This section reports the equations of the core model. It closely follows the description of the EAGLE model Gomes et al. (2012), and the reader should refer for all the details to that paper.
There are two types of households in the model, Ricardian (unconstrained) and non-Ricardian (hand-to-mouth, HtM, or unconstrained). Their utilities are shown in equation 10:
Us,t = (1/(1-σ)((cs,t+k-κ cs,t+k-1)/(1-κ))1-σ-1/(1+ζ)hs,t+k1+ζ)
(10)
where cs,t is consumption of type s households, hs,t is their hours worked, σ is the (inverse of) intertemporal elasticity of substitution, κ is the degree of habit formation and ζ is the (inverse of) the Frisch labour supply elasticity.
Ricardian households hold capital, invest in internationally traded bonds denominated in US dollars Bt*, that pay the interest rate Rt* set by the Federal Reserve. They also hold domestic government bonds Bt that pay the rate Rt, which is determined by the European Central Bank. Investment in capital is subject to investment-adjustment costs that give rise to the standard Tobin's marginal q. The Euler equation with respect to domestic government bonds is
uprime(cs,t) = β(uprime(cs,t+1)(Rt(1-ΓB,t+1))/(πt+1)),
(11)
where ΓB,t+1 is the transaction cost for the euro area traded bonds, which depends on the intra-EA bond holdings.²⁴ There is an analogous Euler equation for internationally traded bonds:
uprime(cs,t) = β(uprime(cs,t+1)(Rt*(1-ΓB*,t+1))/(πt+1*)(rert+1)/(rert)),
(12)
where rert is the real exchange rate and ΓB^ * is the international transaction premium for US-denominated bonds, where StH,US is the nominal exchange rate, expressed in terms of units of Home currency per unit of US dollars:
ΓB^ *((StH,USBt+1*)/(PY,tYt);rpt) ≡ γB^ *(exp((StH,USBt+1*)/(PY,tYt)- BY*)-1)-rpt
(13)
where γB^ * is a parameter, B Y* is the steady-state net foreign asset position, rpt is a risk premium shock, PY,t is the GDP deflator, Yt is the GDP in real terms. The term ΓB,t+1 in the Euler equation for euro area bonds is analogous and depends on the target for intra-EA bond holdings. For further details see the Appendix in Gomes et al. (2010).
Ricardian households also invest It in physical capital Kt, which involves as: