Chapter 15 took monetary policy as a Taylor rule — a feedback function from inflation and the output gap to the interest rate. This chapter goes deeper. Why do people hold money? What determines the optimal quantity of money? Why do central banks consistently produce too much inflation (time inconsistency)? And how does fiscal policy interact with monetary policy through the government budget constraint?
The chapter's culmination is the Fiscal Theory of the Price Level (FTPL) — the radical claim that, under certain conditions, fiscal policy, not monetary policy, determines the price level.
The CIA constraint assumes agents must hold money to purchase consumption goods:
Money is valued because it is required for transactions. When the nominal interest rate $i > 0$, holding money has an opportunity cost (foregone interest), creating a wedge that distorts consumption decisions.
An alternative: money directly enters the utility function, capturing the liquidity services it provides:
The first-order condition equates the marginal utility of real balances to the opportunity cost of holding money:
where $m = M/P$ is real balances and $i$ is the nominal interest rate.
The marginal cost of producing money is essentially zero. Efficiency requires that the price of each good equal its marginal cost. The "price" of holding money — the opportunity cost — is the nominal interest rate $i$. Since the marginal cost of money is zero, the efficient price is $i = 0$.
Since the Fisher equation gives $i = r + \pi$, and the real rate $r$ is determined by fundamentals, the Friedman rule implies:
The optimal inflation rate is the negative of the real interest rate — the central bank should deflate at the rate of time preference, making the nominal rate zero and eliminating the distortion from money holding.
The central bank minimizes a loss function:
where $y^*$ is natural output, $k > 0$ reflects the central bank's desire to push output above natural, and $a$ is the weight on inflation. An expectations-augmented Phillips curve links output and inflation:
Under commitment: The central bank announces $\pi = 0$ and sticks to it. The loss is $k^2$.
Under discretion: In rational expectations equilibrium ($\pi = \pi^e$), the inflation bias emerges:
The loss under discretion is $L_{disc} = k^2(1 + b^2/a)$ — strictly worse than commitment. The inflation bias is all cost and no benefit: output stays at $y^*$ under both regimes, but discretion adds gratuitous inflation.
The inflation bias under discretion is $\pi^* = bk/a$. Adjust the central bank's preferences and the Phillips curve slope to see how the bias and losses change.
Figure 16.1. Loss under commitment vs. discretion. The gap is the cost of the central bank's inability to commit. A more conservative banker (higher $a$) shrinks the inflation bias. Drag sliders to explore.
Solutions to time inconsistency: (1) Central bank independence (Rogoff, 1985): appoint a "conservative central banker" with higher $a$. (2) Inflation targeting: explicit numerical commitment. (3) Reputation: in repeated interactions, the long-run credibility cost exceeds the short-run gain. (4) Performance contracts (Walsh, 1995): penalties for missing targets.
Consider utility $u(c, m) = \ln c + \gamma\ln m$ with budget constraint and Fisher equation $i = r + \pi$.
Step 1: FOC for real balances: $\gamma/m = i \cdot (1/c)$, so $m/c = \gamma/i$.
Step 2: Marginal utility of money: $u_m = \gamma/m$. Marginal utility of consumption: $u_c = 1/c$. Optimality: $u_m/u_c = \gamma c/m = i$.
Step 3: The social cost of producing money is zero. Efficiency requires $u_m/u_c = $ marginal cost $= 0$. Therefore $i^* = 0$.
Step 4: From Fisher equation: \$1 = r + \pi^*$, so $\pi^* = -r$. With $r = 4\%$: optimal inflation is $-4\%$/year (deflation). The central bank should shrink the money supply at the rate of time preference.
Parameters: Phillips curve slope $b = 0.5$, output ambition $k = 0.02$, inflation weight $a = 1.0$.
Step 1: Inflation bias under discretion: $\pi^* = bk/a = 0.5 \times 0.02 / 1.0 = 0.01$ (1% per year).
Step 2: Loss under commitment ($\pi = 0$): $L_c = k^2 = 0.0004$.
Step 3: Loss under discretion: $L_d = k^2(1 + b^2/a) = 0.0004(1 + 0.25) = 0.0005$.
Step 4: Cost of discretion: $L_d - L_c = 0.0001$. Society gets 1% gratuitous inflation with zero output benefit.
Step 5: If a "conservative banker" has $a = 4$: $\pi^* = 0.5 \times 0.02/4 = 0.0025$ (0.25%). The bias shrinks by 75%, justifying central bank independence.
The government's flow budget constraint:
The intertemporal government budget constraint (IGBC) in real terms:
where $R_t = \prod_{j=0}^{t-1}(1+r_j)$ is the cumulative discount factor and $s_t = T_t - G_t$ is the primary surplus. Real government debt equals the present value of future primary surpluses.
The theorem requires strong assumptions. Key failures: (1) Finite horizons / OLG: current generation benefits, future generation pays. (2) Liquidity constraints: credit-constrained households spend windfall tax cuts. (3) Distortionary taxes: timing of income taxes changes relative incentives. (4) Uncertainty about future fiscal policy. (5) Behavioral biases: present-biased agents overconsume windfalls.
