A government faces an exogenous stream of government expenditures {gt} that it must finance. Total government expenditures at t, consist of two components:

Where gTt is ‘transitory’ expenditures and gPt is ‘permanent’ expenditures. At the beginning of period t, the government observes the history up to t of both gTt and gPt. Further, it knows the stochastic laws of motion of both, namely,

Gaussian vector process with mean zero and identity covariance matrix. The government finances its budget with a distorting taxes. If it collects Tt total revenues at t, it bears a dead weight loss of W(Tt) where W(T ) = The government’s loss functional is

The government can purchase or issue one-period risk free loans at a constant price q . Therefore, it faces a sequence of budget constraints

Where q−1 is the gross rate of return on one period risk-free government loans. Assume that b0 = 0. The government also faces the terminal value condition

Which prevents it from running a Ponzi scheme. The government wants to design a tax collection strategy expressing Tt as a function of the history of gTt, gPt, bt that minimizes (3) subject to (1), (2), and (4).

a. Formulate the government’s problem as a dynamic programming problem. Please carefully define the state and control for this problem. Write the Bellman equation in as much detail as you can. Tell a computational strategy for solving the Bellman equation. Tell the form of the optimal value function and the optimal decision rule.

b. Using objects that you computed in part a, please state the form of the law of motion for the joint process of gTt, gPt, Tt, bt+1 under the optimal government policy. Some background: Assume now that the optimal tax rule that you computed above has been in place for a very long time. A macroeconomist who is studying the economy observes time series on gt, Tt, but not on bt or the breakdown of gt into its components gTt, gPt. The macroeconomist has a very long time series for [gt, Tt] and proceeds to computing a vector auto regression for this vector.

c. Define a population vector auto regression for the [gt, Tt] process. (Feel free to assume that lag lengths are infinite if this simplifies your answer.)

d. Please tell precisely how the vector auto regression for [gt, Tt] depends on the parameters [ρ, β, µ, q, w1, w2, c1, c2] that determine the joint [gt, Tt] process according to the economic theory you used in part a.

e. Now suppose that in addition to his observations on [Tt, gt], the economist gets an error ridden time series on government debt bt: