## _wp13120

## Source details

**Canonical URL:** [_wp13120](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13120.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13120.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13120.pdf.json)

---

### Introduction — Empirical puzzle
- Emerging economies have exhibited an increase in consumption volatility relative to output volatility after financial development (financial sector reform).
- Table 1 (selected emerging economies) — selected pre → post relative consumption volatility:
  - Mean pre-reform relative consumption volatility: 1.15
  - Mean post-reform relative consumption volatility: 1.29
  - Std. dev. pre-reform: 0.44
  - Std. dev. post-reform: 0.30
  - Country examples:
    - Chile: 1.10 → 1.26 ⇑
    - Mexico: 0.94 → 1.45 ⇑
    - India (1992): 0.83 → 1.23 ⇑
    - Indonesia: 2.45 → 1.01 ⇓
    - Korea: 0.93 → 1.69 ⇑
    - Poland: 0.92 → 1.45 ⇑
    - South Africa (1994): 1.42 → 1.40 ⇓
- Financial access indicators show increases over the 2000s (World Development Indicators): examples include growth in commercial bank branches and depositors with commercial banks.
- Figure 1 illustrates rising bank deposits to GDP for a set of emerging economies and several individual countries.

### Central hypothesis and mechanism
- Two key features of the proposed explanation:
  1. Presence of credit constraints: a fraction λ of households cannot save or borrow (liquidity-constrained).
  2. Shocks to trend growth of productivity: anticipated permanent increases in growth (g) lead unconstrained households to raise current consumption more than current income via borrowing or reduced saving.
- Prediction: financial development that reduces λ (more households unconstrained) increases aggregate consumption responses to anticipated permanent-growth shocks, raising relative consumption volatility.

### Contribution and organization
- Provides a model combining liquidity constraints and AR(1) shocks to trend growth of labour productivity to explain the observed rise in relative consumption volatility following financial development.
- Calibrates the model to Indian data for pre- and post-reform years, changing only λ and holding other parameters constant.
- Paper structure:
  - Section 2: evidence on relative consumption volatility and financial development.
  - Section 3: model and predictions.
  - Section 4: calibration (India case).
  - Section 5: conclusion.
  - Appendixes: solution methods and proofs.

### Theoretical framework — setup
- Economy:
  - Closed economy with continuum of households and firms (measure unity).
  - Fraction λ of households are liquidity-constrained (L); remaining 1−λ are Ricardian (R).
- Preferences:
  - U(C_i t) = ln(C_i t), i = R, L
  - Ricardian objective: V_t = E_t Σ_{t=0}^∞ β^t log(C^R_t), β ∈ (0,1)
- Budget, capital, and constraints:
  - Ricardian: C^R_t + I^R_t = R_t K^R_t + W_t
  - Capital accumulation with quadratic adjustment costs as specified:
    - K^R_{t+1} = (1−δ)K^R_t + I^R_t − (φ/2)((K^R_{t+1}/K^R_t − μ_g)^2 / K^R_t)
  - Liquidity-constrained: C^L_t = W_t
- Aggregates:
  - C_t = λ C^L_t + (1−λ) C^R_t
  - K_t = (1−λ) K^R_t
  - I_t = (1−λ) I^R_t
- Production and productivity:
  - Y_t = e^{a_t} [(1−λ)K_t]^{1−α} Γ_t^{α}, α ∈ (0,1)
  - a_t = ρ_a a_{t−1} + ε^a_t, ε^a_t ~ N(0, σ_a)
  - Γ_t dynamics:
    - Γ_t = g_t Γ_{t−1}
    - ln(g_t / μ_g) = ρ_g ln(g_{t−1} / μ_g) + ε^g_t, ε^g_t ~ N(0, σ_g^2)
- Resource constraint: C_t + I_t = Y_t
- Detrending: X_t detrended by Γ_{t−1}: x_t = X_t / Γ_{t−1}

### Model dynamics and solution form
- Log-linearized cyclical components ̃x_t = ln x_t − ln x* lead to an approximate system including an Euler condition for Ricardian consumers and the resource constraint adjusted for constrained consumption.
- Solution by undetermined coefficients:
  - ̃k^R_{t+1} = a_1 ̃k^R_t + b_1 a_t + d_1 ̃g_t
  - ̃c^R_t = a_2 ̃k^R_t + b_2 a_t + d_2 ̃g_t
- Unknown coefficients (a_1, b_1, d_1, a_2, b_2, d_2) depend on (β, α, δ, λ, μ_g).
- Stability requires a_1 < 1.

### Qualitative predictions
- For positive permanent shocks to g:
  - Ricardian households increase current consumption by more than the contemporaneous income rise (borrowing/dis-saving).
  - Liquidity-constrained households can only increase consumption by the contemporaneous income rise.
- When λ falls (financial development), a larger share behaves like Ricardian households, increasing aggregate consumption volatility relative to output volatility.
- Main theoretical prediction: easing credit constraints in presence of shocks to trend growth can increase relative consumption volatility.

### Analytical volatility relationships and Proposition 1
- Consumption-to-output volatility decomposition:
  - σ^2_{̃c} / σ^2_{̃y} = (c^R_* / c_*)^2 (1−λ)^2 σ^2_{̃c_R} / σ^2_{̃y} + (c^L_* / c_*)^2 λ^2 . (equation (16))
- Expressions for σ^2_{̃c_R} and σ^2_{̃y} depend on solution coefficients and σ^2_a, σ^2_{̃g} where:
  - σ^2_a = σ^2_{ε_a} / (1−ρ_a^2)
  - σ^2_{̃g} = σ^2_{ε_g} / (1−ρ_g^2)
- Proposition 1 (key implications):
  - (i) For liquidity-constrained households under σ_{ε_a} > 0 and σ_{ε_g} > 0:
    - σ_{̃c_L} / σ_{̃y} = 1.
  - (ii) For transitory income shocks only (σ_{ε_a} > 0; σ_{ε_g} = 0):
    - σ_{̃c_R} / σ_{̃y} < 1 and σ_{̃c} / σ_{̃y} < 1 for λ ∈ [0,1).
    - Numerical illustration (Table 11):
      - λ = 0.900: σ_{̃c_R}/σ_{̃y} = 0.745 ; σ_{̃c}/σ_{̃y} = 0.807
      - λ = 0.000: σ_{̃c_R}/σ_{̃y} = 0.406 ; σ_{̃c}/σ_{̃y} = 0.406
  - (iii) For trend growth shocks only (σ_{ε_a} = 0; σ_{ε_g} > 0):
    - σ_{̃c_R} / σ_{̃y} > 1 always; σ_{̃c} / σ_{̃y} may be <, =, or > 1 depending on λ.
    - Numerical illustration (Table 12):
      - λ = 0.900 : σ_{̃c_R}/σ_{̃y} = 1.493 ; σ_{̃c}/σ_{̃y} = 0.834
      - λ = 0.786 : σ_{̃c_R}/σ_{̃y} = 1.992 ; σ_{̃c}/σ_{̃y} = 1.018
      - λ = 0.500 : σ_{̃c_R}/σ_{̃y} = 3.086 ; σ_{̃c}/σ_{̃y} = 1.897
      - λ = 0.331 : σ_{̃c_R}/σ_{̃y} = 3.556 ; σ_{̃c}/σ_{̃y} = 2.585
      - λ = 0.000 : σ_{̃c_R}/σ_{̃y} = 4.223 ; σ_{̃c}/σ_{̃y} = 4.223
  - (iv) Under trend growth shocks, both σ_{̃c_R} / σ_{̃y} and σ_{̃c} / σ_{̃y} increase as λ falls:
    - ∂(σ_{̃c_R} / σ_{̃y}) / ∂λ < 0
    - ∂(σ_{̃c} / σ_{̃y}) / ∂λ < 0

### Case study: India — calibration and data construction
- Sample split:
  - Pre-reform period: 1951-1991
  - Post-reform period: 1992-2009 (analysis uses data up to 2012 for some series)
- Data and constructions:
  - Output, consumption, investment: real gdp at factor cost, private consumption expenditure, gross fixed capital formation (1951-2012).
  - Business cycle moments from HP-filtered cyclical components of per capita series.
  - Total Factor Productivity (tfp): sectoral tfp weighted by sectoral gdp, net fixed capital stock, and employment (1980-2009).
  - Labour productivity growth: sectoral real gdp and labour force; gross growth rate g_t = 1 + ∆ log(Γ_t).
  - Detrending: quadratic trend for tfp; trend regression for labour productivity growth; AR(1) for cyclical components.
- Benchmark parameter values (Table 4):
  - Discount factor β = 0.98
  - Rate of Depreciation δ = 5%
  - Share of labour α = 0.7
  - Adjustment cost parameter φ = 2.82
  - Mean trend growth rate of labour productivity μ_g −1 = 4.7%
  - Persistence in tfp shock process ρ_a = 0.495
  - Volatility in tfp σ_a = 0.015
  - Persistence in labour productivity growth shock ρ_g = 0.261
  - Volatility in labour productivity growth shock σ_g = 0.020
- Financial development proxy and λ calibration (benchmark: bank accounts per population):
  - 1980: 21.4% of population had access to banking ⇒ λ = 0.786 (pre-reform).
  - 2010: 66.9% of population had access to banking ⇒ λ = 0.331 (post-reform).
  - Mean trend net growth rate over 1980-2009: μ_g −1 = 4.7%

