## 1.   Substitution in Input Use.

## Source details

**Canonical URL:** [1.   Substitution in Input Use.](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15199.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15199.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15199.pdf.json)

---

### Substitution in Input Use: core derivation and interpretation
- Key linearized equation (from substitution of (6),(7),(8) into market clearing (9)):
  - ˆQ = S_F ˆF + S_X [−ρM_1 ˆp + ρM_2 ˆp_x + M_2(−γˆp_x + γˆp + ˆQ)]. (13)
- Rearranged to isolate ˆQ using ˆp_x = W_x ˆp:
  - ˆQ = [I − S_X M_2]^{−1} S_F ˆF + [I − S_X M_2]^{−1} S_X [−ρM_1 ˆp + ρM_2 W_x ˆp − γM_2 W_x ˆp + γM_2 ˆp]. (14)
- Interpretation highlights:
  - First term: maps bilateral final goods shipments (ˆF) through input-output structure into gross output changes.
  - Second term: captures response of input choices and input-output structure to gross output price changes; input re-optimization governed by γ and ρ.
  - Presence of an input-output loop since gross output appears on both sides.

### Substitution across Final Goods
- Substituting for ˆF yields:
  - ˆQ = [I − S_X M_2]^{−1} S_F M_2 ˆF
    − σ[I − S_X M_2]^{−1} S_F (M_1 − M_2 W_f) ˆp
    − ρ[I − S_X M_2]^{−1} S_X (M_1 − M_2 W_x) ˆp
    + γ[I − S_X M_2]^{−1} S_X M_2 (I − W_x) ˆp. (15)
- Interpretation of terms:
  - First: role of changes in real final expenditure levels (ˆF).
  - Second: substitution in final goods purchases (elasticity σ).
  - Third: substitution within the input bundle (elasticity ρ).
  - Fourth: substitution between real value added and inputs (elasticity γ).
- Conclusion: price pass-through to gross output demand depends on supply-side elasticities (γ, ρ) and demand-side elasticity (σ).

### Demand for Value Added: definition and decomposition
- Value-added expressed via production shares and input substitutions:
  - ˆV = ˆQ − γ[diag(s_{x i}/s_{v i})](I − W_x) ˆp. (16)
- Combining (15) and (16):
  - ˆV = [I − S_X M_2]^{−1} S_F M_2 ˆF
    − [I − S_X M_2]^{−1} [σ S_F (M_1 − M_2 W_f) + ρ S_X (M_1 − M_2 W_x) − γ S_X M_2 (I − W_x)] ˆp
    − γ[diag(s_{x i}/s_{v i})](I − W_x) ˆp. (17)
- Summary: demand for real value added depends on final expenditure levels (ˆF) and gross price changes (ˆp), with coefficients set by input-output structure and elasticities.

### Linking Value-Added to Gross Output Prices
- Gross output price changes as function of value-added price changes:
  - ˆp = [I − Ω']^{−1} [diag(s_{v i})] ˆp_v, (18)
  - Ω' = diag(s_{x i}) W_X; element Ω'_{ij} = share of inputs from i purchased by j in total gross output of j.
- Two key points:
  - Gross output price changes are weighted averages of value-added price changes across countries.
  - Mapping from value-added to gross-output prices involves only production input shares (no elasticities).

### Value-Added Real Effective Exchange Rates (REER): aggregation
- Stylized mapping of value added to value-added prices:
  - ˆV = −[σ T_σ + ρ T_ρ + γ T_γ] ˆp_v + ˆF_w. (19)
  - Definitions:
    - T_σ ≡ [I − S_X M_2]^{−1} S_F (M_1 − M_2 W_f)[I − Ω']^{−1}[diag(s_{v i})],
    - T_ρ ≡ [I − S_X M_2]^{−1} S_X (M_1 − M_2 W_x)[I − Ω']^{−1}[diag(s_{v i})],
    - T_γ = [[diag(s_{x i}/s_{v i})] − [I − S_X M_2]^{−1} S_X M_2](I − W_x)[I − Ω']^{−1}[diag(s_{v i})],
    - ˆF_w ≡ [I − S_X M_2]^{−1} S_F M_2 ˆF.
- REER construction (hold ˆF = 0 and normalize):
  - For country i, define T_{ii} ≡ σ T_{ii}^σ + ρ T_{ii}^ρ + γ T_{ii}^γ. Then:
    - ˆV_i = − T_{ii} ∑_{j ≠ i} [−(σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ)/T_{ii}] (ˆp_{v i} − ˆp_{v j}). (20)
  - Define:
    - ̂REER_i ≡ ∑_{j ≠ i} [−(σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ)/T_{ii}] (ˆp_{v i} − ˆp_{v j}). (21)
    - ˆV_i = − T_{ii} ̂REER_i.
- Signs and weights:
  - σ T_{ii}^σ + ρ T_{ii}^ρ + γ T_{ii}^γ > 0.
  - Typically σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ for j ≠ i will be negative (foreign price increases raise demand for domestic value added), but can be positive under some elasticities, producing negative REER weights.
- Three conceptual points:
  1. Treat value-added prices (ˆp_v) as primitives and aggregate to a multilateral index.
  2. Weights depend on interaction of supply and demand elasticities with input-output structure.
  3. T_{ii} (trade structure and elasticities) scales mapping from REER into demand for value added.

### Three substitution margins in the framework
- Among inputs from different source countries (elasticity ρ).
- Between inputs and value added in production (elasticity γ).
- Among final goods from alternative sources (elasticity σ).

---

### III.  The Mechanics of Demand for Value Added

### A. Equal elasticities: Value-Added Armington-CES model
- Set ε ≡ γ = ρ = σ. Then:
  - ˆV_i = − ε (ˆp_{v i} − ˆP_{w i}) + ˆF_{w i}, (22)
  - with ˆP_{w i} and ˆF_{w i} as weighted aggregates over destinations.
- Interpretation:
  - Each country faces a CES demand for its value added vis-à-vis a composite world market at elasticity ε.
  - ˆP_{w i} is perceived world price faced by country i; weights derive from value-added export shares.
  - Demand falls when own value-added price rises relative to ˆP_{w i} with elasticity ε.

### 1. The VAREER under equal elasticities
- With ˆF_j = 0:
  - ˆV_i = − ε T_{ii}^{VA} ̂REER_{VA i},
  - ̂REER_{VA i} ≡ ∑_{j ≠ i} [1 / T_{ii}^{VA} ∑_k (p_{v i} V_{i k} / p_{v i} V_i) (p_{v j} V_{j k} / P_k F_k)] (ˆp_{v i} − ˆp_{v j}), (23)
  - T_{ii}^{VA} = 1 − ∑_k (p_{v i} V_{i k} / p_{v i} V_i) (p_{v i} V_{i k} / P_k F_k).
- Notes:
  - VAREER is based on value-added data alone (double value-added export weights).
  - T_{ii}^{VA} captures degree of openness in value-added terms; as openness increases T_{ii}^{VA} tends to increase.

### B. Heterogeneous elasticities: IOREER and implications
- When σ, γ, ρ differ:
  - REER weights require full global input-output framework.
  - Weights depend on relative magnitudes of σ, γ, ρ.
  - Higher σ raises weight on final-goods substitution (T^σ); higher γ or ρ raise weights on input substitution (T^γ, T^ρ).
- IOREER emphasizes dependence on input-output structure and heterogeneous elasticities.

