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

## 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.

- Authors:

- Bielefeld University, Center for Mathematical Economics (Germany)

- Publication Date:

- OSTI Identifier:
- 22617043

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 75; Journal Issue: 3; Other Information: Copyright (c) 2017 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTROL; EQUATIONS; EQUILIBRIUM; INTERACTIONS; MATHEMATICAL SOLUTIONS; STOCHASTIC PROCESSES; SYMMETRY; UNCERTAINTY PRINCIPLE

### Citation Formats

```
Ferrari, Giorgio, E-mail: giorgio.ferrari@uni-bielefeld.de, Riedel, Frank, E-mail: frank.riedel@uni-bielefeld.de, and Steg, Jan-Henrik, E-mail: jsteg@uni-bielefeld.de.
```*Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach*. United States: N. p., 2017.
Web. doi:10.1007/S00245-016-9337-5.

```
Ferrari, Giorgio, E-mail: giorgio.ferrari@uni-bielefeld.de, Riedel, Frank, E-mail: frank.riedel@uni-bielefeld.de, & Steg, Jan-Henrik, E-mail: jsteg@uni-bielefeld.de.
```*Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach*. United States. doi:10.1007/S00245-016-9337-5.

```
Ferrari, Giorgio, E-mail: giorgio.ferrari@uni-bielefeld.de, Riedel, Frank, E-mail: frank.riedel@uni-bielefeld.de, and Steg, Jan-Henrik, E-mail: jsteg@uni-bielefeld.de. Thu .
"Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach". United States.
doi:10.1007/S00245-016-9337-5.
```

```
@article{osti_22617043,
```

title = {Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach},

author = {Ferrari, Giorgio, E-mail: giorgio.ferrari@uni-bielefeld.de and Riedel, Frank, E-mail: frank.riedel@uni-bielefeld.de and Steg, Jan-Henrik, E-mail: jsteg@uni-bielefeld.de},

abstractNote = {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.},

doi = {10.1007/S00245-016-9337-5},

journal = {Applied Mathematics and Optimization},

number = 3,

volume = 75,

place = {United States},

year = {Thu Jun 15 00:00:00 EDT 2017},

month = {Thu Jun 15 00:00:00 EDT 2017}

}