## 1. Growth Multiplier Under Cobb‑Douglas Production Function

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---

### A. Growth slowdown in Chile: facts and magnitudes
- Chile’s average annual real GDP growth rates:
  - 1991-99: 6.2 percent
  - 2000-09: 4.2 percent
  - 2010-19: 3.3 percent
  - 2020-23: 1.9 percent (close to the Central Bank of Chile projected potential growth rate of around 2 percent for the next ten years)
- Main proximate driver: marked slowdown in total factor productivity (TFP) growth.
  - According to CNEP estimates, compared to the 1990s, the average annual contribution of TFP to growth declined by 2.5 percentage points in the 2010s.
  - Contributions of labor and capital each fell by 0.2 percentage points in the 2010s relative to the 1990s.
- Solow model illustration:
  - With a capital share of 0.5, a 2.5 percentage point drop in TFP growth would predict a 5-percentage point reduction in capital accumulation along the balanced growth path; actual annual capital growth decelerated only by 0.2 percentage points between the 1990s and 2010s.
- Sectoral studies (summarized):
  - CNEP (2017): declining productivity in copper mining; recommends improvements in permit timelines and standards for suppliers/contractors and labor training.
  - CNEP (2020a): inefficiencies in construction sector—planning of public works, uncertainty around building permits, low adoption of digital tools.
  - CNEP (2020b): suboptimal utilization of surgical rooms, contributing to longer wait times for surgeries.
  - Asturias et al. (2023): entry barriers impact manufacturing sector productivity growth.
- Author’s focus: input-output linkages and intermediate input use (share of gross output), and mapping of gross output-to-value added (Q/Y) ratios as summary measures of production network connectedness.

### B. Theoretical background: Cobb‑Douglas growth multiplier and Domar weights
- Producer-level Cobb-Douglas setup (single intermediate input):
  - q_i = a_i l_i^{1−σ_i} x_i^{σ_i}
  - Log-differentiation: Δlog(q_i) = Δlog(a_i) + (1−σ_i) Δlog(l_i) + σ_i Δlog(x_i)  (Equation (1.1))
  - With value-added y_i and fixed intermediate share σ_i: Δlog(q_i) = (1−σ_i) Δlog(y_i) + σ_i Δlog(x_i)  (Equation (1.2))
  - Combining: Δlog(y_i) = Δlog(a_i)/(1−σ_i) + Δlog(l_i)
  - Producer TFP growth (conventional growth accounting): ΔTFP_i = Δlog(a_i)/(1−α_i)
- Aggregate TFP growth with N producers:
  - ΔTFP = Σ (Y_i/Y) ⋅ ΔTFP_i = Σ (Q_i/Y) ⋅ Δlog(a_i)  (Equation (1.3))
  - Interpretation: aggregate TFP growth is weighted by gross output shares (Domar weights λ_i = Q_i/Y), not value-added shares; Σ λ_i = Q/Y > 1 when intermediate inputs are used, creating an amplification/multiplier effect.
- Generalized result (non-Cobb-Douglas, Leontief framework):
  - Δlog(TFP) = Σ λ_i Δlog(a_i) where λ_i = Q_i/Y (Domar weight) and Σ λ_i = Q/Y (Equation (1))
  - Gross output-to-GDP ratio (Q/Y) measures the production network multiplier.

### C. Cross‑country empirical relationship and elasticities
- Cross-country OLS (dependent variable: average real GDP per capita growth 2014-19):
  - Coefficient on gross output-to-VA (Q/Y): 3.04*** in Column (1); 2.90*** in Column (2) after controlling for log(GDP per capita) in 2014.
  - Coefficient on Log(GDP per capita) in 2014 (PPP): -1.13** in Column (2).
  - Constant terms: -4.29 (Column (1)); 7.79 (Column (2)).
  - Note on statistical notation: *** (**) stands for p value<1% (5%).
- Interpretation: a 0.1 increase in the gross output-to-GDP ratio is associated with an around 0.3 percentage point increase in real GDP per capita growth (using Column (1) elasticity).
- Cross-country variation in Q/Y (2014 examples):
  - Korea and several Eastern European countries: Q/Y ranging from 2.4 to 2.7.
  - Chile and G7 countries: Q/Y around 2.0.
  - Mexico and Costa Rica: Q/Y around 1.8.
- Robustness: McNerny et al. (2022) find positive correlation between Q/Y in 1995 and real GDP per capita growth during 1995-2000 using WOID sample.

