Results from the Dixit/Stiglitz monopolistic competition model

Transcription

1 Results from the Dxt/Stgltz monopolstc competton model Rchard Foltyn February 4, 2012 Contents 1 Introducton 1 2 Constant elastcty sub-utlty functon Preferences and demand General case Cobb-Douglas case ( Dxt-Stgltz lte ) Frms and producton Market equlbrum Introducton The Dxt/Stgltz monopolstc competton model has been wdely adopted n varous felds of economc research such as nternatonal trade. The Dxt and Stgltz (1977) paper actually contans three dstnct models, yet economc lterature has mostly taken up only the frst one (constant elastcty case) and ts market equlbrum soluton. The man results of ths subset of Dxt and Stgltz (1977) are derved and explaned below n order to ad n understandng ths wdespread model. 2 Constant elastcty sub-utlty functon 2.1 Preferences and demand Assumpton 2.1. Preferences are gven by a (weakly-)separable, convex utlty functon u = U(x 0, V (x 1, x 2,..., x n )) where U( ) s ether a socal ndfference curve or the multple of a representatve consumer s utlty. x 0 s a numerare good produced n one sector whle x 1, x 2,..., x n are dfferentated goods produced n another sector. 1 1 Wth weakly-separable utlty functons, the MRS (and thus the elastcty of substtuton) of two goods from the same group s ndependent of the quanttes of goods n other sub-groups (see Gravelle and Rees 2004, 67). 1

2 Dxt and Stgltz (1977) treat three dfferent cases n whch they alternately mpose two of the three followng restrctons: 1. symmetry of V ( ) w.r.t. to ts arguments; 2. CES specfcaton for V ( ) 3. Cobb-Douglas form for U( ) However, throughout the lterature many authors have mposed all three restrctons together n what Neary (2004) calls Dxt-Stgltz lte. The next secton examnes the more general case n whch only restrctons 1 and 2 are mposed. Subsecton brefly looks and the more specal case of Dxt-Stgltz lte General case Frst-stage optmzaton. Assumpton 2.2. In the CES case the utlty functon s gven as [ ] 1/ρ u = U x 0, wth ρ (0, 1) to allow for zero quanttes and ensure concavty of U( ). ρ s called the substtuton or love-of-varety parameter. 2 Furthermore, U( ) s assumed to be homothetc n ts arguments. Snce U( ) s a separable utlty functon, the consumer optmzaton problem can be solved n two separate steps: frst the optmal allocaton of ncome for each subgroup s determned, then the quanttes wthn each subgroup. Defnton 2.1. Let y be a quantty ndex presentng all goods x 1, x 2,..., x n from the second sector such that [ ] 1/ρ y x ρ. (1) Fgure 1 shows some llustratve examples for the two-dmensonal case. As requred, the utlty functon s concave and x 1, x 2 are nether complements nor perfect substtutes f ρ (0, 1). The frst-stage optmzaton problem s gven as follows: x ρ max. U(x 0, y) s.t. x 0 + q y = I where I s the ncome n terms of the numerare and q s the prce ndex of y. 3 2 Wth ρ = 1, x 1,..., x n are perfect substtutes as the subutlty functon smplfes to V (x 1, x 2,..., x n) = x and thus t does not matter whch x s consumed. For ρ < 0 they are complements (see Brakman et al. 2001, 68). 3 The ncome conssts of an ntal endowment whch s normalzed at 1, plus frm profts dstrbuted to consumers or mnus a lump sum requred to cover frm losses. However, snce the dscusson here s lmted to the market equlbrum case, frms make zero proft so I = 1. 2

3 Fgure 1: CES functons wth two-dmensonal doman and varyng ρ parameter (from left to rght: ρ = {-10, 0.5, 0.99}). The frst case s excluded n ths model due to the restrctons on parameter ρ. From the Lagrangan L = U(x 0, y) λ[x 0 qy I] we obtan the frst-order condtons From (2) and (3) we get the necessary condton = U 0 λ = 0 x 0 (2) y = U y λq = 0 (3) λ = x 0 + qy I = 0 U y U 0 = q = p y p 0 (4) whch s the famlar U /U = p /p as q s the prce ndex for y and p 0 = 1 due to the numerare defnton. Snce U( ) was assumed homothetc, ths unquely dentfes the share of expendture for x 0 and y because these solely depend on relatve margnal utltes. Denote the share of expendture on y as s(q) and that on x 0 as (1 s(q)). Then the optmal quanttes for each sector are x 0 = (1 s(q))i y = s(q)i q (5) Second-stage optmzaton. s Gven the defnton of y and s(q), the second-state problem max. y = s.t. [ x ρ ] 1/ρ p x = s(q)i 3

