Katz (1986) - An Analysis of Cooperative Research and Development

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Katz, Michael L. (1986), "An Analysis of Cooperative Research and Development", The RAND Journal of Economics, Vol. 17, No. 4 (Winter), pp. 527-543 pdf

Abstract

I analyze the effects of cooperative research, whereby member firms agree to share the costs and fruits of a research project before they undertake it. In this model industrywide agreements tend to have socially beneficial effects when the degree of product market competition is low, when there are R&D spillovers in the absence of cooperation, when a high degree of sharing is technologically feasible, and when the agreement concerns basic research rather than development activities. I show that a royalty-free cross-licensing agreement among any number of firms lowers the equilibrium level of innovation even though it increases the efficiency of R&D through sharing

The Model

Three Approaches in the Literature

There are general problems arising from market failure, opportunism and asymmetric information which affect incentives to innovate. These include:

  • Spillovers reduce private incentives
  • Problems price discriminating lead to incomplete surplus extraction and reduced incentives
  • Duplication of research is inefficient
  • Arrow's paradox prevents the sale of information goods

These have lead to three approaches:

  1. Use strong IP protection, like patents, to maintain incentives - but this may actually reduce the efficient sharing of R&D as per Spence (1984).
  2. 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.
  3. 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:

  • It eliminates wasteful duplication
  • It restores at least some incentives to conduct R&D, by internalizing the externalities of spillovers.
  • 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

The model is a four stage games solved by backwards induction using perfect Nash equilibrium as the solution concept, as well as symmetry. The stages are:

  1. Membership Stage - Decide to join a research coop or stay out
  2. Agreement Stage - Chose the cost and output sharing rules
  3. Development Stage - Choose R&D
  4. Production Stage - Choose output in product markets

There are n firms, indexed by [math]i\;[/math], with constant marginal costs of production [math]c_i\;[/math], and equilibrium profits [math]V^i(c)\;[/math], where [math]c = (c_1,\ldots,c_n)\;[/math]. [math]c_{-i}\;[/math] is the vector of costs excluding [math]c_i\;[/math]. The partial derivitive of [math]V^i(c)\;[/math] with respect to the change in costs of firm [math]j\;[/math] are denoted [math]V_j^i(c)\;[/math].

The Prodcution Stage

By assumption:

  • For every [math]i\;[/math], [math]V^i(c) = V(c_i,\Omega(c_{-i}))\;[/math] where [math]\Omega(\cdot)\;[/math] is a symmetric function
  • For every [math]c\;[/math] such that [math]i\;[/math] is an active producer: [math]V_i^i(c) \lt 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:


The Development Stage

Suppose that:

  • [math]k\;[/math] firms join the coop, that is a member is in set [math]K\;[/math], where the spillover rate is [math]\phi^k\;[/math], which has an upperbound of [math]\overline{\phi}\;[/math]
  • [math]n-k\;[/math] firms stay out, that is a non-member is in the set [math]N-K\;[/math], where the spillover rate is [math]\underline{\phi}\;[/math]

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]


A firm's effectively level of R&D is both it's own expenditures and the spillovers that it gets.


For members these are:

[math]z_i(r) = r_i + \phi^k \sum_{j \in K-\{i\}} r_j + \underline{\phi} \sum_{h \in N-K} r_h\;[/math]


For non-members these are:

[math]z_i(r) = r_i + \underline{\phi} \sum_{j \in N-\{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' \lt 0\;[/math]
    • [math]c''\gt 0\;[/math]
    • [math]\lim_{z \to 0}c'(z) = -\infty\;[/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},\ldots,\overline{c})\;[/math]

That is equilibrium profits are positive when no one does any R&D.


Because we are looking at the symmetric equilibrium:

  • All member firms have effective R&D of [math]z^k\;[/math], and a cost vector with [math]c(z^k)\;[/math] in the [math]i\;[/math] th position is [math]c^k\;[/math]
  • All non-member firms have effective R&D of [math]z^{-k}\;[/math], and a cost vector with [math]c(z^{-k})\;[/math] in the [math]i\;[/math] th position is [math]c^{-k}\;[/math]

Define:

[math]\rho(c^k)= \frac{V_j^i(c^k)}{V_i^i(c^k)} \quad \forall i \ne j \;[/math]

where both [math]i\;[/math] and [math]j\;[/math] are members for [math]k \gt 0\;[/math], and for any [math]i\;[/math] and [math]j\;[/math] if [math]k=0\;[/math]


By the assumption above, [math]\rho(c^k) \le 0\;[/math].