### D. Input‑output linkage in Chile: levels, trends, and decomposition
- Level comparisons:
  - Chile’s Q/Y averaged around 2.0 in the 2010s (BCCh input-output tables); OECD average: 2.1 during the same period.
  - In 2018, Chile’s Q/Y was lower than Korea by 0.47 and lower than Czech Republic by 0.62.
  - Using the elasticity 2.9 (Column (2) of Text Table 1), the lower Q/Y in Chile would translate into slower annual real GDP per capita growth than Korea (Czech Republic) by 1.4 (1.8) percentage points.
- Time trend (Chile):
  - Q/Y fell from 2.16 in 2008 to 1.92 in 2021 (BCCh input-output tables).
  - The elasticity in Text Table 1 implies the lower intensity of intermediate input use in 2021 compared to 2008 would translate into a deceleration in annual real GDP per capita growth by 0.7 percentage points.
  - To isolate commodity price effects: a hypothetical non-mining and non-utility economy shows Q/Y declined by 0.19 between 2008 and 2013, compared to a 0.18 decline in the actual economy — implying copper and oil price fluctuations were not the primary drivers of the aggregate Q/Y decline.
- Sources of Chile’s lower Q/Y relative to Korea and Czech Republic:
  - Smaller manufacturing share in Chile’s economy.
  - Lower intermediate input intensity across several industries (except agriculture).
  - Manufacturing has the highest sectoral Q/Y, but Chile’s manufacturing sector is much smaller relative to Korea and Czech Republic.
- Supply-side decomposition (change in aggregate Q/Y between t and t+1):
  - Δ(Q/Y) = Σ Δ(Q_i/Y_i)_{i} ⋅ (Y_{i,t+1}/Y_{t+1}) + Σ (Q_{i,t}/Y_{i,t}) ⋅ Δ(Y_i/Y)
  - Empirical result for Chile, 2008-2019: aggregate Q/Y change = -0.24
    - Change driven mainly by fewer intermediate input use at sector level: -0.52
    - Composition shift effect increased aggregate Q/Y by +0.28
  - Conclusion: weakening of input-output linkage over time mainly reflects reduced intermediate input use within sectors rather than sectoral compositional shifts.

### E. Demand‑side decomposition: drivers of Domar weights and counterfactual policy levers
- Demand-side identity for Domar weights (vector form):
  - λ = Ψ (β + ẽ δ)  (Equation (3))
    - λ = {λ_i} (Domar weights), β = {β_i} where β_i = C_i/C (industry share in total final domestic demand), δ = {δ_i} where δ_i = E_i/E (export composition), ẽ = E/Y (economy-wide export-to-VA ratio).
    - Ψ = (I − Ω′)^{-1} = I + Ω′ + Ω′^2 + Ω′^3 + ⋯ (Leontief inverse), with Ω_{ij} = X_{ij}/Q_i (input-output coefficients).
- Decomposition attributes cross-country Q/Y differences to:
  - Differences in domestic input-output linkages (Ψ),
  - Differences in final domestic demand distribution across industries (β),
  - Differences in total exports and export composition (ẽ δ).
- Advantages of demand-side decomposition:
  - Breaks down aggregate Q/Y into domestic final demand, domestic intermediate demand, and exports — elements linked to exogenous factors and policy levers.
  - Accounts for industries’ upstream spillovers via Ψ.
  - Enables counterfactual policy experiments (e.g., changes in final demand composition or export composition and their effects on Q/Y through Ψ).
- Empirical implication for Chile:
  - Chile’s lower Q/Y relative to peers arises from weaker domestic production linkages (smaller Ψ effects) and less export contribution to Domar weights, in addition to demand composition differences.
- Policy-relevant mechanisms flagged:
  - Strengthening domestic producer linkages and diversifying exports (away from concentrated mining dependence) could raise the Q/Y ratio and amplify aggregate productivity gains from firm- or industry-level technology improvements.
  - Better contract enforcement is identified as a potential policy to strengthen production network linkages.

