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Market structure of multi-product firms under free entry
Economics Letters 61 (1998) 159–163
Market structure of multi-product firms under free entry a, b David Hanly *, Keith C.K. Cheung a
b
Department of Economics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7 N 5 A5 Department of Finance, Faculty of Business Administration, University of Windsor, Windsor, Ontario, Canada Received 22 April 1998; accepted 25 June 1998
Abstract Three sources of advantages to multi-product firm are considered: economies of scope, risk reduction and demand complementarity. Each source if of sufficient magnitude leads to a market equilibrium dominated by multi-product firms. As the market becomes sufficiently large the multi-product Cournot-Nash model is shown to converge to an efficient outcome. 1998 Elsevier Science S.A. All rights reserved. Keywords: Cournot-Nash; Free entry; Multi-product; Economies of scope JEL classification: C72; D43
1. Introduction Many models exist of industry equilibrium under free entry with contributions including Novshek (1980) considering Cournot-Nash equilibrium, Hart (1985) examining Chamberlinian monopolistic competition, and Maskin and Tirole (1988) looking at Edgeworth cycles. With the exception of contestable market theory (Baumol et al., 1982) and work by Okuguchi and Szidarovskzky (1990) on multi-product oligopoly, attention has tended to focus on single product firms. Calem (1988); Cairns and Mahabir (1988) and Wegberg and Witteloostuijn (1992) argue that multi-product firms provide the best source for potential competition necessary to ensure markets are contestable. Rather than treating multi-product firms as a source of potential competition, we focus on how the actual presence of firms in many different product markets influences market performance. More generally, we identify factors which are the source of competitive advantage of multi-product firms and determine how these factors affect market performance.
2. The model The model extends Cheung’s (Cheung, 1995, Ch. 1) restricted entry Cournot-Nash duopoly under price uncertainty model to a free entry market. Uncertainty enters as an additive error in market *Corresponding author. Tel.: 11-306-966-5213; fax: 11-306-966-5232; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00160-8
D. Hanly, K.C.K. Cheung / Economics Letters 61 (1998) 159 – 163
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prices. Risk averse firms maximize the expected utility of income, using a mean-variance utility function. 1 max EU(p(q1 ,q2 )) 5 P1 q1 1 P2 q2 2 C(q1 ,q2 ) 2 ]RA (q 12 s 12 1 q 22 s 22 1 2q1 q2 s12 ), q 1, q 2 2
(1)
where Pi 5 a i 2 b i Q i 1 h ij q j for i 5 1,2 and 2
2
1/2
1/2
C(q1 ,q2 ) 5 (c 1 q1 1 c 2 q2 1 d 1 q 1 1 d 2 q 2 1 eq1 q2 2 gq 1 q 2 1 F 1 f1 1 f2 ).
(2)
All terms except s12 are assumed to be positive. Pi , Q i and qi -market price, market quantity and firm output of product i; d i -diseconomies of scale product i; e-diseconomies of scope; g-unit economies of scope; F-non-product specific indivisible costs; fi -indivisible cost of product i; h ij -demand complementarity of product j on product i; RA -level of risk aversion; si 2 -price variance of product i and s12 -covariance of prices. Three sources of potential advantage to diversified firms are present: economies of scope in two forms-unit costs and indivisible non-product specific costs, risk reduction through negative covariance and complementarity in product demands. The utility of expected income equation has a number of common terms in the relationship between q1 and q2 which have the same type of impact and can be represented by a single parameter. All firms are assumed to be identical. The analysis is further simplified by assuming the two products have the same demand, costs, and risk. max a 1 (q1 1 q2 ) 1 gq 11 / 2 q 12 / 2 2 a 2 (Q 1 q1 1 Q 2 q2 ) 2 a 3 (q 21 1 q 22 ) 2 a 4 q1 q2 2 a 5 q ,q 1
