Lecture 4 (continued): The Sutton Model of the Size Distribution

Transcription

1 Lecture 4 (continued): The Sutton Model of the Size Distribution Goal is to understand structure of industry, how many firms, how big. What are the basic economic forces we want to work in here. Have handbook of IO chapter posted, look to that for details. As an aside, Sutton started career as a theorist. Finds industry case studies unappealing, wants to be a scientist, look for general patterns. Respects what old-time IO guys like Joe Bain from the 960s were up to as far as ambition, just thinks they were on the wrong track. Sutton looking for robust predictions. Thinks he has one. Contrasts

2 Exogenous fixed cost case (Berry Waldfogel example: Restaurants) Endogenous fixed cost case (Berry Waldfogel example: Newspapers)

3 Exogenous Fixed Cost Case : Model = size of market (number of consumers) Assume each consumer has unit income, get industry demand (per consumer) equals = Unless 0 (and then go to outside good). cost and entry cost is. Suppose marginal Work out Cournot

4 Taking as given that otherguysdo,lookatproblemof firm (maximize profit per consumer) max ( +( ) ) FONC +( ) [ +( ) ] 2 = = 0 2 = 0 µ = So industry output is µ = =

5 and industry profit per firm per consumer is ( ) = Ã! = ³ µ = So Cournot profit perfirm(takingintoaccount consumers)is Cournot Profit = 2 Now work out free entry Zero profit condition = 2

6 = Ã! = = ³ where is concentration ratio (share of output acounted for by largest firms) Key point is that falls and goes to zero as gets large Suppose change model so firms collude with entry. = 0 So industry profit per consumer is Then ( 0 ) 0

7 And profit per firm Collusive Profit =( 0 ) 0 So entry solves =( 0 ) 0 Get same point that market share goes to zero as gets large

8 Connection to Bresnahan and Reiss (99) Connect to broader agenda in empirical IO, beginning with Breshnahan, Rob Porter. How does entry affect profitability? Illustrate Breshnahan and Reiss approach with an example. Suppose Two worlds out there. Cournot competition world

9 = Ã! Collusive world (ex post) Noncooperative in entry. (Lorschian Competition) solve equation above =( 0 ) 0 How distinguish empirically? Compare markets that vary by

10 Back to Sutton: Endogenous fixed cost case: =() Suppose have set ( 2 3 ), a vector of qualities. assume +. W.L.O.G. Consumers maximize. (pick best deal.)firms compete in output in a Cournot fashion each with marginal cost Can derive the following reduced form profit functions : = 2 P µ

11 with threshold quality and firms with survive. Rest produce no output. This threshhold is defined implicitly as follows (add a hypothetical +firm and equate profit tozero = + = = X + X (can recast this as follows. W.L.O.G. suppose. Problem isomorphic to one where set =, =,and =.Sonow asymmetric cost Cournot oligopoly with homogeneous products. Easy to see if have a Cournot oligopoly with ( 2 3 )and all in, easy to see what the marginal cuy would be.

12 Now make it a three-stage game Stage : enter pay 0 Stage 2: incur additional sunk costs () = 0, 0 0. (think of 0 0 to be advertising or R&D to get quality above the minimal level). What happens: if S small, straight Cournot, =. Make big,

13 General Structure: Profit is: Π( ) ( ), for some location choice for entrant. ( 6=, means enter and pick some quality, etc.) Assume Π( 0 ) ( 0 ), some 0 (excludes nonviable markets)and that there is always enough potential entrants Assumption 2: with each firm i an integer. at any point Free to enter Equilibrium configurations (i) Viability Π( ) ( ) 0

14 (ii) Stability there is no + ( + ) ( + ) 0 Proposition: any outcome that can be supported as a perfect Nash equilibrium in pure stateges is an equilibrium configuation

15 Escalation Mechanism for Endogenous Fixed Cost Define ˆ() =max Π = ( ) () is industry output per consumer given, () =() total output. Define for a given, () = ((ˆ )()

16 Theorem : given any pair ( ()), a necessary condition for any configuation to be an equilibrium configuration is that a firm offering the highest level of quality has a share of industry sales revenue exceeding () Proof Consider the net profit of an entran who attains quality ˆ. () (ˆ) =() (ˆ) Stability implies:

17 Viability implies... whence its market share...

18 Define =sup () Reformulate the theorem: For Cournot with (ˆ ) = the expression cannot be less than ½ ¾ 2 2 P µ +

19 So So () = µ 2,k ³ 2 =sup Straightforward to show that in the solution to the above problem So = 2+ () =4 (2 + ) (2+)

20 0(limit) Can show: low in this model, equilibrium fraction of expenditure on advertising/rd is high.

21 Picture from 99 book

22 Figure :

23 Berry and Waldfogel, newspapers versus restaurants Bronnerberg, Dhar and Dube, Mayonaise in local markets. Are these ideas applicable? What about implication that consumer utility ( gets unbounded?)

24 Notes Look at maximization problem yields FONC max µ 2 2 = 2 µ 2 µ 2 µ µ 2 2 Divide through by ³ 2 yields µ 2 = 2 = 2

25 So And equals = max à For for example if =,then = 2+ µ 2! 2 à 2+ = 2+ Ã! 2 à 2 2+ = 2+ = 4 (2 + ) (2+) = 4(3) 3 = 4 9.!!