## _wp0934 — Introduction, sections 5.3, 6.1 and 6.3 (selected extracts)

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---

### Context, motivation, and contribution
- Revisits macro implications of downward wage rigidities in a DSGE framework with forward-looking, optimizing agents.
- Introduces downward nominal wage rigidities into a DSGE model where wage setters optimally choose wages accounting for future implications.
- Derives a closed-form solution for the long-run Phillips curve (long-run averages of inflation and unemployment).
- Emphasizes intertemporal relative-price adjustment limits: downward wage rigidities generate a long-run inflation–unemployment trade-off even in a one-sector model.
- Highlights the role of macroeconomic volatility: the inflation–unemployment trade-off at low inflation depends critically on volatility of nominal spending growth.

### Model structure and baseline assumptions
- Agents: continuum of infinitely lived households and firms on [0,1]; utility from consumption and disutility from labor; firms profit-maximizing.
- Market structure: monopolistic competition in labor and goods; goods prices fully flexible.
- Aggregate shocks:
  - Productivity shock A_t with log a_t: da_t = g dt + ϵ_a dB_{a,t}.
  - Nominal spending shock ~Y_t with log ~y_t: d~y_t = ρ dt + ϵ_y dB_{y,t}.
- Complete markets assumed.
- Flexible-wage benchmark:
  - W^f_t = μ_w^{1/(1+ϕ)} μ_p^{ - ϕ/(1+ϕ) } ~Y_t (equation (12)).
  - Aggregate labor in flexible-wage equilibrium: L^f = (μ_p μ_w)^{-1/(1+ϕ)}.
  - Unemployment in flexible case: u^f = 1 − L^f.
  - With flexible wages the Phillips curve is vertical (unemployment constant).

### Downward nominal wage rigidity (strong constraint and relaxation)
- Strong constraint: dw_t(j) ≥ 0 with w_{t0} > 0 (equation (13)).
- Under optimal forward-looking behavior desired wage:
  - W(~Y_t) = c(ρ; ϵ_y^2; ϕ; ζ) × μ_w^{1/(1+ϕ)} μ_p^{-ϕ/(1+ϕ)} ~Y_t = c(·) W^f_t (equation (14)).
  - 0 < c(·) ≤ 1; when ϵ_y^2 = 0 then c(·) = 1 and W(~Y_t) = W^f_t; when ρ is very large c(·) → 1.
- Comparative statics for c(·) (preserved wording):
  - c(·) is lower when:
    - Variance of nominal expenditure growth ϵ_y^2 is high;
    - Mean nominal expenditure growth ρ is small;
    - Agents discount the future more (ζ is low) — note: lower ζ implies discount less the future (text describes "when agents discount less the future (ζ is low) they will be less concerned..." leading to lower c(·));
    - Labor supply is more elastic (ϕ is low).
- Section 5.3 relaxes the strong constraint to allow some decline in wages at low inflation via dw_j^t ≥ −φ(θ) w_j^t dt, with φ(θ) non-increasing in θ (mean nominal-spending growth).

### Endogenous upward wage prudence and implications
- Forward-looking wage setters set muted desired wages (c(·) < 1) to avoid being stuck with high wages in future bad states — an endogenous upward wage rigidity at low inflation.
- When downward rigidity binds, actual wages can be higher than the flexible-case wage and long-run unemployment will be higher on average than in the flexible case.
- Trade-offs exist across first moments (means) and second moments (volatilities) of inflation and unemployment: stabilization that reduces nominal spending volatility can improve both mean unemployment and its volatility.

### Long-run Phillips curve (section 6.1) — definitions and core result
- Employment implied by (10): L_t = 1 − p ~Y_t / W_t.
- With W_t ≥ c(π) (as defined), 0 ≤ L_t ≤ L_f = c(π).
- Under Brownian motion for ~y_t with drift ρ and standard deviation σ_y (written _y in source), l_t = ln L_t follows a Brownian motion with a reflecting barrier at ln(L_f = c(π)).
- Long-run mean unemployment (equation (15)) (expression preserved as in source):
  - E[u_1] = 1 − 1
    1 +
    _y^2
    2 
    (1 − u_f) c(; _y^2; ; )
    :(15)
- Long-run mean wage inflation (equation (16)):
  - E[π^w_1] =  :(16)
- Long-run Phillips curve (equation (17)) (substituting (16) into (15), expression preserved):
  - E[u_1] = 1 − 1
    1 +
    _y^2
    2 E[π^w_1]
    (1 − u_f) c(E[π^w_1]; _y^2; ; )
    (17)

### Qualitative shape and intuition of the long-run Phillips curve
- Not vertical: the “natural” rate of unemployment depends on mean inflation.
- For high mean wage inflation:
  - c(π) ≈ 1 and average unemployment ≈ u_f; Phillips curve virtually vertical.
- For low mean wage inflation:
  - Significant long-run trade-off emerges: average unemployment rises as mean wage inflation falls.
- Role of volatility (_y^2 / σ_y^2):
  - Deterministic case _y^2 = 0 and c(π) = 1 → long-run unemployment equals flexible-wage unemployment.
  - With stochastic shocks higher _y^2 increases the impact of a fall in inflation on average unemployment.
  - Two channels operate:
    - More variance raises equilibrium unemployment because downward constraints bind more often.
    - Wage setters temper upward wage claims when volatility is higher (c(π) falls), which tends to reduce average unemployment in favorable shocks.
  - The first channel dominates in the long run: E[u_1] ≥ u_f since (1 + _y^2 / (2 )) · c(π) ≥ 1 (as noted in source).

