Effects of Entry Restriction on Free Entry General Competitive Equilibrium. Mitsuo Takase

Transcription

1 CAES Working Pper Series Effects of Entry Restriction on Free Entry Generl Competitive Euilirium Mitsuo Tkse Fculty of Economics Fukuok University WP Center for Advnced Economic Study Fukuok University (CAES) Nnkum Jonn-ku Fukuok JAPAN

2 1 Effects of Entry Restriction on Free Entry Generl Competitive Euilirium Mitsuo Tkse Septemer Fculty of Economics Fukuok University Jonn-ku Nnkum Fukuok JAPAN Emil: mtkse@fukuok-u.c.jp

3 2 Astrct This pper exmines effects of entry restriction in one mrket on the economy. I uild generl competitive euilirium model under free entry s se model nd show tht the restriction reduces the household utility totl output mesured y prices t which free entry is llowed nd the lor productivity of the restricted mrket. This pper suggests tht the competitive euilirium cn chieve Preto efficiency if profits of ll firms ecome zero. As long s ny firms hve positive profits there is room for improving the economic welfre through free entry. As mcroeconomic impliction the restriction my cuse infltion under recession. Key wrds: Entry restriction Generl competitive euilirium model Totl output JEL code: A10 C30 D50 D51.

4 3 INTRODUCTION The ojective of this pper is to exmine effects of entry restriction in generl euilirium model under perfect competition. Entry restriction inevitly entils resource shifts from one mrket to the others. If resource endowments re fixed nd fully employed entry restriction reduces the numer of firms in the ffected mrket. Prtil euilirium nlyses ssume wy these shifts of resources mong mrkets. A generl euilirium nlysis shows effects of the entry restriction on the economy s whole nd these my hve the long-run perspective. According to the trditionl prtil euilirium nlysis n entry restriction shifts the mrket supply curve leftwrd decreses the untity of output increses the price nd decreses socil welfre or socil surplus consisting of consumer nd producer surplus. This pper exmines whether we cn confirm these conventionl conclusions of prtil euilirium model even in generl euilirium setting nd shows differences from the prtil euilirium. This pper is sed on the generl competitive euilirium model which Arrow nd Dereu (1954) hve developed. There re some studies which discuss effects of new entry on mrkets under imperfect competition. Mnkiw nd Whinston (1986) find excessive entry or inefficiency of free entry under imperfect competition. Hopenhyn (1992) considers n nlyticl model where entry nd exit re determined endogenously in long-run prtil euilirium setting where ggregte demnd is given nd the udget constrint of household sector is not included in the model. Hopenhyn nd Rogerson (1993) consider model where entry nd exit re determined endogenously. Although their model consists of two mrkets which re one output mrket nd lor mrket they consider the sustitution etween goods consumption nd leisure time(unemployment). The utility function is

5 4 given nd they hve not exmined how different prmeters of the utility function ffect simultion results. This pper exmines welfre effects of entry restriction in generl competitive euilirium model y using simultion method. In order to seek the roustness of the simultion results four vlues of the elsticity of sustitution re exmined in the use of utility function. Tkse (2011) nlyticlly shows tht the entry restriction reduces the household utility in competitive generl euilirium model. This implies tht Preto efficiency cnnot e chieved in competitive generl euilirium model unless free entry is llowed. He ssumes tht the production set is convex where the production function exhiits monotoniclly incresing with lor input nd strictly concve downwrd from the origin. This pper ssumes tht the production set is non-convex where the production function exhiits monotoniclly incresing concve upwrd to n inflection point nd concve downwrd from tht point. Becuse the production set is non-convex it ecomes hrd to nlyze the model nlyticlly. Then this pper nlyzes the model y simultion nd presents wht the simultion shows including the issue whether the entry restriction reduces the economic welfre even if competitive euilirium is ttined. In order to discuss these issues this pper is orgnized s follows. Section II presents generl euilirium model of two goods nd one fctor. Section III shows the results of simultion nlysis of the model. Section IV summrizes the min results nd drws conclusions. Ⅰ. MODEL The model is sed on generl competitive euilirium model developed y Arrow nd Dereu (1954) where they hve proved the existence of euilirium in the model of mny goods

