## _wp11253

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---

### Abstract and key result
- Common assumption: agents are risk-averse and prudent, and prudence is necessary for precautionary savings.
- Main finding: Prudence is not necessary to generate precautionary savings in small open economy models with more than two periods.
- Introduces a class of preferences that isolates the effect of risk aversion on precautionary savings.
- Qualitative effects of changes in risk aversion, interest rates, and persistence and volatility of shocks on average asset holdings match those for standard constant-elasticity-of-substitution preferences.
- Concludes the literature’s almost universal assertion—that only prudent consumers can generate positive levels of precautionary savings—is incorrect.

### Conceptual distinctions and preference class
- Risk aversion vs. prudence:
  - Risk aversion: dislike of risk and desire to smooth consumption; determined by concavity of u.
  - Prudence: preparation for very bad outcomes via buffer-stock assets; determined by convexity of marginal utility (third derivative).
- Preference class introduced:
  - Utility: u(c_t) = a c_t - b c_t^3 (equation (1)).
  - Properties: increasing, strictly concave, and u′′′ < 0 (imprudent).
  - Constant and invariant coefficient of relative prudence independent of parameters defining risk aversion.
  - Allows isolation of changes in risk aversion holding prudence fixed.
- Contrast with CES preferences:
  - CES has constant relative prudence tied to the same parameter as risk aversion; cannot separate effects.

### Section II — utility function and exact derivatives
- u(c_t) = a c_t - b c_t^3.
- Derivatives:
  - u′(c_t) = a - 3 b c_t^2
  - u′′(c_t) = -6 b c_t
  - u′′′(c_t) = -6 b
- Coefficient of relative prudence (CRP), η_R:
  - η_R = - c u′′′(c) / u′′(c)
  - For this class: η_R = 6 (constant).
- Coefficient of relative risk aversion (RRA), θ_R:
  - θ_R = - c u′′(c) / u′(c)
  - For this class: θ_R = [6 b c^2] / [a - 3 b c^2].
  - Condition: as long as [a - 3 b c^2] > 0, a lower a implies higher coefficient of relative risk aversion.

### Two-period model: setup, benchmark, and key result
- Environment:
  - Small open economy, one tradable good, perfect capital integration, r ≥ 0.
  - Budget constraints:
    - Bad state: c2L + (1+r)b1 = y2L (equation (2)).
    - Good state: c2H + (1+r)b1 = y2H (equation (3)).
    - Period 1: c1 + b1 = y1 + (1+r)b0 (equation (4)).
  - Preferences: U = u(c1) + βE[u(c2)] with β ∈ [0,1] (equation (5)).
- Benchmark parameterization (Table 1):
  - a = 6, b = 1/3, r = 0.01, β = 0.98, y1 = 1.5, y2L = 1, y2H = 2, b0 = 0.
  - Probabilities: 0.5 for low and high states.
- No-uncertainty outcome (Table 2):
  - Period 1: Endowment 1.5, Assets 0, Consumption 1.5064, E(assets) = 0
  - Period 2: Endowment 1.5, Assets -0.0064, Consumption 1.4936, E(assets) = -0.0064
- Uncertainty outcome (Table 3):
  - Period 1: Endowment 1.5, Assets 0, Consumption 1.5476, E(assets) = 0
  - Period 2: Endowment 1.0, Assets -0.0476, Consumption 0.9519, E(assets) = -0.0476
  - Period 2: Endowment 2.0, Assets -0.0476, Consumption 1.9519
- Key finding:
  - With u′′′ < 0 (imprudence), average asset holdings are negative under uncertainty → precautionary savings are negative (i.e., save less / hold more debt under uncertainty).
- Mechanism:
  - Euler equation (6): E[u′(c2)] = (1+ r)β u′(c1).
  - Jensen-type comparisons: sign of u′′′ determines whether E[u′(c2)] relative to u′(E[c2]) leads to positive or negative precautionary saving.

