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---

### 2 Brief History of TIP and Literature
- Early proposals and microfoundations:
  - Wallich and Weintraub (1971): permanent tax on wage increases levied on wage increases in excess of a pre-announced target, paid by employers.
  - Kotowitz and Portes (1974): tax on wage increases reduces rate of change of wages (robust to myopic or forward-looking unions).
  - Latham and Peel (1977): tax on wage increases less effective under monopsony.
  - Seidman (1978a): TIP akin to a pollution tax to internalize price-setting externalities.
- Macroeconomic implications:
  - Peel (1979): TIP could reduce likelihood of business cycles.
  - Scarth (1983): employer TIP based on wage increases is stabilizing; other designs may be destabilizing.
  - Oswald (1984), Jackman and Layard (1989): conditions under which TIP raises employment and welfare.
- Implementation debate and design choices:
  - Debated elements: tax base (price vs. wage increases), payer (employers vs. employees), coverage, continuity, penalty vs. reward, exemptions, hurdles vs. continuous systems.
  - Dildine and Sunley (1978) favored taxing wage changes at large private corporates, penalty over reward, against exemptions, auditable hurdle.
  - Seidman (1978a,b; 1979) favored continuous system with penalties and rewards.
  - Lerner (1978) and Lerner and Colander (1980): market-based permit system for wage increases.
- Historical implementations:
  - France (1974–1977): "prélèvement conjoncturel" covering largest 1500 firms (representing 60% of the economy).
  - Other versions: Mexico, Belgium, Italy, the Netherlands.
  - U.S. near-implementation in 1978: "real wage insurance" proposal; failed due to fiscal concerns and association with price controls.
  - Transition economies (Bulgaria, Poland, Romania): TIP variants with mixed evidence.
- Authors’ contributions highlighted:
  - Use sticky-price models to reassess TIP effectiveness.
  - TIP and monetary policy (MP) are complementary: TIP focuses on markup and inflation expectation shocks; MP focuses on demand shocks.
  - TIP in steady-state increases welfare by decreasing markup and increasing employment; optimal TIP should vary over time—rise when inflation increases and fall as inflation reaches long-term target.
  - Analyze robustness to alternative instruments, targeting rules, and tax bases.
- Summary insight:
  - TIP is implementable, effective, and not too costly to administer (Paci, 1988).

### 3 A Model with Sticky Prices and a Tax on Inflation
- Model structure:
  - New Keynesian framework with a tax on inflation (TIP) τ_t proportional to price increases (P_{ti} − P_{t−1,i}) per unit sold; Rotemberg (1982) quadratic price-adjustment costs with parameter θ.
  - Households: continuum of mass one; preferences with parameters β∈(0,1), σ, ψ; budget constraint P_t C_{th} + Q_t B_{th} = B_{t−1}^h + W_{th} N_{th} + T_t; solvency lim_{T→+∞} Π_{j=0}^T Q_j B_{Th} ≥ 0.
  - Final goods: CES aggregator with elasticity ε_t.
  - Intermediate firms: monopolistic competition, production Y_{ti} = A_t N_{ti}^{1−α}, marginal cost MC_t defined; ideal markup M_t = ε_t/(ε_t−1).
- Firms’ optimization and pricing FOC:
  - Aggregate first-order condition (with firms’ expectation shock u^E_t):
    - (ε_t − 1)(M_t/MC_t − 1) + E_t[Q_t Y_{t+1}/Y_t (τ_{t+1} + θ(π_{t+1}+1)π_{t+1})] + E_t[Q_t Y_{t+1}/Y_t θ u^E_t (2 π_{t+1} + 1)] = τ_t [(1−ε_t) π_t/(1+π_t)] + θ π_t (π_t + 1).
- Linearized Phillips-like relation (Lemma 1):
  - Euler (log-linear): ˆy^e_t = E_t ˆy^e_{t+1} − (1/σ) (i_t − E_t π_{t+1} − r^e_t)
  - Phillips curve (log-linear): π_t = β E_t π_{t+1} + κ ˆy^e_t + (1/θ) [β E_t τ_{t+1} − τ_t + ũ_t]
  - Definitions:
    - κ = (ε − 1)/θ [σ + (ψ + α)/(1 − α)]
    - ũ_t = u_t + u^w_t + θ u^E_t
- Robustness: Calvo-type frictions deliver same first-order approximation and macro dynamics; optimal TIP results robust to using Calvo friction.

### 3.3.3 The Mechanics of TIP at the Micro-level
- Linearized firm FOC around zero-inflation steady state:
  - (ε − 1) ˆmc_t + u_t + θ u^E_t + β E_t[τ_{t+1} + θ π_{t+1}] = τ_t + θ π_t
- Illustrative case:
  - If τ_0 > 0 and τ_t = 0 for t > 0; π_{t+1} = 0; ˆmc_t = 0 ⇒ p_0 = p_{−1} + (u_0 + θ u^E_0 − τ_0)/θ.
  - Interpretation:
    - Higher TIP τ_0 induces firms to lower log price p_0—TIP attenuates pass-through of positive markup shocks u_0 and inflation expectation shocks u^E_0.
    - If u_0 = u^E_0 = 0 and τ_0 > 0, TIP may lead firms to lower prices to receive a subsidy (possible deflation).
    - Smaller θ (lower adjustment cost) ⇒ stronger disinflationary effect from a given τ_0.

### 4 Optimal Tax on Inflation and First-Best Allocation (preview of results)
- Objective: characterize MP and TIP combination implementing first-best allocation (no inflation, no output gap).
- Key qualitative result:
  - TIP should increase with markup shocks and inflation expectation shocks because these create externalities in firms’ pricing.
  - TIP realigns private incentives with social valuations and corrects excessive aggregate inflation.

### 4.1 First Best: Complete Stabilization and Specialization
- Social planner’s first best:
  - Optimal condition: (C^e_t)^σ (N^e_t)^ψ = (1−α) A_t (N^e_t)^−α.
  - Optimal consumption and output: C^e_t = (1−α)^(1−α) (1−α)^{−((1−α)σ+ψ+α)} A^{1+ψ}_{t}^{(1−α)σ+ψ+α}, Y^e_t = C^e_t. (Equation (18))
- Limits of MP alone:
  - MP that tracks neutral rate Q^e_t can implement first best only when no markup or inflation expectation shocks.
  - Distributional conflict and inflation-expectation shocks create a private–social wedge:
    - Private − Social = (ε_t − 1) (M_t / M^w_{t−1}) + E_t Q^e_t (A_{t+1}/A_t)^{(1+ψ)/((1−α)σ+ψ+α)} θ u^E_t. (Equation (20))
  - Conclusion: MP alone cannot implement first best when these externalities exist; instrument directly affecting firms’ price-setting is required.
- First-best implementation with TIP (Proposition 1):
  - For any paths of M_t, M^w_t, u^E_t, β_t, A_t, there exists combination of TIP and MP implementing first best.
  - Nominal policy interest rate given by (19).
  - TIP given by:
    - τ_t = (ε_t − 1) (M_t / M^w_{t−1}) + E_t Q^e_t (A_{t+1}/A_t)^{(1+ψ)/((1−α)σ+ψ+α)} [ τ_{t+1} + θ u^E_t ]. (Equation (21))
  - Implications:
    - Complete stabilization of output gap and inflation possible with TIP and MP.
    - Specialization: MP focuses on aggregate-demand and TFP shocks; TIP specializes in markup and inflation-expectation shocks.
    - TIP most effective when expected to decrease in future: temporarily raise TIP when shocks hit and announce later decrease.
- Linearized corollary (Corollary 1):
  - Complete stabilization: ŷ^e_t = π_t = 0.
  - Specialization in linear model:
    - i_t = r^e_t
    - τ_t = β E_t τ_{t+1} + u_t + u^w_t + θ u^E_t
  - AR(1) shocks with coefficient ρ_u < 1:
    - Example 1: τ_t = (u_t + u^w_t + θ u^E_t) / (1 − β ρ_u). (Equation (22))
  - Determinacy: require central bank φ_π > 1 in i_t = r^e_t + φ_π π_t to ensure uniqueness.

- Remarks:
  - Efficient steady-state: TIP can correct steady-state distortion:
    - τ = (ε − 1) / (1 − β) (M M^w − 1). (Equation (23))
  - Time-consistency: MP and TIP combination is time-consistent; discretion suffices.
  - Policy coordination: policies in Proposition 1 form a Nash equilibrium between separate authorities (shared welfare objective).
  - Second-best and TFP shocks: TIP has no role when inflation stems purely from TFP shocks under base model, but can supplement MP when additional frictions exist.

### 6.2 Determinacy with TIP
- Determinacy and the Taylor principle:
  - Baseline NK model requires Taylor principle for uniqueness.
  - Theoretical possibility: determinacy can be obtained if TIP reacts strongly to output gap while MP reacts passively to inflation (Appendix A.4).
  - Practical conclusion: necessary strength of TIP reaction to guarantee determinacy is beyond realistic implementation; Taylor principle remains only practical way to ensure determinacy.
- Key implication:
  - Taylor principle remains necessary in practice to avoid indeterminate, self-fulfilling equilibria.

### 6.3 Stabilization Properties of TIP — Approach and Baseline Calibration
- Methods:
  - Examine IRFs and welfare implications for markup, monetary, and productivity shocks.
  - Report variables: annualized inflation, output gap, annualized policy rate, TIP rate τ_t, TIP as percent of total sales.
  - Stochastic simulations compute standard deviations and welfare losses under one shock type at a time.
- Baseline MP: standard Taylor rule (31) with φ_π = 1.5 and φ_y = 0.125. TIP targets inflation only (φ_y = 0) unless otherwise noted.

### 6.3.1 Markups and Inflation Expectation Shocks — IRFs and mechanisms
- Shock analyzed: 0.25pp distributional or expectation shock (ũ_t^θ = 0.25pp). Without TIP:
  - On impact inflation initially rises to 1.3% and output drops slightly over 0.51%.
- Moderate TIP (φ_π = 150):
  - Initial inflation response falls to 1.0%.
  - Output decline to −0.40%.
  - Imposes a 38% tax on price changes in the first quarter.
- Strong TIP (φ_π = 300):
  - Initial inflation response falls to 0.84%.
  - Output gap to −0.34%.
  - Imposes a 63% tax on price changes.
- Mechanism:
  - TIP reduces need for policy-rate hikes, mitigating negative output effects.
  - Baseline (no TIP): policy rate increases by 170 basis points on impact.
  - Strong TIP (φ_π = 300): policy rate increase reduced to 110 basis points.
- Tax burden on firms small because tax applies only to price changes:
  - Strong TIP tax burden = 0.13% of sales in first quarter (0.04% in first year).
- Nonlinearities and profits:
  - τ_t π_t/(1+π_t) ≈ φ_π π_t^2 and E π_t τ_t = φ_π V(π_t).
  - As φ_π quadruples from 150 to 600, π_t^2 and V(π_t) decrease, yielding small increase in taxes relative to sales.
  - Profit effects depend on shock source: TIP moderates profits when inflation driven by firm markups; TIP can worsen profitability when inflation driven by wage markups.

### Sensitivity to price flexibility
- TIP’s effectiveness increases as Rotemberg parameter θ decreases (more price flexibility).

### Stochastic simulations: markup and inflation expectation shocks (selected entries)
- Shock standard deviation u_t^θ = 0.25pp. TIP strength scenarios: No TIP, Moderate TIP, Strong TIP, Extreme TIP with φ_π = 0, 150, 300, 600; φ_π (MP) = 1.5; φ_y = 0.125.
- Selected standard deviations and metrics (all standard deviations and E(π_t τ_t) multiplied by 100; L^* defined using quarterly variables):
  - σ(π_ann_t): 0.74, 0.59, 0.49, 0.36 for φ_π = 0,150,300,600.
  - σ(ŷ^e_t): 0.59, 0.47, 0.39, 0.29.
  - σ(i_ann_t): 0.96, 0.76, 0.63, 0.47.
  - σ(τ_t): 0, 44.1, 73.0, 108.8.
  - E(π_t τ_t): 0, 0.13, 0.18, 0.20.
  - L^* × 10^4: 1.22, 0.77, 0.53, 0.29.
- Welfare: L declines by 47% from 7.91 (no TIP, text) to 4.38 with φ_π = 300 in reported simulations; consistent reductions across many Taylor rules.

### 6.3.2 Aggregate Demand, Productivity and Monetary Shocks — IRFs and trade-offs
- For aggregate demand or productivity shocks sized to match baseline inflation response:
  - Strong TIP (φ_π = 300) lowers inflation by 0.34pp and reduces policy-rate hike by about 57 bps.
  - TIP amplifies initial widening of output gap by 0.18pp relative to no TIP — a trade-off arises because these shocks are ideally addressed by MP.
- Stochastic simulations (Table F1 referenced):
  - Increasing φ_π to 300 lowers variance of inflation from 2.52 to 1.87 but raises variance of output gap from 0.94 to 1.27.
  - Because inflation is more volatile than output in baseline, reduction in inflation variance outweighs increase in output variance, yielding higher welfare (lower L).

### 6.4 Optimal Monetary Policy with TIP
- Question: How should introduction of TIP change Taylor-rule design? Should MP specialize on output gap?
- Method: Simulate economies varying φ_π, φ_y, φ_π^TIP, φ_y^TIP; focus on TIP rules with φ_y = 0. Shocks: demand and markup shocks with standard deviation .25.
- Main results (Table 3, selected):
  - Specialization desirable: MP should focus more on output and less on inflation when TIP is present.
  - Optimal monetary policy weights minimizing L^* conditional on TIP:
    - No TIP: φ^*_π = 1.500, φ^*_y = 0.125.
    - Moderate TIP: φ^*_π = 0.990, φ^*_y = 0.126.
    - Strong TIP: φ^*_π = 0.983, φ^*_y = 0.162.
    - Extreme TIP: φ^*_π = 0.980, φ^*_y = 0.134.
  - Outcome metrics:
    - σ(π_ann_t): 0.77, 0.71, 0.58, 0.41.
    - σ(ŷ^e_t): 0.66, 0.47, 0.43, 0.44.
    - σ(i_ann_t): 1.03, 0.67, 0.55, 0.41.
    - σ(τ_t): 0, 53.3, 86.3, 124.0.
    - E(π_t τ_t): 0, 0.19, 0.25, 0.26.
    - L^* × 10^4: 1.38, 0.84, 0.56, 0.31.
- Interpretation and robustness:
  - Interest rate directly affects aggregate demand/output; TIP directly affects firms’ price-change incentives and inflation.
  - Specialization more desirable as variance of markup shocks increases.
  - φ_y in TIP need not increase monotonically with φ_π^TIP; optimal φ_y may vary non-monotonically due to loss-function trade-offs.
- On φ_y:
  - For markup and inflation expectation shocks, TIP should react strongly to inflation and strongly but negatively to output gap; φ_y < 0 is unlikely in practice—restrict φ_y ≥ 0 and set φ_y = 0 as reasonable.
  - For aggregate demand or productivity shocks, TIP with φ_y = 0 can be feasible when reaction to inflation is strong enough.

### 7 TIP and Relative Price Distortions — Concern and Extended Model (7.1)
- Concern: TIP could impede adjustment of relative prices across sectors, causing misallocation.
- Extended model:
  - Continuum of sectors s ∈ [0,1], sector weights γ_s with ∫_0^1 γ_s ds = 1; final good aggregator Y_t = exp(∫_0^1 γ_s ln Y_ts ds).
  - Sector-specific TFP shocks A_ts and sector-specific Rotemberg parameters θ_s.
  - Welfare second-order approximation (Lemma 2):
    - L = Σ_{t=0}^∞ β^t [ (σ + (α+ψ)/(1−α)) (ŷ^e_t)^2 + E[θ_s π^2_st] + E[(p̃^e_st)^2]/(1−α) ] where E denotes cross-sectional mean under γ_s.

### 7.2 Analytical Independence Result (Proposition 5)
- Assumptions: θ_s common across sectors (θ_s = θ) and steady-state TIP = 0.
- Proposition 5 conclusions:
  - Relative prices across sectors are independent of TIP; they depend only on θ, ε, α, β, and sector-specific productivity shocks ã_st.
  - Responses of output gap and aggregate inflation π_t to aggregate shocks and TIP are same as linearized single-sector economy.
  - Welfare varies with TIP only through output gap and average inflation because TIP does not affect cross-sector deviations E[(p̃_st − p̃^e_st)^2].
- Intuition:
  - Linearity and symmetry of TIP yield offsetting sectoral forces; with θ_s = θ and τ̄ = 0, relative prices remain exactly unchanged.
  - Price controls (nonlinear) would distort relative prices.

### 7.3 Independence Result: Numerical Simulations
- Calibration and experiment:
  - Sectoral θ_s calibrated from Nakamura and Steinsson (2008): θ_s = {20.4, 62.5, 689.7} for durable, non-durable, services; sector sizes {0.4, 0.2, 0.4}.
  - Sectoral log TFP follows AR(1): log(A_ts) = (1−ρ)logA_s + ρ log(A_{t−1,s}) + ν_st with ν_st i.i.d.; std dev of ν_st = .05 and persistence ρ = .5.
  - One-period markup shock: ε_0 falls to 7.5 at t=0 (from steady-state 9) and returns to 9 for t>0.
  - Compare τ_0 = 0 to τ_0 = 10% and τ_0 = 15%.
- Numerical results:
  - Average distortion of relative prices E[(p̃_st − p̃^e_st)^2] remains broadly unaffected by TIP size and may slightly decrease with TIP—confirming Proposition 5.
  - Higher TIP ⇒ lower aggregate inflation, lower average squared price changes E[θ_s π^2_st], and lower output gap ⇒ welfare improves in simulations.
  - Intuition for slight decrease in average distortion: second-order channel in τ_t[(1−ε_t) π_t/(1+π_t)] can facilitate relative price adjustments for firms raising prices more.
- Robustness:
  - Homogeneous stickiness (θ = 372.8): similar results—TIP has no effect on relative-price distortions and improves welfare.
  - Different steady-state TIP τ̄ = 15%: qualitatively similar—TIP reduces inflation and output-gap volatility without worsening relative-price distortions.
- Extensions:
  - Taxing only large corporates (section 8.1):
    - Two types of firms: small (S) no TIP, large (L) pay TIP. Calibrated so top 1% firms account for 75% of sales.
    - TIP nearly as effective when applied only to 1% of firms as when covering all firms.
    - Example: with φ_π = 300, initial inflation response falls to 1.3% (compared to 1.2% when TIP covers all firms); output gap falls to −.65 (compared to −.6%).
  - Sticky wages and tax on wage increases (section 8.2):
    - Wage-ROTemberg adjustment costs with ε_N = 4.5; θ_w set to match wage Phillips slope.
    - TIP on prices remains effective; TIP on wages effectiveness depends on shock source:
      - When shock is wage Phillips curve, TIP on wages with φ_π = 300 reduces inflation and output gap effectively.
      - When shock is price Phillips curve, TIP on wages is ineffective.
    - Effectiveness sensitive to elasticity across labor types ε_L.

### Key implications for policy and interpretation
- Numerical simulations corroborate Proposition 5: linear TIP does not worsen relative-price distortions and can reduce aggregate inflation and output-gap volatility.
- TIP complements MP: combining TIP with MP significantly lowers volatility of inflation and output relative to MP alone.
- Specialization policy prescription: TIP should address markups and inflation-expectation shocks; MP should address demand shocks.
- Robustness: results hold under heterogeneous/homogeneous price stickiness, limited coverage (large firms), and sticky wages.
- Implementation considerations for further research: taxing wages vs. prices, tax avoidance risks (relabeling products, quality shrinking), firm-level monitoring, political-economy risks regarding TIP governance and independence.

### Quantitative and Simulation Highlights (selected exact figures)
- Baseline Taylor rule: φ_π = 1.5, φ_y = 0.125.
- Markup/inflation-expectation shock: standard deviation u_t^θ = 0.25pp; without TIP: inflation rises to 1.3%, output drops slightly over 0.51%.
- Strong TIP (φ_π = 300) outcomes for markup-type shock:
  - Initial inflation response: 0.84%.
  - Output gap: −0.34%.
  - Tax on price changes first quarter: 63%.
  - Tax burden first quarter as percent of sales: 0.13% (first year: 0.04%).
- Stochastic simulation table (markup/inflation-expectation shocks, std dev ×100):
  - σ(π_ann_t): 0.74, 0.59, 0.49, 0.36 for φ_π = 0,150,300,600.
  - σ(ŷ^e_t): 0.59, 0.47, 0.39, 0.29.
  - σ(i_ann_t): 0.96, 0.76, 0.63, 0.47.
  - σ(τ_t): 0, 44.1, 73.0, 108.8.
  - E(π_t τ_t): 0, 0.13, 0.18, 0.20.
  - L^* × 10^4: 1.22, 0.77, 0.53, 0.29.
- Fiscal comparison (numerical):
  - To match impulse responses of strong TIP (φ_π = 300), wage subsidies would amount to 4% of total payrolls (or 3.5% of output) in first period; strong TIP costs below 0.15% of output in same period.
- Determinacy calibration note:
  - For quantitative calibration, φ_y necessary to ensure determinacy via TIP equals θκ/(1−β) ≈ slightly above 6300 (impractically large).
- Heterogeneous-sector one-period markup shock (Panel A.1 selected entries; all columns ×100):
  - τ0 = 0%: initial outcomes: 2.75 (inflation), −2.02 (output gap), 1.52, 0.33, 0.29, 2.14 (L metric).
  - τ0 = 10%: 1.05, −1.89, 0.55, 0.28, 0.29, 1.12.
  - τ0 = 15%: 0.18, −1.66, 0.39, 0.22, 0.29, 0.90.
- Homogeneous sectors θ = 372.8 (Panel B.1):
  - τ0 = 0%: 0.87, −1.87, 0.35, 0.28, 0.39, 1.02.
  - τ0 = 10%: 0.34, −1.76, 0.20, 0.25, 0.39, 0.84.
  - τ0 = 15%: 0.03, −1.67, 0.17, 0.22, 0.39, 0.79.