### F. Key analytical takeaways
- The production network amplifies micro-level technological improvements into larger aggregate TFP gains through Domar-weighted gross output shares (Σ λ_i = Q/Y > 1).
- Chile’s Q/Y is slightly below OECD average and significantly below high-Q/Y peers (Korea, Czech Republic), contributing materially to slower real GDP per capita growth.
- The recent decline in Chile’s Q/Y (2008–2021) is driven mainly by reduced intermediate input intensity at the sector level rather than by compositional shifts; export and commodity price fluctuations are not the primary drivers of the decline.
- Demand-side decomposition (λ = Ψ (β + ẽ δ)) provides a framework to evaluate how:
  - Domestic input-output linkages (Ψ),
  - Domestic final demand composition (β),
  - Export scale and composition (ẽ δ),
  jointly determine the Q/Y multiplier and therefore how policy changes could alter the production network’s amplification of productivity.

---

### 15. A cross‑country comparison indicates that Chile’s lower intermediate input use mainly

### Cross‑country decomposition of gross output‑to‑value added (Q/Y) differences
- Korea’s economy‑wide Q/Y in 2018 = 2.42, about 0.47 higher than Chile.
  - Of this 0.47 differential:
    - 0.34 is attributed to variations in the intensity of domestic demand for domestically produced intermediate input (captured by Ω / Ψ).
    - 0.16 is due to Chile’s lower total export‑to‑value added and smaller share of manufacturing exports.
    - -0.03 reflects the final demand composition (Chile’s final demand distribution is slightly more favorable to intermediate input use than Korea, driven by Chile’s smaller public consumption share concentrated in low‑linkage service sectors).
- Comparison with Czech Republic in 2018:
  - Total Q/Y differential = 0.62.
    - Explained by domestic IO linkages (ψ) = 0.20.
    - Explained by exports and export composition (eδ) = 0.61.
    - Explained by final demand distribution (β) = -0.20.

### Changes over time in Chile (2008–2021)
- Aggregate decline in Q/Y between 2008 and 2021 = -0.23.
  - Of the -0.23 decline:
    - -0.09 is attributed to changes in less domestic intermediate input demand (changes to Ψ / domestic input‑output linkages).
    - -0.13 is explained by exports, as export‑to‑value added (ẽ) declined by 9 percentage points.
    - -0.01 is due to shifts in the distribution of final demand over industries.
- Export composition shifts during 2008–2021:
  - Mining’s share in total exports increased by 12 percentage points.
  - Manufacturing’s export share decreased by 5 percentage points.

### Policy implications: enhancing input‑output linkages to raise productivity
- Three components determine intermediate input use (Equation (3)): domestic input‑output linkage (Ψ), exports and export composition (푒훿), and composition of final demand (β).
- Policies can target:
  - Exports (푒) and trade costs — e.g., transportation infrastructure and trade facilitation.
  - Domestic input‑output linkage (Ψ) — e.g., enhancing competition, reducing entry barriers (streamlining permit processes).
  - Contract enforcement — likely influences Ψ.
  - Export composition (훿) — directly affects the multiplier effects via sectoral linkages.

### Contract enforcement and domestic input‑output linkage
- Theoretical channel: improved contract enforcement reduces transaction costs and lowers incentives for in‑house production, facilitating intermediate input use.
- World Bank Doing Business Survey (2020) flags de jure weaknesses for Chile in "court structure and proceedings" and "mediation and conciliation".
- Empirical approach (following Boehm (2022)):
  - Construct industry‑specific litigation frequency index; classify industries into high (H) and low (L) litigation groups.
  - Compute Δ = (λ_H^US − λ_H^CHL) − (λ_L^US − λ_L^CHL). A positive Δ indicates contract enforcement lowers intermediate input intensity in Chile relative to the U.S.
- Result:
  - Economy‑wide, Chile’s gross output‑to‑GDP ratio in 2018 is higher than the U.S. by 0.09.
  - Using 24 industries ranked by litigation index (top 12 = high; bottom 12 = low), Δ = 0.21: Chilean producers in high‑litigation industries have a gross output‑to‑GDP ratio gap 0.21 lower than those in low‑litigation industries.

