Changes

*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.

The firm discounts the future reciepts at a rate <math>r\;</math> (note that using continuous compounding, <math>PV = FV \cdot e^{-rt})\;</math>, but apparently this paper values the future price at <math>\frac{V}{r}\;</math>).

:<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>

:<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>

:<math>\Pi = \int_0^{\infty} \left ( \underbrace{pr(\tau(x_i) 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. 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 }{\partial a}\cdot (h(x^*) + (n-1)h'(x^*)) + \frac{\partial \Pi}{\partial x} \frac{\partial x}{\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}{\partial x}((n-1)h(x),x) + (n-1)h'(x) \cdot \frac{\partial \pi}{\partial a}((n-1)h(x),x) = 0\;</math> Whereas the individual firms choose an <math>x^*\;</math> characterized by: :<math>\frac{\partial \pi}{\partial x}((n-1)h(x),x)= 0\;</math>  Since <math>\frac{\partial \pi}{\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.