Changes

===Loury's Model=== The basis for this model is identical to that in[[Loury (1979) - Market Structure And Innovation |Loury (1979)]]. The following are defined the same:*<math>h(x_i)\;</math>*<math>F_{\tau}\;</math>, <math>F_{\hat{\tau}}\;</math> and <math>a_i\;</math>*<math>V\;</math> and <math>r\;</math> (though true continuous discounting in used here, and there seems to be difference in the math) The expected benefits are (supposed the same as in [[Loury (1979) - Market Structure And Innovation |Loury G.C.(1979)]]): :<math>\mathhbb{E}B = \int_0^{\infty} pr(\hat{\tau_i} = t) \left ( \int_0^t pr(\tau=s) V e^{-sr} ds \right) dt\;</math> :<math>\therefore \mathhbb{E}B = \int_0^{\infty} a e^{-at} \left ( \int_0^t h e^{-hs} V e^{-sr} ds \right) dt \;</math> :<math>\therefore \mathhbb{E}B = \frac{Vh}{a+h+r}\;</math> ===Modelling Costs=== However, now the costs are incurred in two parts:*A fixed cost that is paid upfront (as in Loury)*A flow cost that is paid continously until the first firm innovates. Expected costs are thus: :<math>\mathhbb{E}C = \int_0^{\infty} \left \int_0^{t} x e^{-rs} ds \right \cdot pr(\hat{\tau_i} = t or \tau_i = t) dt + F\;</math> :<math>\therefore \mathhbb{E}C = \int_0^{\infty} \left \int_0^{t} x e^{-rs} ds \right \cdot (a+h) e^{-(a+h)t} dt + F\;</math> :<math>\therefore \mathhbb{E}C = \frac{x}{a+h+r} + F\;</math> Expected profit is expected benefit minus expected cost: :<math>\mathhbb{E}\pi = \frac{Vh - x}{a+h+r} - F\;</math>