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**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} }
Lee,Wilde (1980) - Market Structure And Innovation A Reformulation (view source)
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*This page is referenced in [[PHDBA602 (Innovation Models)]]
==Reference(s)==
==Abstract==
iv. When there are initial increasing returns to scale in the R & D technology, competitive entry leads to more than the socially optimal number of firms in the industry.
It turns out that conclusions (i) and (ii) are sensitive to Loury's specification of the costs of R & D. In this paper we investigate the effects of an alternative specification.
==The Model==
===Loury's Model===
The basis for this model is identical to that in [[Loury (1979) - Market Structure And Innovation |Loury (1979)]].
The following are defined the same:
*<math>h(x_i)\;</math>
*<math>F_{\tau}\;</math>, <math>F_{\hat{\tau}}\;</math> and <math>a_i\;</math>
*<math>V\;</math> and <math>r\;</math> (though true continuous discounting in used here, and there seems to be difference in the math)
The expected benefits are (supposed the same as in [[Loury (1979) - Market Structure And Innovation |Loury (1979)]]):
:<math>\mathbb{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 \mathbb{E}B = \int_0^{\infty} a e^{-at} \left ( \int_0^t h e^{-hs} V e^{-sr} ds \right) dt \;</math>
:<math>\therefore \mathbb{E}B = \frac{Vh}{a+h+r}\;</math>
===Modelling Costs===
However, now the costs are incurred in two parts:
*A fixed cost that is paid upfront (as in Loury)
*A flow cost that is paid continously until the first firm innovates.
Expected costs are thus:
:<math>\mathbb{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 \mathbb{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 \mathbb{E}C = \frac{x}{a+h+r} + F\;</math>
Expected profit is expected benefit minus expected cost:
:<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.
