2.1 Social Welfare Function

Typically a SWF is a function of the instantaneous utility of consumption of current and future generations. When specifying a SWF we specify the relative importance of current and future consumption, and the consumption goods considered in the instantaneous utility. In doing so we set a key ingredient to determine both how much future generations benefit from oil wealth and how much redistribution of private wealth across generations takes place.

2.2 Policy Instruments

A variety of policy instruments may be available to governments when implementing fiscal policies. Savings and debt, taxation, investment, and stabilization funds are among those most relevant for the problem considered in this paper.

2.3 Private Sector

An important issue regarding private sector behavior is whether there is a bequest motive or not. The assumption of no bequest motive (or, more generally, of a weak bequest motive) is implicit in the intergenerational equity question central to this paper, for otherwise no government intervention would be needed to ensure that future generations benefit government owned wealth. If current generations do not care for their descendants, the private sector will not save for future generations and, given the opportunity to do so, will spend all the government owned wealth.⁹

The private sector also participates in the production of goods and services in markets which are assumed competitive. These goods and services may be consumed locally or exported. The private sector also has access to international finance for investment projects within the country.

The private sector also maximizes a welfare function, which even though qualitatively similar to the SWFs considered earlier in this section, may differ in some fundamental ways. An important difference we will encounter in most cases is that the time horizon considered by private agents is considerably shorter than that considered by the government’s SWF. This is due to our assumption that private agents do not want to leave inheritance to their descendants.

We will see in Section 4 that the interaction between the objectives posed by the government’s SWF and the private sector’s behavior may lead to surprising results. For example, if it wishes, the government may use fiscal policy to have the currently alive private sector care for future generations.

3.1 Consumption Smoothing

Individuals dislike variations in consumption and are generally willing to sacrifice some welfare to avoid such fluctuations. For this reason, in the absence of income uncertainty, optimal fiscal policy often requires that per capita consumption levels remain constant over time. With income uncertainty this intuition needs to be modified, and current consumption levels are equal to permanent income, so that, in expectations or on average consumption is constant over time (Friedman [1957]).

For example, after discovering a new exhaustible natural resources, say natural gas in Qatar, consumption should increase by the annuity value of the corresponding increase in wealth. The country acts as if it deposited in a bank abroad the present discounted value of the profits it expects to make from selling the natural resource, and spends every year the interest payments it receives. Consumption should increase immediately after the natural gas is discovered, if the country can borrow against future incomes there is no reason to wait until production begins. Thus the current account deteriorates immediately after the discovery of natural gas and recuperates once actual production begins.

As we shall see in Section 4, the main assumption underlying consumption smoothing in the case without income uncertainty is that βR = 1, where we recall that β denotes the subjective discount rate and R the gross interest rate. Even though it may be argued that in the long run βR will be close to one, in the short and medium run (e.g., over the next couple of decades) there is no reason why this should be the case. If βR < 1, which may be interpreted as society being relatively impatient, per capita consumption falls over time at a constant rate. Alternatively, if βR > 1, per capita consumption grows at a constant rate.

The extension of the consumption smoothing intuition to the case with uncertain income—certainty equivalence—assumes that the instantaneous utility function is quadratic. This assumption is popular precisely because it preserves this intuition, even though it has some unappealing properties, such as a degree of risk aversion that increases with consumption levels and the implication that the optimal consumption path does not depend on the variance of income.

Another intuition that follows from consumption smoothing with uncertain income is that the government should react differently to transitory and permanent changes in income. A transitory positive shock to income should increase consumption only by the annuity value of the income shock, By contrast, a permanent increase should be met by a one-for-one reduction of consumption. For example, the increase in the price of oil following the invasion of Kuwait by Iraq in August of 1990 was clearly transitory. By the time the oil price had returned to its pre-invasion levels (in mid 1991), the rule described above can be used to spend the windfall generated by the price increase.

More generally, if income follows an autoregressive process with first order correlation ψ, which therefore also captures the degree of persistence of income shocks, the fraction of the current shock to income that should be spent is (R − l)/(R − ψ).¹⁰ The case ψ = 0 corresponds to i.i.d. (and therefore transitory) shocks while ψ = 1 corresponds to the case where income follows a random walk (permanent shocks).

In practice it is often not easy to determine the extent to which a change in income is permanent or transitory. Most shocks can be thought of as having both a permanent and a transitory component. In Section 6 we review recent econometric developments that can be used to accomplish this decomposition, concluding that a geometric random walk appears as a sensible description for the oil price.

Furthermore, because oil is an exhaustible resource, even permanent price shocks have only a transitory effect on income. The transitory component of the shock is more important the shorter the expected duration of the resource.

3.2 Precautionary Saving

A fundamental intuition underlying savings behavior is that an increase in risk should increase current savings and decrease current consumption. This is known as the precautionary saving motive, see Leland (1968). The consumption smoothing intuition does not incorporate this idea, since it prescribes that the current annuity value of expected wealth should be spent every year, regardless of the degree of uncertainty associated with this wealth.

To capture the precautionary savings motive, we must consider more realistic instantaneous utility functions than the quadratic case. This typically comes at the price of not having an explicit expressions for optimal consumption,¹¹ and numerical methods must be used to determine the optimal plan (as in Zeldes [1989], Deaton [1991] and Carroll [1992]).

¿From a policy perspective, these numerical procedures have limited applicability. Implementing solutions is cumbersome and the results are not as transparent as the political process requires. In this paper we derive approximations to the optimal consumption plan that are closed-form and can be easily interpreted. Their simplicity should be a great advantage in terms of applicability.

We consider two sources of uncertainty: the usual income uncertainty and what we call budget uncertainty, which attempts to capture the uncertainty that governments face when designing next year’s budget. In particular, we consider the effect of not knowing the income level that will prevail during the coming year. This type of uncertainty is different from the one that originates the standard precautionary savings because it focuses on the level of prices only one period ahead.¹²

3.3 Adjustment Costs

In the presence of adjustment costs as those described in section 2.1.3 (convex adjustment costs) governments typically adjust their per capita expenditures slower than they would in the absence of such costs. For example, following the discovery of gas reserves, the government should increase its spending on the public good only slowly until it achieves its new and, in the absence of income fluctuations, constant level. The larger the adjustment costs, the slower the process by which consumption increases and the higher the steady state level of consumption.

Below we derive a closed-form solution for a partial adjustment model in which the adjustment coefficient is a function of the size of the adjustment cost (that could be asymmetric). Moreover, we present a procedure by which this adjustment cost can be approximated.

3.4 Separability of the Investment Problem

Under the assumptions we made for the private sector, namely that there are no constraints to international borrowing, we have that all projects with positive net present value can and will be financed. Of course, this result stops holding, say, when moral hazard or adverse selection problems limit the availability of credit for local entrepreneurs. If the government faces fewer informational asymmetries than international lenders, there may be a role for government support of investment projects.

3.5 Tax Smoothing

In a fundamental result, Barro (1974) provided conditions under which the optimal consumption path does not depend on how the government finances its expenditures (debt vs. taxes). This result is known as Ricardian equivalence. When taxes are distorting, Ricardian equivalence docs not hold and all sources of finance should be used in such a way that the marginal distortion they introduce is the same over time and across financing instruments. This result is referred to as “tax-smoothing”, see, for example, Barro (1979).

4.1 Benchmark Model

The following model will be a useful benchmark throughout this section.

1. Social welfare function: Utilitarian with constant elasticity of substitution across time (1/ρ). The initial population is normalized to one and grows at a constant rate n. The time horizon is infinite and there is no income uncertainty. Then (4) becomes:

   $\begin{array}{ccc}u={\displaystyle \sum _{t=0}^{\infty }{\beta }^t{(1+n)}^t{u}_t^{1-\rho }.}& & (14)\end{array}$

   The instantaneous utility has consumption of the public and private goods as separate arguments and the elasticity of substitution between both consumption goods is constant (1/γ) as in (2).

2. Policy Instruments: The government is the only provider of the public good, which it finances with taxes, debt and proceeds from the sales of the government owned natural resource (oil in what follows). Oil income in period t is denoted by Y_G,t; it is known with certainty and determined exogenously.

   The government collects taxes and makes transfers to the private sector without generating any distortions in doing so. The government may also save and borrow at the international gross rate R. The only constraint it faces in setting taxes and borrowing is its intertemporal budget constraint (13). Initially it holds financial assets equal to F_G,0.

3. Private Sector:

   Consumers live for one period and have no bequest motive; it follows that the private sector holds no assets or debt. Private sector production in period t is exogenous and equal to Υ_P,t.

Total production in period t is denoted by Y_t and equal to Y_G,t + Y_P,t. in general, we denote aggregate variables by upper case letters, and per capita variables by lower case letters.

Constant elasticity of substitution between both goods implies that in the solution to the problem posed above the ratio of their consumption levels remains constant over time (see Lemma A.1 in the Appendix):

Denoting

we have that u_t is proportional to c_t (Lemma A.1), so that we may write (14) as:

We denote society’s initial wealth by:

We define:

and assume $\tilde{\alpha }<1$.

In the Appendix (Proposition A.1) we show that the solution to this problem is given by:

If βR = 1, the right hand side of (18) is society’s permanent (total) income (Friedman, [1957]), that is, it is the highest per capita consumption level that can be maintained indefinitely.

Equation (15) determines how c_t is split between consumption of the private and public good, thereby determining government expenditures.

The evolution of total financial assets can be determined as follows: F₁ is calculated using the dynamic budget constraint (11), the expression for C₀ given above and the (exogenously given) values of F₀ and Y₀. The dynamic budget constraint can then be used recursively to obtain F₁, F₃,....

The current account is given by (see Proposition A.1 in the Appendix):

The absence of bequests and the assumption that individuals live for one period imply that the private sector will accumulate no assets. Hence F_g,t = F_t and the current account surplus is equal to the government’s total (including interest receipts) surplus. Furthermore, optimal per capita taxes, τ_t, are equal to:

4.1.1 Examples

Example 4.1 (Constant Non-oil Production)

We assume no population growth (n = 0), R = 1.06, βR = 1,¹³ and no initial financial assets (F₀ = 0). The optimal mix of the public and private goods requires that the former represent 20% of total consumption.¹⁴

Initial oil production, which accrues to the government, accounts for 80% of GDP, while the remaining 20% is produced by the private sector. Oil production remains constant (in real terms) for 25 periods, moment at which oil reserves are exhausted. Production in the non-oil sector remains constant indefinitely.

Figure 4.1 shows the evolution of consumption, financial assets (as a fraction of non-oil GDP), and the current account (also as a fraction of non-oil GDP). The first two series are divided by 100 and 50, respectively. It can be seen that consumption remains constant and equal to the annuity value of initial wealth (both from the oil and non-oil sectors). During the “boom years” of oil production, assets are accumulated (by the government) to maintain a level of consumption above production once oil is exhausted. During the boom years we also observe a positive and, due to interest payments, increasing current account surplus, which turns into a constant deficit once oil is exhausted. Since oil revenues can finance more than the optimal level of the public good, the government transfers a fixed amount (not shown in the figure) to every generation.

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Consumption, current account and financial assets with constant non-oil production

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 4.1: The figure shows the optimal paths of normalized consumption (- - -), normalized financial assets (^***) and the current account as a fraction of GDP (—) under the assumptions of the benchmark model.The following assumptions are made: no population growth (n = 0), R = 1.06, βR = 1, no initial financial assets (W₁ = 0), the optimal mix of the public and private goods requires that the former represent 20% of total consumption, initial oil production, which accrues to the government, accounts for 80% of GDP, while the remaining 20% is produced by the private sector. Oil production remains constant (in real terms) for 25 periods, moment at which oil reserves are exhausted. Production in the non-oil sector remains constant indefinitely.

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It is interesting to note that if βR < 1 (impatient individuals), the consumption path will be downwards sloping instead of constant, since individuals want to consume more and save less today. If this effect is large enough, there may be no initial current account surplus, as individuals spend more than the sum of their private income and the current oil income. ∎

Example 4.2 (Increasing Non-oil Production)

Assume now that, instead of remaining constant, non-oil production grows 2% per period forever. The remaining assumptions are the. same as in the previous example.

Figure 4.2 shows the evolution of the same three variables considered in Figure 4.1, with the same normalizing constants. It also shows the path of optimal taxes (as a fraction of non-oil GDP). Consumption is constant, at a level 12.3% higher than in Figure 4.1, reflecting the fact that non-oil production increases over time, instead of remaining constant, as in the case of Figure 4.1. Assets increase during the years when oil is produced and are depleted thereafter, eventually approaching a constant (and negative) fraction of GDP. There is an increasing current account surplus during the boom years and a slightly decreasing current account deficit after oil is exhausted.

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Consumption, current account, financial assets and taxes with increasing non-oil production

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 4.2: The figure shows the optimal paths of normalized consumption (—), normalized financial assets as a fraction of non-oil GDP (^***), the current account as a fraction of GDP (—) and taxes as a fraction of non-oil GDP (…) under the assumptions of the benchmark model. The normalizing constants and the parameters are the same as in Figure 4.1, with the exception that non-oil production increases at an annual rate of 2%.

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In contrast with Example 4.1, in this case taxes, as a fraction of non-oil GDP, do not remain constant. Initially the private sector receives large government transfers. These transfers decrease steadily, and individuals must begin paying taxes in period 67. From then onwards taxes increase significantly, so as to pay back the debt incurred by the government during the oil boom. ∎

Example 4.2 shows that the Benchmark Model (BM) may lead to significant wealth transfers from future to current generations. The government may borrow against incomes from future generations to increase current consumption. It is interesting to note that the model has this implication even in the absence of oil wealth, as long as per capita private sector production increases over time. Since such large intergenerational transfers are rarely observed, this raises the issue of whether the BM provides an adequate criterion for deciding how to spend the revenue generated by oil production. We return to this issue in section 4.3.

4.2 Permanent Oil Income Model

The BM prescribes that permanent total income should be constant over time. Since this may lead to large wealth transfers across generations, it may be better to focus on permanent oil income instead:

“Because most export revenue from oil and natural gas accrues to the public sector, the central government usually decides through the budgetary process how much of this revenue will be saved and consumed. To make this decision based on inter generational equity considerations, it is necessary to determine the permanent rent available from hydrocarbon exploitation. This rent represents the level of public consumption that can be currently enjoyed without increasing the country’s debt and depleting its wealth.”¹⁵

This approach can be rationalized within the framework of Section 2 as follows:

1. Social welfare function: The difference with the BM is that the instantaneous utility function only depends on consumption of the public good.¹⁶

2. Policy Instruments: The difference with the BM is that the government cannot collect taxes.

3. Private Sector: The private sector does not appear, at least explicitly, in the problem.

The Permanent Oil Income Model (POIM) considers the problem of spending the government owned oil as if it were totally unrelated to the private sector’s consumption of private goods. The solution to the problem is obtained by substituting total initial government wealth for total wealth in (17):

We then have:

If βR = 1, the right hand side of (23) (divided by period 1 population) is permanent oil income, that is, the highest per capita consumption level from oil resources that can be maintained indefinitely, thereby justifying the name of the model.

The POIM can be used to rationalize the often mentioned criterion of intergenerational fairness according to which oil wealth (either in absolute or per capita terms) should be kept constant. Equations (23) and (24) imply that per capita government wealth, which in this model corresponds to oil wealth, remains constant along the optimal consumption path only if βR = 1.¹⁷ If βR < 1, it is optimal for society to deplete oil wealth as time goes by. It also follows from (23) and (24) that total oil wealth remains constant along the optimal consumption path only if βR(1 +n)^ρ = 1. If n > 0 this requires a relatively impatient society, since βR < 1.

An advantage of the POIM, compared with the BM, is that it avoids intergenerational wealth transfers of non-oil assets. It does so by assuming that private income and consumption of the private good do not interact at all with the government’s income and consumption of the public good. Next we present two unattractive consequences of this limitation, one that can be accommodated with a straightforward extension of the model and one that cannot.

A first limitation is that the mix of privately and publicly produced goods will usually be suboptimal. The optimal path of the POIM determines the level of consumption of the public good without taking account of consumption of the private good chosen by consumers. This objection can be accommodated by assuming that consumers live for one period and have no bequest motive, and introducing a limited role for taxation: in every period the government sets taxes/transfers so as to ensure that the optimal mix of the public and private consumption goods is provided. That is, if we denote by ${\widehat{C}}_{G,t}$ the consumption of the public good derived from the POIM, total consumption during period t will satisfy:

where we have used the fact that consumers do not save.

A second example of the limitations of the POIM is illustrated by the following example. Assume that private income and oil income are perfectly negatively correlated.¹⁸ When oil income is high, private income is low and viceversa, so that total income (GDP) remains constant over time. Consumers live one period and do not save. The (certainty-equivalence version of the) POIM implies that only consumption of the public good will be smoothed out over time, so that total consumption will be high in years with high private income and low in years with low private income. Even though this is the optimal solution within this framework, common sense suggests that all generations would be better off if the government smoothed total consumption. Before knowing whether oil income or private income will be high during their lifetime, a generation prefers receiving its total permanent income for sure to receiving the sum of permanent oil income and private income. Also note that the private sector cannot mitigate this limitation since, having ruled out taxation for intergenerational purposes, improvements of the sort described above are not possible. We conclude that in this example there exists a consumption path that is better (as measured by the BM) for all generations than the solution from the POIM. Furthermore, ex-ante, this improvement involves no intergenerational transfers on average.

4.3 A New Approach

Both models discussed in the previous subsections have serious shortcomings. The Benchmark Model allows for intergenerational wealth transfers which we do not observe even in the absence of oil wealth. On the other hand, the POIM avoids intergenerational transfers by ruling out government policies that benefit all generations (as viewed from the BM). The Benchmark Model’s SWF is more appealing than that of the POIM, since individuals benefit both from consumption of the private and public goods, Regarding instruments, the BM has more than we would like, while the POIM eliminates unattractive instruments (intergenerational wealth transfers) at the cost of ruling out appealing policy alternatives.

The challenge therefore is to limit the policy instruments available to the government in the BM in such a way that the attractive properties of both models can be recovered. We propose the following approach. Add to the BM the restriction that no generation can be worse off than it would have been in the absence of oil wealth, where the counterfactual with no oil wealth should be determined by positive considerations.

The approach we propose, which we describe as conditionally normative, does not undo what society would have done in the absence of oil. Instead it spreads the wealth of oil across generations optimally, not by giving every generation the same amount of the public good, as in the simplest POIM, but by choosing among all possible policies that are Pareto improving, the one that increase the SWF the most. The additional constraint imposed by the Conditionally Normative Model (CNM) on the BM ensures that no intergenerational transfers of non-oil related wealth take place while allowing for an efficient allocation of oil wealth.