### IOREER: illustrative three-country example
- Stylized setup produces:
  - ˆV_1 = − T_{11} [((γ − σ s_{23} w) s_v / T_{11}) (ˆp_{v 1} − ˆp_{v 2}) + (σ s_{23} w / T_{11}) (ˆp_{v 1} − ˆp_{v 3})] ≡ − T_{11} ̂REER_{IO 1}, (24)
  - T_{11} = γ s_v + σ s_{23} w (1 − s_v).
- Two features:
  1. IOREER depends on prices in country 3 even without direct trade between 1 and 3 (indirect competition via embodied inputs).
  2. Sign of weight on country 2 prices is ambiguous:
     - If σ = 0 (Leontief preferences): fall in ˆp_{v 2} is bad for country 1 (REER appreciates).
     - If γ = 0 (Leontief production): fall in ˆp_{v 2} depreciates country 1’s IOREER (competitiveness spillovers).
  - Net response depends on relative elasticities; low input elasticities can produce negative REER weights and damp beggar-thy-neighbor expenditure-switching.

---

### 2.  Value-Added Elasticities

### Role of elasticities in mapping IOREER into demand
- General mapping:
  - ˆV_i = −T_ii ˆREER_IO_i, where T_ii = σ T_ii^σ + ρ T_ii^ρ + γ T_ii^γ.
- Effective value-added elasticity:
  - ˜ε_i(σ,ρ,γ) ≡ T_ii / T_ii^VA = [σ (T_ii^σ / T_ii^VA) + ρ (T_ii^ρ / T_ii^VA) + γ (T_ii^γ / T_ii^VA)]. (25)
  - So ˆV_i = −˜ε_i(σ,ρ,γ) T_ii^VA ˆREER_IO_i.
- Example: a uniform 1% multilateral appreciation in ˆp^v_i implies change in demand for value added = ˜ε_i(σ,ρ,γ) T_ii^VA percent.

### Comparison with conventional REERs (Armington)
- Armington-REER uses double gross sales weights and typically CPI or product-price proxies.
- Value-added REERs differ by:
  - Using double value-added export weights (VAREER) or IO-based weights (IOREER).
  - Using value-added price measures (GDP deflators) rather than CPI.
  - Measuring competitiveness for value added rather than for gross output.

### Data and elasticity parameterization
- Data:
  - WIOD: 40 countries, 1995-2011.
  - Johnson and Noguera (2014): 37 countries, 1970-2009.
  - GTAP Version 7 (one figure): 94 countries, 19 regions, year 2004.
  - Price data from IMF World Economic Outlook (annual averages).
- Elasticity scenarios:
  - Homogeneous: σ = γ = ρ = 1.
  - Leontief production extreme: ρ = γ = 0; σ calibrated so GDP-weighted mean value-added elasticity = 1:
    - σ = 3.00 (WIOD),
    - σ = 2.25 (Johnson-Noguera),
    - σ = 2.90 (GTAP).
- Observations:
  - σ and ρ matter most for REER weights and value-added elasticities; γ is less important in practice.
  - Value-added REER weights depend on relative elasticities (ρ/σ) and (γ/σ).

### Building blocks of demand for value added
- Three elements:
  1. REER index (VAREER, IOREER) and bilateral weight differences from Armington-REER.
  2. Value-added elasticity ˜ε_i(σ,ρ,γ) with cross-country variation.
  3. Value-added openness T_ii^VA (scales elasticity effects).

### Empirical ranges and patterns
- T_ii^VA varies across countries ≈ 0.15 to 0.65.
- Typical country: T_ii^VA ≈ 50% larger than Armington gross-openness analogue S_i.
- Median T_ii^VA rose from 0.26 in 1970 to 0.42 in the sample period.
- VAREER and IOREER reassign weights relative to Armington-REER:
  - Supply-chain partners typically receive smaller weights in VAREER/IOREER than in Armington-REER.
  - IOREER accentuates supply-chain partner weight reductions when production elasticities are low.
- Example magnitudes:
  - A 20% fall in Chinese prices would induce a 4.6% (20×0.23) appreciation in Taiwan’s Armington-REER, but only a 0.08% (20×0.04) appreciation in IOREER.
  - For a typical Asian country, total weight attached to Asian partners is 15 percentage points lower in IOREER than in Armington-REER.

### Value-added elasticities and openness (empirical facts)
- Under IOREER with Leontief production (ρ = γ = 0; σ calibrated):
  - ˜ε_i(3,0,0) shows significant downward adjustments below 1 for supply-chain-intensive countries (e.g., Taiwan, Ireland, Indonesia, Hungary).
  - Strong positive correlation between final-goods share of trade and ˜ε_i: more input-intensive countries have lower effective value-added elasticities.
- Value-added openness T_ii^VA:
  - Cross-country range ≈ 0.15 to 0.65.
  - For same REER change, impact on demand for value added can be up to 4 times larger in most open vs most closed countries.
  - Median T_ii^VA increased from 0.26 (1970) to 0.42 (present sample).

### Value-added REERs and time series behavior
- Construction: year-on-year bilateral value-added price changes aggregated with year-t weights and chained (Federal Reserve Board approach).
- Weight distances:
  - City-block distance between VAREER and Armington-REER weights (d_VAREER_it) rose slowly 1970-1990, then more rapidly 1990-2010.
  - Distance between IOREER and VAREER weights shows no clear long-term trend.
- Price differences:
  - Large and persistent differences between GDP deflators and CPI across countries help explain VAREER vs Armington-REER gaps.
- Historical examples (1995-2011):
  - Germany: VAREER and IOREER depreciate more strongly than Armington-REER (German value-added REER depreciated by 15 percentage points post-2000 while Armington-REER was little changed).
  - China: VAREER and IOREER appreciate by over 20 percentage points during the 2000s despite little trend in Armington-REER.
- Cross-country annual gaps |(ˆREER_IO − ˆREER_Arm)/ˆREER_Arm| quartiles:
  - For Figures 9 and 10 countries: {0.13, 0.37, 1.06}.
  - For all available countries: {0.16, 0.42, 1.25}.

### Decomposition of value-added vs Armington REER deviations
- Two components: weight differences and price differences.
- For VAREER vs Armington-REER:
  - Weight differences play small role for median country at 1-year and 10-year horizons; price differences (shift from CPI to value-added prices) explain most deviation.
- For IOREER vs Armington-REER:
  - Weight differences account for ~30% of gap at 1-year horizon (median country) and ~15% over 10-year horizon.
  - Price differences still account for majority of deviations overall.

### Value-added expenditure switching (mapping REER changes to demand)
- Formulas:
  - IOREER: ˆV_IO_i = −˜ε_i(σ,ρ,γ) T_ii^VA ˆREER_IO_i.
  - VAREER (ε = 1): ˆV_VA_i = −T_ii^VA ˆREER_VA_i.
- Hybrid measure (isolate REER shift effect): ˆV_IO_i(˜ε = 1) = −T_ii^VA ˆREER_IO_i.
- Empirical magnitudes (example year 2005):
  - Korea: demand for value added decreased by 3.0–3.5 percentage points depending on measure.
  - Germany: demand for value added increased by 1.6 percentage points.
  - China: IOREER vs VAREER demand impact difference −0.35 percentage points (2005 decomposition).
- Elasticity deviations explain about 1/2 of deviations between ˆV_IO_i and ˆV_VA_i for median country over 1970-2009.

### Key quantitative takeaways (preserve numbers exactly)
- Homogeneous elasticity baseline: σ = γ = ρ = 1.
- Leontief production extreme: ρ = γ = 0; σ calibrated so GDP-weighted mean value-added elasticity = 1:
  - σ = 3.00 (WIOD),
  - σ = 2.25 (Johnson-Noguera),
  - σ = 2.90 (GTAP).
- Value-added openness T_ii^VA range: ≈ 0.15 to 0.65.
- Median T_ii^VA rose from 0.26 in 1970 to 0.42 in recent data.
- Example demand impacts:
  - Korea: −3.0 to −3.5 percentage points (2005).
  - Germany: +1.6 percentage points (2005).
  - China IOREER vs VAREER demand impact: −0.35 percentage points (2005).
- Cross-country distribution of annual relative IOREER vs Armington-REER gaps (all countries) quartiles: {0.16, 0.42, 1.25}.

