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### Stylized heterogeneous-agent model: structure and mechanisms
- Households: isoelastic non-homothetic CES preferences (à la Matsuyama (2019)) over food (C_F) and non-food (C_NF); households differ by income y(h).
- Definitions:
  - Food insecurity (PoU): fraction of households whose food consumption falls below an exogenous threshold C_F (subsistence requirement).
  - Poverty (POV): real income falling below R.
- Key model determinants of responsiveness of food insecurity to food price changes:
  - (i) shape of the income distribution;
  - (ii) magnitude of the substitution elasticity (η) relative to the income elasticity of food demand (ε_F);
  - (iii) degree of income redistribution (κ).
- Urban vs rural distinction:
  - Urban: food prices do not affect household nominal income.
  - Rural: higher food prices raise nominal income proportionally to households’ claims on agricultural output (farmers receive windfall gains).
- Explicit threshold expressions:
  - y_PoU = ˜α p_F Q − κ μ_y  / (1 − κ)
  - y_POV = P R + p_F C_F − κ μ_y  / (1 − κ)
  - 1/˜α is the share of nominal income that affords the minimum food basket Q.
- Assumptions for comparative statics:
  - Assumption 1: growth shifts the income CDF right such that dF/dμ_y < 0.
  - Assumption 2: growth parameter θ_i does not affect Gini (∂G/∂θ_i = 0).
  - Assumption 3: higher global food prices increase income only in rural economies; dμ_y/dp_F = 1(Rural) R_F.
  - Assumption 4: eligible CDFs homogeneous of degree zero in y and μ_y.
- Example CDFs satisfying assumptions:
  - Log-normal: F(y; μ, σ) = Φ((ln(y) − μ)/σ); Gini = 2Φ(σ/√2) − 1.
  - Pareto-like: F(y; μ, ρ) = (1/(1+ρ) y/μ)^{1/ρ}; Gini = ρ/(ρ + 2).

### Comparative statics: analytical results and intuition
- Proposition 1: semi-elasticities (equations (14)–(16)):
  - dPoU/dpF / pF = ( f[yPoU] dyPoU/dpF (±) + ∂F/∂μy dμy/dpF (0/−) ) pF
  - dPoU/dμy / μy = [ ∂F/∂μy − f[yPoU] κ/(1−κ) ] μy < 0
  - dPoU/dκ / κ = f[yPoU] ( yPoU − μy/(1−κ) ) κ ≷ 0 for yPoU ≷ μy
- Interpretations:
  - Higher food prices change the nominal income needed for minimum calories; effect on PoU depends on mass f[yPoU] at the threshold and on rural income shifts.
  - Higher μy lowers PoU via CDF shift and larger social benefits for households below mean.
  - Higher κ lowers PoU if marginally food insecure household income is below mean.
- Proposition 2 (CES case, εF = εNF = 1, κ = 0): closed-form PoU and POV in urban and rural economies (eqs. (17)–(20)):
  - PoU_U and PoU_R expressions show urban PoU price elasticity is typically positive; rural PoU price elasticity can be positive, zero, or negative depending on η (positive only for η > 1; negative for η < 1; zero for η = 0).
  - POV elasticities: urban POV price elasticity positive (if λ > 0); rural POV both income and price elasticities negative.
- Three sub-effects govern food price elasticity of PoU: (i) price effect (positive), (ii) expenditure share effect (≷0 for η ≷ 1), (iii) negative income effect for farmers (rural only).
- Inequality effects (Proposition 3):
  - Log-normal: dPoU/dσ / σ = − [ ln(yPoU) − μ ] / σ  Φ′( [ ln(yPoU) − μ ] / σ ) ≷ 0 ⇔ yPoU ≷ e^μ. PoU increases with inequality if yPoU is below the median income.
  - Pareto: dPoU/dρ / ρ ≷ 0 ⇔ PoU ≷ PoŪ ≡ e^{−1/(1+ρ)}. PoU increases with inequality iff initial inequality is high enough (or initial PoU small enough).
- Structural responsiveness (Proposition 4, CES Pareto Example):
  - dPoU/dμy / μy sign depends on whether ρκ ≶ ṽα pF Q / μy.
  - Semi-elasticity can vary with income and inequality; two opposing effects determine responses (fewer households near threshold vs. larger redistribution).

### Calibration and quantitative model predictions
- Calibration approaches:
  - Stone-Geary Cobb-Douglas: εF = εNF = η = 1; Pareto income distribution; region-specific parameter matching (Gini, α, κ, sq, λ).
  - Isoelastic non-homothetic CES: Pareto income distribution; parameters from Nath (2023): η = 0.27; εF = 0.29; εNF = 1.08; m_NF informed by Smith and Subandoro (2007).
- Calibration semi-elasticity predictions:
  - Stone-Geary CD: semi-elasticity of PoU to income ranges from -0.10 pp to -0.17 pp; average ≈ -0.12 pp.
  - Implication: a doubling of income per capita should lead to a potential reduction of the PoU by 12 pp (calibration interpretation).
  - Under CD with κ = 0: urban food price elasticity equals income elasticity in magnitude but opposite in sign; rural food price elasticity = 0. Country-level food price semi-elasticity range: 0 to 0.17 percentage points depending on urban/rural representation.
  - Isoelastic CES yields same semi income elasticity; price elasticities remain close to opposite of income semi-elasticity given chosen parameters (η − εF / εF near zero).

### Empirical strategy, data, and instruments
- Dataset: longitudinal covering 142–143 countries over 2001–2021 (sample described as 142 countries in some sections, 143 in others); final estimation samples vary due to missing data and exclusions.
- Main outcome: Prevalence of Undernourished (PoU) constructed using a log-normal probability density for dietary energy intake; PoU = ∫_{x<MDER} f(x|Θ) dx.
- Additional outcomes: diet composition (share of dietary energy from cereals, tubers and roots; average supply of animal protein grams per capita per day).
- Key controls: GDP per capita (constant 2015 dollars), social protection expenditure (% of GDP), food inflation (year-on-year change in food CPI component from Ha et al. (2023)).
- Estimation approach:
  - First-differenced specification with FD-IV to address endogeneity and reverse causality: ∆y_it = β_0 + θ y_i0 + β_1 ∆X_it + ∆ǫ_it.
  - Instruments for income growth:
    - Average growth rate of trading partners (three-year export-weighted).
    - Change in commodities terms-of-trade (IMF CTOT index; three-year weights).
  - Instruments for food inflation:
    - External cereal harvest shocks (regional and global): percentage deviations from per capita trend production of wheat, corn, soybeans, rice using HP filter (smoothing parameter 6.25); regional shocks exclude country i.
- Identification checks: first-stage F statistics, Hansen tests; evidence of instrument relevance and validity reported in appendix tables.

