Consider a version of the Sidrauski money-in-the-utility-function model in which nominal government debt is used to carry out transactions instead of money. There are two stores of value: capital, which has real return r, and nominal debt, which has nominal return z. Let π denote the rate of inflation and assume that z < r="">+ π. The government also purchases a constant amount g per capita of goods and services; this spending does
not contribute to utility.

The spending and debt service is financed by per-capita lump-sum taxes in the amount τ and issuance of new nominal debt. Assume agents have discount rate θ, and that there is no population growth or technical progress. Let k denote the per-capita capital stock and b the per-capita stock of nominal debt. Production is given by y = f (k).

(a) Write down the consumer's optimization problem.

(b) Write down the equation of motion for nominal debt.

(c) Solve the optimization problem to give the joint behavior of con- sumption, capital and debt.

(d) Under what conditions does nominal debt have no effect on consump- tion or capital accumulation?

(e) Suppose there is a permanent rise in per-capita taxes, τ . Describe the transition path of and steady-state effects on capital, consumption and debt.

(f) Suppose there is a permanent rise in per-capita government pur- chases, g. Describe the transition path of and steady-state effects on capital, consumption and debt.

(g) What happens if z = r + π? If z > r + π?