Advanced Macroeconomic Analysis - FTPL Notes

FTPL: Introduction

FTPL (Fiscal Theory of the Price Level) explains price level determination through government surpluses.
The theory will be developed using a two-period model initially, then extended to an intertemporal version.
Generalizations, such as long-term debt, risk, risk aversion, and money, will be assessed if time allows.
Some derivations are available in the online appendix.

Two-Period Model

Flexible prices and constant interest rates.
Short-term debt with no risk premia.
Two periods: Day 0 and Day 1.
Debt $B_0$ is pre-determined in Day 1.
The government pays bondholders by printing money at the start of Day 1.
The government collects taxes, $P1s1$ , at the end of Day 1.
Equilibrium condition: Printed money must equal taxes, $B0 = P1s1$ , which implies $\frac{B0}{P1} = s1$ (1.1)
$P_1$ adjusts to satisfy the equilibrium condition.

Intuition and Mechanism

If $P1$ is too low, B0 > P1s1, indicating excessive money supply relative to taxes.
Excess money leads to increased demand, driving $P_1$ up.
Mechanism can be viewed as a wealth effect of government debt.
High government debt relative to surpluses acts like net wealth, stimulating spending and increasing $P_1$ .

Two-Period Model and Present Value

Day 0 equilibrium condition: $B{-1} = P0s0 + Q0B_0$ (1.2)
Money supply must equal surpluses and sales of bonds.
Bond price is given by $Q0 = \frac{1}{1 + i0} = βIE0\left(\frac{P0}{P1}\right) = \frac{1}{R} IE0\left(\frac{P0}{P1}\right)$ (1.3) (Fisher equation).
Using $B0 = P1s1$ and (1.3) in (1.2), we get $\frac{B{-1}}{P0} = s0 + βIE0(s1)$ (1.5)
$P_0$ adjusts to equate the real value of nominal debt with the present value of real primary surpluses.
Equation (1.5) is the government debt valuation equation, similar to stock valuation.

Monetary Policy, Fiscal Policy, and Inflation

Government policy levers: $B0$ and ${s0, s_1}$ .
Price levels, $P0$ and $P1$ , determined by $\frac{B{-1}}{P0} = s0 + βIE0(s1)$ and $\frac{B0}{P1} = s1$
Selling more debt $B0$ without changing surpluses increases $P1$ but keeps $P_0$ the same.
Selling more debt without changing surpluses acts like a share split.
Monetary policy involves buying/selling bonds or setting a fixed interest rate.
Interest rate target, $i0$ , sets expected inflation via $\frac{1}{1+i0} = βIE0 \left( \frac{P0}{P_1} \right)$
Increasing $i0$ has no effect on contemporenous inflation $P0/P_{-1}$ (Fisherian response).
Fiscal policy determines unexpected inflation. Taking innovations in equation (1.1), we get $\frac{B0}{P0} (IE1 − IE0) \left(\frac{P0}{P1}\right) = (IE1 − IE0)(s_1)$ (1.10)
Monetary policy determines expected inflation, fiscal policy determines unexpected inflation.

Fiscal Policy Debt Sales

Debt sales (↑ $B0$ ) paired with increasing surpluses (↑ $s0, s_1$ ) are like an equity issue.
From $\frac{B0}{P1} = s1$ (1.11), if the government raises $B0$ and $s1$ proportionally, $P1$ remains unchanged.
Options for financing a deficit at Day 0, s_0 < 0, include:
1. Lowering $s0$ by $∆s0$ but increasing $s1$ by $-R∆s0$ has no effect on $P0$ or $P1$ .
2. Inflating away the debt: $s0$ falls, $s1$ stays constant, then $P_0$ increases.
3. If $s1$ falls when $s0$ does (serially correlated deficits), time 0 inflation is larger than in 2. The time 0 deficit comes with a decline in the value of end of period debt: ↓ $Q0B0/P0$ =↓ $βIE0(s_1)$ .
Option 3 is not typical in advanced economies with 's-shaped' surplus processes.

Debt Reactions and a Price Level Target

Consider a surplus (or fiscal rule) at time 1 that responds to $B0$ according to $s1 = \frac{B0}{P1^}$ (1) where $P1^$ is a price level target; then in equilibrium $P1 = P_1^*$
Financing a deficit, $s0 < 0$ , with debt, $B0$ , implies a commitment to raise surplus $s1$ to repay that debt at the target price level, $P1^*$ .

Fiscal Policy Changes Monetary Policy

The fiscal policy rule, $s1 = \frac{B0}{P_1^*}$ , changes the effect of monetary policy.
Before, with fixed ${s0, s1}$ , an increase in $B0$ increased $P1$ .
Now $P1$ is fixed at the target, $P1 = P1^*$ , so an increase in $B0$ lowers $P_0$ (bond sales soak up cash at time 0).
A higher interest rate, ↑ $i0$ , also increases expected inflation but now through a lower $P0$ , $\frac{1}{1 + i0} = βIE0 \left(\frac{P0}{P1}\right)$
Now, in response to an interest rate increase, current inflation, $P0/P{-1}$ , falls!
The fiscal rule matters for the effect of monetary policy on inflation. Higher interest rates induce a future fiscal contraction.

Budget Constraints and Active versus Passive Policies

The equation $\frac{B0}{P1} = s_1$ (1.14) is an equilibrium condition, not a government budget constraint.
Budget constraints hold on and off-equilibrium prices. Equilibrium conditions do not hold off-equilibrium prices. In other words, $B0/P1 = s1$ does not hold for every price $P1$ .
What holds for every price is $B0 = P1s1 + M1$ (1.15) and consumer optimization implies $M_1 = 0$ .

Active versus Passive Policies

Suppose the government follows a fiscal rule at time 1 $s1 = τ1y1 = \frac{B0}{P1}$ (1.16), lowering the tax rate, $τ1$ , as the price level, $P_1$ , rises and vice versa.
This is a "passive" fiscal rule.
If the government follows (1.16), then (1.14) no longer pins down the price level. $P_1$ cancels from both sides of the equation.
A government that lets the price level be set by means other than (1.14) follows a passive fiscal policy.
Active fiscal policy excludes the one-for-one case $s1(P1) = B0/P1$ , so that (1.14) has a unique solution for $P_1$ .

Active versus Passive with a Debt Rule

Active versus passive policy is often framed in terms of responses to debt, but this is not precise.
For example, $s1 = \frac{B0}{P_1^*}$ is an active policy that responds one-to-one to debt.
Tests of $γ$ from a regression $s1 = a + γ(\frac{B0}{P1}) + u1$ could not distinguish passive from active. Under both cases one gets $γ = 1$ .
Active-passive, passive-active regimes can be observationally equivalent on some dimensions.
Active and passive are understood in relation to (1.14), to the mechanism for pinning down the price level.

The Intertemporal Model

Government budget constraint: $M{t-1} + B{t-1} = Ptst + Mt + QtB_t$ (2.1)
Household maximizes $max IE \sum{t=0}^∞ β^tu(ct)$ subject to $M{t-1} + B{t-1} + Pty = Ptct + Ptst + Mt + QtBt$ (2.2) with $Bt ≥ 0$ and $Mt ≥ 0$ .
In equilibrium, $ct = y$ and $Qt = \frac{1}{1 + it} = \frac{1}{R} IE \left( \frac{Pt}{P{t+1}} \right) = βIE \left( \frac{Pt}{P_{t+1}} \right)$ (2.3)
When $it > 0$ , demand for money is $Mt = 0$ . If $it = 0$ , money and bonds are perfect substitutes, so we let $Bt$ stand for both.
Then (2.1) is $B{t-1} = Ptst + QtB_t$ (2.4)
Divide by $Pt$ and use FOC for $Qt$ : $\frac{B{t-1}}{Pt} = st + βBtIEt \left( \frac{1}{P_{t+1}} \right)$ (2.5)
The household transversality condition $lim{T→∞} IEt \left( \frac{β^T B{T-1}}{P_T} \right) = 0$ (2.6)
Implies $\frac{B{t-1}}{Pt} = IEt\sum{j=0}^∞ β^j s{t+j}$ (2.7)

Dynamic intuition

Inflation in the fiscal theory has the feel of a run.
(2.7) suggests that demand for government debt falls on bad news about $s_{t+j}$ even if in the far future.
A complementary interpretation is that government debt falls today because of fears the government won’t be able to roll over the debt: $\frac{B{t-1}}{Pt} = st + Qt\frac{Bt}{Pt}$
Short-term debt constantly rolled over is the classic ingredient of a sovereign debt crisis.
Key difference is that the government can devalue via inflation rather than by explicit default.

Equilibrium Formation

What force pushes the price level to its equilibrium value?
If the price level is too low then $B{t-1} = Ptst + QtBt + Mt$ (2) money printed up ( $B{t-1}$ ) exceeds money soaked up in taxes and bond sales ( $Ptst + QtBt$ ). The extra money chases goods and drives $Pt$ up.
Alternatively, government bonds soak too much money when the price level is too low. Debt sales generate more revenue than the present value of surpluses.
Consumers could hold less debt and increase consumption, again putting upward pressure on the price level.
Intuition generalises to off-equilibrium price sequences, ${Pt}$ ; note that $Qt = βEt(Pt/P{t+1})$ , so off-equilibrium price sequences imply off-equilibrium bond price sequences.

Fiscal and Monetary Policy

The price level is pinned down by $\frac{B{t-1}}{Pt} = IEt\sum{j=0}^∞ β^j s{t+j}$ (2.7)
Advance (2.7) by one period: $\frac{Bt}{P{t+1}} = IEt+1\sum{j=0}^∞ β^j s{t+1+j}$ (2.12)
Define innovations as $∆IE{t+1} = IE{t+1} − IEt$ and apply to (2.12): $\frac{Bt}{Pt} ∆IE{t+1} \left( \frac{Pt}{P{t+1}} \right) = ∆IE{t+1}\sum{j=0}^∞ β^j s{t+1+j}$ (2.13)
At t + 1, $\frac{Bt}{Pt}$ is pre-determined ⇒ unexpected inflation is determined by changing expectations of the present value of surpluses.
Using (2.12), divide by $Pt$ , use the FOC for bond holdings and take expected value at t to get $\frac{Bt}{Pt} \frac{1}{1 + it} = \frac{Bt}{Pt} \frac{1}{R} IEt \left( \frac{Pt}{P{t+1}} \right) = IEt\sum{j=1}^∞ β^j s{t+j}$ (2.15)
If the government sells more debt with no changes in ${s{t+j}}$ , expected inflation must move one-for-one with debt sales. The government can control the interest rate, $\frac{1}{1+it} = Qt$ , and expected inflation, by changing the amount of debt, $Bt$ , with no changes in ${s_{t+j}}$ Monetary policy determines expected inflation.
If there are no changes in ${s{t+j}}$ , then $\frac{QtBt}{Pt}$ is a constant.
The government faces a unit elastic demand for nominal debt.
Monetary policy can use as its instrument either a price, $Qt$ , or a quantity, $Bt$ , taking demand for bonds as given.
In the absence of shocks, monetary policy determines expected inflation which equals realized inflation $IEt \left( \frac{Pt}{P{t+1}} \right) = \frac{Pt}{P{t+1}}$
Monetary policy determines the growth rate of prices.
Fiscal policy determines the level of prices via (2.7) $\frac{B{-1}}{P0} = IE0\sum{t=0}^∞ β^tst$
The time path, ${P0, P1, P_2, . . .}$ , is uniquely pinned down by monetary and fiscal policy.

