## 1. Evolutionof Wealth Under Certainty (wpiea2019108)

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### Introduction: central messages and implications
- Prevailing paradigm: permanent income hypothesis (PIH) used to smooth government spending and ensure long-term sustainability and intergenerational equity.
- Under certainty, PIH is optimal and analogous to Hartwick’s rule: invest exhaustible-resource income in productive capital to maintain a constant consumption stream.
- Under uncertainty (commodity price or interest rate fluctuations) PIH optimality is not guaranteed.
- Illustrative numerical example:
  - Initial endowment: $100 earning a five percent annual dividend (dividend = $5).
  - One-off loss of principal: $20 (remaining principal = $80).
  - PIH post-shock consumption recommendation: $4 (five percent of $80).
  - Implication: following PIH after a shock preserves the post-shock wealth level, making transitory shocks permanent and causing wealth to behave as a random walk.
- Normative framing:
  - Intergenerational equity evaluated via Ramsey’s utilitarian framework (maximizing a discounted sum of expected utilities).
  - Long-term sustainability requires probabilistic assessment of long-term wealth and consumption, including tail risks.
- Main conceptual findings previewed:
  - Under uncertainty PIH skims returns to smooth short-term spending but makes wealth and spending non-stationary and more volatile over the long term.
  - Optimal "prudent" policy requires a precautionary premium—additional savings that raise expected wealth and compensate future generations for risk.
  - Volatility tradeoff: anchoring wealth requires higher short-term volatility of government spending; prioritizing consumption smoothing (PIH) makes wealth less stable.
  - Proposal: Prudent Wealth Stabilization (PWS) balances the tradeoff, builds precautionary savings, and anchors long-term sustainability while remaining implementable.

### Analytical framework: setting, uncertainty, and wealth dynamics
- Definitions and key equations (as in source):
  - w_t = a_t + Y^n_t + Q_t
  - a_t = (1 + r_{t-1}) a_{t-1} + y_t + y^n_t − g_t
  - Y^n_t = E_t sum_{i=1 to ∞} y^n_{t+i} / (1 + r_{t+i})^i
  - Q_t = E_t sum_{i=1 to T−t} y_{t+i} / (1 + r_{t+i})^i
  - y_t = (p_t − v_t) q_t, for t ≤ T; y_t = 0, for t > T
- Oil price uncertainty:
  - log(p_t) = μ + ρ log(p_{t−1}) + ε_t
  - ε_t iid normal with zero mean and standard deviation σ
  - Other variables held constant: r_t = r, y^n_t = y^n, v_t = v, q_t = q
- Recursive wealth dynamics:
  - Y^n_t = (1 + r_{t−1}) Y^n_{t−1} − y^n_t
  - Q_t = (1 + r_{t−1}) Q_{t−1} − E_{t−1} y_t + φ_t (φ_t captures changes in expected discounted future oil revenue)
  - w_t = (1 + r_{t−1}) w_{t−1} − g_t + φ̃_t
    - φ̃_t = y_t − E_{t−1} y_t + φ_t = E_t( sum_{i=0 to T−t} y_{t+i} / (1 + r)^i ) − E_{t−1}( sum_{i=0 to T−t} y_{t+i} / (1 + r)^i ), t < T
- Preferences and objective:
  - U_t = U(g_t), U′(·) > 0, U′′(·) ≤ 0, U′′′(·) > 0
  - Max E_t sum_{i=0 to ∞} β^{t+i} U(g_{t+i}); β (1 + r) = 1 for simplicity
  - Consistency/intergenerational equity condition: U′(g_t) = β(1 + r) E_t U′(g_{t+i}), ∀ i > 0
- Calibration note: model calibrated to match basic features of the United Arab Emirates’ (UAE) economy; preferences assume constant relative risk aversion.

### PIH under uncertainty: intuition and non-stationarity
- PIH under certainty:
  - g_t = g implies g = r w (skimming returns and preserving principal).
- PIH with shocks:
  - PIH rule with shocks: g_t = r w_{t−1}
  - Combining with wealth law of motion yields w_t = w_{t−1} + φ̃_t
  - Wealth follows a random walk and is non-stationary.
- Implications:
  - Short-term smoothing masks high long-term volatility in wealth and spending.
  - Consumption smoothing transfers risk to future generations, undermining intergenerational equity and long-term sustainability.

### Non-stationarity of wealth under PIH: implications and quantification
- Non-stationarity arises whenever oil price is stochastic; long-run equilibrium is indeterminate under PIH.
- Contrast with Hartwick: Hartwick pins a unique equilibrium via marginal product of capital; PIH lacks linkage between wealth and marginal rate of intertemporal substitution.
- Simulation insights:
  - PIH reduces spending procyclicality relative to oil price but transfers volatility onto saving.
  - Series of negative oil price shocks cause precipitous decline in wealth.
  - Consumption is smoothed yet non-stationary; long-term volatility passed to future generations is larger for more volatile oil prices and longer resource horizons.
- Intergenerational equity findings (Monte Carlo simulations):
  - Expected wealth and consumption (ex ante) similar to starting values, but actual inherited wealth and consumption for future cohorts exhibit large mean-preserving spread.
  - PIH consistent with intergenerational equity only if cohorts are risk neutral.
- Utilitarian precautionary premium (Kimball, 1990):
  - Defined by U′(g_t) = E_t U′(g_{t+T}(1+x)).
  - Calibration result: x = 9 percent.
  - This x implies post-oil generation would need an additional $300 billion in wealth (14 percent additional financial wealth if non-oil income excluded or 78 percent of UAE’s 2017 GDP).
  - Equivalent precautionary premium increases proportionally with risk.
- Conclusion: PIH transfers risk forward without compensating future generations under risk aversion.