### Export composition, multipliers, and growth implications
- Chile’s export concentration in mining (~50 percent of total exports) reduces production network multipliers due to weaker input‑output linkages.
- 2021 simulated domestic and foreign intermediate input shares (per dollar of sector output):
  - Mining: domestic intermediate inputs = $0.31, imported intermediate inputs = $0.04.
  - Manufacturing: domestic intermediate inputs = $0.49, imported intermediate inputs = $0.23.
- Multipliers from 2018 input‑output table:
  - A one‑dollar increase in manufacturing demand → increase of $2.60 in total domestic output.
  - A one‑dollar increase in mining demand → increase of $1.90 in total domestic output.
- Counterfactual (general equilibrium model with input‑output linkages; Annex III):
  - Shifting export composition from actual 2021 (60 percent mining, 27 percent manufacturing) to 100 percent manufacturing would:
    - Increase Q/Y by 0.24.
    - Correspond to a 0.7 percentage point difference in annual real GDP per capita growth rate (based on coefficients in Table 1).
    - Raise manufacturing output‑to‑aggregate value‑added ratio by 0.39.
    - Increase export‑to‑value added ratio by 0.07.
  - If all exports shifted to mining, Q/Y would decline by 0.07.
- Service exports:
  - Shifting all exports to financial and business services would reduce Q/Y by 0.4 and reduce export‑to‑VA ratio by 0.26.
  - A one‑dollar increase in financial and business services yields $1.50 in aggregate output.
  - Services can still raise incomes via labor reallocation to higher‑skill, higher‑salary professions, but a weakly‑linked service export concentration implies lower growth than manufacturing‑focused export structures on the balanced growth path.

### Counterfactual scenarios (Text Table 3 highlights)
- Actual Data in 2021 (Mining 60%, Manufacturing 27%, Business Services 1%):
  - Q/Y = 1.88.
  - E/Y = 34%.
  - Sectoral output shares: Manufacturing 34%, Mining 34%, Business services 24%, Other 14%.
- Full manufacturing exports (Manufacturing 100%):
  - Q/Y = 2.12.
  - E/Y = 43%.
  - Sectoral output shares: Manufacturing 73%, Mining 3%, Business services 14%, Other 14%.
  - Implied change to annual real GDP per capita growth = 0.7 (percentage point).
- Full mining exports (Mining 100%):
  - Q/Y = 1.80.
  - E/Y = 31%.
  - Sectoral output shares: Manufacturing 23%, Mining 35%, Business services 13%, Other 13%.
  - Implied change = -0.2 (percentage point).
- Full business service exports (Business service 100%):
  - Q/Y = 1.76.
  - E/Y = 31%.
  - Sectoral output shares: Manufacturing 22%, Mining 31%, Business services 1%, Other 45%.
  - Implied change = -0.3 (percentage point).

### Data sources, model, and decomposition details
- Input‑output tables:
  - BCCh: annual input‑output tables for 2008–2021 with 111 industries and 12 sectors (used for Chile cross‑time analyses).
  - OECD Statistics: input‑output tables aggregated into 45 industries (used for Chile–Korea–Czech comparisons).
  - WIOD: input‑output tables for 40 largest economies (56 industries), used for gross output‑to‑GDP ratios and bilateral cross‑border transaction detail.
- Decomposition (Annex II):
  - Domar weights λ_i = β_i + ∑Ω_ji λ_j + ẽ δ_i with Ψ = (I − Ω′)−1; economy‑wide Q/Y = ∑λ_i.
  - Cross‑country decomposition separates differences into variation in Ψ (domestic IO linkages), variation in exports and export composition (ẽδ), and variation in final demand composition (β).
- Structural model for counterfactuals (Annex III):
  - Production network with final goods and intermediate goods sectors; calibrated to Chile 2018 moments.
  - Balanced trade assumption; counterfactuals alter export distribution σ_f while holding Ω and β fixed; e_y endogenous.

*Source: IMF staff chapter “1. Growth Multiplier Under Cobb‑Douglas Production Function” and section “15. A cross‑country comparison indicates that Chile’s lower intermediate input use mainly” (sipea2025010).*

### 1. Growth Multiplier Under Cobb-Douglas Production Function ________________________ 4

### 1. Growth Multiplier Under Cobb-Douglas Production Function

### A. Growth slowdown in Chile: facts and magnitudes
- Chile’s average annual real GDP growth rates:
  - 1991-99: 6.2 percent
  - 2000-09: 4.2 percent
  - 2010-19: 3.3 percent
  - 2020-23: 1.9 percent (close to the Central Bank of Chile projected potential growth rate of around 2 percent for the next ten years)
- Main proximate driver: marked slowdown in total factor productivity (TFP) growth.
  - According to CNEP estimates, compared to the 1990s, the average annual contribution of TFP to growth declined by 2.5 percentage points in the 2010s.
  - Contributions of labor and capital each fell by 0.2 percentage points in the 2010s relative to the 1990s.
- Solow model illustration:
  - With a capital share of 0.5, a 2.5 percentage point drop in TFP growth would predict a 5-percentage point reduction in capital accumulation along the balanced growth path; actual annual capital growth decelerated only by 0.2 percentage points between the 1990s and 2010s (outperforming that prediction).
- Sectoral studies cited (summarized findings):
  - CNEP (2017): declining productivity in copper mining; recommends improvements in permit timelines and standards for suppliers/contractors and labor training.
  - CNEP (2020a): inefficiencies in construction sector—planning of public works, uncertainty around building permits, low adoption of digital tools.
  - CNEP (2020b): suboptimal utilization of surgical rooms, contributing to longer wait times for surgeries.
  - Asturias et al. (2023): entry barriers impact manufacturing sector productivity growth.
- Author’s focus: input-output linkages and intermediate input use (share of gross output), and mapping of gross output-to-value added (Q/Y) ratios as summary measures of production network connectedness.

### B. Theoretical background: Cobb-Douglas growth multiplier and Domar weights
- Producer-level Cobb-Douglas setup (single intermediate input):
  - q_i = a_i l_i^{1−σ_i} x_i^{σ_i}
  - Log-differentiation yields: Δlog(q_i) = Δlog(a_i) + (1−σ_i) Δlog(l_i) + σ_i Δlog(x_i)  (Equation (1.1) in source)
  - With value-added y_i and fixed intermediate share σ_i, Δlog(q_i) = (1−σ_i) Δlog(y_i) + σ_i Δlog(x_i)  (Equation (1.2) in source)
  - Combining gives Δlog(y_i) = Δlog(a_i)/(1−σ_i) + Δlog(l_i)
  - Producer TFP growth (conventional growth accounting): ΔTFP_i = Δlog(a_i)/(1−α_i)
- Aggregate TFP growth with N producers:
  - ΔTFP = Σ (Y_i/Y) ⋅ ΔTFP_i = Σ (Q_i/Y) ⋅ Δlog(a_i)  (Equation (1.3) in source)
  - Interpretation: aggregate TFP growth is weighted by gross output shares (Domar weights λ_i = Q_i/Y), not value-added shares; Σ λ_i = Q/Y > 1 when intermediate inputs are used, creating an amplification/multiplier effect.
- Generalized result (non-Cobb-Douglas, Leontief framework):
  - Δlog(TFP) = Σ λ_i Δlog(a_i) where λ_i = Q_i/Y (Domar weight) and Σ λ_i = Q/Y (Equation (1) in source).
  - Gross output-to-GDP ratio (Q/Y) measures the production network multiplier (Hulten 1978; Baqee and Farhi 2020; Basu and Fernald 2002 noted).

### C. Cross-country empirical relationship and elasticities
- Cross-country OLS (dependent variable: average real GDP per capita growth 2014-19):
  - Coefficient on gross output-to-VA (Q/Y): 3.04*** in Column (1); 2.90*** in Column (2) after controlling for log(GDP per capita) in 2014.
  - Coefficient on Log(GDP per capita) in 2014 (PPP): -1.13** in Column (2).
  - Constant terms: -4.29 (Column (1)); 7.79 (Column (2)).
  - Note on statistical notation: *** (**) stands for p value<1% (5%).
- Interpretation from Column (1): a 0.1 increase in the gross output-to-GDP ratio is associated with an around 0.3 percentage point increase in real GDP per capita growth.
- Cross-country variation in Q/Y (2014 examples):
  - Korea and several Eastern European countries: Q/Y ranging from 2.4 to 2.7.
  - Chile and G7 countries: Q/Y around 2.0.
  - Mexico and Costa Rica: Q/Y around 1.8.
- Robustness and prior work:
  - McNerny et al. (2022) find positive correlation between Q/Y in 1995 and real GDP per capita growth during 1995-2000 using WOID sample.

