Optimal Consumption in a Stochastic Ramsey Model with Cobb-Douglas Production Function
A stochastic Ramsey model is studied with the Cobb-Douglas production function maximizing the expected discounted utility of consumption. We transformed the Hamilton-Jacobi-Bellman (HJB) equation associated with the stochastic Ramsey model so as to transform the dimension of the state space by changing the variables. By the viscosity solution method, we established the existence of viscosity solution of the transformed Hamilton-Jacobi-Bellman equation associated with this model. Finally, the optimal consumption policy is derived from the optimality conditions in the HJB equation.
In financial decision-making problems, Merton’s [1, 2] papers seemed to be pioneering works. In his seminal work, Merton  showed how a stochastic differential for the labor supply determined the stochastic processes for the short-term interest rate and analyzed the effects of different uncertainties on the capital-to-labor ratio. The existence and uniqueness of solutions to the state equation of the Ramsay problem  is not yet available. In this study, we turned to Merton’s  original problem that is revisited considering the growth model for the Cobb-Douglas production function in the finite horizon. Let us define the following quantities:=, = capital stock at time , = labor supply at time , = constant rate of depreciation, , = consumption rate at time , = totality of consumption rate per labor; = with and is a constant, production function producing the commodity for the capital stock and the labor supply ,= rate of labor growth (nonzero constant), = non-zero constant coefficients,= discount rate, = utility function for the consumption rate ,= one-dimensional standard Brownian motion on a complete probability space endowed with the natural filtration generated by .
Let us assume that is a consumption policy per capita such that is nonnegative , a progressively measurable process, and we denote by the set of all consumption policies per capita.
The utility function is assumed to have the following properties: Following Merton , we make the following assumption on the Cobb-Douglas production function :
We are concerned with the economic growth model to maximize the expected discount utility of consumption per labor with a horizon over the class subject to the capital stock , and the labor supply is governed by the stochastic differential equation
This optimal consumption problem has been studied by Merton , Kamien and Schwartz , Koo , Morimoto and Kawaguchi , Morimoto , and Zeldes . Recently, this kind of problem is treated by Baten and Sobhan  for one-sector neoclassical growth model with the constant elasticity of substitution (CES) production function in the infinite time horizon case. The studies of Ramsey-type stochastic growth models are also available in Amilon and Bermin , Bucci et al. , Posch , and Roche ; comprehensive coverage of this subject can be found, for example, in the books of Chang , Malliaris et al., , Turnovsky [15, 16], and Walde [17–19]. Continuous-time steady-state studies under lower-dimensional uncertainty carried out, for example, by Merton  and Smith  within a Ramsey-type setup, and, for example, by Bourguignon , Jensen and Richter , and Merton  within a Solow-Swan-type setup. But these papers did not deal with establishing the existence of viscosity solution of the transformed Hamilton-Jacobi-Bellman equation, and they did not derive the optimal consumption policy from the optimality conditions in the HJB equation associated with the stochastic Ramsey problem, which we have dealt with in this paper.
On the other hand, Oksendal  considered a cash flow modeled with geometric Brownian motion to maximize the expected discounted utility of consumption rate for a finite horizon with the assumption that the consumer has a logarithmic utility for his/her consumption rate. He added a jump term (represented by a Poissonian random measure) in a cash flow model. The problem discussed in Oksendal  is related to the optimal consumption and portfolio problems associated with a random time horizon studied in Blanchet-Scalliet et al., , Bouchard and Pham , and Blanchet-Scalliet et al., . However, our paper’s approach is different.
By the principle of optimality, it is natural that solves the general (two-dimensional) Hamilton-Jacobi-Bellman (in short, HJB) equation where and , , and are partial derivatives of with respect to and .
The technical difficulty in solving the problem lies in the fact that the HJB equation (6) is a parabolic PDE with two spatial variables and . We apply the viscosity method of Fleming and Soner  and Soner  to this problem to show that the transformed one-dimensional HJB equation admits a viscosity solution and the optimal consumption policy can be represented in a feedback from the optimality conditions in the HJB equation.
This paper is organized as follows. In Section 2, we transform the two-dimensional HJB equation (6) associated with the stochastic Ramsey model. In Section 3, we show the existence of viscosity solution of the transformed HJB equation. In Section 4, a synthesis of the optimal consumption policy is presented in the feedback from the optimality conditions. Finally, Section 5 concludes with some remarks.
2. Transformed Hamilton-Jacobi-Bellman Equation
In order to transform the HJB equation (6) to one-dimensional second-order differential equation, that is, from the two-dimensional state space form (one state for capital stock and the other state for labor force), it has been transformed to a one-dimensional form, for the ratio of capital to labor. Let us consider the solution of (6) of the form Clearly Setting and substituting these above in (6), we have the HJB equation of of the following form: where , and .
We found that (9) is the transformed HJB equation associated with the stochastic utility consumption problem so as to maximize over the class , subject to where denotes the class with replacing . We choose and rewrite (9) as
The value function can be defined as a function whose value is the maximum value of the objective function of the consumption problem, that is, where and is the element of the class consisting of progressively measurable processes such that
3. Viscosity Solutions
In this section, we will show the existence results on the viscosity solution of the HJB equation (9).
Let and . Then is called a viscosity solution of the reduced (one-dimensional) HJB equation (9) if the following relations hold: where and are defined by
Proof. Itô’s formula gives
and by considering , then (11) becomes
by the comparison theorem of Ikeda and Watanabe ; we see that , . Hence, by applying the existence and uniqueness theorem for (11), we have
Therefore, from (24), we have
which yields that is a martingale, and again by (11), we can take sufficiently small such that
Therefore, by (20), (11) and taking expectation on both sides of (23), we obtain
from which we deduce (21).
We set and by (11), it is clear that Since by (3), is Lipschitz continuous and concave and , then we have Take such that , and we can find from (18) and (20) that Using Itô’s formula and by (20), we have Letting , and by Fatou’s lemma, we obtain which implies (22).
Proof. Following (13) and (21) we have for any stopping time . By (13) and for any , there exists such that Since is Lipschitz continuous, it follows that By (11), we can consider that is the solution of So by the comparison theorem Ikeda and Watanabe , we have Since for all , now by (11) we have Letting and then , we obtain so that Passing to the limit to (37) and applying (43), we obtain which implies . Thus, . So by the standard stability results of Fleming and Soner , we deduce that is a viscosity solution of (9).
4. Optimal Consumption Policy
Proof. By (17) and (18), we take such that Take and such that , and by (33) and (46) Setting , where , we have by (6) and (47) By Itô’s formula and (48), we obtain Letting , we have which implies . By (21), we have which completes the proof.
Proof. Let us consider , and since . are continuous and , there exists an progressively measurable solution of
Now we shall show a.s. Suppose . By (9), we have for all , since if . Moreover, by L’Hospital’s rule this gives
Letting in (9), we have , and this is contrary with (2). Therefore, we get , which implies . We note by the concavity of that for . Then applying the comparison theorem to (54) and
we obtain for all . Further, in case , we have at
Therefore, solves (52) and . To prove uniqueness, let , be two solutions of (52). Then satisfies
. We have
Note that the function is decreasing. Hence,
By Gronwall’s lemma, we have
So, the uniqueness of (52) holds.
Now by (6), (52), and Itô’s formula, we have By the HJB equation (6), we have from which where is a sequence of localizing stopping times for the local martingale. From (11), (51), and Doob’s inequalities for martingales, it follows that Letting and , hence, we obtain by the dominated convergence theorem We deduce by Lemma 3 Following the same calculation as above, we have Again by the HJB equation (6), we can obtain from which for any . The proof is complete.
Remark 5. From the proof of Theorem 4, it follows that Thus, under (2), we observe that the smooth solution of of the HJB equation (6). Furthermore, let be the solution of (9) on the entire domain with . Setting and , by (8), we have that satisfies (6). Therefore, we obtain the uniqueness of .
5. Concluding Remarks
In this paper we have studied the optimal consumption problem of maximizing the expected discounted value of consumption utility in the context of one-sector neoclassical economic growth with Cobb-Douglas production function. We have derived a transformed (one-dimensional) Hamilton-Jacobi-Bellman equation associated with the optimization problem. By the technique of viscosity method we established the viscosity solution to the transformed (one-dimensional) Hamilton-Jacobi-Bellman equation. Finally we have derived the optimal consumption feedback form from the optimality conditions in the two-dimensional HJB equation.
The authors wish to thank the anonymous referees for their comments that have led to an improved version of this paper.
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