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Lee,Wilde (1980) - Market Structure And Innovation A Reformulation (view source)
Revision as of 19:15, 29 September 2020
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{{Article
|Has page=Lee,Wilde (1980) - Market Structure And Innovation A Reformulation
|Has bibtex key=
|Has article title=Market Structure And Innovation A Reformulation
|Has author=Lee,Wilde
|Has year=1980
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*This page is referenced in [[PHDBA602 (Innovation Models)]]
==Reference(s)==
*Lee, T. and L.L. Wilde (1980), "Market structure and innovation: A reformulation", Quarterly Journal of Economics, 94, pp. 429-436. [http://www.edegan.com/pdfs/Lee%20Wilde%20(1980)%20-%20Market%20structure%20and%20innovation%20A%20reformulation.pdf (pdf)]
@article{lee1980market,
title={Market structure and innovation: a reformulation},
author={Lee, T. and Wilde, L.L.},
journal={The Quarterly Journal of Economics},
pages={429--436},
year={1980},
publisher={JSTOR}
}
==Abstract==
The expected benefits are (supposed the same as in [[Loury (1979) - Market Structure And Innovation |Loury (1979)]]):
:<math>\mathhbbmathbb{E}B = \int_0^{\infty} pr(\hat{\tau_i} = t) \left ( \int_0^t pr(\tau=s) V e^{-sr} ds \right) dt\;</math>
:<math>\therefore \mathhbbmathbb{E}B = \int_0^{\infty} a e^{-at} \left ( \int_0^t h e^{-hs} V e^{-sr} ds \right) dt \;</math>
:<math>\therefore \mathhbbmathbb{E}B = \frac{Vh}{a+h+r}\;</math>
===Modelling Costs===
Expected costs are thus:
:<math>\mathhbbmathbb{E}C = \int_0^{\infty} \left ( \int_0^{t} x e^{-rs} ds \right ) \cdot pr(\hat{\tau_i} = t or \tau_i = t) dt + F\;</math>
:<math>\therefore \mathhbbmathbb{E}C = \int_0^{\infty} \left ( \int_0^{t} x e^{-rs} ds \right ) \cdot (a+h) e^{-(a+h)t} dt + F\;</math>
:<math>\therefore \mathhbbmathbb{E}C = \frac{x}{a+h+r} + F\;</math>
Expected profit is expected benefit minus expected cost:
:<math>\mathhbbmathbb{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.
