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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
{{Article
|Has page=Katz (1986) - An Analysis of Cooperative Research and Development
|Has bibtex key=
|Has article title=An Analysis of Cooperative Research and Development
|Has author=Katz
|Has year=1986
|In journal=
|In volume=
|In number=
|Has pages=
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*This page is referenced in [[PHDBA602 (Innovation Models)]]
These have lead to three approaches:
#'''Use strong IP protection''', like patents, to maintain incentives - but this may actually reduce the efficient sharing of R&D as per Spence (1984).#Have lax property rights but '''use subsidies ''' to restore incentives - but this has dissemination problems when spillovers are weak (again see Spence), introduces monitoring problems as firms may spuriously report R&D expenses, and may have deadweight losses in the tax system.#'''Encourage cooperative R&D''', by permitting joint-ventures under anti-trust law.
The third approach is the topic of this paper and may be successful because:
*Can help avoid problems of opportunism and asymmetric information that arise in the sale of the innovations
*Monitoring R&D inputs is easy for firms
There is a moral hazard (essential a Team's Problem) effect potentially working in the other direction though. The strength of this depends upon the product market competition in the markets where the resulting innovation will be used. If firms were Bertrand competitors (in Constant Returs to Scale market) then they have no incentive to innovate (this way), however, if the innovation were used in unrelated product markets, this effect is zero.
===The Basic Set-up===
*For every <math>c\;</math> such that <math>i\;</math> is an active producer: <math>V_i^i(c) < 0\;</math>
*For every <math>c\;</math> such that <math>i\;</math> and <math>j\;</math> are active producers: <math>V_j^i(c) \ge 0, \; i\ne j\;</math>
These assumptions are satisfied by a number of standard oligopoly models including:
Where <math>0 \le \underline{\phi} \le \overline{\phi} \le 1\;</math>
A firm's R&D effort is denoted <math>r_i\;</math>, with <math>r=(r_1,\ldots,r_n)\;</math> being the vector. As we are using symmetry we use linear sharing rules, where <math>s^k\;</math> is given below. If firm <math>i \;</math> is a member then their total expenditure on R&D is:
:<math>s^k r_I + \frac{(1-s^k)}{k-1} \sum_{j \in K-\{i\}} r_j\;</math>
There is a deterministic relationship between a firm's marginal cost <math>c_i\;</math> and its effective level of R&D <math>z_i\;</math> as follows:
*<math>c \in \(\underline{c},\overline{c}\]\;</math>, where <math>\underline{c},\overline{c} \ge 0, \;\; \underline{c},\overline{c} \le \infty\;</math>
*<math>c\;</math> is C2 such that:
**<math>c' <0\;</math>
**<math>\lim_{z \to \infty}c'(z) = 0\;</math>
That is <math>c \;</math> is positive, decreasing and convex, starting from a high value of <math>\overline{c}\;</math> at <math>z = 0\;</math>, and declining asymptotically to <math>\underline{c}\;</math> as <math>z \to \infty\;</math>.
Also, we assume that: <math>V^i(\overline{c}) \ge 0, \;</math> where <math>\overline_c overline{c} = (\overline{c},\ldots,\overline{c})\;</math>
That is equilibrium profits are positive when no one does any R&D.
With a strict inequality in each case iff <math>r_i = 0\;</math>
There are two equilibria, one in the corner where <math>z^0 = 0 = z^n\;</math> (when the term in the brackets is less than or equal to zero), and one in the interior where the sign of <math>z^n - z^0\;</math> is given by:
:<math>(1-s^n)(1 + (n-1) \phi^n \rho(c^0)) + (\phi^n - \underline{\phi}s^n(n-1)\rho(c^0)\;</math>
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>
At this stage, firms can choose:
*<math>\phi \in [\underline{\phi},\overline{\phi}]\;</math>
*<math>s \in [0,\overline{s}]\;</math>, where it is possible that <math>\overline{s} \ge 1\;</math>
Assuming an interior solution to the development stage for all values of <math>c\;</math>, then for an industry wide cooperative agreement it must be the case that:
:<math>\phi^n = \overline{\phi} \;\mbox{and}\; s^n = \min (\overline{s}, \frac{1 + (n-1) \overline{\phi}\rho(c^n)}{(1+(n-1)\rho(c^n))(1+(n-1)\overline{\rho})}\;</math>
The proof for the first part is by strict dominance. With a positive sharing rule, and <math>\phi < \overline{\phi}\;</math>, it is always possible to raise <math>\phi\;</math> and simultaneously lower <math>s^n\;</math>, to make more profits and hold the effective level of R&D constant.
:<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:
:<math>V_i^i(c^n) (1 + (n-1)\overline{\phi} \rho(c^n))c'(z^n) - s^n = 0\;</math>
And again if full sharing is possible (i.e. <math>\overline{\phi} = 1\;</math>) then members will set <math>s^k \le \frac{1}{k}\;</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\;</math> 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>\epsilon\;</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.
