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Question : Debt dynamics .

a. Derive an expression for how the primary budget balance as a share of GDP and the existing debt-to-GDP ratio affect the change in the current debt-to-GDP ratio.

b. Use the equation derived in question a. to discuss what is required to stabilize the debt ratio. Explain the trade-off that governments face when contemplating debt stabilization. (4 points)

Answer 1

a.

Let the government budget constraint as follows:

B_t - B_t-1 = rB_t-1 + G_t - T_t -----------equation 1

where Bt = current year debt level

Bt-1 = existing or previous year debt level

Tt = current year taxes

r = interest rate

Assume Yt = current year income

Then Bt / Yt = Current Debt-to-GDP Ratio

using equation 1, we have

B_t - B_t-1 = rB_t-1 + G_t - T_t

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

Dividing both sides by Yt

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

B_t / Y_t = (1+r) B_t-1 / Y_t * (Y_t-1 / Y_t-1) + G_t - T_t / Y_t

B_t / Y_t = (1+r) B_t-1 / Y_t-1 * (Y_t-1 / Y_t) + G_t - T_t / Y_t ---------------equation 2

Now, Let g be the percentage change in income Y

g = Y_t - Y_t-1 / Y_t-1

1 + g = Y_t / Y_t-1

Y_t-1 / Y_t = 1 / 1+g ----------equation 3

Put eq 3 in equation 2

B_t / Y_t = ((1+r) / (1+g)) B_t-1 / Y_t-1 + (G_t - T_t / Y_t ) ---------equation 4

G_t - T_t / Y_t = Primary Budget Balance as share of GDP

B_t / Y_t = Current Debt-to-GDP ratio

B_t-1 / Y_t-1 = Existing Debt-to-GDP ratio

Equation 4 shows how the current Debt to GDP ratio is dependent upont the existing raio and the primary budget balance as share of GDP.

As Primary Deficit increases debt to GDP will increase

As previous year debt to GDP ratio increases, the current ratio will also increase.

b.

Following are the factors that should be controlled to stabilize the ratio:

- interest rate: to be ensured that the interest rates on the debt should not exceed a permissible limit.

- growth rate: a sustainable GDP growth rate has to be maintained.

- Primary Budget Balance: to be ensured that the primary budget should not go into deficit. If this happens, then the interest payments will increase that can cause increase debt to GDP ratio

Trade-off:

The government faces a trade-off between managing the budgetary risk (fluctuations in the budget) and minimizing the cost of debt service (reduce interest payments and controlling interest rates to rise).