4 The Lagrangan L = ( xρ )1/ρ λ( p x s(q)i) yelds the frst-order condtons = y 1 ρ x ρ 1 λp = 0 x (6) λ = p x s(q)i = 0 (7) Solvng (6) for x gves x = y(λp ) 1/(ρ 1) (8) Insertng ths nto (7) and solvng for λ we get p y(λp ) 1/(ρ 1) = s(q)i λ 1/(ρ 1) y = s(q)i λ 1/(ρ 1) = s(q)i) y [ ] 1 (9) Fnally, pluggng (9) back nto (8) we get the prelmnary demand functon facng a sngle frm n the second sector: 4 x = s(q)ip 1/(ρ 1) 1 (10) To further smplfy ths expresson, take (10) to the power of ρ and sum over : 5 x ρ = [ ] 1/ρ y = x ρ = From (5) and (11) we obtan x ρ = (s(q)i)ρ ρ 1 (s(q)i) ρ s(q)i [ ρ ] 1/ρ s(q)i y = [ ] (ρ 1)/ρ (11) pρ/(ρ 1) Result 2.1. For the utlty functon gven n Assumpton 2.2 and the composte quantty ndex from Defnton 2.1 the correspondng prce ndex s q = [ ] (ρ 1)/ρ (12) 4 The summaton ndex has been changed to to reflect that t n s unrelated to. 5 The last step follows from the fact that both sums are dentcal, regardless whether the ndex or s used. 4

5 Usng Result 2.1, (10) can be smplfed to arrve at Result 2.2. For the utlty functon gven n Assumpton 2.2, the resultng demand functon facng a sngle frm s [ ] 1/(1 ρ) q x = y (13) p wth y and q defned n (1) and (12), respectvely. Some remarks regardng the demand functon and CES preferences are n order. Frst, by pluggng y = s(q)i/q nto (13) and takng logs, t can easly be seen that the varetes x 1, x 2,..., x n have unt ncome elastctes log x / log I. Second, assumng a suffcently large number of varetes so that prcng decsons of a sngle frm do not affect the general prce ndex, the prce elastcty of demand for x s ɛ d = log x log p = 1 (14) q const. ρ 1 At ths pont t s convenent to defne σ 1/(1 ρ) so that ɛ d = σ. 6 Thrd, to get the elastcty of substtuton between two varetes, from (6) we see that x x = [ ] 1/(1 ρ) p and hence the elastcty of substtuton can be obtaned as Ths can be summarzed as p ɛ s = log(x /x ) log(p /p ) = log(p /p ) 1/(1 ρ) = 1 log(p /p ) 1 ρ = σ (15) Result 2.3. Dxt-Stgltz preferences gven n Assumpton 2.2 result n constant demand and substtuton elastctes gven by ɛ d = 1 ρ 1 = σ ɛ s = 1 1 ρ = σ Often the model s specfed drectly n terms of σ nstead of ρ, 7 wth u = U(x 0, y), y [ x 1 1/σ ] 1/(1 1/σ), q [ p 1 σ ] 1/(1 σ) Fourth, to see why the CES utlty specfcaton s called varety-lovng, nspect the largesubgroup case wth many varetes n wth smlar prce levels,.e. p p and hence x = x. Then expendture s equally dvded over all varetes x 1, x 2,..., x n snce they symmetrcally 6 Here σ s dfferent from σ(q) n Dxt and Stgltz (1977), but reflects the notaton of many other Dxt- Stgltz-based models. 7 For example, see Baldwn et al. (2005, 38). 5

