## 1. Optimal Investment Rates with Different Productivity Parameters

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### Introduction and background
- Study period and data: averages and estimates based on 1970-2008.
- Policy question: how oil exporters should allocate volatile tradable income among safe liquid assets, domestic investment, and consumption over a long horizon given volatile commodity (especially oil) revenues.
- Empirical patterns:
  - Large oil exporters face high income volatility, sizable saving, but relatively low investment.
  - In the late 1970s (high real oil price) oil exporters on average invested about 30 percent of GDP; in the 2000s investment fell to about 20 percent of GDP.
  - Current accounts and foreign reserve buildups fluctuated; broadly balanced in the 1990s and large surpluses in the 2000s.

### Model framework (the "silo" model)
- Key model features:
  - Two goods: tradable (X) and nontradable (Z).
  - Tradable output Yt = Pt εt with permanent component Pt and temporary shock εt (log-normal i.i.d.).
  - Permanent component evolves: P_{t+1} = (1 + τ ξ) P_t η_{t+1}, with τ = constant investment rate (share of permanent tradable output) and ξ a productivity parameter; η is a log-normal i.i.d. permanent shock.
  - Nontradable output follows deterministic process Ỹ_{t+1} = (1 + τ ξ̃) Ỹ_t when linked to tradable investment, with ξ̃ and τ̃ generally different from ξ and τ.
  - Two assets: a safe asset (e.g., sovereign wealth fund) and risky capital (investment irreversible and risky).
  - Social planner interpretation: maximize expected separable utility over T periods with CRRA preferences, relative risk aversion ρ, discount factor β, and relative weight α on tradable consumption.
  - Budget constraint (normalized variables): W_{t+1} = (1 + r) W_t + Y_t − τ P_t − X_t, with nontradable production clearing Z_t = Ỹ_t and tradable good numeraire.
  - Solution via Bellman equation normalized by permanent output and numerical solution using the endogenous grid method.

### Calibration and stochastic specification
- Preferences and discounting:
  - Coefficient of relative risk aversion ρ = 2.
  - Discount rate = 4 percent.
- Technology and productivity:
  - Baseline ξ and ξ̃ set to 0.1, implying that a country with an investment rate of 20 percent would grow on average at 2 percent per year.
- Shocks and volatility:
  - Permanent and temporary shocks assumed unit-mean log-normal.
  - Calibrated standard deviations of permanent and temporary shocks for tradable output averaged over a group of nine oil exporters: (0.23, 0.04). Most income fluctuations come from permanent shocks.
  - Rare disaster assumption: probability of 1.7 percent of a temporary production drop of 30 percent.
- Other parameters:
  - Initial wealth assumed equal to zero (baseline).
  - Initial tradable and nontradable output normalized to 1.
  - Time horizon in baseline: 50 years.
  - Return on wealth r assumed zero for normalization.

### Main quantitative results and scenarios
- Baseline optimal allocations (excluding nontradable production link):
  - Precautionary saving (buffer-stock in safe liquid assets) ≈ 30 percent of initial income.
  - Golden-rule optimal investment rate ≈ 15 percent of permanent income.
  - Initial consumption ≈ 55 percent of initial income.
  - SWF (planner’s “cash-on-hand”) in the optimal lower-investment case reaches a maximum of about 4.5 times the level of initial income.
- Tradeoffs and comparative dynamics:
  - At a higher investment rate than the optimal (example: 20 percent vs. optimal 15 percent), average consumption is higher but consumption volatility and confidence bands are larger.
- Time-horizon and initial-wealth sensitivity:
  - Time horizon = 60 years → optimal investment rate rises to 20 percent (from 15 percent at baseline 50 years).
  - Time horizon = 40 years → optimal investment rate falls to 9 percent.
  - Initial wealth = initial income → investment rate increases to 16 percent.
  - Initial wealth = 3 × initial income → investment rate increases to 20 percent.
  - Buffer-stock saving declines with higher initial wealth:
    - Baseline buffer-stock ≈ 30 percent of initial income.
    - With initial wealth = initial income, buffer-stock ≈ 20 percent of initial income.
    - With initial wealth = 3 × initial income, buffer-stock ≈ 5 percent of initial income and initial consumption increases to about 75 percent of initial income.
- Linking tradable and nontradable production (τ̃ = τ):
  - Optimal investment rate depends primarily on investment productivity ξ.
  - With productivity parameters at 0.1 and nontradable production affected by tradable investment, optimal investment rate increases from 15 percent to 47 percent of permanent output.
  - Halving the productivity parameter to 0.05 reduces optimal investment rate to 12 percent.
  - Even with an optimal investment rate of 47 percent, buffer-stock saving remains sizable, about 10 percent of income in the initial period.
  - Varying nontradable-sector productivity has limited effect on optimal investment relative to changes in tradable productivity.
- Comparison with perfect-foresight predictions:
  - Perfect foresight model predicts large borrowing rather than saving; precautionary-saving model predicts substantial initial accumulation of safe assets and lower initial consumption.

### Marginal propensity to consume (MPC) results
- MPC definitions:
  - MPC out of wealth (and temporary shocks/windfalls): change in optimal consumption at the end of the time horizon due to a small change in wealth (averaged across 5000 simulations).
  - MPC out of permanent shocks: computed separately (full numerical value not included in the supplied excerpt).
- Baseline numerical MPC estimates:
  - MPC out of wealth (baseline calibration) = 0.07.
  - With initial wealth = initial income, MPC out of wealth = 0.08.
  - With initial wealth = 3 × initial income, MPC out of wealth = 0.13.
  - Perfect foresight model MPC = 0.03 (reference value).
- Interpretation:
  - Under precautionary saving, consumption is expected to increase by 7 cents rather than 3 cents if income is raised by a dollar for one period in the baseline calibration.

### Model evaluation against data: aggregate patterns for oil exporters
- Data basis: government revenue, public investment, and government spending for a large group of oil exporters (Algeria, Angola, Azerbaijan, Bahrain, Ecuador, Equatorial Guinea, Gabon, Iran, Iraq, Kazakhstan, Libya, Nigeria, Norway, Oman, Qatar, Russia, Saudi Arabia, Turkmenistan, UAE, Venezuela, and Yemen). Data source: World Economic Outlook database.
- Empirical observations aligned with the model:
  - The investment rate is about 25 percent (noted as somewhat higher than the optimal investment rate with lower productivity).
  - Consumption (computed as public expenditure less investment) amounts to about 50-60 percent of total income before 2009, corresponding to the model’s optimal initial consumption level.
  - The average growth rates of income and consumption are similar, consistent with the precautionary saving model.
- Key empirical statistics (selected table excerpt):
  - Consumption/Income:
    - 2002-2006 (average): 0.60
    - 2007: 0.55
    - 2008: 0.52
    - 2009: 0.72
  - Investment/Income:
    - 2002-2006 (average): 0.25
    - 2007: 0.26
    - 2008: 0.25
    - 2009: 0.36
  - ΔConsumption/Consumption:
    - 2002-2006 (average): 0.21
    - 2007: 0.24
    - 2008: 0.41
    - 2009: 0.03
  - ΔIncome/Income:
    - 2002-2006 (average): 0.29
    - 2007: 0.14
    - 2008: 0.43
    - 2009: -0.25
  - MPC Average:
    - 2002-2006 (average): 0.33
    - 2007: -0.03
    - 2008: 0.51
    - 2009: -0.10
  - MPC Median:
    - 2002-2006 (average): 0.62
    - 2007: 0.52
    - 2008: 0.45
    - 2009: 0.00
- Empirical interpretation:
  - Median MPC before the global output collapse in 2009 is in the range of about 0.5-0.6, consistent with the MPC out of permanent shocks in the precautionary saving model.
  - In 2009, the median MPC fell to 0.00 and the average MPC was -0.10, suggesting policymakers treated the 2009 shock as temporary and used buffer-stock savings to smooth consumption.
  - Large variance across oil exporters in reactions to income changes (average versus median MPC differences).

### Conclusions and policy implications
- Theoretical conclusions:
  - Precautionary saving should be sizable, about 30 percent of initial income, when faced with high permanent shocks to oil income.
  - Tradable-sector productivity crucially determines precautionary saving and optimal investment; higher tradable productivity can justify much larger investment rates and reduce buffer-stock needs.
  - If tradable-sector productivity is high enough, the investment rate can increase substantially from about 15-20 percent to about 50 percent of initial income.
  - Productivity in the nontradable sector is not important for aggregate saving and investment dynamics.
- Policy recommendations:
  - If productivity in the tradable sector is low:
    - Build up sizable buffer-stock savings of safe and liquid assets to mitigate negative persistent income shocks.
    - Accept a growth-risk tradeoff that calls for relatively lower optimal investment.
  - If tradable productivity is high:
    - Lower buffer-stock savings and higher investment would be optimal.
  - Spending policy should be conservative:
    - Optimal MPC out of permanent shocks is below one.
    - MPC out of temporary shocks is much lower.
  - Policy focus: improve productivity in the tradable sector and reduce volatility through developing and diversifying this sector to lower precautionary saving needs, increase investment, raise consumption, and improve utility.
- Additional interpretations:
  - Large current account surpluses in the 2000s are consistent with high volatility of permanent shocks and low tradable productivity.
  - Buffer-stock savings accumulated before the global financial crisis enabled more consumption smoothing when income collapsed in 2009.

### Empirical caveats and avenues for future research
- Caveats:
  - The model is not expected to reproduce oil exporters’ behavior exactly due to other factors (including political economy developments) affecting allocation decisions.
  - Oil exporters may not know ex-ante the exact nature of an income shock and must respond based on available information and estimated parameters at the time.
- Suggested future research:
  - Study whether differences in consumption, saving, and investment patterns across oil exporters can be explained by different levels of initial wealth, productivity, and volatility.

*Source: _wp1204 — 1. Optimal Investment Rates with Different Productivity Parameters — IMF working paper excerpt provided in the supplied PDF.*

### 1. Optimal Investment Rates with Different Productivity Parameters ....................................14

### 1. Optimal Investment Rates with Different Productivity Parameters

### Sections and page references
- 1. Optimal Investment Rates with Different Productivity Parameters ....................................14
- 2. Statistics Based on Government Revenue, Expenditure, and Public Investment ................16
- References ................................................................................................................................19

### Figures listed in the content unit
- Figure 1. Volatility and Saving (1970-2008 Averages) ........................................................................3
- Figure 2. Investment and Saving (1970-2008 Averages) ......................................................................4
- Figure 3. Investment, Saving, and Real Oil Price for a Group of Oil Exporters ...................................4
- Figure 4. Simulated Time Paths of Average Consumption and Income with 80 Percent Confidence Bands and Perfect Foresight Consumption at Average Income ..........................................12
- Figure 5. Consumption and "Cash-on-Hand" (SWF) at Optimal (15 Percent) and Higher (20 Percent) Investment Rates .............................................................................................13

### Key themes implied by headings and figure titles
- Comparative analysis of optimal investment rates under varying productivity parameters (section heading indicates focus on optimal investment rates).
- Use of historical averages (1970-2008) to examine volatility, saving, and investment relationships (Figures 1 and 2).
- Interaction between investment, saving, and real oil prices for oil-exporting countries (Figure 3).
- Simulation-based assessment of consumption and income dynamics, including uncertainty quantification using 80 percent confidence bands and perfect-foresight benchmarks (Figure 4).
- Evaluation of consumption and sovereign wealth fund ("Cash-on-Hand (SWF)") outcomes at an optimal investment rate of 15 percent and at a higher investment rate of 20 percent (Figure 5).
- Descriptive and statistical treatment of government revenue, expenditure, and public investment (Section 2).

### Document structure cues
- Main analytical content begins at page 14 with the principal section on optimal investment rates.
- Supporting empirical statistics and government fiscal details commence at page 16.
- Visual and simulation evidence spans pages 3, 4, 12, and 13 as indicated by figure placements.
- References located on page 19.

*Source: _wp1204 - 1. Optimal Investment Rates with Different Productivity Parameters ....................................14 — https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1204.pdf*

### Appendix Table

### Appendix Table

### Introduction and background
- Study period and data: averages and estimates based on 1970-2008.
- Policy question: how oil exporters should allocate volatile tradable income among safe liquid assets, domestic investment, and consumption over a long horizon given volatile commodity (especially oil) revenues.
- Empirical patterns highlighted:
  - Large oil exporters face high income volatility, sizable saving, but relatively low investment.
  - In the late 1970s (high real oil price) oil exporters on average invested about 30 percent of GDP; in the 2000s investment fell to about 20 percent of GDP.
  - Current accounts and foreign reserve buildups fluctuated; broadly balanced in the 1990s and large surpluses in the 2000s.