Denote instantaneous utility in period t by u_t, and instantaneous utility in the absence of oil income by $u_t^{*}$. Applying the CNM in period 1 requires choosing a among all possible consumption paths that satisfy $u(c_t)\ge u_t^{*},t\ge 0$, the one that maximizes the SWF considered in the BM. If there is income uncertainty, then the constraint becomes ${{E}}_0[u(c_t)]\ge {{E}}_0[u_t^{*}]$, where E₀ denotes expectations based on information available in period 0.

4.3.1 Examples

We consider three examples to illustrate the CNM.

Example 4.5 (Constant Non-Oil Income)

Assume that non-oil GDP remains constant over time and βR = 1 (see Example 4.1). In this case the three approaches considered in this paper, the BM, the POIM and the CNM, imply the same constant path for consumption. ∎

Almost any departure from the simple case described above will result in different consumption paths for the three models. The following two examples consider changes in future non-oil income.

Example 4.6 (Increasing Non-Oil Income)

Figure 4.3 shows the consumption path associated with the three models when non-oil income grows 2% per period, for the first 50 periods, and remains constant thereafter.²⁰ Optimal (total) consumption in the BM is constant. In the POIM it grows together with non-oil income, the difference between both series being equal to the annuity value of oil wealth. Optimal consumption in the CNM is constant during the first 18 periods and follows the path of non-oil income thereafter.

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Optimal Consumption Path for Alternative Models with Increasing Non-Oil GDP

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 4.3: The figure shows the optimal paths of consumption for the Benchmark Model (xxx), the Permanent Oil Income Model (—) and the Conditionally Normative Model (—). Parameter values: no population growth; R = 1.04, βR = 1, initial oil wealth: 100; initial non-oil GDP: 30; non-oil GDP grows 2% per period for 50 periods and then remains constant forever.

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Compared with the POIM, those living in the first 12 periods are better off under the CNM while those living thereafter are worse off. Since marginal utility of consumption in the absence of oil is higher during the initial periods, the CNM spreads the oil wealth among those living in these periods.

Those that benefit the most are those that would have been poorest without oil wealth—generations that expected relatively high private incomes do not benefit at all. ∎

Example 4.7 (Decreasing Non-Oil Income)

Figure 4.4 shows what happens when non-oil income decreases by 2% during the first 50 periods, and remains constant thereafter.²¹ The behavior of the. optimal consumption path in the BM and POIM art qualitatively similar to those described in the previous example. In the case of the CNM. optimal consumption decreases initially, being equal to non-oil income during this phase. Eventually (period 13 in the figure) it stops decreasing and remains constant thereafter. By contrast with Example 4.6, in this case the optimal consumption path of the CNM is fiscally more conservative than that of the POIM. It prescribes not spending oil related wealth during early years, saving it to help those who expect to be worse off in the future. Only in period 13 the CNM recommends to begin spending oil wealth to help maintain the highest consumption level compatible with the restriction of not leaving any generation worse off than it would have been without oil. It is also interesting to note that in this example the consumption path of the Benchmark Model is the one that is most conservative from a fiscal point of view. It taxes heavily the initial generations to finance a constant level of consumption for everybody. ∎

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Optimal Consumption Path for Alternative Models with Decreasing Non-Oil GDP

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 4.4: The figure shows the optimal paths of consumption for the Benchmark Model (xxx), the Permanent Oil Income Model (—) and the Conditionally Normative Model (—). Parameter values: the only difference with Figure 4.3 is that non-oil GDP decreases 2% per period during the first 50 periods.

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The following general result, for the optimal consumption path under CNM is presented in the Appendix (Propositon A.2). It assumes no income uncertainty and βR = 1. Under these assumptions, the optimal consumption path for the CNM can be found as follows: First, the generations are ordered according to their utility in the non-oil scenario. Next, oil wealth is used to raise the income of the poorest generation until it equals that of the second poorest. If this does not exhaust the oil wealth, the income of the two poorest generations is raised until it equals that of the third poorest. And so on until no oil wealth remains to be distributed. If oil wealth is large enough so that the income of all generations can be brought to the level of the richest generation (in the scenario without oil), the constraint that differentiates the CNM from the BM is not be binding and both optimal consumption paths are the same (constant, equal to the annuity value of total wealth). Otherwise, the richest generations do not benefit from the oil wealth.

5.1 Previous Literature

In this subsection we present a short review of the recent literature of oil price forecasts based on time-series models. It is understood throughout that models under consideration arc for the logarithm of the oil price.

The benchmark model to forecast oil prices (as well as other commodity prices) at medium run horizons, say 1 or 2 years, is a random walk, with and without drift. In this case the best prediction of future prices is the spot price (probably plus a drift). Furthermore, every shock to prices is permanent, affecting all expected future prices. The intuition for having this simple process follows from thinking about oil as an asset. Arbitrage prevents the existence of predictable price jumps for they offer an opportunity of making (potentially) unlimited profits. A drift reflects a fixed broad opportunity cost of maintaining the asset.²³

The idea that oil prices follow a random walk, however, is at odds with the presumption that production of both oil and its substitutes should increase at higher oil prices. At the same time, oil production should decrease if prices are below marginal costs. By contrast, if prices follow a random walk, they could increase without bound and/or approach arbitrarily close to zero.²⁴ Despite this notion, it is not easy to reject the random walk hypothesis. Researchers have either used extremely long samples to find mean reversion or have had to resort to less standard approaches, where by “standard approaches” we mean the Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests.

For example, with several years of data, Videgaray (1998) finds mean reversion after allowing for a structural break in 1973.²⁵ Pindyck (1999) rejects the random walk null hypothesis using an ADF unit root test only after considering more than 70 years of data. Interestingly, he concludes that even with 120 years of data, permanent shocks do exist (although their size is considerably smaller than that of the transitory shocks). Finally, Bessembinder et al. (1995) find evidence of mean reversion using the future prices term structure.

The difficulty in rejecting the random walk hypothesis has led to more sophisticated models to describe oil prices. Rather than assuming reversion to a constant trend, Pindyck (1999) proposes a model in which both the constant and the trend are, in turn, non observable mean reverting stochastic processes. He estimates this model with a long sample of annual data using a Kalman Filter, and predicts prices 20 years ahead. Although no formal tests are provided, the forecasts appear to be better than those of a fixed trend AR(1) process. Of course, there is always the question of whether it is valid to use pre 1973 data to forecast future prices given the large structural break that took place at that time.

