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Katz (1986) - An Analysis of Cooperative Research and Development (view source)
Revision as of 15:53, 7 December 2010
, 15:53, 7 December 2010no edit summary
Using this we can say that:
*When <math>s^n <1\;</math> and <math>\phi^n=\underline({\phi)}\;</math> (that is firms share costs, but knowledge overspills are unaffected by cooperation) there exist equilibria with <math>z^n > z^0\;</math>
*When <math>\phi^n > \underline{\phi}\;</math> and <math>s^n = 1\;</math> (there is no cost sharing but knowledge overspills are greater in the coop) then there exists equilibria with <math>z^n < z^0\;</math>
Under some circumstances it is useful to have everyone participate. This can be shown to be true if <math>\underline{\phi}=0\;</math> and <math>z^{n-1} > z^0\;</math>. Under very special circumstances it can also be true if <math>\underline{\phi} = 1 = \overline{\phi}\;</math>, as then both members and non-members do no research.
===Output and Welfare Effects===
To discuss welfare when every firm participates in the coop we need to know:
*The sign of <math>z^n - z^0\;</math> (whether the effective level of R&D per firm is higher under membership or not)
*The agreed value of <math>\phi\;</math> (the agreed spillover inside the partnership)
*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 depends on <math>\rho(c^n)\;</math>.
Supposing that <math>\rho\;</math> is a constant for all values of <math>c\;</math>, it is possible to make predictions for both the full membership case and, to a lesser degree, for the partial membership case. For full membership the following holds:
*If <math>\rho\;</math> is large in absolute value: the would-be gain in profit to a firm from the reduction in its costs is almost fully offset by the reduction in its rivals costs - thus gains accrue largely to consumer surplus.
*For industries in which industry-wide cost reduction raises profits (such as those doing basic research), higher spillovers (again such as those doing basic research) result in more effective R&D.
*Generally an increase in effective R&D leads to an increase in welfare, however, welfare can also increase when effective R&D does not increase, simply because of the cost-savings arising from efficiency from sharing when <math>\phi^n > \underline{\phi}\;</math>.
==Specific Models of the Product Market==
The next step is to consider specific models of product market competition and the effects that they would have on the model above. These effects enter through the value of <math>\rho(c)\;</math>.
===Independent Product Markets===
if the output markets are unrelated then <math>\rho(c) = 0\;</math>. Since <math>\phi^k = \overline{\phi}\;</math>, cooperation raises the efficiency of R&D whenever <math>\overline{\phi} > \underline{\phi}\;</math>, and hence raises welfare.
===Homogeneous Good Markets===
This section uses an '''n-firm conjectural variations''' model, which is not a game-theoretic construct. The model is simply that when a firm increases its output by <math>x\;</math>, its competitors increase their outputs by <math>\delta x\;</math>. Thus for an inverse demand function <math>P(X)\;</math>, where <math>X\;</math> is the aggregate demand, the FOCs for an equilibrium are:
:<math>x_i (1+\delta) \cdot P'(X) + P(X) - c_i \le 0 \quad \forall i \in N\;</math>
(with strict inequality iff <math>x_i = 0\;</math>).
In this model <math>\delta \in (-1,n-1)\;</math>, representing the extremes of competition and collusion. When:
*<math>\delta = -1\;</math>: The firm is a price taker and there is Bertrand competition
*<math>\delta = 0\;</math>: This is a Cournot competition model
*<math>\delta = n-1\;</math>: There is joint-profit maximization
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>.
It is then possible to get an equation for <math>\rho\;</math> in terms of <math>n,\delta,\epsilon\;</math>. Furthermore, it is then possible to write the equation for the sign of <math>z^n - z^0\;</math> in terms of these parameters, noting that:
*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>\epsilion\;</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.
===Imperfect Substitutes===
This corresponds to the Spence (1976) or [[Dixit Stiglitz (1977) - Monopolistic Competition And Optimum Product Diversity| Dixit and Stiglitz (1977)]] models, where firms produce goods that are imperfect substitutes for one another. The paper notes that the derivation is so complicated that the author was forced to use simulations to determine the effects. However, as products become less substitutable (i.e. competition weakens), or as there is less crowding out in the market, the set of values of parameters that support industrywide cooperatation that raises effective R&D increases.
==Imperfect Competition==
The paper comments on a fear regarding joint-ventures - that they might serve as a (collusive) mechanism for retarding innovation. Providing that each firm can also conduct independent R&D, this would not appear to be a problem.