### Three-period model and short-horizon results
- Parameterization (Table 4):
  - a = 6, b = 1/3, r = 0.01, β = 0.98, y1 = 1.5, y2L = 1, y2H = 2, y3L = 1, y3H = 2, b0 = 0.
- No-uncertainty (Table 5) key entries:
  - Period 1: Endowment 1.5, Assets 0, Consumption 1.5, E(assets)=0
  - Period 2: Endowment 1.5, Assets -0.0085, Consumption 1.4957, E(assets)=-0.0085
  - Period 3: Endowment 1.5, Assets -0.0043, Consumption 1.4957, E(assets)=-0.0043
- Uncertainty (Table 6) selected entries:
  - Period 1: Consumption 1.5767
  - Period 2: Endowment 1.0 → Assets -0.0767, Consumption 1.2613; Endowment 2.0 → Consumption 1.7487
  - Period 3: Endowment 1.0 → Assets -0.3387, Consumption 0.6579; Endowment 2.0 → Assets 0.01739, Consumption 2.1757
- Key finding:
  - Introducing uncertainty still makes the economy save less (hold more debt) than under no uncertainty; average assets negative in periods 2 and 3 driven by u′′′ < 0.

### Comparative-statics experiments (Section V)
- Benchmark: Table 4 parameters.
A. Volatility
- Mean-preserving spread: y2L = 0.7, y2H = 2.3, y3L = 0.7, y3H = 2.3 (Table 7).
- Result (Table 8):
  - Period 1: Consumption 1.6841, E(assets)=0
  - Period 2: E(assets) = -0.1841
  - Total average asset holdings change from -0.1591 (Table 6) to -0.3999 (Table 8).
- Conclusion: Higher volatility implies lower savings (more debt) for these preferences because u′ is concave.

B. Intertemporal distortions
- Set y3H = y3L = 0 (lower endowment in period 3).
- No-uncertainty (Table 9) and Uncertainty (Table 10):
  - No-uncertainty total average assets = 1.4648
  - Uncertainty total average assets = 1.4341
- Conclusion: Intertemporal distortion increases incentives to save; imprudence effect can be dominated but both deterministic and uncertain cases see higher saving incentives.

C. Risk Aversion
- Change a = 5 (higher relative risk aversion).
- No-uncertainty (Table 11): total average assets = -0.0093.
- Uncertainty (Table 12): total average assets = -0.1557.
- Comparison: higher relative risk aversion implies higher savings.

D. Interest Rates
- r = 0.02.
- No-uncertainty (Table 13): near-zero average assets (-0.0003 total).
- Uncertainty (Table 14): total average assets = -0.1471 (higher than -0.1591 in Table 6).
- Conclusion: Higher interest rates increase average asset holdings (reduce debt).

- Overall message: Except for volatility (which lowers savings here), increases in risk aversion, interest rates, and intertemporal distortions have standard qualitative effects similar to CES preferences. The mean-preserving spread leads to lower savings because of concavity of u′.

### Cases where uncertainty raises savings despite imprudence
- Constructed example: intertemporal distortion with y3H = y3L = 0 and time-varying a: a1 = 49, a2 = 49, a3 = 3.
- No-uncertainty (Table 15) vs Uncertainty (Table 16) highlights:
  - Table 15 Period 1: Endowment 2.5, Consumption 4.8123, Assets 0; Period 2 assets -2.3123.
  - Table 16 Period 1: Consumption 4.4757; Period 2 assets -1.9756; Period 3 E(assets) positive components (0.2514, 0.5048).
- Result: Savings are higher under uncertainty in this example.
- Conclusion: To overturn the imprudence effect (u′′′ < 0) and obtain positive precautionary saving requires combining a drastic increase in risk aversion and a negative endowment shock in the last period.

### Infinite-horizon model: borrowing constraints and positive precautionary savings
- Preferences: Σ_{t=0}^∞ β^t E[u(c_t)] (equation (21)), period utility as in Section II.
- Resource constraint: c_t + (1+r)b_t+1 = y_t + (1+r)b_t + A (equation (22)), A ensures c_t ≤ a/b^(1/3).
- Constraints: c_t ≥ 0; borrowing limit b_t ≥ φ.
- Income: first-order Markov chain.
- Parameterization (Table 17):
  - β = 0.94, r = 0.059, a = 8.35, b = 0.333, σ_e = 0.026, ρ = 0.597, φ = -1.34, A = -0.77.
- Quantitative results (Table 18) selected:
  - Baseline precautionary savings = 0.92; NFA Ratios = 0.31.
  - Alternative columns: rho=0.7 → Precautionary Savings 0.94; sd=5% → 1.05; sd=2.5% → 0.88; a = 6 → 0.98; r=6.3% → 1.12.
  - Means: Output 1.00, Consumption 0.69, Foreign Assets -0.42 (baseline).
- Intuition:
  - Infinite horizon + borrowing constraint → positive probability of very long bad shock sequences → optimal buffer-stock accumulation → positive precautionary savings independent of u′′′ sign.
  - Higher persistence, higher volatility, higher risk aversion, higher interest rates generally increase precautionary savings.
  - Notable sensitivity: an interest rate increase of less than one percentage point can change debt from 42 percent of GDP to 22 percent of GDP in this calibration.

### Finite-horizon, long-horizon experiments
- Parameterization (Table 19):
  - β = 0.98, r = 0.02, a = 6, b = 0.333, σ_e = 0.05/0.35, ρ = 0.6, φ = -2.
- Experiments: T = 40 and T = 200 under low (0.05) and high (0.35) variance.
- Results:
  - Forty-period, low volatility (Table 20):
    - Y mean 1.4999; NFA mean -0.0756; C mean 1.4988; CA mean -0.0004.
  - Forty-period, high volatility (Table 21):
    - Y mean 1.5002; NFA mean -0.3501; C mean 1.4941; CA mean -0.001.
  - Two-hundred-period, low volatility (Table 22):
    - Y mean 1.4997; NFA mean -0.6772; C mean 1.4863.
  - Two-hundred-period, high volatility (Table 23):
    - Y mean 1.5033; NFA mean -0.5447; C mean 1.4925.
- Interpretation:
  - For short finite horizons (T = 40) under these preferences, higher volatility can reduce incentives to save (NFA more negative).
  - For long finite horizons (T = 200), higher volatility increases incentives to save (NFA less negative), mirroring infinite-horizon behavior: longer horizon raises probability of hitting borrowing-constraint sequences so uncertainty effect dominates imprudence.

### Conclusions and policy-relevant implications
- Introduces a new class of preferences: risk-averse but imprudent, with constant relative prudence independent of risk aversion parameters.
- Key conclusions:
  - In two-period models, sign of u′′′ determines sign of precautionary savings: u′′′ < 0 → negative precautionary savings.
  - In multi-period (sufficiently long) and infinite-horizon models with borrowing constraints, precautionary savings can be positive despite u′′′ < 0.
  - Interaction of uncertainty, horizon length, and borrowing constraints is the crucial determinant of precautionary saving in infinite-horizon settings.
  - Isolation of risk aversion is possible here because relative prudence is parameter-independent.
- Policy implications:
  - Effects of volatility, persistence, risk aversion, interest rates, and intertemporal distortions on precautionary saving depend on horizon and borrowing constraints.
  - Caution advised when generalizing two-period results to longer horizons or aggregate settings.

*Content based solely on the provided PDF excerpt for _wp11253 - references.*

### references

### _wp11253 - references

### Abstract and key result
- A common assumption is that agents are risk-averse and prudent, and that prudence is necessary to generate precautionary savings.
- The paper shows prudence is not necessary to generate precautionary savings in small open economy models with more than two periods.
- A new class of preferences is introduced that enables the isolation of the effect of risk aversion on precautionary savings.
- The effects of changes in risk aversion, interest rates, and persistence and volatility of shocks on average asset holdings are qualitatively identical to those observed for standard constant-elasticity-of-substitution preferences.
- The paper concludes that the literature’s almost universal assertion—that only prudent consumers can generate positive levels of precautionary savings—is incorrect.