### Model framework (the "silo" model)
- Key features:
  - Two goods: tradable (X) and nontradable (Z).
  - Tradable output Yt = Pt εt with permanent component Pt and temporary shock εt (log-normal i.i.d.).
  - Permanent component evolves: P_{t+1} = (1 + τ ξ) P_t η_{t+1}, with τ = constant investment rate (share of permanent tradable output) and ξ a productivity parameter; η is a log-normal i.i.d. permanent shock.
  - Nontradable output follows deterministic process Ỹ_{t+1} = (1 + τ ξ̃) Ỹ_t when linked to tradable investment, with ξ̃ and τ̃ generally different from ξ and τ.
  - Two assets: a safe asset (e.g., sovereign wealth fund) and risky capital (investment irreversible and risky).
  - Social planner interpretation: maximize expected separable utility over T periods with CRRA preferences, relative risk aversion ρ, discount factor β, and relative weight α on tradable consumption.
  - Budget constraint (normalized variables): W_{t+1} = (1 + r) W_t + Y_t − τ P_t − X_t, with nontradable production clearing Z_t = Ỹ_t and tradable good numeraire.
  - Solution uses Bellman equation normalized by permanent output and numerical solution via the endogenous grid method.

### Calibration and stochastic specification
- Preferences and discounting:
  - Coefficient of relative risk aversion ρ = 2.
  - Discount rate = 4 percent.
- Technology and productivity:
  - Baseline ξ and ξ̃ set to 0.1, implying that a country with an investment rate of 20 percent would grow on average at 2 percent per year.
- Shocks and volatility:
  - Permanent and temporary shocks assumed unit-mean log-normal.
  - Calibrated standard deviations of permanent and temporary shocks for tradable output (exports proxied in constant USD) averaged over a group of nine oil exporters: (0.23, 0.04). Most income fluctuations come from permanent shocks.
  - Rare disaster assumption: probability of 1.7 percent of a temporary production drop of 30 percent.
- Other parameters:
  - Initial wealth assumed equal to zero (baseline).
  - Initial tradable and nontradable output normalized to 1.
  - Time horizon in baseline: 50 years.
  - Return on wealth r assumed zero for normalization.

### Main quantitative results and scenarios
- Baseline optimal allocations (excluding nontradable production link):
  - Precautionary saving (buffer-stock in safe liquid assets) ≈ 30 percent of initial income.
  - Golden-rule optimal investment rate ≈ 15 percent of permanent income.
  - Initial consumption ≈ 55 percent of initial income (much less than initial income).
  - Under perfect foresight life-cycle model, policymakers would borrow substantially; under uncertainty, policymakers build buffer-stock savings instead.
  - SWF (planner’s “cash-on-hand”) in the optimal lower-investment case reaches a maximum of about 4.5 times the level of initial income.
- Tradeoff between investment and buffer-stock savings:
  - At a higher investment rate than the optimal (example: 20 percent vs. optimal 15 percent), average consumption is higher but consumption volatility and confidence bands are larger; planner faces a tradeoff: higher growth/higher volatility versus lower growth/lower volatility.
- Time-horizon and initial-wealth sensitivity:
  - With time horizon = 60 years, optimal investment rate rises to 20 percent (from 15 percent at baseline 50 years).
  - With time horizon = 40 years, optimal investment rate falls to 9 percent.
  - With initial wealth = initial income, investment rate increases to 16 percent.
  - Tripling initial wealth increases investment rate to 20 percent.
  - Buffer-stock saving declines with higher initial wealth:
    - Baseline buffer-stock ≈ 30 percent of initial income.
    - With initial wealth = initial income, buffer-stock ≈ 20 percent of initial income.
    - With initial wealth = 3 × initial income, buffer-stock ≈ 5 percent of initial income and initial consumption increases to about 75 percent of initial income; investment rate goes to 20 percent.
- Linking tradable and nontradable production (τ̃ = τ):
  - Optimal investment rate depends primarily on investment productivity ξ.
  - With productivity parameters at 0.1 and nontradable production affected by tradable investment, optimal investment rate increases from 15 percent to 47 percent of permanent output.
  - Halving the productivity parameter to 0.05 reduces optimal investment rate to 12 percent.
  - Even with an optimal investment rate of 47 percent, buffer-stock saving remains sizable, about 10 percent of income in the initial period.
  - Varying nontradable-sector productivity has limited effect on optimal investment relative to changes in tradable productivity.
- Comparison with perfect-foresight predictions:
  - Perfect foresight model predicts large borrowing rather than saving; precautionary-saving model predicts substantial initial accumulation of safe assets and lower initial consumption.

