How Should Control Theory Be Used to Calculate a Time-Consistent Government Policy?

We study different solutions to a simple one-dimensional linear quadratic game with a large number of private agents and a government. A “time-consistent” solution is defined as a solution to the Hamilton-Jacobi-Bellman equation, i.e. as a policy for which the government has noprecommitment capability. This solution is compared to a policy where the government has an “instantaneous” pre-commitment, i.e. an equilibrium in which the government has a period by period leadership. In both cases, we show how control theory should be applied to calculate the equilibrium and how to relate these equilibria to the differential game literature.