## Annex I. Solving the model in Section V

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### Value Functions
- Value of an unemployed worker:
  - U = λK + β[θ q(θ) W + (1 − θ q(θ)) U]. (1)
- Value of employment (worker):
  - W = w + λK + β[s U + (1 − s) W]. (2)
- Value of an unmatched firm (vacancy):
  - V = −c + β[q(θ) J + (1 − q(θ)) V]. (3)
- Capital demand and output per firm/worker:
  - α k^{α−1} = λ.
  - y = k^{α} = (λ/α)^{α/(α−1)}.
  - α share of output y is paid as capital rentals; firm retains the rest and pays wage w.
- Value of a filled vacancy (J):
  - J = (1 − α) y − w − λ + β[s V + (1 − s) J]. (4)
- Free entry (V = 0) implies:
  - c / [β q(θ)] = [(1 − α) y − w] / [1 − β(1 − s)].
  - With β = (1 + r)^{−1}, job creation curve:
    - w = (1 − α) y − (r + s) / q(θ) c. (5)

### Nash Bargaining
- Match-surplus for unemployed worker:
  - S_U = W − U.
  - W = (1 + r) / (r + s) [w + λK + s U / (1 + r)].
  - S_U = (1 + r) / (r + s) [w + λK − r/(1 + r) U]. (from W expression)
- Match surplus for firm:
  - S_f = J − V = J = (1 + r) / (r + s) ((1 − α) y − w).
- Total surplus:
  - S_tot = S_U + S_f = [(1 − α) y + λK − r/(1 + r) U].
- Nash solution S_U = ϕ S_tot and S_f = (1 − ϕ) S_tot implies:
  - (1 − ϕ)[w + λK − r/(1 + r) U] = ϕ((1 − α) y − w).
  - Hence:
    - w = ϕ(1 − α) y + (1 − ϕ)( r/(1 + r) U − λK ). (6)
- Eliminating U:
  - S_U = W − U = (1 − β) U − λK / [β θ q(θ)] = r U − (1 + r) λK / [θ q(θ)]. (7)
  - Also S_U = ϕ S_tot = ϕ/(1 − ϕ) S_f = ϕ/(1 − ϕ) J = ϕ/(1 − ϕ) c / [β q(θ)]. (8)
  - Equating (7) and (8) yields:
    - r U − (1 + r) λK / [θ q(θ)] = ϕ/(1 − ϕ) c / β,
    - delivering U in terms of θ and parameters:
      - U = (1 + r) / r [ ϕ/(1 − ϕ) c θ + λK ]. (9)
  - Replacing U in (6) gives the wage curve:
    - w = ϕ((1 − α) y + c θ). (10)
- The job creation curve (5) and wage curve (10) determine w*, θ*.

### Stationary Equilibrium
- Capital rental market clearing:
  - k(1 − u) L = (λ/α)^{1/(α−1)} (1 − u) L = K.
  - Hence λ = α (K / [(1 − u) L])^{α−1}.
- Output per firm/worker:
  - y = (λ/α)^{α/(α−1)} = [ K / ((1 − u) L) ]^{α}. (11)
- Flow equilibrium (separation vs. job finding):
  - s (1 − u) L = u L θ q(θ).
  - Equilibrium unemployment:
    - u = s / [s + θ q(θ)]. (12)
- The job creation curve (5), wage curve (10), output (11), and unemployment (12) form a system determining stationary equilibrium values of θ, u, y, and w.

### Comparative statics
- Eliminating w using (5) and (10), replacing y via (11) and u via (12) yields a single equation in θ:
  - (1 − ϕ)(1 − α)[ (s / [θ q(θ)] + 1) K / L ]^{α} = [ (r + s) / q(θ) + ϕ θ ] c. (13)
- Effect of labor force participation rate L:
  - dθ / dL < 0.
  - The derivative expression given in the text:
    - dθ/dL = (1 − ϕ)(1 − α)[(s θ q(θ) + 1) K / L]^{α} × α / [ L (−σ) (1 − ϕ)(1 − α)^{α} [(s θ q(θ) + 1) K / L]^{α−1} K / L s (1/μ θ − σ − 1) − (r + s) c (1/μ (1 − σ) θ − σ) − c ϕ ] < 0.
  - Sign of derivative is negative because numerator positive and denominator negative.
- Effect of matching efficiency μ:
  - q(θ) = μ θ^{σ−1}.
  - dθ / dμ > 0.
  - The text shows the numerator of the RHS is proportional to:
    - (1 − ϕ)(1 − α)^{α}[(s θ q(θ) + 1) K / L]^{α} s − (r + s) c θ (s θ q(θ) + 1).
  - Using (13) this reduces to:
    - (α − 1) (r + s) / q(θ) s c + [α s ϕ − (r + s)] c θ,
    - which is negative because α < 1 and α s ϕ < s (since ϕ < 1).
  - Thus dθ/dμ > 0.
- Implications:
  - Since θ q(θ) is increasing in θ and dθ/dμ > 0, equation (12) implies d u / d μ < 0.
  - Equation (11) then implies y is decreasing in μ.
  - With equation (5), this implies d w / d μ < 0.

### Wage comparison
- Consider two economies i = 1, 2 with L1 < L2 and μ1 < μ2, all other parameters identical.
- If θ1 = θ2 = θ, then w1 > w2.
- From (5) and (10):
  - (1 − ϕ)(1 − α) y_i = [ (r + s) / (μ_i θ^{σ−1}) + ϕ θ ] c. (14)
  - Since θ1 = θ2 and μ1 < μ2, (14) implies y1 > y2.
  - Recall (5):
    - w_i = (1 − α) y_i − (r + s) / q(θ) c.
  - Thus w1 > w2.

*IMF Working Paper No. WP/2025/082 — Annex I. Solving the model in Section V*

### Annex I. Solving the model in Section V ................................................................................

### Annex I. Solving the model in Section V

### Content summary
- Annex I. Solving the model in Section V ......................................................................................................... 17

*Source: wpiea2025082-print-pdf - Annex I. Solving the model in Section V (page listing as provided)*

### References .............................................................................................................

### wpiea2025082-print-pdf - References

### Overview and key findings
- Korea entered the COVID-19 pandemic with sound fundamentals; the paper defines "post-pandemic" as since 2021Q3 and "pre-pandemic" as before 2020Q1.
- Unemployment peaked at 4.8 percent in January 2021 and then declined steadily; since 2021Q2 the unemployment rate has remained below 3.0 percent, compared with a pre-pandemic norm of 3.5 – 4.0 percent.
- The Beveridge curve for Korea shifted inward since the pandemic began, suggesting higher labor market matching efficiency.
- Using a standard labor market flows framework with a Cobb-Douglas matching function, the paper finds that sustained improvements in matching efficiency are the main driver of Korea’s low post-pandemic unemployment.
- Matching efficiency improvements began at least two years before the pandemic, fell sharply in 2020Q2, rebounded within a quarter, and remained at higher levels through 2024Q4.

### Labor market developments (supply and demand)
- Labor supply:
  - Labor force participation rate rose from 62.4 percent in January 2014 to 64.7 percent in December 2024.
  - Female and elderly (60+) participation rose; the elderly unemployment rate fell from as high as 4.5 percent in early-2021, but unemployment for those below 60 also trended down and stayed below 3 percent since 2021H2.
  - The share of regular workers increased from about 65 percent in January 2014 to 75 percent by end-2024.
- Labor demand:
  - The job vacancy rate averaged 0.7 percent in 2018-2019.
  - Job vacancies fell sharply in 2020, rebounded strongly in 2021, peaked in the first half of 2022, then moderated from 2022H2.
  - Job openings rate patterns differ by sector: manufacturing showed more volatility than non-manufacturing.
- Interpretation:
  - The recovery in labor supply to its pre-COVID rising trend implies labor force participation is unlikely to be the main cause of low post-pandemic unemployment.
  - The upward trend in the share of regular workers contradicts the hypothesis that substitution of regular by temporary workers drove lower unemployment.

### Quantifying matching efficiency and decomposition methodology
- Model framework:
  - Law of motion for employment and a matching function Ht = μt Vt^σ Ut^(1−σ).
  - Variables converted to rates relative to the labor force: ut (unemployment rate), vt (vacancy rate), ht (gross hires ratio), nt (labor force growth).
  - Parameter σ chosen as 0.4 (midpoint of literature range 0.3–0.5); results robust across the range σ = 0.3, 0.4, 0.5.
- Estimation approach:
  - Two-step procedure: compute ht from the law of motion, then back out μt from ht, vt, ut.
  - Data sources: job separation rates from Ministry of Employment and Labor (monthly from Jan-2018), other variables from Haver Analytics.
- Empirical estimates:
  - Matching efficiency shows a clear rising trend through 2022Q1 and stabilization thereafter.
  - The matching efficiency fell significantly in 2020Q2 but rebounded strongly within a quarter.
  - Improvements in matching efficiency began well before the pandemic and sustained at much higher values relative to pre-pandemic.