Schwartz (1997) also presents Kalman Filter estimates and formally compares the forecast capability of three alternative models for future and forward prices using high frequency data spanning 11 years. He considers a one factor model in which the (logarithm of the) oil price follows an AR(1) process, a two factor model in which the convenience yield is stochastic, and a three factor model in which a stochastic interest rate is also included. The estimation procedure he uses takes into account that the spot price, the convenience yield and the interest rate are not perfectly observable—thus the need of the Kalman Filter. The results he obtains indicate that including a second factor (the convenience yield) improves substantially the forecast capability of the model.

A simple random walk, an AR(1), and the models presented in Pindyck (1999) and Schwartz (1997) can be thought of as special cases of the following model:

where p_t is the log of the real oil price, α_t, δ_t and ψ_t are possibly stochastic parameters, Trend_t is a time trend and ε_t is a stochastic stationary shock.

A random walk with drift assumes α_t constant, δ_t = 0, and ψ_t = 1 (as well as ε white noise). An AR(1) assumes a constant α_t, a constant ψ_t < 1 and (possibly) a positive δ_t.

More interestingly, Pindyck (1999) considers that both α_t and δ_t follow unobservable AR(1) stochastic processes with uncorrelated innovations. These processes are meant to represent reduced forms for the effects of demand, cost of extraction and available reserves shocks. Prices then would revert to a changing trend (level). Also, Schwartz (1997) considers the possibility that in his two factor model ψ_t follows a stochastic process (possibly mean reverting) with innovations that can be correlated with innovations of the current spot price. The economic interpretation of this model is that the convenience yield follows a process itself. The intuition for why this variable affects current spot prices is simple: If oil represents an asset, then the current and future spot prices are linked through the current interest rate, storage costs and the convenience yield. Thus, for a given future spot price, a higher convenience yield will increase the current spot price.

We will use these alternative models below to evaluate the extent to which oil prices can be forecast.

5.2 Revisiting the Random Walk Hypothesis

A key issue that we face is the question of to what extent future oil price changes can be predicted. In one extreme, it is possible to think that oil prices follow a simple random walk. If that were the case, then the best prediction for all future periods is the current value, while the standard deviation of this prediction grows linearly with time. In the other extreme, one could think of oil prices following a stationary process, where it is possible to forecast future prices with greater precision.

In order to evaluate the forecast ability of oil prices we present below three group of tests: standard ADF and PP, Variance Ratio, and non-linear adjustment. In all cases we consider quarterly observations of the log of the real price of Brent oil (using the US WPI as the deflator).

5.2.1 ADF and PP Tests

Augmented Dickey-Fuller and Phillips-Perron tests are the standard procedures to evaluate whether a series follows a stationary process. Intuitively, these tests measure the strength of the forces that tend to move the series back to a constant trend after suffering a shock. If the strength of these forces is low, then one concludes that the process is non stationary (that there is no mean reversion).

Table 5.1 presents the results for three alternative samples of quarterly data: 1957.I-1999.II, 1974.I–1999.II. and 1986.I–1999.II, and two specifications with and without trend. The test shows that when the larger sample is considered, the process appears to be non stationary. In contrast, the shorter samples, particularly 1986.I–1999.II suggest a stationary process.

TABLE 5.1

ADF and PP Tests

Note: ^*, ^**, and ^*** = significant at 10, 5, and 1% respectively.

TABLE 5.1

ADF and PP Tests

	1957.I–1999.II	1974.I–1999.II	1986.I–1999.II
ADF no trend	−1.77	−2.60*	−3.42**
ADF with trend	−1.69	−3.83**	−3.52**
PP no trend	−1.65	−2.56	−3.93***
PP with trend	−1.52	−4.57**	−4.25**

Note: ^*, ^**, and ^*** = significant at 10, 5, and 1% respectively.

View Table

TABLE 5.1

ADF and PP Tests

	1957.I–1999.II	1974.I–1999.II	1986.I–1999.II
ADF no trend	−1.77	−2.60*	−3.42**
ADF with trend	−1.69	−3.83**	−3.52**
PP no trend	−1.65	−2.56	−3.93***
PP with trend	−1.52	−4.57**	−4.25**

Note: ^*, ^**, and ^*** = significant at 10, 5, and 1% respectively.

This evidence shows that when one excludes large changes in regime, oil prices appear to be stationary. However, when these regime shifts are considered, price shocks tend to have relevant permanent effects. In terms of forccastability, these results show that assuming a stationary process is a valid procedure as long as one assumes that the current regime will prevail with probability one. More generally, however, one could improve the forecast by considering and modeling the transitory or permanent components of a shock.

5.2.2 Variance Ratio Test

The second type of test we consider to evaluate whether oil prices follow a non-stationary process is the Variance-Ratio (VR) Test. This test makes use of the linearly increasing volatility of a non-stationary process and evaluates whether the standard deviation measured at different horizons increases as predicted under the null of random walk. Furthermore, it gives a measure of the relative importance of transitory and permanent shocks.

In particular, the VR test calculates a statistic J(s), s = 1,2, …, S that has the following properties.²⁶ As the sample size becomes large and s increases the ratio J(s)/s should converge to zero if the true process is stationary. If it does not converge to zero the process is non-stationary. Moreover, the value to which J (s) converges represents the standard error for long term forecasts. These properties hold as long as the sample size is large and s is considerably smaller than this sample size.

Figures 5.1 and 5.2 present the results of VR tests for the log of the oil price for two samples: 1957.1–1998.IV and 1974.I–199S.IV. In both cases the statistic J(s)/s does not converge to zero, showing that the shocks to the true process probably have some permanent effects. The size of these effects appears clearly smaller than the standard deviation of the innovations of a simple random walk estimated for each sample. This fact shows that shocks also have some transitory effects on prices, suggesting that it should be possible to do better, in terms of forecasting, than with a random walk.

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Variance Ratio Test: 1957–1998

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 5.1: The figure shows the results of the Variance Ratio tests for the sample 1957–1998 [solid line (—)]. The dashed lines (- - -) show the results of a Montecarlo exercise (with 1000 replications) assuming that the true process is a geometric random walk and a AR(1) with autoregressive coefficient equal to the sample estimate.

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Variance Ratio Test: 1974–1998

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 5.2: The figure shows the results of the of the Variance Ratio tests for the sample 1974-1998 [solid line (—)]. The dash lines (—) show the results of a Montecarlo exercise (with 1000 replications) assuming that the true process is a geometric random walk and a AR(1) with autoregressive coefficient equal to the sample estimate.

