Product Di¤erentiation1 IntroductionWe have seen earlier how pure external IRS can lead tointra-industry trade.Now we see how product di¤erentiation can provide abasis for trade due to consumers valuing variety.When trade occurs due to product di¤erentiation, evenidentical c ountries will trade by exchanging di¤erent va-rieties of the same good.The value consumers place on variety generates anothersource of gains from trade.IRS due to a …xed cost of producing each variety limitsthe number of varieties produced by a country.Two key versions of modeling preferences for the di¤er-entiated products are the love of variety approach andthe ideal variety (bliss point or spatial) approach.Both provide a subutility function that increases in thenumber of varieties available, but the love of variety a p-proach is easier to employ.2 Product Di¤erentiation (Helpmanand Krugman 1985)The CES utility function has proved very useful in modelsof product di¤erentiation.The typical form of modeling preferences is to assume anupper-tiered utility functionu(x0; V ) = U (x0; V (x1; :::; xn))where x0is consumption of some homogeneous numerairegood, x1; :::; xnare consumptions of n di¤erentiatedgoods, and V is a sub-utility function for a set of dif-ferentiated products.Utility is separable between the set of di¤erentiated goodsand the numeraire.Apply a two stage budgeting procedure to allocate spend-ing across di¤erentiated products and then between theset of di¤erentiated products and the numeraire.We assume that preferences are homothetic between thenumeraire good x0and the set of di¤erentiated goodsx1: : : xnso consumers spend a …xed share of their in-come on the two categories of goods.Suppose the upper tier utility function is Cobb-Douglasin the numeraire good and the set of di¤erentiated goodsu(x0; V ) = x0V1so the elasticity of substitution between the di¤erentiatedgoods and the numeraire good equals one.Normalizing the price of the numeraire good to one p0=1, the consumer’s budget constraint sets expenditure equalto incomex0+nXi=1pixi= Iwhere piis the price of good i and I is income in termsof the numeraire good.When preferences are homothetic, the consumer spendsa …xed proportion of income I on the two set of goods:x0= I on the numeraire good andPni=1pixi= (1 )I on all the di¤erentiated goods.LetE  I  x0= (1  )Ibe expenditure on the set of di¤erentiated products.Budget constraint for spending on di¤erentiated goods isnXi=1pixi= E2.1 Love of VarietySuppose the sub-utility function is a symmetrical CESfunctionV =0@nXi=1xi1A1;  < 1This subutility function has several nice properties: Every pair of varieties is equally s ubstitutable: =11  > 1 !  = 1 1 Degree of substitution does not depend upon thelevel of consumption of the goods. Variety has value. Suppose n varieties are availableat the same price p. Then the consumer buys equalamounts of all goods. Subutility can be written asV =0@Xixi1A1= (nx)1="n Enp!#1= n1Enp= n11Epand increases as the number of varieties n increases.@V@n= 1 1!Vn> 0If the number of di¤e rentiated goods is large, the set ofdi¤erentiated goods may be represented by a continuum,so the sum is then replaced by an integral in the subutilityfunction,V =Zn0xidi1The goal is to maximize subutility V subject to the bud-get constraintZn0pixidi = EGiven an expenditure E on the di¤erentiated goods (fromthe …rst stage), the consumer’s problem becomesmaxZn0xidi1+ E Zn0pixidiThe …rst order condition for good i is1Zn0xidi11x1i pi= 0and similarly for another good j1Zn0xidi11x1j pj= 0The above two …rst order conditions implyxixj="pjpi#11="pjpi#If goods are equally priced, then they will be equally de-mandedpi= pj!pjpi= 1 !xixj= 1 ! xi= xjFor a CES utility function, the elasticity of substitutionbetween two di¤erent varieties is  =11.Using demand functions,E =Zn0xjp1ipjdi = xjpjZn0p1idiimpliesxj=EpjRn0p1idi=Ep11jRn0p1idiAn individual …rm viewsRn0p1idi as …xed and thusfaces a constant elasticity dema nd curvexj= kpjwherek ERn0p1idiwith demand elasticity equal to the elas ticity of substitu-tion  between a pair of the di¤erentiated goods.Each …rm chooses the price of its variety to maximize itspro…ts, taking as given the price charged by other …rms.Assume that every variety is produced with the same pro-duction function.Focus on a representative …rm (producing a unique vari-ety), whose problem is to pick its price to maximize itspro…ts = px  C (x)Suppose the cost function takes the forms of a …xed costplus a constant marginal costC(x) = b + cxThen pro…ts are =(p  c) x  b = (p  c) kp bThe …rst order condition for pro…t m aximization isp1 1= cwhich implies that all varieties are priced equally atp =c1 1and in the limit as elasticity bec omes in…nite, price equalscost and lim!1p = c.With all varieties priced equally, the budget constraintnxp = E implies that consumers evenly spread consump-tion over all available varietiesx =EnpA zero pro…t condition  = 0 then pins down the measureof varieties available.n =EbThe measure of varieties available decreases in the elas-ticity  and the …xed cost of variety b@n@= Eb2,@n@b= Eb2as intuitively should happen based on cost and bene…t.3 Krugman AER 1980 Explanation for trade between countries with similar(even identical) factor endowments (same technol-ogy and tastes too). Ha ving a large domestic market operates as a sourceof comparative advantage (export goods for whichhave a larger domestic market than other goods rel-ative to ROW). Model derived from the famous Dixit and Stiglitz(1977) model of monopolistic c ompetition (horizon-tal product di¤erentiation).3.1 Consumers A large number of potential goods enter symmetri-cally into utility according to the utility functionU =nXi=1ci; 0 <  < 1 (1)where ciis consumption of good i. Utility exhibits love of variety. Preference s exhibit a constant elasticity of substitu-tion between any two goods. Consumers choose their consumptions cito maxi-mize their utility (1) subject to the budget constraintnXi=1pici= Iwhere piis the price of good i and I is income.