1.Introduction Md.AzizulBaten andAntonAbdulbasahKamil OptimalConsumptioninaStochasticRamseyModelwithCobb-DouglasProductionFunction ResearchArticle

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Volume 2013, Article ID 684757,8pages http://dx.doi.org/10.1155/2013/684757

Research Article

Optimal Consumption in a Stochastic Ramsey Model with Cobb-Douglas Production Function

Md. Azizul Baten

¹

and Anton Abdulbasah Kamil

²

1Department of Decision Science, School of Quantitative Sciences, Universiti Utara Malaysia (UUM), Sintok, 06010 Kedah, Malaysia

2Mathematics Program, School of Distance Education, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia

Correspondence should be addressed to Md. Azizul Baten; baten [email protected] Received 9 November 2012; Revised 7 January 2013; Accepted 16 January 2013 Academic Editor: Aloys Krieg

Copyright © 2013 Md. A. Baten and A. A. Kamil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. 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:

𝜏_𝑘 = inf{𝑡 ≥ 0 : 𝐾_𝑡= 0}, 𝐾_𝑡= capital stock at time𝑡 ≥ 0, 𝐿_𝑡= labor supply at time𝑡 ≥ 0,

𝜆= constant rate of depreciation,𝜆 ≥ 0, 𝑐_𝑡𝐾_𝑡= consumption rate at time𝑡 ≥ 0, 0 ≤ 𝑐_𝑡≤ 1, 𝑐_𝑡𝐾_𝑡/𝐿_𝑡= totality of consumption rate per labor;

𝐹(𝐾, 𝐿)=𝐴𝐾^𝛼𝐿^1−𝛼 with0 < 𝛼 < 1 and 𝐴 is a constant, production function producing the commodity for the capital stock𝐾 > 0and the labor supply𝐿 > 0, 𝑛 = rate of labor growth (nonzero constant),

𝜎= non-zero constant coefficients, 𝜌 = discount rate𝜌 > 0,

𝑈(𝑐)= utility function for the consumption rate𝑐 ≥ 0, 𝑊_𝑡 = one-dimensional standard Brownian motion on a

complete probability space(Ω,F, 𝑃)endowed with the natural filtrationF_𝑡generated by𝜎(𝑊_𝑠, 𝑠 ≤ 𝑡).

Let us assume that𝑐 = {𝑐_𝑡}is a consumption policy per capita such that 𝑐_𝑡 is nonnegative F_𝑡 = 𝜎(𝑊_𝑠, 𝑠 ≤ 𝑡), a progressively measurable process,

∫^𝑡

0𝑐_𝑠𝑑𝑠 < ∞ a.s. ∀𝑡 ≥ 0, (1) and we denote byAthe set of all consumption policies{𝑐_𝑡} per capita.

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The utility function𝑈(𝑐)is assumed to have the following properties:

𝑈 ∈ 𝐶 [0, ∞) ⋂ 𝐶²(0, ∞) , 𝑈 (𝑐) :strictly concave on[0, ∞) , 𝑈^󸀠(𝑐) :strictly decreasing, 𝑈^󸀠󸀠< 0for𝑐 > 0, 𝑈^󸀠(∞) = 𝑈^󸀠(0+) = 0, 𝑈^󸀠(0+) = 𝑈 (∞) = ∞.

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Following Merton [2], we make the following assumption on the Cobb-Douglas production function𝐹(𝐾, 𝐿):

𝐹 (𝛾𝐾, 𝛾𝐿) = 𝛾𝐹 (𝐾, 𝐿) , for𝛾 > 0, 𝐹_𝐾(𝐾, 𝐿) > 0, 𝐹_𝐿(𝐾, 𝐿) > 0, 𝐹_𝐾𝐾(𝐾, 𝐿) < 0

𝐹_𝐿𝐿(𝐾, 𝐿) < 0,

𝐹 (0, 𝐿) = 𝐹 (𝐾, 0) = 0, 𝐹_𝐾(0+, 𝐿) < ∞, 𝐹_𝐾(∞, 𝐿) = 0, 𝐿 > 0.

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We are concerned with the economic growth model to maximize the expected discount utility of consumption

𝐽 (𝑐) = 𝐸 [∫^𝑇

0 𝑒^−𝜌𝑡𝑈 (𝑐_𝑡𝐾_𝑡

𝐿_𝑡 ) 𝑑𝑡] (4)

per labor with a horizon𝑇over the class𝑐 ∈Asubject to the capital stock𝐾_𝑡, and the labor supply𝐿_𝑡is governed by the stochastic differential equation

𝑑𝐾_𝑡= [𝐹 (𝐾_𝑡, 𝐿_𝑡) − 𝜆𝐾_𝑡− 𝑐_𝑡𝐾_𝑡] 𝑑𝑡 𝐾₀= 𝐾, 𝐾 > 0, 𝑑𝐿_𝑡= 𝑛𝐿_𝑡𝑑𝑡 + 𝜎𝐿_𝑡𝑑𝑊_𝑡, 𝐿₀= 𝐿, 𝐿 > 0. (5) 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 elas- ticity 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 [17–19]. Continuous-time steady-state studies under lower-dimensional uncertainty carried out, for example, by Merton [2] and Smith [20] within a Ramsey- type 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 that𝑢solves the general (two-dimensional) Hamilton-Jacobi-Bellman (in short, HJB) equation

