# 2005-02-15, Updated 2005-10-31

## Boldrin and Levine: Perfectly Competitive Innovation

### Summary

1. More accessible - not too much math - but this also means one is less convinced by overall thrust (especially given the problems with some of the claims).

2. Their distinction between fixed costs and indivisibilities appears to me to be obscure and ultimately groundless. See, in particular, their figure 1 on page 8. The fact is that, in either case, the production set is not convex.

3. The approximation of small to zero for replication costs is not an innocent one (pp. 8-9):

If replication costs are truly so small, would it not be a reasonable approximation to set them equal to zero and work under the assumption that ideas are nonrivalrous? Maybe. As a rule of scientific endeavor, we find approximations acceptable when their predictions are unaffected by small perturbations. Hence, conventional wisdom would be supported if perturbing the nonrivalry hypothesis did not make a difference with the final result. As we show, it does: even a minuscule amount of rivalry can turn standard results upside down.

4. While theoretically ideas may be purely nonrival this is irrelevant as what matters is the embodiment of ideas in rival goods.

5. Increased ease (reduced cost) of reproduction may raise rents to an innovator rather than lowering them (p.12).

6. It is a model of scarcity without market power. In each period the information good has non-zero price because the good is scarce (there isn't an infinite amount of it) yet at the same time there are no strategic issues relating to this limited ownership. In this sense it is analogous to all the standard models with their capital K that produces itself and which though finite in amount conveys no monopoly power.

### Analysis

1. Rejection of 'traditional' nonrivalry (p. 6):

One model of the production and distribution of ideas is to assume that they take place with an initial fixed cost. The technical description is that ideas are nonrivalrous: once they exist they can be freely appropriated by other entrepreneurs. Since at least Shell [1966, 1967], this is the fundamental assumption underlying the increasing returns-monopolistic competition approach: "technical knowledge can be used by many economic units without altering its character" (Shell [1967, p. 68]). Our use of the fundamental theorem of calculus cannot prevent innumerable other people from using the same theorem at the same time. While this observation is correct, we depart from conventional wisdom because we believe it is irrelevant for the economics of innovation. What is economically relevant is not some bodyless object called the fundamental theorem of calculus, but rather our personal knowledge of the fundamental theorem of calculus. Only ideas embodied in people, machines or goods have economic value. To put it differently: economic innovation is almost never about the adoption of new ideas. It is about the production of goods and processes embodying new ideas. Ideas that are not embodied in some good or person are not relevant. This is obvious for all those marvelous ideas we have not yet discovered or we have discovered and forgotten: lacking embodiment either in goods or people they have no economic existence. Careful inspection shows the same is also true for ideas already discovered and currently in use: they have economic value only to the extent that they are embodied into either something or someone. Our model explores the implications of this simple observation leading to a rejection of the long established wisdom, according to which "for the economy in which technical knowledge is a commodity, the basic premises of classical welfare economics are violated, and the optimality of the competitive mechanism is not assured." (Shell [1967, p.68]). In short, we reject the idea of unpriced "spillovers."

2. The traditional ex post / ex ante problem for innovation (p.3):

The central feature of any story of innovation is that rents, arising from marginal values, do not fully reflect total social surplus. This may be due to non rivalry or to an indivisibility or to a lack of full appropriability. Nonrivalry we discuss thoroughly in the next section. Appropriability, or lack of it thereof, depends on whether ideas can be obtained without paying the current owner. Romer [1990a] argues convincingly that appropriability (excludability in his terminology) has no bearing on the shape of the feasible technology set. Since we do not believe that ideas are easily obtained without paying at least for goods that embody them, we do not believe that appropriability is an important problem. In our analysis we assume full appropriability of privately produced commodities and concentrate on the presence of an indivisibility in the inventive process.

Note the sleight of hand here. We all agree that ideas may not be obtained "without paying ... for goods that embody them" but this does not mean that appropriability is not a problem. The first claim in no way implies the second (since while one might appropriate some return one more might not appropriate enough). It may undermine the traditional argument (which consists of a parallel implication of the converse of both statements: ideas are nonrival so without excludability people can get idea for free once created and hence there is a problem of appropriability).

## Notes on Quah: Almost Efficient Innovation By Pricing Ideas

### Overview of Results

Core Quah Results:

1. Even though ex-post have competition (no monopoly) prices are bounded away from zero. This is because reproduction of a idea only occurs slowly so the supply of an idea in any given period is finite hence non-zero price. Recursing this back to initial period gives non-zero price for initial, single, instance of the good.
2. Structure of rights (i.e. whether consumers are allowed to make their own copies or not) does not matter and so IP does not matter (p. 22,29). This is a consequence of CRS production, homogeneity of firms and consumers, and most importantly an assumption that firms exercise no market power. In Quah's setup, when consumers can make their own copies just assume that firms buy back from consumers all of these copies (because of CRS production ownership structure doesn't matter). Firms exercise no market power and prices are those generated from the std GE process. In this setup the extra costs for the firm of buyback are precisely compensated by the greater amount consumers are willing to pay because of buyback.
3. Given an innovation set standard welfare theorems imply we achieve efficient outcomes under competition.
4. Under some more assumption can prove the existence of a non-trivial (finite?) competitive innovation equilibrium (CIE) - definition and details below.
5. However there is a difference between socially efficient innovation and competitive innovation. Can interpret this difference in terms of Dupuit triangles.

### Setup

1. Two types of good. One "normal good" that can be used both for consumption and production of the second type of good: innovations. By convention refer to this good at good 0 (g0).
2. Have possible innovation set $$\mathbb{M}$$ indexed by the natural numbers $$\mathbb{N}$$ (why use a new symbol: to emphasize this is an arbitray set with no particular structure unlike the natural numbers)
3. The act of innovation takes place at the start of period zero. It can be summarized as an innovation/instantiation set $$M \in \mathbb{M}$$ of possible innovations that are actualized/activated. This has some fixed cost in terms of g0.
4. Once a good is activated it can then reproduced from itself by a bounded, convex production function (usually CRS). It may also be possible to simultaneousely consume it. The production function will display greater than unit returns to scale. To summarize: $$\infty > f(x) > x$$. This is the BL/Quah version of nonrivalry.

### Social Efficiency

Rather confusingly Quah has two distinct but related definitions of social efficiency.

1. (The preliminary defn). An allocation $$(s_{mt}),(c_{mt})$$ is efficient given $$(s,M)$$ if it maximizes welfare. This is traditional efficiency in GE but it takes pattern of innovation as given.
2. (True) An $$M \in$$ M is (socially) efficient if it maximizes welfare across all possible innovations (and is feasible).

The main result of the section is then Thm 3.11 which, given some auxiliary technical assumptions, characterizes efficiency by the following condition:

\begin{theorem} Theorem: [informal] Finite innovation $$M \ne \emptyset$$ is efficient iff:

$$$m \in M \Leftrightarrow V_{m}(s_{m}) - V_{m}(0) \geq V'_{0}(s_{00})\Psi_{m}$$$

Where $$V_{m}$$ are the standard value functions that one obtains ..., $$s_{m}$$ is the minimal amount of good $$m$$ produced initially (could assume = 1), $$s_{00}$$ is the amount of good 0 remaining after production of innovation goods, $$\Psi_{m}$$ is the amount of good 0 needed to produce the initial amout of good $$m$$. \end{theorem}

Intuitively the inequality is requiring that the utility from instantiating good $$m$$ is greater than the cost in terms of foregone utility from good $$0$$

### Competitive Innovation Equilibrium

A competitive innovation equilibrium is defined (non-technically) as follows:

1. Take an innovation set M
2. This gives a set of equilibrium asset prices and good prices for all periods (once innovation has occurred all production and exchange takes place with convex production and consumptions sets and complete markets
3. Given these prices define a set of innovations $$M'$$ that would be activated given these prices by checking whether $$q_{m0} > c_{m}$$. where $$q_{m0}, c_{m}$$ are respectively the first period money value and cost of instantiation of innovation $$m$$. So we have a mapping: $$M' = C(M)$$.
4. An equilibrium is then an innovation set $$M$$ and resulting set of prices such that $$M = C(M)$$ i.e. a fixed point of the above mapping

In order to examine the effect of different rights regimes Quah divides IGs into 2 types:

1. Unrestricted dissemination: you can copy the good you buy and do what you want with it (resell, lend, give away etc)
2. Restricted dissemination: you can only use the good for your own use

We then get 3 main results. The first is the standard GE solutions for the CE given the innovation pattern. This is Quah's Thm 4.4 and is really just writing out results of Bellman Eqn work. I will not quote it here. The important results are Thm 4.9 and Thm 4.10 which respectively provide existence of a CIE and conditions for this to be socially (in)efficient.