*Italic source attribution: wpiea2023254-print-pdf*

### Section 7 analyzes the implications of TIP for relative price distortion. Section 8

### wpiea2023254-print-pdf - Section 7 analyzes the implications of TIP for relative price distortion. Section 8

### 2 Brief History of TIP and Literature
- Early proposals:
  - Wallich and Weintraub (1971) proposed a permanent tax on wage increases (initially “Tax-based Incomes Policies”) levied on wage increases in excess of a pre-announced target and paid by employers.
  - Kotowitz and Portes (1974): microeconomic model with a union setting wages and firms setting prices; a tax on wage increases reduces the rate of change of wages (robust to myopic or forward-looking unions).
  - Latham and Peel (1977): tax on wage increases less effective under monopsony.
  - Seidman (1978a): firms don’t internalize economy-wide inflationary effects; TIP akin to a pollution tax to internalize price-setting externalities.
- Macroeconomic implications:
  - Peel (1979): TIP could reduce the likelihood of business cycles.
  - Scarth (1983): employer TIP based on price increases and employee TIP based on wage increases are destabilizing; employer TIP based on wage increases is stabilizing.
  - Oswald (1984): conditions for inflation tax to be equivalent to an employment subsidy and raise employment.
  - Jackman and Layard (1989): TIP could reduce NAIRU and increase welfare even if affecting workers’ effort.
- Implementation debate (Brookings Panel, 1978; Okun and Perry, 1978):
  - Design choices debated: tax base (price vs. wage increases), payer (employers vs. employees), coverage (all firms vs. large firms), continuity (continuous vs. discontinuous), penalty vs. reward, exemptions (overtime, bonuses), hurdles vs. continuous systems.
  - Dildine and Sunley (1978): favor taxing wage changes at large private corporates, penalty over reward, against exemptions, favor an auditable hurdle.
  - Seidman (1978a,b; 1979): favor continuous system with penalties and rewards.
  - Layard (1982), Jackman and Layard (1982): favor TIP on wage increases based on average hourly earnings of a firm.
- Alternative designs:
  - Lerner (1978) and Lerner and Colander (1980): market-based permit system for wage increases (permits issued and traded; price set by market).
- Contributions highlighted by the authors:
  - Leverage progress in sticky price models to reassess TIP effectiveness.
  - Find TIP and monetary policy (MP) are complementary: TIP should focus on markup and inflation expectation shocks; MP should focus on demand shocks.
  - TIP in steady-state increases welfare by decreasing markup and increasing employment, but optimal TIP should vary over time—rise when inflation increases and fall as inflation reaches long-term target.
  - Analyze robustness of TIP to alternative instruments (feebate, market for inflation permits), targeting rules for MP and TIP, and alternative tax bases (tax on wage increases or tax on large firms).
- Historical implementations and experiences:
  - France (1974–1977): "prélèvement conjoncturel" covering largest 1500 firms, representing 60% of the economy; based on excess increase in value-added in nominal terms relative to announced threshold with adjustments for fast-growing firms.
  - Other versions implemented in Mexico, Belgium, Italy (Paci, 1988) and the Netherlands (OECD, 1975).
  - U.S. near-implementation in 1978: Carter administration proposed "real wage insurance" tied to voluntary wage-price guidelines (7% nominal wage increase and 5.75% nominal price increase limits); program would give tax-credit to workers in groups whose earnings rose by less than 7% proportional to the difference between realized inflation and 7% (Colander, 1981). Reasons for failure: Congressional concern over federal-budget costs, association with price controls, and Volcker’s successful anti-inflation policy establishing MP dominance.
  - Transition economies (Bulgaria, Poland, Romania): TIP variants implemented; mixed evidence on inflation-reducing effects (Bogetic and Fox, 1993; Enev and Koford, 2000; Crombrugghe and de Walque, 2011).
- Summary insight:
  - TIP is implementable, effective, and not too costly to administer (Paci, 1988).

### 3 A Model with Sticky Prices and a Tax on Inflation
- Model setup:
  - New Keynesian framework with a tax on inflation (TIP).
  - Households: continuum of mass one, identical infinitely-lived. Preferences:
    - U(B_{t−1}^h) = max_{C_{th},N_{th},B_{th}} (C_{th}^{1−σ}/(1−σ) − N_{th}^{1+ψ}/(1+ψ) + β_t E_t U(B_t))
    - Variables: C_{th} consumption, N_{th} labor supply, B_{th} nominal wealth. Parameters: β∈(0,1), σ, ψ.
  - Household budget constraint:
    - P_t C_{th} + Q_t B_{th} = B_{t−1}^h + W_{th} N_{th} + T_t
    - Q_t is price of one-period bonds, W_{th} nominal wages, T_t transfers and firms’ profits.
  - Solvency condition: lim_{T→+∞} Π_{j=0}^T Q_j B_{Th} ≥ 0.
  - Labor supply takes firms’ demand into account:
    - N_{th} = (W_{th}/W_t)^{−ε_N} N_t.
  - First-order conditions:
    - W_{th}/P_t = M_{w,t} C_{th}^σ N_{th}^ψ, where M_{w,t} = ε_{Nt}/(ε_{Nt}−1) is time-varying wage markup (worker-side distributional conflict shock).
    - Q_t = E_t[β_t (C_{t+1}^h/C_{th})^{−σ} P_t/P_{t+1}] (Euler).
- Final goods firms:
  - CES aggregator with elasticity ε_t:
    - Y_t = [∫_0^1 Y_{ti}^{1−1/ε_t} di]^{ε_t/(ε_t−1)}
  - Demand for variety i: Y_{ti} = (P_{ti}/P_t)^{−ε_t} Y_t.
  - Price index: P_t = [∫_0^1 P_{ti}^{1−ε_t} di]^{1/(1−ε_t)}.
- Intermediate goods monopolistic firms:
  - Production technology with decreasing returns:
    - Y_{ti} = A_t N_{ti}^{1−α}, A_t common TFP, 1−α elasticity of output to labor.
  - Firms face Rotemberg (1982) quadratic price-adjustment costs:
    - C_t(P_{t−1,i},P_{ti}) = (θ/2) [(P_{ti}/P_{t−1,i} − 1)^2] P_t Y_t.
  - Novelty: firms pay a tax proportional to price increases τ_t (P_{ti} − P_{t−1,i}) per unit sold, so total tax scales with output Y_{ti}.
  - Net profits:
    - Π(P_{t−1,i},P_{ti}) = P_{ti} Y_{ti} − W_t N_{ti} − τ_t (P_{ti} − P_{t−1,i}) Y_{ti} − C_t(P_{t−1,i},P_{ti}).
  - Firms maximize discounted profits V(P_{t−1,i}) = max_{P_{ti}} Π + E_f [Q_t V(P_{ti})].
  - Define inflation π_t = P_t/P_{t−1} − 1. Marginal cost MC_t = W_t/P_t (1−α) Y_t^{α/(1−α)} A_t^{−1/(1−α)}. Ideal markup M_t = ε_t/(ε_t−1).
  - First-order condition (aggregate identical firms) before expectation shock specification:
    - (ε_t − 1)(M_t/MC_t − 1) + E_f [Q_t Y_{t+1}/Y_t (τ_{t+1} + θ(π_{t+1}+1)π_{t+1})] = τ_t [(1−ε_t) π_t/(1+π_t)] + θ π_t (π_t + 1).
- Expectations specification:
  - Firms’ expectations E_f may deviate from rational expectations E. Inflation expectation shock u^E_t defined so that for any g:
    - E_f_t[g(π_{t+1})] = E_t[g(π_{t+1} + u^E_t)].
  - First-order condition rewritten with u^E_t:
    - (ε_t − 1)(M_t/MC_t − 1) + E_t[Q_t Y_{t+1}/Y_t (τ_{t+1} + θ(π_{t+1}+1)π_{t+1})] + E_t[Q_t Y_{t+1}/Y_t θ u^E_t (2 π_{t+1} + 1)] = τ_t [(1−ε_t) π_t/(1+π_t)] + θ π_t (π_t + 1).

### 3.3.3 The Mechanics of TIP at the Micro-level
- Linearized first-order condition around zero-inflation steady state:
  - (ε − 1) ˆmc_t + u_t + θ u^E_t + β E_t[τ_{t+1} + θ π_{t+1}] = τ_t + θ π_t
  - Where u_t is the firm’s markup shock.
- Simple illustrative case:
  - Suppose τ_0 > 0 and τ_t = 0 for t > 0; π_{t+1} = 0; ˆmc_t = 0.
  - Then p_0 = p_{−1} + (u_0 + θ u^E_0 − τ_0)/θ.
  - Interpretation:
    - A higher TIP τ_0 induces firms to lower their log price p_0 (incentive to moderate price increases).
    - TIP attenuates pass-through of positive markup shocks u_0 and inflation expectation shocks u^E_0.
    - If u_0 = u^E_0 = 0 and τ_0 > 0, TIP leads firms to lower prices to receive a subsidy, possibly leading to deflation.
    - The smaller the adjustment cost θ, the stronger the disinflationary effect of a given TIP τ_0.

### 3.4 Equilibrium and Linearized System
- Definitions:
  - Output gap deviation: ˆy^e_t = log(y_t / y^e_t), where y^e is efficient flexible-price equilibrium output (θ = 0, no markup or inflation expectation shocks).
  - Shocks in the sticky-price economy: u_t (firm markup), u^w_t (wage markup), u^E_t (inflation expectation).
- Lemma 1 — linearized equilibrium equations:
  - Euler equation (log-linear):
    - ˆy^e_t = E_t ˆy^e_{t+1} − (1/σ) (i_t − E_t π_{t+1} − r^e_t)
  - Phillips curve (log-linear):
    - π_t = β E_t π_{t+1} + κ ˆy^e_t + (1/θ) [β E_t τ_{t+1} − τ_t + ũ_t]
  - Definitions:
    - κ = (ε − 1)/θ [σ + (ψ + α)/(1 − α)]
    - r^e_t = − log β_t + σ E_t[ (1+ψ)/( (1−α)σ + ψ + α) (a_{t+1} − a_t ) ]
    - ũ_t = u_t + u^w_t + θ u^E_t
- Robustness:
  - Appendix C shows Calvo-type frictions deliver the same first-order approximation and macro dynamics; optimal TIP results are robust to using Calvo friction.

