Generalized Kuhn--Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources

Chiarolla, Maria B.; Ferrari, Giorgio; Riedel, Frank

doi:10.1137/120870360

Cited by 28 publications

(28 citation statements)

References 18 publications

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“…Problem (2.6) takes the form of a singular stochastic control problem, i.e. of a problem in which control processes may be singular with respect to the Lebesgue measure, as functions of time (see [44] for an introduction; [30] and [31] as classical references in Markovian settings; [2], and [11], among others, for studies in not necessarily Markovian frameworks). Remark 2.4.…”

Section: The General Model and The Control Problemmentioning

confidence: 99%

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Ferrari¹

2018

SIAM J. Control Optim.

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Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such link to characterise the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein-Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.Public Debt Control 2 of a crisis. This negative effect on economic growth from high debt levels has been observed in 18 different advanced economies (see [8]).In this paper we propose a continuous-time stochastic model for the control of the debtto-GDP ratio. The problem we have in mind is that of a government aiming to answer the question: How much is too much? 1 Following classical macroeconomic theory (see, e.g., [5]), in any given period the debt ratio stock grows by the existing debt stock multiplied by the difference between real interest rate and GDP growth, less the primary budget balance 2 . We assume that the government can reduce the level of the debt-to-GDP ratio by adjusting the primary budget balance, e.g. through fiscal interventions like raising taxes or reducing expenses. We therefore interpret the cumulative interventions on the debt ratio as the government's control variable, and we model it as a nonnegative and nondecreasing stochastic process. Uncertainty in our model comes through the GDP growth rate of the country, and its inflation and nominal interest rate which, by Fisher law [21], directly affect the growth rate of the debt ratio. We first assume that they are general not necessarily Markovian stochastic processes whose dynamics is not under government control. Indeed, the level of these macroeconomic variables is usually regulated by an autonomous Central Bank, whose action, however, is not modelled in this paper (see, e.g., [10] and [25] for problems related to the optimal control of inflation).Since high debt-to-GDP ratios can constrain economic growth making it more difficult to break the burden of the debt, we assume that debt ratio generates an instantaneous cost/penalty. This is a general nonnegative convex function of the debt ratio level, that the government would like to k...

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Section: The General Model and The Control Problemmentioning

confidence: 99%

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Ferrari¹

2018

SIAM J. Control Optim.

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“…This approach is very powerful in solving general singular control problems as it has been shown in a quite recent literature. We refer to Bank andRiedel (2001, 2003) for an intertemporal utility maximization problem with Hindy, Huang and Kreps preferences; to Bank (2005), Chiarolla and Ferrari (2014), Ferrari (2015) and Riedel and Su (2011) for the irreversible investment problem of a monopolistic firm with both limited and unlimited resources; to Chiarolla et al (2013) for the social planner problem in a market with N firms and limited resources; to Steg (2012) for a capital accumulation game.…”

Section: Introductionmentioning

confidence: 99%

Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach

Ferrari¹,

Riedel²,

Steg³

2016

Appl Math Optim

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Abstract. In this paper we study continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner's optimal policy, we characterize it by necessary and sufficient stochastic Kuhn-Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation. We finally also provide a detailed analysis of the so-called free rider effect. MSC2010 subject classification: 93E20, 91B70, 91A15, 91A25, 60G51 JEL subject classification: C02, C61, C62, C73

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“…These include dynamic programming techniques (see [31], [36] and [40], among others), stochastic first-order conditions and the Bank-El Karoui's Representation Theorem [4] (see, e.g., [5], [14], [21] and [44]), connections with optimal switching problems (cf. [25], among others).…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Reversible Investment Problem on a Finite-Time Horizon: Free Boundary Analysis

Angelis

Ferrari

2013

SSRN Journal

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Abstract. We study a continuous-time, finite horizon optimal stochastic reversible investment problem for a firm producing a single good. The production capacity is modeled as a onedimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment-disinvestment strategy. We associate to the investmentdisinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment-disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.

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Generalized Kuhn--Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources

Cited by 28 publications

References 18 publications

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach

A Stochastic Reversible Investment Problem on a Finite-Time Horizon: Free Boundary Analysis

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