\begin{theorem} Quah Thm 4.9 [informal] Given a group of assumptions then as the initial amount of good 0, $$\mathbf{E} \rightarrow \infty$$ a greatest CIE, $$M_{G}^{e}$$ exists, with:

$$$m \in M_{G}^{e} \Leftrightarrow V_{m}^{'}(s_{m}) \geq V_{0}^{'}(s_{00})\psi_{m}$$$

Where $$V_{m}$$ are the standard value functions that one obtains ..., $$s_{m}$$ is the minimal amount of good $$m$$ produced initially (could assume = 1), $$s_{00}$$ is the amount of good 0 remaining after production of innovation goods, $$\psi_{m}$$ is the amount of good 0 needed to produce a single unit of good $$m$$. (NB: error in Quah's statement of above eqn in his paper) \end{theorem}

Thm 4.10 provides a condition under which at least one good instantiated under social optimality is not instantiated in the CIE:

\begin{theorem} Theorem 4.10: Under same conditions as above then $$M_{G}^{e} \subset M^{\textrm{star} }$$ and necessarily, $$\forall m \in M^{*}$$.

$$$V_{m}(s_{m}) - V_{m}(0) > V_{m}^{'}(s_{m})$$$

Therefore if there exists $$n$$ such that:

$$$V_{n}(s_{n}) - V_{n}(0) > V_{0}^{'}(s_{00}) x \Psi_{n} > V_{n}^{'}(s_{n})$$$

Then the set inclusion is strict and the CIE is not socially efficient. \end{theorem}

Fig. 1 p.36 provides graphical illustration of this equality and points out that strict concavity of $$V_{m}$$ will ensure the existence of such an $$n$$. Thm 4.11 meanwhile shows that if there is no minimum instantiation amount (i.e. it can be instantiated to any $$s_{m0}$$ though with a possible maximum) then the CIE and the social optimum coincide.

## Comments on Nonrivalry, Infinite Expansibility and Intellectual Goods

The central 'innovation' of the new innovation literature is to the concept and definition of (non-)rivalry. In particular it is shown that the traditional assumption of pure nonrivalry, either as approximation or actuality, is neither correct nor innocent. Here is the usual informal definition for nonrivalry:

A good is nonrival if my use of it does not prevent your simultaneous use of it1.

Let us define Infinite Expansibility as follows: A good is infinitely expansible if possession of 1 unit of the good is equivalent to possession of arbitrarily many units of the good - i.e. one unit may be expanded infinitely. Note that this implies that the good may be "expanded" both infinitely in extent and infinitely quickly.

Formalizing the definition of nonrival we have: A good G is nonrival if its use in the production or utility function of one firm or consumer does not prevent its use in that of another. We immediately see that (perfect) nonrivalry is equivalent to infinite expansibility.

Now all goods, including intellectual works, must be embodied physically and/or transmitted and/or comprehended to be copied and such activities involve delays as well as utilizing rival goods whose cost is non-zero (though perhaps very small). Thus the act of copying has non-zero cost, the good is not infinitely expansible and is therefore not purely nonrival. For example digital data such as a CD or essay will require resources either to be stored or to be transmitted across a network2.

Instead the real issue here is that the first producer of an intellectual good must suffer sunk costs that other producers (who utilize existing copies of the good) will not. This is what distinguishes the fixed (or sunk) costs in the production of intellectual goods from such costs in other areas, and it stems from the fact that intellectual goods have the unusual property of being self-reproducing (just like capital, K, in macroeconomics).

This has important consequences for the nature and regulation of competition in these markets. For while a production function for intellectual good production will resemble those involving 'ordinary' fixed costs for a given firm, the production function of those who compete with the initial firm may be different from that of the original firm. The imperfections of competition that will result for intellectual goods will be different, and more complex, than that from 'ordinary' fixed costs, and will require one to model simultaneously several production functions, the linkages between them and the resulting strategic behaviour of the agents who possess them.

## Conclusion

### Critique of BLQ ideas

While BLQ make an important point in highlighting the limits of the nonrivalry concept there are, in turn, several problems with their alternative. On the one hand, while it is undoubtedly true that new ideas must be embodied, be it in goods, services or human capital in order to be useful this does not necessarily remove the nonrivalry of the good. Suppose, for example, that we have a new design for a hard disk drive (cf. Romer 1992) and that, once one has the design, they can be produced at a marginal cost of 10 units. Now, while it is clear that only the disk drives themselves have value to end consumers, nevertheless if the design can be copied at less than the cost of its original development we still have all the traditional problems: competition will drive price to marginal cost of production plus the cost of copying but since the cost of copying is less than the cost of the original development of the design the originator will make a loss.