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One important limitation of these results is that the sample sizes we consider are not very large compared to s. In order to evaluate how this issue may affect the results the figures also present the results of a Montecarlo experiment considering a sample of equal size to what we consider in the calculations. These Montecarlo experiments are based on 1000 replications of a process that has the same standard deviation and parameters as the true data.

The results of these experiments show that, indeed, the small sample affects the performance of the test (for the sample sizes we consider). The statistic J(s)/s for a true random walk decreases instead of converging to a flat value. At the same time, a true AR(1) does not converge to zero for the values of s we consider (although it does not converge to a positive value either). These results, however, do not change our general interpretation of the process. Because the sample statistic decreases faster than for the random walk, we conclude that shocks do not have full permanent effects. And because it tends to converge to a positive value, we conclude that shocks do not have transitory effects only.

5.2.3 Non-linear Adjustment

One potential explanation for finding evidence of non-stationarity when the true process is actually stationary is the existence of non-linearities in the adjustment process. It could be the case, for example, that oil prices follow a random walk within certain range. Outside this range, however, there could be forces that bring oil prices back. The intuition that prices cannot permanently be below marginal costs and that above certain threshold oil substitutes enter the market is in line with this interpretation.

More generally, oil prices may be viewed as the sum of two processes, with the relative importance of both processes dependent on the price level, Prices follow a unit root or even an explosive process for small deviations from a stationary trend, but the process becomes mean-reverting for large deviations. This is the case, for example, of exponential and logistic smooth-transition autoregressive (ESTAR and LSTAR) models. In this case it is assumed that mean-revering forces appear gradually as the actual oil price deviates from its long run equilibrium value.²⁷ Threshold autoregressive models (TAR) are another type of models in which the transition from unit-root to mean-reverting occurs suddenly at a fixed threshold.

In order to test the hypothesis of linearity in the oil price process we follow the procedures described in Michael et al. (1997). In particular, we test the null hypothesis of linearity against a smooth-transition model by using OLS to estimate the model:

$p_t+{\beta }_{00}{\displaystyle \sum _{j=1}^k({\beta }_{0j}{p}_{t-j}+{\beta }_{1j}{p}_{t-j}{p}_{t-d}+{\beta }_{2j}{p}_{t-j}{p}_{t-d}^2)+{\epsilon }_t}$

for alternative values of d. The null hypothesis is β_1j = β_2j = 0 (j= l,…, k). Linear adjustment is rejected if for any of the values of d the p-value of this test is insignificant.

Table 5.2 presents the p-values that result from testing the null hypothesis of linearity of log real oil prices using different samples and three alternatives values for d. It also shows the value of k, the lags required to have white-noise innovations in each case. The results show that the linear adjustment hypothesis is rejected only in the sample 1974-1999 using k = d = 1. We find one rejection in three as relatively weak evidence in favor of non-linear adjustments. In what follows we focus mainly on linear models, but keep as a competing alternative the non-linear model with d = 1.

TABLE 5.2

Ρ-Values Non-Linear Adjustment Test

Note: In parenthesis the value of k that yields white noise.

TABLE 5.2

Ρ-Values Non-Linear Adjustment Test

	1957.I–1999.II	1974.I–1999.II	1986.I–1999.II
d = 1	0.12 (1)	0.02 (1)	0.10 (1)
d = 2	0.56 (1)	0.14 (1)	0.19 (2)
d = 3	0.40 (1)	0.40 (1)	0.12 (2)

Note: In parenthesis the value of k that yields white noise.

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TABLE 5.2

Ρ-Values Non-Linear Adjustment Test

	1957.I–1999.II	1974.I–1999.II	1986.I–1999.II
d = 1	0.12 (1)	0.02 (1)	0.10 (1)
d = 2	0.56 (1)	0.14 (1)	0.19 (2)
d = 3	0.40 (1)	0.40 (1)	0.12 (2)

Note: In parenthesis the value of k that yields white noise.

5.3 Evaluation of Alternative Models

The usefulness of a forecasting model has to be measured out of sample. Ultimately it is the ability to forecast future unknown prices that should discriminate among competing models. In this subsection we evaluate the out of sample forecast capabilities of 12 alternative linear models, a non-linear model, market future prices, and market forecasts.

We consider two alternative samples, one starting in 1974 and the other starting in 1986, and calculate the root mean square error (RMSE) of forecasts at 1 and 2 year horizons proceeding as follows. We estimate repeatedly each model using quarterly data (and weekly data in one case) ending in the second quarter of the years 1994 to 1998 and forecast out of the estimating sample. Then we compute the RMSE using the forecast errors at 1 and 2 years horizons. For each model we have 5 one-year ahead and 4 two-year ahead forecast errors.

The linear models we consider (for the logarithm of the real price of oil) arc the following:

1. A random walk without drift.

2. A random walk with drift.

3. An ARIMA(2,1,2). This model is the equivalent of a random walk augmented by a stationary process for the error term ε_ι.

4. Same as above with a dummy variable that take the value 1 during the invasion of Kuwait in 1991.

5. An AR(1) without drift (assuming that the process is stationary).

6. The permanent value of a Beveridge and Nelson decomposition of the series.²⁸

   Models 7 through 11 consider an AR(1) model with stochastic first-order autocorrelation, ψ_t, which is estimated using the Kalman Filter. The models differ in the assumptions they make on the process followed by ψ_t and whether they include a linear trend or not for the price process.

7. The price process has no trend and ψ_t follows a random walk with innovations orthogonal to those of the price process.

8. As 7 but with a trend in the price process.

9. The price process has no trend and ψ_t follows an AR(1) process with innovations that are orthogonal to oil price innovations (this model resembles model 2 of Schwartz, 1997).

10. As 9 but with a trend in the price process.

11. Both the constant and the trend parameters of the price process follow AR(1) processes with innovations that are orthogonal to oil price innovations and between them (this model is based on Pindyck, 1999).

12. As 7 but using weekly data to estimate the model (this model is based on Schwartz, 1997). The forecast is also weekly and we average the forecasts to calculate MRSEs. The data we consider in this model is slightly different because prices are not deflated.

We also consider three other forecasts in the out-of-sample evaluation. We estimate repeatedly the non-linear ESTAR model discussed above with the same quarterly data and forecast prices one and two year ahead using the estimated model. Finally, we consider the one-year ahead future price from Bloomberg (for June delivery) and the average surveyed one year ahead forecast informed by the June issue of Consensus Forecast.²⁹

Table 5.3 presents the results of this exercise. Notwithstanding the fact that the RMSE are calculated with small samples, the results show that more sophisticated models do not have a better out-of-sample performance. Indeed, the models with stochastic autoregressive parameter are clearly outperformed by a simple random walk. The model with stochastic trend and constant appear to be as good as the random walk. Overall, only the ARIMA models (with and without dummies for the Kuwait invasion) appear to perform somewhat better than the random walk without drift. When we use. the longer sample to estimate each model the best performance corresponds to the ARIMA model without dummies. For the short sample the best performance corresponds to the ARIMA model with dummies. Yet if we consider both samples jointly, it is hard to argue than any model does significantly better than the random walk without drift.³⁰ Furthermore, this model appears to be only marginally less accurate than surveyed forecasts.

TABLE 5.3

One and Two Year Ahead Forecast RMSE

Note: Root mean square error of one and two year ahead forecasts of a rolling sample with one year increment. One year includes 5 forecast points whereas two year includes 4 forecast points. Short sample refers to 1986.I-1999.II whereas long sample refers to 1974.I-1999.II. Model 11 has problems in converging in the small sample.

TABLE 5.3

One and Two Year Ahead Forecast RMSE

		Short sample		Long sample
		1 year	2 years	1 year	2 years
1	Random walk, no drift	15.6%	20.4%	15.6%	20.4%
2	Random walk, with drift	17.2%	20.3%	16.0%	22.5%
3	ARIMA(2,1,2)	17.0%	21.8%	12.7%	13.3%
4	ARIMA(2,1,2). with dummy	13.8%	16.2%	16.3%	23.6%
5	AR(1)	18.2%	21.6%	25.5%	21.6%
6	BN Decomposition	15.0%	21.0%	17.1%	21.6%
7	Kalman ψt RW, no trend	22.7%	34.6%	15.8%	21.2%
8	Kalman ψt RW, with trend	39.1%	77.0%	27.7%	48.3%
9	Kalman ψt AR, no trend	18.5%	28.1%	16.7%	20.4%
10	Kalman ψt AR, with trend	30.7%	61.7%	19.0%	24.5%
11	Kalman αt and δt AR	-	-	15.0%	21.0%
12	Kalman ψt RW weekly data	23.1%	22.3%		-
13	ESTAR d = 1	18.5%	22.2%	19.9%	21.1%
14	Future Prices	22.0%	-	22.0%	-
15	Survey Data	14.0%	-	14.0%	-

Note: Root mean square error of one and two year ahead forecasts of a rolling sample with one year increment. One year includes 5 forecast points whereas two year includes 4 forecast points. Short sample refers to 1986.I-1999.II whereas long sample refers to 1974.I-1999.II. Model 11 has problems in converging in the small sample.

View Table

TABLE 5.3

One and Two Year Ahead Forecast RMSE

		Short sample		Long sample
		1 year	2 years	1 year	2 years
1	Random walk, no drift	15.6%	20.4%	15.6%	20.4%
2	Random walk, with drift	17.2%	20.3%	16.0%	22.5%
3	ARIMA(2,1,2)	17.0%	21.8%	12.7%	13.3%
4	ARIMA(2,1,2). with dummy	13.8%	16.2%	16.3%	23.6%
5	AR(1)	18.2%	21.6%	25.5%	21.6%
6	BN Decomposition	15.0%	21.0%	17.1%	21.6%
7	Kalman ψt RW, no trend	22.7%	34.6%	15.8%	21.2%
8	Kalman ψt RW, with trend	39.1%	77.0%	27.7%	48.3%
9	Kalman ψt AR, no trend	18.5%	28.1%	16.7%	20.4%
10	Kalman ψt AR, with trend	30.7%	61.7%	19.0%	24.5%
11	Kalman αt and δt AR	-	-	15.0%	21.0%
12	Kalman ψt RW weekly data	23.1%	22.3%		-
13	ESTAR d = 1	18.5%	22.2%	19.9%	21.1%
14	Future Prices	22.0%	-	22.0%	-
15	Survey Data	14.0%	-	14.0%	-

Note: Root mean square error of one and two year ahead forecasts of a rolling sample with one year increment. One year includes 5 forecast points whereas two year includes 4 forecast points. Short sample refers to 1986.I-1999.II whereas long sample refers to 1974.I-1999.II. Model 11 has problems in converging in the small sample.

6.1 Income and Budget Uncertainty

Income uncertainty—the risk about future income realizations—can be incorporated easily into consumption models. If the instantaneous utility is quadratic, we have certainty-equivalence, and the results obtained in Section 4.2 need to be modified only slightly. For example, equations (18) and (19) become:

where E_t denotes expectations based on information available in period t. That is, all that changes is that uncertain quantities are replaced by their expected values. Of course, as mentioned in Section 3.2, this solution has the awkward property that current savings do not depend on the variance of future income.

In the more appealing case of a CES instantaneous utility, there does not exist a simple expression for c₀. The solution has to be found resorting to numerical methods. We propose instead an approximation to the optimal solution that is transparent and easily implementable. Of course, because it is an approximation it does not correspond exactly to the optimal solution.

Our procedure is based on a counterfactual experiment in which consumption decisions are made knowing that oil risk is diversified away in the near future, say that the oil industry will be privatized. This procedure allows us to simplify the consumption problem by collapsing all future periods in a single period and treating the overall problem as a two-period problem. Furthermore, assuming that the variance of oil price shocks is small, we can write a closed-form solution for consumption as a function of that variance and initial conditions.

More precisely, consider the period t optimal consumption decision knowing that the oil industry will be privatized in period t+1. Because in period t+1 all income uncertainty is resolved, from that moment onwards the consumption problem is trivial; under the assumption βR = 1 the solution is to consume the sum of the annuity values of the privatization proceeds and the financial assets available at that time. Assume, further, that oil risk is fully diversifiable in the world economy, so that the privatization proceeds equal the expected NPV of oil GDP conditional of the oil price observed in t + 1. As of period t, the privatization proceeds is a random variable that depends on the oil price process. Moreover, it depends on the expected path of future oil prices.

Consider now the comparison of the optimal consumption decision of period t, knowing the oil price of that period, both under certainty equivalence (CE) and the optimal consumption level (given the actual volatility of the price process). The plan under CE corresponds to the POIM solution. The difference between the two consumption levels measures the precautionary savings motive.