6 enter nto the subutlty functon. smplfy to If there exst n varetes, the expressons for y and q [ n ] 1/ρ y = x ρ = xn 1/ρ (16) [ n ] (ρ 1)/ρ q = = pn (ρ 1)/ρ (17) Pluggng (5) and (17) nto (13) gves a smplfed demand functon for the large-subgroup case: x = s(q)i np. (18) Substtutng ths for x n the subutlty functon (1), we obtan [ n [ ] ] ρ 1/ρ s(q)i V (n) = y = np (1/ρ) s(q)i = n np = n 1/ρ 1 (nx) whch s ncreasng n n as ρ (0, 1) by assumpton. The last equalty provdes some ntutve nsghts: snce (nx) s the actual quantty produced, the term n 1/ρ 1 > 1 can be seen as an addtonal bonus, so varety represents an externalty or the extent of the market. Increasng the market sze nx has a more than proportonal effect on utlty due to ths term (Brakman et al. 2001, 68). That utlty ncreases wth varety can also be seen by recallng that V (x) = y = s(q)i/q and examnng how q from (17) changes wth n, as shown n Fgure 2. Fgure 2: Prce ndex q as a functon of the number of varetes n (assumng p = 1) q Ρ Ρ Ρ n It s evdent that for constant expendture, the prce ndex falls rapdly, wth utlty rsng as a consequence. Ths effect s more pronounced for ρ closer to 0, whch can ntutvely be explaned usng Fgure 1: for ρ close to 1, all varetes are close substtutes and hence ntroducng another smlar varety only moderately ncreases utlty. The converse s true for ρ close to 0. 6

7 2.1.2 Cobb-Douglas case ( Dxt-Stgltz lte ) In ths secton we nspect a specal case of the model n whch all three of the ntally mentoned restrctons on utlty are mposed. Assumpton 2.3. If U( ) s Cobb-Douglas and V ( ) s CES, the resultng utlty functon s gven by u = U(x 0, y) = x 1 α 0 y α. Agan a two-state optmzaton approach s applcable. Frst-stage optmzaton. The maxmzaton problem s stated as follows: max. s.t. u = x 1 α 0 y α (19) x 0 + qy = I As n (4), the necessary condton from the Lagrangan s U y /U 0 = q, whch together wth the budget constrant yelds the well-known result for Cobb-Douglas utlty: x 0 = (1 α)i (20) y = αi q (21) Second-stage optmzaton. From here the second-stage optmzaton proceeds exactly as n the general case, wth α replacng s(q). Usng the defnton of q one arrves at the demand functon gven n Result 2.2. Wth the Cobb-Douglas / CES case t can easly be verfed that a sngle-stage optmzaton process yelds the same results. The Lagrangan n ths case s L = x 1 α 0 wth the relevant frst-order condton beng [ = x (1 α) α 0 x ρ [ ] α/ρ [ x ρ λ x 0 + x ρ p x I ] ] (α ρ)/ρ ρx ρ 1 λp = 0 Dvdng the frst-order condtons for x and x, multplyng by p and summng over the 7

8 demand functon can be obtaned: I x 0 = x x = [ ] 1/(ρ 1) p p p x = x p = p 1/(1 ρ) p 1/(1 ρ) x x x = (I x 0)p 1/(ρ 1) pρ/(ρ 1) = I x 0 q [ q = y p q 1/(1 ρ) p 1/(1 ρ) ] 1/(1 ρ) [by def. of q] 2.2 Frms and producton It s assumed that all frms producng varetes of x have dentcal fxed and margnal costs. Snce consumers demand all exstng varetes symmetrcally, any new frm enterng the market wll choose to produce a unque varety and explot monopolstc prcng power nstead of enterng nto a duopoly wth an exstng producer. Also, every frm wll choose to produce one varety only (see Baldwn et al. (2005, 42) on how to derve ths result). Producton for each frm exhbts (nternal) ncreasng returns to scale. Ths s mpled by ntroducng fxed costs n addton to (constant) margnal costs as stated above. Hence the cost functon has the form C(x) = cx + F (22) where c s the margnal costs and F the fxed cost per varety (there are no economes of scope) Market equlbrum Equlbrum n ths model s determned by two condtons: frst, frms maxmze profts consstent wth the demand functon (13); second, as ths creates pure proft whch nduces new frms to enter the market, quanttes of x adust untl the margnal frm ust breaks even (free entry condton). Proft maxmzaton. Snce each frm produces a unque varety, monopolstc prcng apples and each frm faces the maxmzaton problem max. π = p(x)x cx F (23) It s assumed that each frm takes prce settng behavor of other frms as gven (other frms do not adapt ther prces as a reacton to the frm s prce) and that frms gnore effects of ther prcng decsons on the prce ndex q. Agan, ths assumpton s only plausble wth a suffcently large number of frms. 8 As all frms have dentcal cost functons, face dentcal demand functons and all varetes enter symmetrcally nto the utlty functon, subscrpts wll be omtted from now on,.e. x = x, p = p. 8