− 𝜌𝑢 (𝐾, 𝐿) +1

2𝜎²𝐿²𝑢_𝐿𝐿(𝐾, 𝐿) + 𝑛𝐿𝑢_𝐿(𝐾, 𝐿) + {𝐹 (𝐾, 𝐿) − 𝜆𝐾} 𝑢_𝐾(𝐾, 𝐿)

+ 𝑈^∗(𝑢_𝐾(𝐾, 𝐿) , 𝐿) = 0,

𝑢 (0, 𝐿) = 0, 𝐾 > 0, 𝐿 > 0,

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where𝑈^∗(𝑢_𝐾, 𝐿) =max_𝑐>0{𝑈(𝑐𝐾/𝐿)−𝑐𝐾𝑢_𝐾}and𝑢_𝐾,𝑢_𝐿, and 𝑢_𝐿𝐿are partial derivatives of𝑢(𝐾, 𝐿)with respect to𝐾and𝐿.

The technical difficulty in solving the problem lies in the fact that the HJB equation (6) is a parabolic PDE with two spatial variables𝐾and𝐿. 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 solutionVand 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. InSection 4, a synthesis of the optimal consumption policy is presented in the feedback from the optimality conditions. Finally,Section 5concludes 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 state𝐾for capital stock and the other state 𝐿 for labor force), it has been transformed to a one-dimensional form, for(𝑥 = 𝐾/𝐿)the ratio of capital to labor. Let us consider the solution𝑢(𝐾, 𝐿) of (6) of the form

𝑢 (𝐾, 𝐿) =V(𝐾

𝐿) , 𝐿 > 0. (7) Clearly

𝐿𝑢_𝐾=V_𝐾, 𝐿𝑢_𝐿= −𝐾V_𝐾,

𝐿²𝑢_𝐿𝐿= 𝐾²V_𝐾𝐾+ 2𝐾V_𝐾. (8)

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Setting𝑥 = 𝐾/𝐿and substituting these above in (6), we have the HJB equation ofVof the following form:

− 𝜌V(𝑥) +1

2𝜎²𝑥²V^󸀠󸀠(𝑥) + (𝑓 (𝑥) − ̃𝜆𝑥)V^󸀠(𝑥) + 𝑈^∗(𝑥,V^󸀠(𝑥)) = 0,

V(0) = 0, 𝑥 > 0,

(9)

where ̃𝜆 = 𝑛 + 𝜆 − 𝜎², 𝑓(𝑥) = 𝐹(𝑥, 1), and 𝑈^∗(𝑥, 𝑞) = max_0≤𝑐≤1{𝑈(𝑐𝑥) − 𝑐𝑥𝑞}, 𝑞 ∈R.

We found that (9) is the transformed HJB equation associated with the stochastic utility consumption problem so as to maximize

̃𝐽 (𝑐) = 𝐸 [∫^𝑇

0 𝑒^−𝜌𝑡𝑈 (𝑐_𝑡𝑥_𝑡) 𝑑𝑡] (10) over the class𝑐 ∈ ̃A, subject to

𝑑𝑥_𝑡= [𝑓 (𝑥_𝑡) − ̃𝜆𝑥_𝑡− 𝑐_𝑡] 𝑑𝑡 − 𝜎𝑥_𝑡𝑑𝑊_𝑡, 𝑥₀= 𝑥, 𝑥 ≥ 0, (11) where𝑐 ∈ ̃Adenotes the classAwith{𝑥_𝑡}replacing{𝐾_𝑡}. We choose𝛿 > 0and rewrite (9) as

− (𝜌 +1

𝛿)V(𝑥) +1

2𝜎²𝑥²V^󸀠󸀠(𝑥) + (𝑓 (𝑥) − ̃𝜆𝑥)V^󸀠(𝑥) + 𝑈^∗(𝑥,V^󸀠(𝑥)) +1

𝛿V(𝑥) = 0, V(0) = 0, 𝑥 > 0.

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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,

𝑉 (𝑥_𝑡) =sup

𝑐∈̃A

𝐸 [∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑡}{𝑈 (𝑐_𝑡𝑥_𝑡) +1

𝛿V(𝑥_𝑡)} 𝑑𝑡] , (13) where 𝜏 = 𝜏(𝑥) = inf{𝑡 ≥ 0 : 𝑥_𝑡 = 0} and 𝑐 = {𝑐_𝑡} is the element of the class̃Aconsisting ofF_𝑡progressively measurable processes such that

∫^𝑡

0𝑐_𝑠𝑑𝑠 < ∞ a.s. ∀𝑡 ≥ 0. (14)

3. Viscosity Solutions

In this section, we will show the existence results on the viscosity solutionVof the HJB equation (9).