### Counterfactual scenarios and decomposition results (2020–2024)
- Simulation framework: use nonlinear unemployment dynamics (equation (5)) to simulate ut paths holding one key variable constant at its pre-pandemic (2019Q4) or reference level:
  - Variables considered: matching efficiency μt, labor force participation rate (LFPR), working-age population growth (WAP growth), vacancy rate vt.
- Constant matching efficiency scenario:
  - Holding μt at its 2019Q4 level would have produced a simulated unemployment path significantly higher than actual (except in 2020Q2), implying higher μt materially lowered unemployment since 2020H2.
- Constant LFPR scenario:
  - Holding LFPR at 2019Q4 shows the initial pandemic decline in LFPR temporarily reduced unemployment; after 2020Q4, LFPR recovery had little effect on unemployment.
- Constant working-age population growth scenario:
  - Holding WAP growth at 2019Q4 levels produces a somewhat higher simulated unemployment path for much of the post-pandemic period, indicating slower WAP growth moderately lowered unemployment since 2021.
- Constant vacancy rate scenario:
  - Holding vt at its 2018–2019 average (0.7 percent) shows that lower vacancies during the pandemic increased unemployment; after vacancies recovered, their contribution to unemployment was moderate.
- Aggregate decomposition (annual, 2020–2024):
  - 2020–2021: reduction in job vacancies was the largest upward force on unemployment; higher matching efficiency and decline in labor force pushed unemployment down.
  - 2022–2024: higher matching efficiency was the dominant contributor to lower unemployment; recovery of vacancies and slower WAP growth also played roles; rising LFPR exerted some upward pressure.
  - Job separation rates remained relatively stable and had little effect (see Figure 12).

### Medium-term structural unemployment projection (steady-state analysis)
- Steady-state equation (6) used to compute structural unemployment u̅ given steady-state parameters.
- Steady-state parameter assumptions and values:
  - Job vacancy rate v̅ = 0.68% (average level in 2023–2024).
  - Job separation rate s̅ = 5.1% (mean in the last three years).
  - Matching efficiency μ̅ = 3.2 (quarterly average in 2024).
  - Labor force growth rate n̅ = 0.02% (average q/q labor force growth between 2025 and 2030 from projections).
  - Pre-COVID (2018–2019) reference values: v = 0.70%, s = 5.0%, μ = 2.6, n = 0.2% (Table I).
- Labor force and population projections:
  - United Nations projects WAP growth will decline from 0.31 percent in 2025 to 0.10 percent by 2030.
  - Labor force participation projected to peak in 2025 and gradually decline thereafter (cohort approach updating Swiston (2021)); combining these gives labor force growth that slows and starts shrinking in 2028.
  - Resulting average q/q labor force growth is 0.02 percent between 2025 and 2030; labor force projection plotted in Figure 13.
- Structural unemployment results:
  - Calculated pre-COVID structural unemployment (2018–2019) = 3.8 percent (in line with actual).
  - Projected medium-term structural unemployment = 2.7 percent (close to actual unemployment in 2023 and 2024: 2.7 percent and 2.8 percent respectively).
  - Reduction relative to pre-COVID structural unemployment = 1.1 percentage points.
  - Decomposition: nearly all of the decline is accounted for by higher matching efficiency; slower labor force growth contributes moderately.
  - Robustness check: with v̅ = 0.6, structural unemployment = 2.9 percent.
- Key steady-state parameter table (from Table I):
  - v (Job vacancy rate): pre-COVID 0.70% ; steady-state assumption 0.68%
  - s (Job separation rate): pre-COVID 5.0% ; steady-state assumption 5.1%
  - μ (Labor market matching efficiency): pre-COVID 2.6 ; steady-state assumption 3.2
  - n (Labor force growth rate, quarterly): pre-COVID 0.2% ; steady-state assumption 0.02%

