Loury (1979) - Market Structure And Innovation

Reference(s)

• Loury G.C.(1979), "Market structure and innovation", Quarterly Journal of Economics, 93, pp. 395-410. (pdf)

Abstract

In the application of conventional economic theory to the regulation of industry, there often arises a conflict between two great traditions. Adam Smith's "invisible hand" doctrine formalized in the First Fundamental Theorem of Welfare Economics supports the prescription that monopoly should be restrained and competitive market structures should be promoted. On the other hand, Schumpeter, in his classic Capitalism, Socialism and Democracy, takes a dynamic view of the economy in which momentary monopoly power is functional and is naturally eroded over time through entry, imitation, and innovation. Indeed the possibility of acquiring monopoly power and associated quasi rents is necessary to provide entrepreneurs an incentive to pursue innovative activity. As Schumpeter put it, progress occurs through a process of "creative destruction." An antitrust policy that actively promotes static competition is not obviously superior to laissez faire in such a world. This leads one to ponder what degree of competition within an industry leads to performance that is in some sense optimal. This question has been extensively studied in the literature concerning the relationship between industrial concentration and firm investment in research and development.' Both theoretical and empirical studies have suggested the existence of a degree of concentration intermediate between pure monopoly and atomistic (perfect) competition that is best in terms of R & D performance...

The Model

Basic Setup and Assumptions

The basic setup is as follows:

• There are [math]n\;[/math] identical firms, indexed by [math]i\;[/math]
• Each firm invests [math]x_i\;[/math] to buy a random variable [math]\tau(x_i)\;[/math] which gives a completion date
• The firm with the earliest realised completion date wins [math]V\;[/math]
• [math]\tau \sim F_{\tau}(h(x_i))\;[/math] where [math]F_{\tau}\;[/math] is the CDF for the exponential distribution: [math]F_{\tau}(h(x_i)) = 1 - e^{-h(x_i)t}\;[/math]
• [math]h(x_i)\;[/math] is the rate parameter, or the instantaneous probability of the innovation occuring.

[math]h(x_i)\;[/math] is assumed to have the following properties:

• [math]h(0) = 0 = \lim_{x \to \infty} h'(x)\;[/math]
• For some [math]\overline{x} \ge 0\;[/math], [math]h''(x) \ge 0\;[/math] for [math]x \le \overline{x}\;[/math], and [math]h''(x) \le 0\;[/math] for [math]x \ge \overline{x}\;[/math]. This says that h is weakly convex prior to some point (possibly zero, so never convex) and concave after that point. If the point is away from zero then there is an initial range of increasing returns to scale, but after the point there is always diminishing returns to scale.
• [math]\tilde{x}\;[/math] is defined as the point where [math]\frac{h(x)}{x}\;[/math] is greatest

Let [math]\hat{\tau_i}\;[/math] be an random variable giving the date of the earliest other firm:

[math]\hat{\tau_i} = \min_{j \ne i} \{ \tau(x_j) \}\;[/math]

Assuming iid tau's (no externalities in innovation!), then we can use a nice feature of the exponential distribution which is that if [math]X_1,\ldots,X_N\;[/math] are iid exponential with rates [math]\lambda_1,\ldots,\lambda_N\;[/math], then [math]\min(X_1,\ldots,X_N)\;[/math] is distributed exponential with rate [math]\sum_1^N \lambda_i\;[/math].

Therefore [math]\hat{\tau_i} \sim F_{\hat{\tau}}\;[/math], where

[math]F_{\hat{\tau}} = 1 - e^{-\left( \sum_{j\ne i} h(x_j) \right) t}\;[/math].

For convenience we denote

[math]a_i= \sum_{j\ne i} h(x_j)\; [/math]

The firm discounts the future reciepts at a rate [math]r\;[/math] (note that using continuous compounding, [math]PV = FV \cdot e^{-rt})\;[/math].

