For an easy-to-read summary, see "Fiscal Theory of the Price Level" Chapter 1 Summary
Chapter 1:
A. one-period model:
I. For a period of time T:
a. at the beginning of the period, the government issues B_{T-1} zero-coupon bonds that each pay 1$ at expiration
b. at the end of the period (i.e. at time T) the government issues taxes equal to the product P_T * s_T, where P_T is the price level (CPI inflation) and s_T is the primary surplus (government deficit)
c. to maintain equilibrium, the bond payouts must equal taxes B_{T-1} = P_T * s_T
d. therefore, the price level is equal to the bonds issued divided by the primary surplus P_T = B_{T-1} / s_T
II. Intuition: this is distinct from the monetary model, since supply and demand doesn't drive the price level, and distinct from the keynesian model, since interest rates and the phillips curve do not drive the price level
B. two-period model: The government collects taxes AND sells NEW bonds to cover the old bonds' expiration payouts
I. B_{T-1} = P_T * s_T + Q_T * B_T
II. Q_T is the price at which the new bonds are sold
a. Q_T = 1 / (1 + i_T) where i_T is the nominal interest rate
b. Fisher Effect: since nominal interest equals real interest times expected inflation[1], Q_T also equals E_T ( P_T / P_{T+1} ) / R where E_T ( P_T / P_{T+1} ) is the expected inflation rate at time T and R is the real interest rate
c. the above equations imply that the price of a single bond is proportional to expected inflation
III. B_T = P_{T+1} * s_{T+1} using the 1-period model
IV. using substitution, B_{T-1} becomes P_T * s_T + \beta * E_T( s_{T+1} ), where \beta is 1 / R.
V. the new price level P_T will change to satisfy the equation B_{T-1} / P_T = s_T + \beta * E_T( 1 / P_{T+1} ) * B_T.
a. though it seems easier to write it as B_{T-1} / P_T = s_T + B_T / ( R * E_T( P_{T+1} ) )
VI. i.e. the real value of government debt B_{T-1} / P_T equals the present value of real primary surplus s_T + \beta * E_T( s_{T+1} )
VII. Intuition: The government can run large deficits (negative s_0) without any inflation, if expected future primary surplus is high. Also, high real interest rates will decrease \beta and induce inflation
C. monetary policy, fiscal policy, and inflation
I. in above equations, government mainly controls debt B_T (monetary policy) and surplus s_T (fiscal policy).
II. raising B_T, the quantity of bonds sold, lowers bond price Q_T, similar to a stock share split (assuming no change in surplus s_T)
III. alternatively the gov't can fix bond price Q_T and allow the market to buy as many bonds B_T as they want.
IV. since bond price Q_T equals 1 / (1 + i_T), the target price is equivalent to a target interest rate.
V. target interest rate therefore sets the expected rate of inflation since Q_T is proportional to E_T ( P_T / P_{T+1} ). this is monetary policy
VI. revisiting the Fisher Effect: increasing the nominal interest rate increases expected inflation
VII. fiscal policy: government surplus s_T controls unexpected inflation i.e. the error in expected inflation. this can be derived from B_T / P_{T+1} = s_{T+1}.
D. Fiscal policy debt sales
I. B_0 / P_1 = s_1 implies that increasing debt B_0 will not cause inflation if primary future surplus s_1 is increased proportionally
II. if the government runs a deficit s_0 < 0, there are 3 ways to finance it:
a. borrow more, increasing B_0. the government must also promise larger future surpluses s_1 in order to raise the real revenue from selling bonds Q_0 * B_0
b. inflate away the debt. P_0 rises if s_0 declines (assuming no change in s_1) since B_-1 / P_0 = s_0 + \beta * E_0( s_1 ).
c. AR(1): s_0 and expected s_1 both drop. this increases inflation P_0 and lowers the real debt payed off Q_0 * B_0 / P_0
III. in historical data "deficits are not strongly correlated with inflation... [this tells] us that fiscal policy does not routinely inflate away the debt" so deficits are typically offset by promising larger future surplus (e.g. higher taxes)
E. Debt Reactions and a Price Level Target
I. Assuming there's a target price level we want to achieve P*_1
a. if inflation breaks out and P_1 > P*_1, you run a larger real surplus
b. if deflation breaks out and P_1 < P*_1, you refuse to pay surplus to pay "an unexpected windfall to bondholders"
c. borrowing from two-period model B_-1 / P_0 = s_0 + \beta * E_0( 1 / P*_1 ) * B_0, you can increase B_0, borrowing more, to delay a larger real surplus. "In an intertemporal model, the government does not need to be specific about just when the surpluses will arrive."
d. or it doesn't increase B_0 and the debt is naturally inflated away when money is printed to repay B_-1 (AR(1) price model)
F. Fiscal Policy changes Monetary Policy
I. selling more debt -> increasing B_0
a. if fiscal policy dictates a fixed s_0 and s_1, then P_0 remains the same and P_1 increases (to-do: sketch the proof of this, because I'm not sure)
b. on the other hand, if it dictates fixed s_0 and the rule s_1 = B_0 / P*_1 (target price level), then P_0 lowers (another proof needed)
II. fixed s_0 and s_1 -> government bonds have a fixed real revenue, since \beta * E_0 ( s_1), the expected real returns, remains the same -> fixed real bond price Q_0 / P_0 per bond, more revenue with increased B_0
a. fiscal policy: increased debt sales B_0 is done to raise the s_0 deficit -> no inflation
b. monetary policy: increased debt sales B_0 and fixed s_0 -> lower P_0, deflation
III. fixed s_0 and s_1 with higher interest target i_0 and lower price level P_0 *decreases* current inflation while *increasing* expected inflation in 1 / ( 1 + i_0 ) = \beta E_0 ( P_0 / P_1 ), contradicting the general Fisher Effect intuition
IV. this "highlights one of the central mechanisms in many models for producing a negative inflation response to interest rates: Higher interest rates induce a future fiscal contraction."
Chapter 2: Inter-Temporal Model
A. Inter-Temporal Model: "The basic fiscal theory equation quickly generalizes to say that the real value of nominal debt equals an infinite present value of surpluses"
I. B_{T-1} / P_T = E_T * \\sum_{j=0}^{\infinity} \beta^j * s_{T+j}
II. this inter-temporal model will later be analyzed in "fiscal" in "monetary" aspects, just as the two-period model was. it will be revealed that this resembles the "New-Keynesian" central banking models much closer.
III. Chapter 2, like Chapter 1, assumes one-period debt, flexible prices, endowment model, constant real interest
VI. later chapters add price stickiness, discount rate variation, risk premiums, and other "realistic complications"
B.
Footnotes:
[1] Cochrane uses the *gross*, or *discrete*, form of the Fisher equation, while the *net* form is what's traditionally taught. Most textbooks will give the Fisher equation for *compounding* rates of interest.
gross form:1 + i = R*E(\pi)
in net form we use compounding interest (as opposed to simple. see: Interest Rate Equations), so it's ln(1 + i) = ln(R*E(\pi)) = lnR + lnE(\pi)