(3)
2
and a 1 5a2c; a 2 5b; a 3 5d 1RA s 2 / 2; a 4 5e1RA s12 22h and a 5 5F 12f. A market equilibrium (Q *1d ,Q *2d , n *d ) exists when: EU(p (q *1 ,q *2 ;Q 1* ,Q 2* ,n d* )) $ 0 and EU(p (q 1* ,q 2* ;Q 1* ,Q 2* ,n d* 1 1)) , 0,
(4)
where n *d is the number of firms and Q *id 5n *d q *id . Instead of producing many different products an alternative is to produce a single product type. The utility of expected profit function is simpler for the single product firm with a 4 50, g50 and a 5 5a 5u 5F 1f. Since the two markets are identical and * and Q 1u * 5Q 2u *. symmetrical, Q *1d 5Q 2d Proposition 1. Three types of market equilibria exist: ( i) only multi-product firms if Q *d .Q *u 1 qd (Q u* ); ( ii) only single product firms if Q u* .Q d* 1qu (Q *d ) and ( iii) an indeterminate mixture of multi-product and single product firms if Q d* ,Q u* 1qd (Q *u ) and Q *u ,Q *d 1qu (Q *d ). Proof. If Q d* .Q *u 1qd (Q u* ) then multi-product firms can invade the market equilibrium and drive out single product firms since EUd (p(Q u* 1q d* )).0. An identical argument can be made for single product firms when Q *u .Q *d 1qu (Q *d ). In the third case, pure equilibria of only single product or multi-product firms are possible since once established they cannot be entered profitably but equilibria with combinations of the two types of firms may also exist. Proposition 2. If diseconomies of scope, e50 then the presence of any one of the following factors
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alone in sufficient magnitude leads to an equilibrium with only multi-product firms: ( i) unit economies of scope ( g), ( ii) non-product specific fixed costs (F ), ( iii) negative covariance of product prices (s12 ) and demand complementarities (h) reflected in a 4 5RA(s12 )-2h. Proof. By Proposition 1 if Q *d .Q *u 1qd (Q *u ) then the market equilibrium will be composed of only multi-product firms. Each one of the following three conditions by itself ensures Q d* .Q u* 1qd (Q u* ) is satisfied ] a 2œf fa 2 (3a 2 1 4a 3 ) a 2 (3a 2 1 4a 3 ) (i) g* . ]]] (iii) a *4 , 2 ]]]]]. (6) ]]] (ii) F* . ]]]]] 2 2 2(a 2 1 a 3 ) 2a 3 2 a 2 œa 2 1 a 3 F* will only be defined for a 3 .a 2 / œ2. The second order conditions puts a lower bound on a 4min . 2(2a 3 12a 2 ). It can be easily shown that a 4* .a 4min .
3. Impact on market structure Increasing a 1 or g, decreasing a 2 or a 5 , all increase the number of firms in the market. Changing a 3 or a 4 has an ambiguous impact. Both a 3 and a 4 influence the number of firms in the same manner and their impact can be summarized by a 34 52a 3 1a 4 . Proposition 3. A critical a *34 5(a 1 1g)2 /(4a 5 ) exists where the equilibrium number of firms in an industry is at a maximum for a given a 1 , g, a 2 and a 5 . Proof. Solve for the optimal n d* when EU(p (n *d ))50 and differentiate with respect to a 34 . Solving 2 * 5(a 1 1g)2 /(4a 5 )22a 2 which is a maximum since ≠ 2 n *d / ≠a 34 ≠n *d / ≠a 34 50 yields a 34 ,0. In addition max 2 to guarantee n$1, a 34 ,(a 1 1g) /(a 5 )22a 2 . For a 34 ,a *34 , decreasing demand complementarity (h) and increasing risk, variance, covariance, diseconomies of scale and scope all lead to more firms in the market. As a 34 is increased it causes individual firms to lower their output which tends to lower profit, however, the general reduction in industry output acts to increase firm profit and when a 34 ,a *34 the second effect dominates the first effect making it possible for additional firms to enter the industry.