### 4 Optimal Tax on Inflation and First-Best Allocation (preview of results)
- Objective:
  - Characterize MP and TIP combination that implements the first-best allocation (no inflation and no output gap).
- Key qualitative result:
  - TIP should increase with markup shocks and inflation expectation shocks because these shocks introduce externalities in firms’ pricing decisions.
  - TIP realigns private incentives with social valuations and corrects excessive aggregate inflation.

*Source: Excerpt from the provided PDF content unit wpiea2023254-print-pdf.*

### 4.1    First Best: Complete Stabilization and Specialization

### 4.1    First Best: Complete Stabilization and Specialization

### Social Planner’s First Best Allocation
- The social planner maximizes household utility subject to resource constraints, technology, and relevant conditions. The optimal condition equates the marginal rate of substitution between labor and consumption to the marginal product of labor:
  - (C^e_t)^σ (N^e_t)^ψ = (1−α) A_t (N^e_t)^−α.
- Substituting N^e_t from the technology yields optimal consumption and output:
  - C^e_t = (1−α)^(1−α) (1−α)^{−((1−α)σ+ψ+α)} A^{1+ψ}_{t}^{(1−α)σ+ψ+α}
  - Y^e_t = C^e_t. (Equation (18))

### Externalities of Price Changes and Limitations of Monetary Policy
- Monetary policy (MP) that tracks the neutral rate i_t = (Q^e_t)^{−1} − 1 with
  - Q^e_t = E_t [ β^t (A_{t+1}/A_t)^{−σ(1+ψ)/((1−α)σ+ψ+α)} ] (Equation (19))
  - can implement the first best when there are no markup or inflation expectation shocks: it closes the output gap Y_t = Y^e_t and maintains price stability P_t = P_{t−1}.
- Distributional conflict and inflation-expectation shocks create a wedge between private and social returns to price increases; monetary policy alone cannot correct this externality.
- The private–social wedge (setting N_t and C_t equal to first-best and π_{t+1}=0) is:
  - Private − Social = (ε_t − 1) (M_t / M^w_{t−1}) + E_t Q^e_t (A_{t+1}/A_t)^{(1+ψ)/((1−α)σ+ψ+α)} θ u^E_t. (Equation (20))
- The wedge is positive for M_t > 1, M^w_t > 1, or u^E_t > 0; positive expected future inflation π_{t+1} > 0 amplifies current inflation.
- Conclusion: MP alone cannot implement the first best or fully stabilize inflation and the output gap when these externalities exist; an instrument that directly affects firms’ price-setting is required.

### First-Best Implementation with TIP (Tax on Inflationary Price changes)
- TIP gives direct incentives to moderate price increases, realigning private and social valuations of price changes.
- Using the Euler equation and the Phillips curve, necessary conditions on MP and TIP for decentralized equilibrium to coincide with the first best (no inflation, no output gap) are derived:
  - Proposition 1 (First Best Policies): For any paths of distributional conflict M_t, M^w_t and inflation expectation u^E_t, aggregate demand β_t and TFP shocks A_t, there exists a combination of TIP and MP that can implement the first best.
  - Nominal policy interest rate: given by (19).
  - TIP is given by:
    - τ_t = (ε_t − 1) (M_t / M^w_{t−1}) + E_t Q^e_t (A_{t+1}/A_t)^{(1+ψ)/((1−α)σ+ψ+α)} [ τ_{t+1} + θ u^E_t ]. (Equation (21))
- Key implications from Proposition 1:
  - With TIP and MP, policymakers can completely stabilize the output gap and inflation—contrast with MP-only setting.
  - Specialization: MP should focus on shocks that affect the neutral rate of interest (aggregate demand β_t and TFP shocks A_{t+1}/A_t); TIP should specialize in markup shocks M_t, M^w_t and inflation expectation shocks u^E_t.
  - TIP is most effective when expected to decrease in the future: temporarily raise TIP when shocks hit and announce a later decrease to incentivize firms to postpone price increases. Permanence of the policy would weaken incentives.

### First-best in the Linearized Model; Corollary and Example
- Corollary 1 (First Best Policies in Linearized Model):
  - Complete stabilization of the output gap and inflation: ŷ^e_t = π_t = 0.
  - Complete specialization of MP and TIP:
    - i_t = r^e_t
    - τ_t = β E_t τ_{t+1} + u_t + u^w_t + θ u^E_t
- For AR(1) shocks u_t, u^w_t, u^E_t with identical autoregressive coefficient ρ_u < 1 and disturbances mean zero:
  - Example 1: Optimal TIP:
    - τ_t = (u_t + u^w_t + θ u^E_t) / (1 − β ρ_u). (Equation (22))
  - The optimal TIP increases with the current level (u_t + u^w_t + θ u^E_t) and with persistence ρ_u. If shocks are i.i.d., τ_t = u_t + u^w_t + θ u^E_t.

- Determinacy and uniqueness:
  - The set of linearized policies may admit multiple equilibria; to ensure uniqueness, require central bank reaction to inflation with φ_π > 1 in i_t = r^e_t + φ_π π_t, which yields π_t = 0 and i_t = r^e_t.

### Remarks
- Efficient Steady-state:
  - Market power of firms and workers introduces wedges that lower output and consumption in steady state.
  - TIP can correct the steady-state distortion as an alternative to production/payroll subsidies, with:
    - τ = (ε − 1) / (1 − β) (M M^w − 1). (Equation (23))
  - Mechanism: permanent increase in TIP incentivizes firms to lower prices and expand production, raising real wages and output until the new steady state equals the efficient outcome.

- Time-consistency:
  - Combination of MP and TIP is time-consistent: best discretionary MP and TIP coincide with best policies under commitment and deliver full stabilization—discretion suffices because TIP eliminates the short-run output–inflation trade-off.

- Policy Coordination:
  - Although solved jointly, coordination is not required: the policies in Proposition 1 constitute a Nash equilibrium between separate authorities controlling MP and TIP (assuming shared welfare objective).

- Second-best and TFP Shocks:
  - TIP has no role when inflation stems purely from TFP shocks under the base model. However, with additional frictions (labor market imperfections, financial stability constraints) TIP could supplement MP even for TFP and demand shocks—for example, when the central bank is constrained from raising interest rates sufficiently due to financial stability concerns.

*Italic source attribution: 4.1 First Best: Complete Stabilization and Specialization — wpiea2023254-print-pdf*

### 6.2    Determinacy with TIP

### 6.2    Determinacy with TIP

### Determinacy and the Taylor principle
- In the baseline New Keynesian model, monetary policy (MP) must implement the Taylor principle to ensure uniqueness of the equilibrium path.
- Without determinacy, the economy is subject to coordination failures: e.g., if all firms expect high inflation and high output gap, the economy could shift to a self-fulfilling equilibrium with excessive inflation. These coordination failures are distinct from inflation expectations shocks, u^E_t, and are consistent with rational expectations.
- Theoretical possibility: Appendix A.4 shows determinacy can be obtained if TIP reacts strongly to the output gap while MP reacts passively to inflation.
- Practical conclusion: in the calibrated model, the necessary strength of TIP’s reaction to guarantee determinacy is beyond what could be realistically implemented. Therefore, the Taylor principle remains the only practical way to ensure determinacy.

### Key implication
- Taylor principle remains necessary in practice to avoid indeterminate, self-fulfilling equilibria despite theoretical conditions under which TIP could substitute.

---

### 6.3    Stabilization Properties of TIP

### Approach and baseline calibration
- Impulse response functions (IRFs) and welfare implications are examined for markup, monetary, and productivity shocks.
- Reported variables: annualized inflation, output gap, annualized policy rate, TIP rate τ_t, and TIP as percent of total sales (tax normalized by sales).
- Stochastic simulations evaluate standard deviations of inflation, output gap, policy rate, TIP, and total welfare losses under one shock type at a time.
- Baseline monetary policy: standard Taylor rule (31) with φ_π = 1.5 and φ_y = 0.125. TIP only targets inflation (φ_y = 0) unless otherwise mentioned.

### 6.3.1 Markups and Inflation Expectation Shocks — IRFs and mechanisms
- Shock analyzed: 0.25pp distributional or expectation shock (ũ_t^θ = 0.25pp). Inflation and policy rate are annualized.
- Without TIP: on impact inflation initially rises to 1.3% and output drops by slightly over 0.51%.
- Moderate TIP with φ_π = 150:
  - Initial inflation response falls to 1.0%.
  - Decline in output to -0.40%.
  - Imposes a 38% tax on price changes in the first quarter.
- Strong TIP with φ_π = 300:
  - Initial inflation response falls to 0.84%.
  - Output gap to -0.34%.
  - Imposes a 63% tax on price changes.
- Mechanism: TIP’s impact on inflation reduces the need to raise the policy rate, mitigating negative output effects.
  - Baseline (no TIP): policy rate increases by 170 basis points on impact.
  - Strong TIP (φ_π = 300): policy rate increase reduced to 110 basis points.
- Tax burden on firms as percent of sales is small because tax applies only to price changes:
  - Strong TIP tax burden represents 0.13% of sales in the first quarter (or 0.04% in the first year).
- Nonlinearities in tax burden:
  - τ_t π_t/(1+π_t) ≈ φ_π π_t^2 and E π_t τ_t = φ_π V(π_t).
  - As φ_π quadruples from 150 to 600, both π_t^2 and V(π_t) decrease, yielding a small increase in taxes relative to sales.
- Profits:
  - When inflation driven by firms’ desire to increase markups, firms obtain higher profits as share of output; TIP moderates this increase.
  - When inflation driven by wage markups, profits decrease; TIP further worsens firms’ profitability.
  - When inflation driven by inflation expectation shocks u^E_t, firms expect ex ante profit share to fall (like wage markup shock) but ex post profits actually rise (like firm markup shock), reflecting firms’ expectations not fully rational.

### Sensitivity to price flexibility
- TIP’s effectiveness increases in the degree of price flexibility (lower Rotemberg parameter θ). Lower θ ⇒ firms find it less costly to adjust prices ⇒ TIP is a relatively stronger obstacle to price changes.

### Stochastic simulations: markup and inflation expectation shocks (Table 2)
- Shock standard deviation: u_t^θ = 0.25pp.
- Baseline Taylor rule parameters reported alongside TIP strength scenarios: No TIP, Moderate TIP, Strong TIP, Extreme TIP with φ_π = 0, 150, 300, 600 respectively; φ_π (MP) = 1.5; φ_y = 0.125.
- Selected results (all standard deviations and E(π_t τ_t) multiplied by 100; L^* defined using quarterly variables):
  - σ(π_ann_t): 0.74, 0.59, 0.49, 0.36 for φ_π = 0,150,300,600.
  - σ(ŷ^e_t): 0.59, 0.47, 0.39, 0.29.
  - σ(i_ann_t): 0.96, 0.76, 0.63, 0.47.
  - σ(τ_t): 0, 44.1, 73.0, 108.8.
  - E(π_t τ_t): 0, 0.13, 0.18, 0.20.
  - L^* × 10^4: 1.22, 0.77, 0.53, 0.29.
- Welfare losses (L) shrink by 47% from 7.91 (no TIP, reported in text) to 4.38 with φ_π = 300; consistent reductions observed across many Taylor rules.

---

### 6.3.2    Aggregate Demand, Productivity and Monetary Shocks

### IRFs and trade-offs
- For aggregate demand or productivity shocks sized to match baseline inflation response, TIP mitigates the inflationary effect to a similar extent as for markup/inflation expectation shocks.
- With strong TIP (φ_π = 300):
  - Inflation is 0.34pp lower.
  - Hike in the policy rate is about 57 bps smaller.
  - However, TIP amplifies the initial widening of the output gap by 0.18pp relative to no TIP — indicating a trade-off between inflation and output under these shocks.
- Interpretation:
  - Aggregate demand and productivity shocks ideally addressed by monetary policy.
  - When both TIP and MP follow targeting rules, TIP can complement MP in stabilizing inflation but faces a trade-off.

### Stochastic simulations (Table F1 referenced)
- Increasing φ_π to 300 lowers variance of inflation from 2.52 to 1.87 but raises variance of the output gap from 0.94 to 1.27.
- Because inflation is more volatile than output in baseline without TIP, the reduction in inflation variance outweighs the increase in output variance, resulting in higher welfare (lower L).

---

### 6.4    Optimal Monetary Policy with TIP

### Question
- How should the introduction of TIP change the design of the Taylor rule? Should monetary policy specialize and emphasize the output gap?

### Method
- Simulate many economies differing in the four weights in the rules (φ_π, φ_y, φ_π^TIP, φ_y^TIP). Focus reported results on TIP rules with φ_y = 0.
- Economies stochastically hit by demand and markup shocks with standard deviation equal to .25.

### Main results (Table 3)
- Specialization is desirable: monetary policy should focus relatively more on output and less on inflation when TIP is present.
- Reported scenarios: No TIP, Moderate TIP, Strong TIP, Extreme TIP with φ_π^TIP = 0,150,300,600 and baseline φ_π = 1.5.
- Optimal monetary policy weights (φ^*_π, φ^*_y) minimizing L^* conditional on TIP:
  - No TIP: φ^*_π = 1.500, φ^*_y = 0.125.
  - Moderate TIP: φ^*_π = 0.990, φ^*_y = 0.126.
  - Strong TIP: φ^*_π = 0.983, φ^*_y = 0.162.
  - Extreme TIP: φ^*_π = 0.980, φ^*_y = 0.134.
- Outcome metrics:
  - σ(π_ann_t): 0.77, 0.71, 0.58, 0.41.
  - σ(ŷ^e_t): 0.66, 0.47, 0.43, 0.44.
  - σ(i_ann_t): 1.03, 0.67, 0.55, 0.41.
  - σ(τ_t): 0, 53.3, 86.3, 124.0.
  - E(π_t τ_t): 0, 0.19, 0.25, 0.26.
  - L^* × 10^4: 1.38, 0.84, 0.56, 0.31.

### Interpretation and robustness
- Monetary policy should shift weight from inflation to output because:
  - Interest rate directly affects aggregate demand/output; TIP directly affects firms’ price change incentives and thus inflation.
- Specialization becomes more desirable as variance of markup shocks increases (φ_π decreases further and φ_y rises when markup shock variance doubles).
- φ_y in TIP need not increase monotonically with φ_π^TIP; e.g., going from strong to extreme TIP can slightly lower optimal φ_y due to lower cost of i_t^2 in the loss function.

### On φ_y (weight on the output gap in TIP)
- For markup and inflation expectation shocks, unreported simulations indicate TIP should react strongly and positively to inflation and strongly but negatively to the output gap; but φ_y < 0 is unlikely in practice, so restrict to φ_y ≥ 0 and set φ_y = 0 as reasonable.
- For aggregate demand or productivity shocks, a TIP that reacts strongly to inflation and weakly to the output gap can be designed to be equivalent to one that reacts strongly to the output gap and weakly to inflation; thus TIP with φ_y = 0 is feasible when reaction to inflation is sufficiently strong.

---

### 7    TIP and Relative Price Distortions

### Concern
- TIP could impede adjustment of relative prices across sectors, potentially causing misallocation of resources.

### Extended model with multiple sectors (7.1)
- Economy: continuum of sectors s ∈ [0,1]; sector weights γ_s with ∫_0^1 γ_s ds = 1; final goods aggregator Y_t = exp(∫_0^1 γ_s ln Y_ts ds).
- Within-sector CES combining varieties with elasticity ε_t.
- Sector-specific TFP shocks: production for variety i in sector s: Y_tis = A_ts N_tis^(1−α).
- Define log relative productivity ã_st = log(A_st/A_t) and log relative prices p̃_st = p_st − p_t.
- Firms face sector-specific quadratic adjustment costs; sector-specific price stickiness parameter θ_s. Relative price distortions from nominal frictions reduce welfare.
- Welfare second-order approximation (Lemma 2):
  - L = Σ_{t=0}^∞ β^t [ (σ + (α+ψ)/(1−α)) (ŷ^e_t)^2 + E[θ_s π^2_st] + E[ (p̃^e_st)^2 ]/(1−α) ]
  - E denotes cross-sectional mean under density γ_s.

### Analytical independence result (7.2, Proposition 5)
- Assumptions: θ_s common across sectors (θ_s = θ) and steady-state TIP is zero.
- Proposition 5 conclusions:
  - Relative prices across sectors are independent of TIP. They depend only on θ, other parameters (ε, α, β), and sector-specific productivity shocks ã_st.
  - Responses of the output gap and aggregate inflation π_t to aggregate shocks and TIP are the same as in the linearized economy of Section 4. Output gap and inflation π_t are sufficient statistics to evaluate welfare impacts of TIP.
- Intuition:
  - Linearity and symmetry of TIP imply offsetting forces across sectors: firms hit by negative TFP shocks moderate price increases due to TIP; firms that would not change prices lower prices to gain subsidy. When θ_s = θ and steady-state TIP = 0, relative prices remain exactly unchanged.
  - By contrast, price controls are convex and distort relative prices.
- Consequences:
  - Aggregate economy behaves like linearized model; corollary and results in Sections 6 and 8 remain quantitatively exactly the same.
  - Welfare varies with TIP only through output gap and average inflation because TIP does not affect cross-sector deviations of relative prices from efficient levels E[(p̃_st − p̃^e_st)^2], and it changes cost due to price adjustments θE[π^2_st] only through average effect θ π_t^2.

*Italic source: wpiea2023254-print-pdf - 6.2    Determinacy with TIP*

### 7.3    Independence Result: Numerical Simulations

### 7.3    Independence Result: Numerical Simulations

### Calibration and experiment design
- Common parameters across sectors follow the calibration in section 6 (Table 1).
- (log) TFP in each sector follows an AR(1) process: log(A_ts) = (1−ρ)logA_s + ρ log(A_{t−1,s}) + ν_st, with ν_st i.i.d. across time and sectors.
- Standard deviation of ν_st set to .05 and persistence ρ set to .5.
- Price-stickiness heterogeneity θ_s calibrated following Nakamura and Steinsson (2008): industries grouped into durable, non-durable and services; average duration of regular prices computed and Rotemberg parameter θ_s computed using the empirical average duration and equation (34).
- All sectors in a given category share the same Rotemberg parameter θ_s; sectors split into three segments proportional to expenditure weights from Table II in Nakamura and Steinsson (2008).
- Main exercise: simulate an unexpected one-period markup shock in the fully non-linear model.
  - Starting from a steady-state with no TIP, elasticity of substitution decreases to ε_0 = 7.5 at t=0 and goes back to its steady-state value ε_t = 9 for t>0.
  - Compare baseline scenario where TIP doesn’t respond (τ_0 = 0) to two alternative scenarios where TIP responds moderately (τ_0 = 10%) and more strongly (τ_0 = 15%).
  - Table F5 reports the values of the three terms in the loss function (35) at time t = 0.

### Numerical results and interpretation
- Effect on relative-price distortions:
  - The average distortion of relative prices across sectors, E[(p̃_st − p̃^e_st)^2], remains broadly unaffected by the size of TIP and if anything decreases very slightly with TIP.
  - This confirms Proposition 5: because TIP is a linear tax, TIP does not exacerbate price distortions across sectors, in contrast with price controls.
- Effect on inflation, price changes, output gap, and welfare:
  - Higher TIP leads to lower aggregate inflation.
  - Higher TIP leads to lower average squared price changes, E[θ_s π^2_st].
  - Higher TIP leads to a lower output gap.
  - Lower output gap and lower inflation unambiguously improve welfare in these simulations.
- Intuition for possible decrease in average distortion:
  - TIP influences the pricing decision through two channels in the term τ_t[(1−ε_t) π_t/(1+π_t)] in optimal pricing condition (12): a linear first-order channel (raising one’s price increases taxes owed) and a second-order channel (increasing price reduces firm demand which lowers tax bill).
  - The second-order channel is proportional to the firm’s price increase, giving firms that want to increase price more a stronger reduction in tax bill as demand falls—this can facilitate relative price adjustments and lower price distortions.

### Robustness exercises
- Homogeneous stickiness:
  - First robustness exercise sets all sectors to the same adjustment cost parameter θ = 372.8 (as in Table 1).
  - Results: similar to main exercise—TIP has no effect on relative price distortions and improves welfare by reducing inflation and the output gap.
- Different steady-state TIP:
  - Second robustness exercise sets steady-state TIP τ̄ = 15%.
  - Results: similar qualitative outcomes—TIP reduces inflation and output-gap volatility and does not worsen relative-price distortions.

### Extensions: taxing only large corporates (section 8.1)
- Model extension: two types of firms—small firms (S) don’t pay TIP and large firms (L) pay TIP; full equilibrium described in Appendix D.
- Calibration: mass of large firms set to match top 1% of firms accounting for 75% of all sales in the U.S. (Crouzet and Mehrotra, 2020).
- Findings:
  - TIP is almost as effective when applied only to 1% of firms as when it covers all firms.
  - Example: with φ_π = 300, initial inflation response falls from 1.7% to 1.3% (compared to 1.2% when TIP covers all firms); output gap falls to −.65 (compared to −.6% when TIP covers all firms).
  - Effectiveness stems from both coverage of 75% of sales by the top 1% and a slightly higher effective tax incidence: 98% of price changes taxed versus 93% in baseline, driven by a slightly higher inflation rate in the two-tier calibration.

### Extensions: sticky wages and tax on wage increases (section 8.2)
- Model extension: households face Rotemberg-type adjustment costs when changing nominal wages; full equations in Appendix E.
- Calibration: elasticity of substitution across labor types ε_N = 4.5; wage adjustment cost θ_w set to match slope of wage Phillips curve implied by Calvo parameter for wages of 0.75.
- TIP on prices when wages are sticky:
  - TIP on prices remains as effective as in the baseline case.
  - Example: with φ_π = 300 following a shock to the price Phillips curve, initial inflation response falls from 2.5% to 1.6% and the output gap from −1.1% to −.65%.
  - TIP on prices controls wage inflation only when the shock is to the price Phillips curve; ineffective at controlling wage inflation when the shock is to the wage Phillips curve because the price-to-wage feedback is small relative to the direct shock on wages.
- TIP on wages:
  - Effectiveness depends on the source of the shock.
  - When shock is in the wage Phillips curve, TIP on wages with φ_π = 300 is as effective as TIP on prices: initial inflation response falls from .94% to .75% and output gap from −.65% to −.55%.
  - Mechanism: TIP on wages offsets transmission of wage shock to wage inflation; lower wage inflation reduces price inflation because wages are the main component of firms’ costs.
  - When shock is in the price Phillips curve, TIP on wages is ineffective at controlling price inflation or reducing the output gap—wage-price spiral is weak relative to direct price shock.
  - Sensitivity: the weak wage-price spiral depends on calibration of elasticity of substitution across labor types ε_L; relatively low ε_L implies a high wage markup and a slow response of wages.

### Key implications for policy and interpretation
- Numerical simulations corroborate Proposition 5: TIP as a linear tax does not worsen relative-price distortions across sectors and can reduce aggregate inflation and output-gap volatility.
- TIP complements monetary policy (MP): combining TIP with MP can significantly lower volatility of inflation and output relative to MP alone.
- TIP should specialize in addressing markups and inflation-expectation shocks; MP should specialize in addressing demand shocks.
- Robustness: results hold under heterogeneous and homogeneous price stickiness, when TIP covers only very large firms, and when wages are sticky.
- Implementation considerations highlighted for further research: taxing wages vs. taxing prices, tax avoidance risks (relabeling products, quality shrinking), monitoring at firm-level, and political-economy risks regarding TIP governance and independence.

*Source: 7.3 Independence Result: Numerical Simulations (from wpiea2023254-print-pdf)*

### References

### References

### Bibliographic references (selected)
- Ascari, G. and Sbordone, A. M. (2014). The Macroeconomics of Trend Inflation. Journal of Economic Literature, 52(3):679–739.
- Calvo, G. A. (1983). Staggered Prices in a Utility-maximizing Framework. Journal of Monetary Economics, 12(3):383–398.
- Clarida, R., Galí, J., and Gertler, M. (1999). The Science of Monetary Policy: A New Keynesian Perspective. Journal of Economic Literature, 37(4):1661–1707.
- Correia, I., Farhi, E., Nicolini, J. P., and Teles, P. (2013). Unconventional Fiscal Policy at the Zero Bound. American Economic Review, 103(4):1172–1211.
- Eggertsson, G. B. (2011). What Fiscal Policy Is Effective at Zero Interest Rates? In NBER Macroeconomics Annual 2010, volume 25, NBER Chapters, pages 59–112. National Bureau of Economic Research, Inc.
- Galí, J. (2015). Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework and Its Applications Second edition. Number 10495 in Economics Books. Princeton University Press.
- Giannoni, M. P. (2014). Optimal Interest-Rate Rules and Inflation Stabilization versus Price-Level Stabilization. Journal of Economic Dynamics & Control, 41:110–129.
- Nakamura, E. and Steinsson, J. (2008). Five Facts about Prices: A Reevaluation of Menu Cost Models. The Quarterly Journal of Economics, 123(4):1415–1464.
- Werning, I. and Lorenzoni, G. (2023). Inflation is conflict. Technical report.
- Woodford, M. (2011). Simple Analytics of the Government Expenditure Multiplier. American Economic Journal: Macroeconomics, 3(1):1–35.
(Note: the full references list appears in the source document.)

### Appendix A — Baseline Model with TIP: main elements and equations
- Market clearing conditions:
  - Final goods market: Y_t = C_t + Z_0^1 θ/2 ( (P_ti/P_{t-1,i}) − 1 )^2 Y_t di. (Equation (36) form)
  - Labor market: ( ∫_0^N (N_th)^{1−1/σ} dh )^{σ/(σ−1)} = N_t = ∫_0^1 N_ti di. (Equation (37) form)
  - No government debt: B_t = 0.
  - Transfers to households: T_t = ∫_0^1 Π_it + τ_t (P_ti − P_{t-1,i}) Y_ti di. (Equation (38) form)

- Equilibrium definition:
  - Equilibrium is a path {C_th, N_th, B_th, W_th}_{h},{Y_ti, N_ti, P_ti}_{i}, W_t, P_t, Q_t, τ_t such that:
    - Intermediate firms maximize discounted profits (taking TIP as given) subject to profit definition, technology and demand.
    - Final good firms maximize profits (taking prices as given).
    - Households maximize utility (taking prices and transfers as given) subject to budget constraint and no-Ponzi.
    - Markets for final goods, intermediate goods, labor and bonds clear.

### Linearization and key reduced-form relations
- Firms’ first-order conditions lead, under symmetry and linearization around zero inflation steady state (π = 0, τ = 0, Y = Y′, MC_t = 1/ M̄), to a Phillips-curve–type condition:
  - π_t = β E_t π_{t+1} + (ε−1)/θ · ĥmc_t + 1/θ [ β E_t τ_{t+1} − τ_t + u_t ]
    - where ĥmc_t = mc_t − mc is the gap between effective marginal cost and desired marginal cost under flexible prices, and u_t = (ε−1)(μ_t − μ) is the markup shock.
  - κ = (ε−1)/θ · ( σ + (ψ+α)/(1−α) ).

- Household first-order conditions and market clearing yield:
  - w_t − p_t = ( σ + ψ/(1−α) ) y_t − ψ/(1−α) log a_t
  - Real marginal cost: mc_t = ( σ + (ψ+α)/(1−α) ) y_t − (1+ψ)/(1−α) a_t − log(1−α)

- Euler equation and output gap relation:
  - ˆy^e_t = E_t ˆy^e_{t+1} − 1/σ ( i_t − E_t π_{t+1} − r^e_t )
  - Phillips curve restated: π_t = β E_t π_{t+1} + κ ˆy^e_t + 1/θ [ β E_t τ_{t+1} − τ_t + u_t ]

- Definitions:
  - Natural rate and neutral interest rate equal their flexible-price values:
    - y^n_t = mc^n_t + log(1−α)·[ 1/( σ + (ψ+α)/(1−α) ) ] + (1+ψ)/(1−α)·[ 1/( σ + (ψ+α)/(1−α) ) ] a_t
    - r^n_t = ρ + σ E_t ( y^n_{t+1} − y^n_t )

### Firms’ profits and profit-share linearization
- Real profits per unit of price level:
  - Π(P_{t-1},P_t)/P_t = Y_t − W_t/P_t · N_t − τ_t π_t/(1+π_t) Y_t − θ/2 π_t^2 Y_t
- Steady-state profit share:
  - Π_PY = 1 − ( ε_N / (ε_N − 1) ) · ( ε−1 )/ε · (1−α)
- Log-linearized deviation of the profit share:
  - Π_t/(P_t Y_t) − Π_PY ≈ − [ ( ε_N / (ε_N − 1) ) · ( ε−1 )/ε · (1−α) ] [ (σ−1 + (ψ+1)/(1−α)) ˆy^e_t + u_{w,t}/(ε−1) ] − τ_t π_t − θ/2 π_t^2

### Fiscal cost of wage subsidies versus TIP (numerical results)
- To match the impulse responses of inflation and output gap from a strong TIP (φ_π = 300), the model implies:
  - Wage subsidies would amount to 4% of total payrolls (or 3.5% of output) in the first period following a markup shock.
  - A strong TIP costs below 0.15% of output in the same period.
- Persistence and scaling:
  - Fiscal cost of wage subsidies (proportional to π_t) decreases much slower than the cost of TIP (proportional to π_t^2).
  - As instrument strength increases, wage subsidy fiscal costs grow linearly, while TIP fiscal costs increase at a decreasing speed — because a stronger TIP moderates price changes and thus reduces its own tax base.

### Determinacy and Proposition 6 (uniqueness conditions)
- Substituting the monetary policy (Taylor) rule and TIP rule into the Euler and Phillips equations yields a two-equation linear system in π_t and ˆy^e_t. Determinacy requires both eigenvalues inside the unit circle (Blanchard and Kahn, 1980).
- Proposition 6. The equilibrium path is unique if one of the following conditions holds:
  - φ_π > 1 and φ_y < min [ θ κ /(1−β) , θ φ_π /( κ + θ(1−β)( σ(1−β)+φ_y ) ) · ( (1+φ_π/θ) φ_π − β )^{-1} ]
  - β < φ_π < 1 and θ κ /(1−β) < φ_y < θ φ_π /( κ + θ(1−β)( σ(1−β)+φ_y ) ) · ( (1+φ_π/θ) φ_π − β )^{-1}
  - φ_π < β and θ κ /(1−β) < φ_y

- The appendix derives the matrix representation, computes A^{-1}B, and provides expressions for the trace and determinant of the system matrix to assess the eigenvalues and thus determinacy.

*Italic: Source: wpiea2023254-print-pdf - References (extracted content).*

### 0. These restrictions are consistent with any empirically reasonable parametriza-

### wpiea2023254-print-pdf - 0. These restrictions are consistent with any empirically reasonable parametriza-

### Determinacy conditions for the 2-variable system
- The system with two non predetermined variables is determinate iff both eigenvalues are within the unit circle (Blanchard and Kahn, 1980).
- Two necessary and sufficient conditions for eigenvalues within the unit circle:
  - det A′ < 1, which leads to the condition
    - φ_y θ (φ_π − β) < φ_π κ + (σ(1 − β) + φ_y) [1 + φ_π/θ]
  - Tr A′ < 1 + det A′, which leads to the more complex inequality reported in the text (multiple terms involving φ_y, φ_π, κ, σ, β, θ).
- If φ_π < β and φ_y ≥ 0, the det A′ < 1 condition is always true.
- If φ_π > β, det A′ < 1 requires φ_y to be sufficiently small (inequality re-stated in source).

### Interpretation and policy-relevant implications
- Traditional Taylor principle case:
  - Reaction of monetary policy to inflation is strong enough that the path of inflation is always unique.
  - Under this case, the reaction of TIP to the output gap cannot be too strong.
- New principle cases (second and third cases):
  - Monetary policy reaction to inflation is weak but the reaction of TIP to the output gap is strong.
- Intuitive mechanism for high φ_y ensuring determinacy:
  - If agents expect high inflation and high output gap, a strong TIP response (high φ_y) leads policymakers to set a very high tax on inflation, producing deflation and recession that contradict initial expectations, restoring uniqueness of equilibrium.
- Practical calibration constraint:
  - For the calibration used in the quantitative section, the φ_y necessary to ensure determinacy is θκ/(1−β), which is slightly above 6300.
  - This φ_y implies a reaction function that is too strong to be realistically implemented.
  - Therefore, the remainder of the paper restricts attention to targeting rules where the Taylor principle applies and the reaction of TIP to the output gap is not too strong.

### Full system used for simulations (system of equations)
- Key equations used (as given in source; variables preserved exactly):
  - ˆy^e_t = E_t ˆy^e_{t+1} − 1/σ [ρ + φ_π π_t + φ_y ˆy^e_t − E_t π_{t+1} − r^e_t]
  - [1 + φ_π/θ] π_t = β [1 + φ_π/θ] E_t π_{t+1} + [κ − φ_y/θ] ˆy^e_t + φ_y/θ β E_t ˆy^e_{t+1} + 1/θ u_t
  - u_t = ρ_u u_{t−1} + ε_{u,t}
  - a_t = ρ_a a_{t−1} + ε_{a,t}
  - r^e_t = ρ + σ E_t [ u_{t+1} − u_t (ε − 1) [σ + (ψ + α)/(1 − α)] + (1 + ψ)/(1 − α) [σ + (ψ + α)/(1 − α)] (a_{t+1} − a_t) ]
- These equations form the system solved in simulations.

### Distortion of relative prices — model setup and Phillips curve
- Model extension:
  - Continuum of sectors s ∈ [0, 1] and continuum of monopolistically competitive firms within each sector.
  - Sector good C_s aggregated into final good ln Y_t = ∫ γ_s ln C_{t s} ds with ∫ γ_s = 1.
  - Sector aggregators: Y_{t s} = [∫ y_{t i s}^{1−1/ε} di]^{ε/(ε−1)}.
  - Market clearing in each sector: Y_{t s} = C_{t s}; adjustment costs paid in final goods; aggregate goods market: Y_t = C_t + ∫ θ_s/2 π_{s t}^2 Y_{t s} ds.
- Firms’ FOCs and linearization:
  - Define MC_{s t}, markup M_{s t}, ˜P_{s t} = P_{s t}/P_t, steady-state markup ̄M.
  - Linearized sector Phillips curve (around zero inflation steady state):
    - π_{s t} = β E_t π_{s, t+1} + (ε − 1)/θ_s ˆmc_{s t} + 1/θ_s [β E_t τ_{t+1} − τ_t + u_t]
  - Relation between sector marginal cost gap and aggregate/relative terms:
    - ˆmc_{s t} = ˆmc_t + ψ ˆδ_t − [1/(1 − α)] ˆ˜p^e_{s t}, with ˆδ_t = 0 at first order.
  - Aggregating across sectors (γ_s-weighted) yields aggregate Phillips curve:
    - π_t = β E_t π_{t+1} + ˆmc_t ∫ (ε − 1)/θ_s γ_s ds − [1/(1 − α)] ∫ (ε − 1) γ_s θ_s ˆ˜p^e_{s t} ds + [u_t + β E_t τ_{t+1} − τ_t] ∫ γ_s θ_s ds

### Welfare second-order approximation and loss decomposition
- Household welfare loss (second-order) depends on:
  - Dispersion of relative prices around efficient levels.
  - Average inflation rates across sectors.
- Welfare loss expression (as given):
  - L = ∑_{t=0}^∞ β^t [ (σ + (α + ψ)/(1 − α)) (ˆy^e_{s t})^2 + E[θ_s π_{s t}^2] + E[(˜p_{s t} − ˜p^e_{s t})^2]/(1 − α) ]
- Second-order approximation steps and final welfare expression (preserving source algebra):
  - U_t − U = −1/2 E[θ_s π_{s t}^2] − 1/2 [σ + (α + ψ)/(1 − α)] (ˆy^e_t)^2 − E[(˜p_{s t} − ˜p^e_{s t})^2]/[2(1 − α)] + t.i.p.
  - The term E[θ_s π_{s t}^2] can be decomposed as:
    - E[θ_s π_{s t}^2] = π_t^2 E[θ_s] + E[θ_s ˜π_{s t}^2]
- Interpretation:
  - Welfare losses arise from aggregate inflation (π_t^2 E[θ_s]), sectoral dispersion of price changes (E[θ_s ˜π_{s t}^2]), and deviations of relative prices from efficient levels.

### Independence of relative-price dynamics from TIP under homogeneous θ_s
- Proposition (proof sketch as given):
  - If θ_s = θ for all s and elasticities ε are homogeneous across sectors, relative-price dynamics (and their distribution) at first order depend only on ε, α, β, θ and sector-specific productivity shocks A_{s t}, and are independent of TIP.
  - Derived relation for relative-price evolution:
    - ˜p_{s t} − ˜p_{s, t−1} = β E_t (˜p_{s, t+1} − ˜p_{s t}) − (ε − 1)/θ * 1/(1 − α) (˜p_{s t} + ˜a_{s t})
  - Hence, ˜p_{s t} dynamics are independent of aggregate inflation and TIP; therefore E[θ_s ˜π_{s t}^2] is independent of aggregates and of TIP, while E[θ_s π_{s t}^2] varies with TIP only through aggregate inflation π_t^2 E[θ_s].

### Numerical simulations — algorithm and steady-state computation
- Steady-state normalization: nominal wage W normalized to 1.
- Algorithm steps to solve steady-state and simulate a one-period markup shock:
  1. Start from a guess for steady-state output Y^∗ and price level P^∗.
  2. Solve for firms’ pricing rule ˜P_s(A_s, ˜P_{s,−1}) given aggregate output and price index using the sector FOC (equation provided).
     - π_s = log(˜P_s) − log(˜P_{s,−1}) + log(P^∗) − log(P^∗_{−1})
     - MC_s = W ˜P_s / P^∗ (1 − α) Y_s^{α/(1−α)} A_s^{1/(1−α)}
     - M_s = ε_s/(ε_s − 1), ˜P_s = P_s/P^∗, Y_s = γ_s Y^∗ ˜P_s
  3. Update output and price index via:
     - Y^∗ = A^{(1+ψ)k} (1 − α)^{(1−α)k} (Γ^∗)^{−σ(1−α)k} (∆^∗)^{−ψ(1−α)k} (M^∗)^{−(1−α)k}
     - k = 1/[ψ + (1 − α) σ + α]
     - Γ^∗ = 1 − ∫ γ_s 1/˜P_s θ_s π_s^2 ds
     - ∆^∗ = ∫ [γ_s A ˜P_s A_s]^{1/(1−α)} ds
     - M^∗ = exp(∫ γ_s log(M_s) ds), M_s = 1/MC_s
     - In steady-state, W/P = (Y^∗)^{σ + ψ/(1 − α)} A^{−ψ/(1 − α)} (∆^∗)^{ψ} (Γ^∗)^{σ}
  4. Iterate until Y^∗ and price index converge.
- One-period markup shock procedure (shock at t = 0, return to steady state at t = 1):
  1. Set Y_1 = Y^∗; inflation lasts one period so P_1 = P_0; set ˜P_s(A_s, ˜P_{s,−1}) at t = 1 equal to steady-state rule.
  2. Guess P_0 and Γ_0.
  3. Compute Y_0 using Euler equation and Taylor rule:
     - Y_0 / Y^∗ = [Γ^∗ / Γ_0 (P_0 / P^∗)^{−φ_π}]^{1/(1 + φ_y)}
  4. Solve for firms’ optimal pricing at t = 0 given P_0, Y_0.
  5. Use optimal pricing to derive π_0 and update P_0.
- The numerical implementation follows these steps to evaluate the effects of a one-period increase in ε_0.

*Italic: Content derived from wpiea2023254-print-pdf - 0. These restrictions are consistent with any empirically reasonable parametriza-*

### 6.  Go back to 2 and iterate until convergence

### 6.  Go back to 2 and iterate until convergence

### Calibration and Results
- Calibration (see Section 7.3): three sectors non-durable, durable, and services with sector-specific parameters θ_s = {20.4, 62.5, 689.7} and sector sizes {0.4, 0.2, 0.4}.
- Results: Panel (A) in Table F5 shows effects of TIP in an economy with heterogeneous θ's; Panel (B) shows effects when all sectors share the same θ.

### Time-dependent Calvo pricing — derivation and equivalence
- Firm optimization when granted opportunity to reset price P^*_{t,i} includes:
  - Expected discounted sum of period profits net of adjustment tax τ_t and expected future losses when rivals reset prices.
  - Firm F.O.C. involves terms in Q_{t,t+k}, Y_{t+k|t_i}, Ψ'(·), τ_t, and future optimal prices P^{**}_{t+k,i}.
- After manipulation and linearization around steady state (P^*_t / P_{t-1} = 1, steady-state output Y, τ = 0), the model yields a Phillips-curve representation identical at first order to the Rotemberg-adjustment-cost economy.
- Definitions and parameters used in linearization:
  - Θ = (1−α) / (1−α+αε)
  - λ = (1−βθ)(1−θ)θ^{-1} Θ
  - ζ = λ (ε−1)
  - κ = λ [(σ+ψ+α)/(1−α)]
- First-order Phillips curve under Calvo pricing:
  - π_t = β E_t π_{t+1} + κ \hat{y}_t − ζ [τ_t − β E_t τ_{t+1}]
- Implication: using time-dependent (Calvo) rather than state-dependent (Rotemberg) frictions leads to the same first-order representation and macro dynamics; optimal-policy results from the linearized model (corollary 1, section 6, section 8, proposition 5) are robust to Calvo frictions.

### Relative price distortions and independence from TIP (proposition 5)
- Sectoral Phillips curve for heterogeneous sectors:
  - π_{s,t} = β E_t π_{s,t+1} + λ_s [\hat{mc}_t − (1/(1−α)) \widehat{\tilde p}^e_{s,t}] + ζ_s [β E_t τ_{t+1} − τ_t] + u_t
  - λ_s = (1−βθ_s)(1−θ_s) θ_s^{-1} Θ
  - ζ_s = λ_s (ε−1)
- Sufficient conditions for relative prices to be independent of aggregate variables and of TIP: λ_s = λ and ζ_s = ζ, which imply θ_s = θ (i.e., identical Calvo parameters across sectors). The second bullet of proposition 5 follows directly from the first.

### Taxing large corporates — model extension and calibration
- Model extension: two firm types, small (S) and large (L). Large firms face TIP and a potentially different consumer taste parameter \bar ξ (with \bar ξ ≥ ξ).
- Utility aggregator:
  - C = [∫_{i∈S} ξ^{1/ε_t} C_{t,i}^{1−1/ε_t} di + ∫_{i∈L} \bar ξ^{1/ε_t} C_{t,i}^{1−1/ε_t} di]^{ε_t/(ε_t−1)}
- Demand for varieties:
  - For i ∈ L: C_{t,i} = \bar ξ (P_{t,i}/P_t)^{−ε_t} C_t
  - For i ∈ S: C_{t,i} = ξ (P_{t,i}/P_t)^{−ε_t} C_t
- Aggregate price index:
  - P_t = [∫_{i∈S} ξ P_{S,t}^{1−ε_t} + ∫_{i∈L} \bar ξ P_{L,t}^{1−ε_t}]^{1/(1−ε_t)}
- Focus on symmetric equilibria for large firms: P_{t,i} = P^L_t for i ∈ L; define real price of L goods \tilde P^L_t = P^L_t / P_t.
- Log-linearization around τ = π = π^L = 0 and constant output produces sectoral pricing equations; marginal cost MC_t and relative prices enter the Phillips-like equations for L and S firms.
- Calibration for large vs small firms:
  - Empirical motivation: Crouzet and Mehrotra (2020) — U.S. top 1% firms account for 75% of total sales and 86% of total assets.
  - Model assumption: large firms account for 75% of nominal output in steady state ⇒ μ_L = 0.75, μ_S = 0.25. To avoid calibrating \bar ξ and ξ separately, set \bar ξ = ξ = 1 and interpret μ_L/μ_S as the ratio of the number of products produced by large firms relative to small firms.

### Equilibrium implications for multi-size firms
- Relative sales expression in the model:
  - P_L C_L / (P_S C_S) = μ_L / μ_S × ( \bar ξ / ξ )^{1/(1−α+αε_t) }
- With \bar ξ = ξ = 1, relative sales differences arise from μ_L ≠ μ_S (different product counts).
- Linearized relations connect:
  - Sectoral inflation π^L_t and π^S_t, tax τ_t, marginal cost deviations \hat{mc}_t, output gap \hat{y}^e_t, and relative-price change \hat δ_t.
- Compact system (definitions):
  - θ_S = θ ( \tilde P_S )^{ε−1 }, θ_L = θ ( \tilde P_L )^{ε−1 }
  - Key equations include the IS/Euler condition for \hat y^e_t, sectoral Phillips equations with τ_t terms, the law of motion for relative price δ̂_t, aggregate inflation π_t = χ_L π^L_t + χ_S π^S_t, and AR(1) processes for shocks u_t and a_t.
- Sectoral Phillips coefficients κ_L, γ_L and κ_S, γ_S are provided as functions of model primitives and \tilde P_{L,S}.

### Sticky wages — household problem and wage Phillips curve
- Household utility and consumption aggregator (heterogeneous varieties); labor types indexed by i with elasticity across labor types ε_N.
- Wage adjustment costs assumed quadratic: C( W_{t-1,i}, W_{t,i} ) = (θ/2) ( W_{t,i} / W_{t-1,i} − 1 )^2 W_t N_t.
- F.O.C.s yield a wage-setting first-order condition (Rotemberg-type wage adjustment), under symmetry and after log-linearization around constant consumption, labor supply, zero inflation and zero tax on inflation.
- Wage Phillips curve (log-linearized):
  - π^w_t = β E_t π^w_{t+1} + (ε_N−1) θ^{−1} [ (ψ/(1−α) + σ) \hat y^e_t − \hat ω^e_t ] + u^w_t − τ^w_t θ^{−1} + θ^{−1}[β τ^w_{t+1} − τ^w_t]
  - where ω_t = w_t − p_t, \hat ω^e_t = ω_t − ω^e_t, and u^w_t is the wage-markup shock (proportional to μ̂^w_t).
- Price Phillips curve with wage effects:
  - π_t = β E_t π_{t+1} + (ε−1) θ^{−1} [ \hat ω^e_t + (α/(1−α)) \hat y^e_t ] + u_t − τ_t θ^{−1} + θ^{−1}[β E_t τ_{t+1} − τ_t]
- Link between wage and price dynamics:
  - \hat ω^e_t evolves with past gap and contemporaneous wedge: \hat ω^e_t = \hat ω^e_{t-1} + π^w_t − π_t − Δ ω^e_t
  - Δ ω^e_t expressed in terms of productivity shocks and parameters: Δ ω^e_t = ψ_{ayn} Δ a_t, where ψ_{ayn} = (1+ψ)/(σ(1−α)+ψ+α).
- Complete system (log-linearized):
  - π_t, π^w_t, \hat ω^e_t, \hat y^e_t with IS/Euler equation, and monetary-policy rule (i_t = ρ + φ_π π_t + φ_y \hat y^e_t).
- Calibration note: θ and θ_w are chosen to match slopes of price and wage Phillips curves in Galí (2015):
  - θ = (ε−1) \bar φ / [ (1−\bar φ)(1−\bar φ β) ] × (1−α+αε) / (1−α)
  - θ_w = (ε_N−1) \bar φ / [ (1−\bar φ)(1−\bar φ β) ] × (1+ψ ε_N)

### Simulations and additional graphs (high-level)
- Figures F1–F10 present impulse responses and comparative statics across:
  - Productivity shocks (Figure F1)
  - Monetary policy shocks (Figure F2)
  - Markup and inflation-expectation shocks with varying price stickiness (Figure F3)
  - Time-varying wage subsidies vs TIP comparisons (Figures F4–F6)
  - Effects of TIP targeted to large firms under markup and inflation-expectation shocks (Figure F7)
  - Sticky-wage model responses for price and wage markup shocks (Figures F8–F9)
  - TIP effects on profits in sticky-wage model (Figure F10)
- Tables F1–F4 evaluate policy rules under different shocks and TIP calibrations:
  - Table F1: Evaluation under productivity shocks (statistics ×100 reported for various TIP strengths; sample entries: σ(π_ann_t) = 0.74, 0.59, 0.49, 0.36; σ(\hat y^e_t) = 1.09, 1.21, 1.29, 1.39; σ(i_ann_t) = 1.38, 1.18, 1.05, 0.89; σ(τ_t) = 0, 44.17, 3.01, 108.8; E(π_t τ_t) = 0, 0.13, 0.18, 0.20; L^* × 10^4 = 2.00, 1.47, 1.19, 0.90)
  - Table F2: Evaluation under monetary-policy shocks (sample entries: σ(π_ann_t) = 0.20, 0.16, 0.13, 0.10; σ(\hat y^e_t) = 0.30, 0.33, 0.35, 0.38; σ(i_ann_t) = 0.20, 0.25, 0.29, 0.33; σ(τ_t) = 0, 12.12, 0.12, 9.9; E(π_t τ_t) = 0, 0.01, 0.01, 0.01; L^* × 10^4 = 0.08, 0.08, 0.09, 0.10)
  - Table F3: Optimal monetary policy with higher markup-shock variance (sample entries: φ^*_π = 1.50, 0.987, 0.960, 0.930; φ^*_y = 0.125, 0.155, 0.380, 0.460; σ(π_ann_t) = 1.49, 1.39, 1.16, 0.83; σ(\hat y^e_t) = 1.49, 1.39, 1.16, 0.83; σ(i_ann_t) = 1.96, 1.24, 0.94, 0.68; σ(τ_t) = 0, 104.5, 173.8, 249.4; E(π_t τ_t) = 0, 0.73, 1.01, 1.04; L^* × 10^4 = 5.05, 3.04, 1.97, 1.03)
  - Table F4: Optimal monetary policy with higher demand-shock variance (sample entries: φ^*_π = 1.50, 0.991, 0.992, 0.992; φ^*_y = 0.125, 0.105, 0.076, 0.050; σ(π_ann_t) = 0.84, 0.78, 0.64, 0.46; σ(\hat y^e_t) = 0.84, 0.84, 0.89, 0.96; σ(i_ann_t) = 1.23, 0.83, 0.68, 0.50; σ(τ_t) = 0, 58.79, 95.31, 138.8; E(π_t τ_t) = 0, 0.23, 0.30, 0.32; L^* × 10^4 = 1.83, 1.16, 0.81, 0.49)
- Table of statistics (×100) reports baseline heterogeneous-sector θ_s ∈ {20.4, 62.5, 689.7}.

*Source: wpiea2023254-print-pdf — Section 6 and Appendices C–F (time-dependent Calvo pricing, taxing large corporates, sticky wages, simulations).*

### 1. One-period markup shock when

### 1. One-period markup shock when

### A. Heterogeneous sectors
- θ unspecified in panel heading (implicitly heterogeneous)
- Panel A.1: ̄τ = 0%
  - Steady-state: 0.00 -1.58 0.44 0.20 0.28 0.92
  - τ0 = 0%: 2.75 -2.02 1.52 0.33 0.29 2.14
  - τ0 = 10%: 1.05 -1.89 0.55 0.28 0.29 1.12
  - τ0 = 15%: 0.18 -1.66 0.39 0.22 0.29 0.90

- Panel A.2: ̄τ = 15%
  - Steady-state: 0.00 -1.58 0.47 0.20 0.28 0.94
  - τ0 = ̄τ: 2.87 -2.00 1.66 0.32 0.29 2.27
  - τ0 = 25%: 1.10 -1.89 0.59 0.29 0.28 1.16
  - τ0 = 30%: 0.23 -1.67 0.42 0.22 0.28 0.93

### B. Homogeneous sectors: θ = 372.8
- Panel B.1: ̄τ = 0%
  - Steady-state: 0.00 -1.65 0.21 0.22 0.39 0.81
  - τ0 = 0%: 0.87 -1.87 0.35 0.28 0.39 1.02
  - τ0 = 10%: 0.34 -1.76 0.20 0.25 0.39 0.84
  - τ0 = 15%: 0.03 -1.67 0.17 0.22 0.39 0.79

- Panel B.2: ̄τ = 15%
  - Steady-state: 0.00 -1.64 0.21 0.22 0.39 0.81
  - τ0 = ̄τ: 0.89 -1.87 0.36 0.28 0.39 1.03
  - τ0 = 25%: 0.33 -1.76 0.20 0.25 0.39 0.84
  - τ0 = 30%: 0.02 -1.66 0.17 0.22 0.39 0.78

### Notes and interpretation
- In panels A and B, the economy is hit by an unexpected markup shock in the initial period.
- TIP is used in the initial period and returns to its steady-state value ̄τ after the shock.
- L is defined in equation 35.
- In panels A.1 and B.1, the steady-state level of TIP is set to 0.
- In panels A.2 and B.2, the steady-state level of TIP is set equal to 15%.
- All columns are multiplied by 100.
- The figures on the lines "Steady-state" correspond to steady-state values. All other figures correspond to the initial period t = 0.

*Optimal Taxation of Inflation Working Paper No. WP/23/254*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023254-print-pdf.pdf_