# Lee,Wilde (1980) - Market Structure And Innovation A Reformulation

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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)

## Contents

## Reference(s)

- Lee, T. and L.L. Wilde (1980), "Market structure and innovation: A reformulation", Quarterly Journal of Economics, 94, pp. 429-436. (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 relationship between market structure and innovative activity has attracted a greal deal of attention from economists over the last two decades. One of the most interesting recent additions to the literature has been provided by Glenn Loury [1979]. He analyzes "a world in which ... firms compete for the constant, known, perpetual flow of rewards ... that will become available only to the first firm that introduces [some given] innovation" [Loury, 1979, p. 397]. Following Kamien and-Schwartz [1976], he assumes that individual firms face a stochastic relationship between investment in R & D and the time at which a usable innovation (a "new" technology) is produced. The interaction of firms competing to introduce the innovation is then modeled as a noncooperative game [Scherer, 1967]. Loury's major conclusions are as follows: i. As the number of firms in the industry increases, the equilibrium level of firm investment in R & D declines. ii. When there are initial increasing returns to scale in the R & D technology, then a zero expected profit industry equilibrium with a finite number of firms always involves "excess capacity" in the R & D technology. iii. Given a fixed market structure, industry equilibrium will have each firm investing more in R & D than is socially optimal. 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).

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)):

- [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 \gt \hat{x}h'\;[/math], which is opposite to Loury, so [math]h''\lt 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''} \gt 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} \gt 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.