UC Santa Barbara

We study energy markets in game theoretic framework. The energy markets consist of two types of energy producers: exhaustible producer and renewable producer. An exhaustible producer produces energy with exhaustible resources, such as oil. The resource reserves of each exhaustible producer diminish due to production, and also get replenished with costly effort to explore for new resources. This exploration activity is modeled through a controlled point process that leads to stochastic increments to reserves level. A renewable producer uses renewable resources, such as solar power, to produce energy. The renewable resources are infinite, but costly in production. Each producer chooses optimal controls of production quantity and exploration effort (exhaustible producers only), in order to maximize individual profit that equals his quantity of production multiplied by market price, minus costs of production and exploration. The producers interact with each other through the energy price that is a function of aggregate production, as one's profit does not only depend on his own production quantity, but also depends on the total quantity of all other producers.

We aim to study the equilibrium total production and price.

In Chapter 2 we study the game between an exhaustible producer and a renewable producer under stochastic demand that switches between different regimes. We study how the regime changes and the relative cost of production, which is a proxy for market competitiveness, affect game equilibria, and compare with the case of deterministic demand. A novel feature driven by stochasticity of demand is that production may shut down during low demand to conserve reserves.

In Chapter 3 we study game with a continuum of homogeneous exhaustible producers. Mean field game approach is employed to solve for an approximate Markov Nash equilibrium of the game. We develop numerical schemes to solve the resulting system of partial differential equations: a backward Hamilton-Jacobi-Bellman (HJB) equation for the game value function of a representative producer and a forward transport equation for the distribution of the reserves levels among all producers.

In Chapter 4 we study a time-stationary mean field game model, in which the reserves level remains invariant due to the counteracting effects of production and exploration. We also study the impact of uncertainty in the regime that the exploration process becomes asymptotically deterministic, so that discovery of new resources happens at high frequency with small amount of each discovery.