---

### Conclusions and implications
- Input-output linkages and heterogeneous elasticities reshape REER construction and interpretation:
  - Supply-chain linkages can allow competitiveness improvements of partners to benefit domestic value added, counteracting beggar-thy-neighbor expenditure-switching.
  - Value-added REERs (VAREER, IOREER) differ systematically from conventional Armington-REERs in weights, prices used, and interpretation.
- Mapping REER changes into demand requires:
  - Measuring country-specific value-added elasticities ˜ε_i(σ,ρ,γ).
  - Measuring value-added openness T_ii^VA, which has increased over time and amplifies REER effects on demand.
- Practical implications for measurement and policy:
  - Use value-added-consistent weights and price measures (e.g., GDP deflators) rather than mixing gross trade weights with CPI or unit-cost proxies.
  - Account for country-specific value-added elasticities when using REERs to assess competitiveness; REER alone is incomplete without elasticity.
  - Conventional CPI-based, gross-export-weighted REERs can misstate changes in demand for domestic value added, especially for supply-chain-integrated countries.
- Suggested research directions noted in source:
  - Incorporate input linkages into general equilibrium macro models.
  - Relax Armington-CES assumptions (allow time-varying markups or alternative production structures).
  - Improve empirical estimates of production-side substitution elasticities.

*Italic source: _wp15199 - 1.  Substitution in Input Use*

### 1.   Substitution in Input Use.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    12

### 1.   Substitution in Input Use.

### Table of contents entries (exact headings and page numbers)
- 2.   Substitution across Final Goods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
- D. Demand for Value Added. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
- E. Linking Value-Added to Gross Output Prices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
- F. Value-Added Real Effective Exchange Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
- III. The Mechanics of Demand for Value Added. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
  - A. Equal Elasticities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
    - 1. The Value-Added Armington-CES Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
    - 2. The VAREER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
  - B. Heterogeneous Elasticities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
    - 1. The IOREER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
    - 2. Value-Added Elasticities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
  - C. Conventional Real Effective Exchange Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
- IV. Data and Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
  - A. Global Input-Output and Price Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
  - B. Elasticity Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
- V. Building Blocks of Demand for Value Added. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
  - A. Value-Added REER Weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
  - B. Value-Added Elasticities and Openness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
- VI. Value-Added REERs and Expenditure Switching. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
  - A. Value-Added REERs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
  - B. Value-Added Expenditure Switching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
- VII. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

*Source: _wp15199 - 1.   Substitution in Input Use. (PDF chapter/section)*

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

### _wp15199 - References.

### Major contributions and high-level findings
- Introduces a demand-for-value-added framework that incorporates trade in both final goods and intermediate inputs to analyze how relative price changes affect demand for domestic value added.
- Defines a value-added Real Effective Exchange Rate (REER) index that aggregates bilateral value-added price changes into a multilateral price of domestic relative to foreign value added.
- Demonstrates that global supply chains generate supply-side channels that can counteract conventional demand-side expenditure-switching effects (e.g., a yen depreciation can lower downstream production costs for trading partners and stimulate their competitiveness).
- Shows that the value-added REER differs conceptually from conventional REERs in three ways:
  - REER weights depend on both global input-output structure and relative elasticities in production versus consumption; weights can be negative when price declines in supply-chain partners improve domestic competitiveness.
  - Mapping from the value-added REER to demand for value added depends on a country-specific value-added elasticity (a weighted average of primitive gross elasticities) that is heterogeneous across countries.
  - Provides a theoretically consistent way to combine REER weights and prices using value-added concepts (e.g., GDP deflators) rather than mixing gross trade weights with disparate price measures.

### Framework structure and key analytical elements
- Economic environment specified by:
  - Preferences over final goods (CES aggregator with elasticity σ).
  - Production functions for gross output that separate gross output Q_i, real value added V_i, and a composite intermediate input X_i with CES nesting: elasticity γ between V_i and X_i, and elasticity ρ among inputs.
  - Market clearing: Q_j = sum_k [F_jk + X_jk].
- Linearized system (first-order conditions, price indexes, market clearing) stacks variables across N countries and N^2 bilateral shipments:
  - Final goods linearization: ˆF = −σ M1 ˆp + σ M2 ˆP + M2 ˆF, with ˆP = W_f ˆp.
  - Intermediate inputs and production linearizations: ˆX = −γ ˆp_x + γ ˆp + ˆQ; ˆX = −ρ M1 ˆp + ρ M2 ˆp_x + M2 ˆX; ˆp_x = W_x ˆp.
  - Market clearing: ˆQ = S_F ˆF + S_X ˆX.
  - Production and price decompositions: ˆQ = [diag(s_v_i)] ˆV + [diag(s_x_i)] ˆX; ˆp = [diag(s_v_i)] ˆp_v + [diag(s_x_i)] ˆp_x.
- Three substitution margins in the framework:
  - Among inputs from different source countries (elasticity ρ).
  - Between inputs and value added in production (elasticity γ).
  - Among final goods from alternative sources (elasticity σ).

### Parameter cases and empirical approach
- Two illustrative elasticity cases used for parameterization:
  - Equal elasticities in production and final demand: framework behaves as if consumers have CES-Armington preferences over real value added; value-added REER weights computable using only value-added trade and production data.
  - Low production elasticities (Leontief production, reflecting inflexible global supply chains): highlights how input linkages dampen beggar-thy-neighbor effects and lower value-added elasticities for countries heavily integrated into global supply chains.
- Empirical construction:
  - Uses global input-output data and observed price changes to compute historical value-added REERs over the 1970-2009 period.
  - Finds substantial differences between value-added REERs and conventional REERs driven by differences in weights and price measures.
  - Example empirical findings:
    - China’s value-added REER appreciated by 20% during the 2000’s, while its conventional REER was roughly unchanged.
    - Value-added REERs capture competitiveness shifts in run-up to the Eurozone crisis (e.g., Germany experienced a substantially stronger depreciation of its value-added REER than its conventional REER; Ireland and Spain showed stronger appreciations in value-added REERs).
  - Historical spans and series cited: 1970-2009, 1990-2009, 1995-2011, and 2005 as a cross-section year used in sensitivity analyses.

### Empirical building blocks and sensitivity analyses (as reported)
- Weight on supply-chain partners in value-added REER typically smaller than in conventional indexes; effect amplified when production elasticities are low.
- Low production elasticities yield lower value-added elasticities for countries with strong global supply-chain integration.
- Both value-added REERs and country-specific value-added elasticities significantly influence demand for value added.
- Sensitivity and robustness exercises mentioned in appendices and figures:
  - Sensitivity of the value-added elasticity and REER weights (Appendix B.1).
  - Leontief final demand case (Appendix B.2).
  - Deviations between prices of value added and CPI (Appendix B.3).
  - Quantification of the role of value-added elasticities (Appendix B.4).
  - Figures exhibiting time series and cross-country patterns (e.g., Figures 1–19) and additional sensitivity figures (Figures 13–19).

### Conceptual implications for policy and measurement
- Value-added perspective aligns competitiveness measurement with macro-policy objectives tied to factor-market demand (employment, inflation), because demand for value added is directly linked to factor demand.
- REER construction guidance:
  - Use value-added-consistent weights and price measures (e.g., GDP deflators for value-added prices) rather than mixing gross trade weights with consumer prices or unit-cost indexes.
  - Account for country-specific value-added elasticities when using REERs to assess competitiveness; the value-added REER alone is an incomplete statistic without the elasticity.
- Implication for international policy analysis:
  - Cross-border input linkages can undermine standard beggar-thy-neighbor channels; currency depreciations in supply-chain partners can improve domestic competitiveness via lower input costs.
  - Policy assessments that rely on conventional REERs may mischaracterize competitiveness trends when input linkages are important.