The firm wins the prize at time [math]t\;[/math] with probability:

[math]pr(\tau(x_i) \le \min(\hat{\tau_i},t) = e^{-a_i t}(1-e^{-h)x_i)t}) + a_i \int_0^t (1-e^{-h(x_i)s})e^{-a_i s})ds\;[/math]

[math]\therefore pr(\tau(x_i) \le \min(\hat{\tau_i},t) = \frac{h(x_i)}{a_i + h(x_i)} (1-e^{-(a_i+h(x_i))t})\;[/math]

This is directly comparable to a contest success function:

[math]pr(\tau(x_i) \le \min(\hat{\tau_i},t)) = \underbrace{\left( \frac{h(x_i)}{\sum_{i=1}^{n} h(x_i)} \right) }_{\mbox{Firm i relative effort}} \cdot \underbrace{ \left ( 1-e^{-\left(\sum_{i=1}^{n} h(x_i)\right)t}\right ) }_{\mbox{Prob of innov at t}}\;[/math]

Solution concept

The model is not actually solved, but comparative statics can be performed on an implicit solution. The implicit solution is arrived at by noting that:

1. If a firms expectations are rational then the beliefs about the fastest competing firm are indeed formed using [math]\hat{\tau_i}\;[/math]
2. [math]a_i\;[/math] can be taken as constant by firm [math]i\;[/math] (i.e. in equilibrium [math]a_i\;[/math] will be correct)
3. [math]V\;[/math] and [math]r\;[/math] are exogenously given
4. As the firms are identical we can look for a symmetric solution!

Each firm maximizes profit:

[math]\max_x \Pi (a_i,x,V,r) = \max_x \left (\frac{V h(x_i)}{r(a_i + r +h (x_i))} - x \right)\;[/math]

This is presumably constructed by taking:

[math]\Pi = \int_0^{\infty} \left( \underbrace{pr(\tau_i \le \min(\hat{\tau_i},t)}_{\mbox{Prob of winning at t}} \cdot \underbrace{PV_t (V)}_{\mbox{PV of V at t}} \right ) dt - \underbrace{x}_{\mbox{cost}}\;[/math]

The FOC for the profit maximization implicitly defines the equilibrium solution.

[math]\frac{h'(\hat{x})(a+r)}{(a+r+h(\hat{x}))^2} - \frac{r}{V} = 0\;[/math]

The SOC must also hold (the paper has the first term missing)

[math]\frac{a+r}{(a+r+h(\hat{x}))^3} \cdot \left ( h''(\hat{x}) (a+r+h(\hat{x})) - 2h'(\hat{x})^2 \right) \le 0\;[/math]

However, this only defines the partial equilibrium. To complete the equilibrium we need to use the symmetry (which is also why the subscripts are dropped above):

[math]a = \sum_{j \ne i} h(x_j) = (n-1)h(x^*)\;[/math]

This equilibrium exists providing R&D is profitable absent rivalry (otherwise their may be a corner, not an internal solution).

Comparative Statics

With the partial equilibrium result the greater rivalry could lead to greater or lesser R&D:

[math]h(\hat{x}) \ge a + r \implies \frac{\partial \hat{x}}{\partial a} \ge 0\;[/math]

[math]h(\hat{x}) \le a + r \implies \frac{\partial \hat{x}}{\partial a} \le 0\;[/math]

However, the full equilibrium result is unambiguous:

[math]h(\hat{x}) \le a, \;\mbox{ as }\;a = (n-1)h(x^*)\quad \therefore \frac{\partial \hat{x}}{\partial a} \le 0 \quad\mbox{ if} n \ge 2\;[/math]

The date of innovation (by the first firm) is always earlier as more firms compete, even though each firm is expending less, because (as the mean of the exponential distribution is the inverse of the rate parameter):

[math]\mathbb{E} \tau(n) = (n h(x^*(n)))^{-1}\;[/math]

This holds providing a reasonable stability condition holds: That a marginal increase in R&D by one firm causes a corresponding small drop in R&D by all other firms. This is proved easily in proposition 2 in the paper, and is intuitive.

Competitive Entry

Rearranging the FOC which characterizes the equilibrium for [math]\frac{V}{r}\;[/math], and subbing into the profit equation we get:

[math]\Pi(a,x) = \frac{h(x^*)}{h'(x^*)} \left ( \frac{a+r+h(x^*)}{(a+r)} \right ) - x^* \quad \mbox{where}\; a = (n-1)h(x^*)\;[/math]

Now if [math]h\;[/math] is concave (i.e. diminishing returns to scale throughout) then [math]\frac{h(x)}{x} \ge h'{x}\;[/math] and expected profits are always positive. They are only driven to zero in the limit of an infinite number of firms.

With an initial range of increasing returns to scale then returns can go to zero with a finite number of firms.