6 5 nd mny fctors. Arrow nd Hhn (1971) prove the uniueness of the euilirium under vrious sufficient conditions where the production set is convex or pproximtely convex. Tkse (2011) dopts their model for comprtive sttics in chnging the numer of firms in one mrket sed on the generl competitive euilirium model where the production set is strictly convex. He shows tht reducing the numer of firms in one mrket y the restrictive policy reduces the household utility level nd the totl output mesured y the prices which re those efore strengthening the restriction. This pper extends his model to free entry generl euilirium model where the production set is non-convex. The production function in this pper exhiits monotoniclly incresing concve upwrd to n inflection point nd concve downwrd from tht point. Then ech firm hs U-shped mrginl cost curve nd its positive minimum verge cost. And this mkes it possile for the numer of firms to e endogenous in the model. As se cse in this simultion nlysis the numer of firms in ech goods mrket ecomes endogenous vrile. This free entry condition is given y constrints tht the profits of ll firms re zero. The model consists of two goods nd one input. All of mrkets in the economy re perfectly competitive nd then ech economic entity is price tker. Ech firm is ssumed to mximize its profits. Ech household is ssumed to mximize its utility level suject to the udget constrint. I ssume tht firms in ech mrket hve the sme size ut my differ etween mrkets. I lso ssume tht ll households hve the sme income level from given lor hours nd the sme dividend income. Here the dividend income comes from firm's profits. The totl profits re ssumed to e shred eully mong ll households.

7 6 To show difference from the prtil euilirium nlysis I illustrte effects of entry restriction on oth mrkets in the following Figures 1 nd 2. P D D S S P D D S S P P * P * O Q * Q O Q Q Figure 1: Good A Mrket Figure 2: Good B Mrket Q * Q Figure 1 nd Figure 2 show mrket supply nd demnd curves in Good A mrket nd Good B mrket respectively. When oth mrkets re under free entry nd perfect competition where the mrket supply nd demnd curves denoted S nd D in the Good A mrket nd S nd D in the Good B mrket respectively the euilirium price nd untity of ech mrket re given s P * nd Q * in Good A mrket nd P * nd Q * in Good B mrket respectively. After imposing restriction on the numer of firms in Good A mrket the mrket supply curve of Good A shifts leftwrd to S. Since this price chnge cuses reduction of household purchsing power nd lowering reltive price of the other mrket the demnd curve for Good A shift leftwrd to D. The new euilirium price nd untity ecome P nd Q. This reltive price chnge lso ffects the demnd curve for Good B. The demnd curve for Good B my shifts rightwrd or leftwrd. It depends on positive sustitution effect nd negtive income effect of the price rise. Figure 2 illustrtes cse of leftwrd shift of the demnd curve

8 7 to D. Since Good B mrket is under free entry the increse of demnd for Good B ttrcts new entry nd ccordingly the mrket supply curve shifts rightwrd to S. The new euilirium price nd untity ecome P nd Q. One of the differences from prtil euilirium model is tht generl euilirium model tkes ccount of effects of the other mrket. In this cse the entry restriction my not only reduce output of the trget mrket ut lso increse output of the other mrket. A generl euilirium model plys n importnt role for understnd externl effects on the economy s whole. Since the production set is ssumed to e non-convex there re firm's shutdown points where the verge cost reches the minimum. Without the entry restriction ll firms operte t zero profits where the price is eul to the minimum of verge cost curve. This simultion nlysis exmines the cses where the numer of firms in one mrket is restricted to certin numer nd shows effects of this entry restriction on the economy. In the simultion I try to chnge the prmeters of the utility function for exmining the roustness of the simultion results. A. Firms Ech firm is perfectly competitive nd employs lor s fctor of production. The production function ssumed to e monotoniclly incresing convex function in the smll production level nd ecomes concve function in the lrger production level so tht its mrginl cost curve shows U-shped curve. There re two goods nd one fctor in the economy nd these re presented s the good A the good B nd lor respectively. 1. A Firm in the Good A Mrket

9 8 given s The reltion etween lor input nd output of the good A stisfies the following eution l 2 (1) where is the untity of the good A produced y ech firm l is the mount of lor employed y ech firm to produce the good A. Figure 3 shows the reltion etween lor input nd output of the good A. O l Figure 3: The reltion etween input nd output The mrginl cost curve nd the verge cost curve re U-shped given s Figure 4 nd ll firms produce outputs t the minimum point of verge cost curve under free entry. MC AC MC AC O Figure 4: MC nd AC curves

10 9 The totl untity of the good A in the mrket s whole is given s Q n (2) where n represents the numer of firms producing the good A nd it is ssumed tht ll firms in the mrket of the good A re eul in size. The profit of ech firm producing the good A is defined s p wl (3) where is the profit of ech firm producing the good A p is the price of the good A nd w is the wge rte. The totl cost is otined y multiplying oth sides of Eution (1) y the wge rte. We get the totl cost function s C 3 w 2 2. By differentiting this with we get the mrginl cost. Ech firm produces output to mximize its profit. function. Then the mrginl cost is eul to the price of output. The first order condition for profit mximiztion is given s C p or p w (4) s The totl profit of the mrket s whole is given s n (5) where is the totl profit of the mrket for the good A.

11 10 2 A Firm in the Good B Mrket The reltion etween lor input nd the good B stisfies the following eution s given s l 2 (6) where is the untity of the good B produced y ech firm l is the mount of lor employed y ech firm to produce the good B. The totl output of the good B in the mrket s whole is given s Q n (7) where n is the numer of firms producing the good B nd it is ssumed tht ll firms in the mrket re eul in size. The profit of ech firm producing the good B is given s p wl (8) where is the profit of ech firm in the mrket nd p is the price of the good B. The totl cost is otined y multiplying oth sides of Eution (6) y the wge rte. We get the totl cost function s C 3 w 2 2. By differentiting this with we get the mrginl cost. Ech firm produces output to mximize its profit. function. Then the mrginl cost is eul to the price of output. The first order condition for profit mximiztion is given s C p or

12 11 p w (9) where the vlue of mrginl product of lor is eul to the wge rte. The totl profit of ll firms in the mrket s whole is given s n (10) where is the totl profit of the mrket for the good B. B. Households Ech household is ssumed to purchse the good A nd the good B so tht it mximizes its utility level suject to its udget constrint under given lor hours. For simplicity ech household is ssumed to hve the sme lor hours nd the sme dividend income nd therefore it hs the sme income level. The utility function of ech household is given s u u i ) (11) ( i where u is the utility level i is the untity of the good A nd j is the untity of the good B demnded y ech household. In this simultion I exmine four cses where the elsticity of sustitution tkes nd 2. When the elsticity of sustitution is eul to 1 the Co-Douglss utility function is dopted. For the rest of three cses I dopt the CES utility function given s u The Co-Douglss utility function is given s u The udget constrint of ech household is given s.

13 12 p i p wl (12) i j where l is the initil endowment of lor hours j is the dividend or profit income held y ech household. given y The first order condition for the utility mximiztion suject to the udget constrint is u j u / (13) j p p where the left hnd side of the eution is the mrginl rte of sustitution expressed y the rtio of the mrginl utility etween the good A nd the good B. C. Mrket Euilirium Conditions The economy consists of two goods mrkets nd one input mrket. The goods mrkets re mrkets of the good A nd the good B. The input mrket is the lor mrket. These three mrkets re ssumed to e in euilirium. The mrket euilirium condition for the good A is given s n m (14) i where m represents the numer of households in the economy. The left hnd side of the eution is the untity of the good A supplied y firms nd the right hnd side is the untity of the good A demnded y households in the economy. Similrly the mrket euilirium condition for the good B is given s n m (15) i where the left hnd side is the untity of the good B supplied y firms nd the right hnd side is the untity of the good B demnded y households. The euilirium condition for the lor mrket is given y

14 13 n l n l ml (16) where the left hnd side is the mount of lor demnded y firms nd the right hnd side is the initil endowments of lor hours in the economy s whole. The right hnd side of the eution lso mens totl mount of lor hours supplied y m households. The totl numer of firms in the economy s whole is n n tht is the sum of the numer of firms in the two goods mrkets. For simplicity the sum of profits of ll firms is ssumed to e distriuted to every household y the sme mount. Then it holds tht n n m i. (17) Under free entry ll firms' profits ecome zero nd ech firm produces outputs t the minimum of verge cost. This is given y nd 0 (18) 0. (19) When the numer of firms in Good A mrket is restricted to certin numer then Eution (18) does not hold nd only zero profits condition given y Eution (19) holds in the Good B mrket. Ⅱ. SIMULATION ANALYSIS This section egins with cse of free entry s se cse nd then shows cses where smller numers of firms re llowed to operte in one mrket y entry restrictions. This section