3.1. Definition. LetV∈ 𝐶[0, ∞)andV(0) = 0. ThenVis called a viscosity solution of the reduced (one-dimensional) HJB equation (9) if the following relations hold:

− 𝜌V(𝑥) +1

2𝜎²𝑥²𝑞 + 𝑝 (𝑓 (𝑥) − ̃𝜆𝑥) + 𝑈^∗(𝑥, 𝑝) ≥ 0,

∀ (𝑝, 𝑞) ∈ 𝐽^2,+V(𝑥) , ∀𝑥 > 0,

− 𝜌V(𝑥) +1

2𝜎²𝑥²𝑞 + 𝑝 (𝑓 (𝑥) − ̃𝜆𝑥) + 𝑈^∗(𝑥, 𝑝) ≤ 0,

∀ (𝑝, 𝑞) ∈ 𝐽^2,−V(𝑥) , ∀𝑥 > 0, (15)

where𝐽^2,+and𝐽^2,−are defined by 𝐽^2,+V(𝑥)

= { (𝑝, 𝑞) ∈R²:

lim sup

̃𝑥 → 𝑥

V(̃𝑥)−V(𝑥)−𝑝 (̃𝑥 − 𝑥)−(1/2) 𝑞|̃𝑥 − 𝑥|²

|̃𝑥 − 𝑥|² ≤ 0} ,

𝐽^2,−V(𝑥)

= { (𝑝, 𝑞) ∈R²:

lim inf

̃𝑥 → 𝑥

V(̃𝑥)−V(𝑥)−𝑝 (̃𝑥 − 𝑥)−(1/2) 𝑞|̃𝑥 − 𝑥|²

|̃𝑥 − 𝑥|² ≥ 0} .

(16) We assume that

𝜌 + ̃𝜆 > 0, (17)

−𝜌 + (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨) +1

2𝜎²< 0. (18) Take0 < 𝛼 < 𝜌, and we choose𝑃₁> 0by concavity such that

𝑓 (𝑥) − ̃𝜆𝑥 ≤ 𝑃₁. (19)

Taking sufficiently large𝑃₀ > 𝑃₁, we observe by (2) and (19) that𝜙(𝑡, 𝑥) = 𝑒^𝜏−𝑡(𝑥 + 𝑃₀)fulfills

− 𝛼𝜙 (𝑥)+1

2𝜎²𝑥²𝜙^󸀠󸀠(𝑥)+(𝑓 (𝑥) − ̃𝜆𝑥) 𝜙^󸀠(𝑥)+𝑈^∗(𝑥, 𝜙^󸀠(𝑥))

≤ 𝑒^𝜏−𝑡(−𝑥 − 𝑃₀+ 𝑃₁) + 𝑈𝑜(𝑈^󸀠)⁻¹𝑒^𝜏−𝑡

< −𝑥 − 𝑃₀+ 𝑃₁+ 𝑈𝑜(𝑈^󸀠)⁻¹(1) < 0,

(20) for some constant𝑃₀> 0.

Lemma 1. One assumes(2),(17),(11), and(20), then the value function𝑉(𝑥)fulfills

0 ≤ 𝑉 (𝑥_𝑡) ≤ 𝜙 (𝑥_𝑡) , (21) sup

𝑐∈̃A

𝐸[𝑒^{−(𝜌+1/𝛿)𝜏}|𝑥 (𝜏) − ̃𝑥 (𝜏)|] ≤ |𝑥 − ̃𝑥| (22)

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for any stopping time𝜏, where{̃𝑥(𝜏)}is the solution of (11)to 𝑐 ∈Awith̃𝑥(0) = ̃𝑥.

Proof. Itˆo’s formula gives 0 ≤ 𝑒−(𝜌+1/𝛿)(𝑡∧𝜏)𝜙 (𝑥_{(𝑡∧𝜏)})

= 𝜙 (𝑥) + ∫^𝑡∧𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑠}

× {− (𝜌 +1 𝛿) 𝜙 (𝑥_𝑠)

+ (𝑓 (𝑥) − ̃𝜆𝑥_𝑠− 𝑐_𝑠𝑥_𝑠) 𝜙^󸀠(𝑥_𝑠) +1

2𝜎²𝑥²_𝑠𝜙^󸀠󸀠(𝑥_𝑠)} 𝑑𝑠

− ∫^𝑡∧𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^󸀠(𝑥_𝑠) 𝑑𝑊_𝑠.

(23)

Since 𝐸 [∫^𝑡

0󵄨󵄨󵄨󵄨󵄨𝑥^𝑠𝜙^󸀠(𝑥_𝑠)󵄨󵄨󵄨󵄨󵄨²𝑑𝑠] ≤ 𝑒^{2(𝜏−𝑡)}𝐸 [∫^𝑡

0󵄨󵄨󵄨󵄨𝑥^𝑠󵄨󵄨󵄨󵄨²𝑑𝑠]

≤ 𝑒^{2(𝜏−𝑡)}𝐸 [∫^𝑡

0(1 + 󵄨󵄨󵄨󵄨𝑥^𝑠󵄨󵄨󵄨󵄨²) 𝑑𝑠]

= 𝑒^{2(𝜏−𝑡)}𝑡 + 𝑒^{2(𝜏−𝑡)}𝐸 [∫^𝑡

0󵄨󵄨󵄨󵄨𝑥^𝑠󵄨󵄨󵄨󵄨²𝑑𝑠] , (24) and by considering𝑐_𝑡= 0, for all𝑡 ≥ 0, then (11) becomes

𝑑 ̆𝑥_𝑡= [𝑓 ( ̆𝑥_𝑡) − ̃𝜆 ̆𝑥_𝑡] 𝑑𝑡 − 𝜎 ̆𝑥_𝑡𝑑𝑤_𝑡, ̆𝑥₀= 𝑥, 𝑥 > 0, (25) by the comparison theorem of Ikeda and Watanabe [29]; we see that0 < 𝑥_𝑡 ≤ ̆𝑥_𝑡, for all 𝑡 ≥ 0. Hence, by applying the existence and uniqueness theorem for (11), we have

𝐸 [∫^𝑡

0󵄨󵄨󵄨󵄨𝑥^𝑠󵄨󵄨󵄨󵄨²𝑑𝑠] < ∞. (26) Therefore, from (24), we have

𝐸 [∫^𝑡

0󵄨󵄨󵄨󵄨󵄨𝑥^𝑠𝜙^󸀠(𝑥_𝑠)󵄨󵄨󵄨󵄨󵄨²𝑑𝑠] < ∞, (27) which yields that∫₀^𝑡𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^󸀠(𝑥_𝑠)𝑑𝑊_𝑠 is a martingale, and again by (11), we can take sufficiently small𝛿 > 0such that

𝐸 [sup

𝑡 ∫^𝑡

0𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^󸀠(𝑥_𝑠) 𝑑𝑊_𝑠]

≤ 2 |𝜎| 𝐸[sup

𝑡 ∫^∞

0 𝑒^{−(𝜌+1/𝛿)𝑠}𝑥²_𝑠𝑑𝑠]^1/2< ∞.

(28)

Hence,

𝐸 [∫^𝑡∧𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^󸀠(𝑥_𝑠) 𝑑𝑤_𝑠] = 0. (29)

Therefore, by (20), (11) and taking expectation on both sides of (23), we obtain

𝐸 [∫^𝑡∧𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑠}{𝑈 (𝑐_𝑠𝑥_𝑠) + 1

𝛿V(𝑥_𝑠)} 𝑑𝑠] ≤ 𝜙 (𝑥) , (30) from which we deduce (21).

We set𝑧_𝑡= 𝑥_𝑡− ̃𝑥_𝑡and by (11), it is clear that 𝑑𝑧_𝑡= 𝑑 (𝑥_𝑡− ̃𝑥_𝑡) = [𝑓 (𝑥_𝑡) − 𝑓 (̃𝑥_𝑡) − ̃𝜆 (𝑥_𝑡− ̃𝑥_𝑡)] 𝑑𝑡

− 𝜎 (𝑥_𝑡− ̃𝑥_𝑡) 𝑑𝑊_𝑡. (31)

Since by (3),𝑓(𝑧)is Lipschitz continuous and concave and 𝑓(0) = 0, then we have

𝑑𝑧_𝑡≤ (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)𝑧^𝑡𝑑𝑡 − 𝜎𝑧 (𝑡) 𝑑𝑊_𝑡, 𝑧 (0) = 𝑥 − ̃𝑥.

(32) Take𝜇 > 0such that𝑔_𝜇(𝑧) = (𝑧²+ 𝜇)^1/2, and we can find from (18) and (20) that

− (𝜌 +1

𝛿) 𝑔_𝜇(𝑧) +1

2𝜎²𝑧²𝑔^󸀠󸀠_𝜇(𝑧) + (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)󵄨󵄨󵄨󵄨󵄨𝑧𝑔^󸀠^𝜇(𝑧)󵄨󵄨󵄨󵄨󵄨

≤ (𝑧²+ 𝜇)^1/2{− (𝜌 +1 𝛿) + 1

2𝜎²

+ (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)} < 0, ∀𝑧 ∈R.

(33) Using Itˆo’s formula and by (20), we have

𝐸[𝑒^{−(𝜌+1/𝛿)𝜏}𝑔_𝜇(𝑧_𝜏)]

= 𝑔_𝜇(𝑥 − ̃𝑥) + 𝐸 [∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑠}

× {− (𝜌 +1

𝛿) 𝑔_𝜇(𝑧_𝑠)

+ (𝑓 (𝑥_𝑠)−𝑓 (̃𝑥_𝑠)−̃𝜆𝑧_𝑠) 𝑔_𝜇^󸀠(𝑧_𝑠) +1

2𝜎²𝑧²_𝑠𝑔_𝜇^󸀠󸀠(𝑧_𝑠)} 𝑑𝑠

− ∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑧_𝑠𝑔_𝜇^󸀠(𝑧_𝑠) 𝑑𝑊_𝑠]

≤ 𝑔_𝜇(𝑥 − ̃𝑥) .

(34)

Letting𝛿 → 0, and by Fatou’s lemma, we obtain

𝐸 [𝑒^{−(𝜌+1/𝛿)𝜏}󵄨󵄨󵄨󵄨𝑧^𝜏󵄨󵄨󵄨󵄨] ≤ |𝑥 − ̃𝑥|, (35) which implies (22).

Theorem 2. One assumes(2),(3),(17), and(18), then the value function is a viscosity solution of the reduced (one-dimension) HJB equation(9)such that0 ≤V(𝑥) ≤ 𝜙(𝑥).

(5)

Proof. Following (13) and (21) we have V(𝑥) =sup

𝑐∈̃A

𝐸 [∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑡}{𝑈 (𝑐_𝑡𝑥_𝑡) +1

𝛿V(𝑥_𝑡)} 𝑑𝑡] ≤ 𝜙 (𝑥) ,

∀𝑥 ≥ 0, (36) for any stopping time𝜏. By (13) and for any𝜀 > 0, there exists 𝑐 ∈ ̃Asuch that

V(𝑥) − 𝜀 < 𝐸 [∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑡}{𝑈 (𝑐_𝑡𝑥_𝑡) +1

𝛿V(𝑥_𝑡)} 𝑑𝑡]

= 𝐸 [∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑡}𝑈 (𝑐_𝑡𝑥_𝑡) 𝑑𝑡]

+ 𝐸 [∫^𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑡}1

𝛿V(𝑥_𝑡) 𝑑𝑡] .

(37)

Since𝑓(𝑧)is Lipschitz continuous, it follows that

𝑑𝑧_𝑡= [𝑓 (𝑥_𝑡) − 𝑓 (̃𝑥_𝑡) − ̃𝜆 (𝑥_𝑡− ̃𝑥_𝑡)] 𝑑𝑡 − 𝜎 (𝑥_𝑡− ̃𝑥_𝑡) 𝑑𝑊_𝑡

≤ (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)𝑧^𝑡𝑑𝑡 − 𝜎𝑧_𝑡𝑑𝑊_𝑡, 𝑧 (0) = 𝑥 − ̃𝑥.

(38) By (11), we can consider that̃𝑧_𝑡is the solution of

𝑑̃𝑧_𝑡= (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)̃𝑧^𝑡𝑑𝑡 − 𝜎̃𝑧_𝑡𝑑𝑊_𝑡, ̃𝑧 (0) = 𝑥 > 0. (39) So by the comparison theorem Ikeda and Watanabe [29], we have

𝜏_𝑧↓ ̃𝜏, 𝑧_𝑡≤ ̃𝑧_𝑡↓ 0, a.s. as𝑧 ↓ 0. (40) Since𝐸[sup_{0≤𝑡≤𝐿}̃𝑧²_𝑡] < ∞for all𝐿 > 0, now by (11) we have

𝐸[𝑧_𝜏∧𝐿] = 𝑧 + 𝐸 [∫^𝜏∧𝐿

0 {𝑓 (𝑧_𝑡) − ̃𝜆𝑧_𝑡− 𝑐_𝑡} 𝑑𝑡] . (41) Letting𝑧 ↓ 0and then𝐿 → 0, we obtain

𝐸 [∫^̃𝜏

0 𝑐_𝑡𝑑𝑡] = 𝐸[∫^̃𝜏

0 {𝑓 (0) − 𝑐_𝑡} 𝑑𝑡] ≥ 0, (42) so that

𝐸 [∫^̃𝜏

0 𝑒^{−(𝜌+1/𝛿)𝑡}𝑈 (𝑐_𝑡𝑥_𝑡) 𝑑𝑡] = 0. (43) Passing to the limit to (37) and applying (43), we obtain

V(0+) − 𝜀 < 𝐸 [∫^∞

0 𝑒^{−(𝜌+1/𝛿)𝑡}1

𝛿V(0+) 𝑑𝑡]

= V(0+) 𝜌𝛿 + 1,

(44)

which impliesV(0+) = 0. Thus, V ∈ 𝐶[0, ∞). So by the standard stability results of Fleming and Soner [27], we deduce thatVis a viscosity solution of (9).

4. Optimal Consumption Policy

Under the assumption (1) and (2),Lemma 3has revealed that the value function of the representative household assets must approach zero as time approaches infinity.

Lemma 3. One assumes(2),(3), and(17). Then for any(𝑐_𝑡) ∈ A. One has

lim inf

𝑡 → ∞ 𝐸[𝑒^−𝜌𝑡𝑢 (𝐾_𝑡, 𝐿_𝑡)] = 0. (45)

Proof. By (17) and (18), we take𝜇 ∈ (0, 𝜌)such that

−𝜇 +1

2𝜎²+ (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨) − ̃𝜆 < 0. (46) Take𝜇 > 0and𝑥 = 𝐾/𝐿 > 0such that𝑔_𝜇(𝑥) = (𝑥²+ 𝜇)^1/2, and by (33) and (46)

−𝜇𝑔_𝜇(𝑥) +1

2𝜎²𝑥²𝑔^󸀠󸀠_𝜇(𝑥) + (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)󵄨󵄨󵄨󵄨󵄨𝑥𝑔^󸀠^𝜇(𝑥)󵄨󵄨󵄨󵄨󵄨

≤ (𝑥²+ 𝜇)^1/2{−𝜇 +1

2𝜎²+ (𝑓^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)} < 0. (47) SettingΦ(𝐾, 𝐿) = 𝐴(𝐾/𝐿)^𝛾, where𝐴, 𝛾 > 0, we have by (6) and (47)

− 𝜇Φ (𝐾, 𝐿) +1

2𝜎²𝐿²Φ_𝐿𝐿+ 𝑛𝐿Φ_𝐿+ Φ_𝐾(𝐹 (𝐾, 𝐿) − 𝜆𝐾) + 𝑈^∗(𝐿Φ_𝐾) < 0 𝐾, 𝐿 > 0.

(48)

By Itˆo’s formula and (48), we obtain

𝑒^−𝜌𝑡Φ (𝐾_𝑡, 𝐿_𝑡)

= Φ (𝐾, 𝐿) + ∫^𝑡

0𝑒^−𝜌𝑠

× { (−𝜌 + 𝜇) Φ

+ Φ_𝐾(𝐹 (𝐾, 𝐿) − 𝜆𝐾 − 𝑐_𝑠𝐿) + 𝑛𝐿Φ_𝐿 +1

2𝜎²𝐿²Φ_𝐿𝐿− 𝜇Φ}󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨^(𝐾=𝐾^𝑠^,𝐿=𝐿^𝑠⁾𝑑𝑠

(6)