The Fiscal Theory of Monetary Policy

Linearizing the FOC for bond holdings we get $it = r + IEtπ{t+1}$ (2.16)
Define $Vt ≡ Bt/Pt$ and $s˜t ≡ st/V$ , and linearizing we can write (2.13) at t + 1 as $∆IE{t+1}π{t+1} = −∆IE{t+1}\sum{j=0}^∞ β^j s˜{t+1+j} ≡ −εΣ_{s,t+1}$ (2.17)
Notation: $εΣ{s,t+1}$ is a shock to the present value of surpluses and $ε{s,t+1} = ∆IE{t+1}s˜{t+1}$ is a shock to the t + 1 surplus itself.
The solution to this model, (2.16) and (2.17), is $π{t+1} = it − εΣ_{s,t+1}$ (2.19)
A permanent increase in $i_t$ permanently increases inflation (Fisherian response).
To a fiscal shock there is a one-time price level jump. The timing of the announcement, matters and not the date of implementation.

Monetary-Fiscal Interactions

Inflation can decline after a monetary policy shock if combined with a fiscal contraction. (non-Fisherian response)
The fiscal rule $s1 = \frac{B0}{P_1^*}$ from the simple two-period.
Same idea works here: if the fiscal authority adjusts surpluses to hit ${p^{t+1}, p^{t+2}, . . .}$ , then $pt$ must decline today to satisfy $it = IEt(p^* − p_t)$
In this dynamic model, monetary-fiscal coordination is key.
If the fiscal authority were to have a target for $p^*t$ , there would be a violent monetary-fiscal dispute, perhaps resulting in non-existence of equilibria.
The response of inflation to a monetary policy shock depends on fiscal policy.

Interest Rate Rules

With a Taylor-type rule, the model now becomes: $it = Etπ{t+1}$ (2.20), $∆IE{t+1}π{t+1} = −εΣ{s,t+1}$ (2.21), $it = θπt + ut$ (2.22), $ut = ηu{t−1} + ε_{i,t}$ (2.23)
The solution for inflation now is $π{t+1} = θπt + ut − εΣ{s,t+1}$ (2.25)
The AR(1) shock and the interest rate rule now generate a hump-shaped response to a monetary shock.
The fiscal shock produces inflation which becomes persistent due to the monetary policy rule.

Fiscal Policy and Debt

Monetary policy changes $Bt$ without changing surpluses. Fiscal policy may change $Bt$ while changing surpluses.
To understand fiscal policy in this model consider this version of (2.15): $\frac{B{t−1}}{Pt} = st + \frac{1}{1 + it} \frac{Bt}{Pt} = st + IEt\sum{j=1}^∞ β^j s_{t+j}$ (2.26)
Assume the government raises $Bt$ and raises expected subsequent surpluses. The real value of the debt rises: $\frac{1}{1 + it} ↑ \frac{Bt}{Pt} = IEt\sum{j=1}^∞ β^j ↑ s{t+j}$
This extra money soaked up by bond sales can finance a deficit, lower $st$ , or could generate a disinflation, lower $Pt$ .
The case where future surpluses, $IEt \sum{j=1}^∞ β^j s{t+j}$ , exactly increase to offset a current deficit, $s_t$ , is important because it generates no unexpected inflation and implies an s-shaped process for the surplus.

The Central Bank and the Treasury

Both monetary and fiscal policies are about debt sales. Communication is key for expectations management.
Separation is imperfect: inflation produces seigniorage revenue and impacts the tax system.
Room for institutional innovation: could we modify institutional arrangements to commit not to back some debt issues (partial backing)?

Flat Supply Curve and Fiscal Stimulus

Government fixes the interest rate and offers nominal debt in a flat supply curve.
The Treasury and Central Bank operating together generate a flat supply curve.
Central Bank sets $i_t$ , Treasury auctions a given quantity of debt, and Central Bank then defends it buying or selling debt as required.
Fiscal loosening in this model creates inflation. But this simple model is an endowment economy so no impact on output, y, at this stage.

