2 Alternative or supplementary dispersion force can be built into the model

2 alternative or supplementary dispersion force can

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2.Alternative (or supplementary) dispersion force can be built into the model:idiosyncratic preference for location, with again similar ”reduced-form”expressions for the spatial equilibrium.ECON280D. Spring 2018. C. GaubertLecture 2Quantitative Spatial Equilibrium Models Toolkit26 / 45
Aside - CES and LOV refresherSuppose demand for varieties of a good is driven by CES preferences:U=(nXi=1[xi]σ-1σ)σσ-1ICES demand leads tolove of varietyIConsumers prefer to diversify their consumptionProof:ISuppose all goods have the same pricep.IThen total expenditure of a consumer isI=pnxx=IpnIIn turn, indirect utility is:u="nXi=1(x)σ-1σ#σσ-1=nσσ-1x=n1σ-1IpIAt constant expenditure: utility is increasing inn, moreso with lowσ.IDecomposedlogudlogn=dlogxdlogn+σσ-1:Iless consumption of each variety (intensive margineffectdlogxdlogn<0)Ibut consumption of an additional variety (extensive margineffectσσ-1>0)Iextensive margin more than offset intensive marginECON280D. Spring 2018. C. GaubertLecture 2Quantitative Spatial Equilibrium Models Toolkit27 / 45
Micro-founded agglomeration: sharing the gains fromvarietyIIn this version of a spatial equilibrium model, we do not assume thatproductivity in cityiisTiLαiIRather, the model delivers endogenously that cities become more productiveas they get largerIA often-used and convenient micro-founded theory of agglomerationIAssumeIIntermediates are non traded (ie, traded only within cities)IFinal good is freely traded between citiesIDrop the city subscript for nowIFocus on one city in isolationECON280D. Spring 2018. C. GaubertLecture 2Quantitative Spatial Equilibrium Models Toolkit28 / 45
Intermediate good production:IFirms are monopolistically competitive, as in Dixit and Stiglitz (1977)IEach firm produces a differentiated goodIFirms have some market power, but they are small (large number ofproducers)ISet prices to maximize profits ...I...but do not take into account the effect of their price on the local price indexIThere are increasing returns to scale at the firm levelIFixed costF(in labor) must be paid to produceIThen constant marginal cost: produceTunits of good with 1 unit of laborECON280D. Spring 2018. C. GaubertLecture 2Quantitative Spatial Equilibrium Models Toolkit29 / 45
Final goods productionIFinal good market is perfectly competitiveIFreely traded across cities, price taken as numeraireIFinal output combines intermediate goods through a constant elasticity ofsubstitution CES production function function, with elasticityσ:Y=ZΩ[x(ω)]σ-1σdωσσ-1ECON280D. Spring 2018. C. GaubertLecture 2Quantitative Spatial Equilibrium Models Toolkit30 / 45
Intermediate firm’s price:IFirm sets price to maximize profits:Maxpπ(p) =D(p)p-wT-FwIWith CES, demand (in quantity) isD(p(ω)) =p(ω)-σP-σYIwithYtotal quantity demanded in the cityIPis the CES price index in the cityP=ZΩ[p(ω)]1-σdω11-σIFindp:p=σσ-1wTIMonopolistic competition + CES demandconstant markup over marginalcostECON280D. Spring 2018. C. GaubertLecture 2Quantitative Spatial Equilibrium Models Toolkit31 / 45

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