### Hallmarks of optimal policy — Prudence
- Precautionary saving is required when agents are risk averse (Arrow 1965; Pratt 1964; Leland 1968; Kimball 1990).
- Numerical solution approach: solve equilibrium conditions recursively using Tauchen (1986) discretization.
- Prudential policy features:
  - Precautionary savings early in extraction by generations facing most uncertainty.
  - These savings raise financial wealth and dividend income; as uncertainty declines, precautionary motive weakens and government spending rises, potentially exceeding PIH in later periods.
- Calibration results:
  - Precautionary premium 퐸_0 w_T / w_0 = 4 percent (7 percent if non-oil revenues are excluded).
- Prudence implies higher-than-initial target post-oil wealth and spending, shifting distributions of w_T and g_T upward.
- Note: Prudence improves intergenerational equity but does not eliminate non-stationarity or large tail risks.

### Hallmarks of optimal policy — Anchoring
- Stationarity can be induced by embedding stabilization incentives linking wealth and intertemporal marginal substitution (examples: wealth- or debt-dependent discount rates, endogenous risk premiums, portfolio adjustment costs).
- Model modification: add penalty φ/2 (w̅ − w_t)^2 into budget constraint; adjusts optimality condition to:
  - U′(g_t) = E_t U′(g_{t+i}) / (1 − φ(w̅ − w_t)), ∀ i > 0
- Mechanics:
  - If w_t < w̅, adjustment acts like an interest rate hike—reducing consumption growth and raising saving so wealth gravitates toward w̅.
  - φ captures stabilization strength.
- Simulation calibration example:
  - Precautionary premium w̅ / w_0 set at 2 percent and φ at 0.00003.
- Outcomes:
  - Wealth anchoring reduces uncertainty passed to future generations and induces mean reversion of wealth and spending.
  - Anchoring reduces standard deviation of post-oil wealth.
  - Tradeoff: stronger anchoring (higher φ) lowers tail risk and speeds reversion but raises contemporaneous spending volatility.

### Volatility tradeoff and policy trilemma
- Fundamental identity: w_t = (1 + r) w_{t−1} − g_t + φ̃_t
- Oil price shocks φ̃_t must be absorbed by wealth or government spending or both:
  - Smoothing g_t forces wealth to absorb shocks (raising intergenerational risk).
  - Stabilizing wealth requires more volatile g_t (short-term spending volatility).
- Polar scenarios:
  - Constant government spending forces wealth depletion under negative shocks.
  - Constant wealth requires government spending to neutralize shocks, producing high spending volatility.
- Conceptual “impossible trinity”: expenditure smoothing, wealth preservation (sustainability), and intergenerational equity cannot all be achieved simultaneously.
- Policy choice parameters:
  - Higher wealth stabilization (higher φ) or higher precautionary premium (higher w̅ / w_0) can pursue intergenerational equity.
  - Tradeoffs: higher w̅ / w_0 lets current generations enjoy smoother but lower average consumption; lower precautionary premium raises current consumption but increases future volatility.

### Fiscal framework design recommendations
- Focus fiscal frameworks on wealth targets rather than expenditure targets:
  - Expenditure benchmarks can be misleading if not tied to a wealth strategy.
  - PIH expenditure benchmark pegs expenditure to past wealth, is forward-poor, volatile, and non-stationary.
- Wealth stabilization implies some procyclicality of spending relative to oil price:
  - Negative shocks call for lower consumption to recover wealth.
  - Countercyclical saving in booms leads to procyclical and volatile wealth but improves long-term equity and sustainability.
- Medium-term benchmarks should aim to raise wealth:
  - Target a higher level of wealth under uncertainty rather than merely maintaining wealth constant.
  - Wealth should increase concavely over time because early generations bear most precautionary saving burdens.

### Proposed policy: Prudent Wealth Stabilization (PWS)
- PWS rule (equation (14)):
  - g_t = f_t + α(w_{t−1} − γ w_0)
  - Under consumption-smoothing assumption f_t = r w_{t−1}:
    - g_t = r w_{t−1} + α(w_{t−1} − γ w_0)
- Interpretation:
  - f_t: short-term objectives (set to r w_{t−1} for consumption smoothing discussion).
  - α (wealth stabilization factor), α > 0: reduces spending when wealth below γ w_0 to induce gravitation toward target.
  - γ (precautionary premium): embeds risk compensation; γ > 1 when oil-extracting generations should bear less relative risk.
- Calibration and implementation guidance:
  - Scale α using average government spending relative to wealth; for countries near PIH, α = r is a reasonable initial ballpark.
  - Simulate rule to obtain post-oil wealth for various γ; choose γ to approximately satisfy U′(g_0) = β(1+ r) E_0 U′(g_T) using simulation output.
  - For any realistic α, a unique γ ensures intergenerational equity between initial and post-oil generations; weaker stabilization (low α) requires larger γ.
- Numerical calibration examples:
  - When α ≈ 0.1, government expenditure became nearly as volatile as the oil price.
  - Baseline in paper: α = 0.03 and γ = 1.05.
    - γ = 1.05 implies additional wealth accumulation of $173 billion during extraction years (45 percent of UAE’s 2017 GDP).
- Performance versus PIH:
  - PWS achieves prudence, anchoring, and a balanced volatility tradeoff while retaining PIH simplicity.
  - Compared to PIH, PWS:
    - Induces initial-generation savings, raising average consumption for future generations.
    - Ensures stationarity of wealth despite negative oil price shocks.
    - Produces higher and more certain post-oil wealth (based on 100,000 oil price simulations).
  - Statistical properties (100,000 Monte Carlo simulations):
    - Under PIH: ex ante expected wealth and expenditure constant but volatility rises over time.
    - Under PWS: rising trend in wealth converging to target (most convergence early), reduced coefficient of variation of wealth, higher short-term spending volatility but lower long-term spending volatility due to stationarity.

