Problem #1 Assume that in a given economy the debt to GDP ratio is initially equ

Question

Problem #1 Assume that in a given economy the debt to GDP ratio is initially equal to 1. Assume that the GDP in this economy typically grows at rate g 0.03 per year and the interest rate r 0.02. Furthermore, assume that the government chooses to run a primary surplus of x = 0.01 of GDP per year i) Find the vale of the debt to GDP ratio after ten years. ii) Find the value of the debt to GDP ratio after ten years if the rate of growth of output increases to g 0.04 per year ii) What happens to the debt to GDP ratio after ten years if the interest rate falls to 0.01? iv) How big should the surplus/deficit be if one is interested in halving the debt to GDP ratio in 10 years? IV

Explanation / Answer

i).

Consider the given problem here the “debt to GDP” is “1”, => the size of “GDP” and “debt” is same. Now, the “GDP” is growing at the rate “g=0.03” and the interest rate is “r=0.02”, => the debt is growing at the rate “r”, => if the initial “debt” is “X”, then we have.

=> debt to GDP = X*(1+r)^10 / X*(1+g)^10 = (1+r/1+g)^10 = (1.02/1.03)^10 = 0.91.

So, after “10 years” the “debt to GDP” decreases to “0.91”.

ii).

Now, if the growth rate increases to “0.04”, => the “debt to GDP” ratio is given by.

=> debt to GDP = X*(1+r)^10 / X*(1+g)^10 = (1+r/1+g)^10 = (1.02/1.04)^10 = 0.82. So, as the rate of output growth increases to “0.04” the “debt GDP ratio” decreases to “0.82”.

iii).

Now, if the interest rate decreases to “0.01”, => the “debt to GDP” ratio is given by.

=> debt to GDP = X*(1+r)^10 / X*(1+g)^10 = (1+r/1+g)^10 = (1.01/1.03)^10 = 0.83. So, as the rate of output growth increases to “0.04” the “debt GDP ratio” decreases to “0.83”.

4).

Let’s assume that “r=0.02” and “g=0.03” and the initial “GDP” and “debt” is “X”. So, after “10 years” the GDP will be “X*(1+g)^10 = 1.34*X” and the debt will be “X*(1+r)^10 = 1.22*X”. So, here surplus will be “1.34*X – 1.22*X = 0.12*X”. So, here the surplus will be, “0.12*X”.

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