The FOCs for an equilibrium in the R&D stage are:


For members:

[math]V_i^i(c^k) (1 + (k-1)\phi^k \rho(c^k))c'(z^k) + \underline{\phi} c'(z^{-k}) \cdot \sum_{j \in N-K} V_j^i(c^k) - s^k \le 0\;[/math]


For non-members:

[math]V_i^i(c^k) c'(z^{-k}) + \underline{\phi} \sum_{j \in N-K-\{i\}} V_j^i c'(z^{-k}) + \sum_{h \in K} V_h^i(c^k)c'(z^k) - 1 \le 0\;[/math]


With a strict inequality in each case iff [math]r_i = 0\;[/math]


When every firm is a member, the FOC for [math]z^n\;[/math] is:

[math]V_i^i(c^n) (1 + (n-1)\phi^n \rho(c^n))c'(z^n) - s^n = 0\;[/math]


Likewise if every firm is a non-member then the FOC for [math]z^0\;[/math] is:

[math]V_i^i(c^0) c'(z^{0})(1 + (n-1) \underline{\phi} \rho(c^0)) - 1 = 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 \lt 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 \gt z^0\;[/math]
  • When [math]\phi^n \gt \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 \lt z^0\;[/math]


The Agreement Stage

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]

The paper briefly considers the case where R&D is not profitable (as in Seade (1983)). We focus on the more natural case where R&D is profitable.

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 \lt \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.

The proof for the second part comes from maximizing the industry wide surplus (because of symmetry):

\pi(r) = \sum_{i \in N} V^i(c(z_1(r)),\ldots,c(z_n(r))) - r_i\;</math>


Taking the FOC wrt [math]r_i\;[/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:

[math]V_i^i (c^n)(1 + (n-1)\overline{\phi} \rho(c^n))c'(z^n) - s^n = 0\;[/math]


We get that [math]s\;[/math] is chosen to induce the profit maximizing level of R&D. It internalizes the spillover and pecuniary externalities.

Note that:

  • if spillovers are always zero (i.e. [math]\underline{\phi} = 0 =\overline{\phi}\;[/math]), then cost sharing is still feasible, and firms can use the cooperative to contract on the amount of R&D to be done by setting [math]s^n \gt 1\;[/math].
  • if [math]\overline{s} \le 1\;[/math], then there exist a range of values of [math]\overline{\phi}\;[/math] for which the firms will sign royalty free cross licensing agreements.
  • if [math]\overline{\phi} =1\;[/math], then from above [math]s^n =\frac{1}{n}\;[/math] and the joint venture will conduct the joint profit maximizing amount of R&D.

If there is less than complete participation (i.e. [math]k \ne n\;[/math]), then strict dominance still results in the cooperating firms choosing [math]\phi = \overline{\phi}\;[/math]. Furthermore if [math]\underline{\phi} = 0\;[/math], then it can be shown that:

[math]s^k \le \frac{1+(k-1)\overline{\phi}\rho(c^k)}{(1+(k-1)\rho(c^k))(1+(k-1)\overline{\phi}}\;[/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].


The Membership Stage

The arguement for the membership loosely follows D'Aspremont et al (1983).

To determine the optimum membership size we denote:

  • [math]\pi^{in}(k)\;[/math] as the profits to those inside the coop
  • [math]\pi^{out}(k)\;[/math] as the profits to those outside the coop

The equilibrium for membership size [math]k^*\;[/math] is then given by:

[math]\pi^{in}(k^*) \ge \pi^{out}(k^* +1) \quad \mbox{and} \quad \pi^{out}(k^*) \ge \pi^{in}(k^*+1)\;[/math]


Since the coop could act as if there were no agreement (i.e. [math]s=1, \phi=\underline{\phi}\;[/math]), it must be that [math]\pi^{in}(2) \ge \pi^{out}(3)\;[/math], with strict inequality if [math]\phi \lt \overline{\phi}\;[/math]. Therefore:

[math]2 \le k^* \le n\;[/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} \gt 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 \gt \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} \gt \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 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.

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