Optimal Consumption in a Growth Model with the Cobb–Douglas Production Function

Morimoto, Hiroaki

doi:10.1137/070709153

Cited by 11 publications

(32 citation statements)

References 4 publications

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“…Proof Letting

τ_{n}^{*} = \inf false{t \geq 0 : k^{*} false(t false) \leq \frac{1}{n} false}

and

τ^{*} = \inf false{t \geq 0 : k^{*} false(t false) = 0 false}

, it is obvious that

τ_{n}^{*} \to τ^{*}

. From Equation and the concavity of the utility function, we get

\begin{matrix} V false[ξ k_{0} + false(1 - ξ false) {\overset{k}{true}}_{0} false] & \geq E ([], \int_{0}^{τ} e^{- ρ s} U false[ξ c false(s false) + false(1 - ξ false) ĉ false(s false) false] d s) \\ \geq ξ E ([], \int_{0}^{τ} e^{- ρ s} U false(c false(s false) false) d s) + false(1 - ξ false) E ([], \int_{0}^{τ} e^{- ρ s} U false(ĉ false(s false) false) d s) \\ \geq ξ V false(k_{0} false) + false(1 - ξ false) V false({\overset{k}{true}}_{0} false) - ε . \end{matrix}

Letting ε →0, we obtain that the value function V ( k 0 ) is concave (see the work of Morimoto and Zhou), as well as v ( k ∗ ). Due to the concavity of v ( k ∗ ), we know that v ( k ∗ ) is bounded above by its first‐order Taylor approximation v (0) ≤ v ( k ∗ ) + v ′ ( k ∗ )(0 − k ∗ ), which means v ′ ( k ∗ ) k ∗ ≤ v ′ ( k ∗ ).…”

Section: Optimal Consumptionmentioning

confidence: 99%

“…Theorem 1. If (4), (7), and (15) hold, then the value function V(k 0 ) is a viscosity solution of Equation (13).…”

Section: Definition 1 (See the Work Of Morimoto 6 )mentioning

confidence: 99%

“…Optimal consumptions in economic growth models have been investigated by Morimoto and Morimoto and Zhou . Such optimal consumption problems are also discussed by Baten et al and Liu .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal consumption analysis for a stochastic growth model with technological shocks

Yuan

Lai

2018

Appl Stoch Models Bus & Ind

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An optimal consumption problem of maximizing the expected discounted value of utility is discussed for an economic growth model with the random technological shocks. Applying the technique of dynamic programming principle, we derive the Hamilton‐Jacobi‐Bellman equation corresponding to the optimization problem and prove that the value function is a unique viscosity solution to the equation. Moreover, the optimal consumption policy is given in a feedback form under weak assumptions.

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“…Proof Letting

τ_{n}^{*} = \inf false{t \geq 0 : k^{*} false(t false) \leq \frac{1}{n} false}

and

τ^{*} = \inf false{t \geq 0 : k^{*} false(t false) = 0 false}

, it is obvious that

τ_{n}^{*} \to τ^{*}

. From Equation and the concavity of the utility function, we get

\begin{matrix} V false[ξ k_{0} + false(1 - ξ false) {\overset{k}{true}}_{0} false] & \geq E ([], \int_{0}^{τ} e^{- ρ s} U false[ξ c false(s false) + false(1 - ξ false) ĉ false(s false) false] d s) \\ \geq ξ E ([], \int_{0}^{τ} e^{- ρ s} U false(c false(s false) false) d s) + false(1 - ξ false) E ([], \int_{0}^{τ} e^{- ρ s} U false(ĉ false(s false) false) d s) \\ \geq ξ V false(k_{0} false) + false(1 - ξ false) V false({\overset{k}{true}}_{0} false) - ε . \end{matrix}

Section: Optimal Consumptionmentioning

confidence: 99%

“…Theorem 1. If (4), (7), and (15) hold, then the value function V(k 0 ) is a viscosity solution of Equation (13).…”

Section: Definition 1 (See the Work Of Morimoto 6 )mentioning

confidence: 99%

See 1 more Smart Citation

Optimal consumption analysis for a stochastic growth model with technological shocks

Yuan

Lai

2018

Appl Stoch Models Bus & Ind

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“…This paper is concerned with dividend payments of a firm in the stochastic Ramsey model studied by Merton [12], Liu and Morimoto [10], Morimoto and Zhou [13]. The firm grows by investing in capital stock and the production technology is represented by the Cobb-Douglas function:…”

Section: Introductionmentioning

confidence: 99%

Optimal dividend payments in the stochastic Ramsey model

Morimoto

2010

Stochastic Processes and their Applications

Self Cite

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We consider the dividend payments of a self-financing firm in the stochastic Ramsey model. The firm invests in capital stock and its production technology is given by the Cobb-Douglas function. Our objective is to maximize the expected present value of future real dividends subject to a positive constraint on the capital stock. We use the penalization method to obtain a solution for the variational inequality associated with the optimal growth problem and give a synthesis of the optimal dividend policy.

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“…It shows that Xĉ is indeed a well-defined strictly positive process, on the strength of Feller's test for explosion and a mixture of probabilistic arguments in Nakao [11] and Yamada [14]. Now, with Xĉ well-defined and V solving a nonlinear elliptic equation, a standard verification argument establishes the optimality ofĉ.Note that the construction of Xĉ was done with much more ease in [10], through a change of measure. This works, however, only with bounded consumptions and finite time horizon.…”

mentioning

confidence: 99%

Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints

Huang¹,

Khalili²

2019

SIAM J. Control Optim.

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This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.MSC (2010): 91B62 93E20.1 Surprisingly, many of these works require an a priori uniform upper bound, usually the constant 1, for consumption processes {c t } t≥0 . This is implicitly suggested in the problem formulation of [7], and explicitly stated as 0 ≤ c t ≤ 1 in [10] and [6]. While this uniform upper bound provides technical conveniences, it can not be fully justified economically in continuous time. After all, for each t ≥ 0, c t represents the consumption ratio per unit of time instantly at time t, which does not admit any natural upper bound. This is in contrast to the discrete-time setting where the upper bound 1 can be easily justified. Morimoto [8,9] consider general, unbounded consumption processes, but not without a cost. There, the production function in the Ramsey model is required to have finite first derivatives, along a boundary of its domain. This particularly rules out the standard Cobb-Douglas production function, commonly used in economic modeling.In other words, a tradeoff exists between the viscosity solutions approach in [10, 6] and Banach's fixed-point argument in [8,9]. The former accommodates the classical Cobb-Douglas production function, but is limited to uniformly bounded consumption processes; the latter allows for general consumptions, but fails to cover the Cobb-Douglas production function. We aim to resolve this tradeoff: this paper considers both unbounded consumption processes and the Cobb-Douglas production function, in the stochastic Ramsey problem. The goal is to characterize the associated value function V , as well as a (possibly unbounded) optimal consumption processĉ.The upfront challenge of our studies is the non-standard stochastic differential equation (SDE) of the state process X, which represents capital per capita; see (2.8) below. On the one hand, the Cobb-Douglas production function renders the drift coefficient of X non-Lipschitz (see Section 5.1 for...

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Optimal Consumption in a Growth Model with the Cobb–Douglas Production Function

Cited by 11 publications

References 4 publications

Optimal consumption analysis for a stochastic growth model with technological shocks

Optimal consumption analysis for a stochastic growth model with technological shocks

Optimal dividend payments in the stochastic Ramsey model

Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints

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