COMPETITIVE GROWTH MODEL

Transcription

1 COMPETITIVE GROWTH MODEL I Assumpions We are going o now solve he compeiive version of he opimal growh moel. Alhough he allocaions are he same as in he social planning problem, i will be useful o compare he wo moels. Afer his poin, mos moels will have exernaliies or oher mare failures, which means we will have o solve he compeiive moel irecly. Furher, by fining prices we can gain furher insighs no possible from he social planning problem, which only gives allocaions. A Reain Assumpions All moels from his poin ou will be varians of he opimal growh moel. Therefore, we will reain all he assumpions of he opimal growh moel, excep for he changes noe below. B Firms 1 Capial ownership We will assume househols own capial an labor an ren hem o firms. I is equivalen o assume firms own capial, an househols own shares of soc in he firms. Noe ha firms rening capial is cerainly consisen wih sole proprieorships an also public corporaions. The share of soc esablishes ownership of he firm s capial. 2 Number of Firms Assume here exiss one firm per househol. Wih CRS, he number of firms is ineerminie. Definiion 13 A variable x is ineerminae if any value of x is consisen wih he moel. To see his, change he size of he firm capial an labor by a facor λ. Given consan reurns o scale, proucion will change by a facor λ. In aiion, coss will cerainly change by he same facor λ, since coss are capial an labor coss. Thus profis are unchange. Since firms maximize profis, firms o no care abou he size of he firm. Thus firm size is ineerminae. 49

2 On he oher han, wih increasing ecreasing reurns, an incenive exiss o reuce he number of firms hrough mergers spinoffs. Thus one can view CRS as an equilibrium oucome. 3 Wha are Firms? Firms are enowe wih a proucion echnology F. Oher han ha, firms are simply he way worers an capial owners are organize. C Prices 1 Consumpion Goos Le he price of he consumpion goo, P be normalize o one. Consumpion goos shmoos are hus he numeraire goo. We quoe all oher prices in erms of he numeraire goo ie. my wage is w shmoos per hour. Typically, money is he numeraire, bu we have no money in his moel. 2 Renal raes The renal rae of labor is he wage: w unis of consumpion goos per uni of labor L. The renal rae of capial is he ne reurn on capial r unis of consumpion goos per uni of capial. D Compeiive Economy The moel ha of a compeiive economy: firms an househols ae prices as given. Prices, in urn, equae supply an eman. II Firm Problem A Profi maximizaion Firms maximize profis. I is sraighforwar o show ha owners of he firm prefer he firm o maximize profis. In he moel where firms own he capial, profi maximizaion is equivalen o maximizing he presen iscoune value of a share. 50

3 1 Firm Problem { Π = max 1 F K,L K r K w L},L Here superscrip inicaes firm eman. 2 Firs orer coniions The problem of he firm is saic: all of he ynamic savings are one by he househols. The firs orer coniions are hus: r = F K,L w = F l K,L These equaions eermine an l as a funcion of he prices. They are eman curves. From he firm s poin of view, i is only worh hiring an aiional uni of a facor if he oupu ha uni prouces excees he cos. If hiring one worer resuls in he proucion of 10 more wiges an he worer coss $8 per hour hen hire: profis will rise by $2. As wih he opimal growh moel, we use he properies of CRS o simplify. Noe ha: K F K,L = L F L, Hence: K 1 K F K,L = L F L,1 L = F L,1 = f K K K F l K,L = F L,1 LF L,1 L = f f Hence le = K /L, hen: r = f w = f f

4 3 Zero Profis We have: Π = L f f K f f L = f L +f L = So profis are equal o zero. QUESTION: ARE ECONOMIC PROFITS OR ACCOUNTING PROFITS EQUAL TO ZERO? WHERE ARE ACCOUNTING PROFITS? 4 Equilibrium In equilibrium, supply equals eman: = s. Furhermore, suppose ha L firms exis. Then since all firms are ienical, hey choose ienical amouns of he aggregae capial soc K an labor soc L. Hence equilibrium labor per firm is L /L = 1. Equilibrium oal capial per firm is similarly K /L = K. Therefore in equilibrium: = K L = K L 1 = Hence in equilibrium: r = f w = f f These equaions eermine he prices r an w given he sae. They are equilibrium coniions. Equilibrium coniions eermine prices. III Househol Problem A Normalize Problem Househols maximize: U = max uc =0 The maximizaion is subjec o he buge consrain. The buge consrain ses income 52

5 equal o expenses. Consier a paricular househol, labele h. Normalize: r K h +w L h +1 δkh = C +K h r h +w +1 δ h = ch +h +1 1+η Noice ha he househol will supply all i s capial. There is no sense in no rening a facory. Similarly, he househol has no preference for leisure an hence will ren ou all i s labor. Hence L h = 1 one can hin of his as woring 100% of available ime an ch = C. RULE: for maximizaion you can maximize he conrol variables in any orer, an subsiue in he soluion if you wan. B Recursive Problem 1 Problem We have a new in of variable: prices. Prices are aen as given, in of lie a sae, bu also eermine, lie a conrol bu eermine by equilibrium coniions, no firs orer coniions. Since we have no easy way o eal wih hese, le us ry subsiuing ou. RULE: you can always subsiue one variable eermine in equilibrium for anoher. Le s subsiue ou using he firm firs orer coniions in equilibrum. f h +f f +1 δ h = c ++1 h 1+η Noe ha we have reuce he space by one: if we now he aggregae capial soc, hen we now he wage an ineres rae. We now ivie sae variables ino wo caegories, iniviual saes an aggregae saes. Aggregae saes are similar o iniviual saes in ha hey are given oay bu can change over ime. However, iniviuals o no consier heir effec on aggregae saes over ime, since an iniviual is oo small o maerially affec he aggregae capial soc. Now we have aggregae sae an iniviual sae h. Hence: v h, = max h { u f +f h +1 δ h 1+η h ]+βv h, } Checing hrough he value funcion, is no a sae or conrol. I is everyone else s invesmen ecision, which househol h cares abou because i will affec fuure wage an 53