9 The necessary frst-order condton resultng from (23) s the well-known ] p [1 + 1ɛd = c [ p 1 1 ] = c σ where ɛ d s the elastcty of demand, whch was shown to equal σ = 1/(ρ 1) n Result 2.3. Solvng for p, we obtan Result 2.4. In equlbrum, the optmal prce s gven by p e = c ρ, where p e s calculated as a constant mark-up over margnal cost c. Free entry condton. As the model assumes free entry, new frms wll enter the market and produce a new varety as long as ths yelds postve proft. When a frm enters the market and starts producng a new varety, consumers dvert some of the expendture prevously spent on exstng varetes to purchase the new good. The quantty of each varety sold decreases, as does proft due to rsng average costs. As a consequence, the free entry condton states that n equlbrum the margnal frm (ndexed by n) ust breaks even,.e. operatng proft equals fxed cost: 9 (p n c)x n = F (24) Wth symmetry and dentcal frms, condton (24) holds for all ntramargnal frms as well. Solvng (24) for x, we get Result 2.5. The free entry condton dctates that n equlbrum the quantty of each varety produced s x e = F p e c = F (σ 1). (25) c Naturally, n equlbrum the number of varetes produced has to be consstent wth the demand functon from (18), and therefore s(p e n e (ρ 1)/ρ ) F = p e n e (p e c) (26) must hold. Ths unquely dentfes an equlbrum f the left-hand sde s a monotonc functon of n, whch s the case f the elastcty w.r.t. n has a determnate sgn. It s assumed to be negatve as the quantty of each varety consumed decreases when more varetes are avalable. See Dxt and Stgltz (1977, 300) for a formal condton for ths to hold. Before fnshng ths secton, some further remarks regardng the equlbrum are necessary. Frst, from (25) t can be seen that equlbrum quanttes are constant and depend on the two cost parameters, F and c, and on one demand parameter, σ, all of whch are exogenously determned. They are ndependent of other factors such as the number of varetes produced. Therefore, aggregate manufacturng output can only ncrease by ncreasng the number of 9 Ignorng nteger constrants, the number of frms n s assumed to be large enough that ths can be stated as an equalty. 9

10 varetes. Ths determnes the outcome of models such as Krugman (1980), where ncreasng the market sze va trade lberalzaton results n more varetes, not hgher quanttes per frm. 10 Second, calculatng equlbrum operatng proft (gnorng fxed costs) as π e = (p e p e ρ)x e π e = (1 ρ)p e x e π e = p ex e σ we see that operatng profts are determned as a constant proft margn 1/σ of revenue p e x e. 10 However, ths result depends on several assumptons, vz. ce-berg trade costs, homothetc cost functons and mll prcng,.e. nvarant mark-ups (see Baldwn et al. 2005, 42). 10

11 References Baldwn, R. E., R. Forsld, P. Martn, G. I. P. Ottavano, and F. Robert-Ncoud (2005). Economc geography and publc polcy. Prnceton, N.J.: Prnceton Unv. Press. Brakman, S., H. Garretsen, and C. van Marrewk (2001). An ntroducton to geographcal economcs. Cambrdge: Cambrdge Unv. Press. Dxt, A. K. and J. E. Stgltz (1977). Monopolstc competton and optmum product dversty. Amercan Economc Revew 67 (3), Gravelle, H. and R. Rees (2004). Mcroeconomcs (3rd ed.). Harlow: Pearson. Krugman, P. R. (1980). Scale economes, product dfferentaton, and the pattern of trade. Amercan Economc Revew 70 (5), Neary, J. P. (2004). Monopolstc competton and nternatonal trade theory. In S. Brakman and B. J. Hedra (Eds.), The Monopolstc Competton Revoluton n Retrospect, pp Cambrdge: Cambrdge Unv. Press. 11