Long-term Debt

$B(t+j)t$ is the quantity of zero-coupon bonds outstanding at t − 1 that come due at t + j. From the FOCs we now get that bond prices satisfy $Q(t+j)t = IEt \left(β^j \frac{Pt}{P{t+j}} \right)$ (3.1)
The flow budget constraint is now: $B(t){t−1} = Ptst + \sum{j=1}^∞ Q(t+j)t \left[B(t+j)t − B(t+j)_{t−1} \right]$ (3.2)
The present value condition now is: $\sum{j=0}^∞ Q(t+j)t \frac{B(t+j){t−1}}{Pt} = IEt\sum{j=0}^∞ β^j s{t+j}$ (3.3)
Now a fiscal shock can be met with lower bond prices; a shock that raises nominal interest rates with no change in ${st}$ lowers $Pt$ .

Risk and Discounting

Introduce risk by letting the endowment $yt$ vary so the SDF is $\frac{Λ{t+1}}{Λt} = β \frac{u′(c{t+1})}{u′(c_t)}$
This alters the bond holding condition: $Qt = IEt \left(\frac{Λ{t+1}}{Λt} \frac{Pt}{P_{t+1}} \right)$
And gives the stochastically-discounted valuation formula: $\frac{B{t−1}}{Pt} = IEt\sum{j=0}^∞ \frac{Λ{t+j}}{Λt} s{t+j}$ (3.5)
Now the present value of surpluses can change with changes in the SDF, or changes in real returns

Money

If people hold cash, $Mt$ , then the flow constraint becomes $B{t-1} + M{t-1} = Ptst + \frac{1}{1 + it} Bt + Mt$ (3.9)
Iterating forward we obtain the valuation equation $\frac{B{t−1} + M{t−1}}{Pt} = IEt\sum{j=0}^∞ \frac{Λ{t+j}}{Λt} \left[s{t+j} + \frac{i{t+j}}{1 + i{t+j}} \frac{M{t+j}}{P_{t+j}} \right]$ (3.10)
Total government debt is the sum of overall debt, $B{t−1} + M{t−1}$ .
The household can hold one unit less of money, purchasing instead a bond which yields i. The real value of this payment is $i/(1 + π)$ , but as it is received in the next period, its present value is $i/[(1 + π)(1 + r)]$
Since money pays no interest $i^m = 0$ , the opportunity cost of holding money is determined by $i$ .
$i/(1 + i)$ is a cost of holding money to households but is revenue to the government $\frac{it}{1 + it} \frac{Mt}{Pt}$
When money pays interest (i.e. reserves or ES balances), the expression generalises to $\frac{B{t−1} + M{t−1}}{Pt} = IEt\sum{j=0}^∞ \frac{Λ{t+j}}{Λt} \left[s{t+j} + \frac{i{t+j} − i^m{t+j}}{(1 + i{t+j})(1 + i^m{t+j})} \frac{M{t+j}}{P_{t+j}} \right]$ (3.10)

The Zero Bound

If $Qt = 1$ , so $it = 0$ , money and bonds are perfect substitutes and (3.10) is $\frac{B{t−1} + M{t−1}}{Pt} = IEt\sum{j=0}^∞ \frac{Λ{t+j}}{Λt} s_{t+j}$ (3.16) so the price level can be determined at the zero bound.
The same result holds if $it = i^mt$ .
Money and bonds are perfect substitutes in both cases, and fiscal theory delivers a unique price level.

