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Bolton Farrell (1990) - Decentralization Duplication And Delay (view source)
Revision as of 14:46, 19 November 2010
, 14:46, 19 November 2010no edit summary
*If both firms enter they each get <math>\mu\,</math>, for a payoff of <math>\mu - S\,</math>
*<math>S \sim F(\cdot)\,</math>, with support such that <math>\mu < S < \lambda\,</math> for all realizations
*There are infinite periods (<math>t\,</math>), discounted by <math>\delta\,</math>, and the strategy space is <math>\{Enter,Wait\}\,</math>
*The game ends when one or more firms <math>Enter\,</math>
*a probability that <math>B\,</math> will not have entered prior to <math>t\,</math> of <math>\alpha(t)\,</math>, where we denote <math>a(t) = \delta^t \alpha(t)\,</math>.
Then <math>A\,</math>'s expected payoff from entering at <math>t\,</math> is:
:<math>a(t)(\lambda - S) + a(t)(h(t)(\mu - \lamdbalambda)) = a(t)(\lambda - S - h(t)(\lamdba lambda -\mu))\,</math>
For <math>t_1\,</math> to be prefered with <math>S_A^1\,</math>, and <math>t_2\,</math> to be prefered with <math>S_A^2\,</math>, it must be true that:
:<math>a(t_1)(\lambda - S_A^1 - h(t)(\lamdba lambda -\mu)) \ge a(t_2)(\lambda - S_A^1 - h(t)(\lamdba lambda -\mu))\,</math>
and
:<math>a(t_2)(\lambda - S_A^2 - h(t)(\lamdba lambda -\mu)) \ge a(t_1)(\lambda - S_A^2 - h(t)(\lamdba lambda -\mu))\,</math>
So:
:<math>a(t_1)(\lambda - S_A^1 - h(t)(\lamdba lambda -\mu)) + a(t_2)(\lambda - S_A^2 - h(t)(\lamdba lambda -\mu)) \ge a(t_2)(\lambda - S_A^1 - h(t)(\lamdba lambda -\mu)) + a(t_1)(\lambda - S_A^2 - h(t)(\lamdba lambda -\mu))\,</math>
:<math>\therefore (a(t_1) - a(t_2))(S_A^1 - S_A^2) \le 0\,</math>
Th paper then sets up the '''Fundamental Difference Equation''', where is supposes cutoffs <math>S_1\,</math>, <math>S_2\,</math> and so forth such that firms with costs between <math>S_{t-1}\,</math> and <math>S_t\,</math> will enter in <math>t\,</math> (providing no previous entry has occurred). Using the indifference of such a firm between periods <math>t\,</math> and <math>t+1\,</math> we have:
:<math>\lambda - S_t -h(t)(\lambda - \mu) = \delta(1-\h(t))(\lamdba lambda - S_t - h(t+1)(\lambda - \mu)\,</math>
We work with the equilibria where low cost types enter in <math>t=1\;</math>, and high cost types enter strictly afterwards.
One Bayesian equilibrium is then:
To calculate the expected social surplus we need the probability that nothing happens until <math>t\,</math> and then one firm enters, <math>q_t\,</math>, and the probability that nothing happens until <math>t \,</math> and then both firms enter, <math>r_t\,</math>:
:<math>q_t = (1-p)^{2(t-1)}\cdot 2p(1-p)\,</math>