15 14 exmines effects of entry restriction in one mker on economic welfre nd shows how the simultion results re ffected y different utility functions. In this simultion nlysis I use MicroSoft Excel dd-in softwre clled Solver. A. A Bse Cse: Perfect Competition Under Free Entry This is free entry cse where ll mrkets re under perfect competition nd ech output price is eul to the minimum of verge cost curve. In the simultion the utility function given s Eution 11 is mximized suject to 18 constrints which re Eution 1 to Eution 10 nd Eution 12 to Eution 19. There re 18 endogenous vriles which re Q l Q l j j j u w p n nd n. The price of the Good B p is ssigned to e eul to 1 nd is treted s numerire. The utility function in the se cse is CES type utility function where the elsticity of sustitution etween two goods is prmeter tking the vlue of or 2.0 nd Co-Douglss type utility function where the elsticity of sustitution is eul to 1. In the se cse simultion l l j j w nd p ecome ll eul to 1. Q Q n nd n ecome ll eul to 50. j re ll eul to zero. By the Wlrs' lw one of three mrket euilirium conditions is not independent from the rest of the other mrket clering eutions. Then the euilirium condition for the mrket of the good B given s eution (15) is removed from the system of eutions theoreticlly. However in the numericl simultion virtully ll constrints re not fully ut pproximtely stisfied nd then no mrket euilirium conditions re removed from the simultion. The demnd nd supply functions in the mrkets re homogenous of degree zero in prices nd profits nd one of the unknowns of prices nd profits is indeterminte theoreticlly.

16 15 Although this homogeneity condition is not fully ut pproximtely stisfied in the simultion nlysis the price of the good B is ssumed to e constnt s numerire in order to mke the simultion uickly converge to the solutions. There re 7 prmeters denoted s l nd m. Except the vlue of ll prmeters re kept constnt through this simultion. nd re set to 0.5. nd re set to 1.0. l is set to 2.0. And m is set to 50. B. Effects of Entry Restriction In the se cse the solutions for n nd n re oth 50. This mens tht the numers of firms in two goods mrkets ecome oth 50 under free entry. Under entry restriction I exmine four cses which restrict the numer of firms in Good A mrket n to nd 10 respectively. Good B mrket is lwys under free entry. Now n ecomes policy vrile given s n constnt (20) In the simultion the utility function given s Eution (11) is mximized suject to 18 constrints which re Eution (1) to Eution (10) Eution (12) to Eution (17) Eution (19) nd Eution (20). There re 18 endogenous vriles which re Q l Q l j j j u w p n nd n. Tle 1 presents the simultion results when the elsticity of sustitution 2. First column shows list of vriles nd the min economic indictors. The second column shows vlues of these vriles nd indictors when oth goods mrkets re perfectly competitive under free entry. From the third column to the six column shows vlues of vriles nd indictors

17 16 when the numer of firms in the Good A mrket is restricted to nd 10 respectively. The seventh column summrizes 4 cses of simultion for nd 0.5 on effects of entry restriction in Good A mrket. (minus sign) in the column represents negtive effect nd the mgnitude of effect gets lrger s the restriction is strengthened. + (plus sign) in the column represents the mgnitude of positive effect gets lrger s the restriction is strengthened. As gets smller the mgnitude of effect gets lrger in (the 3rd row) (the 5th row) l (the 6th row) (the 7th row) j (the 14th row) p (the 16th row) totl output (or p Q + Q ) (the 18th row) totl output mesured y competitive prices under free entry (or Q + Q ) (the 19th row) lor productivity of Good A mrket (or Q l ) (the 20th row) lor shre of totl output (or wl (p Q + Q )) (the 24th row) growth rte of totl output mesured y prices efore restriction strengthened (the 25th row) nd growth rte of totl output mesured y prices efore restriction relxed (the 26th row). As gets smller the mgnitude of effect gets smller in Q nd j. As for the shre of Good A mrket in totl output (the 22nd row) the effects of restriction is negtive nd weker s 1 nd gets smller. If 1 then the shre ecomes 0.5 regrdless of restriction. If 0. 5 the shre gets lrger s the entry restriction gets strengthened. I conduct the simultions for nd For Q (the 9th row) j (the 13th row) nd the lor shre of Good A mrket (23rd row) the entry restriction hs negtive income effect on the demnd for Good B. This effect is stronger when the vlue of gets smller nd ecomes lrger thn the sustitution effect of increse in p in the cse tht 0. 5 nd the numer of firms in the Good A mrket is limited to 10. The decrese of demnd for Good B mkes the individul demnd for Good B j the mrket supply of Good B Q nd the mrket shre of Good A ( or

18 17 p Q (p Q + Q ) ) increse. Those results re presented in Tle 2 Tle 3 nd Tle 4 in the Appendix. As for (the 8th row) (the 10th row) l (the 11th row) w (the15th row) nd lor productivity of Good B mrket (or Q l ) (the 21st row) ll these re constnt through the simultion. The reson is tht Good B mrket is under perfect competition with free entry nd ll firms in the mrket produce t the minimum of verge cost with zero profits. Ⅲ.CONCLUSION This pper exmines effects of entry restriction on n economy. The generl competitive euilirium model in this pper is sed on Arrow nd Dereu (1954) lthough they hve not considered free entry in the model. I uild two-goods one-fctor generl competitive euilirium model under free entry. The numer of firms is endogenously determined. Lor is only fctor endowment in the economy nd is fixed. This pper exmines effects of policy which restricts the numer of firms in one mrket to certin numer. The entry restriction in Good A mrket rises the price of Good A (p ) totl profits of firms in the Good A mrket (Π ) n individul firm's output ( ) profit (π ) nd employment (l ) in the Good A mrket n individul household's dividend income from profits (π j ) nd (nominl) totl output (or p Q + Q ) evluted current prices. On the other hnd this restriction in Good A mrket reduces the household utility level (u) (rel) totl output (or Q + Q ) evluted with the prices under free entry n individul household's consumption of Good A ( j ) the totl output of Good A (Q ) nd the lor productivity of Good A (Q /l ).. This pper suggests tht the competitive euilirium under free entry cn chieve Preto efficiency. As long s ny firms hve positive profits there is room for improving the

19 18 economic welfre through free entry. In other words this pper suggests tht the competitive euilirium cnnot e to Preto efficient unless free entry is llowed. Eulity of the mrginl rte of sustitution etween different goods for every household is chieved nd eulity etween the mrginl rte of trnsformtion nd the mrginl rte of sustitution etween different goods re chieved. Those rtes re ll eul to the reltive price etween different goods in this model. Perfect competition previls in ll mrkets including goods mrkets nd fctor mrket. Lor cn freely move etween industries. This pper implies tht the first fundmentl theorem of welfre economics does not lwys hold under the conventionl ssumptions. If the ssumption of free entry or zero profits is stisfied then the theorem holds true under the conventionl ssumptions. The model dopted in this pper is generl euilirium model nd then hs mcroeconomic implictions. The entry restriction reduces lor productivity of Good A mrket lor shre of totl income nd (rel) totl output (or Q + Q ). The entry restriction rises the reltive price of Good A nd my ffect the Consumer Price Index nd the GNP defltor. The price of Good B is used s numerire in this model nd ny price chnge is reltive price chnge. If money mrket is included in this model one cn discuss the generl (nominl) price chnge. Since the entry restriction is shown to reduce totl output in the model it reduces the trnsction demnd for money. If money supply is fixed this reduction of money demnd my result in infltion. If neutrlity of money is ssumed then the infltion rte is eul to the reduction rte of rel totl output. The model developed in this study cn serve s se model to tckle with mny economic policy issues nd give some useful insights. By dding or deleting some endogenous vriles

20 19 nd constrints from the model one my derive mny fruitful policy implictions nd provide some mcroeconomic insights. However these will e left for further reserch.