### India — empirical facts and model simulation results
- Stylised business cycle facts (selected reported figures):
  - Real gdp Std. dev.: 2.25 (pre); 1.87 (post)
  - Private Consumption Std. dev.: 1.88 (pre); 2.14 (post)
  - Relative consumption volatility: 0.83 (pre); 1.14 (post)
  - Investment Std. dev.: 5.42 (pre); 5.58 (post)
- Financial development trends:
  - Bank account in total population: ~20% in 1980 → above 70% in 2010 (with decline 1990-2005).
  - Bank branches per 100,000 population in 2010 > doubled 1970 value.
  - Deposit to GDP and private credit to GDP ratios increased over time.
- Relative consumption volatility evolution:
  - Five-year rolling relative consumption volatility increased.
  - Reported values: 0.83 (1951-1991) → 1.14 (1992-2012).
- Simulated model moments (selected, as reported):
  - Data (selected):
    - Pre-reform: Std. dev. YC ICI Data: 2.25; Private Consumption Std. dev.: 1.88; Relative consumption volatility: 0.83
    - Post-reform: Std. dev.: 1.87; Private Consumption Std. dev.: 2.14; Relative consumption volatility: 1.14
  - Model (selected reported entries):
    - Pre-reform: Std. dev. 3.25; Rel. std. dev. 3.21; First ord. corr. 5.67; Rel. std. dev. 0.99; Std. dev. 1.74
    - Post-reform: Std. dev. 3.22; Rel. std. dev. 3.23; First ord. corr. 4.85; Rel. std. dev. 1.00; Std. dev. 31.51
  - Correlations and autocorrelations (selected reported):
    - Data pre-reform: Correlation CIYCI 0.70; Auto-correlation 0.14; First order autocorr. 0.47; other entries 0.79
    - Data post-reform: Correlation 0.90; Auto-correlation 0.75; First order 0.67; other entries 0.598; 0.425
    - Model pre-reform: Correlation 0.99; Auto-correlation 0.69; First order 0.62; other entries 0.667; 0.163
    - Model post-reform: Correlation 0.98; Auto-correlation 0.73; First order 0.61; other entries 0.749; 0.183
- Interpretation:
  - Model reproduces the direction of observed increase in relative consumption volatility after financial development.
  - Magnitude: model’s rise is smaller than observed; possible reason: closed-economy framework (no foreign borrowing channel), which may under-predict observed rise.
  - Model broadly matches qualitative changes in correlations and persistence between pre- and post-reform periods.

### Sensitivity analysis — alternative financial development measure
- Alternative proxy: bank deposit to GDP ratio ⇒ calibrated λ:
  - λ = 0.687 (pre-reform)
  - λ = 0.305 (post liberalisation)
- Model moments using deposit-to-GDP-based λ are similar to benchmark results.
- Key implication: main results are robust to using bank deposit to GDP ratio as alternative measure of financial development.

### Conclusion — key takeaways and limitations
- Main analytical result: liquidity constraints and the nature of income shocks (transitory vs. permanent) determine whether consumption volatility relative to output is <1, =1, or >1 for different household types.
- Mechanism: financial development (lower λ) raises the share of households able to act on permanent income shocks by borrowing/dis-saving, increasing aggregate consumption volatility relative to output.
- India empirical application:
  - Access to banking: 21.4% in 1980 → 66.9% in 2010 ⇒ λ: 0.786 → 0.331.
  - Relative consumption volatility: 0.83 (1951-1991) → 1.14 (1992-2012).
  - Calibrated model reproduces direction of change; magnitude under-predicted in closed-economy setup.
- Limitations and avenues for further research:
  - Closed-economy assumption excludes foreign financial access and external borrowing channels.
  - Household sector as sole channel for financial development; including credit-constrained firms could improve investment business cycle replication.
  - How household access to foreign financial instruments would affect consumption-smoothing remains an open question.

*Source: IMF Working Paper — Introduction, theoretical framework, model predictions, empirical calibration, sensitivity, and conclusion sections from _wp13120 (pages 3–14 and appendixes highlights).*

### Introduction.............................................................................................. 3

### Introduction

### Empirical puzzle
- Emerging economies have exhibited an increase in consumption volatility relative to output volatility after financial development (financial sector reform).
- This behavior contrasts with traditional models and evidence from advanced economies, which suggest consumption should become smoother after credit constraints are reduced.
- Table 1 (selected emerging economies) documents mixed outcomes but a general rise in relative consumption volatility:
  - Mean pre-reform relative consumption volatility: 1.15
  - Mean post-reform relative consumption volatility: 1.29
  - Std. dev. pre-reform: 0.44
  - Std. dev. post-reform: 0.30
  - Selected country examples (pre → post, direction):
    - Chile 1.10 → 1.26 ⇑
    - Mexico 0.94 → 1.45 ⇑
    - India (1992) 0.83 → 1.23 ⇑
    - Indonesia 2.45 → 1.01 ↓
    - Korea 0.93 → 1.69 ⇑
    - Poland 0.92 → 1.45 ⇑
    - South Africa (1994) 1.42 → 1.40 ↓
- Financial access indicators show increases over the 2000s (Table 2; World Development Indicators): examples include growth in commercial bank branches and depositors with commercial banks in many countries.
- Figure 1 illustrates a rising trend in bank deposits to GDP for a set of emerging economies and several individual countries.

