therefore , which for small percentage changes is very
closely approximated by
where is the budget deficit and is equal to. The
second term is the percentage change in nominal GDP, which is
approximately equal to , where g is the growth rate of real
GDP and is the rate of inflation. We therefore rewrite the
expression as
Now, multiply both sides by d to get:
We can now let x be the primary budget deficit as a share
of GDP. The equation then becomes
Finally, note that the real interest rate on government bonds is
, and so our final equation becomes
Δd/dΔd/d=ΔD/D−ΔGDP/GDPΔD G−T+iDg+π
πΔd/d=(G−T+iD)/D−(g+π)Δd=(G−T+iD)/GDP−(g+π)d(G−T)Δd=x+(i−π−g)dr=i−π
Δd=x+(r−g)d