Ralph s Strategic Disclosure 1

Transcription

1 Ralph s Strategic Disclosure Ralph manages a firm that operates in a duopoly Both Ralph s (privatevalue) production cost and (common-value) inverse demand are uncertain Ralph s (constant marginal) production cost equals the rival s constant marginal cost of c = 0 plus a shock, ε 2 c 0 = 0 + ε 2 where ε 2 { e, e} are equally likely Inverse demand is linear in total output P = 5 + ε q 0 q where ε { e, e} are equally likely and independent of ε 2, e =, q 0 is Ralph s production quantity, and q is the rival s production quantity The two firms compete by selecting their respective production quantities (this is often referred to as Cournot competition) It is common knowledge that Ralph privately learns ε and ε 2 prior to producing and the rival does not Therefore, Ralph has multiple disclosure strategies that potentially impact his competitive advantage: full disclosure, common-value (price) disclosure of ε, private-value (cost) disclosure of ε 2, or no disclosure Any disclosures are auditor-vetted and truthful The timeline is depicted below Ralph establishes Ralph learns Ralph discloses Firms choose his disclosure his cost and according to production levels, policy industry demand his policy demand and profits are realized timeline Ralph selects his disclosure policy to maximize expected firm profit Π 0 where both firms choose their quantities as best responses to the other firm s best response production choice given the information in hand Π 0 = max 4 [P c 0] q 0 (ε, ε 2 ) q 0(ε,ε 2) ε,ε 2 Rival s profit maximizing production choice depends on Ralph s disclosure strategy For full disclosure, rival s objective function mirrors Ralph s π (ε, ε 2 ) = max [P c] q (ε, ε 2 ) q (ε,ε 2) For common-value (demand) disclosure, rival s objective function is π (ε ) = max q (ε ) [P c] q (ε ) The example is borrowed from Arya and Mittendorf (forthcoming), "The interaction between corportate tax structure and disclosure policy," Annals of Finance

2 For private-value (cost) disclosure, rival s objective function is π (ε 2 ) = max q (ε 2) [P c] q (ε 2 ) And for no disclosure, rival s objective function is Full disclosure π ( ) = max q ( ) [P c] q ( ) For full disclosure, first order conditions for the two firms are foc 0 ε,ε 2 : 5 + ε 0 ε 2 2q 0 (ε, ε 2 ) q (ε, ε 2 ) = 0 foc ε,ε 2 : 5 + ε 0 q 0 (ε, ε 2 ) 2q (ε, ε 2 ) = 0 Solving the two linear equations for q 0 (ε, ε 2 ) and q (ε, ε 2 ) yields the firms best production responses given full disclosure by Ralph and q 0 (ε, ε 2 ) = 5 + ε 0 2ε 2 q (ε, ε 2 ) = 5 + ε 0 + ε 2 Expected profits for the two firms are Π 0 = 4 = (5 0)2 + 5e 2 Π = 4 = (5 0)2 + 2e 2 max [P c 0] q 0 (ε, ε 2 ) q 0(ε,ε 2) max [P c] q (ε, ε 2 ) q (ε,ε 2) Producer surplus is a measure of producer welfare and equals the sum of expected firm profits P S = Π 0 + Π Consumer surplus is a measure of consumer s welfare It s equal to the amount each marginal consumer benefits based on the market price and industry production level That is, consumer surplus is the excess of consumers willingness to pay over prevailing market conditions Since, demand is linear in this setting, consumer surplus is the area of a triangle whose base equals production q 0 + q and whose height equals the difference between the price at 2

3 which there is no demand P 0 and the market price Hence, consumer surplus equals the expected net benefit to consumers CS = 4 = 4 given full disclosure = 4 (5 0)2 + 5e (P 0 P ) [q 0 (ε, ε 2 ) + q (ε, ε 2 )] 2 [q 0 (ε, ε 2 ) + q (ε, ε 2 )] 2 Common-value (demand) disclosure For common-value (demand) disclosure, first order conditions for the two firms are foc 0 ε : 5 + ε 0 ε 2 2q 0 (ε, ε 2 ) q (ε ) = 0 foc ε : 5 + ε 0 q 0 (ε, e) + q 0 (ε, e) 2 2q (ε ) = 0 Solving the three linear equations for q 0 (ε, e), q 0 (ε, e), and q (ε ) yields the firms best production responses given common value (demand) disclosure by Ralph q 0 (ε, e) = 2 (5 + ε 0) e q 0 (ε, e) = 2 (5 + ε 0) + e q (ε ) = 5 + ε 0 For common-value (demand) disclosure, expected profits for the two firms are and Π 0 = 4 = 4 (5 0)2 + e 2 Π = 4 = (5 0)2 + e 2 max [P c 0] q 0 (ε, ε 2 ) q 0(ε,ε 2) max [P c] q (ε ) q (ε )

4 Producer surplus equals the sum of expected firm profits Consumer surplus is CS = 4 = 4 P S = Π 0 + Π = (5 0)2 + 25e 2 72 given common-value (demand) disclosure 2 (P 0 P ) [q 0 (ε, ε 2 ) + q (ε )] 2 [q 0 (ε, ε 2 ) + q (ε )] 2 Private-value (cost) disclosure For private-value (cost) disclosure, first order conditions for the two firms are foc 0 ε 2 : 5 + ε 0 ε 2 2q 0 (ε, ε 2 ) q (ε 2 ) = 0 foc ε 2 : 5 0 q 0 (e, ε 2 ) + q 0 ( e, ε 2 ) 2 2q (ε 2 ) = 0 Solving the three linear equations for q 0 (e, ε 2 ), q 0 ( e, ε 2 ), and q (ε 2 ) yields the firms best production responses given private-value (cost) disclosure by Ralph and q 0 (e, ε 2 ) = 2 (5 0 2ε 2) + e q 0 ( e, ε 2 ) = 2 (5 0 2ε 2) e q (ε 2 ) = ε 2 For private-value (cost) disclosure, expected profits for the two firms are Π 0 = 4 = 4 (5 0)2 + 25e 2 Π = 4 = (5 0)2 + e 2 max [P c 0] q 0 (ε, ε 2 ) q 0(ε,ε 2) 4 max [P c] q (ε 2 ) q (ε 2)

5 Producer surplus equals the sum of expected firm profits Consumer surplus is CS = 4 = 4 P S = Π 0 + Π = (5 0)2 + e 2 72 given private-value (cost) disclosure No disclosure 2 (P 0 P ) [q 0 (ε, ε 2 ) + q (ε 2 )] 2 [q 0 (ε, ε 2 ) + q (ε 2 )] 2 For no disclosure, first order conditions for the two firms are foc 0 : 5 + ε 0 ε 2 2q 0 (ε, ε 2 ) q ( ) = 0 foc : 0 = 5 0 q 0 (e, e) + q 0 ( e, e) + q 0 (e, e) + q 0 ( e, e) 4 2q ( ) Solving the four linear equations for q 0 (e, e), q 0 ( e, e), q 0 (e, e), q 0 ( e, e), and q ( ) yields the firms best production responses given no disclosure by Ralph q 0 (e, e) = q 0 ( e, e) = q 0 (e, e) = q 0 ( e, e) = q ( ) = 2 (5 0) + ( ) e 2 (5 0) + ( + ) e 2 (5 0) + ( + ) e 2 (5 0) + ( ) e 5 0 For no disclosure, expected profits for the two firms are Π 0 = 4 = 2 (5 0)2 + e 2 8 max [P c 0] q 0 (ε, ε 2 ) q 0(ε,ε 2) 5

6 and Π = 4 max [P c] q ( ) q ( ) = (5 0)2 Producer surplus equals the sum of expected firm profits Consumer surplus is CS = 4 given no disclosure Welfare = 4 P S = Π 0 + Π = 8 (5 0)2 + e 2 2 (P 0 P ) [q 0 (ε, ε 2 ) + q ( )] 2 [q 0 (ε, ε 2 ) + q ( )] 2 Total welfare is the sum of consumer surplus and producer surplus W = CS + P S Required: Verify the welfare expressions 2 Determine consumer surplus, producer surplus, and total welfare for each of Ralph s disclosure strategies: full disclosure, common-value (demand) disclosure, private-value (cost) disclosure, and no disclosure Rank Ralph s disclosure strategies from Ralph s perspective, from consumers perspective, and from a total welfare perspective 4 Why might this product market disclosure setting invite regulation?