### Conceptual distinctions and preference class
- Distinction between "risk aversion" and "prudence":
  - Risk aversion: agents dislike risk and like to smooth consumption across time; determined by concavity of the utility function.
  - Prudence: agents prepare for very bad outcomes by holding buffer-stock assets; determined by the convexity of marginal utility (the third derivative).
- The introduced class of preferences:
  - Characterized by risk aversion and imprudence (i.e., a negative third derivative).
  - Has a constant and invariant coefficient of relative prudence which is completely independent of the coefficient of relative risk aversion.
  - Allows isolation of how changes in risk aversion (without changing prudence) affect precautionary savings.
- Contrast with CES preferences:
  - CES preferences are characterized by a constant coefficient of relative prudence that is the same parameter affecting risk aversion, making it hard to disentangle effects.
  - The new class separates relative prudence from risk aversion.

### Model results and scope
- Two-period vs. multi-period/infinite-horizon:
  - In two-period models (e.g., Leland, 1968), prudence (positive third derivative) is necessary and sufficient for positive precautionary savings.
  - In infinite-horizon models with heterogeneous agents and i.i.d. shocks (Huggett and Ospina (2001)), the degree of prudence is irrelevant.
  - The paper demonstrates that in small open economy models with more than two periods (finite or infinite horizon), prudence is not necessary for positive precautionary savings.
- Role of borrowing constraints and horizon:
  - Following Aiyagari (1994), the key determinants of precautionary savings are the infinite horizon and a borrowing constraint, not the sign of the third derivative.
  - When uncertainty, an infinite horizon, and a borrowing constraint are present, precautionary savings are always higher than under perfect foresight, regardless of prudence or imprudence.
  - For finite-horizon models, given a structure of shocks, positive precautionary savings with imprudent agents arise provided the time horizon is sufficiently long.

### Mechanisms and intuition
- Risk aversion vs. imprudence incentives:
  - Risk aversion provides incentives to increase savings.
  - Imprudence provides incentives to save less in the face of uncertainty.
  - The presented preference class permits risk aversion and imprudence to coexist, enabling analysis of net effects.
- Buffer-stock and fear of prolonged bad shocks:
  - With borrowing constraints and long horizons, agents face the risk of long sequences of bad shocks that could force them to the borrowing constraint and consume only income.
  - This risk induces precautionary accumulation of assets irrespective of the sign of prudence.

### Comparative statics emphasized
- The study analyzes how the following affect precautionary savings (using the new preference class):
  - Volatility of shocks.
  - Persistence of shocks.
  - Risk aversion.
  - Interest rates.
  - Intertemporal distortions.
- Qualitative effects on average asset holdings mirror those observed under CES preferences.

_Italic: Content based solely on the provided PDF excerpt for _wp11253 - references._

### Section II presents the utility function corresponding to the particular class of preferences

### _wp11253 - Section II presents the utility function corresponding to the particular class of preferences

### Invariant Relative Prudence
- Utility function: u(c_t) = a c_t - b c_t^3 (equation (1)).
- Properties:
  - Increasing, strictly concave, and u′′′ < 0.
  - Constant and invariant coefficient of relative prudence, independent of parameters defining risk aversion.
  - Note: for sufficiently high a relative to b, marginal utility of consumption is positive.
- Conceptual point: Preferences are risk-averse but imprudent (negative third derivative); risk aversion does not necessarily imply prudence.

### Two-Period Model (setup and key result)
- Environment:
  - Small open economy, one tradable good, perfect capital integration, r ≥ 0.
  - Budget constraints:
    - Bad state: c2L + (1+r)b1 = y2L (equation (2)).
    - Good state: c2H + (1+r)b1 = y2H (equation (3)).
    - Period 1: c1 + b1 = y1 + (1+r)b0 (equation (4)).
  - Preferences: U = u(c1) + βE[u(c2)] with β ∈ [0,1] (equation (5)); period utility as in Section II.
- Benchmark parameterization (Table 1):
  - a = 6, b = 1/3, r = 0.01, β = 0.98, y1 = 1.5, y2L = 1, y2H = 2, b0 = 0.
  - Probabilities used: 0.5 for low and high states.
- No-uncertainty outcome (Table 2):
  - Period 1: Endowment 1.5, Assets 0, Consumption 1.5064, E(assets) = 0
  - Period 2: Endowment 1.5, Assets -0.0064, Consumption 1.4936, E(assets) = -0.0064
- Uncertainty outcome (Table 3):
  - Period 1: Endowment 1.5, Assets 0, Consumption 1.5476, E(assets) = 0
  - Period 2: Endowment 1.0, Assets -0.0476, Consumption 0.9519, E(assets) = -0.0476
  - Period 2: Endowment 2.0, Assets -0.0476, Consumption 1.9519
- Key finding:
  - With u′′′ < 0 (imprudence), average asset holdings are negative under uncertainty → precautionary savings are negative (i.e., save less / hold more debt under uncertainty).
- Mechanism:
  - Euler equation (6): E[u′(c2)] = (1+ r)β u′(c1).
  - Jensen-type comparisons show sign of u′′′ determines whether E[u′(c2)] relative to u′(E[c2]) leads to positive or negative precautionary saving.

### Three-Period Model
- Parameterization (Table 4):
  - a = 6, b = 1/3, r = 0.01, β = 0.98, y1 = 1.5, y2L = 1, y2H = 2, y3L = 1, y3H = 2, b0 = 0.
- No-uncertainty (Table 5):
  - Period 1: Endowment 1.5, Assets 0, Consumption 1.5, E(assets)=0
  - Period 2: Endowment 1.5, Assets -0.0085, Consumption 1.4957, E(assets)=-0.0085
  - Period 3: Endowment 1.5, Assets -0.0043, Consumption 1.4957, E(assets)=-0.0043
- Uncertainty (Table 6) (selected entries):
  - Period 1: Consumption 1.5767
  - Period 2: Endowment 1.0 → Assets -0.0767, Consumption 1.2613; Endowment 2.0 → Consumption 1.7487
  - Period 3: Endowment 1.0 → Assets -0.3387, Consumption 0.6579; Endowment 2.0 → Assets 0.01739, Consumption 2.1757
- Key finding:
  - Introducing uncertainty still makes the economy save less (hold more debt) than under no uncertainty; average assets negative in periods 2 and 3. Cause: negative u′′′.

### Volatility, Intertemporal Distortions, Risk Aversion, and Interest Rates (experiments)
- Benchmark parameterization used: Table 4 parameters.
A. Volatility
- Mean-preserving spread changed endowments (Table 7):
  - y2L = 0.7, y2H = 2.3, y3L = 0.7, y3H = 2.3.
- Result (Table 8):
  - Period 1: Consumption 1.6841, E(assets)=0
  - Period 2: E(assets) = -0.1841
  - Period 3: E(assets) entries lead to totals: total average asset holdings change from -0.1591 (Table 6) to -0.3999 (Table 8).
- Conclusion: Higher volatility implies lower savings (more debt) for these preferences because u′ is concave.

B. Intertemporal Distortions
- Set y3H = y3L = 0 (lower endowment in period 3).
- No-uncertainty (Table 9) and Uncertainty (Table 10) results:
  - No-uncertainty total average assets = 1.4648
  - Uncertainty total average assets = 1.4341
- Conclusion: Intertemporal distortion increases incentives to save (smoothing), but imprudence effect can dominate; still, a lower y3 raises saving incentives across both deterministic and uncertain cases.

C. Risk Aversion
- Change a = 5 (higher relative risk aversion).
- No-uncertainty (Table 11): total average assets = -0.0093.
- Uncertainty (Table 12): total average assets = -0.1557.
- Comparison to benchmark (Table 5 and 6): higher relative risk aversion implies higher savings.