21 Tle 1: simultion results when σ = 2. vriles Free entry n = 40 n = 30 n = 20 n = 10 Effect of restriction u for 4 vlues of σ for 4 vlues of σ Q for 4 vlues of σ π for 4 vlues of σ l for 4 vlues of σ Π for 4 vlues of σ no chnge Q inconclusive π no chnge l no chnge j for 4 vlues of σ j inconclusive π j for 4 vlues of σ w no chnge p for 4 vlues of σ n inconclusive (nominl) Totl output (p Q + Q ) for 4 vlues of σ (rel) Totl output (Q + Q ) for 4 vlues of σ Lor productivity of Good A mrket (Q /l ) for 4 vlues of σ Lor productivity of Good B mrket (Q /l ) no chnge Shre of Good A mrket in totl output ( p Q (p Q + Q ) ) depends on σ shre of lor in Good A mrket ( wl (p Q )) inconclusive Lor shre of totl output ( wl (p Q + Q ) ) for 4 vlues of σ Growth rte of totl output mesured y prices efore restriction strengthened -0.2 % -1.7% -3.9% -7.5% - for 4 vlues of σ Growth rte of totl output mesured y prices efore restriction relxed 1.2 % 3.2 % 6.5% 13.9% + for 4 vlues of σ

22 APPENDIX Tle 2: simultion results when σ = 1.5. vriles Free entry n = 40 n = 30 n = 20 n = 10 u Q π l Π Q π l j j π j w p n (nominl) Totl output (p Q + Q ) (rel) Totl output (Q + Q ) Lor productivity of Good A mrket (Q /l ) Lor productivity of Good B mrket (Q /l ) Shre of Good A mrket in totl output (p Q (p Q + Q ) ) shre of lor in Good A mrket ( wl (p Q )) Lor shre of totl output ( wl (p Q + Q ) ) Growth rte of totl output mesured y prices efore restriction strengthened -0.3 % -2.2 % -5.0 % -9.8 % Growth rte of totl output mesured y prices efore restriction relxed 1.4 % 4.0 % 8.2 % 18.6 %

23 Tle 3: simultion results when σ = 1. vriles Free entry n = 40 n = 30 n = 20 n = 10 u Q π l Π Q π l j j π j w p n (nominl) Totl output (p Q + Q ) (rel) Totl output (Q + Q ) Lor productivity of Good A mrket (Q /l ) Lor productivity of Good B mrket ( p Q (p Q + Q ) ) Shre of Good A mrket in totl output (p Q (p Q + Q ) ) shre of lor in Good A mrket ( wl (p Q )) Lor shre of totl output ( wl (p Q + Q ) ) Growth rte of totl output mesured y prices efore restriction strengthened -0.5 % -3.0 % -7.0 % % Growth rte of totl output mesured y prices efore restriction relxed 1.7 % 5.0 % 11.0 % 27.0 %

24 Tle 4: simultion results when σ = 0.5. vriles Free entry n = 40 n = 30 n = 20 n = 10 u Q π l Π Q π l j j π j w p n (nominl) Totl output (p Q + Q ) (rel) Totl output (Q + Q ) Lor productivity of Good A mrket (Q /l ) Lor productivity of Good B mrket (Q /l ) Shre of Good A mrket in totl output ( p Q (p Q + Q ) ) shre of lor in Good A mrket Lor shre of totl output ( wl (p Q + Q ) ) Growth rte of totl output mesured y prices efore restriction strengthened -1.0 % -4.7 % % % Growth rte of totl output mesured y prices efore restriction relxed 2.2 % 6.8 % 15.4 % 39.1 %

25 References Arrow Kenneth.J. nd Gerrd Dereu (1954). Existence of n Euilirium for Competitive Economy Econometric Arrow Kenneth J. nd F. H. Hhn (1971). Generl Competitive Anlysis Holden-Dy Inc. Sn Frncisco. Hopenhyn Hugo A. (1992). "Entry Exit Firm Dynmics in Long Run Euilirium" Econometric Hopenhyn Hugo A. nd Richrd Rogerson (1993). "Jo Turnover nd Policy Evlution: A Generl Euilirium Anlysis" Journl of Politicl Economy Mnkiw N. Gregory nd Michel D. Whinston (1986). "Free Entry nd Socil Inefficiency" Rnd Journl of Economics Tkse Mitsuo (2011). "Effects of New Entry in Generl Euilirium Model" Center for Advnced Economic Study Working Pper WP Fukuok University.