### Wage growth, matching efficiency, and the DMP model analysis
- Empirical puzzle:
  - Despite elevated labor market tightness, nominal and real wage growth slowed since 2021 and persisted at lower rates vs pre-pandemic norms (Figure 15).
  - Korea’s economic growth reached 2.7 percent in 2022, above potential growth (~2 percent), yet wage growth slowed that year.
- Model development and mechanisms:
  - A variant of the Diamond-Mortensen-Pissarides (DMP) model is developed with two extensions: (1) labor force participation margin; (2) physical capital in production.
  - Key model elements:
    - Measure 1 of working-age population; labor force participation rate L; unemployment rate u.
    - Matching function M(u,v) = μ v^σ u^(1−σ); tightness θ = v/u.
    - Probability vacancy filled q(θ) = μ θ^(σ−1); job finding rate θ q(θ).
    - Firms rent capital K (aggregate fixed), produce y = k^α with one worker; wage determined by Nash bargaining with worker share φ.
    - Stationary equilibrium conditions include job creation curve and wage curve:
      - Job creation curve: w* = (1−α) y* − (r+s)/q(θ*) c.
      - Wage curve: w* = φ[(1−α) y* + c θ*].
      - Stationary unemployment: u* = s / (s + θ* q(θ*)).
  - Comparative statics:
    - Both a decline in labor force participation L and an increase in matching efficiency μ lead to higher labor market tightness θ*.
    - A decline in L raises capital per employed worker and output per firm, inducing more vacancy creation (tighter labor market) and higher wages.
    - An increase in μ increases job finding rates and reduces unemployment but does not raise output per worker; at the same level of θ*, wages are lower when tightness stems from higher μ than when it stems from lower L.
- Interpretation:
  - Higher matching efficiency can explain the co-existence of tighter labor markets and subdued wage growth: tightness driven by improved matching efficiency generates lower wages than tightness driven by lower labor supply.

### Conclusion and implications
- Main conclusion:
  - Sustained increases in labor market matching efficiency are the principal driver of Korea’s low post-pandemic unemployment and are likely to keep medium-term structural unemployment below 3 percent, substantially lower than pre-pandemic norms.
- Additional points:
  - The timing and persistence of matching efficiency gains suggest structural rather than purely cyclical drivers (e.g., digitalization of job search, online hiring platforms, information policies, sectoral shifts toward services).
  - The paper highlights the need for future research to unpack the sources of increased matching efficiency.
- Uncertainty:
  - There is considerable uncertainty around the medium-term vacancy rate and other parameters; alternative steady-state assumptions (e.g., v̅ = 0.6) yield slightly higher structural unemployment (2.9 percent) but the qualitative conclusion of lower structural unemployment remains.

*Source: IMF Working Paper "Labor Market Matching Efficiency and Korea’s Low Post-Pandemic Unemployment" (content unit: wpiea2025082-print-pdf - References).*

### Annex I. Solving the model in Section V

### Annex I. Solving the model in Section V

### Value Functions
- Value of an unemployed worker:
  - U = λK + β[θ q(θ) W + (1 − θ q(θ)) U]. (1)
- Value of employment (worker):
  - W = w + λK + β[s U + (1 − s) W]. (2)
- Value of an unmatched firm (vacancy):
  - V = −c + β[q(θ) J + (1 − q(θ)) V]. (3)
- Capital demand and output per firm/worker:
  - α k^{α−1} = λ.
  - y = k^{α} = (λ/α)^{α/(α−1)}.
  - α share of output y is paid as capital rentals; firm retains the rest and pays wage w.
- Value of a filled vacancy (J):
  - J = (1 − α) y − w − λ + β[s V + (1 − s) J]. (4)
- Free entry (V = 0) implies:
  - c / [β q(θ)] = [(1 − α) y − w] / [1 − β(1 − s)].
  - With β = (1 + r)^{−1}, job creation curve:
    - w = (1 − α) y − (r + s) / q(θ) c. (5)

### Nash Bargaining
- Match-surplus for unemployed worker:
  - S_U = W − U.
  - W = (1 + r) / (r + s) [w + λK + s U / (1 + r)].
  - S_U = (1 + r) / (r + s) [w + λK − r/(1 + r) U]. (from W expression)
- Match surplus for firm:
  - S_f = J − V = J = (1 + r) / (r + s) ((1 − α) y − w).
- Total surplus:
  - S_tot = S_U + S_f = [(1 − α) y + λK − r/(1 + r) U].
- Nash solution S_U = ϕ S_tot and S_f = (1 − ϕ) S_tot implies:
  - (1 − ϕ)[w + λK − r/(1 + r) U] = ϕ((1 − α) y − w).
  - Hence:
    - w = ϕ(1 − α) y + (1 − ϕ)( r/(1 + r) U − λK ). (6)
- Eliminating U:
  - S_U = W − U = (1 − β) U − λK / [β θ q(θ)] = r U − (1 + r) λK / [θ q(θ)]. (7)
  - Also S_U = ϕ S_tot = ϕ/(1 − ϕ) S_f = ϕ/(1 − ϕ) J = ϕ/(1 − ϕ) c / [β q(θ)]. (8)
  - Equating (7) and (8) yields:
    - r U − (1 + r) λK / [θ q(θ)] = ϕ/(1 − ϕ) c / β,
    - delivering U in terms of θ and parameters:
      - U = (1 + r) / r [ ϕ/(1 − ϕ) c θ + λK ]. (9)
  - Replacing U in (6) gives the wage curve:
    - w = ϕ((1 − α) y + c θ). (10)
- The job creation curve (5) and wage curve (10) determine w*, θ*.

### Stationary Equilibrium
- Capital rental market clearing:
  - k(1 − u) L = (λ/α)^{1/(α−1)} (1 − u) L = K.
  - Hence λ = α (K / [(1 − u) L])^{α−1}.
- Output per firm/worker:
  - y = (λ/α)^{α/(α−1)} = [ K / ((1 − u) L) ]^{α}. (11)
- Flow equilibrium (separation vs. job finding):
  - s (1 − u) L = u L θ q(θ).
  - Equilibrium unemployment:
    - u = s / [s + θ q(θ)]. (12)
- The job creation curve (5), wage curve (10), output (11), and unemployment (12) form a system determining stationary equilibrium values of θ, u, y, and w.

### Comparative statics
- Eliminating w using (5) and (10), replacing y via (11) and u via (12) yields a single equation in θ:
  - (1 − ϕ)(1 − α)[ (s / [θ q(θ)] + 1) K / L ]^{α} = [ (r + s) / q(θ) + ϕ θ ] c. (13)
- Effect of labor force participation rate L:
  - dθ / dL < 0.
  - The derivative expression given in the text:
    - dθ/dL = (1 − ϕ)(1 − α)[(s θ q(θ) + 1) K / L]^{α} × α / [ L (−σ) (1 − ϕ)(1 − α)^{α} [(s θ q(θ) + 1) K / L]^{α−1} K / L s (1/μ θ − σ − 1) − (r + s) c (1/μ (1 − σ) θ − σ) − c ϕ ] < 0.
  - Sign of derivative is negative because numerator positive and denominator negative.
- Effect of matching efficiency μ:
  - q(θ) = μ θ^{σ−1}.
  - dθ / dμ > 0.
  - The text shows the numerator of the RHS is proportional to:
    - (1 − ϕ)(1 − α)^{α}[(s θ q(θ) + 1) K / L]^{α} s − (r + s) c θ (s θ q(θ) + 1).
  - Using (13) this reduces to:
    - (α − 1) (r + s) / q(θ) s c + [α s ϕ − (r + s)] c θ,
    - which is negative because α < 1 and α s ϕ < s (since ϕ < 1).
  - Thus dθ/dμ > 0.
- Implications:
  - Since θ q(θ) is increasing in θ and dθ/dμ > 0, equation (12) implies d u / d μ < 0.
  - Equation (11) then implies y is decreasing in μ.
  - With equation (5), this implies d w / d μ < 0.

### Wage comparison
- Consider two economies i = 1, 2 with L1 < L2 and μ1 < μ2, all other parameters identical.
- If θ1 = θ2 = θ, then w1 > w2.
- From (5) and (10):
  - (1 − ϕ)(1 − α) y_i = [ (r + s) / (μ_i θ^{σ−1}) + ϕ θ ] c. (14)
  - Since θ1 = θ2 and μ1 < μ2, (14) implies y1 > y2.
  - Recall (5):
    - w_i = (1 − α) y_i − (r + s) / q(θ) c.
  - Thus w1 > w2.

*IMF Working Paper No. WP/2025/082 — Annex I. Solving the model in Section V*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025082-print-pdf.pdf_