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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.     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 [http://en.wikipedia.org/wiki/Exponential_distribution#Distribution_of_the_minimum_of_exponential_random_variables 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 With an initial range 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:#If a firms expectations are rational increasing returns to scale then the beliefs about the fastest competing firm are indeed formed using <math>\hat{\tau_i}\;</math>#<math>a_i\;</math> returns can be taken as constant by firm <math>i\;</math> (i.e. in equilibrium <math>a_i\;</math> will be correct)#<math>V\;</math> and <math>r\;</math> are exogenously given#As the firms are identical we can look for go to zero with 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 finite number 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)firms.