− ∫^𝑡

0𝑒^−𝜌𝑠𝜎𝐿_𝑠Φ_𝐿𝑑𝑤_𝑠

≤ Φ (𝐾, 𝐿) + ∫^𝑡

0𝑒^−𝜌𝑠{(−𝜌 + 𝜇) Φ − 𝑐_𝑠𝐿Φ_𝐾

−𝑈^∗(𝐿Φ_𝐾)} |_(𝐾=𝐾_𝑠_,𝐿=𝐿_𝑠₎𝑑𝑠

− ∫^𝑡

0𝑒^−𝜌𝑠𝜎𝐿_𝑠Φ_𝐿𝑑𝑤_𝑠

≤ Φ (𝐾, 𝐿) + ∫^𝑡

0𝑒^−𝜌𝑠{(−𝜌 + 𝜇) Φ

−𝑈 (𝑐_𝑠)} |_(𝐾=𝐾_𝑠_,𝐿=𝐿_𝑠₎𝑑𝑠

− ∫^𝑡

0𝑒^−𝜌𝑠𝜎𝐿_𝑠Φ_𝐿𝑑𝑤_𝑠,

0 ≤ 𝐸 [𝑒^−𝜌𝑡Φ (𝐾_𝑡, 𝐿_𝑡)] ≤ Φ (𝐾, 𝐿) + 𝐸 [∫^𝑡

0𝑒^−𝜌𝑠(−𝜌 + 𝜇) Φ (𝐾_𝑠, 𝐿_𝑠) 𝑑𝑠] .

(49) Letting𝑡 → ∞, we have

𝐸 [∫^∞

0 𝑒^−𝜌𝑡Φ (𝐾_𝑡, 𝐿_𝑡) 𝑑𝑡] ≤ 1

𝜌 − 𝜇 Φ (𝐾, 𝐿) < ∞, (50) which implies lim inf

𝑡 → ∞ 𝐸[𝑒^−𝜌𝑡Φ(𝐾_𝑡, 𝐿_𝑡)] = 0. By (21), we have 𝑢 (𝐾, 𝐿) ≤ Φ (𝐾, 𝐿) , (51) which completes the proof.

We give a synthesis of the optimal policy𝑐^∗= {𝑐_𝑡^∗}for the optimization problem (4) subject to (5).

Theorem 4. Under(2)and(3), there exists a unique solution 𝐾_𝑡^∗≥ 0of

𝑑𝐾_𝑡^∗= [𝐹 (𝐾_𝑡^∗, 𝐿_𝑡) − 𝜆𝐾_𝑡^∗− 𝑐_𝑡^∗𝐾_𝑡^∗] 𝑑𝑡 𝐾₀^∗= 𝐾, 𝐾 > 0, (52) and the optimal consumption policy𝑐_𝑡^∗ ∈Ais given by

𝑐_𝑡^∗= 𝑔 (𝐾_𝑡

𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) . (53) Proof. Let us consider 𝐺(𝐾, 𝐿) = 𝐹(𝐾_𝑡, 𝐿_𝑡) − 𝜆𝐾_𝑡 − 𝑔(𝐾_𝑡/𝐿_𝑡,𝐿_𝑡𝑢_𝐾(𝐾_𝑡, 𝐿_𝑡))𝐾_𝑡 and since 𝐹(𝐾, 𝐿).

𝑔(𝐾_𝑡/𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) are continuous and 𝐺(0, 𝐿) = 0, there exists anF_𝑡progressively measurable solution𝜅_𝑡of

𝑑𝜅_𝑡= 𝐺 (𝐾_𝑡, 𝐿_𝑡) 𝑑𝑡, 𝜅₀= 𝐾 > 0. (54) Now we shall show𝐾^∗_𝑡 ≥ 0for all 𝑡 ≥ 0a.s. Suppose0 <

V^󸀠(0+) < ∞. By (9), we haveV^󸀠(𝑥) ≥ 0for all𝑥 > 0, since

𝑈^∗(V^󸀠(𝑥)) = ∞ifV^󸀠(𝑥) < 0. Moreover, by L’Hospital’s rule this gives

𝑥 → 0+lim 𝑥²V^󸀠󸀠(𝑥) = lim

𝑥 → 0+𝑥²V^󸀠󸀠(𝑥) + 2𝑥V^󸀠(𝑥)

= lim

𝑥 → 0+

𝑥²V^󸀠(𝑥) 𝑥 = 0.

(55)

Letting𝑥 → 0+in (9), we have𝑈^∗(V^󸀠(0+)) = 0, and this is contrary with (2). Therefore, we get V^󸀠(0+) = ∞, which implies𝑔(0, 𝐿𝑢_𝐾(0+, 𝐿)) = 0. We note by the concavity of𝐹 that𝐺(𝐾, 𝐿) ≤ 𝐶₁𝐾 + 𝐶₂𝐿for𝐶₁, 𝐶₂> 0. Then applying the comparison theorem to (54) and

𝑑̃𝐾_𝑡= (𝐶₁𝐾̃_𝑡+ 𝐶₂𝐿_𝑡) 𝑑𝑡, ̃𝐾₀= 𝐾 > 0, (56) we obtain0 ≤ 𝐾_𝑡^∗ ≤ ̃𝐾_𝑡for all𝑡 ≥ 0. Further, in case𝜏_𝐾^∗ = inf{𝑡 ≥ 0 : 𝐾^∗_𝑡 = 0} = 𝜏_𝜅 < ∞, we have at𝑡 = 𝜏_𝐾^∗

𝑑𝐾_𝑡^∗

𝑑𝑡 = 𝐹 (𝐾_𝑡^∗, 𝐿_𝑡) − 𝜆𝐾^∗_𝑡 − 𝑔 (𝐾_𝑡

𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) 𝐿_𝑡= 0.