### Policy tradeoffs, time consistency, and practical considerations
- Choosing α and γ balances:
  - Short-term expenditure volatility costs versus intergenerational wealth volatility.
  - Preferences over future generations and degree of risk aversion.
- Calibration can aim to minimize short-term expenditure volatility costs (requires modeling links to non-oil growth).
- Time consistency and target revisions:
  - Periodic revision of post-oil target may be needed for structural changes (e.g., reserve revisions) but risks time inconsistency.
  - Anchoring with a fixed post-oil target provides discipline, allowing gradual deviations but limiting recursive reoptimization that induces non-stationarity.
- Alternative stationarity devices to explore:
  - Asymmetric targets; imposing a floor on wealth (w_t ≥ w̲, ∀t).

### Public investment and wealth accounting implications
- Public investment changes composition of wealth (physical vs financial) and can earn higher average returns.
- Inter-temporal allocation governed by same consumption-Euler equation; higher average return raises average consumption level across generations but does not alter intergenerational distribution.
- Wealth remains non-stationary even with physical and financial investments; overall equilibrium wealth level remains indeterminate.
- Fiscal accounting interprets public investment as “spending”; analytical wealth accounting treats it as “saving”.
- Metrics:
  - Non-resource primary budget balance useful short-term but inadequate for intergenerational equity and sustainability.
  - Non-resource primary current balance better for long-term analysis (distinguishes spending of wealth from saving via public investment).

### Research directions and concluding insights
- Future research priorities:
  - Explore other intertemporal risks: uncertain size of oil reserves; energy-efficiency improvements reducing long-run demand; productivity and interest rate shocks.
  - Productivity and interest rate shocks become more important as oil exhaustion nears; oil price shocks wane.
  - Need for models robust to multiple uncertainties; attention to intergenerational considerations will grow with resource scarcity and climate change.
  - Develop better concepts of intergenerational equity balancing current and future cohorts.
  - Model linkages between short-term expenditure volatility and non-oil growth to guide α–γ choices.
  - Investigate time-consistent frameworks (e.g., overlapping generations) and alternative instruments to induce stationarity (e.g., asymmetric targets, wealth floors).
- Core conclusion:
  - Pegging expenditure to a notional return on past wealth (PIH) cannot simultaneously deliver short-term smoothing, intergenerational equity, and long-term sustainability under uncertainty.
  - Policies should embed greater prudence and explicit long-term wealth anchors (e.g., PWS) to operationalize intergenerational equity and sustainability.

*Source: wpiea2019108 (IMF working paper excerpts).*

### 1. Evolutionof Wealth Under Certainty ..................................................................................

### 1. Evolutionof Wealth Under Certainty

### Introduction: central messages and implications
- The prevailing fiscal policy paradigm in resource-rich countries is the permanent income hypothesis (PIH), used to smooth government spending and ensure long-term sustainability and intergenerational equity.
- In a world without shocks, PIH is the best possible fiscal strategy and is analogous to Hartwick’s rule: investing income from exhaustible resources in productive capital maintains a constant stream of consumption indefinitely.
- The optimality of both Hartwick’s rule and PIH critically relies on preserving wealth in the absence of shocks; when wealth is affected by commodity price or interest rate fluctuations, optimality is no longer guaranteed.
- Illustrative numerical example in the text:
  - Initial endowment: $100 earning a five percent annual dividend (dividend = $5).
  - One-off loss of principal: $20 (remaining principal = $80).
  - PIH post-shock consumption recommendation: $4 (five percent of $80).
  - Implication: following PIH after a shock preserves the post-shock wealth level, making transitory shocks permanent and causing wealth to behave as a random walk.
- Key normative framing:
  - Intergenerational equity is evaluated using Ramsey’s utilitarian framework—maximizing a discounted sum of expected utilities—recognizing its limitations but using it for analysis.
  - Long-term sustainability requires probabilistic assessment of long-term wealth and consumption, including tail risks.
- Main conceptual findings previewed:
  - Under uncertainty (e.g., oil price shocks), PIH smooths short-term government spending by skimming a small fraction of wealth, but wealth and spending become non-stationary and more volatile over the long term.
  - Optimal "prudent" policy calls for a precautionary premium—additional savings to compensate future generations for risk—raising expected wealth by resource exhaustion and improving intergenerational equity.
  - There exists a volatility tradeoff: anchoring wealth requires higher short-term volatility of government spending (to offset shocks), while prioritizing consumption smoothing (PIH) makes wealth less stable.
  - Proposal of an alternative: prudent wealth stabilization policy (PWS) that balances the volatility tradeoff, builds precautionary savings, and anchors long-term sustainability while being simple to implement.

### Analytical framework: setting, uncertainty, and wealth dynamics
- Setting and definitions:
  - Government wealth at end-period t: w_t = a_t + Y^n_t + Q_t  (equation (1))
    - a_t = financial wealth
    - Y^n_t = NPV of lifetime non-oil income
    - Q_t = NPV of subsoil oil wealth
  - Financial wealth evolution: a_t = (1 + r_{t-1}) a_{t-1} + y_t + y^n_t − g_t  (equation (2))
  - NPV of lifetime non-oil income: Y^n_t = E_t sum_{i=1 to ∞} y^n_{t+i} / (1 + r_{t+i})^i  (equation (3))
  - NPV of subsoil wealth: Q_t = E_t sum_{i=1 to T−t} y_{t+i} / (1 + r_{t+i})^i  (equation (4))
  - Oil revenue specification:
    - y_t = (p_t − v_t) q_t, for t ≤ T
    - y_t = 0, for t > T  (equation (5))
- Uncertainty and simplification:
  - Oil price is the only source of uncertainty, assumed to follow an AR(1) in logs:
    - log(p_t) = μ + ρ log(p_{t−1}) + ε_t  (equation (6))
    - Shocks ε_t are iid normal with zero mean and standard deviation σ.
  - Other variables held constant: r_t = r, y^n_t = y^n, v_t = v, q_t = q.
  - Note: including other shocks (non-oil revenue, interest rate, extraction cost, output) would not change main arguments that wealth is non-stationary and volatility depends on covariances.
- Wealth recursive dynamics:
  - Recursive forms:
    - Y^n_t = (1 + r_{t−1}) Y^n_{t−1} − y^n_t  (equation (7))
    - Q_t = (1 + r_{t−1}) Q_{t−1} − E_{t−1} y_t + φ_t  (equation (8))
      - φ_t captures changes in expected discounted future oil revenue (impact of oil price shocks).
  - Law of motion for total wealth:
    - w_t = (1 + r_{t−1}) w_{t−1} − g_t + φ̃_t  (equation (9))
      - φ̃_t = y_t − E_{t−1} y_t + φ_t = E_t( sum_{i=0 to T−t} y_{t+i} / (1 + r)^i ) − E_{t−1}( sum_{i=0 to T−t} y_{t+i} / (1 + r)^i ), t < T
  - Interpretation: NPV of wealth earns gross return (1 + r_{t−1}); consumption g_t reduces wealth; φ̃_t summarizes uncertainty-driven changes in expected present and future oil income.
- Preferences and objective:
  - Instantaneous utility: U_t = U(g_t), U′(·) > 0, U′′(·) ≤ 0, U′′′(·) > 0.
  - Government maximizes expected discounted sum of utilities: Max E_t sum_{i=0 to ∞} β^{t+i} U(g_{t+i})  (equation (10))
  - Discounting assumption for simplicity: β (1 + r) = 1.
  - Intergenerational equity condition (consistency): U′(g_t) = β(1 + r) E_t U′(g_{t+i}), ∀ i > 0  (equation (11)
- Calibration note:
  - Model is calibrated to match basic features of the United Arab Emirates’ (UAE) economy (see Appendix I).
  - Preferences assumed to display constant relative risk aversion (constant intergenerational inequality aversion).

### PIH under uncertainty: intuition and non-stationarity
- PIH in absence of shocks:
  - Optimality condition reduces to constant government spending across periods: g_t = g_{t−1} = g.
  - With φ̃_t = 0, maintaining constant consumption requires maintaining constant wealth, implying spending equals the return on wealth: g = r w (skimming off returns while preserving principal).
  - Under certainty, PIH ensures intergenerational equity and long-term sustainability.
- PIH with shocks—key consequences:
  - Under PIH rule in presence of shocks: g_t = r w_{t−1}  (equation (12))
    - This effectively preserves the post-shock level of wealth after each price change, treating shocks as if they were permanent.
  - Combining wealth law of motion (equation (9)) with PIH spending rule (equation (12)) yields:
    - w_t = w_{t−1} + φ̃_t
    - Wealth follows a random walk and is non-stationary.
  - Implications:
    - Short-term smoothing of government spending masks high long-term volatility in wealth and spending.
    - Consumption smoothing under PIH transfers risk to future generations, undermining intergenerational equity and long-term sustainability in an uncertain world.
- Conceptual takeaway:
  - A policy (PIH) that ensures equity and sustainability under certainty makes these goals virtually impossible under uncertainty because it preserves post-shock wealth levels and allows transitory shocks to have permanent effects.

*Source: wpiea2019108 - 1. Evolutionof Wealth Under Certainty.*

### 21.      It is important to emphasize that wealth under PIH is non-stationary regardless of

### 21.      It is important to emphasize that wealth under PIH is non-stationary regardless of

### Non-stationarity of wealth under PIH
- Wealth under PIH is non-stationary whenever the oil price is stochastic, regardless of whether the oil price itself is stationary.
- Non-stationarity arises from indeterminacy of the long-run equilibrium in the PIH framework: any level of wealth can be consistent with a steady state so long as consumption follows the PIH consumption rule.
- Contrast with Hartwick’s framework:
  - Hartwick’s model links the marginal product of capital to the real interest rate, pinning down a unique equilibrium level of capital and inducing mean reversion after shocks.
  - Under PIH, the link between wealth and the marginal rate of intertemporal substitution is absent; optimal consumption growth is independent of wealth, so wealth does not gravitate back to an initial value after shocks.
- Appendix II (referred) illustrates non-stationarity under transitory and permanent shocks.

### Implications: Long-term sustainability
- Simulation insights (Figure 2):
  - PIH smooths government spending and reduces its procyclicality relative to oil price, but this transfers volatility onto saving.
  - Each shock has a permanent effect on wealth; a series of negative oil price shocks causes precipitous decline in wealth (see Figure 3, panel 3.1).
  - Consumption is smoothed but similarly non-stationary.
  - Long-term volatility passed onto future generations is larger for more volatile oil prices and longer resource horizons.

### Implications: Intergenerational equity
- Monte Carlo simulations (Figure 3) reveal:
  - Expected wealth and consumption (as of the initial period) are similar to starting values.
  - The non-oil extracting generation faces significant risk in actual inherited wealth and consumption levels (mean-preserving spread).
  - PIH is consistent with intergenerational equity only if cohorts are risk neutral; under risk aversion, future generations are strictly worse off.
- Utilitarian quantification via an equivalent precautionary premium x (Kimball, 1990):
  - Defined by U′(g_t) = E_t U′(g_{t+T}(1+x)).
  - Calibration result: x amounts to 9 percent — the amount by which post-oil government spending would need to increase to achieve intergenerational equity.
  - This would require the post-oil generation inheriting an additional $300 billion in wealth (14 percent in additional financial wealth if non-oil income is excluded or 78 percent of UAE’s 2017 GDP).
  - The equivalent precautionary premium increases proportionally with risk (panel 3.3 in Figure 3).
- Conclusion: PIH fails to compensate future generations for higher uncertainty, producing intergenerational inequity because it transfers risk forward without precautionary compensation.

### Hallmarks of optimal policy — Prudence
- Optimal intertemporal allocation with uncertainty requires precautionary saving when agents are risk averse (Arrow (1965), Pratt (1964), Leland (1968), Kimball (1990)).
- Solution approach:
  - Find a policy satisfying equilibrium conditions (7) and (9) in every period via numerical methods (Tauchen, 1986), solving recursively backward assuming PIH after oil exhaustion.
- Prudent policy features (Figure 4):
  - Precautionary savings early in extraction by generations facing least uncertainty.
  - These savings raise financial wealth and dividend income; as oil uncertainty diminishes, precautionary motive weakens and government spending rises, exceeding PIH in the second half of the simulation.
- Additional insights:
  - Prudence implies targeting a higher-than-initial level of post-oil wealth and spending:
    - Compensatory precautionary premium shifts distributions of w_T and g_T upward (panels 4.4 and 4.5).
    - Calibration: precautionary premium 퐸_0 w_T / w_0 = 4 percent (7 percent if non-oil revenues are excluded).
  - Achieving intergenerational equity is not equivalent to ensuring long-term sustainability:
    - Prudence embeds risk compensation as an upward trend in consumption and wealth (expected increase), while non-stationarity and large tail risks remain (panel 4.3).

### Hallmarks of optimal policy — Anchoring
- Literature on inducing stationarity (e.g., Schmitt-Grohe and Uribe (2003)) uses wealth- or debt-dependent discount rates, endogenous risk premiums, or portfolio adjustment costs to link wealth and intertemporal marginal substitution.
- Policy implication: embed a stabilization incentive to establish a targeted equilibrium wealth level that incorporates a precautionary premium above initial wealth.
- Model modification: add a notional penalty φ/2 (w̅ − w_t)^2 into the government budget constraint (analogous to portfolio adjustment costs). This adjusts optimality condition (11) to:
  - U′(g_t) = E_t U′(g_{t+i}) / (1 − φ(w̅ − w_t)), ∀ i > 0  (equation (13) as in source)
- Mechanism:
  - When w_t < w̅, the adjustment cost acts like an interest rate hike—lowering consumption growth and raising saving so wealth gravitates toward w̅.
  - Parameter φ captures strength of stabilization motive.
- Simulation calibration:
  - Precautionary premium w̅ / w_0 was set at 2 percent and φ at 0.00003.
- Outcomes (Figure 5):
  - Wealth anchoring reduces uncertainty passed onto future generations, spreads volatility more evenly, and induces reversion of wealth and spending to equilibrium—consistent with long-term sustainability.
  - Anchoring reduces standard deviation of post-oil wealth (panel 5.4).
  - Wealth anchoring requires somewhat greater short-term volatility of government expenditure relative to PIH; higher φ increases speed of mean reversion and reduces tail risk but raises contemporaneous spending volatility (volatility tradeoff).

### Rethinking fiscal policy — Recognizing the volatility tradeoff
- Volatility tradeoff summarized from equation (8):
  - w_t = (1 + r) w_{t−1} − g_t + φ̃_t
  - Oil price shocks φ̃_t impact either wealth or government spending or both; smoothing g_t forces wealth to absorb shocks, while stronger wealth anchoring requires more volatile g_t to offset shocks.
- Polar scenarios (Figure 6):
  - Constant government spending (extreme smoothing) forces wealth to absorb all shocks and leads to near-depletion when negative shocks predominate.
  - Constant wealth (extreme anchoring) requires government spending to fully neutralize oil price shocks, producing unrealistically high spending volatility and bias toward future generations.

### Rethinking fiscal policy — Policy implications and framework design
- Conceptual framework:
  - Countries must balance an “impossible trinity”: expenditure smoothing, wealth preservation (long-term sustainability), and intergenerational equity cannot all be simultaneously achieved.
  - The price of smoother spending is higher wealth volatility, increasing intergenerational inequity.
  - Intergenerational equity can be pursued via higher wealth stabilization (higher φ) or higher precautionary premium (higher w̅ / w_0).
    - Higher w̅ / w_0 lets current generations enjoy smoother but on-average lower consumption.
    - Lower precautionary premium allows higher current consumption but increases consumption volatility for resource-rich cohorts to stabilize wealth.
  - The chosen long-term fiscal anchor and short-term fiscal strategy are interdependent; this interdependence is absent under the PIH paradigm.
- Fiscal framework design recommendations:
  - Focus fiscal frameworks on wealth rather than expenditure targets:
    - Expenditure benchmarks may be misleading if not derived from a strategy to achieve targeted wealth.
    - PIH expenditure benchmark pegs expenditure to past wealth, is forward-poor, volatile, and non-stationary—unsuitable as a long-term anchor.
  - Wealth stabilization implies some procyclicality of spending relative to the oil price:
    - Negative shocks require lower consumption to recover wealth; countercyclical consumption (saving more in booms) produces more procyclical and volatile wealth, increasing intergenerational inequity.
    - Countercyclical short-term policies must be weighed against long-term intergenerational equity and sustainability benefits of procyclicality.
- Fiscal targets:
  - Post-oil wealth target is a strong long-term fiscal anchor: forward-looking, facilitates long-term sustainability, and binds the rest of the fiscal framework by implying the precautionary premium (degree of prudence).
  - Medium-term benchmarks should aim at raising wealth:
    - Under uncertainty, policies should target a higher level of wealth rather than simply maintaining wealth constant.
    - Wealth should increase in a concave fashion over time because early generations bear most precautionary saving burdens.
    - An example medium-term wealth target path is depicted (Figure 10, panel 10.3) derived from a simple policy that incorporates these principles.

### Policy rule illustration
- The text proceeds to introduce a Prudent Wealth Stabilization (PWS) rule (section C) as a practical policy incorporating prudence and anchoring principles (material beyond section header not included in supplied content).

*Source: IMF working paper content unit wpiea2019108 (chapter/section provided).*

### 40.      Translating the intuition and insights developed above into policy is complicated by

### wpiea2019108 - 40.      Translating the intuition and insights developed above into policy is complicated by

### Proposed policy: Prudent Wealth Stabilization (PWS)
- PWS rule (equation (14)):
  - 푔ₜ = 푓ₜ + α(푤ₜ₋₁ − γ푤₀)
  - Under the assumption that short-term policy pursues consumption smoothing (as under PIH):  
    푔ₜ = 푟푤ₜ₋₁ + α(푤ₜ₋₁ − γ푤₀)
- Interpretation of terms:
  - 푓ₜ: captures short-term objectives (macroeconomic stabilization, competitiveness, non-energy growth); set to 푟푤ₜ₋₁ in the discussion to reflect consumption smoothing.
  - α (Wealth stabilization factor), α>0: government spending decreases (saving increases) when wealth is below target γ푤₀; induces wealth to gravitate to the equilibrium (wealth-stabilizing).
  - γ (Precautionary premium): embeds risk compensation in the post-oil wealth target; when risk assumed by oil-extracting generations does not exceed that of non-oil generations, γ>1 (prudence).

### Calibration and implementation guidance
- Calibration steps:
  - Use the average level of government spending relative to wealth as a scale for α.
  - For countries close to PIH behavior, α = 푟 is a reasonable initial ballpark.
  - Simulate the rule to obtain post-oil wealth for different γ values.
  - Impose intergenerational equity approximately by finding γ satisfying the optimality condition (11): U′(푔₀) = 훽(1+푟)E₀U′(푔_T), where the right-hand side is computed from simulations.
    - Note: This enforces equity at two end points (initial and post-oil) and does not ensure strict equity across every generation during oil extraction.
- Volatility tradeoff characterization:
  - For any realistic α, there is a unique γ that ensures intergenerational equity between the initial and post-oil generations.
  - Weak wealth stabilization (low α) requires larger γ.
  - As α increases (stronger stabilization), required γ declines.

### Numerical calibration examples and baseline
- When α approached about 0.1, government expenditure became nearly as volatile as the oil price.
- Baseline setting in the paper: α = 0.03 and γ = 1.05.
  - γ = 1.05 implies additional wealth accumulation of $173 billion during the years of oil extraction (45 percent of UAE’s 2017 GDP).

### Performance versus PIH (key findings from simulations)
- PWS replicates key desired features: prudence, anchoring, and balancing the volatility tradeoff while retaining PIH’s simplicity.
- Compared to PIH:
  - Targeting a precautionary premium leads initial generations to save more and allows higher average consumption for future generations.
  - PWS ensures stationarity of wealth despite negative oil price shocks that cause precipitous declines under PIH.
  - Post-oil wealth under PWS is both higher and more certain than under PIH (empirical distribution from 100,000 oil price simulations).
- Statistical properties highlighted (from 100,000 Monte Carlo simulations):
  - Under PIH: ex ante expectation of wealth and government expenditure is constant but volatility rises over time (non-stationarity and intergenerational inequity).
  - PWS remedies this by:
    - Prudence: α(푤ₜ₋₁ − γ푤₀) induces a rising trend in wealth as it converges gradually and asymptotically to the target; most convergence is achieved early.
      - Consequence: initial generations bear the burden of precautionary savings (lower expected consumption than under PIH).
    - Anchoring: negative shocks widen the gap (푤ₜ₋₁ − γ푤₀), inducing additional saving and reducing wealth volatility (coefficient of variation) relative to PIH and making it more stable across generations.
      - Short-term government spending volatility is higher under PWS than under PIH due to wealth-stabilizing responses, but long-term spending volatility is lower because of stationarity.

### Policy tradeoffs and practical considerations
- Choosing α and γ involves balancing:
  - Volatility of government expenditure (short-term smoothing costs) versus volatility of wealth (intergenerational risk).
  - Preferences over future generations’ welfare and risk aversion.
- Calibration can aim to minimize costs of short-term expenditure volatility (e.g., destabilizing effects on non-oil growth); requires explicit modeling of those linkages.
- Time inconsistency and target revisions:
  - Target for post-oil wealth could be periodically revised for structural changes (e.g., revisions to oil reserves or stochastic properties of oil price).
  - Revising the long-term anchor in response to transitory developments (e.g., an oil price shock in period t) risks time inconsistency and volatility of the anchor.
  - Without a fixed long-term anchor, recursive reoptimization is likely to produce non-stationarity.
  - Anchoring the framework with a fixed post-oil target provides discipline analogous to numerical inflation targets: allows gradual deviations but imposes a forward-looking commitment attractive from the timeless perspective.
- Alternative stationarity devices to investigate:
  - Asymmetric targets, imposing a floor on wealth (푤ₜ ≥ 푤̲, ∀t), akin to ceilings in debt sustainability frameworks.

### Public investment and wealth accounting implications
- Incorporating public investment:
  - Divides saving between financial and physical assets; physical capital can earn a higher average return.
  - Intra-temporal portfolio choice determines allocation between physical capital and financial wealth; under diminishing returns, marginal product net of depreciation equals opportunity cost.
  - Inter-temporal allocation of consumption remains governed by the consumption-Euler equation (11) and remains independent of wealth; higher average return raises average consumption level across all generations but does not alter intergenerational distribution.
  - Wealth remains non-stationary even with physical and financial investments; equilibrium overall wealth level remains indeterminate.
- Interpretation of public investment:
  - Fiscal accounting treats public investment as “spending” (reduces budget and current account balances).
  - Analytical wealth-accounting interpretation treats public investment as “saving” (transformation of oil revenue into physical capital).
  - Proper wealth accounting that includes physical and financial assets shows public investment changes composition of wealth but does not amount to frontloading spending of wealth when interpreted analytically.
- Metrics for long-term fiscal analysis:
  - Non-resource primary budget balance is useful for short-term aggregate demand analysis but inadequate for intergenerational equity and sustainability.
  - The non-resource primary current balance is more relevant for long-term issues as it distinguishes spending of wealth (current expenditure) from saving of wealth (public investment).

### Concluding insights and research directions
- Pegging expenditure to a notional return on past wealth (PIH) does not simultaneously deliver short-term smoothing and the long-term goals of intergenerational equity and sustainability.
- Achieving these goals requires greater prudence and better anchoring in fiscal frameworks; explicit long-term wealth targets and policies like PWS help operationalize this.
- Short- and medium-term policies should be consistent with a chosen long-term wealth target.
- More research is needed on:
  - Modeling linkages between short-term expenditure volatility and non-oil growth to guide α–γ choices.
  - Time-consistent frameworks or overlapping generations approaches to reconcile stationarity with long-term optimality.
  - Alternative instruments to induce stationarity and target wealth (e.g., asymmetric targets, floors on wealth).

*Source: wpiea2019108 - PDF chapter content provided*

### 57.      Future research could also explore the implications of other sources of intertemporal

### Future research could also explore the implications of other sources of intertemporal

### Research directions and intergenerational considerations
- Future research could explore implications of other sources of intertemporal risk, including:
  - the uncertain size of oil reserves;
  - the impact of improvements in energy efficiency on long-run oil demand;
  - productivity shocks and interest rate shocks.
- The impact of productivity and interest rate shocks on wealth rises as oil exhaustion nears, while the importance of oil price shocks wanes.
- Exhaustibility of resources increases the premium on policies robust to various sources of uncertainty (see Brock and Hansen, 2018).
- Attention to intergenerational considerations is expected to grow with rising natural resource scarcity and continued climate change.
- More work is needed toward a better concept of intergenerational equity and balancing contemporaries’ interests with those of future generations.

### Policy implications and framework
- Long-term considerations should inform other policy aspects; a holistic policy framework is needed that balances:
  - short-term macroeconomic stability objectives;
  - goals of intergenerational equity;
  - long-term sustainability.
- Commonly used fiscal rules (e.g., limits on non-oil primary balances) are poorly equipped to encapsulate these considerations.
- A richer modeling framework is required with:
  - a role for public investment;
  - capture of key linkages between non-oil growth, oil prices, wealth, and fiscal policy.
- External sustainability frameworks for commodity-exporting countries that rely on the PIH suffer from the same issues highlighted in this paper and warrant follow-up studies.

### Appendix I — Assumptions and calibration (UAE calibration)
- Oil price process (AR(1) in logs): log(pt) = μ + ρ log(pt−1) + εt, estimated using WEO annual oil price data (simple average of Brent, WTI, Dubai Fateh) from 1980 to 2017 deflated by US inflation (2017 base year), yielding:
  - μ = .5625
  - ρ = .8520
  - σ = .2416
  - long-term average oil price = $49.81 per barrel
- Oil wealth and output assumptions:
  - UAE oil reserves = 97.8 billion barrels
  - Oil extraction = 3 million barrels per day
  - Annual output (q) = 1.095 billion barrels
  - Time until resource exhaustion (T) = 89 years
  - Initial valuation of subsoil oil reserves (evaluated at long-term average oil price) = $1.5 trillion
- Financial wealth:
  - Initial stock of financial wealth (a0) = $638 billion (reflecting ADIA and Mubadala net of debt)
- Non-oil revenue:
  - Annual non-oil revenue (yn) = $40 billion (consolidated general government, 2017, excluding sovereign fund dividends)
- Total initial wealth:
  - w0 = $3.5 trillion
- Utility function and preference:
  - Utility: U(gt) = gt^(1−η) / (1−η)
  - η = 3 (coefficient of relative risk aversion)

### Appendix II — Non-stationarity under PIH: illustrative example (USD billion)
- Baseline setup:
  - No non-oil revenue
  - Initial financial assets (a0) = $1 trillion
  - Exports = 1 billion barrels of oil at $50/barrel
  - Financial savings earn dividend income at a fixed interest rate of 3 percent
  - Resource horizon = 31 years
  - Valuation of initial subsoil wealth (Q0) = $1 trillion
  - Total initial wealth (w0) = $2 trillion
- Scenario A — Certainty:
  - In period 1, present values of subsoil and financial components augmented by rate of return ($30 billion each)
  - Value of extracted oil ($50 billion) subtracted from subsoil wealth and added to financial savings
  - PIH-based consumption = 3 percent of initial wealth = ($60 billion)
  - Result: zero net investment maintains wealth at $2 trillion
- Scenario B — One-time oil price decline to $40/barrel in period 2 onward after first-period extraction:
  - Immediate revenue shortfall: ∆w_t = ∆a_t = ∆y_t = −$10 billion relative to scenario A
  - Second-round effect next period: r∆a_t = −$0.3 billion
  - PIH adjustment offsets second-round effect: ∆g_t = r∆w_t = −$0.3 billion
  - The first-round effect is permanently preserved in lower level of wealth despite price recovery
- Scenario C — Permanent oil price decline by $10/barrel:
  - Expected oil revenue lowered by $10 billion in all future periods
  - Present value decline in subsoil wealth: ∆Q = −∑10/(1+0.03)^i_{i=1}^{30} = −$196 billion
  - Total wealth decline: ∆w_t = ∆a_t + ∆Q_t = −$206 billion relative to baseline
  - Each period’s revenue shortfall produces a permanently lower interest income of $0.3 (per period)
  - Cumulative second-round effect present value: ∑0.3/(1+0.03)^i_{i=0}^{30} (interpreted as cumulative second-round effect)
  - PIH expenditure adjusts to offset second-round effect: ∆g_t = r∆w_t = −∑0.3/(1+r)^i_{i=0}^{30} = $6.2 billion
  - Wealth remains permanently lower at $1,794 billion

### Appendix III — Procedure for optimal policy solution and numerical implementation
- Post-oil deterministic optimal policy (PIH): for t > T,
  - g_{T+i}^* = r a_{T+i−1}
- After omitting non-oil revenue, state variable evolves: a_{T+i} = a_{T+i−1}
- Solve for g_T^* given g_{T+1}^* using Euler equation:
  - u'(g_T^*) = E_T[u'(r a_T)] = u'(r a_T)
  - Substituting g_T^* = r a_T into budget constraint yields:
    - g_T^* = r/(1+r) [ (1+r) a_{T−1} + q p_T ]
- Numerical solution using Endogenous Grid Method:
  1. Oil price shocks: discretize AR(1) oil price process using Tauchen (1986)
     - 25 grid points
     - largest shocks = 3 standard deviations from mean
     - produces 25 by 25 transition matrix and 25 by 1 oil price grid
  2. Financial asset grid:
     - period-T financial asset (a_{T−1}) grid uses 10,000 points
     - grid range must cover all possible outcomes while keeping endogenous grids bounded
     - calibration-bound examples for prudent vs wealth-anchoring policies:
       - lower bound = −1.3 trillion USD (both)
       - upper bound = 14.5 trillion USD (prudent policy)
       - upper bound = 6.3 trillion USD (wealth-anchoring policy)
  3. Policy Rule Iteration:
     - For each duplet (a_{T−1,i}, p_j) solve period-T spending G_T^*(a_{T−1,i}, p_j)
     - Use Euler equation to compute G_{T−1}^*(a_{T−1,i}, p_k) via expectation across p_j with transition probabilities P(p_T = p_j | p_{T−1} = p_k) (j = 1,..,25)
     - Compute a_{T−2}^*(a_{T−1,i}, p_k) from budget constraint and obtain 25 endogenous period T−1 asset grids (one per oil price state)
     - Unify endogenous asset grids across oil price shocks by setting common lower and upper bounds:
       - min a_{T−2}^* = max_{a_{T−1,i}}( min_{p_k} a_{T−2}^*(a_{T−1,i}, p_k) )
       - max a_{T−2}^* = min_{a_{T−1,i}}( max_{p_k} a_{T−2}^*(a_{T−1,i}, p_k) )
     - Create one 1 by 10,000 equally spaced asset grid with a_{T−2,1} = min a_{T−2}^* and a_{T−2,10,000} = max a_{T−2}^*
     - Interpolate optimal government spending policy on this grid for each shock state to build a 25 by 10,000 matrix G_t(a_{t−1,i}, p_k)
     - Repeat backward for periods T−2, T−3, ..., until period 1 using analogous expectation and budget constraint formulas:
       - G_t^*(a_{t,i}, p_k) = { ∑ [ G_{t+1}^*(a_{t,i}, p_j) ]^{−η} P(p_{t+1}=p_j | p_t=p_k) }^{−1/η}
       - a_{t−1}^*(a_{t,i}, p_k) = a_{t,i} − y p_k + { ∑ [ G_{t+1}^*(a_{t,i}, p_j) ]^{−η} P(p_{t+1}=p_j | p_t=p_k) }^{−1/η} / (1+r)
     - Unify endogenous asset grids at each step using the analogous min/max expressions
     - Final check: ensure calibrated initial asset lies on the period 1 asset grid

### Appendix IV — Setting with public investment
- Modify non-oil output to be endogenous and partially determined by public capital k_{t−1}:
  - y_t^n = A k_{t−1}^η, η ∈ (0,1)
  - A is a fixed scale parameter (labor and private capital held fixed)
- Public capital accumulation:
  - k_t = (1−δ) k_{t−1} + I_t
- Financial capital evolution now reflects public investment:
  - a_t = (1+r) a_{t−1} + y_t + y_t^n − g_t − I_t
- Oil wealth evolution unchanged:
  - Q_t = (1+r) Q_{t−1} − E_{t−1} y_t + φ_t
- Total wealth redefined:
  - w_t = k_t + a_t + Q_t
- Law of motion for total wealth:
  - w_t = (1+r) w_{t−1} − (r+δ) k_{t−1} + A k_{t−1}^η − g_t + φ̃_t
  - Additional terms reflect gross return on public capital (non-oil output) and opportunity cost of capital
- Optimality conditions from maximization:
  - U'(g_t) = β (1+r) E_t U'(g_{t+1})  (identical to prior Euler condition)
  - η A k_t^{η−1} = r + δ  (optimal public capital equates marginal product to opportunity cost)
- Implications:
  - If oil price is only uncertainty, optimal public capital k_t = k (constant and independent of oil price)
  - Rewriting law of motion using optimal capital yields:
    - w_t = (1+r) w_{t−1} + (r+δ)^{(1−η)/η} k − g_t + φ̃_t
  - The concave production function generates an excess average return on capital relative to other wealth components:
    - average return on capital y^n/k = (r+δ)/η
    - excess return (less depreciation) over average return on other wealth components appears as the additional term
  - Some oil wealth is converted into physical (public) capital rather than financial capital
  - PIH benchmark (maintaining wealth constant in absence of uncertainty φ̃_t = 0) must be revised to account for excess return:
    - g_t = r w_{t−1} + (r+δ)^{(1−η)/η} k
  - Wealth continues to follow a random walk w_t = w_{t−1} + φ̃_t unless all oil savings are invested in physical capital (Hartwick’s framework)

*Source: IMF Working Paper (excerpts from appendices and concluding paragraphs).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019108.pdf_