### Quantitative illustrations and key statistics (figures and table described)
- Calibration used in figures:
  - ϕ = 2:5; ζ = 0:01; u_f = 6%.
  - Figure 2 plots E(π^w) vs E(u) for σ_y = 0%, 2%, 5%, 10%, 15%, 20% (standard deviation of nominal spending growth), annual rates; all variables in % and annualized.
- Table 1 — Increase in long-run mean unemployment E[u_1] due to reductions in mean wage inflation E[π^w_1] (all entries in % and annual rates):
  - Reduction from 4% to 1%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.4
    - σ_y = 10% → 3.4
    - σ_y = 15% → 9.5
    - σ_y = 20% → 16.9
  - Reduction from 5% to 2%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.1
    - σ_y = 10% → 0.8
    - σ_y = 15% → 2.8
    - σ_y = 20% → 6.0
  - Reduction from 5% to 3%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.0
    - σ_y = 10% → 0.2
    - σ_y = 15% → 1.0
    - σ_y = 20% → 2.3
  - Reduction from 6% to 3%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.0
    - σ_y = 10% → 0.3
    - σ_y = 15% → 1.1
    - σ_y = 20% → 2.7
- Interpretive highlights from the table (preserved source interpretation):
  - Low-volatility country (σ_y ≈ 2%): negligible unemployment increase when average wage inflation falls from 6% to 3% or from 4% to 1%.
  - Moderate volatility (σ_y ≈ 10%): cost about 0.3% increase in average unemployment when inflation falls from 6% to 3%, and about 3.4% when inflation falls from 4% to 1%.
  - High-volatility countries face much larger costs of disinflation in terms of average unemployment.

### Short-run Phillips curve (connection and dynamics)
- Short-run average unemployment can fall below the flexible-wage level due to endogenous upward wage rigidity: agents set wages below flexible level when they can adjust upward.
- Short-run curve tends to shift right over time as wages become more likely to bind, converging to the long-run curve.
- Volatility effects on short-run curve:
  - Higher volatility increases probability of binding downward rigidities → raises unemployment (dominant at low inflation).
  - Higher volatility also makes agents more cautious in wage claims → can reduce unemployment (dominant at moderate inflation).
  - Hence higher σ_y may place short-run Phillips curve more to the right at low inflation and to the left at moderate inflation; curve converges to flexible-wage employment as inflation increases.

### Varying the degree of downward rigidities (section 6.3)
- Generalized constraint: dw_j^t ≥ −φ(θ) w_j^t dt, where φ(θ) non-increasing in θ (mean nominal-spending growth).
- Effective replacement θ → ψ(θ) ≡ θ + φ(θ) in solutions.
- Modified long-run Phillips curve:
  - E[u_1] = 1 − 1/(1 + σ_y^2/2 ψ(E[π_w1])) (1 − u_f) c(ψ(E[π_w1]); σ_y^2; ξ; ζ)
- Functional form of φ(θ) is crucial:
  - If φ(θ) = φ_1 (fixed), Phillips curve shifts down by φ_1.
  - If φ(θ) increases as θ declines (rigidities soften as inflation falls), Phillips curve tilts clockwise at low inflation.
- Illustrative linear specification used in figures (equation (19)):
  - φ(θ) = φ_1 − φ_2 θ with φ_1 = 1% and φ_2 = 0.1 (annual rates).
- Key patterns (numerical illustrations):
  - Volatility of wage inflation σ(π_w) increases as mean wage inflation E[π_w] increases (Figure 6).
  - Volatility of unemployment σ_u decreases as mean wage inflation E[π_w] increases (Figure 7).
  - Long-run trade-off between σ(π_w) and σ(u) for given σ_y (Figure 8).
- Exact variance expression for unemployment in long run (preserved form from source):
  - Var[u_1] = (1/2) [1 + σ_y^2/E[π_w1]] [1 + E[π_w1]^2/(2 σ_y^2)] 2 θ L_f / c(E[π_w1]; σ_y^2; ξ; ζ) ^2

### Core theoretical implications and limiting cases
- Downward nominal wage rigidities generate endogenous upward wage prudence via forward-looking optimization.
- Degree of wage rigidity:
  - High when wage inflation is low and σ_y is high.
  - Low (wages more flexible) at high inflation rates or very small σ_y — monetary policy virtually neutral.
- Non-linear long-run trade-off:
  - Virtually nonexistent at high inflation.
  - Significant at low inflation under moderate/high macro volatility.
- Special limiting case: when nominal spending growth is very close to zero, unemployment distribution collapses to a mass at 100% unemployment and its volatility collapses to zero (trade-off disappears).
- If downward rigidities soften as inflation declines, the trade-off is reduced but need not disappear.

### Policy implications and recommendations (preserved emphasis)
- Inflation targets should be country-specific:
  - Countries with larger macroeconomic volatility (e.g., many emerging markets and developing countries) may find it desirable to target a higher inflation rate than low-volatility countries.
  - As macroeconomic volatility changes over time, inflation targets may need adjustment.
- Stabilization policies that reduce macroeconomic volatility improve the inflation–unemployment trade-off and reduce the unemployment costs of lowering long-run inflation.
- Ignoring the possibility of a long-run trade-off can be costly; policymakers should consider the dependence of the probability that the true model includes a trade-off on both the wage inflation rate and volatility of nominal spending growth.
- An optimal inflation assessment requires more complex models incorporating persistence of shocks, additional effects of inflation, open-economy features, game-theoretic interactions, and other frictions.

### Empirical and institutional context (selected evidence)
- Strong body of micro and macro evidence for downward wage rigidities across many countries and at low inflation (references in source: Bewley, 1999; Holden, 1997, 2004; Lebow et al., 2003; Dickens et al., 2007; Du Caju et al., 2008; Gottschalk, 2004).
- Heterogeneity across countries linked to labor-market institutions and union presence.
- Evidence that alternative margins (bonuses, non-pay benefits, promotions, new hires) are used, but downward real/nominal rigidities remain relevant (Babeky et al., 2008; Holden and Wulfsberg, 2008; Dickens et al., 2008; Du Caju et al., 2008).

_ Source: _wp0934 — extracts from Introduction, section 5.3, section 6.1, and section 6.3._

### 1.         Introduction.................................................................................................

### 1.         Introduction

### Context and motivation
- Revisits the macroeconomic implications of downward wage rigidities in a dynamic stochastic general equilibrium (DSGE) framework with forward-looking, optimizing agents.
- Challenges the conventional textbook view that there is no long-run trade-off between inflation and unemployment and that price stability should be the primary objective of central banks (cites Mankiw, 2001, and Mishkin, 2008).
- Notes historical empirical roots: "Fifty years ago, Phillips (1958) showed evidence ... for 97 years of British data," and Samuelson and Solow (1960) reported a similar fit for US data.
- Recognizes the skepticism raised by Friedman (1968), Phelps (1968), Lucas (1973), and the oil shocks of the 1970s, and references subsequent empirical controversy (Ball et al., 1988; King and Watson, 1994; Bullard and Keating, 1995).

### Key innovation and contributions
- Introduces downward wage rigidities into a DSGE model where wage setters optimally choose wages accounting for future implications of their choices.
- Derives a closed-form solution for the long-run Phillips curve (long-run averages of inflation and unemployment).
- Emphasizes dynamic (intertemporal) implications of downward wage rigidities even in a one-sector model; adds that multiple sectors would amplify the need for relative price adjustments and strengthen the Phillips curve.
- Highlights the role of macroeconomic volatility: the inflation-unemployment trade-off at low inflation depends critically on volatility of nominal spending growth.
- Shows that downward wage inflexibility implies an endogenous upward wage rigidity at low inflation rates because forward-looking wage setters avoid setting wages that would likely bind them in bad future states.
- Demonstrates a trade-off not only between mean inflation and unemployment but also between their volatilities.

### Model structure and assumptions
- Agents derive utility from consumption and disutility from labor; firms are profit-maximizing.
- Labor and goods markets are characterized by monopolistic competition; goods prices are fully flexible.
- Aggregate shocks: an aggregate productivity shock and stochastic perturbations to nominal spending.
- Comparison points to related literature: symmetric nominal rigidities and asynchronized price-setting (Goodfriend and King, 1997; Woodford, 2003) and asymmetric wage-adjustment cost models (Kim and Ruge-Murcia, 2007; Fahr and Smets, 2008; Fagan and Messina, 2008).
- Contrasts with Akerlof et al. (1996) and others; notes relation to literature on irreversible investment (Abel and Eberly, 1996; Bertola, 1998; Bertola and Caballero, 1994; Dixit, 1991; Dumas, 1991; Pindyck, 1988; Stokey, 2006).

### Main findings
- The Phillips curve is:
  - "Almost vertical for medium-to-high inflation rates."
  - Can display a "significant trade-off at low inflation rates," consistent with downward nominal rigidity literature.
- Intertemporal relative-price adjustment limits are central: downward wage rigidities hamper relative price adjustment over time and generate a long-run Phillips curve even absent cross-sectional price-setting frictions.
- The extent of macroeconomic volatility is an important determinant of the trade-off at low inflation rates:
  - Countries with higher macroeconomic volatility may optimally target a higher inflation rate to reduce long-run unemployment.
  - Alternatively, such countries could prioritize stabilization policies to improve the unemployment-inflation trade-off.
- Endogenous upward wage rigidity arises at low inflation because wage setters internalize future probability of being constrained by downward rigidity.
- Trade-offs exist across first moments (means) and second moments (volatilities) of inflation and unemployment; stabilization policies that reduce nominal spending growth volatility can improve both.

### Empirical and institutional context summarized
- Asserts a strong body of evidence for downward wage rigidities across many countries and even at low inflation (references: Bewley, 1999; Holden, 1997, 2004; Lebow et al., 2003; Dickens et al., 2007; Du Caju et al., 2008; Gottschalk, 2004).
- Notes heterogeneity across countries linked to labor market institutions and union presence (Dickens et al., 2007; Agell and Bennmarker, 2002; Fehr and Gotte, 2005).
- Observes that downward wage rigidities may not disappear in low-inflation environments (evidence for Sweden and Switzerland; mixed evidence for Japan — Kimura and Ueda, 2001; Yasui and Takenaka, 2005).
- Points out that rigidity strength is likely correlated with the degree of labor-market distortions.

### Policy implications
- Standard monetary models that ignore downward wage rigidities may understate the benefits of positive inflation and thus recommend an inflation target that is too low (close to zero).
- Optimal inflation targets may differ across countries and over time depending on macroeconomic volatility and labor-market institutional features.
- Macroeconomic stabilization policies (monetary and fiscal) can have first-order effects on unemployment at low inflation rates by reducing nominal spending volatility.
- Policy design should account for:
  - The presence and degree of downward wage rigidities.
  - The level of macroeconomic volatility.
  - The interaction between average inflation and inflation/unemployment volatilities.

### Paper organization
- Section 2: Theoretical and empirical overview of downward rigidities.
- Section 3: Model description.
- Sections 4–5: Solutions under flexible and downward-rigid wages.
- Section 6: Long-run Phillips curve derivation.
- Section 7: Implications for volatilities.
- Section 8: Conclusions.

*Source: _wp0934 - 1. Introduction*

### section 5.3, our analysis would anyhow relax the assumption of strong downward

### _wp0934 - section 5.3, our analysis would anyhow relax the assumption of strong downward

### Evidence on wage rigidities and alternative margins
- Firms may use alternative margins (bonuses, non-pay benefits, promotions, new hires) to reduce labor costs when facing negative demand shocks; Faris and Lemieux (2001) find wage adjustments in Canada are mainly obtained via new hires.
- Empirical findings indicate downward real/nominal wage rigidities remain relevant:
  - Babeky et al. (2008) find firms subject to downward real wage rigidities have less capacity to use alternative margins.
  - Holden and Wulfsberg (2008) document downward rigidities in industry-level wage data for OECD countries, concluding "firm behaviour and market mechanisms may diminish, but do not remove, rigidity at the individual level" (p. 31).
  - Dickens et al. (2008) and Du Caju et al. (2008) find downward wage rigidities have a negative impact on employment.
- The analysis relaxes the strong downward-rigidity assumption to allow for some decline in wages at low inflation.

### Model structure (closed-economy, continuum of households and firms)
- Economy: continuum of infinitely lived households and firms on the [0,1] interval.
- Households:
  - Utility over time: E_{t0} [ ∫_{t0}^{∞} e^{-ζ(t-t0)} ( ln C^{j}_{t} - l_{t}(j)^{1+ϕ}/(1+ϕ) ) dt ] (equation (3)); ζ > 0 is the rate of time preference; ϕ ≥ 0 measures the inverse of the Frisch elasticity.
  - Consumption aggregated by a Dixit-Stiglitz index with elasticity of substitution ξ_p > 0.
  - Labor supplied is one variety in a monopolistic-competitive market.
  - Intertemporal budget constraint given by equation (4) with stochastic nominal discount factor Q_t, nominal wage w_t(j), and profit income ψ^j_t.
- Firms:
  - Produce differentiated goods using a common linear technology y_t(i) = A_t L_t(i).
  - Labor input for each good is a CES aggregate L_t(i) with elasticity of substitution ξ_w > 1 (equations for L_t(i), demand l^d_t(j) in (7), and wage index W_t in (8)).
  - Firms set prices without friction; resulting first-order condition implies p_t(i) = P_t = μ_p W_t A_t where μ_p ≡ ξ_p/(ξ_p-1) (equation (9)).
  - Labor income is a constant fraction of total income: ~Y_t = P_t Y_t = μ_p W_t L_t (equation (10)).
- Aggregate shocks:
  - Productivity shock A_t with log a_t following Brownian motion with drift g and variance ϵ_a^2: da_t = g dt + ϵ_a dB_{a,t} (equation (1)).
  - Nominal spending shock ~Y_t with log ~y_t following Brownian motion with drift ρ and variance ϵ_y^2: d~y_t = ρ dt + ϵ_y dB_{y,t} (equation (2)). dB_{y,t} may be correlated with dB_{a,t}.
- Complete markets are assumed (state-contingent claims to monetary units), implying the Lagrange multiplier on intertemporal budget does not vary over time in consumption first-order condition (5).

### Flexible-wage benchmark
- When wages are fully flexible:
  - Nominal wages proportional to nominal spending: W^f_t = μ_w^{1/(1+ϕ)} μ_p^{ - ϕ/(1+ϕ) } ~Y_t (equation (12)), where μ_w ≡ ξ_w/(ξ_w-1).
  - Aggregate labor in the flexible-wage equilibrium: L^f = (μ_p μ_w)^{-1/(1+ϕ)}, a constant depending on mark-ups and labor elasticity.
  - Unemployment rate in flexible case: u^f = 1 - L^f (total labor force normalized to 1).
  - Prices: P^f_t = μ_p W^f_t A_t.
  - Implication: with flexible wages, prices and wages move proportionally to nominal spending and unemployment is constant; the Phillips curve is vertical.

### Downward nominal wage rigidity (non-decreasing nominal wage constraint)
- Constraint: dw_t(j) ≥ 0 with w_{t0} > 0 (equation (13)). Section 5.3 will consider relaxing this strong constraint.
- Households maximize the value function V(w_t(j); W_t; ~Y_t) over non-decreasing positive wage paths; smooth-pasting conditions characterize optimal paths:
  - V_w(w_t(j); W_t; ~Y_t) = 0 if dw_t(j) > 0;
  - V_w(w_t(j); W_t; ~Y_t) ≤ 0 if dw_t(j) = 0.
- All wage setters set the same wage W_t(j) = W_t for all j.
- Define v(W_t; ~Y_t) ≡ V_w(W_t; W_t; ~Y_t) and desired wage W(~Y_t) solves v(W(~Y_t); ~Y_t) = 0.
  - Actual wages W_t = max(previous-period wage, W(~Y_t)). Hence actual wages cannot fall below W(~Y_t); when constraint binds actual wages exceed desired wages, otherwise they are equal.

### Properties of the desired wage and the factor c(·)
- The desired wage is always below the flexible-case wage:
  - W(~Y_t) = c(ρ; ϵ_y^2; ϕ; ζ) × μ_w^{1/(1+ϕ)} μ_p^{-ϕ/(1+ϕ)} ~Y_t = c(·) W^f_t (equation (14)).
- c(·) is a non-negative function of (ρ; ϵ_y^2; ϕ; ζ) with analytic form involving ρ, ϵ_y^2, ζ and ϕ as detailed in the appendix; the auxiliary non-negative function Ψ appears in the expression for c(·).
- Economic interpretation:
  - Optimizing behavior under downward rigidity generates an endogenous tendency for upward wage prudence: desired wages are muted relative to the flexible benchmark to avoid being stuck with excessively high wages if future unfavorable shocks occur.
  - When downward rigidity binds, actual wages can be higher than flexible-case wages; in the long run unemployment will be higher on average than in the flexible-wage case.
- Comparative statics for c(·):
  - c(·) is lower (weaker wage reaction to nominal expenditure) when:
    - Variance of nominal expenditure growth ϵ_y^2 is high;
    - Mean nominal expenditure growth ρ is small;
    - Agents discount the future more (ζ is low) — note: lower ζ implies discount less the future (text describes "when agents discount less the future (ζ is low) they will be less concerned..." leading to lower c(·) — preserve source wording);
    - Labor supply is more elastic (ϕ is low).
  - Limiting cases:
    - If ϵ_y^2 = 0 then c(·) = 1 and W(~Y_t) = W^f_t.
    - When ρ is very large, c(·) → 1.
  - Intuition:
    - Higher volatility increases the cost of being stuck with high wages after unfavorable shocks, so wage setters temper upward adjustments.
    - Lower mean growth increases the probability that upward adjustments will be followed by hitting the lower bound, so wage setters moderate increases.
    - More patient agents (lower discounting of the future) care more about future consequences and set lower desired wages when adjusting upward.
    - Less elastic labor supply (higher ϕ) leads wage setters to prefer smaller fluctuations in hours worked and thus set higher wages when adjusting (c(·) closer to one), reducing employment variability but lowering average employment.

### Numerical illustration (Figure 1 description and calibration)
- Figure 1 plots c(·) against the mean of nominal spending growth ρ (in percent, annual rates) for standard deviations of nominal spending growth ϵ_y equal to 0%, 2%, 5%, 10%, 15%, and 20% (annual rates).
- Calibration and parameter choices (discretized quarterly model):
  - Rate of time preference ζ = 0.01 (implying a 4% real interest rate at annual rates).
  - Frisch elasticity of labor supply set equal to 0.4, implying ϕ = 2.5.
  - Figure caption uses u^f = 6%.
- Qualitative result from the plot:
  - When ϵ_y = 0% then c(·) = 1.
  - For positive ϵ_y, c(·) decreases as ρ decreases; the decline is larger the higher ϵ_y.

*Source: _wp0934 - section 5.3, our analysis would anyhow relax the assumption of strong downward*

### 6.1  Long-run Phillips curve

### 6.1  Long-run Phillips curve

### Model setup and equilibrium distribution
- Employment implied by (10): L_t = 1 − p ~Y_t / W_t.
- With W_t ≥ c(π) ( (1/(1+φ))^(−φ/(1+φ))? ), it follows that 0 ≤ L_t ≤ L_f = c(π).
- Nominal-spending log-level ~y_t follows a Brownian motion with drift  and standard deviation _y.
- l_t = ln L_t follows a Brownian motion with a reflecting barrier at ln(L_f = c(π)).
- When  > 0 the long-run cumulative distribution of L_t, denoted P(·), exists and for 0 ≤ x ≤ L_f = c(π):
  - P(L_1 ≤ x) = 1 − (x / (L_f / c(π)))^{2  / _y^2}
  - (text statement): “the long-run cumulative distribution of L_t, denoted with P(·), is given by P(L_1 ≤ x) = (x / (L_f / c(π)))^{2  / _y^2} for 0 ≤ x ≤ L_f / c(π)” (as in the source derivation reference).

### Long-run mean unemployment and wage inflation
- Unemployment u_t = 1 − L_t.
- Long-run mean unemployment (equation (15)):
  - E[u_1] = 1 − 1
    1 +
    _y^2
    2 
    (1 − u_f) c(; _y^2; ; )
    :(15)
  - (Exact expression preserved as in the source.)
- From equilibrium condition (10): d~y_t = π^w_t + dl_t, where π^w is the rate of wage inflation.
- Long-run mean wage inflation (equation (16)):
  - E[π^w_1] =  :(16)
- Substituting (16) into (15) yields the long-run Phillips curve (equation (17)):
  - E[u_1] = 1 − 1
    1 +
    _y^2
    2 E[π^w_1]
    (1 − u_f) c(E[π^w_1]; _y^2; ; )
    (17)

### Key qualitative findings about the long-run Phillips curve
- The long-run Phillips curve is not vertical: the “natural” rate of unemployment is not unique but depends on the mean inflation rate.
- For high mean wage inflation:
  - c(π) is close to 1 and average unemployment converges to u_f.
  - The Phillips curve is virtually vertical; there is virtually no long-run trade-off between inflation and unemployment.
- For low mean wage inflation:
  - A long-run trade-off emerges; average unemployment rises as mean wage inflation falls.
- The curve’s shape depends critically on parameters , , u_f and _y^2.
  - _y^2 (variance of nominal-spending growth) can be influenced by stabilization policies and business-cycle shocks.
- In the deterministic case (no uncertainty):
  - _y^2 = 0 and c(π) = 1 → long-run unemployment equals flexible-wage unemployment.
- With stochastic shocks:
  - Higher _y^2 increases the impact of a fall in inflation on average unemployment, generating a gap relative to the flexible-wage level.
  - Two opposing channels:
    - Higher variance raises equilibrium unemployment because downward wage constraints bind more often.
    - Wage setters respond by setting lower wages when adjusting (c(π) falls), which tends to reduce average unemployment by allowing employment to exceed the flexible-wage level in favorable shocks.
  - The first channel dominates in the long run; long-run average unemployment is never below u_f, i.e. E[u_1] ≥ u_f, since (1 + _y^2 / (2 )) · c(π) ≥ 1 (as noted in the source).

### Quantitative illustrations and statistics (figures and table described)
- Calibration used in figures:
  -  = 2:5;  = 0:01; u_f = 6%.
  - Figure 2 plots E(π^w) vs E(u) for σ_y = 0%, 2%, 5%, 10%, 15%, 20% (standard deviation of nominal spending growth), annual rates; all variables in % and annualized.
  - Findings from Figure 2:
    - For high inflation rates the Phillips curve is virtually vertical at u_f.
    - For low inflation rates the curve flattens.
    - Higher σ_y raises long-run average unemployment for the same mean wage inflation.
- Table 1: Increase in long-run mean unemployment E[u_1] due to a reduction in mean wage inflation E[π^w_1], for different σ_y (all in % and annual rates). (Authors’ calculations.)
  - Reduction in E[π^w_1] from 4% to 1%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.4
    - σ_y = 10% → 3.4
    - σ_y = 15% → 9.5
    - σ_y = 20% → 16.9
  - Reduction from 5% to 2%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.1
    - σ_y = 10% → 0.8
    - σ_y = 15% → 2.8
    - σ_y = 20% → 6.0
  - Reduction from 5% to 3%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.0
    - σ_y = 10% → 0.2
    - σ_y = 15% → 1.0
    - σ_y = 20% → 2.3
  - Reduction from 6% to 3%:
    - σ_y = 0% → 0.0
    - σ_y = 2% → 0.0
    - σ_y = 5% → 0.0
    - σ_y = 10% → 0.3
    - σ_y = 15% → 1.1
    - σ_y = 20% → 2.7
- Illustrative interpretation (source text):
  - Low-volatility country (σ_y ≈ 2%): negligible unemployment increase when average wage inflation falls from 6% to 3% or from 4% to 1%.
  - Moderate volatility (σ_y ≈ 10%): cost about 0.3% increase in average unemployment when inflation falls from 6% to 3%, and about 3.4% when inflation falls from 4% to 1%.
  - High volatility countries face much larger costs of disinflation in terms of average unemployment.
  - These calculations are illustrative; realistic assessments would require more complex models.

### Role of macro volatility over recent decades
- Empirical ranges referenced in the source:
  - Industrial countries such as the U.S. and U.K.: σ (quarterly nominal and real GDP growth) ≈ 2–3%.
  - Moderate levels ≈ 4–6% (Sweden and Korea).
  - Common 5–10% (Ireland, Thailand).
  - >10% (Israel) or even >20% (Brazil, Mexico, Turkey).
- Note on endogenizing volatility:
  - It is reasonable to expect nominal GDP growth volatility to decline as inflation declines, which would steepen the Phillips curve.
  - The decline in volatility is likely limited and mainly due to reduced inflation volatility rather than reduced output growth volatility; even at zero inflation both inflation volatility and output growth volatility persist.

### Short-run Phillips curve (connection to long run)
- Short-run Phillips curve (relation between average unemployment and average inflation over a short period) can span regions where unemployment is below the flexible-wage level.
- Mechanism: endogenous upward wage rigidity — agents set wages below the flexible-wage level when they can adjust upward, anticipating future binding downward constraints.
  - When wages are low: higher chance of upward adjustment → average unemployment below flexible-case.
  - When wages are high: lower chance of adjustment → average unemployment above flexible-case.
- Short-run curve properties:
  - Tends to shift right over time as wages become more likely to bind, converging to the long-run curve.
  - Implies a significant short-run trade-off between unemployment and wage inflation in low-inflation environments; trade-off magnitude depends on macro volatility.
- Volatility effects on the short-run Phillips curve:
  - Increases probability of binding downward rigidities → raises unemployment (dominant at low inflation).
  - Makes agents more cautious in wage claims → can reduce unemployment (dominant at moderate inflation).
  - Therefore, a higher-volatility country may have a short-run Phillips curve placed more to the right at low inflation and to the left at moderate inflation.
  - As inflation increases the short-run curve converges to the flexible-wage employment level and becomes concave.
- Figure 3 (simulation description in source):
  - Simulations repeat the first 400 observations 10,000 times;  varies in (0,10] in percent annual rates.
  - In the short run average wage inflation is slightly above  for very low  because agents are cautious and set very low wages at the beginning of the horizon, making upward adjustments frequent.

*Source: _wp0934 - 6.1  Long-run Phillips curve (IMF PDF chapter).*

### 6.3  Varying the degree of downward rigidities

### 6.3  Varying the degree of downward rigidities

### Model extension and effect on the long-run Phillips curve
- The assumption dw_j^t ≥ 0 is replaced by rule (18): dw_j^t ≥ −φ(θ) w_j^t dt, allowing nominal wages to fall but constraining the percentage decline to not exceed φ(θ), where φ(θ) is a non-increasing function of the mean of nominal-spending growth, θ.
- The solution of the model is similar to the previous case with θ replaced by ψ(θ) where ψ(θ) ≡ θ + φ(θ).
- The long-run Phillips curve becomes:
  - E[u_1] = 1 − 1/(1 + σ_y^2/2 ψ(E[π_w1])) (1 − u_f) c(ψ(E[π_w1]); σ_y^2; ξ; ζ)
- The functional form of φ(θ) is crucial:
  - If φ(θ) = φ_1 (fixed amount), the Phillips curve shifts down by φ_1 compared to the benchmark.
  - If φ(θ) increases as θ declines (i.e., rigidities soften as inflation falls), the Phillips curve tilts clockwise at low inflation.
- Illustrative linear specification (equation (19)):
  - φ(θ) = φ_1 − φ_2 θ with φ_1 = 1% and φ_2 = 0.1 (annual rates).
- Numerical/contextual parameters used in figures and examples:  = 2:5;  = 0:01; u_f = 6%; φ_1 = 1%; φ_2 = 0:1; all variables in % and at annual rates.

### Long-run probability of wage rigidity and volatilities
- The wage-rigidity region is approximated by P(0 ≤ L_t ≤ L_f/c(α) − ε) for small ε > 0.
- Long-run probability expression (20):
  - P(0 ≤ L_1 ≤ L_f/c(α) − ε) = [θ/(1 − ε c(α)/L_f)] 2θ/σ_y^2 [1 − 2 E[π_w1]/σ_y^2] c(α)/L_f ε
  - Interpretation: when wage inflation is very low, probability wages remain fixed is close to one; probability declines when inflation increases; the decline is faster when macro volatility is lower.
- Figures and numerical comparisons illustrate dependence on σ_y (standard deviation of nominal spending growth) with examples σ_y = 0%, 2%, 5%, 10%, 15%, 20%.
- Key empirical patterns:
  - Volatility of wage inflation (σ(π_w)) increases as mean wage inflation E[π_w] increases (Figure 6).
  - Volatility of unemployment (σ_u) decreases as mean wage inflation E[π_w] increases (Figure 7).
  - There is a long-run trade-off between σ(π_w) and σ(u) for given σ_y (Figure 8).
- Exact variance expression for unemployment in long run:
  - Var[u_1] = (1/2) [1 + σ_y^2/E[π_w1]] [1 + E[π_w1]^2/(2 σ_y^2)] 2 θ L_f / c(E[π_w1]; σ_y^2; ξ; ζ) ^2
  - (Expression preserved as presented in source; variables and functional dependence retained.)

### Core findings and theoretical implications
- Downward nominal wage rigidities generate endogenous upward nominal wage rigidities through forward-looking wage-setter optimization.
- Overall degree of wage rigidity:
  - High when wage inflation rate is low and when variance of nominal spending shocks (σ_y) is high — nominal disturbances have strong real effects.
  - Low (wages more flexible) at high inflation rates or very small σ_y — monetary policy becomes virtually neutral.
- The long-run trade-off between average inflation and unemployment is highly non-linear:
  - Virtually inexistent at high inflation rates.
  - Becomes significant in a low inflation environment, especially under moderate to high macroeconomic volatility.
- Special limiting case: when nominal spending growth is very close to zero, the unemployment distribution collapses to a mass at 100% unemployment and its volatility collapses to zero (trade-off disappears).

### Policy implications and recommendations
- Inflation targets should be country-specific:
  - Countries with larger macroeconomic volatility (e.g., many emerging markets and developing countries) may find it desirable to target a higher inflation rate than low-volatility countries.
  - As macroeconomic volatility changes over time, inflation targets may need adjustment.
- Stabilization policies that reduce macroeconomic volatility improve the inflation–unemployment trade-off and reduce the unemployment costs of lowering long-run inflation.
- Multiple sectors are not necessary for downward rigidities to generate a long-run inflation–unemployment trade-off; an intertemporal stochastic framework suffices. However, sectoral heterogeneity would strengthen the role of inflation as relative-price adjustment "grease."
- Ignoring the possibility of a long-run trade-off can be costly for policy; model uncertainty implies policymakers should consider dependence of the probability that the true model includes a trade-off on both the wage inflation rate and volatility of nominal spending growth.

### Extensions, robustness, and caveats
- If downward rigidities soften as inflation declines, the extent of the trade-off is reduced but does not necessarily disappear.
- Adding symmetric goods-price rigidities, sectoral heterogeneity, or temporary downward rigidities would alter the shape and magnitude of the Phillips curve but would not generally make it vertical.
- The model implies nominal shocks can have persistent real effects at low inflation due to simultaneous downward and endogenous upward wage rigidities; persistence declines as inflation increases.
- An optimal inflation rate assessment requires more complex models incorporating persistence of shocks, additional effects of inflation, open-economy features, game-theoretic interactions, and other real-world frictions.

_ Source: _wp0934 - 6.3  Varying the degree of downward rigidities_

### References

### _wp0934 - References

### References (bibliography)
- Comprehensive list of cited works numbered [1] through [74], covering topics including:
  - Optimal investment, costly reversibility, and irreversible investment: [1], [12], [13], [60].
  - Wage rigidity, downward nominal and real wage rigidity, and wage-setting micro-evidence: [2], [3], [8], [25], [26], [28], [30], [32], [33], [34], [42], [43], [46], [50], [74].
  - Price stickiness and menu costs, Phillips curve, and New Keynesian frameworks: [6], [9], [10], [22], [39], [66], [72].
  - Monetary policy theory and optimal policy, including rules vs. discretion and the role of inflation: [11], [16], [17], [36], [44], [49], [53], [56], [61], [68], [69], [70], [71].
  - Empirical and theoretical analyses of inflation, unemployment, and labor-market frictions: [5], [7], [20], [21], [31], [35], [38], [47], [51], [52], [55], [58], [59], [62], [63], [64], [65], [67], [73].
- Selected methodological and mathematical references: [24] Cox and Miller (The Theory of Stochastic Processes), [27] Dixit (optimal regulation of Brownian motion), [29] Dumas, [67] Stokey (Brownian Models in Economics).

### Appendix A.1 — Derivation of conditions (14)
- Definitions and setup:
  - Let W be the space of non-decreasing non-negative stochastic processes {w_t(j)} satisfying constraint (13).
  - Objective function: E_t [ ∫_t^1 e^{-ζ(ξ-t)} π(w_ξ(j); W_ξ; ˜Y_ξ) dξ ] with π(·) concave in its first argument; hence the objective is concave over the convex set W.
  - Value function: V(w_t(j); W_t; ˜Y_t) = max_{ {w_ξ(j)}∈W } E_t [ ∫_t^1 e^{-ζ(ξ-t)} π(w_ξ(j); W_ξ; ˜Y_ξ) dξ ].
- Bellman equation and first-order conditions:
  - Bellman written as ζV(w_t(j); W_t; ˜Y_t) dt = max_{dw_t(j)} π(w_t(j); W_t; ˜Y_t) dt + E_t[dV(w_t(j); W_t; ˜Y_t)] (A.1) subject to dw_t(j) ≥ 0 (A.2).
  - Ito expansion leads to complementary slackness:
    - V_w(w_t(j); W_t; ˜Y_t) ≤ 0 for each t.
    - V_w(w_t(j); W_t; ˜Y_t) = 0 whenever dw_t(j) > 0.
- Symmetry and uniqueness:
  - Concavity and convex constraints imply unique optimal process; hence w_t(j) = W_t for each j, so dw_t(j) = dW_t and dW_t has finite variation.
  - Super-contact conditions (Dixit, 1991; Dumas, 1991) when dW_t > 0:
    - V_{ww}(W_t; W_t; ˜Y_t) = 0;
    - V_{wW}(W_t; W_t; ˜Y_t) = 0;
    - V_{wy}(W_t; W_t; ˜Y_t) = 0.
- Reduced form and key definitions:
  - Define v(W_t; ˜Y_t) ≡ V_w(W_t; W_t; ˜Y_t).
  - Define ~π_w(W_t; ˜Y_t) ≡ k_w 2/4 1/W_t 1−√(1−μ_w ˜Y_t √W_t )^{1+ϕ} 1/W_t^3 5; with k_w ≡ 1−ξ_w. (expression preserved exactly as in source).
  - Resulting dynamic equation (A.5):
    - ζ v(W_t; ˜Y_t) = ~π_w(W_t; ˜Y_t) + v_y(W_t; ˜Y_t) ˜Y_t 1_{˜Y_t≥0} + 1/2 v_{yy}(W_t; ˜Y_t) ˜Y_t^2 σ_y^2.
- Solution structure:
  - Particular solution v_p(W_t; ˜Y_t) given explicitly in the text (preserves structure and parameters such as k_w, ζ, ξ_0, ϕ, σ_y, and powers in W_t and ˜Y_t).
  - Complementary solution v_c(W_t; ˜Y_t) = k W_t^{-λ} ˜Y_t, where λ solves the characteristic equation 1/2 σ_y^2 λ^2 + λ ξ − ζ = 0 (A.9), i.e. λ = −ξ + √(ξ^2 + 2 ζ / σ_y^2) σ_y^2^{-1} (as presented).
  - General solution: v(W_t; ˜Y_t) = v_p(W_t; ˜Y_t) + k W_t^{-1−λ} ˜Y_t (A.10) for constant k to be determined.
- Boundary and matching conditions:
  - Conditions when dW_t > 0:
    - v(W(˜Y_t); ˜Y_t) = 0 (A.6);
    - v_w(W(˜Y_t); ˜Y_t) = 0 (A.7);
    - v_y(W(˜Y_t); ˜Y_t) = 0 (A.8).
  - These yield three equations (A.13)–(A.15) (two independent) determining k and W_t(˜Y_t).
- Closed-form for W_t(˜Y_t):
  - W_t(˜Y_t) = c ˜Y_t^{1/(1+ϕ)}_w (1−ϕ/(1+ϕ) √˜Y_t) where c ≡ c(ξ; σ_y^2; ϕ; ζ) with
    - c(ξ; σ_y^2; ϕ; ζ) = [ ξ + 1/2 λ(ξ; σ_y^2; ζ) σ_y^2 ]^{1/(1+ϕ)} / [ ξ + 1/2 ( λ(ξ; σ_y^2; ζ) + ϕ + 1 ) σ_y^2 ]^{1/(1+ϕ)} (as given).
  - It is shown that 0 < c(ξ; σ_y^2; ϕ; ζ) ≤ 1.
  - The text notes c(·) is non-decreasing in ϕ and provides the derivative expression (preserved verbatim) showing non-negativity via ln z + 1 − z ≤ 0.
  - Also c(ξ; σ_y^2; ϕ; ζ) = c(σ_y^2 = ξ; ϕ; ζ = ξ) because λ(ξ; σ_y^2; ζ) = λ(σ_y^2 = ξ; ζ = ξ).

### Appendix A.2 — Adding the employment constraint 0 ≤ l_jt ≤ 1
- Imposition of employment upper bound:
  - Previously computed unconstrained optimum yields W_t ≥ c ˜Y_t^{1/(1+ϕ)}_w (1−ϕ/(1+ϕ) √˜Y_t).
  - With L_t = 1 − √˜Y_t / W_t, combining implies L_t ≤ 1 − (1/(1+ϕ)) w ϕ/(1+ϕ) c^{p} = 1 − u_f c (as in source derivation).
  - Therefore c cannot be smaller than 1 − u_f; otherwise L_t > 1.
- Constrained solution:
  - If unconstrained desired c < 1 − u_f, the concavity and uniqueness imply the constrained optimal constant is c̃ ≡ 1 − u_f.
  - Resulting W(˜Y_t) = c̃(ξ; σ_y^2; ϕ; ζ; u_f) ˜Y_t^{1/(1+ϕ)}_w (1−ϕ/(1+ϕ) √˜Y_t) ≡ c̃(·) W_f_t.
  - Definition of c̃(·):
    - c̃(ξ; σ_y^2; ϕ; ζ; u_f) = c(ξ; σ_y^2; ϕ; ζ) if c ≥ 1 − u_f; otherwise c̃ = 1 − u_f.

*Source: _wp0934 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0934.pdf_