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).
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.
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.
While theoretically ideas may be purely nonrival this is irrelevant as what matters is the embodiment of ideas in rival goods.
Increased ease (reduced cost) of reproduction may raise rents to an innovator rather than lowering them (p.12).
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.
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."
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).
Core Quah Results:
Rather confusingly Quah has two distinct but related definitions of social efficiency.
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$$
A competitive innovation equilibrium is defined (non-technically) as follows:
In order to examine the effect of different rights regimes Quah divides IGs into 2 types:
We then get 3 main results. The first is the standard GE solutions for the CE given the innovation pattern. This is Quah's Thm 4.4 and is really just writing out results of Bellman Eqn work. I will not quote it here. The important results are Thm 4.9 and Thm 4.10 which respectively provide existence of a CIE and conditions for this to be socially (in)efficient.
\begin{theorem} Quah Thm 4.9 [informal] Given a group of assumptions then as the initial amount of good 0, $$\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.
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.
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.
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:
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."↩