We provide a new perspective to the literature on social desirability of entry by showing that, if the input supplier has market power, social desirability of entry of the final goods producers depends on returns to scale. Entry in the final goods market can be socially insufficient under constant returns to scale technology, but it can be socially excessive under decreasing returns to scale technologies if the cost of entry is low so that the final goods market is sufficiently competitive. Hence, the anti-competitive entry regulation policies are more justifiable if the final goods market is characterised by decreasing returns to scale technologies

Anderson

Arijit Mukherjee

Banker

Basu

Cabral

Debasmita Basak

Fudenberg

Ghosh

Mankiw

Mukherjee

Saal

Suzumura

Tirole

Townsend

Basak, Debasmita

Mukherjee, Arijit

Nottingham ePrints

Basak, Debasmita and Mukherjee, Arijit (2016) Social efficiency of entry in a vertically related industry. Economics Letters, 139 . pp. 8-10. ISSN 0165-1765 Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/31592/1/Social%20entry-vertical%20structure-technology-EL-Final.pdfCopyright and reuse: The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions.This article is made available under the Creative Commons Attribution Non-commercial No Derivatives licence and may be reused according to the conditions of the licence.  For more details see: http://creativecommons.org/licenses/by-nc-nd/2.5/A note on versions: The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.ukSocial efficiency of entry in a vertically related industry†  Debasmita Basak Swansea University, UK  and  Arijit Mukherjee Nottingham University Business School and CFGE, Loughborough University, UK, CESifo and INFER, Germany, and RCIE, City University of Hong Kong, Hong Kong   November 2015  Abstract: We provide a new perspective to the literature on social desirability of entry by showing that, if the input supplier has market power, social desirability of entry of the final goods producers depends on returns to scale. Entry in the final goods market can be socially insufficient under constant returns to scale technology, but it can be socially excessive under decreasing returns to scale technologies if the cost of entry is low so that the final goods market is sufficiently competitive. Hence, the anti-competitive entry regulation policies are more justifiable if the final goods market is characterised by decreasing returns to scale technologies.    Key Words: Excess entry; Insufficient entry; Decreasing returns to scale JEL Classifications: L13; L40   Correspondence to: Debasmita Basak,  Section of Accounting, Finance & Economics,  School of Management, Swansea University,  Fabian Way, Swansea SA1 8EN, UK. Email: d.basak@swansea.ac.uk ______________________ † We thank an anonymous referee of this journal for helpful comments and suggestions that helped to improve the paper. The usual disclaimer applies.  1  1. Introduction In a seminal paper, Mankiw and Whinston (1986) show that entry is socially excessive in oligopolistic industries with scale economies and this holds irrespective of the type of technology, such as constant returns or non-constant returns. This “excess-entry theorem”, which may justify the anticompetitive entry regulation policies adopted by many countries,1 ignored an important aspect, viz., the market power of the input suppliers creating strategic input price determination.2 However, in real world, the presence of labour unions or certain key input suppliers, such as the computer chip producer like Intel, may justify market power of the input suppliers in many industries. We show that if the input suppliers have market power, social efficiency of entry of the final goods producers depends on their technologies. Considering a competitive input market, Mankiw and Whinston (1986) show that entry is socially excessive. Ghosh and Morita (2007a, b) argue that if the input suppliers have market power, the result of Mankiw and Whinston (1986) may not hold and entry can be socially insufficient. We contribute in this literature by showing that if the input supplier has market power, social desirability of entry depends on returns to scale. Entry is socially insufficient under constant returns but it is socially excessive under decreasing returns3 if the cost of entry is sufficiently low. Hence, the policy implication of Mankiw and Whinston (1986) remains in the presence of significant market power of the input suppliers if there are decreasing returns in the final goods production and the cost of entry is sufficiently low.  2. The model and the results Consider an industry with a large number of symmetric final goods producers, each of whom decides whether or not to enter the market. Entry requires each final goods producer to incur an entry cost 𝐾 > 0. Assume that there exists an input supplier that                                                  1 Preventing excessive entry was a guiding principle in the Japanese industrial policy in the postwar period (Suzumura and Kiyono, 1987 and Suzumura, 1995). 2 One may look at Suzumura and Kiyono (1987), Anderson et al. (1995), Fudenberg and Tirole (2000), Cabral (2004), Ghosh and Saha (2007), Mukherjee (2012), for a representative sample of the papers on social efficiency of entry with competitive input markets. 