*Source: _wp15199 - References. (IMF PDF content unit)*

### 1.  Substitution in Input Use

### 1.  Substitution in Input Use

### Substitution in Input Use: core derivation and interpretation
- Starting point: substitute Equations (6), (7), (8) into gross output market clearing [Equation (9)] to obtain:
  ˆQ = S_F ˆF + S_X [−ρM_1 ˆp + ρM_2 ˆp_x + M_2(−γˆp_x + γˆp + ˆQ)] (13)
- Rearranged to isolate ˆQ and eliminate composite input price using ˆp_x = W_x ˆp:
  ˆQ = [I − S_X M_2]^{−1} S_F ˆF + [I − S_X M_2]^{−1} S_X [−ρM_1 ˆp + ρM_2 W_x ˆp − γM_2 W_x ˆp + γM_2 ˆp]. (14)
- Interpretation:
  - First term: maps bilateral final goods shipments (ˆF) through input-output structure into changes in gross output.
  - Second term: captures how input choices and the input-output structure respond to gross output price changes; input re-optimization governed by γ and ρ alters the mapping from final goods to gross output.
  - Note: presence of an input-output loop since gross output appears on both sides.

### Substitution across Final Goods
- Substituting for ˆF in Equation (14) using Equation (4) yields:
  ˆQ = [I − S_X M_2]^{−1} S_F M_2 ˆF − σ[I − S_X M_2]^{−1} S_F (M_1 − M_2 W_f) ˆp
  − ρ[I − S_X M_2]^{−1} S_X (M_1 − M_2 W_x) ˆp + γ[I − S_X M_2]^{−1} S_X M_2 (I − W_x) ˆp. (15)
- Interpretation of terms:
  - First term: role of changes in real final expenditure levels (ˆF) in altering demand for output.
  - Second term: substitution in final goods purchases (presence of final goods elasticity σ).
  - Third term: substitution within the input bundle.
  - Fourth term: substitution between real value added and inputs.
  - Conclusion: price change pass-through to gross output demand depends on supply-side elasticities (γ, ρ) and demand-side elasticity (σ).

### Demand for Value Added: definition and decomposition
- Rearranging the production function (Equation (10)) and substituting for ˆX (Equations (6) and (8)) gives:
  ˆV = [diag(s_{v i})]^{−1} [ˆQ − [diag(s_{x i})] ˆX] = ˆQ − γ[diag(s_{x i}/s_{v i})](I − W_x) ˆp. (16)
- Interpretation:
  - First line: definition of double-deflated real value added (strip out input use changes from gross output).
  - Second line: producers substitute toward produced inputs when country i’s gross output price rises, lowering the share of real value added relative to gross output.
- Combining (15) and (16) yields:
  ˆV = [I − S_X M_2]^{−1} S_F M_2 ˆF
  − [I − S_X M_2]^{−1} [σ S_F (M_1 − M_2 W_f) + ρ S_X (M_1 − M_2 W_x) − γ S_X M_2 (I − W_x)] ˆp
  − γ[diag(s_{x i}/s_{v i})](I − W_x) ˆp. (17)
- Summary: demand for real value added depends on real final expenditure levels (ˆF) and gross price changes (ˆp), with coefficients determined by input-output structure and elasticities.

### Linking Value-Added to Gross Output Prices
- Combine Equations (12) and (8) to express gross output price changes as function of value-added price changes:
  ˆp = [I − Ω']^{−1} [diag(s_{v i})] ˆp_v, (18)
  where Ω' = diag(s_{x i}) W_X and Ω' is a global input-output matrix with i j elements equal to the share of inputs from i purchased by j in total gross output of country j.
- Two key points:
  - Gross output price changes are a weighted average of value-added price changes in all countries (ˆp_v), with weights reflecting total cost shares.
  - The mapping from value-added to gross output prices involves no elasticities—only production input shares.

### Value-Added Real Effective Exchange Rates (REER): aggregation to an index
- Combine (17) and (18) to express demand for value added in terms of aggregate expenditure levels ˆF and value-added prices ˆp_v. Stylized form:
  ˆV = −[σ T_σ + ρ T_ρ + γ T_γ] ˆp_v + ˆF_w, (19)
  with
  T_σ ≡ [I − S_X M_2]^{−1} S_F (M_1 − M_2 W_f)[I − Ω']^{−1}[diag(s_{v i})],
  T_ρ ≡ [I − S_X M_2]^{−1} S_X (M_1 − M_2 W_x)[I − Ω']^{−1}[diag(s_{v i})],
  T_γ = [[diag(s_{x i}/s_{v i})] − [I − S_X M_2]^{−1} S_X M_2](I − W_x)[I − Ω']^{−1}[diag(s_{v i})],
  ˆF_w ≡ [I − S_X M_2]^{−1} S_F M_2 ˆF.
- Construction of REER index:
  - Set ˆF = 0 (focus on price effects holding final demand constant).
  - Adopt country-specific normalization so weights on relative price changes sum to one.
- For country i, let T_{i j}^x be the i j element of matrix subscripted by x and define T_{ii} ≡ σ T_{ii}^σ + ρ T_{ii}^ρ + γ T_{ii}^γ. Then:
  ˆV_i = −∑_j [σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ] ˆp_{v j}
  = − T_{ii} ∑_{j ≠ i} [−(σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ)/T_{ii}] (ˆp_{v i} − ˆp_{v j}). (20)
- Define the REER index:
  ̂REER_i ≡ ∑_{j ≠ i} [−(σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ)/T_{ii}] (ˆp_{v i} − ˆp_{v j}). (21)
  - Convention: increase in REER index = appreciation; decrease = depreciation.
  - Parameter T_{ii} translates changes in REER index into changes in demand for value added:
    ˆV_i = − T_{ii} ̂REER_i.
- Signs and weights:
  - σ T_{ii}^σ + ρ T_{ii}^ρ + γ T_{ii}^γ is always positive.
  - Typically (but not always) σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ for j ≠ i will be negative (own price increases lower demand; foreign price increases raise demand for domestic value added).
  - Under some elasticity configurations, σ T_{i j}^σ + ρ T_{i j}^ρ + γ T_{i j}^γ can be positive for j ≠ i, producing negative REER weights.

- Three key conceptual points about the index:
  1. Treats value-added prices as primitives and aggregates them into a composite multilateral index.
  2. Weights attached to bilateral prices depend on interaction of supply and demand elasticities with input-output structure.
  3. Trade structure and elasticities, embodied in T_{ii}, influence the mapping from the index into demand for value added.

### III.  THE MECHANICS OF DEMAND FOR VALUE ADDED

### A. Equal elasticities: Value-Added Armington-CES model
- Set ε ≡ γ = ρ = σ. Then demand for real value added from country i:
  ˆV_i = − ε (ˆp_{v i} − ˆP_{w i}) + ˆF_{w i}, with
  ˆP_{w i} = ∑_j (p_{v i} V_{i j} / p_{v i} V_i) ˆP_j,
  ˆP_j = ∑_k (p_{v k} V_{k j} / P_j F_j) ˆp_{v k},
  ˆF_{w i} = ∑_j (p_{v i} V_{i j} / p_{v i} V_i) ˆF_j. (22)
- Interpretation:
  - Each country faces a CES demand schedule for its value added as if selling to a single world market.
  - ˆP_{w i} and ˆF_{w i} are the aggregate price level and final demand level faced by each country in the composite world market.
  - Demand for country i’s value added falls when its own value-added price rises relative to ˆP_{w i}, with elasticity ε.
  - Perceived world price ˆP_{w i} is a weighted average of value-added price changes across countries; weights derive from value-added export shares and link final goods price levels to value-added prices via value-added import weights.

- Alternative interpretation:
  - The same demand structure could be generated by assuming CES preferences directly over value added:
    F_i = (∑_j (ω_{v j i})^{1/η} V_{j i}^{(η−1)/η})^{η/(η−1)}.

### 1. The VAREER (value-added REER) under equal elasticities
- Setting ˆF_j = 0 for all j, manipulate (22) to obtain:
  ˆV_i = − ε T_{ii}^{VA} ̂REER_{VA i},
  with
  ̂REER_{VA i} ≡ ∑_{j ≠ i} [1 / T_{ii}^{VA} ∑_k (p_{v i} V_{i k} / p_{v i} V_i) (p_{v j} V_{j k} / P_k F_k)] (ˆp_{v i} − ˆp_{v j}),
  and
  T_{ii}^{VA} = 1 − ∑_k (p_{v i} V_{i k} / p_{v i} V_i) (p_{v i} V_{i k} / P_k F_k). (23)
- Notes:
  - ̂REER_{VA i} (VAREER) is based on value-added data alone and captures a normalized relative price change ˆp_{v i} − ˆP_{w i}.
  - Impact governed by elasticity ε and scaling parameter T_{ii}^{VA}, which captures degree of openness in value-added terms.
  - Practical approximation: T_{ii}^{VA} ≈ 1 − (p_{v i} V_{ii} / p_{v i} V_i)(p_{v i} V_{ii} / P_i F_i); as openness increases, T_{ii}^{VA} tends to increase, making demand more sensitive to VAREER changes.

### B. Heterogeneous elasticities: IOREER and implications
- When elasticities differ, REER weights differ from the VAREER in two major ways:
  1. Construction requires the full global input-output framework (not just value-added trade flows).
  2. Weights depend on relative magnitudes of elasticities in production versus demand.
- Elasticities determine weights on different margins of substitution:
  - Higher σ: more weight on substitution across final goods (T_{i j}^σ).
  - Higher γ or ρ: more weight on substitution between inputs from different sources (T_{i j}^ρ) and between inputs and value added (T_{i j}^γ).
- The general value-added REER index (Equation (21)) is termed the IOREER to emphasize dependence on input-output structure and heterogeneous elasticities.

### IOREER: illustrative three-country example (stylized)
- Setup:
  - Country 1 exports all output to 2 as intermediate input.
  - Country 2 consumes some own output and exports remainder to 3 as final goods.
  - Country 3 consumes its own output and does not export.
- Key relationships:
  - Market clearing for country 1: ˆQ_1 = ˆX_{12}, with ˆX_{12} = − γ(ˆp_1 − ˆp_2) + ˆQ_2.
  - Country 2: ˆQ_2 = s_{22} ˆF_{22} + s_{23} ˆF_{23}, with ˆF_{23} = − σ(ˆp_2 − ˆP_3) + ˆF_3.
  - Additional assumptions: ˆV_1 = ˆQ_1, ˆp_1 = ˆp_{v 1}, ˆp_2 = (1 − s_v) ˆp_{v 1} + s_v ˆp_{v 2}, ˆP_3 = (1 − w) ˆp_2 + w ˆp_3, and ˆp_3 = ˆp_{v 3}.
- Setting ˆF_2 = ˆF_3 = 0 and normalizing weights on prices, demand for value added in terms of IOREER:
  ˆV_1 = − T_{11} [((γ − σ s_{23} w) s_v / T_{11}) (ˆp_{v 1} − ˆp_{v 2}) + (σ s_{23} w / T_{11}) (ˆp_{v 1} − ˆp_{v 3})] ≡ − T_{11} ̂REER_{IO 1}, (24)
  with T_{11} = γ s_v + σ s_{23} w (1 − s_v).
- Two highlighted features and implications:
  1. IOREER depends (negatively) on prices in country 3 despite no direct trade between countries 1 and 3: country 1 competes indirectly because its inputs to country 2 are embodied in goods exported to country 3; a rise in country 3 prices makes country 1 relatively more competitive, depreciating country 1’s IOREER.
  2. Sign of IOREER weight on country 2 prices is ambiguous:
     - If preferences are Leontief (σ = 0): a fall in ˆp_{v 2} is bad for country 1 (REER appreciates) via expenditure-switching away from country 1 inputs.
     - If production is Leontief (γ = 0): a fall in ˆp_{v 2} depreciates country 1’s IOREER because increased competitiveness of country 2 in final goods raises demand for country 2’s inputs (including from country 1).
  - In general both input expenditure switching (beggar-thy-neighbor channel) and input linkages (competitiveness spillovers) operate; net response is quantitative and depends on relative elasticities.
  - When input elasticities are low, input linkages dominate and negative REER weights can arise.
  - Input linkages tend to lower REER weights for supply chain partners as competitiveness spillovers counteract demand-side expenditure switching.

*Italic source: _wp15199 - 1.  Substitution in Input Use*

### 2.  Value-Added Elasticities

### 2.  Value-Added Elasticities

### Role of elasticities in mapping IOREER into demand
- Demand for value added is given by: ˆV_i = −T_ii ˆREER_IO_i, with T_ii = σ T_ii^σ + ρ T_ii^ρ + γ T_ii^γ. Elasticities enter because T_ii depends on σ, ρ, γ.
- Effective value-added elasticity:
  - ˜ε_i(σ,ρ,γ) ≡ T_ii / T_ii^VA = [σ (T_ii^σ / T_ii^VA) + ρ (T_ii^ρ / T_ii^VA) + γ (T_ii^γ / T_ii^VA)], as in Equation (25).
  - This ˜ε_i(σ,ρ,γ) summarizes aggregate value-added expenditure switching: ˆV_i = −˜ε_i(σ,ρ,γ) T_ii^VA ˆREER_IO_i.
- Interpretation example:
  - If ˆp^v_i = 0.01 and ˆp^v_j = 0 ∀ j ≠ i (uniform 1% multilateral appreciation), both IOREER and VAREER move by 1%. Change in demand for value added equals ˜ε_i(σ,ρ,γ) T_ii^VA percent.
  - In a pure value-added CES-Armington model this corresponds to elasticity ε = ˜ε_i(σ,ρ,γ).

### Comparison with conventional REERs (Armington)
- Armington-REER (McGuirk (1987)) formula (Equation (26)) uses gross sales and product prices; features “double export weights.”
- Major differences between value-added indexes and Armington-REER:
  - VAREER uses double value-added export weights rather than double gross export weights.
  - IOREER does not feature explicit double-weight scheme; represents a new approach.
  - Price measures differ: value-added indexes use value-added price changes (GDP deflators); Armington-REER commonly uses CPI proxies in practice.
  - Interpretation differs: Armington-REER measures competitiveness for gross output under CES demand for gross output; value-added REERs measure competitiveness for value added, accounting for imported intermediates and input-output linkages.

### Data and elasticity parameterization
- Data sources:
  - WIOD: 40 countries, 1995-2011.
  - Johnson and Noguera (2014): 37 countries, 1970-2009.
  - GTAP used in one figure (Version 7), covers 94 countries and 19 composite regions for year 2004.
  - Price data (GDP deflators, CPI, nominal exchange rates) from IMF World Economic Outlook (annual period averages).
- Elasticity parameterizations examined:
  - Homogeneous case: σ = γ = ρ = 1 (value-added elasticity set to one).
  - Heterogeneous (extreme) case: Leontief production with ρ = γ = 0; choose σ so GDP-weighted mean value-added elasticity over 1995-2009 equals one:
    - σ = 3.00 for WIOD data.
    - σ = 2.25 for Johnson-Noguera data.
    - For GTAP data, σ = 2.90.
- Key elasticity observations:
  - σ and ρ are the key elasticities for REER weights and value-added elasticities; γ is less important.
  - Value-added REER weights depend on relative elasticities (ρ/σ) and (γ/σ).
  - Even positive production elasticities matter when they are low relative to final goods elasticity.

### Building blocks of demand for value added
- Three building blocks to map REER changes into demand:
  1. The REER index (VAREER, IOREER) and how bilateral weights differ from Armington-REER.
  2. Value-added elasticity ˜ε_i(σ,ρ,γ) and its cross-country variation when σ, ρ, γ are heterogeneous.
  3. Value-added openness T_ii^VA (scales the effect of expenditure switching).
- Empirical ranges and patterns:
  - T_ii^VA varies across countries from around 0.15 to 0.65 in the sample.
  - Typical country: T_ii^VA is about 50% larger than the Armington gross-openness analogue S_i.
  - Median T_ii^VA rose over time (see below), increasing how much a given REER change affects demand for value added.

### Value-added REER weights: supply chains and reassignments
- VAREER and IOREER reassign bilateral weights relative to Armington-REER:
  - Countries integrated into supply chains with a partner (e.g., Germany) put less weight on that partner in VAREER and IOREER than in Armington-REER.
  - IOREER accentuates shifts away from supply-chain partners relative to VAREER when production elasticities are low (Leontief).
  - Example magnitudes:
    - Moving from Armington-REER to IOREER roughly halves the weight the Czech Republic attaches to Germany; it roughly doubles the weight Ireland attaches to Germany.
    - For China: a 20% fall in Chinese prices would induce a 4.6% (20×0.23) appreciation in Taiwan’s Armington-REER, but only a 0.08% (20×0.04) appreciation in IOREER.
    - For Vietnam: decline in Chinese prices can raise Vietnamese competitiveness in IOREER (negative IOREER weight observed).
- Regional summary:
  - Typical country’s weight attached to regional partners declines in VAREER and IOREER relative to Armington-REER; extra-regional weights rise.
  - For a typical Asian country, total weight attached to Asian partners is 15 percentage points lower in IOREER than in Armington-REER.
- Correlates:
  - Regional trade agreements (RTAs) associated with increased supply chain activity and declines in VAREER versus Armington-REER weights (~4 percentage points lower VAREER weight for partners with RTA in 2005).
  - Distance positively correlated with differences between value-added and Armington weights; large negative gaps concentrated among partners within 5000km.

### Value-added elasticities and openness (empirical facts)
- Value-added elasticity: ˜ε_i(σ,ρ,γ) = [σ (T_ii^σ / T_ii^VA) + ρ (T_ii^ρ / T_ii^VA) + γ (T_ii^γ / T_ii^VA)].
- Under IOREER with Leontief production (ρ = γ = 0, σ chosen so GDP-weighted mean = 1):
  - Significant downward adjustments in ˜ε_i(3,0,0) relative to 1 for many supply-chain-intensive countries (e.g., Taiwan, Ireland, Indonesia, Hungary).
  - Strong positive correlation between final-goods share of trade and ˜ε_i: more input-intensive (supply-chain) countries have lower effective value-added elasticities.
- Value-added openness (T_ii^VA):
  - Wide cross-country variation (≈ 0.15 to 0.65).
  - For a given REER change, impact on demand for value added can be up to 4 times larger in the most open versus the most closed countries.
  - Median T_ii^VA is substantially larger than gross openness S_i (about 50% larger for the typical country).
  - Time trend: median T_ii^VA nearly doubles from 0.26 in 1970 to 0.42 in the present sample (rising importance of REER changes for demand over time).

### Value-added REERs and time series behavior
- Construction:
  - Year-on-year bilateral value-added price changes aggregated with year-t weights, chained to form index levels (following Federal Reserve Board time-varying weights approach).
- Weight distance measures:
  - City-block distance between VAREER and Armington-REER weights (d_VAREER_it) rose slowly 1970-1990, then more rapidly 1990-2010.
  - Distance between IOREER and VAREER weights (d_IOREER−VAREER_it) shows no obvious long-term trend (stability of input share in trade).
- Prices:
  - Large and persistent differences between GDP deflators and CPI across countries; these price differences account for part of gaps between value-added REERs and CPI-based Armington-REERs.
- Historical REER examples (1995-2011):
  - Eurozone:
    - Germany: VAREER and IOREER depreciate more strongly than Armington-REER (German value-added REER depreciated by 15 percentage points post-2000 while Armington-REER was little changed).
    - GIIPS (Greece, Ireland, Italy, Portugal, Spain): VAREER and IOREER appreciate more than Armington-REER (larger losses in value-added competitiveness).
  - China and United States:
    - China: VAREER and IOREER appreciate by over 20 percentage points during the 2000s despite little trend in Armington-REER.
    - United States: VAREER depreciation larger than conventional REER depreciation after 2000.
- Cross-country statistics on IOREER vs Armington-REER annual gaps:
  - For countries in Figures 9 and 10, quartiles of |(ˆREER_IO − ˆREER_Arm)/ˆREER_Arm| = {0.13, 0.37, 1.06}.
  - For all available countries, quartiles = {0.16, 0.42, 1.25}.
- Decomposition of value-added vs Armington REER deviations (Equation (30)):
  - Two components: weight differences and price differences.
  - For VAREER vs Armington-REER:
    - Weight differences play a small role for the median country at one-year and ten-year horizons; price differences (shift from CPI to value-added prices) explain most of the deviation.
    - Exceptions: some large countries (Germany, France, U.S.) show significant short-run weight-role.
  - For IOREER vs Armington-REER:
    - Weight differences account for ~30% of the gap at one-year horizon (median country) and ~15% over 10-year horizon.
    - Price differences still account for more than half of deviations in general.
  - Reasons weight differences are not larger:
    - Weight reassignments sum to zero; broad simultaneous depreciations reduce differential effects.
    - Covariance between weight reassignments and bilateral price changes is typically small historically.
    - Long-run trends in price differences dominate over weight shifts for long horizons.

### Value-added expenditure switching (mapping REER changes to demand)
- Formulas:
  - IOREER case: ˆV_IO_i = −˜ε_i(σ,ρ,γ) T_ii^VA ˆREER_IO_i.
  - VAREER case (with ε = 1): ˆV_VA_i = −T_ii^VA ˆREER_VA_i.
- Hybrid measure to isolate REER shift effect: ˆV_IO_i(˜ε = 1) = −T_ii^VA ˆREER_IO_i.
  - Comparing ˆV_IO_i(˜ε = 1) to ˆV_VA_i isolates the role of shifting REER measure.
  - Comparing ˆV_IO_i to ˆV_IO_i(˜ε = 1) isolates the role of the value-added elasticity deviation.
- Empirical magnitudes (example year 2005):
  - Korea: demand for value added decreased by 3.0-3.5 percentage points depending on measure used.
  - Germany: demand for value added increased by 1.6 percentage points.
  - Gaps between ˆV_IO_i and ˆV_VA_i can be large (e.g., −0.9 percentage points for Turkey, +0.7 percentage points for Brazil); in Hungary and Taiwan the two measures move in opposite directions.
  - For China in 2005, the larger IOREER appreciation implies a negative demand impact of −0.35 percentage points relative to VAREER.
- Elasticity deviations account for about 1/2 of deviations between ˆV_IO_i and ˆV_VA_i for the median country over 1970-2009 (systematic examination in Appendix B.4).
- Time variation in T_ii^VA matters:
  - Median T_ii^VA rose from 0.26 in 1970 to 0.42 in the present sample, implying a given REER change is nearly twice as influential for demand for value added today as in 1970.

