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

Abstract

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 time 𝑡≥0,𝐿_𝑡 = labor supply at time 𝑡≥0,𝜆 = constant rate of depreciation, 𝜆≥0,𝑐_𝑡𝐾_𝑡 = consumption rate at time 𝑡≥0,0≤𝑐_𝑡≤1,𝑐_𝑡𝐾_𝑡/𝐿_𝑡 = totality of consumption rate per labor;𝐹(𝐾,𝐿) = 𝐴𝐾^𝛼𝐿^1−𝛼 with 0<𝛼<1 and 𝐴 is a constant, production function producing the commodity for the capital stock 𝐾>0 and the labor supply 𝐿>0,𝑛= rate of labor growth (nonzero constant),𝜎 = non-zero constant coefficients,𝜌= discount rate𝜌>0,𝑈(𝑐) = utility function for the consumption rate 𝑐≥0,𝑊_𝑡= one-dimensional standard Brownian motion on a complete probability space (Ω,ℱ,𝑃) endowed with the natural filtration ℱ_𝑡 generated by 𝜎(𝑊_𝑠,𝑠≤𝑡).

Let us assume that 𝑐={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, ^𝑡₀𝑐_𝑠𝑑𝑠<∞a.s.∀𝑡≥0, (1) and we denote by 𝒜 the set of all consumption policies {𝑐_𝑡} per capita.

The utility function 𝑈(𝑐) is assumed to have the following properties: 𝑈∈𝐶[0,∞)𝐶²(),𝑈(𝑐)[),𝑈0,∞∶strictlyconcaveon0,∞_′(𝑐)∶strictlydecreasing,𝑈^′′𝑈<0for𝑐>0,_′(∞)=𝑈^′(0+)=0,𝑈^′(0+)=𝑈(∞)=∞. (2) Following Merton [2], we make the following assumption on the Cobb-Douglas production function 𝐹(𝐾,𝐿): 𝐹()𝐹𝛾𝐾,𝛾𝐿=𝛾𝐹𝐾,𝐿,for𝛾>0,_𝐾(𝐾,𝐿)>0,𝐹_𝐿(𝐾,𝐿)>0,𝐹_𝐾𝐾()𝐹𝐾,𝐿<0_𝐿𝐿()𝐹()()𝐾,𝐿<0,0,𝐿=𝐹𝐾,0=0,𝐹_𝐾()𝐹0+,𝐿<∞,_𝐾(∞,𝐿)=0,𝐿>0. (3)

We are concerned with the economic growth model to maximize the expected discount utility of consumption 𝐽(𝑐)=𝐸^𝑇₀𝑒^−𝜌𝑡𝑈𝑐_𝑡𝐾_𝑡𝐿_𝑡𝑑𝑡 (4) per labor with a horizon 𝑇 over the class 𝑐∈𝒜 subject to the capital stock 𝐾_𝑡, and the labor supply 𝐿_𝑡 is governed by the stochastic differential equation 𝑑𝐾_𝑡=𝐹𝐾_𝑡,𝐿_𝑡−𝜆𝐾_𝑡−𝑐_𝑡𝐾_𝑡𝑑𝑡𝐾₀=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 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 [17–19]. 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 that 𝑢 solves the general (two-dimensional) Hamilton-Jacobi-Bellman (in short, HJB) equation ()+1−𝜌𝑢𝐾,𝐿2𝜎²𝐿²𝑢_𝐿𝐿(𝐾,𝐿)+𝑛𝐿𝑢_𝐿()+{𝐹()}𝑢𝐾,𝐿𝐾,𝐿−𝜆𝐾_𝐾(𝐾,𝐿)+𝑈^∗𝑢_𝐾()𝑢()𝐾,𝐿,𝐿=0,0,𝐿=0,𝐾>0,𝐿>0, (6) where 𝑈^∗(𝑢_𝐾,𝐿)=max_𝑐>0{𝑈(𝑐𝐾/𝐿)−𝑐𝐾𝑢_𝐾} and 𝑢_𝐾, 𝑢_𝐿, and 𝑢_𝐿𝐿 are partial derivatives of 𝑢(𝐾,𝐿) with respect to 𝐾 and 𝐿.

The technical difficulty in solving the problem lies in the fact that the HJB equation (6) is a parabolic PDE with two spatial variables 𝐾 and 𝐿. 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.

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 𝐾 for 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 solution 𝑢(𝐾,𝐿) of (6) of the form 𝑢()𝐾𝐾,𝐿=𝑣𝐿,𝐿>0. (7) Clearly 𝐿𝑢_𝐾=𝑣_𝐾,𝐿𝑢_𝐿=−𝐾𝑣_𝐾,𝐿²𝑢_𝐿𝐿=𝐾²𝑣_𝐾𝐾+2𝐾𝑣_𝐾. (8) Setting 𝑥=𝐾/𝐿 and substituting these above in (6), we have the HJB equation of 𝑣 of the following form: (𝑥)+1−𝜌𝑣2𝜎²𝑥²𝑣^′′(𝑥)+𝑓(𝑥)−𝜆𝑥𝑣^′(𝑥)+𝑈^∗𝑥,𝑣^′(𝑥)𝑣(0)=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 𝐽̃(𝑐)=𝐸^𝑇₀𝑒^−𝜌𝑡𝑈𝑐_𝑡𝑥_𝑡𝑑𝑡 (10) over the class 𝑐∈𝒜,   subject to 𝑑𝑥_𝑡=𝑓𝑥_𝑡−𝜆𝑥_𝑡−𝑐_𝑡𝑑𝑡−𝜎𝑥_𝑡𝑑𝑊_𝑡,𝑥₀=𝑥,𝑥≥0, (11) where 𝑐∈𝒜 denotes the class 𝒜with {𝑥_𝑡} replacing {𝐾_𝑡}. We choose 𝛿>0 and rewrite (9) as −1𝜌+𝛿𝑣(𝑥)+12𝜎²𝑥²𝑣^′′(𝑥)+𝑓(𝑥)−𝜆𝑥𝑣^′(𝑥)+𝑈^∗𝑥,𝑣^′(𝑥)+1𝛿𝑣(𝑥)𝑣(0)=0,=0,𝑥>0. (12)

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, 𝑉𝑥_𝑡=sup_𝑐∈𝒜𝐸^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑡}𝑈𝑐_𝑡𝑥_𝑡+1𝛿𝑣𝑥_𝑡𝑑𝑡, (13) where 𝜏=𝜏(𝑥)=inf{𝑡≥0∶𝑥_𝑡=0} and 𝑐={𝑐_𝑡} is the element of the class 𝒜 consisting of ℱ_𝑡 progressively measurable processes such that ^𝑡₀𝑐_𝑠𝑑𝑠<∞a.s.∀𝑡≥0. (14)

3. Viscosity Solutions

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

3.1. Definition

Let 𝑣∈𝐶[0,∞) and 𝑣(0)=0. Then 𝑣 is called a viscosity solution of the reduced (one-dimensional) HJB equation (9) if the following relations hold: (𝑥)+1−𝜌𝑣2𝜎²𝑥²𝑞+𝑝𝑓(𝑥)−𝜆𝑥+𝑈^∗∀𝑥,𝑝≥0,𝑝,𝑞∈𝐽^2,+𝑣(𝑥)(𝑥)+1,∀𝑥>0,−𝜌𝑣2𝜎²𝑥²𝑞+𝑝𝑓(𝑥)−𝜆𝑥+𝑈^∗∀𝑥,𝑝≤0,𝑝,𝑞∈𝐽^2,−𝑣(𝑥),∀𝑥>0, (15) where 𝐽^2,+ and 𝐽^2,− are defined by 𝐽^2,+𝑣(𝑥)=𝑝,𝑞∈𝐑²∶𝑥limsup_→𝑥𝑣(𝑥)−𝑣(𝑥)−𝑝(𝑥−𝑥)−(1/2)𝑞|𝑥−𝑥|²|𝑥−𝑥|²,𝐽≤0_2,−𝑣(𝑥)=𝑝,𝑞∈𝐑²∶𝑥liminf_→𝑥𝑣(𝑥)−𝑣(𝑥)−𝑝(𝑥−𝑥)−(1/2)𝑞|𝑥−𝑥|²|𝑥−𝑥|²≥0. (16)