3 The implications of decreasing returns to scale technologies, which can be motivated by capacity constraints or scarcity of resources, have been discussed extensively in different contexts (Tirole, 1988). Banker et al. (1994) show that decreasing returns may prevail in the software industry. Saal et al. (2007) found constant or decreasing returns in the UK industries. The typical two-digit industries in the US (Basu and Fernald, 1997), three-digit manufacturing industries in Singapore (Kee, 2002), Broadacre industries in Australia (Townsend et al., 1998) appear to have (slightly) decreasing returns to scale.   2  supplies inputs to all the final goods producers. To illustrate our result in a simple fashion we assume that input supplier unilaterally sets the input price. Consider the following game. At stage 1, the final goods producers decide whether or not to enter the market. At stage 2, the input supplier determines the input price. At stage 3, the final goods producers, which entered the market, produce outputs simultaneously by purchasing inputs according to their requirements and the profits are realised. We solve the game through backward induction. For simplicity we assume that the final goods producers have symmetric production technologies  and they can transform the inputs to a homogeneous final good at zero cost. We consider the case of constant returns to scale technology in Section 2.1 and the case of decreasing returns to scale technology in Section 2.2. Assume that the inverse market demand function is P = a – Q, where P is price and Q is the total output.   2.1. Constant returns to scale technology Assume that the production technology of the jth firm, 𝑗 = 1,2, … , ∞, is 𝐿𝑗 = 𝑞𝑗,  where 𝐿𝑗   is the amount of input and 𝑞𝑗 is the output. If n final goods producers enter the market, the ith final goods producer maximises the following expression to determine its output:  𝑀𝑎𝑥  𝜋𝑖𝐶𝑅𝑆 =(𝑃 − 𝑤)𝑞𝑖 − 𝐾.        (1) The equilibrium output of the ith firm is  𝑞𝑖𝐶𝑅𝑆 =𝑎−𝑤1+𝑛.         (2) The input supplier maximises the following expression to determine the input price:  1( )nCRSiwiMaxU w c q          (3) where 𝑐 represents the marginal cost of input production. The equilibrium input price is 𝑤𝐶𝑅𝑆 =𝑎+𝑐2. The equilibrium net profit of the ith final goods producer, i=1,2,…,n, which has entered the market is  𝜋𝑖𝐶𝑅𝑆 =14(𝑎−𝑐1+𝑛)2− 𝐾.                   (4) For analytical convenience, we consider the number of firms as a continuous variable. Hence, entry in the market occurs as long as the net profit of a final goods producer is 3  non-negative. Given the symmetry of the firms, the free entry equilibrium number of final goods producers is given by 0CRSi   or 14(𝑎−𝑐1+𝑛)2= K.                    (5) It follows from (5) that, if the cost of entry (i.e., K ) falls, the free entry equilibrium number of final goods producers, n, increases. Now we determine the welfare maximising number of final goods producers, where welfare is the sum of the total net profits of the final goods producers (∑ 𝜋𝑖𝐶𝑅𝑆), the profit of the input supplier (𝑈𝐶𝑅𝑆) and consumer surplus (𝐶𝑆𝐶𝑅𝑆). Following the literature, we consider the second-best problem of welfare maximisation, i.e., we determine the welfare maximising number of final goods producers subject to Cournot behaviour of the firms. Hence, the social planner can control the number of final goods producers entering the market, but it cannot control the output choice behaviour of the final goods producers. If n final goods producers produce the final good, it follows from (4) that the total net profit of the n firms is  ∑ 𝜋𝑖𝐶𝑅𝑆 =n4(𝑎−𝑐1+𝑛)2− 𝑛𝐾.               (6) The profit of the input supplier is  𝑈𝐶𝑅𝑆 =𝑛(𝑎−𝑐)24(1+𝑛).        (7) Since the total final goods production is 𝑄𝐶𝑅𝑆 = 𝑛 (𝑎−𝑤1+𝑛), the consumer surplus is 𝐶𝑆𝐶𝑅𝑆 =𝑄22=𝑛28(𝑎−𝑐1+𝑛)2.       (8) The social planner chooses 𝑛 to maximise social welfare (which is the sum of (6), (7) and (8)):              Max𝑛  𝑆𝑊𝐶𝑅𝑆 = Max𝑛𝑛(𝑎−𝑐)2(4+3𝑛)8(1+𝑛)2− 𝑛𝐾     (9) The welfare maximising 𝑛 is given by               𝜕(𝑆𝑊𝐶𝑅𝑆)𝜕𝑛≡(𝑎−𝑐)2(2+𝑛)4(1+𝑛)3− 𝐾 = 0                                         (10) It follows from (10) that as the cost of entry (i.e., 𝐾) falls, the welfare maximising number of final goods producers increases.  Proposition 1 If the final goods production technology exhibits constant returns to scale, entry in the final goods market is insufficient. 4  Proof: Evaluating ( )CRSSWn at the free entry equilibrium number of final goods producers, i.e., substituting 14(𝑎−𝑐1+𝑛)2 for K into (10), we get that ( )0CRSSWn since (𝑎−𝑐)2(2+𝑛)4(1+𝑛)3−14(𝑎−𝑐1+𝑛)2=(𝑎−𝑐)24(1+𝑛)3> 0, which proves the result. ■  Entry creates two effects. First, as in Mankiw and Whinston (1986), it creates a “business-stealing effect” by lowering the outputs of the incumbent final goods producers. On the other hand, as observed by Ghosh and Morita (2007a, b), entry creates a “business-creation effect” in the input sector by raising the total input demand. We find that the latter effect dominates the former effect for social welfare and makes entry less attractive than the socially optimum level. This result affirms the finding of Ghosh and Morita (2007a, b), suggesting that entry is insufficient in a vertically related industry with constant returns to scale technologies.  2.2. Production with decreasing return to scale technology We now consider the case of decreasing returns to scale technologies for the final goods production. Assume that the technology of the jth final goods producer, j=1,2,…,∞, is j jL q  or 2j jL q , implying that 2jq  inputs are required to produce jq  units of the output. If n final goods producers enter the market, the ith final goods producer, i=1,2,..,n, maximises             𝑀𝑎𝑥   𝜋𝑖𝐷𝑅𝑆 =(𝑎 − 𝑄)𝑞𝑖 − 𝑤𝑞𝑖2 − 𝐾 .              (12) The equilibrium output of the ith final goods producer is  𝑞𝑖𝐷𝑅𝑆 =𝑎(1+2𝑤)−11+𝑛(1+2𝑤)−1.                 (13) The input supplier maximises the following expression to determine w:  21(w c)nDRSiwiMaxU q   .                (14) The equilibrium input price is  𝑤𝐷𝑅𝑆 =12(1 + 𝑛 + 4𝑐). Hence, the equilibrium profit of the input supplier is  𝑈𝐷𝑅𝑆 =𝑛8(𝑎21+𝑛+2𝑐), the profit of the ith final goods producer 5  that entered the market is  𝜋𝑖𝐷𝑅𝑆 =𝑎2(3+n+4c)8(1+n+2c)2− 𝐾 and the consumer surplus is 𝐶𝑆𝐷𝑅𝑆 =𝑄22=18(𝑛𝑎1+𝑛+2𝑐)2.  The free entry equilibrium number of final goods producer is given by 𝜋𝑖𝐷𝑅𝑆 = 0 or   𝑎2(3+n+4c)8(1+n+2c)2= K.                                                                                            (15) It follows from (15) that a lower K increases the free equilibrium number of final goods producers.  Now determine the welfare maximising number of firms, which is determined by maximising the following expression:              Max𝑛  𝑆𝑊𝐷𝑅𝑆 = Max𝑛𝑎2𝑛(4+3𝑛+6𝑐)8(1+𝑛+2𝑐)2− 𝑛𝐾.                         (16) The welfare maximising number of firms is given by             𝜕(𝑆𝑊𝐷𝑅𝑆)𝜕𝑛≡𝑎2(2+𝑛+3𝑐𝑛+7𝑐+6𝑐2)4(1+𝑛+2𝑐)3− 𝐾 = 0              (17) It follows from (17) that a lower K increases the welfare maximising number of final goods producers.  Proposition 2: If the production technology exhibits decreasing return to scale, entry in the final goods market is excessive (insufficient) if the cost of entry is sufficiently low (high) so that the free entry equilibrium number of firms 𝑛 > (<)√2(1 + 2𝑐 + 2𝑐2) − 1 ≡ 𝑛∗. Proof: We get that  𝜕2(𝑆𝑊𝐷𝑅𝑆)𝜕𝑛2= −𝑎2(5+2𝑛+6𝑐𝑛+16𝑐+12𝑐2)4(1+𝑛+2𝑐)4< 0 and evaluating ( )DRSSWn at the free entry equilibrium number of final goods producers, i.e., substituting  𝑎2(3+n+4c)8(1+n+2c)2 for K into (17), we get that 𝜕(𝑆𝑊𝐷𝑅𝑆)𝜕𝑛< (>)0 for 𝑎2(2+7𝑐+6𝑐2+𝑛+3𝑐𝑛)4(1+𝑛+2𝑐)3< (>)𝑎2(3+n+4c)8(1+n+2c)2  or  * 22 1 2c 2 1( ) cn n    , implying that entry is excessive (insufficient) for  * 22 1 2c 2 1( ) cn n    . Since the free entry equilibrium number of firms is negatively related to the cost of entry, excessive (insufficient) entry occurs, i.e., n > (<)n∗, for sufficiently low (high) cost of entry. ■  6  The intuitions for Proposition 2 also follows from the trade-off between the “business-stealing effect” and “business-creation effect” as discussed in Proposition 1 with an exception that, under decreasing returns to scale technology, the input price increases with the number of final goods producers, thus strengthening the business-stealing effect. If the cost of entry is low so that the final goods market is sufficiently competitive, i.e., n > n∗,  a new entrant creates significant business-stealing effect by reducing the profits of a large number of incumbent final goods producers, thus creating excessive entry. However, if the cost of entry is high, the business-stealing effect is not so strong to outweigh the business-creation effect and entry is insufficient in this situation.  3. Conclusion We show that the production technology plays an important role in determining social efficiency of entry of the final goods producers if the input supplier has market power. Entry in the final goods market can be socially insufficient under constant returns to scale technology but it can be socially excessive if the cost of entry is low. Hence, the anti-competitive entry regulation policies are more justifiable in a vertically related industry if the final goods market is characterised by decreasing returns to scale technologies.  References Anderson, S.P., A. de Palma and Y. Nesterov (1995), “Oligopolistic competition and the optimal provision of products”, Econometrica, 63: 1281-1301.  Banker, R.D., H. Chang and C.F. Kemerer (1994) “Evidence on economies of scale in software development”, Information and Software Technology, 36, 275-282. Basu, S. and J.G. Fernald (1997) “Returns to scale in U.S. production: estimates and implications”, Journal of Political Economy, 105, 249-283. Cabral, L.M.B. (2004) “Simultaneous entry and welfare”, European Economic Review, 48: 943-957. Fudenberg, D. and J. Tirole (2000), “Pricing a network good to deter entry”, The Journal of Industrial Economics, XLVIII: 373-390. Ghosh, A. and A. Morita (2007a) “Social desirability of free entry: a bilateral oligopoly analysis” International Journal of Industrial Organization 25, 925-34. 7  Ghosh, A. and A. Morita (2007b) “Free entry and social efficiency under vertical oligopoly”, Rand Journal of Economics 38, 539-52. Ghosh A. and S. Saha (2007) “Excess entry in the absence of scale economies”, Economic Theory, 30:575-586. Kee, H. L. (2002) “Markups, returns to scale and productivity: a case of Singapore’s manufacturing sector, Mimeo. Mankiw, N.G., Whinston, M.D. (1986) “Free entry and social inefficiency”, Rand Journal of Economics, 17, 1, 48-58. Mukherjee, A. (2012) “Social Efficiency of Entry with Market Leaders”, Journal of Economics & Management Strategy, 21: 431-444. Saal, D.S., Parker, D. and Weyman-Jones, T. (2007) “Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985–2000”. Journal of Productivity Analysis, 28 (1), 127–139. Suzumura, K. (1995) Competition, Commitment and Welfare, New York, USA: Oxford. Suzumura, K. and K. Kiyono (1987) “Entry barriers and economic welfare”, Review of Economic Studies 54, 157–167. Tirole, J. (1988) The theory of industrial organization, MIT Press, Cambridge. Townsend, R.F., J. Kirsten and N. Vink (1998) “Farm size, productivity and returns to scale in agriculture revisited: a case study of wine producers in South Africa”, Agricultural Economics, 19: 175-180. 

