OPTIMAL SIMPLE FISCAL RULES 15 Table 2. Welfare Performance of Optimal and Classical Rules.

|_ψ_ = 0_._01|(1)|(2)|(3)|(4)|(5)|(6)|(7)|
|---|---|---|---|---|---|---|---|
|_ρP_ = 0_._93|**HtMHH**|**BBR**|**SSR**|**BBR CCY**|**SSR CCY**|**OSR**|**OSR Equal**|
|_θ_<sup>_′_</sup><br>_a_|0.10|0.10|0.10|0.10|0.10|0.08|0.09|
|_θ_<sup>_′_</sup><br>_y_|-0.53|0.15|0.00|-0.85|-0.85|-0.80|-0.77|
|_θ_<sup>_′_</sup><br>_p_|1.72|1.00|0.00|1.00|0.00|0.70|0.68|
|_θ_<sup>_′′_</sup><br>_a_|0.00|0.10|0.10|0.10|0.10|0.09|0.09|
|_θ_<sup>_′′_</sup><br>_y_|-0.85|0.15|0.00|-0.85|-0.85|-0.80|-0.77|
|_θ_<sup>_′′_</sup><br>_p_|0.00|1.00|0.00|1.00|0.00|0.70|0.68|
|_sd_(ˆ_c_<sup>_′_</sup>)|0.31|0.16|0.16|0.16|0.16|0.15|0.15|
|_sd_(ˆ_c_<sup>_′′_</sup>)|0.00|0.16|0.18|0.16|0.18|0.16|0.15|
|_sd_( ̃_a_)|0.93|0.03|2.46|0.12|2.47|0.87|0.91|
|_sd_(<sup> ̃</sup>_b_)|1.85|1.57|2.39|1.57|2.39|0.22|0.17|
|**Loss (% of** _Css_**)**|**4.87**|**2.58**|**2.96**|**2.54**|**2.93**|**2.38**|**2.38**|

— costs become larger, there become worries about future solvency) thus reducing the income available for consumption. In the real world, it would also mean the SWF would be exhausted, or the government/agents would eventually reach their debt limit, though with a linear model the occasionally binding constraints are excluded from our analysis.<sup>18</sup>

**The importance of persistence** The reason the BBR outperforms the SSR is that the oil price shocks are very persistent. This means that current income is very close to permanent income and so a BBR where households just consume current income is close to optimal. In Figure 2, we plot the welfare loss of the SSR and BBR against the persistence of the commodity shock. As the persistence of the shock increases, so does its standard deviation (which is equal to _σp/_<sup>�</sup> 1 _− ρ_<sup>2</sup> _p_<sup>).</sup> To isolate the effect of persistence on welfare, we decrease _σp_ as _ρP_ increases to keep the total SD constant. (In Figure 8 in Appendix B we repeat this exercise with constant _σp_ .) In the upper — LHS of Figure 2 one can see that for _ρP <_ 0 _._ 90 the SSR is better which is the result in the literature. However, for _ρP >_ 0 _._ 90, which is the empirically relevant region for many commodity prices like oil, a BBR is preferred. The change in ranking of BBR and SSR is due entirely to the HtM HH (bottom LHS), who prefers a BBR for _ρP >_ 0 _._ 90 (but SSR for _ρP <_ 0 _._ 90) because for persistent shocks current income closely resembles permanent income. One can see that the HtM HH consumption SD is constant for the BBR (as variance of consumption equals that of income, which is constant by construction). Ricardian HHs are indifferent between the two rules because they can smooth income themselves, and so “undo” the effects of a sub-optimal fiscal rule. Note, however, that the Ricardian HH prefers less persistent shocks because they are easier to smooth.

> 18The standard deviation of _a_ and _b_ are relative to non-resource GDP, and are very large. However this large standard deviation comes from high persistence, rather than from large year-to-year variation. In fact, the “error” component of the standard deviation is only around 0.24, but because assets are very persistent, their variance is high. This suggests that it would probably take some years of persistently negative shocks for debt limits to be reached.