BLQ's models avoid this outcome by equating idea production with capital production in standard neoclassical macroeconomic models. Just as new capital is produced from old in those models so new copies of an idea are got from old. But this analogy is simply false, or rather it papers over the fundamental distinction between capital in a neoclassical growth model and ideas in an innovation model: while reproduction of capital can be viewed as a homogenous process (though even this might be dubious) reproduction of ideas is not. Once you have the initial copy of the idea as a producer 'normal' production using capital and labour kicks in and there is no constant returns to scale in the idea. But if that is so, other than the delay (which is important and is the major insight of these models), we are back to our original situation where the original innovator will be out of pocket.

In explicit production function terms: if any copy can be used as a basis for reproduction (as in BLQ) but that (unlike BLQ) once you have one copy you can make additional ones using capital in a CRS production function f(n,k) where n is the number of ideas (think of burning CDs be it as stamped plastic in a factory or as bits on a computer) then: f(0,k) = 0, f(n, k) = f(1,k) for all n \neq 0 and f(1, k) = \alpha k, i.e. there is nonconvexity with respect to ideas. Under competition this then implies that any second period price must be \alpha and profits are zero. But then no-one would be willing to pay more than 0 for a copy of the idea and the originator cannot cover development costs.

### The Way Forward: Imitation vs. Reproduction

Nonetheless BLQ do perform a valuable service in focusing attention on the fact that reproduction is not instantaneous. This ties in closely with the empirical fact of lead-time advantages. However to understand this fully we must introduce the following distinction between imitation and reproduction. Imitation is the making of a first copy – a template – by a new producer who is not the originator. Once a producer has this first copy it may engage in reproduction: the making copies of its own copy in a standard manner.

Armed with this defintion traditional nonrivalry can now be interpreted as the assumption that imitation is the same as reproduction. Conversely, with this definition, we can see the analogy of imitation with the original development, the innovation, as it similarly involves:

1. A fixed cost of creating a first 'copy': imitators have 'development' costs just like originators
2. Time to produce a first 'copy': development be it whether for an imitator or the originator is not instantaneous

### Extensions of BLQ-type Models

1. Model a world of pure ideas. That is possession of the good allows costless reproduction. This continues the standard assumption of the literature that knowledge is entirely explicit and reproduction of ideas is trivial. Clearly this is large - and probably significant - simplication of the real world.
2. Maintain the division of idea/knowledge production into two stages: 1. R&D stage where the idea/good is developed 2. the replication/production stage ([boldrin_ea_2003:5ff.]). Presumes that the idea/knowledge is perfectly codified and there is a clean distinction between reproduction and intial development. What would happen as one allows for tacit knowledge and that replication (implementation) may itself result in new discoveries (e.g. the collapsing of consumer into producer that can occur with software).
3. Modelling innovation process itself. BL do have a model of cumulative innovation but Quah does not (conversely quah has a better worked out GE setup). These are macro models with, at least in Quah, a relatively complete GE backing (albeit with a new equilibrium concept). However most detailed models of innovation itself are from the micro literature. Would like some way to bridge between the 2 sets of models.
4. Examine the robustness of the results (esp. e.g. Quah's "structure of property rights doesn't matter") to variation in structure of model. E.g. dispense with CRS production function and/or representative agent and/or introduce transaction costs. All three of these seem crucial to that result on irrelevance of property rights (though working in different directions).

## Endnotes

1. For example: "... a purely nonrival good has the property that its use by one firm or person in no way limits its use by another" (Romer, 1990_b p. S75). It is interesting to see how Romer understands and uses the concept:

The interesting case for growth theory is the set of goods that are nonrival yet excludable. The third premise cited in the Introduction implies that technology is a nonrival input. ...

... The example of a nonrival input used in what follows is a design for a new good. The vast majority of designs result from the research and development activities of private, profit-maximizing firms. A design is nevertheless, nonrival. Once the design is created, it can be used as often as desired, in as many productive activities as desired.

[romer_1990_b:S75]

Though he acknowledges (p. S75): "Like any scientific concept, nonrivalry is an idealization. It is sometimes observed that a design cannot be a nonrival good because it is itself tied to the physical piece of paper or the physical computer disk on which it is stored. What is unambiguously true about a design is that the cost of replicating it with a drafter, a photocopier, or a disk drive is trivial compared to the cost of creating the design in the first place. This is not true of the ability to add."

2. Note there is a separate but related point that is often confused with this in the debate (see [boldrin_ea_2003]) namely that ideas often require rival goods, such as human capital, to generate any economic value.