Social efficiency of entry in a vertically related industry

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 Cronfa -  Swansea University Open Access Repository   _____________________________________________________________   This is an author produced version of a paper published in :Economics Letters                                     Cronfa URL for this paper:http://cronfa.swan.ac.uk/Record/cronfa25035_____________________________________________________________ Paper:Basak, D. & Mukherjee, A. (2015).  Social efficiency of entry in a vertically related industry. Economics Lettershttp://dx.doi.org/10.1016/j.econlet.2015.12.003        _____________________________________________________________  This article is brought to you by Swansea University. Any person downloading material is agreeing to abide by theterms of the repository licence. Authors are personally responsible for adhering to publisher restrictions or conditions.When uploading content they are required to comply with their publisher agreement and the SHERPA RoMEOdatabase to judge whether or not it is copyright safe to add this version of the paper to this repository. http://www.swansea.ac.uk/iss/researchsupport/cronfa-support/  Accepted ManuscriptSocial efficiency of entry in a vertically related industryDebasmita Basak, Arijit MukherjeePII: S0165-1765(15)00512-1DOI: http://dx.doi.org/10.1016/j.econlet.2015.12.003Reference: ECOLET 7014To appear in: Economics LettersReceived date: 28 September 2015Revised date: 28 November 2015Accepted date: 11 December 2015Please cite this article as: Basak, D., Mukherjee, A., Social efficiency of entry in a verticallyrelated industry. Economics Letters (2015), http://dx.doi.org/10.1016/j.econlet.2015.12.003This is a PDF file of an unedited manuscript that has been accepted for publication. As aservice to our customers we are providing this early version of the manuscript. The manuscriptwill undergo copyediting, typesetting, and review of the resulting proof before it is published inits final form. Please note that during the production process errors may be discovered whichcould affect the content, and all legal disclaimers that apply to the journal pertain.	• This paper examines the social efficiency of entry in vertically related industry. • The social desirability of entry depends on returns to scale. • Entry is socially insufficient under constant returns to scale technology. • Entry can be socially excessive under decreasing returns to scale technology. *Highlights (for review)Social efficiency of entry in a vertically related industry†  Debasmita Basak Swansea University, UK  and  Arijit Mukherjee Nottingham University Business School and CFGE, Loughborough University, UK, CESifo and INFER, Germany, and RCIE, City University of Hong Kong, Hong Kong   November 2015  Abstract: We provide a new perspective to the literature on social desirability of entry by showing that, if the input supplier has market power, social desirability of entry of the final goods producers depends on returns to scale. Entry in the final goods market can be socially insufficient under constant returns to scale technology, but it can be socially excessive under decreasing returns to scale technologies if the cost of entry is low so that the final goods market is sufficiently competitive. Hence, the anti-competitive entry regulation policies are more justifiable if the final goods market is characterised by decreasing returns to scale technologies.    Key Words: Excess entry; Insufficient entry; Decreasing returns to scale JEL Classifications: L13; L40   Correspondence to: Debasmita Basak,  Section of Accounting, Finance & Economics,  School of Management, Swansea University,  Fabian Way, Swansea SA1 8EN, UK. Email: d.basak@swansea.ac.uk ______________________ † We thank an anonymous referee of this journal for helpful comments and suggestions that helped to improve the paper. The usual disclaimer applies.  *Title Page1  1. Introduction In a seminal paper, Mankiw and Whinston (1986) show that entry is socially excessive in oligopolistic industries with scale economies and this holds irrespective of the type of technology, such as constant returns or non-constant returns. This “excess-entry theorem”, which may justify the anticompetitive entry regulation policies adopted by many countries,1 ignored an important aspect, viz., the market power of the input suppliers creating strategic input price determination.2 However, in real world, the presence of labour unions or certain key input suppliers, such as the computer chip producer like Intel, may justify market power of the input suppliers in many industries. We show that if the input suppliers have market power, social efficiency of entry of the final goods producers depends on their technologies. Considering a competitive input market, Mankiw and Whinston (1986) show that entry is socially excessive. Ghosh and Morita (2007a, b) argue that if the input suppliers have market power, the result of Mankiw and Whinston (1986) may not hold and entry can be socially insufficient. We contribute in this literature by showing that if the input supplier has market power, social desirability of entry depends on returns to scale. Entry is socially insufficient under constant returns but it is socially excessive under decreasing returns3 if the cost of entry is sufficiently low. Hence, the policy implication of Mankiw and Whinston (1986) remains in the presence of significant market power of the input suppliers if there are decreasing returns in the final goods production and the cost of entry is sufficiently low.  2. The model and the results Consider an industry with a large number of symmetric final goods producers, each of whom decides whether or not to enter the market. Entry requires each final goods producer to incur an entry cost   . Assume that there exists an input supplier that                                                  1 Preventing excessive entry was a guiding principle in the Japanese industrial policy in the postwar period (Suzumura and Kiyono, 1987 and Suzumura, 1995). 2 One may look at Suzumura and Kiyono (1987), Anderson et al. (1995), Fudenberg and Tirole (2000), Cabral (2004), Ghosh and Saha (2007), Mukherjee (2012), for a representative sample of the papers on social efficiency of entry with competitive input markets. 3 The implications of decreasing returns to scale technologies, which can be motivated by capacity constraints or scarcity of resources, have been discussed extensively in different contexts (Tirole, 1988). Banker et al. (1994) show that decreasing returns may prevail in the software industry. Saal et al. (2007) found constant or decreasing returns in the UK industries. The typical two-digit industries in the US (Basu and Fernald, 1997), three-digit manufacturing industries in Singapore (Kee, 2002), Broadacre industries in Australia (Townsend et al., 1998) appear to have (slightly) decreasing returns to scale.   *ManuscriptClick here to view linked References2  supplies inputs to all the final goods producers. To illustrate our result in a simple fashion we assume that input supplier unilaterally sets the input price. Consider the following game. At stage 1, the final goods producers decide whether or not to enter the market. At stage 2, the input supplier determines the input price. At stage 3, the final goods producers, which entered the market, produce outputs simultaneously by purchasing inputs according to their requirements and the profits are realised. We solve the game through backward induction. For simplicity we assume that the final goods producers have symmetric production technologies  and they can transform the inputs to a homogeneous final good at zero cost. We consider the case of constant returns to scale technology in Section 2.1 and the case of decreasing returns to scale technology in Section 2.2. Assume that the inverse market demand function is P = a – Q, where P is price and Q is the total output.   2.1. Constant returns to scale technology Assume that the production technology of the jth firm,    	  , is   ,  where   is the amount of input and  is the output. If n final goods producers enter the market, the ith final goods producer maximises the following expression to determine its output:       .        (1) The equilibrium output of the ith firm is    ! .         (2) The input supplier maximises the following expression to determine the input price:  1( )nCRSiw iMaxU w c q== − ∑        (3) where " represents the marginal cost of input production. The equilibrium input price is    #$ . The equilibrium net profit of the ith final goods producer, i=1,2,…,n, which has entered the market is   % &# !'$ .                   (4) For analytical convenience, we consider the number of firms as a continuous variable. Hence, entry in the market occurs as long as the net profit of a final goods producer is 3  non-negative. Given the symmetry of the firms, the free entry equilibrium number of final goods producers is given by 0CRSipi =  or % &# !'$ (.                    (5) It follows from (5) that, if the cost of entry (i.e., K ) falls, the free entry equilibrium number of final goods producers, n, increases. Now we determine the welfare maximising number of final goods producers, where welfare is the sum of the total net profits of the final goods producers )∑ +, the profit of the input supplier , and consumer surplus -.. Following the literature, we consider the second-best problem of welfare maximisation, i.e., we determine the welfare maximising number of final goods producers subject to Cournot behaviour of the firms. Hence, the social planner can control the number of final goods producers entering the market, but it cannot control the output choice behaviour of the final goods producers. If n final goods producers produce the final good, it follows from (4) that the total net profit of the n firms is  ∑  /% &# !'$ 0.               (6) The profit of the input supplier is  ,  !#1% ! .        (7) Since the total final goods production is 2  0 & ! ', the consumer surplus is -.  31$ !14 &# !'$.       (8) The social planner chooses 0 to maximise social welfare (which is the sum of (6), (7) and (8)):              567! .8  567! !#1% 9!4 !1  0     (9) The welfare maximising 0 is given by               :);<=>+:! ?#1$ !% !@                                             (10) It follows from (10) that as the cost of entry (i.e., ) falls, the welfare maximising number of final goods producers increases.  Proposition 1 If the final goods production technology exhibits constant returns to scale, entry in the final goods market is insufficient. 4  Proof: Evaluating ( )CRSSWn∂∂  at the free entry equilibrium number of final goods producers, i.e., substituting % &# !'$ for K into (10), we get that ( ) 0CRSSWn∂>∂  since #1$ !% !@ % &# !'$ #1% !@  , which proves the result.   Entry creates two effects. First, as in Mankiw and Whinston (1986), it creates a “business-stealing effect” by lowering the outputs of the incumbent final goods producers. On the other hand, as observed by Ghosh and Morita (2007a, b), entry creates a “business-creation effect” in the input sector by raising the total input demand. We find that the latter effect dominates the former effect for social welfare and makes entry less attractive than the socially optimum level. This result affirms the finding of Ghosh and Morita (2007), suggesting that entry is insufficient in a vertically related industry with constant returns to scale technologies.  2.2. Production with decreasing return to scale technology We now consider the case of decreasing returns to scale technologies for the final goods production. Assume that the technology of the jth final goods producer, j=1,2,…,, is j jL q=  or 2j jL q= , implying that 2jq  inputs are required to produce jq  units of the output. If n final goods producers enter the market, the ith final goods producer, i=1,2,..,n, maximises             A   2  $   .              (12) The equilibrium output of the ith final goods producer is  A  $BC ! $BC.                 (13) The input supplier maximises the following expression to determine w:  21(w c)nDRSiw iMaxU q== − ∑ .                (14) The equilibrium input price is  A  $  D 0 D E". Hence, the equilibrium profit of the input supplier is  ,A  !4 &1 ! $#', the profit of the ith final goods producer 5  that entered the market is  A 19 / %F4 / $F1   and the consumer surplus is -.A  31$ 4 &! ! $#'$.  The free entry equilibrium number of final goods producer is given by A   or   19 / %F4 / $F1  (.                                                                                            (15) It follows from (15) that a lower K increases the free equilibrium number of final goods producers.  Now determine the welfare maximising number of firms, which is determined by maximising the following expression: 567! .8A  567! 1!% 9! G#4 ! $#1  0.                         (16) The welfare maximising number of firms is given by             :);H=>+:! ?1)$ ! 9#! I# G#1+% ! $#@                  (17) It follows from (17) that a lower K increases the welfare maximising number of final goods producers.  Proposition 2: If the production technology exhibits decreasing return to scale, entry in the final goods market is excessive (insufficient) if the cost of entry is sufficiently low (high) so that the free entry equilibrium number of firms 0  JK D " D "$   ? 0L. Proof: We get that  :1);H=>+:!1  1)M $! G#! G# $#1+% ! $#N J  and evaluating ( )DRSSWn∂∂  at the free entry equilibrium number of final goods producers, i.e., substituting  19 / %F4 / $F1 for K into (17), we get that :);H=>+:! J  for 1)$ I# G#1 ! 9#!+% ! $#@ J 19 / %F4 / $F1  or ( )* 22 1 2c 2 1( ) cn n +< +> −≡ , implying that entry is excessive (insufficient) for ( )* 22 1 2c 2 1( ) cn n +< +> −≡ . Since the free entry equilibrium number of firms is negatively related to the cost of entry, excessive (insufficient) entry occurs, i.e., O  JOL, for sufficiently low (high) cost of entry.   6  The intuitions for Proposition 2 also follows from the trade-off between the “business-stealing effect” and “business-creation effect” as discussed in Proposition 1 with an exception that, under decreasing returns to scale technology, the input price increases with the number of final goods producers, thus strengthening the business-stealing effect. If the cost of entry is low so that the final goods market is sufficiently competitive, i.e., O  OL,  a new entrant creates significant business-stealing effect by reducing the profits of a large number of incumbent final goods producers, thus creating excessive entry. However, if the cost of entry is high, the business-stealing effect is not so strong to outweigh the business-creation effect and entry is insufficient in this situation.  3. Conclusion We show that the production technology plays an important role in determining social efficiency of entry of the final goods producers if the input supplier has market power. Entry in the final goods market cane be socially insufficient under constant returns to scale technology but it can be socially excessive if the cost of entry is low. Hence, the anti-competitive entry regulation policies are more justifiable in a vertically related industry if the final goods market is characterised by decreasing returns to scale technologies.  References Anderson, S.P., A. de Palma and Y. Nesterov (1995), “Oligopolistic competition and the optimal provision of products”, Econometrica, 63: 1281-1301.  Banker, R.D., H. Chang and C.F. Kemerer (1994) “Evidence on economies of scale in software development”, Information and Software Technology, 36, 275-282. Basu, S. and J.G. Fernald (1997) “Returns to scale in U.S. production: estimates and implications”, Journal of Political Economy, 105, 249-283. Cabral, L.M.B. (2004) “Simultaneous entry and welfare”, European Economic Review, 48: 943-957. Fudenberg, D. and J. Tirole (2000), “Pricing a network good to deter entry”, The Journal of Industrial Economics, XLVIII: 373-390. Ghosh, A. and A. Morita (2007a) “Social desirability of free entry: a bilateral oligopoly analysis” International Journal of Industrial Organization 25, 925-34. 7  Ghosh, A. and A. Morita (2007b) “Free entry and social efficiency under vertical oligopoly”, Rand Journal of Economics 38, 539-52. Ghosh A. and S. Saha (2007) “Excess entry in the absence of scale economies”, Economic Theory, 30:575-586. Kee, H. L. (2002) “Markups, returns to scale and productivity: a case of Singapore’s manufacturing sector, Mimeo. Mankiw, N.G., Whinston, M.D. (1986) “Free entry and social inefficiency”, Rand Journal of Economics, 17, 1, 48-58. Mukherjee, A. (2012) “Social Efficiency of Entry with Market Leaders”, Journal of Economics & Management Strategy, 21: 431-444. Saal, D.S., Parker, D. and Weyman-Jones, T. (2007) “Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985–2000”. Journal of Productivity Analysis, 28 (1), 127–139. Suzumura, K. (1995) Competition, Commitment and Welfare, New York, USA: Oxford. Suzumura, K. and K. Kiyono (1987) “Entry barriers and economic welfare”, Review of Economic Studies 54, 157–167. Tirole, J. (1988) The theory of industrial organization, MIT Press, Cambridge. Townsend, R.F., J. Kirsten and N. Vink (1998) “Farm size, productivity and returns to scale in agriculture revisited: a case study of wine producers in South Africa”, Agricultural Economics, 19: 175-180. 