### Key quantitative takeaways
- Homogeneous elasticity baseline: σ = γ = ρ = 1.
- Leontief production extreme: ρ = γ = 0; σ calibrated so GDP-weighted mean value-added elasticity = 1:
  - σ = 3.00 (WIOD), σ = 2.25 (Johnson-Noguera), σ = 2.90 (GTAP).
- Value-added openness T_ii^VA range in sample: ≈ 0.15 to 0.65.
- Median T_ii^VA increased from 0.26 in 1970 to 0.42 in recent data.
- Example demand impacts:
  - Korea: −3.0 to −3.5 percentage points (2005).
  - Germany: +1.6 percentage points (2005).
  - China IOREER vs VAREER demand impact: −0.35 percentage points (2005 decomposition).
- Cross-country distribution of annual relative IOREER vs Armington-REER gaps: quartiles {0.16, 0.42, 1.25} (all countries).

### Conclusions and implications
- Input-output linkages and heterogeneous elasticities reshape REER construction and interpretation:
  - Supply chain linkages allow competitiveness improvements of partners to benefit domestic value added, counteracting standard beggar-thy-neighbor effects.
  - Value-added REERs (VAREER, IOREER) differ systematically from conventional Armington-REERs in weights, prices used, and interpretation.
- Mapping REER changes into demand requires both:
  - Measurement of value-added elasticities (˜ε_i(σ,ρ,γ)), which vary across countries when σ, ρ, γ are heterogeneous.
  - Measurement of value-added openness (T_ii^VA), which has increased over time and amplifies REER effects on demand.
- Practical implications:
  - Conventional CPI-based, gross-export-weighted REERs can misstate changes in demand for domestic value added, especially for supply-chain-integrated countries.
  - Policy analysis of competitiveness and external adjustment should account for input linkages, value-added prices, and elasticity heterogeneity.
- Suggested avenues for future research (as noted in the source):
  - Incorporate input linkages into general equilibrium macro models to study channels for identified shocks.
  - Relax Armington-CES assumptions to allow time-varying markups or alternative production structures.
  - Improve empirical estimates of substitution elasticities, particularly on the production side, since they materially affect price spillovers and REER behavior.

*Italic: Source: _wp15199 - 2.  Value-Added Elasticities (IMF Working Paper chapter/section).*

### REFERENCES

### REFERENCES

### Key literature cited
- Listings of foundational and contemporary works on international trade, production fragmentation, effective exchange rates, and value-added trade, including:
  - Ambler, Steve; Cardia, Emanuela; Zimmerman, Christian (2002), “International transmission of the business cycle in a multi-sector model,” European Economic Review, vol. 46, pp. 273–300.
  - Antràs, Pol (2014), “Global Production: A Contracting Perspective,” Unpublished Manuscript, Harvard University.
  - Armington, Paul (1969), “A Theory of Demand for Products Distinguished by Place of Production,” IMF Staff Papers, vol. 16, no. 1, pp. 159–178.
  - Bems, Rudolfs; Johnson, Robert C.; Yi, Kei-Mu (2010), “Demand Spillovers and the Collapse of Trade in the Global Recession,” IMF Economic Review, vol. 58, no. 2, pp. 295–326.
  - Hummels, David; Ishii, Jun; Yi, Kei-Mu (2001), “The Nature and Growth of Vertical Specialization in World Trade,” Journal of International Economics, vol. 54, pp. 75–96.
  - Johnson, Robert C.; Noguera, Guillermo (2012a), “Accounting for Intermediates: Production Sharing and Trade in Value Added,” Journal of International Economics, vol. 86, no. 2, pp. 224–236.
  - Yi, Kei-Mu (2003), “Can Vertical Specialization Explain the Growth of World Trade?” Journal of Political Economy, vol. 111, pp. 52–102.
- (Additional references cover REER construction, measurement of competitiveness, trade elasticities, input-output databases (WIOD), and methods for estimating value-added trade and effective exchange rates.)

### Tables and figures — core empirical patterns and numeric summaries
- Table 1 (Differences in REER Weights in 2005, by Region)
  - Panel A: VAREER Weight minus Armington-REER Weight (changes in weights expressed in percentage points). Examples:
    - From Source Region Asia to Partner Region Asia: -5.9
    - From Source Region EU to Partner Region EU: -4.7
  - Panel B: IOREER Weight minus Armington-REER Weight. Examples:
    - From Source Region Asia to Partner Region Asia: -15.7
    - From Source Region EU to Partner Region EU: -5.7
  - Note: Columns might not sum to zero due to rounding. Data from Johnson and Noguera (2014).
- Table 2 (Contribution of Weights to Differences Between Value-Added and Armington REERs)
  - Reports median contributions (as shares of ̂REER_x it − ̂REER_Armington it) over 1970-2009 at 1-year and 10-year horizons for two comparisons: VAREER minus Armington-REER and IOREER minus Armington-REER.
  - Sample entries (selected):
    - AUS: VAREER 1-year 0.00, 10-year 0.04; IOREER 1-year 0.13, 10-year 0.17
    - DEU: VAREER 1-year 0.33, 10-year 0.00; IOREER 1-year 0.47, 10-year -0.02
  - Median row: VAREER 1-year 0.03, 10-year 0.03; IOREER 1-year 0.28, 10-year 0.16.
- Figures (selected empirical findings summarized)
  - Figure 2–4: REER weights assigned to specific countries (Germany, China, South Korea) demonstrate substantial differences between Armington REER, VAREER, and IOREER depending on elasticity assumptions. Notes preserve elasticity parameter choices used in figures:
    - VAREER weights: {σ,γ,ρ}={1,1,1}
    - IOREER weights: examples include {σ,γ,ρ}={3,0,0}, {3,0,0}, and {2.9,0,0} in different figures.
  - Figure 5 and Figure 16: Cross-country deviations in effective value-added elasticities under inflexible global supply chains show nontrivial deviations from homogeneous elasticity ( ̃ε_i − ε (ε=1) displayed).
  - Figure 6: Value-Added Openness versus Gross Openness (2004) — value-added openness defined as T_ii^VA [Equation (23)] and gross openness as S_i [Equation (26)] (Data from WIOD).
  - Figure 7: Reassignment of REER weights over time, 1970-2009 — distances between weight vectors measured by city-block metric; VAREER-Armington REER and IOREER-VAREER plotted (elasticity choices noted with examples {σ,γ,ρ}={1,1,1} and {2.25,0,0}).
  - Figure 8 and Figure 18: Differences between GDP deflator and CPI and decomposition into VA terms-of-trade and approximation components; notes: log relative price normalized to zero in 2000; data from IMF’s World Economic Outlook and EU KLEMS databases.
  - Figure 9–11: Time series of Armington-REER, VAREER, IOREER for selected countries (EMU countries, China, USA), normalized to zero in 1995; elasticity parameter choices noted in figure notes.
  - Figure 11 & Figure 19: Changes in demand for value added and decomposition of contributions — highlight that the elasticity effect (heterogeneous effective elasticities) can account for a sizable share of deviations in demand for value added.