### Central hypothesis and mechanism
- Two key features characterize the proposed explanation:
  1. Presence of credit constraints in the economy: a fraction of households cannot save or borrow and therefore cannot smooth consumption.
  2. Shocks to trend growth of productivity: anticipated permanent increases in growth lead unconstrained households to raise current consumption more than current income via borrowing or reduced saving.
- When financial sector reform increases access to finance (fewer credit-constrained households), more households can act on anticipated permanent income shocks by borrowing against future income or reducing savings, increasing current consumption responses.
- Key prediction: financial development in an emerging economy can lead to an increase in relative consumption volatility.

### Contribution
- The paper provides a model combining liquidity constraints (some households cannot save or borrow) and AR(1) shocks to trend growth of labour productivity to explain the observed rise in relative consumption volatility following financial development.
- It calibrates the model to Indian data for pre- and post-reform years, holding other parameters constant and changing only the share of credit-constrained consumers, and reports support for the model’s prediction.
- The framework builds on:
  - Liquidity-constrained household modeling: Hayashi (1982); Campbell and Mankiw (1991).
  - Stochastic trend shocks to productivity: Aguiar and Gopinath (2007).
  - Empirical context on limited financial access in emerging economies: Honohan (2006).

### Organization of the paper
- Section 2: Evidence on relative consumption volatility and financial development in emerging economies.
- Section 3: The model and its predictions.
- Section 4: Calibration exercise (including India case study).
- Section 5: Conclusion.
- Appendixes contain solution methods and proofs.

---

### Theoretical framework: credit constraints and consumption volatility

### Population, preferences, and household types
- Closed economy with a continuum of infinitely lived households and firms, both of measure unity.
- A fraction λ of households are liquidity-constrained (no access to banking, cannot save or borrow). These are indexed L (liquidity-constrained); Ricardian households are indexed R.
- Preferences (identical across types) over a single commodity:
  - U(Ci t) = ln(Ci t), i = R, L
- Ricardian household objective:
  - Vt = Et Σt=0∞ βt log(CR t), with β ∈ (0,1)
- Ricardian budget constraint:
  - CR t + IR t = Rt KR t + Wt
- Ricardian capital accumulation:
  - KR t+1 = (1−δ)KR t + IR t − (φ/2)((KR t+1 / KR t − μg)^2 / KR t)
  - Quadratic capital adjustment costs as in Aguiar and Gopinath (2007).
- Liquidity-constrained household constraint:
  - CL t = Wt (consume entire labour income each period).
- Aggregate consumption:
  - C t = λ CL t + (1−λ) CR t
- Aggregate capital and investment:
  - K t = (1−λ) KR t, I t = (1−λ) IR t

### Production and productivity processes
- Representative firm uses Cobb-Douglas technology with one unit of labour and capital:
  - Y t = ea t [(1−λ)K t]^(1−α) Γ t^α, with α ∈ (0,1)
- Total factor productivity (tfp) follows AR(1):
  - a t = ρa a t−1 + εa t, |ρa| < 1, εa t ~ i.i.d. N(0, σa)
- Labour productivity growth:
  - Γ t = g t Γ t−1
  - ln(g t / μg) = ρg ln(g t−1 / μg) + εg t, εg t ~ N(0, σg^2)
- Resource constraint:
  - C t + I t = Y t

### Detrending and equilibrium definition
- Because g has persistent effects on Γ, output and other nominal aggregates are non-stationary; variables X t are detrended by Γ t−1: x t = X t / Γ t−1.
- Competitive equilibrium: sequences of {Rt, Wt, y t, c t, cR t, cL t, i t, k t} that solve households’ and firms’ problems and satisfy the resource constraint, given shocks to tfp and labour productivity growth.

---

### Model dynamics and predictions

### Log-linearized system (after normalization)
- Key log-linearized conditions (cyclical components  ̃x t = ln x t − ln x*):
  - 1 = β Et−1[Ω t cR t−1 / cR t g t]
  - Ω t = (1−α) ea t (1−λ)^(1−α) g t^α + (1−δ)
  - cR t = ((1−αλ)/(1−λ)) e a t [(1−λ) kR t]^(−α) g t^α + (1−δ) kR t − g t kR t+1 − (φ/2)((kR t+1 / g t k t − μg)^2 / kR t)
  - a t = ρa a t−1 + εa t
  - ln(g t / μg) = ρg ln(g t−1 / μg) + εg t
- Log-linearised approximate system (14) highlights the intertemporal Euler condition for Ricardian consumers and the resource constraint after accounting for liquidity-constrained consumption.
- Approximation notation: ̃x t = ln x t − ln x*, x* steady state, steady state labour productivity growth μg.

### Solution form and stability condition
- Solution via undetermined coefficients:
  - ̃kR t+1 = a1 ̃kR t + b1 a t + d1 ̃g t
  - ̃cR t = a2 ̃kR t + b2 a t + d2 ̃g t
- Unknown coefficients (a1, b1, d1, a2, b2, d2) are functions of (β, α, δ, λ, μg).
- Transversality/stability requires a1 < 1 (kR t must converge to 0 following a shock).

### Qualitative predictions
- With a positive permanent (or perceived permanent) shock to g, unconstrained (Ricardian) households increase current consumption more than the contemporaneous income rise by borrowing against future income or dis-saving.
- Liquidity-constrained households can only raise consumption by the increase in current income.
- When financial development reduces λ (more households become unconstrained), a larger share of the population reacts to permanent-growth shocks by magnifying current consumption responses, increasing aggregate consumption volatility relative to output volatility.
- Therefore: easing credit constraints in the presence of shocks to trend growth can increase relative consumption volatility; the model predicts an increase in relative consumption volatility following financial development in emerging economies.

---

### Empirical evidence and calibration (overview)
- The paper measures relative volatility of consumption (consumption volatility relative to output volatility) in pre- and post-financial sector reform periods for a cross-section of emerging economies, with reform dates chosen from prior literature.
- Table 1 and Figure 1 provide evidence of rising financial development (e.g., bank deposits to GDP) and many cases of increased relative consumption volatility after reform.
- Calibration exercise:
  - The model is calibrated to Indian data for pre- and post-reform years, changing only the share of credit-constrained consumers (λ) while holding other parameters constant.
  - The calibration finds support for the model’s key prediction that financial development (lower λ) raises relative consumption volatility in the presence of persistent shocks to trend growth.
- Detailed calibration steps and shock processes (total factor productivity and labour productivity growth) are addressed in Section 4 (subsections 4.2.1 and 4.2.2) and sensitivity checks in subsection 4.4.

---

*Source: IMF Working Paper — Introduction and model sections (pages 3–14 of the supplied content).*

### Appendix II describes the solution method in details.

### _wp13120 - Appendix II describes the solution method in details.

### Volatility relationships and analytical results
- Equation linking consumption volatility to output volatility:
  - σ^2 ̃c / σ^2 ̃y = (c_R^* / c^*)^2 (1−λ)^2 σ^2 ̃c_R / σ^2 ̃y + (c_L^* / c^*)^2 λ^2 . (16)
- Expressions derived from the solution system (15) and log-linearised output (8):
  - σ^2 ̃c_R = [a_2^2 b_1^2 / (1−a_2^1 + b_2^2)] σ^2_a + [a_2^2 d_1^2 / (1−a_2^1 + d_2^2)] σ^2 ̃g
  - σ^2 ̃y = [1 + (1−α)^2 b_1^2 / (1−a_2^1)] σ^2_a + [α^2 + (1−α)^2 d_1^2 / (1−a_2^1)] σ^2 ̃g
  - Volatilities of transitory and permanent income shocks:
    - σ^2_a = σ^2_a / (1−ρ_a^2)
    - σ^2 ̃g = σ^2_g / (1−ρ_g^2)

- Proposition 1 (summary):
  - (i) Liquidity-constrained household consumption volatility relative to output volatility is unity when σ_a > 0; σ_g > 0:
    - σ ̃c_L / σ ̃y = 1.
  - (ii) For transitory income shocks (σ_a > 0; σ_g = 0), volatility of Ricardian household consumption relative to output and total consumption relative to output are both < 1 for λ ∈ [0,1):
    - σ ̃c_R / σ ̃y < 1
    - σ ̃c / σ ̃y < 1
  - (iii) For trend growth shocks (σ_a = 0; σ_g > 0), Ricardian household consumption volatility relative to output always > 1, while total consumption relative to output may be ≷ 1 depending on λ ∈ [0,1):
    - σ ̃c_R / σ ̃y > 1
    - σ ̃c / σ ̃y ≷ 1
  - (iv) Under trend growth shocks (σ_a = 0; σ_g > 0), both σ ̃c_R / σ ̃y and σ ̃c / σ ̃y increase when the share of liquidity-constrained households decreases (λ falls):
    - ∂(σ ̃c_R / σ ̃y)/∂λ < 0
    - ∂(σ ̃c / σ ̃y)/∂λ < 0

- Intuition and mechanism:
  - Liquidity-constrained households consume changes in income one-for-one; hence relative volatility = 1.
  - Ricardian households smooth consumption in response to transitory shocks; hence relative volatility < 1.
  - Ricardian households raise current consumption more than the rise in current income after permanent (trend) shocks by borrowing or reducing savings; hence relative volatility > 1.
  - Financial development (lower λ) increases the share able to behave like Ricardian households, so aggregate consumption volatility relative to output rises under permanent shocks.

- Main theoretical prediction:
  - Other things unchanged, under the occurrence of permanent income shock, financial development leads to a rise in the volatility of consumption in the economy relative to output volatility.

### Case study: India — calibration and data construction
- Sample split:
  - Pre-reform period: 1951-1991
  - Post-reform period: 1992-2009 (analysis uses data up to 2012 for some series)
- Data definitions and transformations:
  - Output, consumption and investment measured by real gdp at factor cost, private consumption expenditure, and gross fixed capital formation for 1951-2012.
  - Business cycle moments obtained from HP-filtered cyclical components of per capita output, consumption and investment.
  - Total Factor Productivity (tfp) constructed as weighted average of sectoral tfp using sectoral gdp, net fixed capital stock, and employment (1980-2009).
  - Sectoral tfp formula: log(A_s^t) = log(Y_s^t) − ∑_j w_j^s log(X_sj^t) with ∑_j w_j^s = 1.
  - Aggregate tfp: log(A_t) = ∑_s γ_s^t log(A_s^t).
  - Labour productivity growth measured from sectoral real gdp and labour force: log(Γ_s^t) = log(Y_s^t) − log(X_sj^t) (j = labour); log(Γ_t) = ∑_s γ_s^t log(Γ_s^t).
  - Gross growth rate of labour productivity: g_t = 1 + ∆ log(Γ_t).
  - Detrending: quadratic trend for tfp; trend regression for labour productivity growth; AR(1) fitted to cyclical components.

- Benchmark parameter values (Table 4):
  - Discount factor β 0.98
  - Rate of Depreciation δ 5%
  - Share of labour α 0.7
  - Adjustment cost parameter φ 2.82
  - Mean trend growth rate of labour productivity μ_g −1 4.7%
  - Persistence in tfp shock process ρ 0.495
  - Volatility in tfp σ_a 0.015
  - Persistence in labour productivity growth shock ρ_g 0.261
  - Volatility in labour productivity growth shock σ_g 0.020

- Financial development proxy and calibration of λ:
  - Proxy: number of bank accounts to population.
  - 1980: 21.4% of population had access to banking → share without access λ = 0.786 (pre-reform).
  - 2010: 66.9% of population had access to banking → λ = 0.331 (post-reform).

- Estimated trend and shock parameters:
  - Mean trend net growth rate over 1980-2009: μ_g −1 = 4.7%

### India: empirical facts and model simulation results
- Stylised business cycle facts (Table 3, data summary):
  - Pre-reform (1951-1991) vs Post-reform (1992-2009) (selected reported figures):
    - Real gdp Std. dev.: 2.25 (pre); 1.87 (post)
    - Rel. std. dev. (consumption?): 1.00 (pre); 1.00 (post) [table headings complex in source]
    - Private Consumption Std. dev.: 1.88 (pre); 2.14 (post)
    - Relative consumption volatility: 0.83 (pre); 1.14 (post)
    - Investment Std. dev.: 5.42 (pre); 5.58 (post)
    - Contemporaneous correlation and autocorrelation: reported increases in post-reform period for consumption and investment with output; persistence in output and consumption cycles higher post-reform.

- Observed financial development trends (Figure 2 description):
  - Bank account in total population rose from 20% in 1980 to above 70% in 2010 (with a period of decline in 1990-2005).
  - Bank branches per 100,000 population in 2010 more than doubled the 1970 value.
  - Deposit to GDP ratio and private credit to GDP ratio increased over time.

- Relative consumption volatility (Figure 3 description):
  - Five-year rolling relative consumption volatility increased; mean of relative consumption volatility higher in post-reform period.
  - Reported values: relative consumption volatility rose from 0.83 (1951-1991) to 1.14 (1992-2012).