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Social efficiency of entry in a vertically related industry†  Debasmita Basak Swansea University, UK  and  Arijit Mukherjee Nottingham University Business School and CFGE, Loughborough University, UK, CESifo and INFER, Germany, and RCIE, City University of Hong Kong, Hong Kong   November 2015  Abstract: We provide a new perspective to the literature on social desirability of entry by showing that, if the input supplier has market power, social desirability of entry of the final goods producers depends on returns to scale. Entry in the final goods market can be socially insufficient under constant returns to scale technology, but it can be socially excessive under decreasing returns to scale technologies if the cost of entry is low so that the final goods market is sufficiently competitive. Hence, the anti-competitive entry regulation policies are more justifiable if the final goods market is characterised by decreasing returns to scale technologies.    Key Words: Excess entry; Insufficient entry; Decreasing returns to scale JEL Classifications: L13; L40   Correspondence to: Debasmita Basak,  Section of Accounting, Finance & Economics,  School of Management, Swansea University,  Fabian Way, Swansea SA1 8EN, UK. Email: d.basak@swansea.ac.uk ______________________ † We thank an anonymous referee of this journal for helpful comments and suggestions that helped to improve the paper. The usual disclaimer applies.  1  1. Introduction In a seminal paper, Mankiw and Whinston (1986) show that entry is socially excessive in oligopolistic industries with scale economies and this holds irrespective of the type of technology, such as constant returns or non-constant returns. This “excess-entry theorem”, which may justify the anticompetitive entry regulation policies adopted by many countries,1 ignored an important aspect, viz., the market power of the input suppliers creating strategic input price determination.2 However, in real world, the presence of labour unions or certain key input suppliers, such as the computer chip producer like Intel, may justify market power of the input suppliers in many industries. We show that if the input suppliers have market power, social efficiency of entry of the final goods producers depends on their technologies. Considering a competitive input market, Mankiw and Whinston (1986) show that entry is socially excessive. Ghosh and Morita (2007a, b) argue that if the input suppliers have market power, the result of Mankiw and Whinston (1986) may not hold and entry can be socially insufficient. We contribute in this literature by showing that if the input supplier has market power, social desirability of entry depends on returns to scale. Entry is socially insufficient under constant returns but it is socially excessive under decreasing returns3 if the cost of entry is sufficiently low. Hence, the policy implication of Mankiw and Whinston (1986) remains in the presence of significant market power of the input suppliers if there are decreasing returns in the final goods production and the cost of entry is sufficiently low.  2. The model and the results Consider an industry with a large number of symmetric final goods producers, each of whom decides whether or not to enter the market. Entry requires each final goods producer to incur an entry cost 𝐾 > 0. Assume that there exists an input supplier that                                                  1 Preventing excessive entry was a guiding principle in the Japanese industrial policy in the postwar period (Suzumura and Kiyono, 1987 and Suzumura, 1995). 2 One may look at Suzumura and Kiyono (1987), Anderson et al. (1995), Fudenberg and Tirole (2000), Cabral (2004), Ghosh and Saha (2007), Mukherjee (2012), for a representative sample of the papers on social efficiency of entry with competitive input markets. 3 The implications of decreasing returns to scale technologies, which can be motivated by capacity constraints or scarcity of resources, have been discussed extensively in different contexts (Tirole, 1988). Banker et al. (1994) show that decreasing returns may prevail in the software industry. Saal et al. (2007) found constant or decreasing returns in the UK industries. The typical two-digit industries in the US (Basu and Fernald, 1997), three-digit manufacturing industries in Singapore (Kee, 2002), Broadacre industries in Australia (Townsend et al., 1998) appear to have (slightly) decreasing returns to scale.   2  supplies inputs to all the final goods producers. To illustrate our result in a simple fashion we assume that input supplier unilaterally sets the input price. Consider the following game. At stage 1, the final goods producers decide whether or not to enter the market. At stage 2, the input supplier determines the input price. At stage 3, the final goods producers, which entered the market, produce outputs simultaneously by purchasing inputs according to their requirements and the profits are realised. We solve the game through backward induction. For simplicity we assume that the final goods producers have symmetric production technologies  and they can transform the inputs to a homogeneous final good at zero cost. We consider the case of constant returns to scale technology in Section 2.1 and the case of decreasing returns to scale technology in Section 2.2. Assume that the inverse market demand function is P = a – Q, where P is price and Q is the total output.   