Empirically, about 20–40% of U.S. households appear liquidity-constrained (Zeldes, 1989). Tax rebates increase spending by about 20–40% of the rebate amount — inconsistent with full Ricardian equivalence.
What fraction of households are liquidity-constrained? At 0%, full Ricardian equivalence holds and a tax cut has zero effect on consumption. At 100%, the full tax cut is spent (pure Keynesian). The real world is somewhere in between.
Figure 16.2. Consumption response to a \$100B tax cut as a function of the fraction of constrained households. At 0% constrained, agents fully internalize future taxes and save the entire cut (Ricardian equivalence). At 100%, the full cut is spent. Empirical estimates (gray band) suggest 20–40% of households are constrained. Drag the slider to explore.
A government cuts lump-sum taxes by \$100B, financed by issuing bonds. Assume $r = 3\%$ and taxes will increase by \$103B$ next year.
Under Ricardian equivalence: Households receive \$100B today but know they owe \$103B next year (PV = \$100B). They save the entire \$100B. Consumption unchanged: $\Delta C = 0$. Bond market absorbs \$100B in new debt with no change in interest rates.
With 40% liquidity-constrained households: Unconstrained (60%) save the full tax cut. Constrained (40%) spend it all. $\Delta C = 0.4 \times 100B = 40B$. The fiscal multiplier is 0.4, not zero.
Empirical evidence: Johnson, Parker, and Souleles (2006) found that U.S. households spent 20–40% of the 2001 tax rebates within the first quarter, consistent with partial Ricardian equivalence failure.
From Eq. 16.9, the IGBC must always hold. In the Ricardian regime, fiscal policy adjusts surpluses to satisfy the IGBC at whatever price level the central bank determines. In the non-Ricardian regime, surpluses are set independently, and the price level adjusts:
If the government increases debt ($B_0$) without adjusting future surpluses, the price level $P_0$ must rise. Inflation is a fiscal phenomenon, not a monetary one.
| Monetary policy | Fiscal policy | Outcome |
|---|---|---|
| Active ($\phi_\pi > 1$) | Passive (adjusts surpluses) | Standard NK: monetary policy determines $\pi$ |
| Passive ($\phi_\pi < 1$) | Active (fixed surpluses) | FTPL: fiscal policy determines $P$ |
| Active | Active | No equilibrium (over-determined) |
| Passive | Passive | Indeterminate (under-determined) |
In a non-Ricardian regime, $P = B / PV(\text{surpluses})$. Watch how the price level responds to changes in nominal debt or expected fiscal surpluses.
Figure 16.3. FTPL price determination. The price level adjusts to equate real government debt with the present value of surpluses. Increasing debt without increasing surpluses causes inflation. Decreasing expected surpluses without reducing debt also causes inflation. Drag the sliders to explore fiscal dominance.
A government has nominal debt $B_0 = 100$ and announces a new fiscal plan.
Scenario A (credible surpluses): Primary surpluses of 5 per year forever, $r = 5\%$. $PV(s) = 5/0.05 = 100$. Price level: $P_0 = 100/100 = 1.00$. No inflation.
Scenario B (lower surpluses): Surpluses fall to 4 per year. $PV(s) = 4/0.05 = 80$. Price level: $P_0 = 100/80 = 1.25$. Inflation: 25%.
Scenario C (war or crisis): Government doubles debt to $B_0 = 200$ with unchanged surpluses ($PV = 100$). $P_0 = 200/100 = 2.00$. Inflation: 100%.
Key insight: Under FTPL, inflation is determined by the gap between government liabilities and the present value of surpluses — independent of money supply growth. The central bank's inflation target is overridden by fiscal dominance.
Seigniorage — the revenue from printing money — is an inflation tax on money holders. Real seigniorage is:
where $\mu$ is the money growth rate and $m(\mu)$ is real money demand (decreasing in $\mu$). At low inflation, higher $\mu$ raises revenue. But at high inflation, the tax base ($m$) erodes faster than the rate rises — a seigniorage Laffer curve.
Real money demand falls exponentially with inflation: $m(\mu) = m_0 \cdot e^{-\alpha \mu}$. Seigniorage revenue $S = \mu \cdot m(\mu)$ is an inverted U. Push inflation too high and you destroy the tax base.
Figure 16.4. The seigniorage Laffer curve. Revenue first rises with inflation, then falls as the real money base is destroyed. Hyperinflation economies (Zimbabwe, Venezuela) operate on the right side of the curve — high inflation, low revenue. Drag the slider to explore.
How should the government structure taxes to minimize distortions? Ramsey's rule (1927): among commodities, tax those with inelastic demand more heavily (the inverse elasticity rule):
Taxes on inelastic goods cause less behavioral distortion (less DWL, recall Chapter 3). The Ramsey rule minimizes total DWL for a given revenue requirement.
Two goods with different demand elasticities. The inverse elasticity rule says tax the inelastic good more. Compare Ramsey optimal rates to a uniform tax — same revenue, less deadweight loss.
Figure 16.5. Ramsey optimal tax rates vs. uniform taxation. The Ramsey rule assigns higher tax rates to the more inelastic good, reducing total DWL while raising the same revenue. The further apart the elasticities, the larger the efficiency gain. Drag sliders to change elasticities.
Normal times ($\phi_\pi > 1$): Fiscal multiplier $\approx 0.5$–\$1.0$. Government spending raises aggregate demand, but the central bank raises rates, crowding out investment.
Zero lower bound ($i = 0$): Fiscal multiplier $> 1$, possibly \$1.5$–\$1.0$. The central bank cannot raise rates, so there is no crowding out. Fiscal policy is more effective precisely when it is most needed (Christiano, Eichenbaum & Rebelo, 2011; Woodford, 2011).
Two goods with elasticities $|\varepsilon_1| = 0.5$ (inelastic, e.g., food) and $|\varepsilon_2| = 2.0$ (elastic, e.g., electronics). Revenue target: $R = 400$.
Step 1: Inverse elasticity rule: $\tau_1/\tau_2 = \varepsilon_2/\varepsilon_1 = 2.0/0.5 = 4$. The inelastic good should be taxed 4x more heavily.
Step 2: Revenue constraint: $\tau_1 Q_1 P_1 + \tau_2 Q_2 P_2 = 400$. With base $Q_0 = 100$, $P_0 = 10$, and demand $Q_i \approx Q_0(1 - \varepsilon_i\tau_i)$:
With $\tau_1 = 4\tau_2$: solve numerically to find $\tau_2 \approx 8.3\%$ and $\tau_1 \approx 33.2\%$.
Step 3: DWL comparison. Ramsey: $DWL = 0.5 \times 0.5 \times 0.332^2 \times 1000 + 0.5 \times 2.0 \times 0.083^2 \times 1000 = 27.6 + 6.9 = 34.5$.
Uniform tax ($\tau_1 = \tau_2 = 0.20$): $DWL = 0.5 \times 0.5 \times 0.04 \times 1000 + 0.5 \times 2.0 \times 0.04 \times 1000 = 10 + 40 = 50$.
Result: Ramsey reduces DWL by 31% relative to uniform taxation. The efficiency gain comes from concentrating the tax burden on the less responsive good.
Zimbabwe hyperinflation and Japan's lost decades: Two extremes of monetary-fiscal interaction.
Zimbabwe (2007–2008): Peak inflation reached approximately 79.6 billion percent per month in November 2008. The government financed massive fiscal deficits (land reform, military spending) by printing money. As inflation accelerated, the real money base collapsed — the economy moved to the wrong side of the seigniorage Laffer curve. The Zimbabwe dollar became worthless; transactions shifted to U.S. dollars and South African rand. This is the textbook case of fiscal dominance: the central bank was subservient to fiscal needs, and the FTPL equation $P = B/PV(s)$ played out with $PV(s) \to 0$.
Japan (1990s–present): The opposite extreme. Government debt exceeded 250% of GDP, yet inflation remained near zero or negative for decades. The Bank of Japan cut rates to zero in 1999 and implemented massive quantitative easing. Neither fiscal nor monetary expansion produced inflation. Possible explanations: (1) Japanese fiscal surpluses are expected to eventually adjust (Ricardian regime despite high debt). (2) The deflationary equilibrium is self-fulfilling — agents expect zero inflation, which validates itself at the ZLB. (3) Demographic decline reduces the natural rate permanently below zero.
The lesson: Zimbabwe and Japan bracket the spectrum of monetary-fiscal regimes. Zimbabwe shows what happens when fiscal policy dominates and surpluses collapse. Japan shows that even enormous debt need not produce inflation if fiscal credibility is maintained — but also that escaping deflationary equilibria is extraordinarily difficult.
Kaelani's government has debt of 85% of GDP. The central bank follows a Taylor rule with $\phi_\pi = 1.5$ (active monetary policy), and the government has announced primary surpluses of 2% of GDP for 15 years.
If the government delivers: Ricardian regime. If surpluses fall short: $P_0 = B_0 / PV(surpluses)$. If surpluses drop from 8.5B KD to 6B KD in PV, prices must rise by \$1.5/6 = 42\%$ — fiscal dominance overrides the inflation target.
About 40% of Kaelani's households are liquidity-constrained, so a tax cut has a positive (but partial) effect on aggregate demand — Ricardian equivalence fails for them.
| Label | Equation | Description |
|---|---|---|
| Eq. 16.1 | $P_tc_t \leq M_t$ | CIA constraint |
| Eq. 16.4 | $\pi^* = -r$ | Friedman rule |
| Eq. 16.7 | $\pi^* = bk/a$ | Inflation bias under discretion |
| Eq. 16.9 | $B_0/P_0 = \sum R_t^{-1}s_t$ | Intertemporal GBC |
| Eq. 16.10 | $P_0 = B_0 / \sum R_t^{-1}s_t$ | FTPL price determination |
| Eq. 16.11 | $\tau_i/\tau_j = \varepsilon_j/\varepsilon_i$ | Ramsey inverse elasticity rule |