### Appendix A — Framework: algebraic derivations and special cases
- A.1 Demand for Value Added with Equal Elasticities (ε ≡ γ = ρ = σ)
  - Key reduced-form expressions (preserved exactly as in source):
    - Equation (31): ̂Q = S_F ̂F + S_X [ −ε M1 ̂p + ε M2 ̂p + M2 ̂Q ] (notation preserved).
    - Equation (33): ̂Q = −ε ̂p + ε[I − S_X M2]^{-1} S_F M2 W_f ̂p + ̂F_w, with ̂F_w defined as in main text.
    - Equation (35): ̂V = −ε(̂p_v − ̂P_w) + ̂F_w with ̂P_w ≡ [I − S_X M2]^{-1} S_F M2 W_f [I − Ω′]^{-1} [diag(s_v i)] ̂p_v and ̂F_w ≡ [I − S_X M2]^{-1} S_F M2 ̂F.
  - Interpretation:
    - ̂P_w are perceived world prices of value added (weighted averages of ̂p_v originating from all countries).
    - Weights in ̂P_w equal shares of value added from source i embodied in final goods in destination j (value-added export shares).
    - This derivation yields Equation (22) in main text — the Armington-CES model basis for the VAREER.
- A.2 Demand for Value Added with Restrictions on Input Trade — three special cases (explanatory interpretations preserved)
  - Case I: no intermediate inputs (S_X, Ω and s_X i zeros)
    - Equation (36): ̂V_i = −σ ̂p_v i + η ∑_j (p_v i F_ij / p_v i V_i) ̂P_j with ̂P_j = ∑_k (p_v k F_kj / P_j F_j) ̂p_v k.
    - Interpretation: value-added REER is a double export-weighted index of bilateral relative value-added prices. Exports equal final goods; can be re-written in gross terms because Q_i = V_i and p_i = p_v i.
    - Discussion: this special-case rationale for interpreting conventional REERs as demand for value added is problematic for three reasons listed in the source (counterfactual, no practical guidance, inconsistent with Armington approach).
  - Case II: domestic inputs only (Ω diagonal with ω_ii = s_X i)
    - Equation (38): ̂V_i = −η ̂p_v + η ∑_j ( (1−ω_ii)^{-1} p_i F_ij / p_i Q_i ) ̂P_j with ̂P_j = ∑_k (p_k F_kj / P_j F_j) ̂p_v k.
    - Interpretation: final goods shares converted into gross-output-needed shares via (1−ω_ii)^{-1}; destination weights equal share of value-added exports in total value added.
  - Case III: restricted input trade and homogeneous elasticities
    - Consider two-country input link (Ω_12 > 0 only) with equal elasticities (Armington VAREER case).
    - Equations (39)–(41) show destination price indexes and country-specific demand expressions:
      - ̂P_j = (p_1 F_1j + p_2 F_2j Ω_12 / P_j F_j) ̂p_v1 + (p_2 F_2j (1−Ω_12) / P_j F_j) ̂p_v2 + ∑_{k≠1,2} (p_k F_kj / P_j F_j) ̂p_v k.
      - ̂V_1 = −η ̂p_v1 + η ∑_j (p_1 F_1j + Ω_12 p_2 F_2j) / p_1 Q_1 ̂P_j.
      - ̂V_2 = −η ̂p_v2 + η ∑_j (p_2 F_2j / p_2 Q_2) ̂P_j = −η ̂p_v2 + η ∑_j ( (1−Ω_12) p_2 F_2j / (1−Ω_12) p_2 Q_2 ) ̂P_j.
    - Interpretation: destination weights equal shares of value-added from source i consumed in destination j, capturing direct and indirect (via other countries’ exports that embody inputs) consumption of value added. Takeaway: value-added trade measures capture production linkages not visible in gross-export or final-goods-only measures.

### Appendix B — Empirical appendix: sensitivity, parametrization, and decompositions
- B.1 Sensitivity of value-added elasticity and bilateral REER weights to σ, γ, ρ
  - Value-added elasticity expression (Equation (42)):
    - ̃ε_i(σ,γ,ρ) = (T_ii^σ σ + T_ii^ρ ρ + T_ii^γ γ) / T_ii^VA  (preserved formula structure).
  - Bilateral REER weight for country i toward partner j:
    - (σ T_σ ij + ρ T_ρ ij + γ T_γ ij) / T_ii(σ,γ,ρ).
  - Key empirical finding (Figure 13 summary):
    - The weight attached to ρ (elasticity of substitution among inputs) is the largest, with a median value of 0.52 in the full sample.
    - The weight attached to σ has a country median of 0.29.
    - The weight attached to γ accounts for the remainder with median value of 0.17.
    - Interpretation: value-added elasticity and REER weights are most sensitive to ρ; γ is least important due to offsetting substitution effects described.
  - Decomposition of T_ii^γ into T_ii,V^γ and T_ii,X^γ:
    - Median sample values: T_ii,V^γ / T_ii^VA = 0.57 and T_ii,X^γ / T_ii^VA = −0.40.
- B.2 Leontief final demand case (σ = 0 and γ = ρ > 0)
  - Parametrization chosen: γ = ρ = 1.50 so that global, GDP-weighted value-added elasticity ≈ 1 over 1995-2009 (WIOD).
  - Main empirical patterns:
    - IOREER weights with Leontief final demand tend to move back toward Armington-REER weights (i.e., undo VAREER adjustments), because low final demand elasticity assigns higher weights to partners with larger final goods shares.
    - Correlation between input trade intensity and IOREER-VAREER weight differences is negative under Leontief demand (contrast with positive correlation under Leontief production).
    - Value-added elasticities under Leontief demand are lower for countries with higher final goods shares in trade. Overall magnitude of deviations from 1 is about half as large as in the Leontief production case, reflecting input trade shares being twice final-goods trade shares.
    - Figure 17: cumulative deviations between IOREER and VAREER indexes (1995-2011) present mirror-image patterns between Leontief production vs Leontief demand parameterizations.
- B.3 Decomposition: value-added versus CPI price differences
  - Decomposition (Equation (44)):
    - ̂p_v − ̂p_cpi = (̂p_v − ̂p) [VA terms of trade] + (̂p − ̂p_cpi) [approximation].
  - Interpretation:
    - ̂p_v − ̂p captures differences between value-added and gross-output prices (value added terms of trade).
    - ̂p − ̂p_cpi captures differences between gross output prices and consumer prices (approximation error when using CPI-based REERs).
  - Empirical patterns (Figure 18):
    - Both components contribute importantly to the gap between value-added and consumer price measures, with country-specific heterogeneity (e.g., Germany: gross output and value-added prices track closely while CPI grows faster; Spain: gross output and CPI track, large gap between value-added and gross output).
    - Data sources: IMF World Economic Outlook and EU KLEMS (gross output price indexes).
- B.4 Quantifying the role of effective value-added elasticities (elasticity effect) in deviations of demand for value added
  - Decomposition (Equation (45)):
    - ̂V_IO it − ̂V_VA it = T_VA ii t (̂REER_IO it − ̂REER_VA it) [REER effect] + T_VA ii t ̂REER_IO it (̃ε_it(σ,ρ,γ) − 1) [Elasticity effect].
  - Interpretation:
    - First term (REER effect): captures differences in IOREER vs VAREER price changes, holding elasticity at 1.
    - Second term (Elasticity effect): captures how country-specific deviations in effective value-added elasticities (̃ε_it − 1) interact with IOREER changes to affect demand for value added; both terms scaled by country-specific openness.
  - Empirical magnitudes (Figure 19 summary):
    - Median one-year Elasticity effect can be economically meaningful; example: Japan median one-year contribution of elasticity effect = 0.13 percentage points.
    - For many countries the elasticity effect contributes substantially; median contribution of elasticity effect to deviations in demand for value added across countries is close to 0.5 (i.e., elasticity effect explains about half of the deviation for the median country over 1970-2009).
    - Left panel of Figure 19: absolute size of median elasticity effect per country (median absolute value across 1970-2009) shown in percentage points.
    - Right panel of Figure 19: median contribution (share of total deviation) of elasticity effect across countries.

*Italic: Content unit _wp15199 - REFERENCES (source PDF: _wp15199 - REFERENCES).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15199.pdf_