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Lee,Wilde (1980) - Market Structure And Innovation A Reformulation (view source)
Revision as of 17:10, 18 November 2010
, 17:10, 18 November 2010no edit summary
:<math>\mathbb{E}\pi = \frac{Vh - x}{a+h+r} - F\;</math>
==Comparative Statics==
The FOC gives:
:<math>\frac{\partial \pi}{\partial x} = \frac{(a+r)(Vh' - 1) - (h-xh')}{(a+h+r)^2} = 0\;</math>
Rearranging for <math>V\;</math> and subbing back in we get:
:<math>\mathbb{E}\pi = \frac{h-xh'}{(a+r)h'} -F\;</math>
Non-negative profits require (at least) <math>h > \hat{x}h'\;</math>, which is opposite to Loury, so <math>h''<0\;</math> at <math>\hat{x}\;</math>.
So when we do the comparative static on investment with respect to the degree of rivalry we find that it is now positive::
:<math>\frac{d \hat{x}}{d a} = \frac{-(Vh'-1)}{((a+r)V-x)h''} > 0\;</math>
Again this differs from Loury.
In the full equilibrium, as a result of symmetry, it is the case that:
:<math>a = (n-1)h(\hat{x})\;</math>
Letting the implicit solution to <math>\frac{\partial \mathbb{E}\pi}{\partial x} = 0\;</math> be denoted <math>\hat{x} = H(a)\;</math>, then in the full equilibrium <math>\hat{x} = H((n-1)h(\hat{x}))\;</math>.
Noting that:
:<math>\frac{d H}{d a} = \frac{d \hat{x}}{da} >0\;</math>
We can see the comparative static with respect to <math>n\;</math> is also exactly opposite to that of Loury (providing an analogous stability condition holds):
:<math> \frac{d \hat{x}}{dn} = \frac{H}{\partial a} h/1 - \left( \frac{\partial H}{\partial a} \right )(n-1)h'\;</math>
(but I get: <math>\frac{d \hat{x}}{dn} = \frac{\partial H}{\partial a}\cdot( h + (n-1)h')\;</math> )
The remainder of the proofs have the same comparative statics as Loury, despite these differences.
