Final goods are consumption goods QC,t and investment goods QI,t (to save space, we list equations only for consumption goods, as the equations for investment goods are analogous). They are assembled from tradable goods, TTC,t and non-tradable goods, NTC,t using the constant elasticity of substitution technology:
QC,t = [νC1/(μC)TTC,t(μC-1)/(μC)+(1-νC)1/(μC)NTC,t(μC-1)/(μC)](μC)/(μC-1),
(20)
where:
TTC,t = [νTC1/(μTC)HTC,t(μTC-1)/(μTC)+(1-νTC)1/(μTC)IMC,t(μTC-1)/(μTC)](μTC)/(μTC-1).
(21)
Tradable goods are a composite bundle of domestic tradable goods, HTC,t, and imports, IMC,t. Imports, in turn, are also a composite of imports from other regions CO, denoted by IMC,CO,t, and H stands for the home country:
IMC,t = [sum over CO ≠ H of (νIM_ CH,CO)1/(μIMC)(IMC,CO,t(1-ΓIM^ CH,CO((IMC,CO,t)/(QC,t))))(μIMC-1)/(μIMC)](μIMC)/(μIMC-1)
(22)
Parameters μC, μTC, and μIMC are intratemporal elasticities of substitution between the inputs in the production functions, while νC, νTC, and νIMC are weights (quasi-shares) of the inputs into the production functions, with sum over CO ≠ H of vIMCH,CO = 1. The function ΓIMCH,CO ((IMC,CO,t)/(QC,tC)) is the adjustment cost for bilateral consumption imports of country H from country CO, where γIMC determines the magnitude of the cost:
ΓIM^ CH,CO((IMC,CO,t)/(QC,t)) ≡ (γIM^ C)/2((IMC,CO,t/QC,t)/(IMC,CO,t-1/QC,t-1)-1)2.
(23)
A final good's firm chooses the combination of the tradable and nontradable bundles that minimizes the expenditure PHT,tHTC,t + PIMC,tIMC,t + PNT,tNTC,t subject to technology constraints (20) and (21), taking the input prices as given, which yields the following demand functions:
HTC,t = νTCνC((PHT,t)/(PTTC,t))-μTC((PTTC,t)/(PC,t))-μ_ CQ_ C,t
IMC,t = (1-νTC)νC(cfrac PIMC,t PTTC,t)-μTC(cfrac PTTC,t PC,t)-μ_ CQ_ C,t
NTC,t = (1-νC)((PNT,t)/(PC,t))-μCQ_ C,t
(24)
(25)
(26)
IMC,CO,t = νIMCH,CO((PIM,tH,CO)/(PIMC,tΓIM^ CH,COdagger(IMC,CO,t/QC,t)))-μ_ IMC(IMC,t)/(1-ΓIM^ CH,CO(IMC,CO,t/QC,t)) (27)
The term ΓIM^ CH,CO^ dagger((IMC,CO,t)/(QC,t)) in the bilateral import bundle is the derivative of the adjustment cost.