### Marginal propensity to consume (MPC) results
- MPC definitions used:
  - MPC out of wealth (and temporary shocks/windfalls): change in optimal consumption at the end of the time horizon due to a small change in wealth (averaged across 5000 simulations).
  - MPC out of permanent shocks: computed separately (discussion truncated in supplied content).
- Baseline numerical MPC estimates:
  - MPC out of wealth (baseline calibration) = 0.07.
  - With initial wealth = initial income, MPC out of wealth = 0.08.
  - With initial wealth = 3 × initial income, MPC out of wealth = 0.13.
  - Perfect foresight model MPC = 0.03 (reference value from Carroll, 1997).
  - Interpretation: under precautionary saving, consumption is expected to increase by 7 cents rather than 3 cents if income is raised by a dollar for one period in the baseline calibration.
- MPC out of permanent shocks: the precautionary saving model yields an MPC below one and markedly different from the perfect-foresight MPC (which is above one); full numerical value for MPC out of permanent shocks is not included in the supplied excerpt.

### Key implications and comparative literature notes
- Precautionary saving motives and investment tradeoffs:
  - High volatility of permanent shocks and relatively low productivity of the tradable sector justify sizable precautionary saving (~30 percent) and relatively low optimal investment (~15 percent) in baseline.
  - Investment is risky; higher investment raises expected growth but increases volatility and reduces required buffer-stock savings.
  - Higher tradable productivity can justify much larger investment rates and reduce the need for large buffer-stock savings.
- Relevance to policy debates and literature:
  - Results contrast with perfect-foresight permanent income hypothesis predictions that would suggest low immediate response to windfalls; precautionary saving model implies larger MPC out of windfalls and greater accumulation of foreign assets.
  - Findings align with literature emphasizing volatility’s positive effect on precautionary external saving and argue that with high volatility and low tradable productivity, large investment spending may not be welfare-improving.
  - Model contributes by jointly studying consumption, saving, and investment decisions under a nonstationary income process with permanent and temporary shocks, introducing investment, and computing MPCs numerically over a finite horizon.

*Source: _wp1204 - Appendix Table*

### 0.57 in the baseline case (see Carroll, 2009, for an extensive discussion). A permanent

### _wp1204 - 0.57 in the baseline case (see Carroll, 2009, for an extensive discussion). A permanent

### Precautionary saving mechanism and consumption response
- A permanent increase in income by a dollar does not result in the same or higher increase in consumption as in the perfect foresight model.
- Consumption increases by less in the presence of potential negative income shocks because part of the increase in income is allocated into buffer-stock saving.
- Reference to buffer-stock saving literature: Carroll (1997, 2001, 2009).

### Model evaluation against data: aggregate patterns for oil exporters
- Data basis: government revenue, public investment, and government spending for a large group of oil exporters (Algeria, Angola, Azerbaijan, Bahrain, Ecuador, Equatorial Guinea, Gabon, Iran, Iraq, Kazakhstan, Libya, Nigeria, Norway, Oman, Qatar, Russia, Saudi Arabia, Turkmenistan, UAE, Venezuela, and Yemen). Data source: World Economic Outlook database.
- Empirical observations that align with the precautionary saving model:
  - The investment rate is about 25 percent (noted as somewhat higher than the optimal investment rate with lower productivity).
  - Consumption (computed as public expenditure less investment) amounts to about 50-60 percent of total income before 2009, corresponding to the model’s optimal initial consumption level.
  - The average growth rates of income and consumption are similar, consistent with the precautionary saving model.
- Prudence increased in the latest decade:
  - The median MPC before the global output collapse in 2009 is in the range of about 0.5-0.6, corresponding to the MPC out of permanent shocks in the precautionary saving model but not the perfect foresight model.
  - In 2009, when income fell substantially, consumption increased slightly in many oil exporters to maintain current consumption levels; the median MPC fell to zero, broadly consistent with the MPC out of temporary shocks for both precautionary saving and perfect foresight models.
  - The average MPC in 2009 was slightly negative at -0.1, suggesting spending increased somewhat despite declining income.
- Interpretation:
  - If oil exporters broadly follow the precautionary saving model, MPC values from the model versus data imply that shocks in most of the 2000s were treated as permanent.
  - The 2009 shock appears to have been considered temporary by policymakers, judging by MPC dynamics.
  - There is large variance across oil exporters in reactions to income changes (average versus median MPC differences).

### Key empirical statistics (table excerpt summary)
- Time periods: 2002-2006 (average), 2007, 2008, 2009
- Consumption/Income:
  - 2002-2006 (average): 0.60
  - 2007: 0.55
  - 2008: 0.52
  - 2009: 0.72
- Investment/Income:
  - 2002-2006 (average): 0.25
  - 2007: 0.26
  - 2008: 0.25
  - 2009: 0.36
- ΔConsumption/Consumption:
  - 2002-2006 (average): 0.21
  - 2007: 0.24
  - 2008: 0.41
  - 2009: 0.03
- ΔIncome/Income:
  - 2002-2006 (average): 0.29
  - 2007: 0.14
  - 2008: 0.43
  - 2009: -0.25
- MPC Average:
  - 2002-2006 (average): 0.33
  - 2007: -0.03
  - 2008: 0.51
  - 2009: -0.10
- MPC Median:
  - 2002-2006 (average): 0.62
  - 2007: 0.52
  - 2008: 0.45
  - 2009: 0.00

### Model conclusions and policy implications
- Main theoretical conclusions:
  - Precautionary saving should be sizable, about 30 percent of initial income, when faced with high permanent shocks to oil income.
  - The tradable sector is crucial: tradable volatility determines the level of precautionary saving and investment; productivity of investment in the tradable sector significantly affects optimal investment rates.
  - If tradable-sector productivity is high enough, the investment rate can increase substantially from about 15-20 percent to about 50 percent of initial income.
  - If investment in the tradable sector affects nontradable production, the optimal investment rate is even higher.
  - Productivity in the nontradable sector is not important for aggregate saving and investment dynamics.
  - Improving productivity in the tradable sector is crucial for sustained growth.
- Policy recommendations and implications:
  - If productivity in the tradable sector is low:
    - Build up sizable buffer-stock savings of safe and liquid assets to mitigate negative persistent income shocks.
    - Accept a growth-risk tradeoff that calls for relatively lower optimal investment.
  - If tradable productivity is high:
    - Lower buffer-stock savings and higher investment would be optimal.
  - Spending policy should be conservative:
    - The optimal MPC out of permanent shocks is below one.
    - The MPC out of temporary shocks is much lower.
  - Policy focus should be on improving productivity in the tradable sector and reducing volatility through developing and diversifying this sector to:
    - Lower precautionary/buffer-stock saving needs.
    - Increase investment.
    - Raise consumption.
    - Improve utility.
- Additional model interpretations:
  - Large current account surpluses in the 2000s are consistent with high volatility of permanent shocks to income and low productivity in the tradable sector.
  - Predictions of the perfect foresight model (large borrowing and MPC out of permanent shocks much above one) contrast sharply with observed behavior in the recent decade.
  - Buffer-stock savings accumulated before the global financial crisis enabled more consumption smoothing when income collapsed in 2009.

### Empirical caveats and avenues for future research
- Caveats:
  - The model is not expected to reproduce oil exporters’ behavior exactly due to other factors (including political economy developments) affecting allocation decisions.
  - Oil exporters may not know ex-ante the exact nature of an income shock and must respond based on available information and estimated parameters at the time.
- Suggested future research:
  - Study whether differences in consumption, saving, and investment patterns across oil exporters can be explained by different levels of initial wealth, productivity, and volatility.

*Source: IMF working paper content provided in the supplied PDF excerpt.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1204.pdf_