Money, Seigniorage and Fiscal Theory

Highlights fiscal-monetary interactions: $\frac{B{t−1} + M{t−1}}{Pt} = IEt\sum{j=0}^∞ \frac{Λ{t+j}}{Λt} \left[s{t+j} + \frac{i{t+j}}{1 + i{t+j}} \frac{M{t+j}}{P_{t+j}} \right]$
Open market operations (changing bonds for money) affects seigniorage and therefore fiscal surpluses and the price level.
Higher nominal interest rates can generate seigniorage revenue and drive down the price level.
The effect of monetary policy depends crucially on fiscal policy.

Linearizations

Linearization simplies the analysis; no multiplicative terms like $(\frac{Λ{t+j}}{Λt})s_{t+j}$
The linearized version of the flow condition is: $ρv{t+1} = vt + r{t+1} − g{t+1} − s˜_{t+1}$ (3.17)
where $v{t+1}$ is the log debt-to-output ratio. $r{t+1}$ is the log real return on the portfolio of government bonds; $g{t+1}$ is output growth; $s˜{t+1}$ is the surplus to output ratio. And define the log real return as $r{t+1} ≡ r^n{t+1} − π_{t+1}$
Iterating (3.17) forward and taking expectations at time t, $vt = IEt\sum{j=0}^∞ ρ^{j−1} s˜{t+j} + IEt\sum{j=0}^∞ ρ^{j−1} g{t+j} − IEt\sum{j=0}^∞ ρ^{j−1} r_{t+j}$ (3.19)

Response to Fiscal Shocks

Start from constant real returns ( $IEtr{t+1} = 0$ ) and one-period debt, $ω = 0$ . Then (3.20) and (3.21) become: $∆IE{t+1}π{t+1} = −\sum{j=0}^∞ ρ^j∆IE{t+1}s˜_{t+1+j}$ , a negative fiscal shock results in a positive shock to inflation.
Add time-varying returns: $∆IE{t+1}π{t+1} = −\sum{j=0}^∞ ρ^j∆IE{t+1}s˜{t+1+j} + \sum{j=0}^∞ ρ^j∆IE{t+1}r{t+1+j}$ , a shock to the present value of surpluses can come from the discount rate as well.
With $ω = 0$ , fiscal shocks give rise to one period inflation.

Response to Fiscal Shocks (cont’d)

Now bring back long-term debt, ω > 0, but keep constant real returns ( $IEtr{t+1} = 0$ ), (3.22) reads $\sum{j=0}^∞ ω^j∆IE{t+1}π{t+1+j} = −\sum{j=0}^∞ ρ^j∆IE{t+1}s˜{t+1+j}$ (3.23)
An unexpected rise in expected future inflation, $∆IE{t+1}π{t+1+j}$ , can soak up a fiscal shock. Current inflation, $∆IE{t+1}π{t+1}$ , need not do all the work.
Future inflation is less effective at absorbing the fiscal shock as ω < 1.

Monetary Policy Responses

From chapter 2: a rise in $i_t$ with no change in surpluses led to a ‘Fisherian’ response
With the assumptions in (3.23), assume no change in surpluses, then $\sum{j=0}^∞ ω^j∆IE{t+1}π{t+1+j} = −\sum{j=0}^∞ ρ^j∆IE{t+1}s˜{t+1+j} = 0$ which implies $∆IE{t+1}π{t+1} = −\sum{j=1}^∞ ω^j∆IE{t+1}π_{t+1+j}$ (3.28)
The central bank can change the timing of inflation. If ↑ $i_t$ , expected future inflation goes up on the RHS and current inflation falls on the LHS of (3.28).
The disinflation is larger if there is more long-term debt outstanding, higher $ω$ .

Review

Monetary policy determines expected inflation, and fiscal policy determines unexpected inflation.
In steady state, monetary policy determines the growth rate of prices, and fiscal policy determines the price level.
The response of inflation to monetary policy depends on the underlying fiscal rule.
The timing of news about the PV of surpluses matters, not when these are implemented.
Increases in current deficits that increase the real market value of debt imply an s-shaped process for surpluses.
Long-term debt matters for the response of inflation to monetary and fiscal policy shocks.