MONOPOLISTIC COMPETITION MODEL

Transcription

1 MONOPOLISTIC COMPETITION MODEL Key gredets Cosumer utlty: log (/ ) log (taste for varety of dfferetated goods) Produto of dfferetated produts: y (/ b) max[ f, ] (reasg returs/fxed osts) Assume that good, the agrultural good, s produed wth perfet ompetto ad the ostat returs to sale produto futo y, but that there are maufatured goods that are produed wth moopolst ompetto ad the reasg returs to sale produto futo spefed above The represetatve osumer solves max log (/ ) log s t p p w u/ p u/ p (MRS equals pre rato) / p / p Produer of good solves the osumer s problem to fd the dret demad futo: p (multplyg by ) p p p (summg over )

2 p p (smplfyg) p w (from prevous equato ad budget ostrat) w p Idret demad futo p w p = Profts of frm : py wby wf (reveue-varable osts -fxed osts) We suppose that the frm hooses ts output y to maxmze ts profts, assumg that the outputs of all other frms are ostat ad that pres wll adust to lear the markets of eah good (Ths s the Courot ompetto assumpto) To maxmze profts, the frm sets MR MC: We set y the dret demad futo (ths s the assumpto that the pre of good adusts to lear the market for good ) ad plug ths futo to expresso for profts: w y y wby wf y To maxmze profts, the frms sets the frst dervatve of ths expresso equal to, that s, MR MC : Set w as umerare w ( y ) y y y wb ( y )

3 Se frms are symmetr, we kow that there s a equlbrum whh y y f y : The profts of typal frm are ( y ) y y y b ( y ) ( ) b y ( ) y b y b p p y y ( ) ( ) pyby f f We assume that there s free etry/ext utl profts equal zero: f ( ) The profts of typal frm are ( ) pyby f f We assume that there s free etry/ext utl profts equal zero: f ( ) ( ) ( ) 4( )( f ) 4 f 3

4 Equlbrum A equlbrum of the moopolst ompetto model s the umber of maufaturg frm ˆ, a pre ˆp for the agrultural good, a pre p ˆ for eah maufaturg frm that operates at a postve level, a wage rate ŵ, a osumpto pla ˆ, ˆ ˆ ˆ,,, ˆ, produto plas, ŷ, ˆ for the agrultural good ad y ˆ, ˆ for eah maufaturg frm that operates at a postve level suh that Gve pˆ, pˆ, pˆ,, p ˆ ˆ, ad ŵ, the osumer hooses ˆ, ˆ, ˆ,, ˆ ˆ to solve pˆ wˆ, f yˆ ˆ max log (/ ) log s t p ˆ p ˆ wˆ Gve the dret demad futo p (,,,, ) that omes from solvg the represetatve osumer s utlty maxmzato problem, frm hooses y ˆ to solve max p ( yˆ,, y,, yˆ ) y wby ˆ wf ˆ ˆ pˆ pˆ ( yˆ,, yˆ,, yˆ ) py ˆ ˆ ˆ ˆ ˆ, f ˆ wby wf y where pˆ ˆ ( ˆ ˆ ˆ p y,, y,, y) ŷ ˆ yˆ (/ b) max[ ˆ f, ],,,, ˆ ˆ yˆ,,,,, ˆ ˆ ˆ ˆ 4

5 Numeral example b, f, /, (45) 4(45)(4) 8 7 y 5, p 3333 p w, y 45 Utlty: / log 45+log(7(5) ) 7496 Homogeous of degree oe represetato of utlty (a real ome dex): / exp[(/ )(log 45+log(7(5) ))] exp[(/ )(7496)] 444 5

6 A tegral umber of frms? There s a problem wth our oept of equlbrum f the umber of frms, ˆ, does ot tur out to be a teger Suppose, for example, that The, whe we solve b, f, /, (45) 4(45)(4), 8 we obta ˆ 634 How do we terpret ths soluto? There are two approahes that we ould take: We ould restrt ˆ to be a teger, ad let t be the largest umber of frms for whh profts are oegatve I ths ase, however, there a be postve profts equlbrum These profts eed to be eared by someoe If we gve them to the represetatve osumer, the the osumer s budget ostrat beomes p ˆ p ˆ ˆ ˆ w ˆ where ˆ are profts Everythg beomes a more omplated eve ths smple model wth oly oe market wth moopolst ompetto Thgs beome muh more omplated appled models wth may suh markets We ould thk of ˆ as beg a teger up utl we ompute the umber of frms, at whh we pot we smply alulate a real umber Ths s the approah that eoomsts typally use applyg ths sort of model 6

7 Reterpretg the model as a model of teratoal trade We a reterpret ths model as a model of teratoal trade amog outres that are detal exept for ther szes as measured by ther labor fores, Cosder the umeral example whh b, f, / ad there are two outres, oe whh 44 ad the other whh 49 (We a thk of these outres as beg the Uted States ad Caada respetvely) I the tegrated equlbrum of the world eoomy p w 45 (45) 4(45)(4) ( ) y 9367 b 634 y b 634 p 37 y y ( ) y 45 p To alulate osumpto of eah varety eah outry, we ust dvde the world produto of the varety y proportoally I outry, for example, 44 y We also dvde the produto ad the osumpto of the agrultural good proportoally: ˆ ˆ (44/ 49) ˆ (44/ 49) (49 / 49) ˆ (49 / 49) , ad yˆ (44/ 49) yˆ (44/ 49)45 5 yˆ (49 / 49) yˆ (49 / 49)45 45 (Strtly speakg, there s othg ths model that ps dow the loato of produto of the agrultural good We are alulatg a symmetr equlbrum) 7

8 Trade Equlbrum ˆ ˆ p p wˆ ˆ y y outry outry ˆ ˆ Utlty: Real ome dex: / uˆ log 5 log 634(743) 433 / uˆ log 45 log 634(937) 989 uˆ / e 5 uˆ / e 3557 (Note that, ot surprsgly, the real ome outry s 9 tmes greater tha that outry ) Gas from Trade To alulate the gas from trade, we a ompute the autarky equlbra for both outres (We have already alulated ths equlbrum for outry Autarky Equlbrum ˆ ˆ p p wˆ ˆ y y outry outry ˆ ˆ Utlty: Real ome dex: / uˆ log 5 log 5675(93) 48 / uˆ log 45 log 7(5) 7496 uˆ / e 5748 uˆ / e 444 The smaller outry, outry, has the most to ga from trade: I outry, real ome goes up by 54 peret (5 / ) I outry, real ome goes up by 94 peret (3557 / ) 8