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]
• [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{ \frac{h(x_i)}{\sum_{i=1}^{n} h(x_i)} }_{\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]

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]
• [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{ \frac{h(x_i)}{\sum_{i=1}^{n} h(x_i)} }_{\mbox{Firm i's 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).