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Bolton Farrell (1990) - Decentralization Duplication And Delay (view source)
Revision as of 14:49, 19 November 2010
, 14:49, 19 November 2010no edit summary
:<math>W^G = \sum_{t=1}^{\infty} \delta^{t-1}(q_t(1-S) + r_t(1-2S)\,</math>
:<math>\therefore W^G = (1-S) - \left (1 - \frac{p(2-p)}{1-\delta(1-p)^2}\right)(1-S) - \frac{p^2}{1-\delta(1-p)^2}S\,</math> :<math>\therefore W^G = (1-S) - \underbrace{\left (1 - \frac{p(2-p)}{1-\delta(1-p)^2)}\right)(1-S)}_{\mbox{Delay Loss}} - \underbrace{\frac{p^2}{1-\delta(1-p)^2)}S}_{\mbox{Duplication Loss}}\,</math>
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.
This can be written as the difference from the first-best:
:<math>W^* - W^D = q^2 S_L + (1-q)^2\delta \cdot\frac{p^2}{1-\delta(1-p}^2)S_H + \left (1-q)^2\delta \cdot(1 - \frac{p(2-p)}{1-\delta(1-p)^2}\right)(1-S_H)\,</math>
:<math>W^* - W^D = \underbrace{q^2 S_L + (1-q)^2\delta \cdot\frac{p^2}{1-\delta(1-p)^2)}S_H}_{\mbox{Duplication Loss}} + \underbrace{\left (1-q)^2\delta \cdot(1 - \frac{p(2-p)}{1-\delta(1-p)^2}\right)(1-S_H)}_{\mbox{Delay Loss}}\,</math>
====Central Planning (with incomplete info)====
'''For non-urgent problems''', when <math>\delta = 1\,</math>, decentralization gives:
:<math>W^* - W^D (\delta=1) = q^2 S_L + (1-q)^2 \cdot\frac{p^2}{1-(1-p)^2)}S_H + \left (1-q)^2 \cdot(1 - \frac{p(2-p)}{1-(1-p)^2}\right)(1-S_H)\,</math>