### D. Input-output linkage in Chile: levels, trends, and decomposition
- Level comparisons:
  - Chile’s Q/Y averaged around 2.0 in the 2010s (BCCh input-output tables); OECD average: 2.1 during the same period.
  - In 2018, Chile’s Q/Y was lower than Korea by 0.47 and lower than Czech Republic by 0.62.
  - Using the elasticity 2.9 (Column (2) of Text Table 1), the lower Q/Y in Chile would translate into slower annual real GDP per capita growth than Korea (Czech Republic) by 1.4 (1.8) percentage points.
- Time trend (Chile):
  - Q/Y fell from 2.16 in 2008 to 1.92 in 2021 (BCCh input-output tables).
  - The elasticity in Text Table 1 implies the lower intensity of intermediate input use in 2021 compared to 2008 would translate into a deceleration in annual real GDP per capita growth by 0.7 percentage points.
  - To isolate commodity price effects: a hypothetical non-mining and non-utility economy shows Q/Y declined by 0.19 between 2008 and 2013, compared to a 0.18 decline in the actual economy — implying copper and oil price fluctuations were not the primary drivers of the aggregate Q/Y decline.
- Sources of Chile’s lower Q/Y relative to Korea and Czech Republic:
  - Smaller manufacturing share in Chile’s economy.
  - Lower intermediate input intensity across several industries (except agriculture).
  - Manufacturing has the highest sectoral Q/Y, but Chile’s manufacturing sector is much smaller relative to Korea and Czech Republic.
- Supply-side decomposition (change in aggregate Q/Y between t and t+1):
  - Δ(Q/Y) = Σ Δ(Q_i/Y_i)_{i} ⋅ (Y_{i,t+1}/Y_{t+1}) + Σ (Q_{i,t}/Y_{i,t}) ⋅ Δ(Y_i/Y)
  - Empirical result for Chile, 2008-2019: aggregate Q/Y change = -0.24
    - Change driven mainly by fewer intermediate input use at sector level: -0.52
    - Composition shift effect increased aggregate Q/Y by +0.28
  - Conclusion: weakening of input-output linkage over time mainly reflects reduced intermediate input use within sectors rather than sectoral compositional shifts.

### E. Demand-side decomposition: drivers of Domar weights and counterfactual policy levers
- Demand-side identity for Domar weights (vector form):
  - λ = Ψ (β + ẽ δ)  (Equation (3) in source)
    - λ = {λ_i} (Domar weights), β = {β_i} where β_i = C_i/C (industry share in total final domestic demand), δ = {δ_i} where δ_i = E_i/E (export composition), ẽ = E/Y (economy-wide export-to-VA ratio).
    - Ψ = (I − Ω′)^{-1} = I + Ω′ + Ω′^2 + Ω′^3 + ⋯ (Leontief inverse), with Ω_{ij} = X_{ij}/Q_i (input-output coefficients).
- Decomposition used to attribute cross-country Q/Y differences to:
  - Differences in domestic input-output linkages (Ψ, i.e., inter-industry linkages among domestic producers),
  - Differences in final domestic demand distribution across industries (β),
  - Differences in total exports and export composition (ẽ δ).
- Key advantages of demand-side decomposition relative to supply-side:
  - Breaks down aggregate Q/Y into domestic final demand, domestic intermediate demand, and exports — elements more readily linked to exogenous factors and policy levers (household consumption patterns, investment composition, government spending).
  - Accounts for industries’ upstream spillovers: a stronger basic metal industry can boost upstream sectors (e.g., electricity, mining) via Ψ.
  - Enables counterfactual policy experiments (e.g., how changes in final demand composition or export composition would affect Q/Y through Ψ).
- Empirical implication for Chile:
  - Chile’s lower Q/Y relative to peers arises from weaker domestic production linkages (smaller Ψ effects) and less export contribution to Domar weights, in addition to demand composition differences.
- Policy-relevant mechanisms flagged in the text:
  - Strengthening domestic producer linkages and diversifying exports (away from concentrated mining dependence) could raise the Q/Y ratio and thereby amplify aggregate productivity gains from firm- or industry-level technology improvements.
  - Better contract enforcement is identified as a potential policy to strengthen production network linkages (statement in executive summary).