2.1. Constant returns to scale technology Assume that the production technology of the jth firm, 𝑗 = 1,2, … , ∞, is 𝐿𝑗 = 𝑞𝑗,  where 𝐿𝑗   is the amount of input and 𝑞𝑗 is the output. If n final goods producers enter the market, the ith final goods producer maximises the following expression to determine its output:  𝑀𝑎𝑥  𝜋𝑖𝐶𝑅𝑆 =(𝑃 − 𝑤)𝑞𝑖 − 𝐾.        (1) The equilibrium output of the ith firm is  𝑞𝑖𝐶𝑅𝑆 =𝑎−𝑤1+𝑛.         (2) The input supplier maximises the following expression to determine the input price:  1( )nCRSiwiMaxU w c q          (3) where 𝑐 represents the marginal cost of input production. The equilibrium input price is 𝑤𝐶𝑅𝑆 =𝑎+𝑐2. The equilibrium net profit of the ith final goods producer, i=1,2,…,n, which has entered the market is  𝜋𝑖𝐶𝑅𝑆 =14(𝑎−𝑐1+𝑛)2− 𝐾.                   (4) For analytical convenience, we consider the number of firms as a continuous variable. Hence, entry in the market occurs as long as the net profit of a final goods producer is 3  non-negative. Given the symmetry of the firms, the free entry equilibrium number of final goods producers is given by 0CRSi   or 14(𝑎−𝑐1+𝑛)2= K.                    (5) It follows from (5) that, if the cost of entry (i.e., K ) falls, the free entry equilibrium number of final goods producers, n, increases. Now we determine the welfare maximising number of final goods producers, where welfare is the sum of the total net profits of the final goods producers (∑ 𝜋𝑖𝐶𝑅𝑆), the profit of the input supplier (𝑈𝐶𝑅𝑆) and consumer surplus (𝐶𝑆𝐶𝑅𝑆). Following the literature, we consider the second-best problem of welfare maximisation, i.e., we determine the welfare maximising number of final goods producers subject to Cournot behaviour of the firms. Hence, the social planner can control the number of final goods producers entering the market, but it cannot control the output choice behaviour of the final goods producers. If n final goods producers produce the final good, it follows from (4) that the total net profit of the n firms is  ∑ 𝜋𝑖𝐶𝑅𝑆 =n4(𝑎−𝑐1+𝑛)2− 𝑛𝐾.               (6) The profit of the input supplier is  𝑈𝐶𝑅𝑆 =𝑛(𝑎−𝑐)24(1+𝑛).        (7) Since the total final goods production is 𝑄𝐶𝑅𝑆 = 𝑛 (𝑎−𝑤1+𝑛), the consumer surplus is 𝐶𝑆𝐶𝑅𝑆 =𝑄22=𝑛28(𝑎−𝑐1+𝑛)2.       (8) The social planner chooses 𝑛 to maximise social welfare (which is the sum of (6), (7) and (8)):              Max𝑛  𝑆𝑊𝐶𝑅𝑆 = Max𝑛𝑛(𝑎−𝑐)2(4+3𝑛)8(1+𝑛)2− 𝑛𝐾     (9) The welfare maximising 𝑛 is given by               𝜕(𝑆𝑊𝐶𝑅𝑆)𝜕𝑛≡(𝑎−𝑐)2(2+𝑛)4(1+𝑛)3− 𝐾 = 0                                         (10) It follows from (10) that as the cost of entry (i.e., 𝐾) falls, the welfare maximising number of final goods producers increases.  Proposition 1 If the final goods production technology exhibits constant returns to scale, entry in the final goods market is insufficient. 4  Proof: Evaluating ( )CRSSWn at the free entry equilibrium number of final goods producers, i.e., substituting 14(𝑎−𝑐1+𝑛)2 for K into (10), we get that ( )0CRSSWn since (𝑎−𝑐)2(2+𝑛)4(1+𝑛)3−14(𝑎−𝑐1+𝑛)2=(𝑎−𝑐)24(1+𝑛)3> 0, which proves the result. ■  Entry creates two effects. First, as in Mankiw and Whinston (1986), it creates a “business-stealing effect” by lowering the outputs of the incumbent final goods producers. On the other hand, as observed by Ghosh and Morita (2007a, b), entry creates a “business-creation effect” in the input sector by raising the total input demand. We find that the latter effect dominates the former effect for social welfare and makes entry less attractive than the socially optimum level. This result affirms the finding of Ghosh and Morita (2007a, b), suggesting that entry is insufficient in a vertically related industry with constant returns to scale technologies.  2.2. Production with decreasing return to scale technology We now consider the case of decreasing returns to scale technologies for the final goods production. Assume that the technology of the jth final goods producer, j=1,2,…,∞, is j jL q  or 2j jL q , implying that 2jq  inputs are required to produce jq  units of the output. If n final goods producers enter the market, the ith final goods producer, i=1,2,..,n, maximises             𝑀𝑎𝑥   𝜋𝑖𝐷𝑅𝑆 =(𝑎 − 𝑄)𝑞𝑖 − 𝑤𝑞𝑖2 − 𝐾 .              (12) The equilibrium output of the ith final goods producer is  𝑞𝑖𝐷𝑅𝑆 =𝑎(1+2𝑤)−11+𝑛(1+2𝑤)−1.                 (13) The input supplier maximises the following expression to determine w:  21(w c)nDRSiwiMaxU q   .                (14) The equilibrium input price is  𝑤𝐷𝑅𝑆 =12(1 + 𝑛 + 4𝑐). Hence, the equilibrium profit of the input supplier is  𝑈𝐷𝑅𝑆 =𝑛8(𝑎21+𝑛+2𝑐), the profit of the ith final goods producer 5  that entered the market is  𝜋𝑖𝐷𝑅𝑆 =𝑎2(3+n+4c)8(1+n+2c)2− 𝐾 and the consumer surplus is 𝐶𝑆𝐷𝑅𝑆 =𝑄22=18(𝑛𝑎1+𝑛+2𝑐)2.  The free entry equilibrium number of final goods producer is given by 𝜋𝑖𝐷𝑅𝑆 = 0 or   𝑎2(3+n+4c)8(1+n+2c)2= K.                                                                                            (15) It follows from (15) that a lower K increases the free equilibrium number of final goods producers.  Now determine the welfare maximising number of firms, which is determined by maximising the following expression:              Max𝑛  𝑆𝑊𝐷𝑅𝑆 = Max𝑛𝑎2𝑛(4+3𝑛+6𝑐)8(1+𝑛+2𝑐)2− 𝑛𝐾.                         (16) The welfare maximising number of firms is given by             𝜕(𝑆𝑊𝐷𝑅𝑆)𝜕𝑛≡𝑎2(2+𝑛+3𝑐𝑛+7𝑐+6𝑐2)4(1+𝑛+2𝑐)3− 𝐾 = 0              (17) It follows from (17) that a lower K increases the welfare maximising number of final goods producers.  Proposition 2: If the production technology exhibits decreasing return to scale, entry in the final goods market is excessive (insufficient) if the cost of entry is sufficiently low (high) so that the free entry equilibrium number of firms 𝑛 > (<)√2(1 + 2𝑐 + 2𝑐2) − 1 ≡ 𝑛∗. Proof: We get that  𝜕2(𝑆𝑊𝐷𝑅𝑆)𝜕𝑛2= −𝑎2(5+2𝑛+6𝑐𝑛+16𝑐+12𝑐2)4(1+𝑛+2𝑐)4< 0 and evaluating ( )DRSSWn at the free entry equilibrium number of final goods producers, i.e., substituting  𝑎2(3+n+4c)8(1+n+2c)2 for K into (17), we get that 𝜕(𝑆𝑊𝐷𝑅𝑆)𝜕𝑛< (>)0 for 𝑎2(2+7𝑐+6𝑐2+𝑛+3𝑐𝑛)4(1+𝑛+2𝑐)3< (>)𝑎2(3+n+4c)8(1+n+2c)2  or  * 22 1 2c 2 1( ) cn n    , implying that entry is excessive (insufficient) for  * 22 1 2c 2 1( ) cn n    . Since the free entry equilibrium number of firms is negatively related to the cost of entry, excessive (insufficient) entry occurs, i.e., n > (<)n∗, for sufficiently low (high) cost of entry. ■  6  The intuitions for Proposition 2 also follows from the trade-off between the “business-stealing effect” and “business-creation effect” as discussed in Proposition 1 with an exception that, under decreasing returns to scale technology, the input price increases with the number of final goods producers, thus strengthening the business-stealing effect. If the cost of entry is low so that the final goods market is sufficiently competitive, i.e., n > n∗,  a new entrant creates significant business-stealing effect by reducing the profits of a large number of incumbent final goods producers, thus creating excessive entry. However, if the cost of entry is high, the business-stealing effect is not so strong to outweigh the business-creation effect and entry is insufficient in this situation.  3. Conclusion We show that the production technology plays an important role in determining social efficiency of entry of the final goods producers if the input supplier has market power. Entry in the final goods market can be socially insufficient under constant returns to scale technology but it can be socially excessive if the cost of entry is low. Hence, the anti-competitive entry regulation policies are more justifiable in a vertically related industry if the final goods market is characterised by decreasing returns to scale technologies.  References Anderson, S.P., A. de Palma and Y. Nesterov (1995), “Oligopolistic competition and the optimal provision of products”, Econometrica, 63: 1281-1301.  Banker, R.D., H. Chang and C.F. Kemerer (1994) “Evidence on economies of scale in software development”, Information and Software Technology, 36, 275-282. Basu, S. and J.G. Fernald (1997) “Returns to scale in U.S. production: estimates and implications”, Journal of Political Economy, 105, 249-283. Cabral, L.M.B. (2004) “Simultaneous entry and welfare”, European Economic Review, 48: 943-957. Fudenberg, D. and J. Tirole (2000), “Pricing a network good to deter entry”, The Journal of Industrial Economics, XLVIII: 373-390. Ghosh, A. and A. Morita (2007a) “Social desirability of free entry: a bilateral oligopoly analysis” International Journal of Industrial Organization 25, 925-34. 7  Ghosh, A. and A. Morita (2007b) “Free entry and social efficiency under vertical oligopoly”, Rand Journal of Economics 38, 539-52. Ghosh A. and S. Saha (2007) “Excess entry in the absence of scale economies”, Economic Theory, 30:575-586. Kee, H. L. (2002) “Markups, returns to scale and productivity: a case of Singapore’s manufacturing sector, Mimeo. Mankiw, N.G., Whinston, M.D. (1986) “Free entry and social inefficiency”, Rand Journal of Economics, 17, 1, 48-58. Mukherjee, A. (2012) “Social Efficiency of Entry with Market Leaders”, Journal of Economics & Management Strategy, 21: 431-444. Saal, D.S., Parker, D. and Weyman-Jones, T. (2007) “Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985–2000”. Journal of Productivity Analysis, 28 (1), 127–139. Suzumura, K. (1995) Competition, Commitment and Welfare, New York, USA: Oxford. Suzumura, K. and K. Kiyono (1987) “Entry barriers and economic welfare”, Review of Economic Studies 54, 157–167. Tirole, J. (1988) The theory of industrial organization, MIT Press, Cambridge. Townsend, R.F., J. Kirsten and N. Vink (1998) “Farm size, productivity and returns to scale in agriculture revisited: a case study of wine producers in South Africa”, Agricultural Economics, 19: 175-180. 

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