Derive a condition that would ensure that the government

Question

Let the stock of government debt at time t be denoted by the stock of government debt evolves according to the relation:

B_t = (1 + r) B_t-1 + G_t - T_t

Where G is government spending (excluding debt interest) in period t and T is the flow of tax revenue.

A sustainable position for government debt is one where the ratio of government debt to GDP does not keep rising; and a stable position for government debt is one where the ratio of debt to GDP is constant.

Suppose that GDP is, on average, growing at a rate that is equal to the interest rate. Derive a condition that would ensure that the government debt is guaranteed to be sustainable because the debt to GDP ratio eventually declines to zero. Comment on the condition and what it implies about how long a primary deficit (G > T) can be sustained.

Now suppose that the growth rate of GDP need not be the same as the interest rate. Derive a condition on the size of the primary fiscal deficit (that is G-T) that ensures that the debt to GDP ratio is stable. How does this condition compare with the condition from question 1?

Would you expect Ricardian equivalence to hold if debt was on a path that many people thought might be unsustainable? Explain.