- Simulated model business cycle volatilities (Table 6):
  - Data:
    - Pre-reform: Std. dev. YC ICI Data: 2.25; Rel. std. dev.: 1.88; First ord. corr. 0.83; Std. dev. 2.41
    - Post-reform: Std. dev. 1.87; Rel. std. dev.: 2.14; First ord. corr. 1.14; Std. dev. 2.98
  - Model:
    - Pre-reform: Std. dev. 3.25; Rel. std. dev. 3.21; First ord. corr. 5.67; Rel. std. dev. 0.99; Std. dev. 1.74
    - Post-reform: Std. dev. 3.22; Rel. std. dev. 3.23; First ord. corr. 4.85; Rel. std. dev. 1.00; Std. dev. 31.51

- Simulated correlation and persistence (Table 7):
  - Data correlations and autocorrelations:
    - Pre-reform: Correlation CIYCI 0.70; Auto-correlation 0.14; First order autocorr. 0.47; 0.79 (other entries in the table)
    - Post-reform: Correlation 0.90; Auto-correlation 0.75; First order 0.67; 0.598; 0.425
  - Model:
    - Pre-reform: Correlation 0.99; Auto-correlation 0.69; First order 0.62; 0.667; 0.163
    - Post-reform: Correlation 0.98; Auto-correlation 0.73; First order 0.61; 0.749; 0.183

- Interpretation of simulation results:
  - The model reproduces the direction of observed increase in relative consumption volatility after financial development, consistent with the main prediction.
  - Magnitude: the model’s rise in relative consumption volatility is small compared with the observed increase; potential reason is closed-economy framework (no foreign borrowing channel), which may under-predict the observed rise.
  - The model broadly matches patterns of changes in absolute output and consumption volatility, and matches qualitatively the changes in correlations and persistence between pre- and post-reform periods.
  - Figure 4 (description) indicates the model replicates cyclical patterns in output and consumption fairly well for pre- and post-reform periods.

### Sensitivity analysis
- Alternative calibration of λ using deposit to GDP ratio (Table 8 and Table 9):
  - Model results using deposit-to-GDP-based λ remain broadly unchanged compared to benchmark calibration based on bank accounts per population.
  - Reported model moments (selected):
    - Model Pre-reform: Std. dev. 3.24; Rel. std. dev. 3.21; First ord. corr. 5.38; Rel. std. dev. 0.99; Std. dev. 1.66
    - Model Post-reform: Std. dev. 3.22; Rel. std. dev. 3.23; First ord. corr. 4.83; Rel. std. dev. 1.00; Std. dev. 31.50
  - Correlations and autocorrelations remain broadly similar to baseline (Table 9).

### Key takeaway findings
- Analytical result: Liquidity constraints and the nature of income shocks (transitory vs. permanent) determine whether consumption volatility relative to output is below, equal to, or above 1 for different household types.
- Mechanism: Financial development reduces λ (share of liquidity-constrained households), increasing the fraction of households that respond to permanent income shocks by raising current consumption (borrowing or reducing savings), thereby raising aggregate consumption volatility relative to output.
- Empirical application to India:
  - Access to banking rose from 21.4% in 1980 to 66.9% in 2010 (λ from 0.786 to 0.331).
  - Relative consumption volatility rose from 0.83 (1951-1991) to 1.14 (1992-2012).
  - The calibrated model reproduces the direction of this change; magnitude under-predicted in a closed-economy setup.
- Robustness: Results are robust to alternative measures of financial development (deposit to GDP ratio).

*Appendix II (solution method) and empirical calibration details as presented in _wp13120 - Appendix II describes the solution method in details.*

### 4.4    Sensitivity to the measure of financial development

### 4.4    Sensitivity to the measure of financial development

### Robustness check: alternative financial development measure
- Financial development was measured alternatively by the bank deposit to GDP ratio to obtain the fraction of liquidity constrained households (λ).
- By this measure:
  - λ = 0.687 in the pre-reform period
  - λ = 0.305 in the post liberalisation period
- Table 8 and Table 9 show that key moments from the business cycle model for the pre and post-reform periods based on this alternative measure of λ are similar to those of the benchmark model.

### Key implication
- The main results are robust to using the bank deposit to GDP ratio as an alternative measure of financial development.

---

### 5    Conclusion

### Empirical puzzle addressed
- Emerging economies witnessed an increase in consumption volatility relative to output volatility after financial development, which is puzzling relative to traditional models and evidence from advanced economies that predict smoother consumption after credit constraints are reduced.

### Explanation provided
- Distinguishing feature: a large share of the population in developing economies lacks access to finance.
- Reforms in the financial sector increased access to financial services for households and firms in the last two decades, yet consumption volatility relative to output volatility increased post-reform.
- The puzzle is explained by a model featuring:
  - credit constraints, and
  - shocks to trend growth of productivity.
- The model predicts that relative consumption volatility will rise when more consumers can smooth consumption.

### Empirical validation
- The model, simulated for India before and after an increase in financial development, broadly replicates the rise in relative consumption volatility observed in the data.
- Other empirical regularities from the data are also replicated by the model.

### Limitations and further research
- The framework represents a closed economy; financial development is limited to domestic financial development.
- Open questions:
  - How increased household access to foreign financial instruments affects consumption-smoothing behavior in this framework for an emerging economy.
  - The model assumes the household sector is the sole channel for financial development; this may explain weak performance in replicating investment business cycle patterns.
  - Including credit-constrained firms in the framework would allow further examination of the role of financial development.

---

### Appendix I — Data spans (selected)
- The paper lists spans of data for GDP and consumption expenditure across regions and countries (table content omitted here).

---

### Appendix II — Solution and numerical implementation (highlights)

### Log-linearised system and method
- The log-linearised system of equations spanning the dynamics of the model economy is given in (B.1).
- The solution is assumed of the form in (B.2) with unknown parameters (a1, b1, d1, a2, b2, d2) that are functions of (β, α, δ, λ, μ_g) and solved by undetermined coefficients.
- Convergence condition requires a1 < 1.
- Key expressions: (B.3), (B.4), (B.5), (B.6)–(B.9), and closed-form for a1 in (B.13).

### Steady state expressions (as referenced)
- c_R* = (1−λα)/(1−λ) (1−λ)(1−α)(k_R*)^{1−α} μ_g^{α} − (μ_g − 1 + δ) k_R*
- y* = (1−λ)(1−α)(k_R*)^{1−α} μ_g^{α}
- k_R* = [ (1−α)(1−λ) / (1−α) μ_g^{α} / μ_g β − (1−δ) ]^{1/α}
- c_L* = α y*
- c* = (1−λ) c_R* + λ c_L*

### Numerical exercise parameter values
- β = 0.98
- α = 0.7
- δ = 0.05
- μ_g = 1.047
- ρ_a = 0.495
- ρ_g = 0.261

### Proofs and numerical illustrations of Proposition 1
- B.2: Proof of Proposition 1.(i)
  - Log-linearisation of c_L_t = W_t = α y_t yields ̃c_L_t = ̃y_t; hence σ^2_{̃c_L_t} = σ^2_{̃y}.

- B.3: Proof of Proposition 1.(ii) — transitory income shock only (σ^2_{ε_g} = 0, σ^2_{ε_a} > 0)
  - Relative consumption volatility of a Ricardian household with respect to output volatility:
    - σ^2_{̃c_R} / σ^2_{̃y} = [ a2^2 + b2^2 / (1−a1^2) + b2^2 ] [ 1 + (1−α)^2 b2^2 / (1−a1^2) ]
  - Evaluated at two values of λ:
    - Scenario 1: λ = 0.900 (share of liquidity-constrained households very high)
    - Scenario 2: λ = 0 (economy only Ricardian households)
  - Table 11 — Relative consumption volatility under transitory income shock:
    - λ = 0.900: σ_{̃c_R}/σ_{̃y} = 0.745 ; σ_{̃c}/σ_{̃y} = 0.807
    - λ = 0.000: σ_{̃c_R}/σ_{̃y} = 0.406 ; σ_{̃c}/σ_{̃y} = 0.406
  - Implication: With any share of Ricardian households, consumption volatility relative to output volatility is less than 1 under a transitory income shock. When λ = 0, total consumption relative volatility equals Ricardian household relative volatility.

- B.4: Proof of Proposition 1.(iii) — permanent income shock only (σ^2_{ε_a} = 0, σ^2_{ε_g} > 0)
  - Relative consumption volatility of a Ricardian household:
    - σ^2_{̃c_R} / σ^2_{̃y} = [ a2^2 + d2^2 / (1−a1^2) + d2^2 ] [ α^2 + (1−α)^2 d2^2 / (1−a1^2) ]
  - Evaluated for a range of λ; results shown in Table 12.
  - Table 12 — Relative consumption volatility under permanent income shock (rows: λ ; σ_{̃c_R}/σ_{̃y} ; σ_{̃c}/σ_{̃y}):
    - λ = 0.900 : 1.493 ; 0.834
    - λ = 0.786 : 1.992 ; 1.018
    - λ = 0.500 : 3.086 ; 1.897
    - λ = 0.331 : 3.556 ; 2.585
    - λ = 0.000 : 4.223 ; 4.223
  - Implications:
    - Relative consumption volatility of a Ricardian household always exceeds 1 under a permanent income shock.
    - Relative volatility of total consumption depends on the level of financial development and may fall below 1 if the economy has a large fraction of liquidity-constrained households.

- B.5: Proof of Proposition 1.(iv)
  - From Table 12, as the share of liquidity-constrained households declines (i.e., financial development increases), the volatility of Ricardian household consumption and total consumption relative to output volatility increases.

---

*Source: _wp13120 - 4.4    Sensitivity to the measure of financial development*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13120.pdf_