D. Interest Rates
- r = 0.02.
- No-uncertainty (Table 13): near-zero average assets (-0.0003 total small negative).
- Uncertainty (Table 14): total average assets = -0.1471, which is higher than -0.1591 (Table 6).
- Conclusion: Higher interest rates increase average asset holdings (reduce debt).

Overall message of section V:
- Except for volatility (which lowers savings here), increases in risk aversion, interest rates, and intertemporal distortions have standard qualitative effects similar to CES preferences. The mean-preserving spread leads to lower savings because of the concavity of u′.

### Imprudence and Possibility of Higher Savings
- Constructed example: intertemporal distortion in period 3 (y3H = y3L = 0) plus time-varying a: a1 = 49, a2 = 49, a3 = 3.
- No-uncertainty (Table 15) vs Uncertainty (Table 16):
  - Table 15 Period 1: Endowment 2.5, Consumption 4.8123, Assets 0; Period 2 assets -2.3123.
  - Table 16 Period 1: Consumption 4.4757; Period 2 assets -1.9756; Period 3 E(assets) positive components (0.2514, 0.5048).
- Result: Savings are higher under uncertainty in this example.
- Conclusion: To overturn the imprudence effect (u′′′ < 0) and obtain positive precautionary saving requires combining a drastic increase in risk aversion and a negative endowment shock in the last period.

### Infinite-Horizon Model
- Main claim: In an infinite-horizon small open economy with a borrowing constraint, precautionary savings can be positive regardless of the sign of u′′′.
- Key model features:
  - Preferences: Σ_{t=0}^∞ β^t E[u(c_t)] (equation (21)), with period utility as in Section II.
  - Resource constraint: c_t + (1+r)b_t+1 = y_t + (1+r)b_t + A (equation (22)), A ensures c_t ≤ a/b^(1/3).
  - Non-negativity constraint c_t ≥ 0 and ad-hoc borrowing limit b_t ≥ φ.
  - Income follows a first-order Markov chain.
- Parameterization (Table 17):
  - β = 0.94, r = 0.059, a = 8.35, b = 0.333, σ_e = 0.026, ρ = 0.597, φ = -1.34, A = -0.77.
- Quantitative results (Table 18, selected entries):
  - Baseline precautionary savings = 0.92; NFA Ratios = 0.31.
  - Alternative columns: rho=0.7 → Precautionary Savings 0.94; sd=5% → 1.05; sd=2.5% → 0.88; a = 6 → 0.98; r=6.3% → 1.12.
  - Means: Output 1.00, Consumption 0.69, Foreign Assets -0.42 (baseline).
  - Standard deviations and correlations reported (see Table 18).
- Intuition:
  - Infinite horizon + borrowing constraint → positive probability of very long bad shock sequences → optimal buffer stock accumulation → positive precautionary savings independent of u′′′ sign.
  - Higher persistence, higher volatility, higher risk aversion, higher interest rates generally increase precautionary savings.
  - Notable sensitivity: an interest rate increase of less than one percentage point can change debt from 42 percent of GDP to 22 percent of GDP in this calibration.

### Finite-Horizon Model (long horizons)
- Question: Can finite-horizon models generate precautionary savings without changing utility across time?
- Parameterization (Table 19):
  - β = 0.98, r = 0.02, a = 6, b = 0.333, σ_e = 0.05/0.35, ρ = 0.6, φ = -2.
- Experiments: Solve for T = 40 and T = 200 under low (0.05) and high (0.35) variance.
- Results:
  - Forty-period, low volatility (Table 20):
    - Y mean 1.4999; NFA mean -0.0756; C mean 1.4988; CA mean -0.0004.
  - Forty-period, high volatility (Table 21):
    - Y mean 1.5002; NFA mean -0.3501; C mean 1.4941; CA mean -0.001.
  - Two-hundred-period, low volatility (Table 22):
    - Y mean 1.4997; NFA mean -0.6772; C mean 1.4863.
  - Two-hundred-period, high volatility (Table 23):
    - Y mean 1.5033; NFA mean -0.5447; C mean 1.4925.
- Interpretation:
  - For short finite horizons (T = 40) under these preferences, higher volatility can reduce incentives to save (NFA more negative).
  - For long finite horizons (T = 200), higher volatility increases incentive to save (NFA less negative), mirroring infinite-horizon behavior: longer horizon increases probability of hitting borrowing constraint sequences, so uncertainty effect dominates imprudence.