### Main empirical findings (FD and FD-IV estimates)
- Economic growth effect:
  - FD (OLS): PoU declines by roughly 0.05 percentage points (pp) for every 1 pp increase in GDP per capita growth (0.37 percent reduction).
  - FD-IV (preferred): 1 pp increase in income per capita growth → approximately 0.11 pp reduction in the share of undernourished (0.79 percent reduction).
  - Calibration-consistent: semi-elasticity ≈ -0.11 pp (empirical IV) vs. model average ≈ -0.12 pp.
- Food inflation effect:
  - FD (OLS): increase of 0.011 pp in undernourishment per 1 pp in food inflation.
  - FD-IV: 1 pp acceleration in food inflation → share of undernourished increases by 0.063 pp (0.46 percent).
  - Model prediction: food price semi-elasticities: rural areas positive yet close to 0; urban areas around 0.12 pp; country-level elasticities within those bounds.
- Standard-deviation scenarios:
  - Sample sd: 0.043 for GDP growth and 0.058 for inflation changes.
  - One sd increase in GDP growth → estimated 3.4% reduction in undernourishment.
  - One sd increase in food inflation → estimated 2.7% increase in undernourishment.
- Convergence:
  - Higher initial PoU associated with faster reductions: starting 1 pp higher PoU implies faster reduction by 0.02 pp per year (significant).
- Inequality interactions:
  - Income semi-elasticity shows slight tendency to decrease as inequality grows; empirical evidence suggests semi-elasticity slightly stronger in more equal societies.
  - Interaction estimates: a 1 pp increase in income share of bottom 20% reduces PoU by 0.1 pp (computed as -0.147 + 1.857*0.024), though imprecisely estimated (se 0.323).
- Diet composition:
  - Income growth reduces share of cereals, roots and tubers (CER) and increases animal protein supply (PROT).
  - FD-IV estimates: ∆ln GDP pc → CER: -0.068* (0.038); PROT: 14.257*** (4.033).
  - Food inflation coefficients on diet composition generally indistinguishable from zero in OLS; FD-IV suggests small/insignificant effects.

### Instrument performance, diagnostics, and sensitivity
- First-stage highlights (Table 5):
  - Trade partner growth strongly predicts domestic GDP growth (coefficients ~0.397*** for GDP stage).
  - ∆ln CTOT affects GDP and inflation in expected directions; global/regional harvest shocks negatively predict food inflation.
  - First-stage F statistics vary by specification: examples include F stat: 8.7 (GDP), 1.8 (inflation) in benchmark first stage; alternative instrument sets yield F stat: 16.5 (GDP), 12.4 (inflation).
- Instrument validity:
  - Hansen test in baseline: statistic 5.1; p value (0.16) — cannot reject instrument exogeneity.
  - Alternative instruments (regional shocks) produce similar second-stage results; Hansen p value (0.17).
  - Hausman test of endogeneity favored FD-IV over FD (reported p value = 0), motivating FD-IV as preferred estimates.
- Sensitivity checks:
  - Replacing global with regional harvest shocks: results similar; regional shock smaller but significant.
  - Controlling for armed conflict (dummy for ≥1000 battle deaths): FD estimate of conflict effect positive but small (sd = 0.35); FD-IV diagnostics remain satisfactory; core income and inflation effects unchanged.
  - Extended controls including social protection expenditure: FD and FD-IV indicate social protection contains undernourishment but evidence is weak due to missing data and weaker first-stage diagnostics.

### Heterogeneity results
- Food inflation × agricultural GDP share:
  - FD-IV total effect β1 + β3 W shows declining adverse effect of food inflation with higher agricultural GDP share (suggestive but interaction statistically insignificant).
- Growth × GDP per capita:
  - Growth reduces undernourishment most strongly at low income levels; point-wise significance up to GDP per capita of USD 20 thousand.
  - LICs: rapid decline in PoU with income growth; HICs: much smaller responsiveness.
- Growth × income distribution:
  - Growth effectiveness increases with income share of bottom 20% (small gradient; significance limited to portions of range).
  - Interaction coefficients: ∆ln GDP pc × Income bottom 20% = 1.857 (2.187) in Table 8 (first-stage F statistics reported).

### Policy-relevant implications and recommendations
- Ranking macro drivers:
  - Real GDP growth is quantitatively more important than food inflation for reducing undernourishment: income semi-elasticity ≈ -0.11 pp per 1 pp growth vs. food-inflation semi-elasticity ≈ 0.06 pp per 1 pp inflation (IV estimates).
- Context-specific design:
  - Account for urban-rural differences: higher food prices can have opposite effects on poverty and food insecurity in rural areas via farmer-income gains.
- Redistribution and social protection:
  - Degree of income redistribution (κ) affects PoU and POV thresholds; targeted social protection can tackle food insecurity at the macro level.
  - Extended-control FD-IV suggests a typical change in social protection expenditure leads to a reduction in undernourishment of 0.01 pp (after scaling), but evidence is weak and sample-limited.
- Measurement and targeting:
  - Distinguish poverty from food insecurity when designing interventions because elasticities differ.
- Policy mix recommended:
  - Promote inclusive growth, redistribution, targeted social protection (within fiscal limits).
  - Allow global food price increases to pass through to domestic prices while enhancing targeted social protection to:
    - Encourage sufficient food production;
    - Increase incomes of food-insecure farmers;
    - Protect purchasing power of vulnerable urban households.
  - Assess policy design case-by-case given variation in countries’ capacity to adjust social protection coverage and transfer sizes.

### Appendix highlights: tables, figures, and instruments
- Figures:
  - Figure 1: Effects of income per capita and Gini on PoU (parameter values: η=ǫF=ǫNF=1, α=0.14, C_F=0.5, κ=0.05, λ=0.97, pF chosen so sq ≡ pF Q μy =0.14 for μy =2.5).
  - Figure 2: Sensitivity of semi-elasticity of PoU to income with respect to income and inequality (same low-income parameterization).
  - Figure 3: Evolution of PoU over time by income group (years: 2000, 2005, 2010, 2015, 2020) and PoU vs GDP per capita by income group.
  - Figure 5: Growth effects variation for undernourishment (GDP per capita 0–100 I$ 000; Income bottom 20% range .04–.19).
- Selected table highlights (exact numeric entries preserved in source tables):
  - Table 1 (CD calibration): region dPoU/dμy /μy examples: Africa -0.137; East Asia & Pacific -0.110; Latin America & Carib. -0.168; U.S. & Canada -0.121.
  - Table 2 (CES calibration): Africa dPoUU/dμy /μy = -0.137; dPoUU/dpF /pF = 0.130; dPoUR/dpF /pF = -0.001.
  - Table 4 (main equation PoU): FD OLS ∆ln GDP pc = -0.045*** (0.007); FD-IV ∆ln GDP pc = -0.107*** (0.033); FD OLS ∆Food inflation = 0.011** (0.005); FD-IV ∆Food inflation = 0.063*** (0.024); Hansen test 5.1; p value (0.16); N: 2626 (OLS), 2011 (IV).
  - Table 5 (first stages): Trade partner growth → GDP stage 0.397*** (0.088); Global harvest shock (t-1) → Food inflation stage -0.003*** (0.001). First-stage F stat examples: 8.7 (GDP), 1.8 (inflation).
  - Table 6 (extended controls): ∆Soc protection: OLS -0.001** (0.000); IV -0.003* (0.002); first-stage F 7.3 (p value 0.000); Hansen test 6.9; p value (0.08).
  - Table 9 (alternative instruments FD-IV): ∆ln GDP pc = -0.096*** (0.035); ∆Food inflation = 0.043* (0.023); Hansen test 4.9; p value (0.17).
- Instrument construction formulas:
  - ∆ln(CTOTi,t) = Σj ∆ln(Pj,t) Ωi,j,t where Ωi,j,t defined by three-year average export net export shares.
  - Export-weighted partner growth gxi,t formula uses three-year moving export shares.
  - Harvest shocks: calorie-weighted sum of wheat, maize, soybeans, rice production deviations from HP-trend per capita; regional shocks exclude country i.
- Diet composition regressions (Table A1 & A3): FD-IV ∆ln GDP pc effects on CER and PROT reported (e.g., PROT FD-IV 14.257*** (4.033)) with first-stage diagnostics reported.

*wpiea2024188-print-pdf - Appendix A: Tables and figures, canonical PDF: wpiea2024188-print-pdf*

### conclusion, we believe these findings are of interest for policymakers seeking to tackle food

### GROWTH, INFLATION, AND FOOD INSECURITY

### Stylized heterogeneous-agent model: structure and mechanisms
- Households have isoelastic non-homothetic CES preferences (à la Matsuyama (2019)) over food (C_F) and non-food (C_NF) and differ by income y(h).
- Food insecurity is defined as the fraction of households whose food consumption falls below an exogenous threshold C_F (subsistence requirement); poverty is defined by real income falling below R.
- Key model ingredients that determine responsiveness of food insecurity to food price changes:
  - (i) the shape of the income distribution;
  - (ii) the magnitude of the substitution elasticity (η) relative to the income elasticity of food demand (ε_F);
  - (iii) the degree of income redistribution (κ).
- Urban vs rural distinction:
  - Urban economy: food prices do not affect household nominal income.
  - Rural economy: higher food prices raise nominal income proportionally to households’ claims on agricultural output (farmers receive windfall gains).
- Important model results and comparative-statics:
  - Higher food prices can increase food insecurity while simultaneously reducing poverty in rural settings due to farmer-income windfalls.
  - The model produces explicit expressions for income thresholds associated with undernourishment (y_PoU) and poverty (y_POV):
    - y_PoU = ˜α p_F Q − κ μ_y  / (1 − κ)
    - y_POV = P R + p_F C_F − κ μ_y  / (1 − κ)
  - ˜α is defined exactly as in the model (see equation (8) in source) and 1/˜α is the share of nominal income that affords the minimum food basket Q.
- Assumptions used to derive comparative statics:
  - Assumption 1: growth (a change in parameter θ_i) shifts the income CDF to the right such that dF/dμ_y < 0.
  - Assumption 2: growth parameter θ_i does not affect Gini inequality (∂G/∂θ_i = 0).
  - Assumption 3: higher global food prices increase income only in rural economies; dμ_y/dp_F = 1(Rural) R_F.
  - Assumption 4: eligible CDFs are homogeneous of degree zero in y and μ_y (F(λ y; λ μ_y) = F(y; μ_y)).
- Example CDFs satisfying assumptions:
  - Log-normal: F(y; μ, σ) = Φ((ln(y) − μ)/σ); Gini = 2Φ(σ/√2) − 1.
  - Pareto-like: F(y; μ, ρ) = (1/(1+ρ) y/μ)^{1/ρ}; Gini = ρ/(ρ + 2).

### Quantitative model predictions (calibrated)
- Global average income semi-elasticity of the PoU: approximately -0.12 percentage points (pp).
  - Interpreted: a doubling of income per capita should lead to a potential reduction of the PoU by 12 pp.
- Food price semi-elasticities predicted by the model:
  - Rural areas: positive yet close to 0.
  - Urban areas: around 0.12 pp.
  - Implication: country-level PoU food price elasticities should lie within these rural and urban bounds.

### Empirical strategy and data
- Novel longitudinal dataset covering 142 countries over 2001-2021.
- Instrumental variables used to isolate exogenous variation in domestic income growth and food inflation:
  - (i) changes in a country’s commodity terms-of-trade;
  - (ii) economic growth in trading partners;
  - (iii) external cereal harvest shocks.
- Rationale: address endogeneity and potential reverse causality (e.g., government responses to deteriorating food security affecting next year’s growth or prices).

### Main empirical findings (IV estimates)
- Economic growth effect:
  - A 1 percentage point increase in economic growth leads to an approximately 0.11 pp reduction in the share of undernourished (0.79%), consistent with model predictions.
- Food inflation effect:
  - The undernourishment semi-elasticity to food inflation is estimated at -0.06 pp (0.46%), about 40% smaller than the income semi-elasticity.
- Inequality interaction:
  - The semi-elasticity of undernourishment to income shows a slight tendency to decrease as inequality grows.
- Convergence:
  - Countries with higher initial undernourishment show higher rates of reduction, but the pace is slow — providing modest evidence of cross-country convergence in undernourishment.
- Diet composition:
  - Supplementary appendix results indicate income changes alter diets: households react to recessions by replacing expensive calorie sources (e.g., proteins) with cheaper ones (e.g., carbs).

### Contributions to literature and methodological advances
- Two primary contributions:
  - First tractable heterogeneous-agent macro model of food insecurity generating testable predictions linking income distribution, prices, and redistribution to PoU.
  - Simultaneous estimation of the effects of income growth and food inflation on food insecurity using an empirical strategy that explicitly addresses endogeneity.
- Relation to prior work:
  - OLS estimates in soriano & garrido-type studies are smaller than the preferred IV estimates here; IV estimates align more closely with model magnitudes.
  - The model reconciles findings where higher food prices reduce poverty (via farmer-income channel) yet increase food insecurity.
- Secondary contributions:
  - Study of both quantitative sufficiency (calories) and qualitative adequacy (diet quality) of diets.
  - First to consider social protection as a fiscal instrument in the macro analysis of food insecurity (previous work focused on access to specific programs).

### Policy-relevant implications (for policymakers)
- Ranking macro drivers:
  - Real GDP growth is quantitatively more important than food inflation for reducing undernourishment (income semi-elasticity ≈ -0.11 pp per 1 pp growth vs. food-inflation semi-elasticity ≈ -0.06 pp).
- Context-specific effects:
  - Policies addressing food insecurity should account for urban-rural differences: higher food prices can have opposite effects on poverty and food insecurity in rural areas through farmer-income gains.
- Redistribution and social protection:
  - Degree of income redistribution (κ) matters for thresholds for undernourishment and poverty; social protection can be an instrument to tackle food insecurity at the macro level.
- Measurement and targeting:
  - Distinguish between poverty and food insecurity when designing interventions because food price elasticities differ for each outcome.
- Research and monitoring:
  - Use of instruments (terms-of-trade, trading partner growth, external cereal shocks) is important for causal assessment of macro policy impacts on food insecurity.

*Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024188-print-pdf.pdf*

### 2.2  Comparative Statics

### 2.2 Comparative Statics

### Key analytical results (Proposition 1)
- Differentiating PoU with respect to pF, μy, and κ yields semi-elasticity expressions (see eqs. (14)-(16)):
  - dPoU/dpF / pF = ( f[yPoU] dyPoU/dpF (±) + ∂F/∂μy dμy/dpF (0/−) ) pF  (eq. (14))
  - dPoU/dμy / μy = [ ∂F/∂μy − f[yPoU] κ/(1−κ) ] μy < 0  (eq. (15))
  - dPoU/dκ / κ = f[yPoU] ( yPoU − μy/(1−κ) ) κ ≷ 0 for yPoU ≷ μy  (eq. (16))
- Interpretations:
  - (i) Higher food prices change the nominal income needed to buy minimum calories; PoU changes in proportion to the share f[yPoU] at the income threshold. In rural economies, higher food prices also shift the income CDF to the right via windfall gains to farmers, so the overall effect of higher food prices on PoU is ambiguous.
  - (ii) Higher GDP per capita (μy) lowers PoU by shifting the CDF to the right and by increasing social benefits for households below average income.
  - (iii) An increase in the degree of income redistribution (higher κ) lowers PoU provided the income of the marginally food insecure household is below the mean.
- Emphasis: Propositions concern the extensive margin (share of households below threshold), so responses depend on both demand parameters and attributes of the income distribution (e.g., mass f[yPoU] at the threshold).

### Sharpening predictions with functional forms (Proposition 2 — CES case)
- With CES preferences (εF = εNF = 1) and κ = 0 (no redistribution channel), closed-form solutions for PoU and POV in urban and rural economies are given (eqs. (17)-(20)):
  - PoU_U = [ 1/(1+ρ)  ṽα (pF/P) Q_R ]^{1/ρ} = f_U(pF +, R −)  (eq. (17))
  - PoU_R = [ 1/(1+ρ)  ṽα (pF/P) Q (pF/P)_R^F ]^{1/ρ} = f_R(pF −/+, R_F) −  (eq. (18))
  - POV_U = [ 1/(1+ρ) ( R + pF/P λ Q_R ) ]^{1/ρ} = g_U(pF +, R −)  (eq. (19))
  - POV_R = [ 1/(1+ρ) ( P pF R + λ Q_R^F ) ]^{1/ρ} = g_R(pF −, R_F −)  (eq. (20))
- Qualitative implications:
  - Food insecurity (PoU):
    - (i) In the urban economy, the positive price elasticity of PoU is ≤ the negative income elasticity in magnitude iff η ≤ 1 (i.e., dPoU/dpF / pF ≤ − dPoU/dR / R = 1/ρ PoU < 0).
    - (ii) In the rural economy, the food price elasticity of PoU is positive only for η > 1, negative for η < 1, and zero for η = 0. The agricultural income elasticity of PoU equals the urban income elasticity (dPoU/dR_F / R_F = − 1/ρ PoU < 0).
  - Poverty (POV):
    - (iii) In the urban economy, the food price elasticity of POV is positive (provided λ > 0) and the income elasticity is negative. For the food price elasticity of PoU to be larger in magnitude than that of POV it is sufficient (but not necessary) that either η ≥ 1 or λ = 0.
    - (iv) In rural economies, both income and price elasticities of POV are negative (higher income and higher food prices lower POV).
- Mechanisms: the food price elasticity of PoU is governed by up to three sub-effects — a positive price effect, an expenditure share effect (≷0 for η ≷ 1), and a negative income effect for farmers (only in rural economies). In urban economies the price effect dominates, making the food price elasticity strictly positive. In rural economies the income and price effects can offset so the expenditure share effect determines the sign.

### Reconciling empirical puzzles
- The framework explains how higher food prices can simultaneously magnify food insecurity (PoU) but reduce poverty (POV) under realistic parameter values (e.g., η ≤ 1). Empirical literature summarized: short-term higher food prices typically exacerbate food insecurity (e.g., Headey (2013)) but can reduce poverty (e.g., Headey and Hirvonen (2023), Headey (2018)).

### Inequality effects (Proposition 3)
- Log-normal CDF (Example 1):
  - dPoU/dσ / σ = − [ ln(yPoU) − μ ] / σ  Φ′( [ ln(yPoU) − μ ] / σ ) ≷ 0 ⇔ yPoU ≷ e^μ.
  - Interpretation: PoU strictly increases with inequality if yPoU is below the median income; decreases otherwise.
- Pareto CDF (Example 2):
  - dPoU/dρ / ρ ≷ 0 ⇔ PoU ≷ PoŪ ≡ e^{−1/(1+ρ)}.
  - Interpretation: PoU increases with inequality iff initial inequality is high enough (or initial PoU small enough), and decreases otherwise.
- Mechanism: a decrease in inequality raises income for all households below an "inflection point" and lowers it for those above; thus reduction in inequality lowers PoU iff the marginally food insecure household is below that inflection point.

### Structural characteristics and semi-elasticity of PoU to income (Proposition 4)
- In CES Pareto Example (Example 2):
  - (i) Responsiveness dPoU/dμy / μy can become greater or weaker at higher income levels; the sign depends on whether ρκ ≶ ṽα pF Q / μy.
  - (ii) Responsiveness can become greater or weaker with higher income inequality; the sign condition is PoU ≷ e^{−(2+ρ)/(1+ρ)}.
- Intuition:
  - Two opposing effects as income rises: fewer households near the threshold (weakening semi-elasticity) vs. larger absolute redistribution (strengthening semi-elasticity).
  - Two opposing effects of inequality: substitution effect (reduces impact of GDP growth on PoU because dPoU/dρ > 0 and income semi-elasticity declines with PoU) vs. inclusivity effect (when inequality is low, GDP growth is more inclusive and benefits poor households more).
- Empirical and calibration notes:
  - Using parameter values representative of a low-income country, the model predicts the reaction of PoU to income growth tends to decline at higher income levels (see figure 2 in source); empirical confirmation is reported in section 6.
  - Empirical evidence in section 6 suggests the semi-elasticity is slightly stronger in more equal societies (implying the inclusivity effect may dominate in observed data), although the calibration exercise predicts the semi-elasticity should be weaker in more equal societies for the low-income parameterization.

### Model calibration and quantitative predictions
- Calibration approaches:
  - Stone-Geary Cobb-Douglas (CD) preferences: εF = εNF = η = 1; Pareto income distribution. Parameters:
    - ρ set to match region Gini using G = ρ/(ρ+2).
    - α (household expenditure share on food) based on Sweden (Haver Analytics).
    - κ set equal to social protection expenditure as share of GDP in 2010 (IMF and World Bank).
    - sq derived using GDP p.c. and food component of 2008 social poverty line (2011 PPP USD/day) from Jolliffe and Prydz (2021).
    - λ set so eq. (17) matches region 2010 PoU.
  - Isoelastic non-homothetic CES preferences: Pareto income distribution; parameter values adopted from Nath (2023):
    - η = 0.27
    - εF = 0.29
    - εNF = 1.08 (equated to average income elasticity of manufacturing and services)
    - m_NF based on Smith and Subandoro (2007) suggestion that most vulnerable households spend in excess of 75 percent of their income on food.
    - ṽα (and thus λ) set so eq. (17) matches each region’s 2010 PoU.
- Calibration predictions (semi-elasticities):
  - Stone-Geary CD calibration predicts a semi-elasticity of PoU to income (urban and rural) that ranges from -0.10 pp to -0.17 pp and on average equals about -0.12 pp.
  - Under CD preferences, Proposition 2 implies urban food price elasticity equals income elasticity in magnitude but opposite in sign; in rural economies the food price elasticity is zero. Thus country-level food price semi-elasticity ranges from 0 to 0.17 percentage points depending on whether urban or rural model better represents aggregate responsiveness.
  - Isoelastic non-homothetic CES calibration yields the same semi income elasticity of PoU as the CD case. Price elasticities depend on εF and η; because η − εF / εF is close to zero for the chosen parameters, urban food price elasticities remain close to the opposite of the income semi-elasticity.
- Note: calibration uses regional aggregates and incorporates Gini coefficients, social protection spending, and external demand parameters to provide alternative estimates and bounds for income and price semi-elasticities, complementing regression results.

*Source: 2.2 Comparative Statics (excerpt) from the provided IMF working paper content.*

### 3.3  Summary

### 3.3  Summary

### Model and key theoretical insights
- Presented a heterogeneous agent model with non-homothetic preferences to study the role of food prices, income per capita, and income inequality in driving food insecurity (and poverty).
- The model rationalizes empirical puzzles:
  - Food insecurity is less sensitive to food prices than to income per capita because farmers receive windfall income gains from higher food prices that offset negative affordability effects.
  - Ceteris paribus, the effect of food prices on poverty is more significant compared to their effect on food insecurity because poverty and food insecurity are different concepts and, for poverty, higher food prices may allow households to make more effective trade-offs between food and non-food needs.
- The model provides a unified and analytically tractable framework to examine these puzzles together.

### Calibration and elasticities (magnitude estimates)
- Semi-elasticity of PoU to income should range between -0.10 and -0.17pp.
  - Implication: a doubling of income should lower the prevalence of undernourishment by 10-17pp.
- Effect of food prices from calibration:
  - When food prices fall by a 100percent this leaves the PoU roughly unchanged in rural economies and lowers it by 9-16 pp in urban economies.

### Data and measurement
- Focus on objective manifestations of food insecurity related to qualitative adequacy and quantitative sufficiency of diets.
- Measures used:
  - Share of dietary energy from cereals, tubers and roots.
  - Average supply of animal protein measured in grams per capita per day.
  - Prevalence of Undernourished (PoU): share of a country’s population whose habitual food intake is insufficient to conduct an active and healthy life.
- PoU construction:
  - Uses a log-normal probability density function for yearly dietary energy intake; requires estimation of two parameters (location and scale).
  - PoU = ∫_{x<MDER} f(x|Θ) dx, where f(x|Θ) is the log-normal distribution and Θ includes a location and a scale parameter.
  - Mean estimated annually from national food utilization accounts or household survey data; other parameter and MDER estimated from micro data less frequently.
- Choice rationale:
  - PoU selected as an objective, aggregate measure with longer time coverage (almost two decades) compared to FIES (three waves).
- Additional data:
  - GDP per capita and social protection expenditure from the World Bank; real GDP per capita in constant 2015 dollars.
  - Food inflation defined as year-on-year change in the food component of a country’s Consumer Price Index from Ha et al. (2023).
- Sample:
  - 143 countries across all continents and income groups, spanning 2001 to 2021.
  - Observations dropped when dependent or independent variables missing, or when food inflation > 50% or in the bottom 10% deflationary episodes.
  - Final estimation sample is an unbalanced panel.
- Descriptive statistics highlights:
  - Low-income countries show a 30% incidence of undernourishment compared to the global average of 10%.
  - Undernourishment decreases with GDP per capita, and responsiveness to income growth declines as countries become richer.
  - At the beginning of the observation period, food inflation in developing countries was almost double that in the rich world.
  - Inflation is least volatile in high-income countries.

### Empirical strategy
- Linear model in levels:
  - y_it = α + (β_0 + γ_i) t + β_1 X_it + u_i + ǫ_it, where y_it are measures of food insecurity and X_it includes log real GDP per capita and food inflation plus global and country time effects.
- Estimation approach:
  - First difference instrumental variables (FD-IV) to address omitted variables and reverse causality.
  - First-differenced specification:
    - ∆y_it = β_0 + θ y_i0 + β_1 ∆X_it + ∆ǫ_it, where γ_i = θ y_i0 to capture initial conditions and potential convergence.
  - Use changes in food inflation (∆ inflation) rather than levels to net out monetary factors and persistent institutional influences.
- Instruments:
  - Income growth instrumented with:
    - Average growth rate of a country’s trading partners (year-specific weights: share in previous three years of exports to each partner).
    - Change in the commodities terms of trade (IMF CTOT index; commodity prices weighted by country’s share of net exports on GDP over previous three years).
  - Food inflation instrumented with:
    - Regional and global ("rest-of-world") harvest shocks: percentage deviations from per capita trend production of a calorie-weighted sum of four grains (wheat, corn, soybeans, rice) using a Hodrick-Prescott filter.
    - Regional shocks exclude country i; regions follow World Bank classification: EAP, ECA, LAC, MENA, NA, SA, SSA.
- Identification checks:
  - Evidence of instrument relevance (E[X_it|Z_it] ≠ 0) and validity (E[∆ǫ_it|Z_is] = 0 for s = 1,2..t..T) provided in the appendix and subsequent sections.

### Main regression results (PoU)
- FD (OLS) estimate:
  - PoU declines by roughly 0.05 percentage points (pp) for every 1 pp increase in GDP per capita growth rate.
  - In relative terms this equals a 0.37 percent reduction (column 1).
  - OLS estimate of food inflation: an increase of 0.011 pp in undernourishment per 1 pp in food inflation (statistically significant but considerably smaller than income effect).
- Instrumental variables (IV) estimates:
  - Signs same as OLS but magnitudes increase by at least a factor of 2.
  - 1 pp increase in income per capita growth → approximately 0.11 pp reduction in the share of undernourished (0.79 percent reduction).
  - 1 pp acceleration in food inflation → share of undernourished increases by 0.06 pp (0.46 percent).
- Standard-deviation scenario:
  - Sample standard deviation: 0.043 for GDP growth and 0.058 for inflation changes.
  - One standard-deviation increase in GDP growth → estimated 3.4% reduction in undernourishment.
  - One standard-deviation increase in food inflation → estimated 2.7% increase in the share of undernourished.
- Convergence result:
  - Countries with higher initial PoU exhibit faster reduction.
  - Starting out with a 1 pp higher PoU implies a faster reduction in PoU by 0.02 pp per year (significant in both estimation approaches).

### Instruments and diagnostics
- First-stage results (table 5) show instruments have expected signs and induce variation in endogenous variables.
  - Current growth in trading partners affects domestic growth.
  - Improvements in CTOT positively affect short-term income growth.
  - F-test rejects null that coefficients of exclusion restrictions are jointly zero.
  - Positive harvest shocks ease pressure on food prices.
- Validity argument:
  - Growth of trading partners and CTOT shifts stem predominantly from external forces, minimizing feedback from domestic factors like undernourishment.
  - First differencing removes time-invariant confounders (e.g., institutional quality).
  - Deviations from grains production trend (via HP filter) likely indicate external shocks (e.g., weather variability) outside farmers’ control.
  - Hansen test statistics for instrument validity and p values reported in table 5.

*Source: 3.3 Summary — wpiea2024188-print-pdf*

### 4. We cannot reject the null hypothesis (p-value≈

### 4. We cannot reject the null hypothesis (p-value≈

### Instrument validity and IV estimation results
- We cannot reject the null hypothesis (p-value≈ 0.16) that our instruments are uncorrelated with unobserved drivers of food insecurity. These test results guarantee the consistency of the IV estimates.
- Instruments: trade partners’ growth and commodity terms of trade changes, consistent with Burke, 2012, and found to be both relevant and valid.
- A Hausman test of endogeneity for ∆X based on comparison of the FD vis-à-vis FD-IV sets of estimates did not provide sufficient evidence in favor of the null hypothesis that the potentially endogenous regressors can actually be treated as exogenous (p value = 0). Considering this and the reliability of the instruments, FD-IV is the preferred set of estimates.
- First stage F tests (reported later for various specifications) often reject the null that coefficients of excluded instruments are jointly zero. Hansen tests in baseline heterogeneity checks cannot reject the null hypothesis of exogeneity, supporting consistency of FD-IV estimates.

### Social protection as an additional control
- Re-estimated equation 23 after adding the absolute change in social protection expenditure (% of GDP) as a control; this variable has a large number of missing values, causing a big reduction in estimation sample size.
- FD and FD-IV results for this specification are shown separately (table 6) and are not comparable to the baseline specification.
- FD estimates:
  - Social protection has a significant containing effect on undernourishment, with a less steep gradient compared to income and food inflation.
- FD-IV estimates:
  - Use same instruments as before with addition of age dependency ratio in the population as a dedicated instrument for social protection expenditure.
  - FD-IV estimate of the social protection effect is considerably larger and still significant.
  - To gauge relative strength, the social protection marginal effect is scaled up by a factor of 4 (the ratio of its standard deviation to that of the other two regressors).
  - With this adjustment, a typical change in social protection expenditure leads to a reduction in undernourishment of 0.01 pp, less than a sixth of the inflation effect.
  - Evidence on social protection effects is weak: first stage F statistics are lower and there is insufficient support for validity of the instruments’ set.

### Unpacking heterogeneity of growth and inflation effects (Section 6.1)
- Estimation framework:
  - Equation (24): ∆y_it = β0 + θ y_i0 + β1 ∆X_it + β2 W_it + β3 ∆X_it W_it + ∆ǫ_it
  - W_it includes mediating variables in levels (e.g., share of agricultural GDP; GDP per capita; income share of bottom 20%).
  - Equation (24) is estimated with the same FD-IV approach as equation (23), but inclusion of level variables requires alternative external instruments for W_it: contemporaneous and one year lag of average seasonal temperature levels and old age dependency ratio in the population, added to instruments for ∆X_it.
- Food inflation × share of agricultural GDP:
  - The FD-IV total effect of food inflation on undernourishment is β3(W) = β1 + β3 W.
  - The declining pattern of this function provides some support for the theoretical prediction (Proposition 2) that the adverse effect of food inflation diminishes with a higher share of agricultural GDP, but the interaction term is statistically insignificant (table 7). Detecting interaction effects requires larger sample size.
- Growth × GDP per capita and growth × income share of bottom 20%:
  - Plotted FD-IV total effect of growth against GDP per capita and against income share held by bottom 20% (figure 5; underlying estimates in table 8).
  - Left graph (GDP per capita):
    - Undernourishment reacts to business cycle fluctuations for a broad range of income values; point-wise estimates are statistically significant through an income level of USD 20 thousand.
    - Economic growth reduces hunger especially at early stages of development but becomes less effective as countries grow richer and undernourishment plunges to low levels. Long-run relationship between undernourishment and income is non-linear.
    - LICs: undernourishment declines rapidly as income grows. HICs: curve is considerably less steep.
  - Right graph (inequality measured by income share bottom 20%):
    - Income growth effectiveness in reducing hunger increases with inequality, although gradient is small and significant only at some portions of the range.
    - Interpretation: when the substitution effect is almost completely offset by the direct effect, more inequality implies a higher PoU, mechanically reducing the growth semi-elasticity of the PoU.
    - Quantitative magnitude reported: an increase of 1 pp in income share of the bottom 20% reduces the PoU by 0.1 pp (-0.147 + 1.857*0.024, where 0.024 is the sample average of growth), although imprecisely estimated (se 0.323).

### Sensitivity analysis (Section 6.2)
- Instrument set robustness:
  - Replaced global harvest shocks with regional ones; second stage (table 9) and first stage (table 10) results are very similar to benchmark specification. Regional shock has a smaller coefficient but is still significant.
  - Cannot reject the null hypothesis of instrument validity when using regional shocks.
- Armed conflict control:
  - Included a dummy equal to 1 if country i was involved in an armed conflict in a given year (conflicts with at least 1000 battle-related deaths in a year; data from Uppsala Conflict Data Program; see Davies et al., 2022).
  - FD estimate of the armed conflict effect is positive and statistically significant but not sizable (sd = 0.35).
  - After controlling for armed conflict, effects of income growth and inflation are similar to baseline for both FD and FD-IV.
  - For FD-IV, diagnostics indicate instruments are well correlated with endogenous regressors and uncorrelated with the error term. Conclusion: armed conflicts do not seem to have a direct effect on aggregate undernourishment at the macro level and may exert influence indirectly through income and food price channels.

### Main empirical conclusions summarized
- FD-IV is preferred given evidence on endogeneity and instrument reliability.
- Economic growth is the most effective strategy for reducing food insecurity; food inflation has a detrimental but smaller impact relative to income growth.
- Social protection shows some containing effect on undernourishment but evidence is weak and sample-limited; scaled FD-IV effect implies a typical change leads to a 0.01 pp reduction in undernourishment.
- Heterogeneity:
  - Food inflation impact tends to diminish with larger agricultural GDP share (suggestive but statistically insignificant).
  - Growth reduces undernourishment most strongly at low income levels; point-wise significance up to USD 20 thousand GDP per capita.
  - Inequality influences effectiveness of growth: a 1 pp increase in bottom 20% income share reduces PoU by 0.1 pp (imprecisely estimated; se 0.323).
- Robustness checks (regional shocks, armed conflict controls) do not overturn main findings; instruments generally pass diagnostic tests in these checks.

### Policy implications and discussion (Section 7)
- Policy community responses to recent food insecurity crises emphasize: (i) boosting agricultural production, (ii) strengthening safety nets, (iii) investing in climate resilient agriculture, and (iv) facilitating international free trade in food. All are consistent with the importance of income growth; all except (ii) are directly linked to reducing food inflation.
- Authors’ interpretation and recommendations:
  - Policymakers should revise mental models of food insecurity beyond a narrow purchasing-power (intensive margin) focus to the PoU metric that targets the fraction of households with insufficient calorie intake.
  - Prioritize policies that promote inclusive growth ensuring equal opportunities, redistribution, and targeted social protection within fiscal limits.
  - Allow global food price increases to trickle down to domestic prices while enhancing targeted social protection to:
    - Encourage sufficient food production.
    - Increase incomes of food-insecure farmers.
    - Protect purchasing power of vulnerable urban households.
  - Policy design should be assessed case-by-case given variation in countries’ capacity to flexibly adjust social protection coverage and transfer sizes.

*Source: Excerpt from the IMF working paper chapter "GROWTH, INFLATION, AND FOOD INSECURITY" provided in the supplied content.*

### Appendix A: Tables and figures

### Appendix A: Tables and figures

### Model calibration and analytical figures
- Figure 1: Effects of income per capita and Gini inequality on the PoU. Figure drawn for η=ǫF=ǫNF=1, α=0.14, CF=0.5, κ=0.05, λ=0.97, and with pF chosen such that sq ≡ pF Q μy =0.14 for μy =2.5.
- Figure 2: Sensitivity of the semi-elasticity of PoU to income with respect to income and inequality. Figure drawn for the same low-income country parameter values selected for figure 1.
- Figure 3:
  - Panel (A): Evolution of PoU over time and by income group (years shown: 2000, 2005, 2010, 2015, 2020).
  - Panel (B): Relationship between undernourishment and GDP per capita by income group.
- Figure 4: Food price effects variation for undernourishment (x-axis: Share of agricultural GDP, range shown 0.05 to 5.5).
- Figure 5: Growth effects variation for undernourishment (top panel x-axis: GDP per capita (I$ 000) 0–100; bottom panel x-axis: Income held by bottom 20% (%) range .04–.19).

### Income and food-price elasticities (theoretical calibration)
- Table 1: Income (and food price) elasticity of the PoU. Cobb-Douglas preferences. Regional summary (columns include: Gini, GDP p.c., ραλκsq, PoU0, dPoU/dμy /μy):
  - Africa: Gini 0.47; GDP p.c. 2,484; ραλκs q 1.77 0.14 0.97 0.05 0.14 0.17; dPoU/dμy /μy = -0.137
  - East Asia & Pacific: Gini 0.41; GDP p.c. 14,406; ραλκs q 1.37 0.14 0.99 0.06 0.13 0.09; dPoU/dμy /μy = -0.110
  - Western Europe: Gini 0.30; GDP p.c. 43,790; ραλκs q 0.84 0.14 0.55 0.18 0.07 0.03; dPoU/dμy /μy = -0.110
  - Eastern Europe & C. Asia: Gini 0.31; GDP p.c. 14,827; ραλκs q 0.91 0.14 0.82 0.12 0.09 0.03; dPoU/dμy /μy = -0.098
  - Latin America & Carib.: Gini 0.45; GDP p.c. 6,773; ραλκs q 1.65 0.14 0.93 0.10 0.11 0.11; dPoU/dμy /μy = -0.168
  - U.S. & Canada: Gini 0.37; GDP p.c. 47,958; ραλκs q 1.16 0.14 0.69 0.12 0.05 0.03; dPoU/dμy /μy = -0.121
- Table 2: Isoelastic non-homothetic CES preferences (same data as CD calibration except for CES parameters from Nath (2023)). Elasticities:
  - Africa: dPoUU/dμy /μy = -0.137; dPoUU/dpF /pF = 0.130; dPoUR/dpF /pF = -0.001
  - East Asia & Pacific: -0.110; 0.104; 0.001
  - Western Europe: -0.110; 0.105; 0.005
  - Eastern Europe & C. Asia: -0.098; 0.093; 0.004
  - Latin America & Carib.: -0.168; 0.159; 0.006
  - U.S. & Canada: -0.121; 0.115; 0.008

### Empirical sample descriptive statistics and controls
- Table 3: Sample averages by income level for initial real GDP per capita in thousands of 2015 $ (GDP0), social protection expenditure share of GDP (SP0), food inflation FI0, and other controls:
  - LIC: GDP0 = 0.702 (std dev 0.559); ∆ln(GDP) = 0.021 (0.058); SP0 = 0.001 (.) ; ∆SP = 0.000 (0.014); ∆FI = 0.082 (0.057); gx = 0.001 (0.088); CTOT = 0.018 (0.057); ∆ln(CTOT) = 95.787 (5.555); ∆ln(CTOT) SE 0.002 (0.017)
  - LMC: GDP0 = 1.519 (0.746); ... Total: GDP0 = 11.560 (16.845); ∆ln(GDP) = 0.023 (0.051); SP0 = 0.105 (0.061); ∆SP = 0.001 (0.011); ∆FI = 0.064 (0.071); gx = -0.000 (0.063); CTOT = 0.020 (0.035); ∆ln(CTOT) = 93.524 (8.319); ∆ln(CTOT) SE 0.004 (0.031)
  - (Note: table shows subgroup stats for UMC, HIC as well; standard errors in parenthesis.)

### Main estimation results: PoU regressions
- Table 4: Estimation results of main equation: PoU (columns: FD OLS; FD-IV (IV)):
  - y0: OLS -0.019*** (0.003); IV -0.016*** (0.004)
  - ∆ln GDP pc: OLS -0.045*** (0.007); IV -0.107*** (0.033)
  - ∆Food inflation: OLS 0.011** (0.005); IV 0.063*** (0.024)
  - constant: OLS 0.002*** (0.000); IV 0.003*** (0.001)
  - N: 2626 (OLS), 2011 (IV)
  - Hansen test: 5.1; p value (0.16)
  - Significance notation: *** p<0.01, ** p<0.05, * p<0.1.

### First-stage and instrument performance
- Table 5: First stages for main equation (dependent variables: ∆ln GDP pc, ∆Food inflation):
  - Global harvest shock (t-1): GDP stage coefficient 0.000 (0.000); Food inflation stage -0.003*** (0.001)
  - Trade partner growth: 0.397*** (0.088) for GDP; 0.164** (0.066) for inflation
  - Trade partner growth (t-1): -0.079*** (0.026) for GDP; -0.010 (0.057) for inflation
  - ∆ln CTOT: 0.076* (0.043) for GDP; 0.122** (0.061) for inflation
  - ∆ln CTOT (t-1): 0.151*** (0.040) for GDP; -0.080 (0.050) for inflation
  - y0: 0.017 (0.015) for GDP; 0.013** (0.005) for inflation
  - constant: 0.011*** (0.003) for GDP; -0.005*** (0.002) for inflation
  - N: 2011 each; F stat: 8.7 (GDP), 1.8 (inflation); p value (0.000)(0.000)

### Extended controls and robustness
- Table 6: Main equation with extended controls (columns: FD OLS; FD-IV; First stages F-test/p-value):
  - y0: OLS -0.034*** (0.006); IV -0.028*** (0.006)
  - ∆ln GDP pc: OLS -0.038*** (0.009); IV -0.077** (0.031); First stage F 18.5 (p value 0.000)
  - ∆Food inflation: OLS 0.005 (0.007); IV 0.035 (0.030); First stage F 4.7 (p value 0.000)
  - ∆Soc protection exp.: OLS -0.001** (0.000); IV -0.003* (0.002); First stage F 7.3 (p value 0.000)
  - constant: OLS 0.002*** (0.000); IV 0.003*** (0.001)
  - N: 1224 (OLS), 920 (IV)
  - Hansen test: 6.9; p value (0.08)

### Heterogeneity: agricultural GDP and food inflation
- Table 7: Food inflation effects heterogeneity by agricultural GDP (FD-IV and first-stage F-statistics):
  - ∆ln GDP pc: FD-IV -0.081** (0.033); First stage F 6.7 (p value 0.000)
  - ∆Food inflation: FD-IV 0.036* (0.021); First stage F 4.9 (p value 0.000)
  - Agricultural GDP: FD-IV 0.023 (0.015); First stage F 2.4 (p value 0.000)
  - ∆Food inflation × Ag. GDP: FD-IV -0.094 (0.145); First stage F 2.2 (p value 0.000)
  - y0: -0.036*** (0.013); constant 0.002** (0.001)
  - N: 1487; Hansen test 14.3; p value (0.57)

### Heterogeneity: growth effects and inequality / income distribution
- Table 8: Interactions equation estimates: growth effects on PoU (FD-IV and first stages):
  - ∆ln GDP pc: FD-IV -0.243 (0.170); First stage F 8.5 (p value 0.000)
  - ∆Food inflation: 0.011 (0.018); First stage F 4.3 (p value 0.000)
  - GDP pc level: -0.000 (0.000); First stage F 8.5 (p value 0.000)
  - Income bottom 20%: -0.147 (0.102); First stage F 3.6 (p value 0.000)
  - ∆ln GDP pc × GDP pc level: 0.003 (0.003); First stage F 4.9 (p value 0.000)
  - ∆ln GDP pc × Income bottom 20%: 1.857 (2.187); First stage F 7.6 (p value 0.000)
  - y0: -0.029*** (0.009); constant 0.014** (0.007)
  - N: 1397; Hansen test 12.9; p value (0.53)

### Alternative instruments and sensitivity checks
- Table 9: Estimates with alternative set of instruments (FD-IV):
  - y0: -0.016*** (0.004)
  - ∆ln GDP pc: -0.096*** (0.035)
  - ∆Food inflation: 0.043* (0.023)
  - constant: 0.002*** (0.001)
  - N: 2011; Hansen test 4.9; p value (0.17)
- Table 10: First stage with alternative instruments:
  - Regional harvest shock (t-1): ∆ln GDP pc coefficient -0.000** (0.000); ∆Food inflation -0.001*** (0.000)
  - Trade partner growth: 0.377*** (0.083) for GDP; 0.177*** (0.068) for inflation
  - Trade partner growth (t-1): -0.070*** (0.025) for GDP; -0.028 (0.055) for inflation
  - ∆ln CTOT: 0.080* (0.042) for GDP; 0.129** (0.062) for inflation
  - ∆ln CTOT (t-1): 0.154*** (0.040) for GDP; -0.089* (0.048) for inflation
  - y0: 0.018 (0.015) for GDP; 0.012** (0.006) for inflation
  - constant: 0.011*** (0.003) for GDP; -0.005*** (0.002) for inflation
  - N: 2,011; F stat: 16.5 (GDP), 12.4 (inflation); p value (0.000)(0.000)

### Sensitivity: controlling for armed conflict
- Table 11: Inclusion of Armed conflict control:
  - y0: OLS -0.021*** (0.003); IV -0.016*** (0.004)
  - ∆ln GDP pc: OLS -0.046*** (0.007); IV -0.106*** (0.033); First stage F 8.5 (p value 0.000)
  - ∆Food inflation: OLS 0.010** (0.005); IV 0.064*** (0.024); First stage F 11.8 (p value 0.000)
  - Armed conflict: OLS 0.002*** (0.001); IV 0.002 (0.001)
  - constant: OLS 0.001*** (0.000); IV 0.002*** (0.001)
  - N: 2615 (OLS), 2011 (IV)
  - Hansen test: 5.1; p value (0.17)

### Mathematical proofs and analytical results (selected)
- Appendix presents proofs of Propositions 1–4 with explicit derivative expressions, sign conditions, and comparative statics:
  - Key analytical expressions include derivatives dyPoU/dpF and dyPOV/dpF with components d ̃α/dpF, dP/dpF, ξ0, ξ1, mF, mNF, and dμy/dpF (equations (25)–(35)).
  - Semi-elasticities for income and price under isoelastic non-homothetic CES preferences and log-normal income distribution (equations (41)–(51)):
    - Log-normal case: dFU/dμy /μy = −φ(·)[1+ κ/(1−κ) μy/yPoU ] = −φ(·)[ ̃α s q /( ̃α s q −κ ) ] (equations (45)–(48)).
    - Pareto income distribution analogue: replace φ(·) with 1/(ρ PoU) (equations (50)–(51)).
  - Conditions and thresholds where derivatives change sign (e.g., dPoU/dσ/σ ≷0 if yPoU ≶ eμ; dPoU/dρ/ρ ≷0 for PoU ≶ e^{−1/(1+ρ)}).

### Instrument construction (details)
- Commodity terms of trade instrument:
  - ∆ln(CTOTi,t) = Σj ∆ln(Pj,t) Ωi,j,t (equation (52))
  - Ωi,j,t = (1/3) Σ_{s=1}^3 (x_{i,j,t−s} − m_{i,j,t−s}) / GDP_{i,t−s} (equation (53))
- Export-weighted partner growth instrument:
  - gxi,t = Σ_{τ=1}^3 Xtot_{i,t−τ} / Σ_{τ=1}^3 GDP_{i,t−τ} [ Σ_{j=1}^N Ω_{i,j,t} g_{j,t} ] with Ω_{i,j,t} defined as three-year moving export shares (equations (54)–(55))
- Harvest shock instruments:
  - Production data for wheat, maize, soybeans, rice (FAO 2023) converted to kcal using Roberts and Schlenker (2013) caloric measures.
  - Hodrick-Prescott filter with smoothing parameter 6.25 applied to production series; cyclical component used as weather-induced yield fluctuation proxy.
  - Regional harvest shock for country i: sum of shocks of all countries in i's region excluding i, expressed per capita and as percentage deviation from per capita trend. Rest-of-world shock constructed analogously.

### Results for qualitative food insecurity indicators (diet composition)
- Figure A1: As countries develop, they substitute cereals and staples with animal products; substitution fast early and slows with income.
- Regression results (Table A1 and A2):
  - Table A1 (FD and FD-IV for diet composition: CER = share of energy from starchy staples; PROT = average animal protein supply):
    - y0: CER OLS -0.006*** (0.001); PROT OLS -0.001 (0.001); CER FD-IV -0.005** (0.002); PROT FD-IV -0.001 (0.002)
    - ∆ln GDP pc: CER OLS -0.049*** (0.008); PROT OLS 9.232*** (1.439); CER FD-IV -0.068* (0.038); PROT FD-IV 14.257*** (4.033)
      - Interpretation: 1 pp increase in GDP per capita growth → 0.049 pp reduction in cereals, roots and tubers consumption (FD OLS) and 0.092 gr/cap/day increase in animal product consumption (FD OLS coefficient reported 9.232 units; text interprets relative impacts: 0.29 vs 0.11 percent).
    - ∆Food inflation coefficients indistinguishable from zero in OLS; FD-IV indicates small/insignificant for cereals and suggestive negative effect on protein (low precision).
    - constants and sample sizes:
      - CER PROT N: OLS columns N = 2318 / 2374? (table reports multiple N: 2318 2374 1781 1783 — see table for exact alignment by column)
      - Hansen tests: 4.48 (p value 0.21) for CER; 0.86 (p value 0.83) for PROT.
  - Table A2: First-stage for diet composition equations:
    - Global harvest shock (t-1): -0.001* (0.000) for ∆ln GDP pc; -0.003*** (0.001) for ∆Food inflation (both equations).
    - Trade partner growth: 0.273*** (0.077) for GDP; 0.180** (0.073) for inflation.
    - ∆ln CTOT: 0.082** (0.041) for GDP; 0.146** (0.074) for inflation.
    - y0: 0.049*** (0.010) for GDP in cereal equation; other entries reported (see table).
    - Observations: 1781 / 1783 depending on column. F stat: 7.45 / 9.16 with p value (0.000).
- Table A3: Interactions equation estimates for diet composition (FD-IV and first stages):
  - CER FD-IV ∆ln GDP pc: -0.253* (0.131); PROT FD-IV ∆ln GDP pc: 28.972** (13.809)
  - Interaction terms: ∆ln GDP pc × GDP pc level 0.005* (0.003) for CER; -0.493 (0.331) for PROT
  - ∆ln GDP pc × Income bottom 20%: 0.962 (1.781) for CER; -132.006 (180.298) for PROT
  - N: 1375 (CER) and 1377 (PROT)
  - Hansen tests reported: large statistics with p values indicating rejection in some equations (2928 p value 0.00; other 28 p value 0.01) — see table for exact alignment.

*Italic source attribution: wpiea2024188-print-pdf - Appendix A: Tables and figures, canonical PDF: wpiea2024188-print-pdf*

### 24. The last columns show the F test statistics from the first stage of each

### wpiea2024188-print-pdf - 24. The last columns show the F test statistics from the first stage of each

### Description of excerpt
- "24. The last columns show the F test statistics from the first stage of each regressor and its p value."
- Page marker: "6"
- Document title line: "How do Economic Growth and Food Inflation Affect Food Insecurity? "
- Working Paper identifier: "Working Paper No. WP/2024/188"

### Key technical point
- The excerpt reports that the last columns present:
  - the F test statistics from the first stage of each regressor
  - and its p value

### Contextual identifiers
- Document: "How do Economic Growth and Food Inflation Affect Food Insecurity?"
- Working Paper: "WP/2024/188"
- Excerpt location indicator: "24."

*Source: wpiea2024188-print-pdf - 24. The last columns show the F test statistics from the first stage of each*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024188-print-pdf.pdf_