(57) Therefore,𝐾_𝑡^∗solves (52) and𝐾_𝑡^∗ ≥ 0. To prove uniqueness, let𝐾^∗_𝑖(𝑡), 𝑖 = 1, 2, be two solutions of (52). Then𝐾₁^∗(𝑡)−𝐾^∗₂(𝑡) satisfies

𝑑 (𝐾^∗₁(𝑡) − 𝐾₂^∗(𝑡)) = [ (𝐹 (𝐾₁^∗(𝑡) , 𝐿_𝑡) − 𝐹 (𝐾₂^∗(𝑡) , 𝐿_𝑡))

− 𝜆 (𝐾^∗₁(𝑡) − 𝐾₂^∗(𝑡))

− (𝑔 (𝐾_𝑡

𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) 𝐿_𝑡

−𝑔 (𝐾_𝑡

𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) 𝐿_𝑡)] 𝑑𝑡, (58) 𝐾^∗₁(0) − 𝐾₂^∗(0) = 0. We have

𝑑(𝐾₁^∗(𝑡) − 𝐾₂^∗(𝑡))²

= 2 (𝐾^∗₁(𝑡) − 𝐾^∗₂(𝑡)) 𝑑 (𝐾^∗₁(𝑡) − 𝐾₂^∗(𝑡))

= 2 (𝐾^∗₁(𝑡) − 𝐾^∗₂(𝑡))

× [ (𝐹 (𝐾₁^∗(𝑡) , 𝐿_𝑡) − 𝐹 (𝐾₂^∗(𝑡) , 𝐿_𝑡))

− 𝜆 (𝐾^∗₁(𝑡) − 𝐾₂^∗(𝑡))

− (𝑔 (𝐾_𝑡

𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) 𝐿_𝑡

−𝑔 (𝐾_𝑡

𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡)) 𝐿_𝑡)] 𝑑𝑡.

(59)

(7)

Note that the function𝑥 → −𝑔(𝐾_𝑡/𝐿_𝑡, 𝐿_𝑡𝑢_𝐾(𝐾_𝑡^∗, 𝐿_𝑡))𝐿_𝑡 is decreasing. Hence,

(𝐾₁^∗(𝑡) − 𝐾₂^∗(𝑡))²

≤ 2 ∫^𝑡

0(𝐾₁^∗(𝑠) − 𝐾₂^∗(𝑠))

× [(𝐹 (𝐾₁^∗(𝑠) , 𝐿_𝑠) − 𝐹 (𝐾₂^∗(𝑠) , 𝐿_𝑠))

−𝜆 (𝐾₁^∗(𝑠) − 𝐾^∗₂(𝑠))] 𝑑𝑠

≤ 2 (𝐹^󸀠(0) +󵄨󵄨󵄨󵄨󵄨̃𝜆󵄨󵄨󵄨󵄨󵄨)∫₀^𝑡(𝐾₁^∗(𝑠) − 𝐾₂^∗(𝑠))²𝑑𝑠.

(60)

By Gronwall’s lemma, we have

𝐾^∗₁(𝑡) = 𝐾₂^∗(𝑡) , ∀𝑡 > 0. (61) So, the uniqueness of (52) holds.

Now by (6), (52), and Itˆo’s formula, we have 𝑒^−𝜌𝑡𝑢 (𝐾_𝑡^∗, 𝐿_𝑡)

= 𝑢 (𝐾, 𝐿) + ∫^𝑡

0𝑒^−𝜌𝑠

× { − 𝜌𝑢 (𝐾, 𝐿) +𝑢_𝐾(𝐾, 𝐿)

× (𝐹 (𝐾, 𝐿)− 𝜆𝐾 − 𝑐_𝑠^∗𝐾) + 𝑛𝐿𝑢_𝐿(𝐾, 𝐿)

+1

2𝜎²𝐿²𝑢_𝐿𝐿(𝐾, 𝐿)}󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨^(𝐾=𝐾^∗^𝑠^,𝐿=𝐿^𝑠⁾𝑑𝑠 + ∫^𝑡

0𝑒^−𝜌𝑠𝜎𝐿_𝑠𝑢_𝐿(𝐾, 𝐿) 𝑑𝑊_𝑠.

(62) By the HJB equation (6), we have

𝑒^−𝜌𝑡𝑢 (𝐾^∗_𝑡, 𝐿_𝑡) = 𝑢 (𝐾, 𝐿) − ∫^𝑡

0𝑒^−𝜌𝑠𝑈 (𝑐_𝑠^∗𝐾_𝑠^∗ 𝐿_𝑠 ) 𝑑𝑠 + ∫^𝑡

0𝑒^−𝜌𝑠𝜎𝐿_𝑠𝑢_𝐿(𝐾, 𝐿) 𝑑𝑊_𝑠,

(63)

from which

𝐸[𝑒^{−𝜌(𝑡∧𝜏}^𝑛⁾𝑢 (𝐾_𝑡∧𝜏^∗ _𝑛, 𝐿_𝑡∧𝜏_𝑛)]

+ 𝐸 [∫^𝑡∧𝜏^𝑛

0 𝑒^−𝜌𝑠𝑈 (𝑐_𝑠^∗𝐾_𝑠^∗

𝐿_𝑠 ) 𝑑𝑠] = 𝑢 (𝐾, 𝐿) , (64)

where{𝜏_𝑛}is a sequence of localizing stopping times for the local martingale. From (11), (51), and Doob’s inequalities for martingales, it follows that

𝐸 [sup

𝑛 𝑒^{−𝜌(𝑡∧𝜏}^𝑛⁾𝑢 (𝐾_𝑡∧𝜏^∗ _𝑛, 𝐿_𝑡∧𝜏_𝑛)]

≤ 𝐸 [sup

0≤𝑟≤𝑡𝑒^−𝛼𝑟𝜙 (𝐾^∗_(𝑟), 𝐿_(𝑟))]

≤ 𝜙 (𝐾, 𝐿) + 𝐸 [sup

0≤𝑟≤𝑡

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫⁰^𝑟𝑒^−𝛼𝑡𝜎𝐾_𝑡^∗𝑑𝑊_𝑡󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨]

≤ 𝜙 (𝐾, 𝐿) + 2 |𝜎| 𝐸[∫^𝑡

0(𝑒^−𝛼𝑡𝐾_𝑡^∗)²𝑑𝑡]^1/2

≤ 𝜙 (𝐾, 𝐿) + 2 |𝜎| 𝐸[∫^𝑡

0𝐾̃²_𝑡𝑑𝑡]^1/2.

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Letting𝑛 → ∞ and 𝑡 → ∞, hence, we obtain by the dominated convergence theorem

𝐸[𝑒^−𝜌𝜏^∗𝑢 (𝐾^∗_𝜏∗, 𝐿_𝜏^∗)]

+ 𝐸 [∫^𝜏

∗

0 𝑒^−𝜌𝑠𝑈 (𝑐_𝑠^∗𝐾_𝑠^∗

𝐿_𝑠 ) 𝑑𝑠] = 𝑢 (𝐾, 𝐿) .

(66)

We deduce byLemma 3 𝐽 (𝑐^∗) = 𝐸 [∫^𝜏

∗

0 𝑒^−𝜌𝑠𝑈 (𝑐_𝑠^∗𝐾^∗_𝑠

𝐿_𝑠 ) 𝑑𝑠] = 𝑢 (𝐾, 𝐿) . (67) Following the same calculation as above, we have

𝑒^−𝜌𝑡𝑢 (𝐾_𝑡, 𝐿_𝑡)

= 𝑢 (𝐾, 𝐿) + ∫^𝑡

0𝑒^−𝜌𝑠

× { − 𝜌𝑢 (𝐾, 𝐿)

+ 𝑢_𝐾(𝐾, 𝐿) (𝐹 (𝐾, 𝐿) − 𝜆𝐾 − 𝑐_𝑠𝐾) +𝑛𝐿𝑢_𝐿(𝐾, 𝐿)1

2𝜎²

×𝐿²𝑢_𝐿𝐿(𝐾, 𝐿) }󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨^(𝐾=𝐾^𝑠^,𝐿=𝐿^𝑠⁾𝑑𝑠 + ∫^𝑡

0𝑒^−𝜌𝑠𝜎𝐿_𝑠𝑢_𝐿(𝐾, 𝐿) 𝑑𝑊_𝑠.

(68) Again by the HJB equation (6), we can obtain

0 ≤ 𝐸 [𝑒^−𝜌𝜏𝑢 (𝐾_𝜏, 𝐿_𝜏)] ≤ 𝑢 (𝐾, 𝐿) − 𝐸 [∫^𝜏

0 𝑒^−𝜌𝑠𝑈 (𝑐_𝑠𝐾_𝑠 𝐿_𝑠 ) 𝑑𝑠]

(69) from which

𝐽 (𝑐) = 𝐸 [∫^𝜏

0 𝑒^−𝜌𝑠𝑈 (𝑐_𝑠𝐾_𝑠

𝐿_𝑠 ) 𝑑𝑠] ≤ 𝑢 (𝐾, 𝐿) (70) for any(𝑐_𝑡) ∈A. The proof is complete.

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Remark 5. From the proof ofTheorem 4, it follows that inf𝑐∈A𝐸 [∫^𝑇

𝑠 𝑒^−𝜌𝑡𝑈 (𝑐_𝑡𝐾_𝑡

𝐿_𝑡 ) 𝑑𝑡] = 𝑉 (𝑠, 𝐾, 𝐿) . (71) Thus, under (2), we observe that the smooth solution of𝑢of the HJB equation (6). Furthermore, letVbe the solution of (9) on the entire domain[0, 𝑇) × (0, ∞)withV(𝑇, 𝑥) = 0, 𝑥 > 0.

Setting 𝑥 = 𝐾/𝐿 and 𝑢(𝑡, 𝐾, 𝐿) = V(𝑡, 𝐾/𝐿), 𝐾, 𝐿 > 0, by (8), we have that𝑢satisfies (6). Therefore, we obtain the uniqueness ofV.

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.

Acknowledgment

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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1.Introduction Md.AzizulBaten andAntonAbdulbasahKamil OptimalConsumptioninaStochasticRamseyModelwithCobb-DouglasProductionFunction ResearchArticle

Research Article

Optimal Consumption in a Stochastic Ramsey Model with Cobb-Douglas Production Function

Md. Azizul Baten

and Anton Abdulbasah Kamil

1. Introduction

2. Transformed Hamilton- Jacobi-Bellman Equation

3. Viscosity Solutions

4. Optimal Consumption Policy

5. Concluding Remarks

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis