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.


Introduction
In financial decision-making problems, Merton's [1,2] papers seemed to be pioneering works.In his seminal work, Merton [2] 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 [2] is not yet available.In this study, we turned to Merton's [2] 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:   = inf{ ≥ 0 :   = 0},   = capital stock at time  ≥ 0,   = labor supply at time  ≥ 0,  = constant rate of depreciation,  ≥ 0,     = consumption rate at time  ≥ 0, 0 ≤   ≤ 1,     /  = totality of consumption rate per labor; (, ) =    1− with 0 <  < 1 and  is a constant, production function producing the commodity for the capital stock  > 0 and the labor supply  > 0,  = rate of labor growth (nonzero constant),  = non-zero constant coefficients,  = discount rate  > 0, () = utility function for the consumption rate  ≥ 0,   = one-dimensional standard Brownian motion on a complete probability space (Ω, F, ) endowed with the natural filtration F  generated by (  ,  ≤ ).
Let us assume that  = {  } is a consumption policy per capita such that   is nonnegative F  = (  ,  ≤ ), a progressively measurable process, and we denote by A the set of all consumption policies {  } per capita.

International Journal of Mathematics and Mathematical Sciences
The utility function () is assumed to have the following properties: Following Merton [2], 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  ∈ A 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 [2], Kamien and Schwartz [3], Koo [4], Morimoto and Kawaguchi [5], Morimoto [6], and Zeldes [7].Recently, this kind of problem is treated by Baten and Sobhan [8] 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 [9], Bucci et al. [10], Posch [11], and Roche [12]; comprehensive coverage of this subject can be found, for example, in the books of Chang [13], Malliaris et al., [14], Turnovsky [15,16], and Walde [17][18][19].Continuous-time steady-state studies under lower-dimensional uncertainty carried out, for example, by Merton [2] and Smith [20] within a Ramseytype setup, and, for example, by Bourguignon [21], Jensen and Richter [22], and Merton [2] 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 [23] 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 [23] is related to the optimal consumption and portfolio problems associated with a random time horizon studied in Blanchet-Scalliet et al., [24], Bouchard and Pham [25], and Blanchet-Scalliet et al., [26].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  * (  , ) = max >0 {(/)−  } 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 [27] and Soner [28] to this problem to show that the transformed one-dimensional HJB equation admits a viscosity solution V 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.

Transformed Hamilton-Jacobi-Bellman Equation
In order to transform the HJB equation ( 6) to onedimensional 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 International Journal of Mathematics and Mathematical Sciences 3 Setting  = / and substituting these above in (6), we have the HJB equation of V of the following form: where λ =  +  −  2 , () = (, 1), and  * (, ) = max 0≤≤1 {() − },  ∈ R.
We found that ( 9) is the transformed HJB equation associated with the stochastic utility consumption problem so as to maximize over the class  ∈ Ã, subject to where  ∈ Ã denotes the class A with {  } replacing {  }.We choose  > 0 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  = () = inf{ ≥ 0 :   = 0} and  = {  } is the element of the class Ã consisting of F  progressively measurable processes such that

Viscosity Solutions
In this section, we will show the existence results on the viscosity solution V of the HJB equation ( 9).
Proof.Following ( 13) and ( 21) we have for any stopping time .By ( 13) and for any  > 0, there exists  ∈ Ã such that Since () is Lipschitz continuous, it follows that By (11), we can consider that z is the solution of So by the comparison theorem Ikeda and Watanabe [29], we have Since [sup 0≤≤ z2  ] < ∞ for all  > 0, now by (11) we have Letting  ↓ 0 and then  → 0, we obtain Passing to the limit to (37) and applying (43), we obtain which implies V(0+) = 0. Thus, V ∈ [0, ∞).So by the standard stability results of Fleming and Soner [27], we deduce that V is a viscosity solution of (9).

Optimal Consumption Policy
Under the assumption ( 1) and ( 2), Lemma 3 has revealed that the value function of the representative household assets must approach zero as time approaches infinity.
We give a synthesis of the optimal policy  * = { *  } for the optimization problem (4) subject to (5).