Consider an economy characterized by the following facts:

i. The debt-to-GDP ratio is \(40 \%\).

ii. The primary deficit is \(4 \%\) of GDP.

iii. The normal growth rate is \(3 \%\).

iv. The real interest rate is \(3 \%\).

a. Using your favorite spreadsheet software, compute the debt-to-GDP ratio in 10 years, assuming that the primary deficit stays at \(4 \%\) of GDP each year; the economy grows at the normal growth rate in each year; and the real interest rate is constant at \(3 \%\).

b. Suppose the real interest rate increases to \(5 \%\), but everything else remains as in part a. Compute the debt-to-GDP ratio in 10 years.

c. Suppose the normal growth rate falls to \(1 \%\), and the economy grows at the normal growth rate each year. Everything else remains as in part a. Calculate the debt-to GDP ratio in 10 years. Compare your answer to part \(b\).

d. Return to the assumptions of part a. Suppose policymakers decide that a debt-to-GDP ratio of more than \(50 \%\) is dangerous. Verify that reducing the primary deficit to \(1 \%\) immediately and that maintaining this deficit for 10 years will produce a debt-to-GDP ratio of \(50 \%\) in 10 years. Thereafter, what value of the primary deficit will be required to maintain the debt-to-GDP ratio of \(50 \%\) ?

e. Continuing with part d, suppose policy makers wait 5 years before changing fiscal policy. For five years, the primary deficit remains at \(4 \%\) of GDP. What is the debt-to GDP ratio in 5 years? Suppose that after five years, policy makers decide to reduce the debt-to-GDP ratio to \(50 \%\). In years 6 through 10, what constant value of the primary deficit will produce a debt-to-GDP ratio of \(50 \%\) at the end of year 10 ?

f. Suppose that policy makers carry out the policy in either parts \(\mathrm{d}\) or e. If these policies reduce the growth rate of output for a while, how will this affect the size of the reduction in the primary deficit required to achieve a debt-to-GDP ratio of \(50 \%\) in 10 years?
g. Which policy-the one in part \(\mathrm{d}\) or the one in part edo you think is more dangerous to the stability of the economy?