Changes
Jump to navigation
Jump to search
Loury (1979) - Market Structure And Innovation (view source)
Revision as of 19:42, 17 November 2010
, 19:42, 17 November 2010no edit summary
*<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- this is the point where a firm is using its full capactity.
:<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. To see this we examine the change in profit with respect the number of firms, remembering that the expenditure each firm will make will depend upon the total number of competitors. :<math>\frac{\d \Pi}{d n} = \frac{\partial \pi }{\frac \partial a}\cdot (h(x^*) + (n-1)h'(x^*)) + \frac{\partial \Pi}{\frac \partial x} \frac{\partial x}{\frac \partial n} < 0\;</math> We know, from the envelope theorem, that <math>\frac{\partial \Pi}{\frac \partial x} = 0\;</math>, and from the original profit function that <math>\frac{\partial \Pi}{\partial a} < 0\;</math>. By rearranging the other terms we can see that equilibrium profits decrease with more competition. There is a proof in the paper that shows that with initial increasing returns to scale the finite number of competitors in a zero profit equilibrium will be below <math>\tilde(x)\;</math>, which is the point where firms are using their capacity. ===Welfare Considerations=== Ignoring the problem that social benefits may not equal private benefits, there are two other inefficiencies. The first arises from duplication of effort. Given a fixed market structure, social welfare is maximized with a choice <math>x^{**}\;</math> characterized by: :<math>\frac{\partial \pi}{\frac \partial x}((n-1)h(x),x) + (n-1)h'(x) \cdot \frac{\partial \pi}{\frac \partial a}((n-1)h(x),x) = 0\;</math> Whereas the individual firms choose an <math>x^*\;</math> characterized by: :<math>\frac{\partial \pi}{\frac \partial x}((n-1)h(x),x)= 0\;</math> Since <math>\frac{\partial \pi}{\frac \partial a} < 0\;</math> it follows that <math>x^*(n) > x^{**}(n)\;</math>. The second inefficiency is that there are too many firms. If <math>\overline{x}\;</math> (the point where increasing returns to scale stop) is at zero then infinite firms enter the competitive race. If <math>\overline{x} > 0\;</math> a finite firms enter, but continue to enter until all profits are dissipated.
