Many economic problems can be formulated as dynamic games in which strategically interacting agents choose actions that determine the current and future levels of a single capital stock. Consider, for example, a single stock of an exhaustible or reproductive resource that is simultaneously exploited by several agents that do not cooperate. Each agent chooses an extraction strategy to maximise the discounted stream of future utility.
The actions taken by agents not only determine their levels of utility but also the level of the capital stock. Alternatively, look at the problem that agents voluntarily contribute to a single public stock of capital, like a park or a church. They choose their contributions (investments in the public stock of capital) to maximise the discounted stream of utility from consuming the public stock net of investment costs. Private investment builds up the public stock of capital that eventually can be consumed by all agents.
Both examples have several things in common. First, the actions taken by agents determine the size of a single capital stock that fully describes the current state of the economic system. Second, if there is no mechanism that forces players to coordinate their actions, they will act strategically and play a non-cooperative game. Third, the equilibrium outcome will critically depend on the strategy spaces available to the agents.
We make use of these features and formulate a differential game in which agents act non cooperatively and use Markov strategies. We provide a detailed analysis of Markov perfect Nash equilibria (MNPE) for this class of differential games with a single capital stock and discuss several economic examples that belong to this class.
In a differential game, strategically interacting agents try to maximise an inter-temporal objective function by choosing a strategy that results in an action at every point in time. Collectively, these actions influence the state of the economic system and its time evolution.
There is a wide choice of possible strategies taken by the agents. They may choose a simple time profile of actions and precommit themselves to these fixed actions over the entire planning horizon: the players are then using open-loop strategies. Alternatively players might choose Markov strategies where they condition their actions on the current state of the system and react immediately every time the state variable changes. When agents use feedback or Markov strategies they are not required to precommit. Instead they play credible strategies if these are derived through backward induction.
To better understand the difference between open-loop and Markov strategies let us look at the following example of several agents strategically exploiting the same renewable resource, like for instance a stock of fish. If the fisheries use open-loop strategies they specify a time path of fishing effort at the beginning of the game and commit themselves to stick to these preannounced actions over the entire planning horizon. Alternatively, if they use Markov or feedback strategies they choose decision rules that determine current actions as a function of the current stock of the resource. Markov decision rules capture the strategic interactions present in a dynamic game. If a rival fishery makes a catch today that necessarily results in a lower level of the fish stock, the opponents react with actions that take this change in the stock into account. In that sense Markov strategies capture all the features of strategic interactions.
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