We assume that 𝑓𝜌+𝜆>0,−𝜌+^′(0)+𝜆+12𝜎²<0. (17)(18) Take 0<𝛼<𝜌, and we choose 𝑃₁>0 by concavity such that 𝑓(𝑥)−𝜆𝑥≤𝑃₁. (19) Taking sufficiently large 𝑃₀>𝑃₁, we observe by (2) and (19) that 𝜙(𝑡,𝑥)=𝑒^𝜏−𝑡(𝑥+𝑃₀) fulfills (𝑥)+1−𝛼𝜙2𝜎²𝑥²𝜙^′′(𝑥)+𝑓(𝑥)−𝜆𝑥𝜙^′(𝑥)+𝑈^∗𝑥,𝜙^′(𝑥)≤𝑒^𝜏−𝑡−𝑥−𝑃₀+𝑃₁+𝑈𝑜𝑈^′−1𝑒^𝜏−𝑡<−𝑥−𝑃₀+𝑃₁+𝑈𝑜𝑈^′−1(1)<0, (20) for some constant 𝑃₀>0.

Lemma 1. One assumes (2), (17), (11), and (20), then the value function 𝑉(𝑥) fulfills 0≤𝑉𝑥_𝑡≤𝜙𝑥_𝑡,sup_𝑐∈𝒜𝐸𝑒^{−(𝜌+1/𝛿)𝜏}|𝑥(𝜏)−𝑥(𝜏)|≤|𝑥−𝑥| (21)(22) for any stopping time 𝜏, where {𝑥(𝜏)} is the solution of (11) to 𝑐∈𝒜 with 𝑥(0)=𝑥.

Proof. Itô’s formula gives 0≤𝑒^{−(𝜌+1/𝛿)(𝑡∧𝜏)}𝜙𝑥_{(𝑡∧𝜏)}=𝜙(𝑥)+₀^𝑡∧𝜏𝑒^{−(𝜌+1/𝛿)𝑠}×−1𝜌+𝛿𝜙𝑥_𝑠+𝑓(𝑥)−𝜆𝑥_𝑠−𝑐_𝑠𝑥_𝑠𝜙^′𝑥_𝑠+12𝜎²𝑥²_𝑠𝜙^′′𝑥_𝑠−𝑑𝑠_0𝑡∧𝜏𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^′𝑥_𝑠𝑑𝑊_𝑠. (23) Since 𝐸^𝑡₀𝑥_𝑠𝜙^′𝑥_𝑠²𝑑𝑠≤𝑒^{2(𝜏−𝑡)}𝐸^𝑡₀𝑥_𝑠²𝑑𝑠≤𝑒^{2(𝜏−𝑡)}𝐸^𝑡₀1+𝑥_𝑠²𝑑𝑠=𝑒^{2(𝜏−𝑡)}𝑡+𝑒^{2(𝜏−𝑡)}𝐸^𝑡₀𝑥_𝑠²𝑑𝑠, (24) and by considering 𝑐_𝑡=0,forall𝑡≥0, then (11) becomes 𝑑𝑥̆_𝑡=𝑓𝑥̆_𝑡−𝜆𝑥̆_𝑡𝑑𝑡−𝜎𝑥̆_𝑡𝑑𝑤_𝑡,𝑥̆₀=𝑥,𝑥>0, (25) by the comparison theorem of Ikeda and Watanabe [29]; we see that 0<𝑥_𝑡≤𝑥̆_𝑡,  forall𝑡≥0. Hence, by applying the existence and uniqueness theorem for (11), we have 𝐸^𝑡₀𝑥_𝑠²𝑑𝑠<∞. (26) Therefore, from (24), we have 𝐸^𝑡₀𝑥_𝑠𝜙^′𝑥_𝑠²𝑑𝑠<∞, (27) which yields that ∫^𝑡₀𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^′(𝑥_𝑠)𝑑𝑊_𝑠 is a martingale, and again by (11), we can take sufficiently small 𝛿>0 such that 𝐸sup_𝑡^𝑡₀𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^′𝑥_𝑠𝑑𝑊_𝑠≤2|𝜎|𝐸sup_𝑡^∞₀𝑒^{−(𝜌+1/𝛿)𝑠}𝑥²_𝑠𝑑𝑠^1/2<∞. (28) Hence, 𝐸₀^𝑡∧𝜏𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑥_𝑠𝜙^′𝑥_𝑠𝑑𝑤_𝑠=0. (29) Therefore, by (20), (11) and taking expectation on both sides of (23), we obtain 𝐸₀^𝑡∧𝜏𝑒^{−(𝜌+1/𝛿)𝑠}𝑈𝑐_𝑠𝑥_𝑠+1𝛿𝑣𝑥_𝑠𝑑𝑠≤𝜙(𝑥), (30) from which we deduce (21).
We set 𝑧_𝑡=𝑥_𝑡−𝑥_𝑡 and by (11), it is clear that 𝑑𝑧_𝑡=𝑑𝑥_𝑡−𝑥_𝑡=𝑓𝑥_𝑡−𝑓𝑥_𝑡−𝜆𝑥_𝑡−𝑥_𝑡𝑥𝑑𝑡−𝜎_𝑡−𝑥_𝑡𝑑𝑊_𝑡. (31) Since by (3), 𝑓(𝑧) is Lipschitz continuous and concave and 𝑓(0)=0, then we have 𝑑𝑧_𝑡≤𝑓^′(0)+𝜆𝑧_𝑡𝑑𝑡−𝜎𝑧(𝑡)𝑑𝑊_𝑡,𝑧(0)=𝑥−𝑥. (32) Take 𝜇>0 such that 𝑔_𝜇(𝑧)=(𝑧²+𝜇)^1/2, and we can find from (18) and (20) that −1𝜌+𝛿𝑔_𝜇(𝑧)+12𝜎²𝑧²𝑔_𝜇^′′(𝑧)+𝑓^′(0)+𝜆𝑧𝑔^′_𝜇(𝑧≤𝑧₂)+𝜇^1/2−1𝜌+𝛿+12𝜎²+𝑓^′(0)+𝜆<0,∀𝑧∈𝐑. (33) Using Itô’s formula and by (20), we have 𝐸𝑒^{−(𝜌+1/𝛿)𝜏}𝑔_𝜇𝑧_𝜏=𝑔_𝜇(𝑥−𝑥)+𝐸^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑠}×−1𝜌+𝛿𝑔_𝜇𝑧_𝑠+𝑓𝑥_𝑠−𝑓𝑥_𝑠−𝜆𝑧_𝑠𝑔^′_𝜇𝑧_𝑠+12𝜎²𝑧²_𝑠𝑔_𝜇^′′𝑧_𝑠𝑑𝑠−^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑠}𝜎𝑧_𝑠𝑔^′_𝜇𝑧_𝑠𝑑𝑊_𝑠≤𝑔_𝜇(𝑥−𝑥). (34) Letting 𝛿→0, and by Fatou’s lemma, we obtain 𝐸𝑒^{−(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 that 0≤𝑣(𝑥)≤𝜙(𝑥).

Proof. Following (13) and (21) we have 𝑣(𝑥)=sup_𝑐∈𝒜𝐸^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑡}𝑈𝑐_𝑡𝑥_𝑡+1𝛿𝑣𝑥_𝑡𝑑𝑡≤𝜙(𝑥),∀𝑥≥0, (36) for any stopping time 𝜏. By (13) and for any 𝜀>0, there exists 𝑐∈𝒜 such that 𝑣(𝑥)−𝜀<𝐸^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑡}𝑈𝑐_𝑡𝑥_𝑡+1𝛿𝑣𝑥_𝑡𝑑𝑡=𝐸^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑡}𝑈𝑐_𝑡𝑥_𝑡𝑑𝑡+𝐸^𝜏₀𝑒^{−(𝜌+1/𝛿)𝑡}1𝛿𝑣𝑥_𝑡𝑑𝑡. (37) Since 𝑓(𝑧) is Lipschitz continuous, it follows that 𝑑𝑧_𝑡=𝑓𝑥_𝑡−𝑓𝑥_𝑡−𝜆𝑥_𝑡−𝑥_𝑡𝑑𝑡−𝜎𝑥_𝑡−𝑥_𝑡𝑑𝑊_𝑡≤𝑓^′(0)+𝜆𝑧_𝑡𝑑𝑡−𝜎𝑧_𝑡𝑑𝑊_𝑡,𝑧(0)=𝑥−𝑥. (38) By (11), we can consider that 𝑧_𝑡 is the solution of 𝑑𝑧_𝑡=𝑓^′(0)+𝜆𝑧_𝑡𝑑𝑡−𝜎𝑧_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) Since 𝐸[sup_{0≤𝑡≤𝐿}𝑧²_𝑡]<∞ for all 𝐿>0, now by (11) we have 𝐸𝑧_𝜏∧𝐿=𝑧+𝐸₀^𝜏∧𝐿𝑓𝑧_𝑡−𝜆𝑧_𝑡−𝑐_𝑡𝑑𝑡. (41) Letting 𝑧↓0 and then 𝐿→0, we obtain 𝐸₀𝜏𝑐_𝑡𝜏𝑑𝑡=𝐸₀𝑓(0)−𝑐_𝑡𝑑𝑡≥0, (42) so that 𝐸₀𝜏𝑒^{−(𝜌+1/𝛿)𝑡}𝑈𝑐_𝑡𝑥_𝑡𝑑𝑡=0. (43) Passing to the limit to (37) and applying (43), we obtain 𝑣(0+)−𝜀<𝐸^∞₀𝑒^{−(𝜌+1/𝛿)𝑡}1𝛿𝑣()=𝑣()0+𝑑𝑡0+,𝜌𝛿+1 (44) which implies 𝑣(0+)=0. Thus, 𝑣∈𝐶[0,∞). So by the standard stability results of Fleming and Soner [27], we deduce that 𝑣 is a viscosity solution of (9).

4. 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.

Lemma 3. One assumes (2), (3), and (17). Then for any (𝑐_𝑡)∈𝒜. One has liminf_𝑡→∞𝐸𝑒^−𝜌𝑡𝑢𝐾_𝑡,𝐿_𝑡=0. (45)

Proof. By (17) and (18), we take 𝜇∈(0,𝜌) such that 1−𝜇+2𝜎²+𝑓^′(0)+𝜆−𝜆<0. (46) Take 𝜇>0 and 𝑥=𝐾/𝐿>0 such that 𝑔_𝜇(𝑥)=(𝑥²+𝜇)^1/2, and by (33) and (46) −𝜇𝑔_𝜇(𝑥)+12𝜎²𝑥²𝑔_𝜇^′′(𝑥)+𝑓^′(0)+𝜆𝑥𝑔^′_𝜇(𝑥≤𝑥₂)+𝜇^1/21−𝜇+2𝜎²+𝑓^′(0)+𝜆<0. (47) Setting Φ(𝐾,𝐿)=𝐴(𝐾/𝐿)^𝛾, where 𝐴,𝛾>0, we have by (6) and (47) ()+1−𝜇Φ𝐾,𝐿2𝜎²𝐿²Φ_𝐿𝐿+𝑛𝐿Φ_𝐿+Φ_𝐾(𝐹(𝐾,𝐿)−𝜆𝐾)+𝑈^∗𝐿Φ_𝐾<0𝐾,𝐿>0. (48) By Itô’s formula and (48), we obtain 𝑒^−𝜌𝑡Φ𝐾_𝑡,𝐿_𝑡=Φ(𝐾,𝐿)+^𝑡₀𝑒^−𝜌𝑠×−𝜌+𝜇Φ+Φ_𝐾𝐹(𝐾,𝐿)−𝜆𝐾−𝑐_𝑠𝐿+𝑛𝐿Φ_𝐿+12𝜎²𝐿²Φ_𝐿𝐿−𝜇Φ_{(𝐾=𝐾𝑠,𝐿=𝐿𝑠)}−𝑑𝑠_𝑡0𝑒^−𝜌𝑠𝜎𝐿_𝑠Φ_𝐿𝑑𝑤_𝑠≤Φ(𝐾,𝐿)+^𝑡₀𝑒^−𝜌𝑠−𝜌+𝜇Φ−𝑐_𝑠𝐿Φ_𝐾−𝑈^∗𝐿Φ_𝐾∣_{(𝐾=𝐾𝑠,𝐿=𝐿𝑠)}−𝑑𝑠_𝑡0𝑒^−𝜌𝑠𝜎𝐿_𝑠Φ_𝐿𝑑𝑤_𝑠≤Φ(𝐾,𝐿)+^𝑡₀𝑒^−𝜌𝑠Φ𝑐−𝜌+𝜇−𝑈_𝑠∣_{(𝐾=𝐾𝑠,𝐿=𝐿𝑠)}−𝑑𝑠_𝑡0𝑒^−𝜌𝑠𝜎𝐿_𝑠Φ_𝐿𝑑𝑤_𝑠,0≤𝐸𝑒^−𝜌𝑡Φ𝐾_𝑡,𝐿_𝑡≤Φ(𝐾,𝐿)+𝐸^𝑡₀𝑒^−𝜌𝑠−𝜌+𝜇Φ𝐾_𝑠,𝐿_𝑠𝑑𝑠. (49) Letting 𝑡→∞, we have 𝐸^∞₀𝑒^−𝜌𝑡Φ𝐾_𝑡,𝐿_𝑡≤1𝑑𝑡Φ()𝜌−𝜇𝐾,𝐿<∞, (50) which implies liminf_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 policy 𝑐^∗={𝑐^∗_𝑡} for the optimization problem (4) subject to (5).

Theorem 4. Under (2) and (3), there exists a unique solution 𝐾^∗_𝑡≥0 of 𝑑𝐾^∗_𝑡=𝐹𝐾^∗_𝑡,𝐿_𝑡−𝜆𝐾^∗_𝑡−𝑐^∗_𝑡𝐾^∗_𝑡𝑑𝑡𝐾^∗₀=𝐾,𝐾>0, (52) and the optimal consumption policy 𝑐^∗_𝑡∈𝒜 is given by 𝑐^∗_𝑡𝐾=𝑔^𝑡𝐿_𝑡,𝐿_𝑡𝑢_𝐾𝐾^∗_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 an ℱ_𝑡 progressively measurable solution 𝜅_𝑡 of 𝑑𝜅_𝑡=𝐺𝐾_𝑡,𝐿_𝑡𝑑𝑡,𝜅₀=𝐾>0. (54) Now we shall show 𝐾^∗_𝑡≥0forall𝑡≥0 a.s. Suppose 0<𝑣^′(0+)<∞. By (9), we have 𝑣^′(𝑥)≥0 for all 𝑥>0, since 𝑈^∗(𝑣^′(𝑥))=∞ if 𝑣^′(𝑥)<0. Moreover, by L’Hospital’s rule this gives lim_𝑥→0+𝑥²𝑣^′′(𝑥)=lim_𝑥→0+𝑥²𝑣^′′(𝑥)+2𝑥𝑣^′(𝑥)=lim_𝑥→0+𝑥²𝑣^′(𝑥)𝑥=0. (55) Letting 𝑥→0+ in (9), we have 𝑈^∗(𝑣^′(0+))=0, and this is contrary with (2). Therefore, we get 𝑣^′(0+)=∞, which implies 𝑔(0,𝐿𝑢_𝐾(0+,𝐿))=0. We note by the concavity of 𝐹 that 𝐺(𝐾,𝐿)≤𝐶₁𝐾+𝐶₂𝐿 for 𝐶₁,𝐶₂>0. Then applying the comparison theorem to (54) and 𝑑𝐾_𝑡=𝐶₁𝐾_𝑡+𝐶₂𝐿_𝑡𝑑𝑡,𝐾₀=𝐾>0, (56) we obtain 0≤𝐾^∗_𝑡≤𝐾_𝑡 for all 𝑡≥0. Further, in case 𝜏^∗_𝐾=inf{𝑡≥0∶𝐾^∗_𝑡=0}=𝜏_𝜅<∞, we have at 𝑡=𝜏^∗_𝐾𝑑𝐾^∗_𝑡𝐾𝑑𝑡=𝐹^∗_𝑡,𝐿_𝑡−𝜆𝐾^∗_𝑡𝐾−𝑔^𝑡𝐿_𝑡,𝐿_𝑡𝑢_𝐾𝐾^∗_𝑡,𝐿_𝑡𝐿_𝑡=0. (57) Therefore, 𝐾^∗_𝑡 solves (52) and 𝐾^∗_𝑡≥0. To prove uniqueness, let 𝐾^∗_𝑖(𝑡),𝑖=1,2, be two solutions of (52). Then 𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡) satisfies 𝑑𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)=𝐹𝐾^∗₁(𝑡),𝐿_𝑡−𝐹𝐾^∗₂(𝑡),𝐿_𝑡−𝜆𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)−𝑔𝐾_𝑡𝐿_𝑡,𝐿_𝑡𝑢_𝐾𝐾^∗_𝑡,𝐿_𝑡𝐿_𝑡𝐾−𝑔^𝑡𝐿_𝑡,𝐿_𝑡𝑢_𝐾𝐾^∗_𝑡,𝐿_𝑡𝐿_𝑡𝑑𝑡, (58)𝐾^∗₁(0)−𝐾^∗₂(0)=0. We have 𝑑𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)²=2𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)𝑑𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)=2𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)×𝐹𝐾^∗₁(𝑡),𝐿_𝑡−𝐹𝐾^∗₂(𝑡),𝐿_𝑡−𝜆𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)−𝑔𝐾_𝑡𝐿_𝑡,𝐿_𝑡𝑢_𝐾𝐾^∗_𝑡,𝐿_𝑡𝐿_𝑡𝐾−𝑔^𝑡𝐿_𝑡,𝐿_𝑡𝑢_𝐾𝐾^∗_𝑡,𝐿_𝑡𝐿_𝑡𝑑𝑡. (59) Note that the function 𝑥→−𝑔(𝐾_𝑡/𝐿_𝑡,𝐿_𝑡𝑢_𝐾(𝐾^∗_𝑡,𝐿_𝑡))𝐿_𝑡 is decreasing. Hence, 𝐾^∗₁(𝑡)−𝐾^∗₂(𝑡)²≤2^𝑡₀𝐾^∗₁(𝑠)−𝐾^∗₂(𝑠)×𝐹𝐾^∗₁(𝑠),𝐿_𝑠−𝐹𝐾^∗₂(𝑠),𝐿_𝑠−𝜆𝐾^∗₁(𝑠)−𝐾^∗₂(𝑠)𝐹𝑑𝑠≤2^′(0)+𝜆^𝑡₀𝐾^∗₁(𝑠)−𝐾^∗₂(𝑠)²𝑑𝑠. (60) By Gronwall’s lemma, we have 𝐾^∗₁(𝑡)=𝐾^∗₂(𝑡),∀𝑡>0. (61) So, the uniqueness of (52) holds.
Now by (6), (52), and Itô’s formula, we have 𝑒^−𝜌𝑡𝑢𝐾^∗_𝑡,𝐿_𝑡=𝑢(𝐾,𝐿)+^𝑡₀𝑒^−𝜌𝑠×−𝜌𝑢(𝐾,𝐿)+𝑢_𝐾()×𝐹()𝐾,𝐿𝐾,𝐿−𝜆𝐾−𝑐^∗_𝑠𝐾+𝑛𝐿𝑢_𝐿()+1𝐾,𝐿2𝜎²𝐿²𝑢_𝐿𝐿(𝐾,𝐿)_{(𝐾=𝐾∗𝑠,𝐿=𝐿𝑠)}+𝑑𝑠_𝑡0𝑒^−𝜌𝑠𝜎𝐿_𝑠𝑢_𝐿(𝐾,𝐿)𝑑𝑊_𝑠. (62) By the HJB equation (6), we have 𝑒^−𝜌𝑡𝑢𝐾^∗_𝑡,𝐿_𝑡=𝑢(𝐾,𝐿)−^𝑡₀𝑒^−𝜌𝑠𝑈𝑐^∗_𝑠𝐾^∗_𝑠𝐿_𝑠+𝑑𝑠_𝑡0𝑒^−𝜌𝑠𝜎𝐿_𝑠𝑢_𝐿(𝐾,𝐿)𝑑𝑊_𝑠, (63) from which 𝐸𝑒^{−𝜌(𝑡∧𝜏𝑛)}𝑢𝐾^∗_{𝑡∧𝜏𝑛},𝐿_{𝑡∧𝜏𝑛}+𝐸^{𝑡∧𝜏𝑛}₀𝑒^−𝜌𝑠𝑈𝑐^∗_𝑠𝐾^∗_𝑠𝐿_𝑠𝑑𝑠=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 𝐸sup_𝑛𝑒^{−𝜌(𝑡∧𝜏𝑛)}𝑢𝐾^∗_{𝑡∧𝜏𝑛},𝐿_{𝑡∧𝜏𝑛}≤𝐸sup_{0≤𝑟≤𝑡}𝑒^−𝛼𝑟𝜙𝐾^∗_(𝑟),𝐿_(𝑟)≤𝜙(𝐾,𝐿)+𝐸sup_{0≤𝑟≤𝑡}^𝑟₀𝑒^−𝛼𝑡𝜎𝐾^∗_𝑡𝑑𝑊_𝑡≤𝜙(𝐾,𝐿)+2|𝜎|𝐸^𝑡₀𝑒^−𝛼𝑡𝐾^∗_𝑡²𝑑𝑡^1/2≤𝜙(𝐾,𝐿)+2|𝜎|𝐸^𝑡₀𝐾²_𝑡𝑑𝑡^1/2. (65) Letting 𝑛→∞ and 𝑡→∞, hence, we obtain by the dominated convergence theorem 𝐸𝑒^{−𝜌𝜏∗}𝑢𝐾^∗_𝜏∗,𝐿_𝜏∗+𝐸^𝜏∗₀𝑒^−𝜌𝑠𝑈𝑐^∗_𝑠𝐾^∗_𝑠𝐿_𝑠𝑑𝑠=𝑢(𝐾,𝐿). (66) We deduce by Lemma 3𝐽𝑐^∗=𝐸^𝜏∗₀𝑒^−𝜌𝑠𝑈𝑐^∗_𝑠𝐾^∗_𝑠𝐿_𝑠𝑑𝑠=𝑢(𝐾,𝐿). (67) Following the same calculation as above, we have 𝑒^−𝜌𝑡𝑢𝐾_𝑡,𝐿_𝑡=𝑢(𝐾,𝐿)+^𝑡₀𝑒^−𝜌𝑠×−𝜌𝑢(𝐾,𝐿)+𝑢_𝐾(𝐾,𝐿)𝐹(𝐾,𝐿)−𝜆𝐾−𝑐_𝑠𝐾+𝑛𝐿𝑢_𝐿()1𝐾,𝐿2𝜎²×𝐿²𝑢_𝐿𝐿(𝐾,𝐿)_{(𝐾=𝐾𝑠,𝐿=𝐿𝑠)}𝑑𝑠+^𝑡₀𝑒^−𝜌𝑠𝜎𝐿_𝑠𝑢_𝐿(𝐾,𝐿)𝑑𝑊_𝑠. (68) Again by the HJB equation (6), we can obtain 0≤𝐸𝑒^−𝜌𝜏𝑢𝐾_𝜏,𝐿_𝜏≤𝑢(𝐾,𝐿)−𝐸^𝜏₀𝑒^−𝜌𝑠𝑈𝑐_𝑠𝐾_𝑠𝐿_𝑠𝑑𝑠 (69) from which 𝐽(𝑐)=𝐸^𝜏₀𝑒^−𝜌𝑠𝑈𝑐_𝑠𝐾_𝑠𝐿_𝑠𝑑𝑠≤𝑢(𝐾,𝐿) (70) for any (𝑐_𝑡)∈𝒜. The proof is complete.

Remark 5. From the proof of Theorem 4, it follows that inf_𝑐∈𝒜𝐸^𝑇_𝑠𝑒^−𝜌𝑡𝑈𝑐_𝑡𝐾_𝑡𝐿_𝑡𝑑𝑡=𝑉(𝑠,𝐾,𝐿). (71) Thus, under (2), we observe that the smooth solution of 𝑢 of the HJB equation (6). Furthermore, let 𝑣 be the solution of (9) on the entire domain [0,𝑇)×(0,∞) with 𝑣(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 that 𝑢 satisfies (6). Therefore, we obtain the uniqueness of 𝑣.

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.