So far we have assumed that period t oil prices are known at the beginning of the period, when consumption decisions are taken. However, when deciding next year’s budget, policymakers do not know the level of oil prices that will finally prevail. This information problem corresponds to budget uncertainty. Although it is closely related to income uncertainty, it represents a different source of uncertainty. In order to derive closed-form solutions for the effects of budget uncertainty we consider that at the moment of writing the budget the price of oil at time 0, P₀, is not known but assumed to follow a log-normal distribution with known mean and variance. This distribution captures all the information available to the government about the price of oil during the budget year being considered. Only from period 1 onwards does the (real) price of oil follows a geometric random walk with drift. The possibility of setting the parameters for the initial price allows us to depart from the pure random walk assumption, thereby allowing the incorporation of some degree of mean reversion.

Besides the counterfactual experiment of privatizing, we use approximations to obtain closed-form solutions for consumption. In particular, we consider a first order Taylor approximation around the case in which the variances of both shocks to future prices and the current year (budget year) price are zero.

6.1.2 Examples

In order to evaluate the importance of precautionary savings in the context of oil producing countries we present four examples. The first one presents a baseline case. The other three present comparative statics.

Example 6.1 (Precautionary Saving Correction Factors)

We assume no population growth (n = 0), one inhabitant, no output growth (g = 0), R = 1.05, βR = 1, ρ = 3, Q₀ = 100, μ_P0 = 25, σ_ν = 0.25, σ_P,0 = 6.25, Τ = 50, and F₀ = 2,500 (equivalent to one year of production).

With these parameters the results are as follows. From an income of 2,500, the certainty equivalence consumption is 2,411. The correction factors due to precautionary motives are Δ_BU = 11.9% and Δ_IU = 10.7%. Thus, optimal consumption is 1,868. ∎

Given the role of volatility in the solutions proposed, the correction factors increase linearly with the variance of the shocks to the price process. Thus, precautionary saving increases at rate 0.5 with volatility.

Example 6.2 (Precautionary Saving and Shocks Persistence)

Figure 6.1 shows the correction factors Δ_BU and Δ_IU for different levels of the AR(1) coefficient of the oil price process and the parameters of the baseline example. In this case ψ ranges from 0.9 to 1. When ψ < 1 we use the formulae described in the Appendix. In all these cases me disregard any income effects arising from volatility by directly applying the correction factors to the zero variance consumption.

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Correction Factors and Shock Persistence

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 6.1: The figure shows the correction factors Δ_BU (—) and Δ_IU (- - -) for different autoregressive coefficients ψ. The rest of the parameters correspond to those of example 6.1.

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The results show that precautionary saving increases sharply with the persistence of shocks. When ψ is around 0.9, correction factors are almost one-tenth of what they are in the case of a random walk. Furthermore, this difference is clearly non-linear. When ψ is around 0.95, correction factors are about one-fourth of what they are when ψ = 1. ∎

This key role for shock persistence in determining the importance of precautionary saving has been noted before (see, e.g., Skinner, 1988). It follows from the high sensitivity of wealth uncertainty to the degree of persistence in shocks, particularly in the neighborhood of a random walk.

Example 6.3 (Precautionary Saving and Financial Assets)

Figure 6.2 shows the correction factors Δ_BU and Δ_IU for levels of initial financial assets F₀ and the parameters of the baseline example. We have scaled F₀ by initial production, so it ranges from -4 to 4.

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Correction Factors and Initial Financial Assets

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 6.2: The figure shows the correction factors Δ_BU (—) and Δ_IU (—) for different levels of initial financial assets (scaled by expected income in the first year). The rest of the parameters correspond to those of example 6.1.

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As expected, financial assets accumulation makes less important precautionary saving. Because a larger portion of future consumption is secure when a country has more financial assets, precautionary saving decreases with F₀. In the example at hand, the correction factors drop by almost one third when financial assets increase from zero to four years of income. A similar pattern arises if one assumes that ψ = 0.9, although in this case correction factors are considerably smaller. ∎

Example 6.4 (Precautionary Saving and Resource Duration)

Figure 6.3 shows the correction factors Δ_BU and Δ_IU for different time horizons for resource exhaustion and the parameters of the baseline example. Τ varies from 5 to 105.

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Correction Factors and Resource Duration

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 6.3: The figure shows the correction factors Δ_BU (—) and Δ_IU (- - -) for different resource duration T. The rest of the parameters correspond to those of example 6.1.

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The results show that the correction factors increase quickly with Τ to stabilize around Τ = 40. The opposite happens if ψ = 0.9 (case not reported). The intuition for the result is the following. Given an extraction rate, a longer duration represents a higher initial reserve level of the resource. This, in turn, represents higher total wealth, and less initial financial assets relative to total wealth. Thus, a longer duration produces an effect that is similar to having less financial assets. When ψ < 1, a longer duration has two effects. One the one hand, it produces the same effect of reducing the share of financial assets in total wealth. On the other hand, because ψ < 1. income that is very far in the future is almost secure income, having the same effect of a higher F₀. Figure 6.4 shows the correction factors for different Τ assuming the “intermediate” case ψ = 0.99. In this case the correction factors increase with Τ between 10 and 20-25 and decrease thereafter. ∎

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Correction Factors and Resource Duration

Citation: IMF Working Papers 2000, 118; 10.5089/9781451854077.001.A001

Note to Figure 6.4: The figure shows the correction factors Δ_BU (—) and Δ_IU (—) for different resource duration Τ and ψ = 0.99. The rest of the parameters correspond to those of example 6.1.

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In deriving precautionary saving correction factors we have so far assumed that there is only one source of income, namely oil production. A more realistic representation of oil exporting countries should incorporate natural gas extraction. In order to do so we assume that the price of gas is linear in the price of oil, that the price of oil follows a geometric random walk, and that both natural resources have their particular and known extraction path. In Proposition B.4 in the Appendix we derive expressions for the correction factors in this case. Based on these expressions one can show that having a second income source related to oil prices produces relatively minor changes in the correction factors.

6.2 Adjustment Costs

When adjustment costs are present, optimal consumption may not be equal to frictionless optimal consumption. Adjusting per capita government expenditures may have welfare consequences that go beyond those captured by standard utility functions. A drastic reduction in government expenditure may lead to political instability, discouraging investment and reducing future growth. A sudden increase in government expenditure may deteriorate the quality of management of government projects because of the lack of adequate supervision. It may also increase the costs of new projects because of bottlenecks in the supply of some inputs. In this section we analyze the effects of a specific form of adjustment costs, namely quadratic costs.