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Spence (1984) - Cost Reduction Competition And Industry Performance (view source)
Revision as of 18:28, 7 December 2010
, 18:28, 7 December 2010no edit summary
==The Symmetric Case==
Taking:
:<math>E_i^i + \theta \sum_{j \ne i} E_j^i = (1-s)\;</math>
And making <math>E\;</math> a function of both a scalar <math>v\;</math> and a vector of ones <math>e\;</math>, we have:
:<math>E_i^i(ve) + \theta \sum_{j \ne i} E_j^i(ve) = (1-s)\;</math>
Then we define a new function:
:<math>R(v) = \int_0^v \left (E_i^i(ve) + \theta \sum_{j \ne i} E_j^i(ve) \right )\;</math>
Which is the '''total benefits from R&D''' as a function of a single parameter <math>v, such that the '''marginal benefits from R&D''' are:
:<math>R'(v) = 1-s\;</math>
And therefore the market acts as if it were maximizing (wrt to <math>v\;</math>):
:<math>R(v) = (1-s)v\;</math>
Using symmetry so that <math>M_i = M\;</math>, and <math>z_i = z\;</math>, we then have:
:<math>v = (1+\theta (n-1))\cdot M = K(\theta,n)\cdot M \quad K = (1+\theta (n-1))\;</math>
'''Expenditures per firm''' are:
:<math>M = \frac{z}{K}\;</math>
And '''total surplus''' is:
:<math>T(M) = H(z,n) + nE(z,n) - n M\;</math>
==Properties of the Market Equilibria==
The function <math>R\;</math> (which I labelled the total benefits from R&D), captures the market incentives with respect to R&D. Assuming <math>E_j^i < 0 , \; i \ne j\;</math>, then:
*<math>R_{\theta} < 0\;</math>: Benefits are decreasing in spillovers
*<math>R_{z \theta) < 0\;</math>: ...