4. Efficiency of markets with multi-product firms An efficient market exists if consumer surplus is maximized. Novshek (1980) has shown that as the market becomes large in the single product firm case that the free entry equilibrium converges to the efficient market outcome. A similar result holds for multi-product firms. Definition 1. Relative Welfare Loss (RWL)5(CSE2CSN) / CSE where CSE is Consumer Surplus at the efficient outcome and CSN is Consumer Surplus at the Cournot-Nash outcome. Definition 2. Relative Efficiency (RE)5(CSN / CSE)1 / 2 . Since RWL512RE 2 , as RE→1 then RWL→0.
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Proposition 4. As the market gets large, either through a 2 →0 or a 5 →0, the RWL →0. Proof. CSE5Q 2E and CSN5Q 2N therefore RE5Q E /Q N with Q N 5Q *d and Q E 5((a 1 1g)2 (a 34 a 5 )1 / 2 ) /a 2 . ]]] (a 1 1 g)œ2a 2 1 a 34 2œ] a 5 (a 2 1 a 34 ) Relative Efficiency (RE) 5 ]]]]]]]]]]] ]]] ] ] œ2a 2 1 a 34 (a 1 1 g 2œa 34œa 5 )
(7)
As a 2 →0 or a 5 →0, Relative Efficiency→1 which implies that Relative Welfare Loss→0. Proposition 5. The impact of the various factors which lead to multi-product firms have different impacts on Relative Welfare Loss of the market: ( i) Increasing g decreases RWL ( ii) Increasing F increases RWL and ( iii) increasing h and decreasing s12 have ambiguous impacts. Proof. The impact on RE is shown by differentiating with respect to g, a 5 5F 12f and a 34 5a 3 1e1 s12 22h. A critical value for a *34 may exist, above which RE is decreasing for increases in a 34 . Below a *34 , RE is increasing in a 34 .
5. Conclusion Unit economies of scope ( g), non-product specific fixed costs (F ), negative covariance (s12 ) and demand complementarity (h), each at sufficient levels leads to market equilibria with only multiproduct firms. Whether the number of firms in an industry is increased or decreased depends on the source of competitive advantage to multi-product firms. Unit economies of scope ( g) lead to increasing numbers of firms in the industry. On the other hand increases in non-product specific fixed cost (F ) decreases the number of firms in an industry and factors such as price covariance (s12 ) and demand complementarity (h) have an ambiguous impact which depends on the level of other cost and demand parameters. Lastly, as the market gets large the equilibrium with multi-product firms and free entry as with single product firms converges to the efficient outcome.
References Baumol, W., Panzar, J., Willig, R., 1982. Contestable Markets and the Theory of Industry Structure, Harcourt Brace Jovanovich, New York. Cairns, R.D., Mahabir, D., 1988. Contestability: a revisionist view. Economica 55, 269–276. Calem, P., 1988. Entry and entry deterrence in penetrable markets. Economica 55, 171–183. Cheung, C. K., 1995, The Economics of Venture Strategies with Correlated Demands, Ph.D thesis, York University, Toronto. Hart, O.D., 1985. Monopolistic competition in the spirit of Chamberlin: a general model. Review of Economic Studies 52, 529–546. Maskin, E., Tirole, J., 1988. A theory of dynamic oligopoly. Econometrica 56, 549–599. Novshek, W., 1980. Cournot equilibrium with free entry. Review of Economic Studies 47, 473–486.
D. Hanly, K.C.K. Cheung / Economics Letters 61 (1998) 159 – 163
163
Okuguchi, K., F. Szidarovskzky, 1990. The Theory of Oligopoly with Mult-Product Firms-Lecture Notes in Economics and Mathematical Systems, No. 342, Springer-Verlag, Berlin. Wegberg, M.V., Witteloostuijn, A.V., 1992. Credible entry threats into contestable markets: a symmetric multi-market model of contestability. Economica 59, 437–452.
Market structure of multi-product firms under free entry a, b David Hanly *, Keith C.K. Cheung a
b
Department of Economics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7 N 5 A5 Department of Finance, Faculty of Business Administration, University of Windsor, Windsor, Ontario, Canada Received 22 April 1998; accepted 25 June 1998
Abstract Three sources of advantages to multi-product firm are considered: economies of scope, risk reduction and demand complementarity. Each source if of sufficient magnitude leads to a market equilibrium dominated by multi-product firms. As the market becomes sufficiently large the multi-product Cournot-Nash model is shown to converge to an efficient outcome. 1998 Elsevier Science S.A. All rights reserved. Keywords: Cournot-Nash; Free entry; Multi-product; Economies of scope JEL classification: C72; D43
1. Introduction Many models exist of industry equilibrium under free entry with contributions including Novshek (1980) considering Cournot-Nash equilibrium, Hart (1985) examining Chamberlinian monopolistic competition, and Maskin and Tirole (1988) looking at Edgeworth cycles. With the exception of contestable market theory (Baumol et al., 1982) and work by Okuguchi and Szidarovskzky (1990) on multi-product oligopoly, attention has tended to focus on single product firms. Calem (1988); Cairns and Mahabir (1988) and Wegberg and Witteloostuijn (1992) argue that multi-product firms provide the best source for potential competition necessary to ensure markets are contestable. Rather than treating multi-product firms as a source of potential competition, we focus on how the actual presence of firms in many different product markets influences market performance. More generally, we identify factors which are the source of competitive advantage of multi-product firms and determine how these factors affect market performance.
2. The model The model extends Cheung’s (Cheung, 1995, Ch. 1) restricted entry Cournot-Nash duopoly under price uncertainty model to a free entry market. Uncertainty enters as an additive error in market *Corresponding author. Tel.: 11-306-966-5213; fax: 11-306-966-5232; e-mail: [email protected] 0165-1765 / 98 / $ – see front matter 1998 Elsevier Science S.A. All rights reserved. PII: S0165-1765( 98 )00160-8
D. Hanly, K.C.K. Cheung / Economics Letters 61 (1998) 159 – 163
160
prices. Risk averse firms maximize the expected utility of income, using a mean-variance utility function. 1 max EU(p(q1 ,q2 )) 5 P1 q1 1 P2 q2 2 C(q1 ,q2 ) 2 ]RA (q 12 s 12 1 q 22 s 22 1 2q1 q2 s12 ), q 1, q 2 2
(1)
where Pi 5 a i 2 b i Q i 1 h ij q j for i 5 1,2 and 2
2
1/2
1/2
C(q1 ,q2 ) 5 (c 1 q1 1 c 2 q2 1 d 1 q 1 1 d 2 q 2 1 eq1 q2 2 gq 1 q 2 1 F 1 f1 1 f2 ).
(2)
All terms except s12 are assumed to be positive. Pi , Q i and qi -market price, market quantity and firm output of product i; d i -diseconomies of scale product i; e-diseconomies of scope; g-unit economies of scope; F-non-product specific indivisible costs; fi -indivisible cost of product i; h ij -demand complementarity of product j on product i; RA -level of risk aversion; si 2 -price variance of product i and s12 -covariance of prices. Three sources of potential advantage to diversified firms are present: economies of scope in two forms-unit costs and indivisible non-product specific costs, risk reduction through negative covariance and complementarity in product demands. The utility of expected income equation has a number of common terms in the relationship between q1 and q2 which have the same type of impact and can be represented by a single parameter. All firms are assumed to be identical. The analysis is further simplified by assuming the two products have the same demand, costs, and risk. max a 1 (q1 1 q2 ) 1 gq 11 / 2 q 12 / 2 2 a 2 (Q 1 q1 1 Q 2 q2 ) 2 a 3 (q 21 1 q 22 ) 2 a 4 q1 q2 2 a 5 q ,q 1
(3)
2
and a 1 5a2c; a 2 5b; a 3 5d 1RA s 2 / 2; a 4 5e1RA s12 22h and a 5 5F 12f. A market equilibrium (Q *1d ,Q *2d , n *d ) exists when: EU(p (q *1 ,q *2 ;Q 1* ,Q 2* ,n d* )) $ 0 and EU(p (q 1* ,q 2* ;Q 1* ,Q 2* ,n d* 1 1)) , 0,
(4)
where n *d is the number of firms and Q *id 5n *d q *id . Instead of producing many different products an alternative is to produce a single product type. The utility of expected profit function is simpler for the single product firm with a 4 50, g50 and a 5 5a 5u 5F 1f. Since the two markets are identical and * and Q 1u * 5Q 2u *. symmetrical, Q *1d 5Q 2d Proposition 1. Three types of market equilibria exist: ( i) only multi-product firms if Q *d .Q *u 1 qd (Q u* ); ( ii) only single product firms if Q u* .Q d* 1qu (Q *d ) and ( iii) an indeterminate mixture of multi-product and single product firms if Q d* ,Q u* 1qd (Q *u ) and Q *u ,Q *d 1qu (Q *d ). Proof. If Q d* .Q *u 1qd (Q u* ) then multi-product firms can invade the market equilibrium and drive out single product firms since EUd (p(Q u* 1q d* )).0. An identical argument can be made for single product firms when Q *u .Q *d 1qu (Q *d ). In the third case, pure equilibria of only single product or multi-product firms are possible since once established they cannot be entered profitably but equilibria with combinations of the two types of firms may also exist. Proposition 2. If diseconomies of scope, e50 then the presence of any one of the following factors
D. Hanly, K.C.K. Cheung / Economics Letters 61 (1998) 159 – 163
161
alone in sufficient magnitude leads to an equilibrium with only multi-product firms: ( i) unit economies of scope ( g), ( ii) non-product specific fixed costs (F ), ( iii) negative covariance of product prices (s12 ) and demand complementarities (h) reflected in a 4 5RA(s12 )-2h. Proof. By Proposition 1 if Q *d .Q *u 1qd (Q *u ) then the market equilibrium will be composed of only multi-product firms. Each one of the following three conditions by itself ensures Q d* .Q u* 1qd (Q u* ) is satisfied ] a 2œf fa 2 (3a 2 1 4a 3 ) a 2 (3a 2 1 4a 3 ) (i) g* . ]]] (iii) a *4 , 2 ]]]]]. (6) ]]] (ii) F* . ]]]]] 2 2 2(a 2 1 a 3 ) 2a 3 2 a 2 œa 2 1 a 3 F* will only be defined for a 3 .a 2 / œ2. The second order conditions puts a lower bound on a 4min . 2(2a 3 12a 2 ). It can be easily shown that a 4* .a 4min .
3. Impact on market structure Increasing a 1 or g, decreasing a 2 or a 5 , all increase the number of firms in the market. Changing a 3 or a 4 has an ambiguous impact. Both a 3 and a 4 influence the number of firms in the same manner and their impact can be summarized by a 34 52a 3 1a 4 . Proposition 3. A critical a *34 5(a 1 1g)2 /(4a 5 ) exists where the equilibrium number of firms in an industry is at a maximum for a given a 1 , g, a 2 and a 5 . Proof. Solve for the optimal n d* when EU(p (n *d ))50 and differentiate with respect to a 34 . Solving 2 * 5(a 1 1g)2 /(4a 5 )22a 2 which is a maximum since ≠ 2 n *d / ≠a 34 ≠n *d / ≠a 34 50 yields a 34 ,0. In addition max 2 to guarantee n$1, a 34 ,(a 1 1g) /(a 5 )22a 2 . For a 34 ,a *34 , decreasing demand complementarity (h) and increasing risk, variance, covariance, diseconomies of scale and scope all lead to more firms in the market. As a 34 is increased it causes individual firms to lower their output which tends to lower profit, however, the general reduction in industry output acts to increase firm profit and when a 34 ,a *34 the second effect dominates the first effect making it possible for additional firms to enter the industry.
4. Efficiency of markets with multi-product firms An efficient market exists if consumer surplus is maximized. Novshek (1980) has shown that as the market becomes large in the single product firm case that the free entry equilibrium converges to the efficient market outcome. A similar result holds for multi-product firms. Definition 1. Relative Welfare Loss (RWL)5(CSE2CSN) / CSE where CSE is Consumer Surplus at the efficient outcome and CSN is Consumer Surplus at the Cournot-Nash outcome. Definition 2. Relative Efficiency (RE)5(CSN / CSE)1 / 2 . Since RWL512RE 2 , as RE→1 then RWL→0.
D. Hanly, K.C.K. Cheung / Economics Letters 61 (1998) 159 – 163
162
Proposition 4. As the market gets large, either through a 2 →0 or a 5 →0, the RWL →0. Proof. CSE5Q 2E and CSN5Q 2N therefore RE5Q E /Q N with Q N 5Q *d and Q E 5((a 1 1g)2 (a 34 a 5 )1 / 2 ) /a 2 . ]]] (a 1 1 g)œ2a 2 1 a 34 2œ] a 5 (a 2 1 a 34 ) Relative Efficiency (RE) 5 ]]]]]]]]]]] ]]] ] ] œ2a 2 1 a 34 (a 1 1 g 2œa 34œa 5 )
(7)
As a 2 →0 or a 5 →0, Relative Efficiency→1 which implies that Relative Welfare Loss→0. Proposition 5. The impact of the various factors which lead to multi-product firms have different impacts on Relative Welfare Loss of the market: ( i) Increasing g decreases RWL ( ii) Increasing F increases RWL and ( iii) increasing h and decreasing s12 have ambiguous impacts. Proof. The impact on RE is shown by differentiating with respect to g, a 5 5F 12f and a 34 5a 3 1e1 s12 22h. A critical value for a *34 may exist, above which RE is decreasing for increases in a 34 . Below a *34 , RE is increasing in a 34 .
5. Conclusion Unit economies of scope ( g), non-product specific fixed costs (F ), negative covariance (s12 ) and demand complementarity (h), each at sufficient levels leads to market equilibria with only multiproduct firms. Whether the number of firms in an industry is increased or decreased depends on the source of competitive advantage to multi-product firms. Unit economies of scope ( g) lead to increasing numbers of firms in the industry. On the other hand increases in non-product specific fixed cost (F ) decreases the number of firms in an industry and factors such as price covariance (s12 ) and demand complementarity (h) have an ambiguous impact which depends on the level of other cost and demand parameters. Lastly, as the market gets large the equilibrium with multi-product firms and free entry as with single product firms converges to the efficient outcome.
References Baumol, W., Panzar, J., Willig, R., 1982. Contestable Markets and the Theory of Industry Structure, Harcourt Brace Jovanovich, New York. Cairns, R.D., Mahabir, D., 1988. Contestability: a revisionist view. Economica 55, 269–276. Calem, P., 1988. Entry and entry deterrence in penetrable markets. Economica 55, 171–183. Cheung, C. K., 1995, The Economics of Venture Strategies with Correlated Demands, Ph.D thesis, York University, Toronto. Hart, O.D., 1985. Monopolistic competition in the spirit of Chamberlin: a general model. Review of Economic Studies 52, 529–546. Maskin, E., Tirole, J., 1988. A theory of dynamic oligopoly. Econometrica 56, 549–599. Novshek, W., 1980. Cournot equilibrium with free entry. Review of Economic Studies 47, 473–486.
D. Hanly, K.C.K. Cheung / Economics Letters 61 (1998) 159 – 163
163
Okuguchi, K., F. Szidarovskzky, 1990. The Theory of Oligopoly with Mult-Product Firms-Lecture Notes in Economics and Mathematical Systems, No. 342, Springer-Verlag, Berlin. Wegberg, M.V., Witteloostuijn, A.V., 1992. Credible entry threats into contestable markets: a symmetric multi-market model of contestability. Economica 59, 437–452.