources revisited,” in Green Growth and Sustainable Development (Springer, Berlin, 2013), Dyn. Model. Econometr. Econ. Finance 14, pp. 3–30.

S. M. Aseev, K. O. Besov, and A. V. Kryazhimskiy, “Infinitehorizon optimal control problems in economics,” Usp. Mat. Nauk 67 (2), 3–64 (2012) [Russ. Math. Surv. 67, 195–253 (2012)].

C. I. Jones, “Timeseries test of endogenous growth models,” Q. J. Economics 110, 495–425 (1995).

C. I. Jones, “Growth: With or without scale effects,” Am. Econ. Rev. 89, 139–144 (1999).