### F. Key analytical takeaways
- The production network amplifies micro-level technological improvements into larger aggregate TFP gains through Domar-weighted gross output shares (Σ λ_i = Q/Y > 1).
- Chile’s Q/Y is slightly below OECD average and significantly below high-Q/Y peers (Korea, Czech Republic), contributing materially to slower real GDP per capita growth.
- The recent decline in Chile’s Q/Y (2008–2021) is driven mainly by reduced intermediate input intensity at the sector level rather than by compositional shifts; export and commodity price fluctuations are not the primary drivers of the decline.
- Demand-side decomposition (λ = Ψ (β + ẽ δ)) provides a framework to evaluate how:
  - Domestic input-output linkages (Ψ),
  - Domestic final demand composition (β),
  - Export scale and composition (ẽ δ),
  jointly determine the Q/Y multiplier and therefore how policy changes could alter the production network’s amplification of productivity.

*Source: IMF staff chapter “1. Growth Multiplier Under Cobb-Douglas Production Function” (sipea2025010).*

### 15.      A cross-country comparison indicates that Chile’s lower intermediate input use mainly

### 15.      A cross-country comparison indicates that Chile’s lower intermediate input use mainly

### Cross‑country decomposition of gross output‑to‑value added (Q/Y) differences
- Korea’s economy‑wide gross output‑to‑value added ratio in 2018 was 2.42, about 0.47 higher than that of Chile.
  - Of this 0.47 differential:
    - 0.34 is attributed to variations in the intensity of domestic demand for domestically produced intermediate input (captured by Ω / Ψ).
    - 0.16 is due to Chile’s lower total export‑to‑value added and smaller share of manufacturing exports that tend to have stronger input‑output linkages than other industries.
    - -0.03 reflects the final demand composition (Chile’s final demand distribution is slightly more favorable to intermediate input use than Korea, driven by Chile’s smaller public consumption share concentrated in low‑linkage service sectors).
- Comparison with Czech Republic in 2018:
  - Total Q/Y differential = 0.62.
    - Explained by domestic IO linkages (ψ) = 0.20.
    - Explained by exports and export composition (eδ) = 0.61.
    - Explained by final demand distribution (β) = -0.20.

### Changes over time in Chile (2008–2021)
- Aggregate decline in gross output‑to‑value added ratio between 2008 and 2021 = -0.23.
  - Of the -0.23 decline:
    - -0.09 is attributed to changes in less domestic intermediate input demand (changes to Ψ / domestic input‑output linkages).
    - -0.13 is explained by exports, as export‑to‑value added (푒̃) declined by 9 percentage points.
    - -0.01 is due to shifts in the distribution of final demand over industries.
- Export composition shifts during 2008–2021:
  - Mining’s share in total exports increased by 12 percentage points.
  - Manufacturing’s export share decreased by 5 percentage points.

### Policy implications: enhancing input‑output linkages to raise productivity
- Three components determine intermediate input use (Equation (3)): domestic input‑output linkage (Ψ), exports and export composition (푒훿), and composition of final demand (β).
- Policies can target:
  - Exports (푒) and trade costs — e.g., transportation infrastructure and trade facilitation.
  - Domestic input‑output linkage (Ψ) — e.g., enhancing competition, reducing entry barriers (streamlining permit processes).
  - Contract enforcement — likely influences Ψ.
  - Export composition (훿) — directly affects the multiplier effects via sectoral linkages.

### Contract enforcement and domestic input‑output linkage
- Theoretical channel: improved contract enforcement reduces transaction costs and lowers incentives for in‑house production, facilitating intermediate input use.
- World Bank Doing Business Survey (2020) flags de jure weaknesses for Chile in "court structure and proceedings" and "mediation and conciliation".
- Empirical approach (following Boehm (2022)):
  - Construct industry‑specific litigation frequency index; classify industries into high (H) and low (L) litigation groups.
  - Compute Δ = (λ_H^US − λ_H^CHL) − (λ_L^US − λ_L^CHL). A positive Δ indicates contract enforcement lowers intermediate input intensity in Chile relative to the U.S.
- Result:
  - Economy‑wide, Chile’s gross output‑to‑GDP ratio in 2018 is higher than the U.S. by 0.09.
  - Using 24 industries ranked by litigation index (top 12 = high; bottom 12 = low), Δ = 0.21: Chilean producers in high‑litigation industries have a gross output‑to‑GDP ratio gap 0.21 lower than those in low‑litigation industries, consistent with contract enforcement affecting intermediate input use.

### Export composition, multipliers, and growth implications
- Chile’s export concentration in mining (~50 percent of total exports) reduces production network multipliers due to weaker input‑output linkages.
- 2021 simulated domestic and foreign intermediate input shares (per dollar of sector output):
  - Mining: domestic intermediate inputs = $0.31, imported intermediate inputs = $0.04.
  - Manufacturing: domestic intermediate inputs = $0.49, imported intermediate inputs = $0.23.
- Multipliers from 2018 input‑output table:
  - A one‑dollar increase in manufacturing demand → increase of $2.60 in total domestic output.
  - A one‑dollar increase in mining demand → increase of $1.90 in total domestic output.
- Counterfactual (general equilibrium model with input‑output linkages; Annex III):
  - Shifting export composition from actual 2021 (60 percent mining, 27 percent manufacturing) to 100 percent manufacturing would:
    - Increase gross output‑to‑value added ratio by 0.24.
    - Correspond to a 0.7 percentage point difference in annual real GDP per capita growth rate (based on coefficients in Table 1).
    - Raise manufacturing output‑to‑aggregate value‑added ratio by 0.39.
    - Increase export‑to‑value added ratio by 0.07.
  - If all exports shifted to mining, gross output‑to‑value added ratio would decline by 0.07.
- Service exports:
  - Shifting all exports to financial and business services would reduce gross output‑to‑VA ratio by 0.4 and reduce export‑to‑VA ratio by 0.26.
  - A one‑dollar increase in financial and business services yields only $1.50 in aggregate output (1.5 dollar increase), reflecting low upstream interconnectedness.
  - Services can still raise incomes via labor reallocation to higher‑skill, higher‑salary professions, but on a balanced growth path weakly‑linked service export concentration implies lower growth than manufacturing‑focused export structures.

### Counterfactual scenarios (Text Table 3 highlights)
- Actual Data in 2021 (Mining 60%, Manufacturing 27%, Business Services 1%):
  - Gross output‑to‑GDP (Q/Y) = 1.88.
  - Export‑to‑GDP (E/Y) = 34%.
  - Sectoral output shares: Manufacturing 34%, Mining 34%, Business services 24%, Other 14%.
- Full manufacturing exports (Manufacturing 100%):
  - Q/Y = 2.12.
  - E/Y = 43%.
  - Sectoral output shares: Manufacturing 73%, Mining 3%, Business services 14%, Other 14%.
  - Implied change to annual real GDP per capita growth = 0.7 (percentage point).
- Full mining exports (Mining 100%):
  - Q/Y = 1.80.
  - E/Y = 31%.
  - Sectoral output shares: Manufacturing 23%, Mining 35%, Business services 13%, Other 13%.
  - Implied change = -0.2 (percentage point).
- Full business service exports (Business service 100%):
  - Q/Y = 1.76.
  - E/Y = 31%.
  - Sectoral output shares: Manufacturing 22%, Mining 31%, Business services 1%, Other 45%.
  - Implied change = -0.3 (percentage point).

### Data sources, model, and decomposition details
- Input‑output tables:
  - BCCh: annual input‑output tables for 2008–2021 with 111 industries and 12 sectors (used for Chile cross‑time analyses).
  - OECD Statistics: input‑output tables aggregated into 45 industries (used for Chile–Korea–Czech comparisons).
  - WIOD: input‑output tables for 40 largest economies (56 industries), used for gross output‑to‑GDP ratios and bilateral cross‑border transaction detail.
- Decomposition (Annex II):
  - Domar weights λ_i = β_i + ∑Ω_ji λ_j + ẽ δ_i with Ψ = (I − Ω′)−1; economy‑wide Q/Y = ∑λ_i.
  - Cross‑country decomposition separates differences into variation in Ψ (domestic IO linkages), variation in exports and export composition (ẽδ), and variation in final demand composition (β).
- Structural model for counterfactuals (Annex III):
  - Production network with final goods and intermediate goods sectors; calibrated to Chile 2018 moments.
  - Balanced trade assumption; counterfactuals alter export distribution σ_f while holding Ω and β fixed; e_y endogenous.

*Source: IMF staff analysis in "15. A cross‑country comparison indicates that Chile’s lower intermediate input use mainly," sipea2025010.*

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_Source: https://www.imf.org/-/media/files/publications/selected-issues-papers/2025/english/sipea2025010.pdf_