Kt+1 = (1-δ)Kt+(1-ΓI,t((It)/(It-1))),
where δ is the depreciation rate and investment-adjustment cost is
ΓI,t((It)/(It-1)) ≡ (γ)/2((It)/(It-1)-1)2.
The first-order condition for investment in capital is:
(14)
(15)
pI,t = qi,t(1-ΓI,t((It)/(It-1))-ΓI,tprime((It)/(It-1))It)+β(uprime(cI,t+1))/(uprime(cI,t))qI,t+1ΓI,tprime((It)/(It-1))(It+12)/(It),
(16)
where primes indicate derivatives, qI,t is Tobin's q, and pI,t ≡ (PI,t)/(PC,t) is the relative price of investment goods.
qI,t = β(uprime(cI,t+1))/(uprime(cI,t))[(1-δ)qI,t+1+(1-τt+1k)rK,t+1ut+1+(τt+1kδ-(1-τt+1k)Γ(ut+1))pI,t+1].
(17)
where rK,t is the return on physical capital, ut is capital utilisation, and Γ(ut) is capital utilisation adjustment cost.
Total capital services in the economy are the product of the capital stock and its utilisation, and are used by tradable KT,t and non-tradable KN,t sectors of intermediate goods firms:
utKt = KN,t+KT,t.
(18)
The labour market in the standard model is identical to what is typical in the literature - a standard staggered wage setting. This is the part where the main difference between the model with and without search frictions lies, namely, the standard part described briefly below is replaced by the fully-fledged search frictions, described in detail in Appendix B.
Households supply differentiated labour services in monopolistically competitive market for each type of labour service. Wages are determined by staggered wage setting in terms of nominal wages, where the probability that a wage contract is reset in a given period is 1 - ξs, where s stands for Ricardian or hand-to-mouth households. Households that can reset wage choose the same wage tilde W s,t, while wages that are not reset are indexed to past wages by a combination of the past CPI inflation, ΠC,t-1 = PC,t-1 / PC,t-2, where PC,t is the consumer price index, and inflation target, \Pi, where the indexation weight is χs colon Wt,i = ΠC,t-1χ_ s Π 1-χ_ s Wt-1,i. The first order condition for the optimal reset wage is
Et[sum from k = 0 to ∞ of (βξs)k(uprime(c)s,t+k(1-τt+kN-τt+kW_ h)(tilde W s,t)/(PC,t+k)((PC,t+k-1)/(PC,t-1))χ_ s Π (1-χs)k-(ηs)/(ηs-1)Nt+kζ)Nt+k] = 0
(19)
For details see the Appendix in Gomes et al. (2010).