|Has page=Bolton Farrell (1990) - Decentralization Duplication And Delay
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|Has article title=Decentralization Duplication And Delay
|Has author=Bolton Farrell
|Has year=1990
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|In volume=
|In number=
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}}
*This page is referenced in [[PHDBA602 (Theory of the Firm)]]
*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>
===The General Result===
Then it must be the case that <math>t_1 < t_2\,</math>
To see this assume that <math>A\,</math> believes that there is:
*a hazard-rate probability that <math>B\,</math> will enter at exactly <math>t\,</math> of <math>h(t)\,</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:
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:
There is an inherent trade-off between delay and duplication - both can be expressed in terms of <math>p\,</math>, or just in terms of each other.
:<math>y = \frac{(x-1)^2}{4x}\,</math>
By changing the parameters <math>\lambda\,</math>, <math>\mu\,</math>, or <math>S\,</math>, a planner could make this trade-off, allowing less delay at the expense of more duplication, or vice versa.
This can be written as the difference from the first-best:
Which is less likely to hold when <math>S_H\,</math> and <math>S_L\,</math> are close (private info is almost unimportant), and more likely to hold when <math>1-S_H\,</math> is small (delay is not costly - not this is opposite of what the paper says) and <math>S_L\,</math> is large (duplication is costly).
'''For non-urgent problems''', when <math>\delta = 1\,</math>, decentralization gives: