OPTIMAL SIMPLE FISCAL RULES 12 3.2. **Equilibrium, stability and an irrelevance result.** We take a first-order Taylor expansion of the system of equations (3.2)-(3.12) around the steady state and consider an equilibrium driven by the two exogenous shocks: a commodity price shock ( _ε_<sup>_p_</sup> _t_<sup>)andanon-resourceGDPshock(</sup><sup>_εy_</sup> _t_<sup>).</sup> The equations of the linear system are presented in section B.1 in the appendix.

Lemma 1 presented in Appendix B provides two conditions for the existence and uniqueness of an equilibrium in the exogenous-income model. To guarantee a stable path for the SWF, the first condition states that governments must transfer to households no less than 4% (the long-run real interest rate) of the deviations of the value of the SWF from its target size each year (but also no more than 104%). The second condition removes the unit root in the consumption of the Ricardian households by imposing a debt-elastic interest rate spread in the model ( _ψ >_ 0), as in Schmitt-Grohe and Uribe (2003).

Lemma 2, also in Appendix B, presents a version of the well-known Ricardian equivalence result in Barro (1974) adapted to our heterogeneous agent framework. It says that if the government commits to a transfers rule to the hand-to-mouth household that does not depend on the size of the SWF ( _θa_<sup>_′′_= 0), then the equilibrium path of consumption of the Ricardian household is completely</sup> independent of the transfer rule coefficients for the Ricardian HH _{θa_<sup>_′, θ_</sup> _y_<sup>_′, θ_</sup> _p_<sup>_′}_.Thereasonisthat</sup> transfers to the Ricardian household do not alter the discounted flow of expected after-tax income of the Ricardian household when _θa_<sup>_′′_= 0.</sup>

# 3.3. **Calibration. General Parameters**

Many of the most important parameters are not country specific, and so we calibrate these to international data or take them from the literature. The most important are the persistence of commodity price shocks, which we calibrate to a weighted average of oil prices and gas prices taken from BJS2013. The overall persistence ( _ρ_ = 0 _._ 93) is a weighted average of oil prices ( _ρ_ = 0 _._ 94) and gas prices ( _ρ_ = 0 _._ 89) as many oil exporters also produce gas.<sup>15</sup> We set _β_ = 0 _._ 96 so that long-run annual real rate of interest is 4%. The coefficient of risk aversion _σ_ = 2, which is a standard parameter in the literature. We follow Galí _et al_ (2007) to set _ω_ = _ωy_ = _ωU_ = 0 _._ 5 (50% of the population is HtM). For simplicity we set _Bss_ = 0 so that steady-state consumption is equal across households.

We set the benchmark debt-elastic interest spread to be _ψ_ = 0 _._ 01, which implies that a 100% of GDP increase in debt (or reduction in assets) increases interest rates by 1%. Schmitt-Grohe and Uribe (2003) set _ψ_ = 0 _._ 001 to match volatility of the observed current-account-to-GDP ratio for Canada. Schmitt-Grohe and Uribe (2016) argue that _ψ_ should be set to match the autocorrelation of the trade-balance-to-GDP ratio, and they estimate _ψ_ = 1 for Argentina. We take _ψ_ = 0 _._ 01 as compromise between these two approaches. In Section 5, we use an alternative calibration of the exogenous income model with _ψ_ = 0 _._ 45 and find broadly similar results.

# **Country-specific calibration to Algeria and Trinidad & Tobago (TTO)**

> 15The actual weights are a combination of country-specific persistence for Trinidad & Tobago (TTO) and Algeria, discussed below.