6 ineres raes. I is eermine in equilibrium. We refer o such variables as aggregae conrols. They are eermine in equilibrium, bu aen as given by he househol he same as prices. Overall: 1. Iniviual saes: given oay, bu he iniviual may change hem over ime. 2. Aggregae saes: given oay, bu everyone s ecisions may change hem over ime. 3. Iniviual conrols: iniviuals may change hese oay. 4. Aggegae conrols or prices: eermine oay via equilibrium coniions. Unaffece by an iniviual. 5. Parameers: consan hrough ime. 6. We have v iniviual saes, aggregae saes. We max over iniviual conrols. For each aggregae conrol/price, we mus have an equilibrium coniion. We nee an equilibrium coniion for each aggregae conrol or price. Here he iniviual ecision will be a funcion h = h h,. Le us suppose anequilibrium in which househols are ienical. If househols are ienical hey hol he same amouns of capial h =, an mae he same invesmen ecisions h =. Thus he equilibrium coniion is: h = implies h = = h, Firs orer coniions an envelopes We have: 1+ηu c f +f h ] +1 δ h 1+η h = βv 1 h, This eermines he iniviual ecision h. Once again, he marginal uiliy of consumpion equals he marginal value of invesmen, ivie by he larger populaion an iscoune bac o oay. The envelope equaion is: v 1 h, = u c f +f h +1 δ h 1+η ]f h +1 δ

7 So again he marginal value of he househol s capial is he gross reurn ne of epreciaion imes he value of he reurn which is he marginal uiliy of consumpion. Noice from he firm firs orer coniion: v 1 h, = u c f +f h +1 δ h 1+η h ]r +1 δ So r in he compeiive moel is a gross ineres rae before epreciaion. The househol pays he epreciaion, i is no aen ou of he firm s ineres paymen. This is jus accouning. If firms hanle he epreciaion an reurne r δ o he househol, he envelope equaion woul be ienical. IV Equilibrium Now impose he equilibrium coniions. RULE: impose equilibrium only afer fining he firs orer coniions an envelope equaions. ] 1+ηu c f +1 δ 1+η = βv 1, v 1, = u c f +1 δ 1+η ]f +1 δ c = f +1 δ 1+η These equaions eermine he aggregae conrols. These compleely eermine he equilibrium allocaions of he compeiive economy. I is raiional o efine he equilibrium: wrie own he sysem of equaions an variables which consiue an equilibrium. The proceure: given all saes, an equilibrium is he se of iniviual ecisions, aggregae ecisions, prices an a value funcion, such ha firms an househols opimize, buge consrains are saisfie, resource consrains are saisfie, an he Bellmans equaion hols. In he efiniion, we shoul see he number of variables eermine equal he number of equaions. This is a goo chec. Here: Definiion 14 A Recursive Compeiive Equilibirum given an iniviual sae h an an aggregae sae, is a se consising of iniviual ecisions h an c h, aggregae ecisions an c, prices r an w, an a value funcion v such ha firms opimize equaions an 55

8 hol, househols opimize equaion hols, he buge consrain is saisfie, he Bellman s equaion hols, an iniviual ecisions are consisen wih aggregae oucomes equaion hols. Le us now verify we have a well-efine sysem of equaions. We have 7 unnowns 2 iniviual ecisions, 2 aggregae ecisions, 2 prices, an v. We have wo firm firs orer coniions, 1 househol firs orer coniion, 1 buge consrain, an 1 Bellman s equaion. So we have 5 equaions so far. The equilibrium coniion gives us wo more equaions, one applie o he buge consrain an anoher o he househol firs orer coniion. So we have 7 equaions for 7 unnowns. V Welfare A Psueo Planning problem PSP Recall he equilibrium allocaions are: ] 1+ηu c f +1 δ 1+η = βv 1, v 1, = u c f +1 δ 1+η ]f +1 δ c = f +1 δ 1+η A Psueo planning problem is a planning problem which generaes he same allocaions as he compeiive equilibrium. We can consruc such a problem as follows: { ] } f +1 δ 1+η ν = max u The firs orer coniion an envelopes are: +βν ] 1+ηu c f +1 δ 1+η = βν ν = u c f +1 δ 1+η ]f +1 δ

9 c = f +1 δ 1+η Le ν be such ha ν = v 1,, hen he PSP an he compeiive equilibrium have he same allocaions. The main use of he PSP is o come up wih a planning problem which is easier o wor wih han he compeiive moel. I ofen illusraes why a compeiive equilibrium is no welfare maximizing. A secon use is for comparaive saics. I is generally easier o compue he erivaive h han h 1,+h 2,. B Welfare heorems Noice ha he PSP is he social planning problem. We have hus shown ha he Compeiive equilibrium wih ienical househols maximizes welfare. There is no role for governmen in his problem. Se up a price sysem an allow househols o mae iniviual ecisions ha benefi hemselves an overall welfare of sociey is maximize. In he fuure we will have exernaliies or isorions an so in general he Social Planning problem iffers from he PSP. In his case, however, ofen we can pu in a ax or subsiy o equae he allocaions. 57