Let d be the percentage change in government debt, g the rate of growth in real GDP, RGDP the real GDP, NGDP the nominal GDP, P the price level, and ð the inflation rate. Let G[X] denote the growth rate in variable X, which is the same thing as the percentage change in X; thus, G[X] = (X2 - X1)/X1 ×100% for small changes in X. Here are two properties of the growth rate operator G: (i) G[X×Y] = G[X] + G[Y], and (ii) G[X/Y] = G[X] - G[Y].
a. Show that the growth rate in NGDP is equal to g + ð, where g is the real GDP growth rate and ð is the inflation rate.
b. Show that d is equal to (Deficit/Debt) × 100%.
c. Show that the percentage change in the Debt/NGDP ratio is equal to d - (g + ð).
d. Show that the condition for the Debt to NGDP ratio not to increase is d = g + ð.

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The Answer of Let d be the percentage change in government debt, g the rate of growth in real GDP, RGDP the real GDP, NGDP the nominal GDP, P the price level, and ð the inflation rate. Let G[X] denote the growth rate in variable X, which is the same thing as the percentage change in X; thus, G[X] = (X2 - X1)/X1 ×100% for small changes in X. Here are two properties of the growth rate operator G: (i) G[X×Y] = G[X] + G[Y], and (ii) G[X/Y] = G[X] - G[Y].
a. Show that the growth rate in NGDP is equal to g + ð, where g is the real GDP growth rate and ð is the inflation rate.
b. Show that d is equal to (Deficit/Debt) × 100%.
c. Show that the percentage change in the Debt/NGDP ratio is equal to d - (g + ð).
d. Show that the condition for the Debt to NGDP ratio not to increase is d = g + ð.

Let D Be the Percentage Change in Government Debt, G