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

Authors: Md. Azizul Baten; Anton Abdulbasah Kamil
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## 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.


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## Body

## 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:τk
=inf{t≥0:Kt=0},Kt
= capital stock at timet≥0,Lt
= labor supply at timet≥0,λ
= constant rate of depreciation,λ≥0,ctKt
= consumption rate at timet≥0,0≤ct≤1,ctKt/Lt
= totality of consumption rate per labor;F(K,L)
=AKαL1-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 ratec≥0,Wt
= one-dimensional standard Brownian motion on a complete probability space(Ω,ℱ,P) endowed with the natural filtration ℱt generated by σ(Ws,s≤t).Let us assume thatc={ct} is a consumption policy per capita such that ct is nonnegative ℱt=σ(Ws,s≤t), a progressively measurable process,
(1)∫0t‍csds<∞a.s.∀t≥0,
and we denote by 𝒜 the set of all consumption policies {ct} per capita.The utility functionU(c) is assumed to have the following properties:
(2)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(∞)=∞.
Following Merton [2], we make the following assumption on the Cobb-Douglas production function F(K,L):
(3)F(γK,γL)=γF(K,L),forγ>0,FK(K,L)>0,FL(K,L)>0,FKK(K,L)<0FLL(K,L)<0,F(0,L)=F(K,0)=0,FK(0+,L)<∞,FK(∞,L)=0,L>0.We are concerned with the economic growth model to maximize the expected discount utility of consumption(4)J(c)=E[∫0T‍e-ρtU(ctKtLt)dt]
per labor with a horizon T over the class c∈𝒜 subject to the capital stock Kt, and the labor supply Lt is governed by the stochastic differential equation
(5)dKt=[F(Kt,Lt)-λKt-ctKt]dtK0=K,K>0,dLt=nLtdt+σLtdWt,L0=L,L>0.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 thatu solves the general (two-dimensional) Hamilton-Jacobi-Bellman (in short, HJB) equation
(6)-ρu(K,L)+12σ2L2uLL(K,L)+nLuL(K,L)+{F(K,L)-λK}uK(K,L)+U*(uK(K,L),L)=0,u(0,L)=0,K>0,L>0,
where U*(uK,L)=maxc>0{U(cK/L)-cKuK} and uK, uL, and uLL 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 Section2, 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
(7)u(K,L)=v(KL),L>0.
Clearly
(8)LuK=vK,LuL=-KvK,L2uLL=K2vKK+2KvK.
Setting x=K/L and substituting these above in (6), we have the HJB equation of v of the following form:
(9)-ρv(x)+12σ2x2v′′(x)+(f(x)-λ~x)v′(x)+U*(x,v′(x))=0,v(0)=0,x>0,
where λ~=n+λ-σ2,f(x)=F(x,1), and U*(x,q)=max0≤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
(10)J~(c)=E[∫0T‍e-ρtU(ctxt)dt]
over the class c∈𝒜~,   subject to
(11)dxt=[f(xt)-λ~xt-ct]dt-σxtdWt,x0=x,x≥0,
where c∈𝒜~ denotes the class 𝒜with {xt} replacing {Kt}. We choose δ>0 and rewrite (9) as
(12)-(ρ+1δ)v(x)+12σ2x2v′′(x)+(f(x)-λ~x)v′(x)+U*(x,v′(x))+1δv(x)=0,v(0)=0,x>0.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,(13)V(xt)=supc∈𝒜~E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt],
where τ=τ(x)=inf{t≥0:xt=0} and c={ct} is the element of the class 𝒜~ consisting of ℱt progressively measurable processes such that
(14)∫0t‍csds<∞a.s.∀t≥0.


## 3. Viscosity Solutions

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


### 3.1. Definition

Letv∈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:
(15)-ρv(x)+12σ2x2q+p(f(x)-λ~x)+U*(x,p)≥0,∀(p,q)∈J2,+v(x),∀x>0,-ρv(x)+12σ2x2q+p(f(x)-λ~x)+U*(x,p)≤0,∀(p,q)∈J2,-v(x),∀x>0,
where J2,+ and J2,- are defined by
(16)J2,+v(x)={v(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2(p,q)∈R2:limsupx~→xv(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2≤0},J2,-v(x)={v(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2(p,q)∈R2:liminfx~→xv(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2≥0}.We assume that(17)ρ+λ~>0,(18)-ρ+(f′(0)+|λ~|)+12σ2<0.
Take 0<α<ρ, and we choose P1>0 by concavity such that
(19)f(x)-λ~x≤P1.
Taking sufficiently large P0>P1, we observe by (2) and (19) that ϕ(t,x)=eτ-t(x+P0) fulfills
(20)-αϕ(x)+12σ2x2ϕ′′(x)+(f(x)-λ~x)ϕ′(x)+U*(x,ϕ′(x))≤eτ-t(-x-P0+P1)+Uo(U′)-1eτ-t<-x-P0+P1+Uo(U′)-1(1)<0,
for some constant P0>0.Lemma 1.
One assumes (2), (17), (11), and (20), then the value function V(x) fulfills
(21)0≤V(xt)≤ϕ(xt),(22)supc∈𝒜~E[e-(ρ+1/δ)τ|x(τ)-x~(τ)|]≤|x-x~|
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(23)0≤e-(ρ+1/δ)(t∧τ)ϕ(x(t∧τ))2=ϕ(x)+∫0t∧τ‍e-(ρ+1/δ)s=ϕ(x)+d×{-(ρ+1δ)ϕ(xs)=ϕ(x)+cc+(f(x)-λ~xs-csxs)ϕ′(xs)=ϕ(x)+cc+12σ2xs2ϕ′′(xs)}ds2-∫0t∧τ‍e-(ρ+1/δ)sσxsϕ′(xs)dWs.
Since
(24)E[∫0t‍|xsϕ′(xs)|2ds]≤e2(τ-t)E[∫0t|xs|2ds‍]≤e2(τ-t)E[∫0t‍(1+|xs|2)ds]=e2(τ-t)t+e2(τ-t)E[∫0t|xs|2ds‍],
and by considering ct=0,forallt≥0, then (11) becomes
(25)dx˘t=[f(x˘t)-λ~x˘t]dt-σx˘tdwt,x˘0=x,x>0,
by the comparison theorem of Ikeda and Watanabe [29]; we see that 0<xt≤x˘t,  forallt≥0. Hence, by applying the existence and uniqueness theorem for (11), we have
(26)E[∫0t|xs|2ds‍]<∞.
Therefore, from (24), we have
(27)E[∫0t|xsϕ′(xs)|2ds‍]<∞,
which yields that ∫0t‍e-(ρ+1/δ)sσxsϕ′(xs)dWs is a martingale, and again by (11), we can take sufficiently small δ>0 such that
(28)E[supt∫0t‍e-(ρ+1/δ)sσxsϕ′(xs)dWs]≤2|σ|E[supt∫0∞‍e-(ρ+1/δ)sxs2ds]1/2<∞.
Hence,
(29)E[∫0t∧τ‍e-(ρ+1/δ)sσxsϕ′(xs)dws]=0.
Therefore, by (20), (11) and taking expectation on both sides of (23), we obtain
(30)E[∫0t∧τ‍e-(ρ+1/δ)s{U(csxs)+1δv(xs)}ds]≤ϕ(x),
from which we deduce (21).
We setzt=xt-x~t and by (11), it is clear that
(31)dzt=d(xt-x~t)=[f(xt)-f(x~t)-λ~(xt-x~t)]dt-σ(xt-x~t)dWt.
Since by (3), f(z) is Lipschitz continuous and concave and f(0)=0, then we have
(32)dzt≤(f′(0)+|λ~|)ztdt-σz(t)dWt,z(0)=x-x~.
Take μ>0 such that gμ(z)=(z2+μ)1/2, and we can find from (18) and (20) that
(33)-(ρ+1δ)gμ(z)+12σ2z2gμ′′(z)+(f′(0)+|λ~|)|zgμ′(z)|≤(z2+μ)1/2{-(ρ+1δ)+12σ2+(f′(0)+|λ~|)1δ}<0,∀z∈R.
Using Itô’s formula and by (20), we have
(34)E[e-(ρ+1/δ)τgμ(zτ)]=gμ(x-x~)+E[∫0τ‍e-(ρ+1/δ)s×{-(ρ+1δ)gμ(zs)+(f(xs)-f(x~s)-λ~zs)gμ′(zs)+12σ2zs2gμ′′(zs)}ds-∫0τ‍e-(ρ+1/δ)sσzsgμ′(zs)dWs]≤gμ(x-x~).
Letting δ→0, and by Fatou’s lemma, we obtain
(35)E[e-(ρ+1/δ)τ|zτ|]≤|x-x~|,
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
(36)v(x)=supc∈𝒜~E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt]≤ϕ(x),v(x)=supc∈𝒜~E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt]∀x≥0,
for any stopping time τ. By (13) and for any ε>0, there exists c∈𝒜~ such that
(37)v(x)-ε<E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt]=E[∫0τ‍e-(ρ+1/δ)tU(ctxt)dt]+E[∫0τ‍e-(ρ+1/δ)t1δv(xt)dt].
Since f(z) is Lipschitz continuous, it follows that
(38)dzt=[f(xt)-f(x~t)-λ~(xt-x~t)]dt-σ(xt-x~t)dWt≤(f′(0)+|λ~|)ztdt-σztdWt,z(0)=x-x~.
By (11), we can consider that z~t is the solution of
(39)dz~t=(f′(0)+|λ~|)z~tdt-σz~tdWt,z~(0)=x>0.
So by the comparison theorem Ikeda and Watanabe [29], we have
(40)τz↓τ~,zt≤z~t↓0,a.s.asz↓0.
Since E[sup0≤t≤Lz~t2]<∞ for all L>0, now by (11) we have
(41)E[zτ∧L]=z+E[∫0τ∧L‍{f(zt)-λ~zt-ct}dt].
Letting z↓0 and then L→0, we obtain
(42)E[∫0τ~‍ctdt]=E[∫0τ~‍{f(0)-ct}dt]≥0,
so that
(43)E[∫0τ~‍e-(ρ+1/δ)tU(ctxt)dt]=0.
Passing to the limit to (37) and applying (43), we obtain
(44)v(0+)-ε<E[∫0∞‍e-(ρ+1/δ)t1δv(0+)dt]=v(0+)ρδ+1,
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).


## 3.1. Definition

Letv∈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:
(15)-ρv(x)+12σ2x2q+p(f(x)-λ~x)+U*(x,p)≥0,∀(p,q)∈J2,+v(x),∀x>0,-ρv(x)+12σ2x2q+p(f(x)-λ~x)+U*(x,p)≤0,∀(p,q)∈J2,-v(x),∀x>0,
where J2,+ and J2,- are defined by
(16)J2,+v(x)={v(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2(p,q)∈R2:limsupx~→xv(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2≤0},J2,-v(x)={v(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2(p,q)∈R2:liminfx~→xv(x~)-v(x)-p(x~-x)-(1/2)q|x~-x|2|x~-x|2≥0}.We assume that(17)ρ+λ~>0,(18)-ρ+(f′(0)+|λ~|)+12σ2<0.
Take 0<α<ρ, and we choose P1>0 by concavity such that
(19)f(x)-λ~x≤P1.
Taking sufficiently large P0>P1, we observe by (2) and (19) that ϕ(t,x)=eτ-t(x+P0) fulfills
(20)-αϕ(x)+12σ2x2ϕ′′(x)+(f(x)-λ~x)ϕ′(x)+U*(x,ϕ′(x))≤eτ-t(-x-P0+P1)+Uo(U′)-1eτ-t<-x-P0+P1+Uo(U′)-1(1)<0,
for some constant P0>0.Lemma 1.
One assumes (2), (17), (11), and (20), then the value function V(x) fulfills
(21)0≤V(xt)≤ϕ(xt),(22)supc∈𝒜~E[e-(ρ+1/δ)τ|x(τ)-x~(τ)|]≤|x-x~|
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(23)0≤e-(ρ+1/δ)(t∧τ)ϕ(x(t∧τ))2=ϕ(x)+∫0t∧τ‍e-(ρ+1/δ)s=ϕ(x)+d×{-(ρ+1δ)ϕ(xs)=ϕ(x)+cc+(f(x)-λ~xs-csxs)ϕ′(xs)=ϕ(x)+cc+12σ2xs2ϕ′′(xs)}ds2-∫0t∧τ‍e-(ρ+1/δ)sσxsϕ′(xs)dWs.
Since
(24)E[∫0t‍|xsϕ′(xs)|2ds]≤e2(τ-t)E[∫0t|xs|2ds‍]≤e2(τ-t)E[∫0t‍(1+|xs|2)ds]=e2(τ-t)t+e2(τ-t)E[∫0t|xs|2ds‍],
and by considering ct=0,forallt≥0, then (11) becomes
(25)dx˘t=[f(x˘t)-λ~x˘t]dt-σx˘tdwt,x˘0=x,x>0,
by the comparison theorem of Ikeda and Watanabe [29]; we see that 0<xt≤x˘t,  forallt≥0. Hence, by applying the existence and uniqueness theorem for (11), we have
(26)E[∫0t|xs|2ds‍]<∞.
Therefore, from (24), we have
(27)E[∫0t|xsϕ′(xs)|2ds‍]<∞,
which yields that ∫0t‍e-(ρ+1/δ)sσxsϕ′(xs)dWs is a martingale, and again by (11), we can take sufficiently small δ>0 such that
(28)E[supt∫0t‍e-(ρ+1/δ)sσxsϕ′(xs)dWs]≤2|σ|E[supt∫0∞‍e-(ρ+1/δ)sxs2ds]1/2<∞.
Hence,
(29)E[∫0t∧τ‍e-(ρ+1/δ)sσxsϕ′(xs)dws]=0.
Therefore, by (20), (11) and taking expectation on both sides of (23), we obtain
(30)E[∫0t∧τ‍e-(ρ+1/δ)s{U(csxs)+1δv(xs)}ds]≤ϕ(x),
from which we deduce (21).
We setzt=xt-x~t and by (11), it is clear that
(31)dzt=d(xt-x~t)=[f(xt)-f(x~t)-λ~(xt-x~t)]dt-σ(xt-x~t)dWt.
Since by (3), f(z) is Lipschitz continuous and concave and f(0)=0, then we have
(32)dzt≤(f′(0)+|λ~|)ztdt-σz(t)dWt,z(0)=x-x~.
Take μ>0 such that gμ(z)=(z2+μ)1/2, and we can find from (18) and (20) that
(33)-(ρ+1δ)gμ(z)+12σ2z2gμ′′(z)+(f′(0)+|λ~|)|zgμ′(z)|≤(z2+μ)1/2{-(ρ+1δ)+12σ2+(f′(0)+|λ~|)1δ}<0,∀z∈R.
Using Itô’s formula and by (20), we have
(34)E[e-(ρ+1/δ)τgμ(zτ)]=gμ(x-x~)+E[∫0τ‍e-(ρ+1/δ)s×{-(ρ+1δ)gμ(zs)+(f(xs)-f(x~s)-λ~zs)gμ′(zs)+12σ2zs2gμ′′(zs)}ds-∫0τ‍e-(ρ+1/δ)sσzsgμ′(zs)dWs]≤gμ(x-x~).
Letting δ→0, and by Fatou’s lemma, we obtain
(35)E[e-(ρ+1/δ)τ|zτ|]≤|x-x~|,
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
(36)v(x)=supc∈𝒜~E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt]≤ϕ(x),v(x)=supc∈𝒜~E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt]∀x≥0,
for any stopping time τ. By (13) and for any ε>0, there exists c∈𝒜~ such that
(37)v(x)-ε<E[∫0τ‍e-(ρ+1/δ)t{U(ctxt)+1δv(xt)}dt]=E[∫0τ‍e-(ρ+1/δ)tU(ctxt)dt]+E[∫0τ‍e-(ρ+1/δ)t1δv(xt)dt].
Since f(z) is Lipschitz continuous, it follows that
(38)dzt=[f(xt)-f(x~t)-λ~(xt-x~t)]dt-σ(xt-x~t)dWt≤(f′(0)+|λ~|)ztdt-σztdWt,z(0)=x-x~.
By (11), we can consider that z~t is the solution of
(39)dz~t=(f′(0)+|λ~|)z~tdt-σz~tdWt,z~(0)=x>0.
So by the comparison theorem Ikeda and Watanabe [29], we have
(40)τz↓τ~,zt≤z~t↓0,a.s.asz↓0.
Since E[sup0≤t≤Lz~t2]<∞ for all L>0, now by (11) we have
(41)E[zτ∧L]=z+E[∫0τ∧L‍{f(zt)-λ~zt-ct}dt].
Letting z↓0 and then L→0, we obtain
(42)E[∫0τ~‍ctdt]=E[∫0τ~‍{f(0)-ct}dt]≥0,
so that
(43)E[∫0τ~‍e-(ρ+1/δ)tU(ctxt)dt]=0.
Passing to the limit to (37) and applying (43), we obtain
(44)v(0+)-ε<E[∫0∞‍e-(ρ+1/δ)t1δv(0+)dt]=v(0+)ρδ+1,
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 (ct)∈𝒜. One has
(45)liminft→∞E[e-ρtu(Kt,Lt)]=0.Proof.
By (17) and (18), we take μ∈(0,ρ) such that
(46)-μ+12σ2+(f′(0)+|λ~|)-λ~<0.
Take μ>0 and x=K/L>0 such that gμ(x)=(x2+μ)1/2, and by (33) and (46)
(47)-μgμ(x)+12σ2x2gμ′′(x)+(f′(0)+|λ~|)|xgμ′(x)|≤(x2+μ)1/2{-μ+12σ2+(f′(0)+|λ~|)}<0.
Setting Φ(K,L)=A(K/L)γ, where A,γ>0, we have by (6) and (47)
(48)-μΦ(K,L)+12σ2L2ΦLL+nLΦL+ΦK(F(K,L)-λK)+U*(LΦK)<0K,L>0.
By Itô’s formula and (48), we obtain
(49)e-ρtΦ(Kt,Lt)=Φ(K,L)+∫0t‍e-ρs×{12(-ρ+μ)Φ+ΦK(F(K,L)-λK-csL)+nLΦL+12σ2L2ΦLL-μΦ12}|12(K=Ks,L=Ls)ds12ds-∫0t‍e-ρsσLsΦLdws≤Φ(K,L)+∫0t‍e-ρs{(-ρ+μ)Φ-csLΦK-U*(LΦK)}∣(K=Ks,L=Ls)ds-∫0t‍e-ρsσLsΦLdws≤Φ(K,L)+∫0t‍e-ρs{(-ρ+μ)Φ-U(cs)}∣(K=Ks,L=Ls)ds-∫0t‍e-ρsσLsΦLdws,0≤E[e-ρtΦ(Kt,Lt)]≤Φ(K,L)+E[∫0t‍e-ρs(-ρ+μ)Φ(Ks,Ls)ds].
Letting t→∞, we have
(50)E[∫0∞‍e-ρtΦ(Kt,Lt)dt]≤1ρ-μΦ(K,L)<∞,
which implies liminft→∞E[e-ρtΦ(Kt,Lt)]=0. By (21), we have
(51)u(K,L)≤Φ(K,L),
which completes the proof.We give a synthesis of the optimal policyc*={ct*} for the optimization problem (4) subject to (5).Theorem 4.
Under (2) and (3), there exists a unique solution Kt*≥0 of
(52)dKt*=[F(Kt*,Lt)-λKt*-ct*Kt*]dtK0*=K,K>0,
and the optimal consumption policy ct*∈𝒜 is given by
(53)ct*=g(KtLt,LtuK(Kt*,Lt)).Proof.
Let us considerG(K,L)=F(Kt,Lt)-λKt-g(Kt/Lt,LtuK(Kt,Lt))Kt and since F(K,L). g(Kt/Lt,LtuK(Kt*,Lt)) are continuous and G(0,L)=0, there exists an ℱt progressively measurable solution κt of
(54)dκt=G(Kt,Lt)dt,κ0=K>0.
Now we shall show Kt*≥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
(55)limx→0+x2v′′(x)=limx→0+x2v′′(x)+2xv′(x)=limx→0+x2v′(x)x=0.
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,LuK(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
(56)dK~t=(C1K~t+C2Lt)dt,K~0=K>0,
we obtain 0≤Kt*≤K~t for all t≥0. Further, in case τK*=inf{t≥0:Kt*=0}=τκ<∞, we have at t=τK*(57)dKt*dt=F(Kt*,Lt)-λKt*-g(KtLt,LtuK(Kt*,Lt))Lt=0.
Therefore, Kt* solves (52) and Kt*≥0. To prove uniqueness, let Ki*(t),i=1,2, be two solutions of (52). Then K1*(t)-K2*(t) satisfies
(58)d(K1*(t)-K2*(t))=[(KtLt,LtuK(Kt*,Lt))(F(K1*(t),Lt)-F(K2*(t),Lt))-λ(K1*(t)-K2*(t))-(g(KtLt,LtuK(Kt*,Lt))Lt-g(KtLt,LtuK(Kt*,Lt))Lt)]dt,K1*(0)-K2*(0)=0. We have
(59)d(K1*(t)-K2*(t))2=2(K1*(t)-K2*(t))d(K1*(t)-K2*(t))=2(K1*(t)-K2*(t))×[(KtLt,LtuK(Kt*,Lt))(F(K1*(t),Lt)-F(K2*(t),Lt))-λ(K1*(t)-K2*(t))-(g(KtLt,LtuK(Kt*,Lt))Lt-g(KtLt,LtuK(Kt*,Lt))Lt)]dt.
Note that the function x→-g(Kt/Lt,LtuK(Kt*,Lt))Lt is decreasing. Hence,
(60)(K1*(t)-K2*(t))2≤2∫0t‍(K1*(s)-K2*(s))×[(F(K1*(s),Ls)-F(K2*(s),Ls))-λ(K1*(s)-K2*(s))]ds≤2(F′(0)+|λ~|)∫0t‍(K1*(s)-K2*(s))2ds.
By Gronwall’s lemma, we have
(61)K1*(t)=K2*(t),∀t>0.
So, the uniqueness of (52) holds.
Now by (6), (52), and Itô’s formula, we have
(62)e-ρtu(Kt*,Lt)=u(K,L)+∫0t‍e-ρs=u(K,L)0+×{12-ρu(K,L)+uK(K,L)=u(K,L)077+×(F(K,L)-λK-cs*K)=u(K,L)0777++nLuL(K,L)=u(K,L)077i++12σ2L2uLL(K,L)}|12(K=Ks*,L=Ls)ds+∫0t‍e-ρsσLsuL(K,L)dWs.
By the HJB equation (6), we have
(63)e-ρtu(Kt*,Lt)=u(K,L)-∫0t‍e-ρsU(cs*Ks*Ls)ds+∫0t‍e-ρsσLsuL(K,L)dWs,
from which
(64)E[e-ρ(t∧τn)u(Kt∧τn*,Lt∧τn)]+E[∫0t∧τn‍e-ρsU(cs*Ks*Ls)ds]=u(K,L),
where {τn} is a sequence of localizing stopping times for the local martingale. From (11), (51), and Doob’s inequalities for martingales, it follows that
(65)E[supne-ρ(t∧τn)u(Kt∧τn*,Lt∧τn)]≤E[sup0≤r≤te-αrϕ(K(r)*,L(r))]≤ϕ(K,L)+E[sup0≤r≤t|∫0r‍e-αtσKt*dWt|]≤ϕ(K,L)+2|σ|E[∫0t‍(e-αtKt*)2dt]1/2≤ϕ(K,L)+2|σ|E[∫0t‍K~t2dt]1/2.
Letting n→∞ and t→∞, hence, we obtain by the dominated convergence theorem
(66)E[e-ρτ*u(Kτ**,Lτ*)]+E[∫0τ*‍e-ρsU(cs*Ks*Ls)ds]=u(K,L).
We deduce by Lemma 3(67)J(c*)=E[∫0τ*‍e-ρsU(cs*Ks*Ls)ds]=u(K,L).
Following the same calculation as above, we have
(68)e-ρtu(Kt,Lt)=u(K,L)+∫0t‍e-ρs×{12-ρu(K,L)+uK(K,L)(F(K,L)-λK-csK)+nLuL(K,L)12σ2×L2uLL(K,L)12}|12(K=Ks,L=Ls)ds+∫0t‍e-ρsσLsuL(K,L)dWs.
Again by the HJB equation (6), we can obtain
(69)0≤E[e-ρτu(Kτ,Lτ)]≤u(K,L)-E[∫0τ‍e-ρsU(csKsLs)ds]
from which
(70)J(c)=E[∫0τ‍e-ρsU(csKsLs)ds]≤u(K,L)
for any (ct)∈𝒜. The proof is complete.Remark 5.
From the proof of Theorem4, it follows that
(71)infc∈𝒜E[∫sT‍e-ρtU(ctKtLt)dt]=V(s,K,L).
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.



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*Source: 684757-2013-03-23.xml*