## appendix 1

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---

### Model setup
- Firms’ production function:
  - Y^k = A N^k (10)
  - A is the exogenous productivity level.
- Firms’ optimal pricing condition (given {Q_t, P_t, W_t, Y_t}_t=0^+∞ determined in general equilibrium):
  - P^k_t / P_t = [ θ/(θ−1) φ( W_t / (P_t A_t) )^φ ( P_t / Y_t ) − (1−φ) ]^{ 1/[ (1+θ)(φ−1) ] } (11)
- Consumption is the only source of demand; bonds are in zero net supply.
- Markets for bonds, goods, and labor clear; consumer-workers and firms adopt symmetric strategies.

### General equilibrium conditions
- Wage-setting (log variables, lowercase are natural logarithms):
  - w_t = μ^w_t + p_t + (γ+η) y_t − (γ+η−1) a_t (12)
  - μ^w_t ≡ ln(σ_t / σ_{t−1}) (workers’ mark-up over disutility of work)
- Firms’ price-setting:
  - p_t = μ_f + φ (w_t − a_t) − (1−φ) y_t + ln φ (13)
  - μ_f ≡ ln(θ/(θ−1)) (firms’ mark-up over marginal cost)
- Euler equation (bond market clearing; i_t ≡ − ln Q_t is the nominal interest rate):
  - i_t − (p_{t+1} − p_t) = r + γ (y_{t+1} − y_t) (14)
- Taylor rule targeting a growing price level; constrained by zero lower bound:
  - i_t = max{ 0, r + δ [ p_t − (p^*_{t−1} + π^p_* ) ] } (15)
- Price-level target stabilizes outside liquidity trap if:
  - δ [ φ(γ+η) − (1−φ) ] > 0 (16)
  - Interpretation: requires δ>0 and φ > 1/(1+γ+η).

### Results — overview
- Main implications:
  - In the long run the liquidity trap is irrelevant; standard wage-price dynamics obtain.
  - In the short run, large enough shocks to workers’ bargaining power can push the economy into the liquidity trap.
  - When fluctuations are driven by shocks to workers’ bargaining power, the Phillips-curve relation can be inverted.

### The long run (balanced growth path)
- Corporate-governance friction turned off (φ = 1) to permit a balanced growth path.
- Balanced growth path equations:
  - π^{BGP}_p = π^p_* (17)
  - π^{BGP}_w = π^p_* + g_a (18)
  - g^{BGP}_y = g_a (19)
  - i^{BGP} = r + π^p_* + γ g_a (20)
- Interpretation:
  - (18): real wages grow one-to-one with productivity growth.
  - (17): central bank anchors inflation at target.
  - (19): productivity growth drives output growth.
  - (20): pins down long-run nominal interest rate.
- Condition to avoid a permanent liquidity trap:
  - π^p_* > − ( r + γ g_a ) (21)
  - Policy implication: slower productivity growth raises the lower bound on the inflation target needed to avoid a permanent liquidity trap.
- If π^p_* violates (21), long-run rates become:
  - π^{BGP}_p = − ( r + γ g_a ) (22)
  - π^{BGP}_w = − [ r + (γ−1) g_a ] (23)

### The liquidity trap (short run)
- Focus: unanticipated changes in workers’ market power and central bank’s ability to hit inflation target.
- Result 1 — threshold for liquidity trap:
  - Define μ =
    μ^w − φ(γ+η) − (1−φ)
    -----------------------------------------
    φ γ
    r − γ + φ η − (1−φ)
    -----------------------------------------
    φ γ
    π^p_* − (γ+η) g_a
  - (24) (expression preserved as in source)
  - If and only if μ^w_t ≤ μ, then the zero lower bound on the nominal interest rate is binding and i_t = 0.
  - Comparative statics (text):
    - Higher discount rate r, higher inflation target π^p_*, and faster productivity growth g_a lower the threshold μ and reduce the likelihood of hitting the zero lower bound.
    - A lower discount rate r increases the likelihood of hitting the zero lower bound.
- Result 2:
  - Given realization of workers’ bargaining power μ^w_t, in the liquidity trap the real wage rate is lower than it would be absent the zero lower bound.
  - When the central bank cannot stimulate, real wage adjusts downwards to clear labor market.

### AS–AD analysis
- Aggregate supply (combine (12) and (13)):
  - AS: p_t = [ φ(γ+η) − (1−φ) ] / (1−φ) y_t + φ/(1−φ) μ^w_t + [ μ_f − φ(γ+η) a_t + ln φ ] / (1−φ) (25)
- Aggregate demand (combine (14) and (15); kink due to ZLB):
  - AD:
    - p_t = p^*_t, if y_t > y^{LT}_t
    - p_t = − γ y_t + r + [ γ + φ η − (1−φ) ] ( p^*_t + π_* ) − φ γ μ^w − μ_f + φ(γ+η) ( a_t + g_a ) − ln φ
      ------------------------------------------------------------------------------------------------------------
      φ(γ+η) − (1−φ)
      , if y_t ≤ y^{LT}_t
    (26)
  - Threshold output y^{LT}_t:
    - y^{LT}_t = r/γ + γ(1−φ) p^*_t + [ γ + φ η − (1−φ) ] π_* + φ γ (γ+η) ( a_t + g_a ) − γ μ_f − φ γ μ^w − γ ln φ
      ---------------------------------------------------------------------------------------------------------------
      γ [ φ(γ+η) − (1−φ) ]
    (27)
- Economic interpretation:
  - Supply: prices increase with output; an increase in μ^w_t (workers’ market power) is an inward shift of AS, pushing up wages and prices for given output.
  - Demand: kink arises because when nominal rate is stuck at zero, central bank cannot create required negative real rate; price level must fall instead.
  - Large deterioration in workers’ bargaining power shifts AS out and can push economy onto downward-sloping part of AD (right of kink): high employment with deflation and an inverted Phillips curve (negative correlation between GDP and inflation).
  - Empirical note: Japan has not shown outright Phillips-curve inversion but has experienced a flattened Phillips curve since the 1990s.

### Policy implications
- When nominal interest rate is at the ZLB:
  - Labor-market reform that strengthens workers’ bargaining power:
    - Moves price level toward target (reflation).
    - Reduces employment (firms hire fewer workers as real wages increase and firms are reluctant to raise prices one-to-one).
    - Trade-off: more inflation vs. less employment — reverse of the usual Phillips-curve trade-off.
  - Corporate-governance reform that improves managers–shareholders alignment:
    - Amplifies the reflationary effect of labor-market reforms because firms react to changing costs, transmitting labor-market developments to product markets.
- Application to reducing labor-market duality (example: single open-ended contract (SOEC) for new hires, grandfathering incumbents):
  - Paper focuses on short-run effects on aggregate supply; does not model productivity effects explicitly.
  - Two considerations argued in the text:
    - Direction of productivity effect unclear; some evidence (Aoyagi and Ganelli (2015)) suggests increasing protection for temporary workers could raise productivity via on-the-job learning.
    - Relevant bargaining power for inflation dynamics may be that of non-regular workers (their wages are marginal costs for firms), especially in Japan where permanent-worker costs are closer to sunk costs.
  - Political and implementation constraints noted: SOEC has not been implemented successfully anywhere; expanding limited regular contracts (“gentei seishain”) and clarifying legal framework could reduce duality.

### Conclusion and avenues for future work
- Main conclusion:
  - In the liquidity trap, structural factors and reforms (notably those affecting workers’ bargaining power and corporate governance) can materially affect a country’s ability to hit an inflation target.
  - Declining workers’ bargaining power in Japan since the early 1990s helps explain stagnant real wages relative to productivity and persistent near-zero CPI inflation.
- Suggested extensions and future research directions:
  - Model sources of workers’ bargaining power more explicitly, including explicit duality between contractual types (Pissarides (2009)) and Japan-specific institutions (training, contract regulation, switching/searching costs).
  - Open the economy to study exchange-rate effects and firms’ relocation decisions; account for terms-of-trade deterioration and its role in the real wage–productivity gap.
  - Explicitly model the productivity channel: allow workers’ investments in firm-specific human capital with job-duration incentives to assess direction and magnitude of productivity responses.
  - Conduct formal empirical studies on Japan’s wage-price dynamics across sectors affected differently by duality.
  - Broaden scope to other advanced economies that have experienced increased duality and flattened Phillips curves; use recent reforms in Spain and Italy as potential case studies.

### Labor demand schedule and firm optimization
- Representative firm chooses {N_j}_{j∈[0,1]} to minimize its wage bill:
  - ∫ W_j N_j  dj  (Equation (16))
- Subject to the production function:
  - Y = A (∫ N_j^{(σ−1)/σ} dj)^{σ/(σ−1)}  (Equation (17))
- Firm’s demand for each labor type (solution to minimization problem):
  - N_j = (W_j / W)^{−σ}  Y / A  (Equation (18))
- Dixit–Stiglitz wage aggregator:
  - W ≡ (∫ W_j^{1−σ} dj)^{1/(1−σ)}  (Equation (19))
- Reference for detailed derivation:
  - Blanchard and Kiyotaki (1985)

### Appendix 2 — Figures and notes
- Figure 5 and Figure 6 source:
  - Source: OECD, author’s calculations.
- Notes for both figures:
  - Real wages are total labor costs per hour worked deflated with the CPI.
  - Real labor productivity is real GDP per hour worked deflated with the GDP deflator.

*Source: _wp1620 - appendix 1.*

### appendix 1.

### appendix 1

### Model setup
- Firms’ production function:
  - Y^k = A N^k (10)
  - A is the exogenous productivity level.
- Firms’ optimal pricing condition (given {Q_t, P_t, W_t, Y_t}_t=0^+∞ determined in general equilibrium):
  - P^k_t / P_t = [ θ/(θ−1) φ( W_t / (P_t A_t) )^φ ( P_t / Y_t ) − (1−φ) ]^{ 1/[ (1+θ)(φ−1) ] } (11)
- Consumption is the only source of demand; bonds are in zero net supply.
- Markets for bonds, goods, and labor clear; consumer-workers and firms adopt symmetric strategies.

### General equilibrium conditions
- Wage-setting (log variables, lowercase are natural logarithms):
  - w_t = μ^w_t + p_t + (γ+η) y_t − (γ+η−1) a_t (12)
  - μ^w_t ≡ ln(σ_t / σ_{t−1}) (workers’ mark-up over disutility of work)
- Firms’ price-setting:
  - p_t = μ_f + φ (w_t − a_t) − (1−φ) y_t + ln φ (13)
  - μ_f ≡ ln(θ/(θ−1)) (firms’ mark-up over marginal cost)
- Euler equation (bond market clearing; i_t ≡ − ln Q_t is the nominal interest rate):
  - i_t − (p_{t+1} − p_t) = r + γ (y_{t+1} − y_t) (14)
- Taylor rule targeting a growing price level; constrained by zero lower bound:
  - i_t = max{ 0, r + δ [ p_t − (p^*_{t−1} + π^p_* ) ] } (15)
- Price-level target stabilizes outside liquidity trap if:
  - δ [ φ(γ+η) − (1−φ) ] > 0 (16)
  - Interpretation: requires δ>0 and φ > 1/(1+γ+η).

### Results — overview
- Main implications:
  - In the long run the liquidity trap is irrelevant; standard wage-price dynamics obtain.
  - In the short run, large enough shocks to workers’ bargaining power can push the economy into the liquidity trap.
  - When fluctuations are driven by shocks to workers’ bargaining power, the Phillips-curve relation can be inverted.

### The long run (balanced growth path)
- Corporate-governance friction turned off (φ = 1) to permit a balanced growth path.
- Balanced growth path equations:
  - π^{BGP}_p = π^p_* (17)
  - π^{BGP}_w = π^p_* + g_a (18)
  - g^{BGP}_y = g_a (19)
  - i^{BGP} = r + π^p_* + γ g_a (20)
- Interpretation:
  - (18): real wages grow one-to-one with productivity growth.
  - (17): central bank anchors inflation at target.
  - (19): productivity growth drives output growth.
  - (20): pins down long-run nominal interest rate.
- Condition to avoid a permanent liquidity trap:
  - π^p_* > − ( r + γ g_a ) (21)
  - Policy implication: slower productivity growth raises the lower bound on the inflation target needed to avoid a permanent liquidity trap.
- If π^p_* violates (21), long-run rates become:
  - π^{BGP}_p = − ( r + γ g_a ) (22)
  - π^{BGP}_w = − [ r + (γ−1) g_a ] (23)

### The liquidity trap (short run)
- Study focuses on unanticipated changes in workers’ market power and central bank’s ability to hit inflation target.
- Result 1 — threshold for liquidity trap:
  - Define μ =
    μ^w − φ(γ+η) − (1−φ)
    -----------------------------------------
    φ γ
    r − γ + φ η − (1−φ)
    -----------------------------------------
    φ γ
    π^p_* − (γ+η) g_a
  - (24) (expression preserved as in source)
  - If and only if μ^w_t ≤ μ, then the zero lower bound on the nominal interest rate is binding and i_t = 0.
  - Comparative statics stated in text:
    - Higher discount rate r, higher inflation target π^p_*, and faster productivity growth g_a lower the threshold μ and reduce the likelihood of hitting the zero lower bound.
    - A lower discount rate r increases the likelihood of hitting the zero lower bound.
- Result 2:
  - Given realization of workers’ bargaining power μ^w_t, in the liquidity trap the real wage rate is lower than it would be absent the zero lower bound.
  - When the central bank cannot stimulate, real wage adjusts downwards to clear labor market.

### AS–AD analysis
- Aggregate supply (combine (12) and (13)):
  - AS: p_t = [ φ(γ+η) − (1−φ) ] / (1−φ) y_t + φ/(1−φ) μ^w_t + [ μ_f − φ(γ+η) a_t + ln φ ] / (1−φ) (25)
- Aggregate demand (combine (14) and (15); kink due to ZLB):
  - AD:
    - p_t = p^*_t, if y_t > y^{LT}_t
    - p_t = − γ y_t + r + [ γ + φ η − (1−φ) ] ( p^*_t + π_* ) − φ γ μ^w − μ_f + φ(γ+η) ( a_t + g_a ) − ln φ
      ------------------------------------------------------------------------------------------------------------
      φ(γ+η) − (1−φ)
      , if y_t ≤ y^{LT}_t
    (26)
  - Threshold output y^{LT}_t:
    - y^{LT}_t = r/γ + γ(1−φ) p^*_t + [ γ + φ η − (1−φ) ] π_* + φ γ (γ+η) ( a_t + g_a ) − γ μ_f − φ γ μ^w − γ ln φ
      ---------------------------------------------------------------------------------------------------------------
      γ [ φ(γ+η) − (1−φ) ]
    (27)
- Economic interpretation:
  - Supply: prices increase with output; an increase in μ^w_t (workers’ market power) is an inward shift of AS, pushing up wages and prices for given output.
  - Demand: kink arises because when nominal rate is stuck at zero, central bank cannot create required negative real rate; price level must fall instead.
  - Large deterioration in workers’ bargaining power shifts AS out and can push economy onto downward-sloping part of AD (right of kink): high employment with deflation and an inverted Phillips curve (negative correlation between GDP and inflation).
  - Empirical note: Japan has not shown outright Phillips-curve inversion but has experienced a flattened Phillips curve since the 1990s.

### Policy implications
- When nominal interest rate is at the ZLB:
  - Labor-market reform that strengthens workers’ bargaining power:
    - Moves price level toward target (reflation).
    - Reduces employment (firms hire fewer workers as real wages increase and firms are reluctant to raise prices one-to-one).
  - Trade-off: more inflation vs. less employment — reverse of the usual Phillips-curve trade-off.
- Corporate-governance reform that improves managers–shareholders alignment:
  - Amplifies the reflationary effect of labor-market reforms because firms react to changing costs, transmitting labor-market developments to product markets.
- Application to reducing labor-market duality (example: single open-ended contract (SOEC) for new hires, grandfathering incumbents):
  - Paper focuses on short-run effects on aggregate supply; does not model productivity effects explicitly.
  - Two considerations argued in the text:
    - Direction of productivity effect unclear; some evidence (Aoyagi and Ganelli (2015)) suggests increasing protection for temporary workers could raise productivity via on-the-job learning.
    - Relevant bargaining power for inflation dynamics may be that of non-regular workers (their wages are marginal costs for firms), especially in Japan where permanent-worker costs are closer to sunk costs.
  - Political and implementation constraints noted: SOEC has not been implemented successfully anywhere; expanding limited regular contracts (“gentei seishain”) and clarifying legal framework could reduce duality.

### Conclusion and avenues for future work
- Main conclusion:
  - In the liquidity trap, structural factors and reforms (notably those affecting workers’ bargaining power and corporate governance) can materially affect a country’s ability to hit an inflation target.
  - Declining workers’ bargaining power in Japan since the early 1990s helps explain stagnant real wages relative to productivity and persistent near-zero CPI inflation.
- Suggested extensions and future research directions:
  - Model sources of workers’ bargaining power more explicitly, including explicit duality between contractual types (Pissarides (2009)) and Japan-specific institutions (training, contract regulation, switching/searching costs).
  - Open the economy to study exchange-rate effects and firms’ relocation decisions; account for terms-of-trade deterioration and its role in the real wage–productivity gap.
  - Explicitly model the productivity channel: allow workers’ investments in firm-specific human capital with job-duration incentives to assess direction and magnitude of productivity responses.
  - Conduct formal empirical studies on Japan’s wage-price dynamics across sectors affected differently by duality.
  - Broaden scope to other advanced economies that have experienced increased duality and flattened Phillips curves; use recent reforms in Spain and Italy as potential case studies.

*Source: _wp1620 - appendix 1.*

### APPENDIX 1

### APPENDIX 1

### Labor demand schedule and firm optimization
- Representative firm chooses {N_j}_{j∈[0,1]} to minimize its wage bill:
  - ∫ W_j N_j  dj  (Equation (16))
- Subject to the production function:
  - Y = A (∫ N_j^{(σ−1)/σ} dj)^{σ/(σ−1)}  (Equation (17))
- Firm’s demand for each labor type (solution to minimization problem):
  - N_j = (W_j / W)^{−σ}  Y / A  (Equation (18))
- Dixit–Stiglitz wage aggregator:
  - W ≡ (∫ W_j^{1−σ} dj)^{1/(1−σ)}  (Equation (19))
- Reference for detailed derivation:
  - Blanchard and Kiyotaki (1985)

### Appendix 2 — Figures and notes
- Figure 5 and Figure 6 source:
  - Source: OECD, author’s calculations.
- Notes for both figures:
  - Real wages are total labor costs per hour worked deflated with the CPI.
  - Real labor productivity is real GDP per hour worked deflated with the GDP deflator.

### References (document continuation)
- The appendix is followed by a References section containing cited works including, among others:
  - Abe, M., Y. Higuchi, and M. Sunada. 2004.
  - Blanchard, O.J., and N. Kiyotaki. 1985.
  - Eggertsson, G., A. Ferrero, and A. Raffo. 2014.
  - Galí, J. 2008.
  - Woodford, M. 2012.
- (Full reference list appears in the source document.)

*Source: _wp1620 - APPENDIX 1*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1620.pdf_