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IT IS NOT CLEAR TO ME HOW WE GOT THESE FOCS!
I DON"T KNOW WHERE THE SECOND BRACKET TERM COMES FROM!
Katz (1986) - An Analysis of Cooperative Research and Development (view source)
Revision as of 19:15, 29 September 2020
, 19:15, 29 September 2020no edit summary
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|Has article title=An Analysis of Cooperative Research and Development
|Has author=Katz
|Has year=1986
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*This page is referenced in [[PHDBA602 (Innovation Models)]]
With a strict inequality in each case iff <math>r_i = 0\;</math>
:<math>\frac{d \pi(r)}{d r_i} = V_i^i(c)c'(z_i)(1+\overline{\phi}(n-1))(1 + (n-1) \rho(c)) - 1\;</math>
Comparing this with the FOC from the development stage, when every firm is a member:
*The agreed valus of <math>s\;</math> (the cost sharing parameter inside the partnership)
We know that <math>\phi=\overline{\phi}\;</math>, but the sign of <math>z^n - z^0\;</math> depends on <math>\rho(c^0)\;</math>, and <math>s^n \;</math> depends on <math>\rho(c^n)\;</math>.
To get <math>\rho\;</math> constant there must be a constant elasticity of demand. There are two possibilities:
*<math>P(X) = \alpha + \beta X ^\gamma\;</math>, which has an elasticity of <math>\epsilon = \gamma - 1\;</math>.
*<math>P(X) = \alpha +\beta \ln X\;</math>, which elasticity of <math>\epsilon = - 1\;</math>.
*Raising <math>\delta\;</math> is increasing the product market competition
*Raising <math>\epsilon\;</math> makes the equilibrium price less responsive to changes in costs
*Raising either <math>\delta\;</math> or <math>\epsilionepsilon\;</math> raises <math>\rho\;</math>, which in turn expands the set of parameters over which industrywide cooperation raises effective R&D.
There are further specific examples in the paper, including Cournot competition.