### Conclusion
- Introduces a new class of preferences: risk-averse but imprudent, with constant relative prudence independent of risk aversion parameters.
- Key findings:
  - In two-period models, sign of u′′′ determines sign of precautionary savings: u′′′ < 0 → negative precautionary savings.
  - In multi-period (sufficiently long) and infinite-horizon models with borrowing constraints, precautionary savings can be positive despite u′′′ < 0.
  - Interaction of uncertainty, horizon length, and borrowing constraints is the crucial determinant of precautionary saving in infinite-horizon settings.
  - Isolating risk aversion is possible here because relative prudence is parameter-independent.
- Policy-relevant implications:
  - Effects of volatility, persistence, risk aversion, interest rates, and intertemporal distortions on precautionary saving depend on horizon and borrowing constraints; caution when generalizing two-period results.

### Appendix: Relative Prudence and Relative Risk Aversion (exact derivatives and coefficients)
- Derivatives of u(c):
  - u′(c_t) = a - 3 b c_t^2
  - u′′(c_t) = -6 b c_t
  - u′′′(c_t) = -6 b
- Coefficient of relative prudence (CRP), η_R:
  - η_R = - c u′′′(c) / u′′(c)
  - For this class: η_R = 6 (constant); expression given: 6 = (1/6) ... (presentation in source).
  - CRP is constant and independent of curvature parameters.
- Coefficient of relative risk aversion (RRA), θ_R:
  - θ_R = - c u′′(c) / u′(c)
  - For this class: θ_R = [6 b c^2] / [a - 3 b c^2]  (source expression: 2 22 66 (...) — preserves source form).
  - Condition: as long as [a - 3 b c^2] > 0, a lower a implies higher coefficient of relative risk aversion.

*Source: _wp11253 - Section II presents the utility function corresponding to the particular class of preferences*

### REFERENCES

### _wp11253 - REFERENCES

### References
- Aiyagari, Rao, 1994, ―Uninsured Idiosyncratic Risk and Aggregate Saving,‖ Quarterly Journal of Economics, Vol. 109 (August), pp. 659–84.
- Browning, Martin and  Annamaria Lusardi, 1996. "Household Saving: Micro Theories and Micro Facts," Journal of Economic Literature, American Economic Association, vol. 34(4), pages 1797-1855, December.
- Chamberlain, Gary and Charles A. Wilson, 2000, ―Optimal Intertemporal Consumption Under Uncertainty,‖ Review of Economic Dynamics, Vol. 3 (July), pp. 365–395.
- Durdu, Ceyhun Bora, Enrique G. Mendoza, and Marco E. Terrones, 2009, ―Precautionary Demand for Foreign Assets in Sudden Stop Economies: An Assessment of the New Mercantilism,‖ Journal of Development Economics, Vol. 89 (July), pp. 194–209.
- Huggett, Mark, and Sandra Ospina, 2001, ―Aggregate Precautionary Savings: When is the Third Derivative Irrelevant?,‖ Journal of Monetary Economics, Vol. 48 (September), pp. 373–396.
- Huggett, Mark, 2004, ―Precautionary Wealth Accumulation,‖ Review of Economic Studies, Vol. 71 (July), pp. 769–781.
- Kimball, Miles S., 1990, ―Precautionary Saving in the Small and in the Large,‖ Econometrica, Vol. 58 (January), pp. 53–73.
- Leland, Hayne E., 1968, ―Saving and Uncertainty: The Precautionary Demand for Saving,‖ Quarterly Journal of Economics, Vol. 82 (August): 465–473.
- Ljungqvist, Lars, and Thomas J. Sargent, 2004, Recursive Macroeconomic Theory (Cambridge, MA: MIT Press).

*Source: _wp11253 - REFERENCES*

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