1. Introduction
Innovators have technological access to many distinct research avenues (ideas).Footnote
1
However, it is often the case that several firms engage in an innovation race for the exact same idea, i.e. research avenues are scarce. In particular, Lemley (Reference Lemley2011) details anecdotal evidence that virtually every major historical innovation (such as the cotton gin, the steam engine, the computer, and the laser) has been simultaneously innovated by several groups of researchers. Perhaps the most famous example is that of the Alexander Bell and Elisha Gray telephone controversy. On February 14, 1876 Bell filed a patent application for the telephone and only hours later Gray submitted a similar application for the exact same innovation. Furthermore, the same empirical regularity is observed for non-major innovations. Cohen and Ishii (Reference Cohen and Ishii2005) find that a positive fraction of patents for the period between
$1988$
and
$1996$
were declared in interference.Footnote
2
More recent examples of simultaneous innovation include companies such as Siemens, Philips, Google Inc., Microsoft Corporation, and Yahoo! Inc.Footnote
3
Furthermore, coordination of research efforts by firms (firm
$A$
directs its effort towards project
$1$
, firm
$B$
towards project
$2$
, and so on) is very unlikely in this setting because of two main reasons. First, the size of the “market” for ideas makes coordination very hard to achieve. Second, such coordination requires each firm to know the portfolio of research projects of all of its rivals. This is particularly implausible in the current context given that firms actively employ secrecy as an intellectual property protection mechanism. For example, Hall et al. (Reference Hall, Helmers, Rogers and Sena2014) report that
$67\%$
of all R&D-doing firms rate trade secrets to be somewhat important or highly important to the company and Cohen et al. (Reference Cohen, Nelson and Walsh2000) find that R&D managers rate secrecy as an effective intellectual property protection mechanism for over
$50\%$
of innovations. For a survey of the evidence see, for example, Hall et al. (Reference Hall, Helmers, Rogers and Sena2014).
Motivated by these observations, we develop an expanding-variety endogenous growth model that features scarce research avenues and lack of research effort coordination in order to study the implications of coordination frictions for growth and welfare. We find that the frictions generate a number of foregone innovations and reduce the economy-wide research intensity. Both effects decrease the growth rate and, as a consequence, endogenously amplify the fraction of duplicative simultaneous innovation. Formalizing the coordination frictions uncovers a novel link between the “stepping on toes” and “standing on shoulders” externalities—their sizes are endogenously determined by the ratio of firms to ideas.Footnote 4 The former externality dominates the latter one for all parameter values, thus, decentralizing the planner’s allocation requires imposing a tax on R&D investment.
In our model, R&D firms direct their research efforts towards a particular project out of an endogenously determined number of ideas (which we also refer to as research avenues or projects). If innovated, each idea is transformed into one new variety. Firms which secure a patent over a variety produce. We focus on the symmetric equilibrium where firms use identical mixed strategies when directing their R&D efforts.Footnote 5 The random realization of these mixed strategies implies that some ideas are simultaneously innovated by many firms while others are not innovated at all. In equilibrium, the number of firms that innovate the same idea simultaneously follows a Poisson distribution with mean equal to the tightness in the market for ideas (the ratio of firms to ideas). The market tightness captures the level of congestion in our economy. When it is high, there are relatively more firms that apply for the same patent, on average, and so the fraction of duplicative innovations is higher, i.e. firms have a higher chance to “step on toes”. Knowledge is cumulative—each innovated idea allows firms to “stand on the shoulders of giants” and gain technological access to a number of new research projects. This intertemporal spillover effect is the ultimate source of growth in our economy—an expanding number of ideas permanently alleviates future congestion problems, thus reducing the cost of discovering new varieties. Along the balanced growth path (BGP henceforth), the growth rate of the economy is determined by the growth rate of the number of ideas, which is in turn endogenously determined by the market tightness and the coordination problems.
Because of simultaneous innovation, some of the R&D effort in the economy is wasteful—when several firms innovate the same idea simultaneously only one secures the patent and the rest make a duplicative innovation.Footnote 6 At the same time the frictions imply that some profitable ideas will be left uninnovated because, by chance, no firm will choose to direct its research effort towards them. This latter possibility generates a number of foregone innovations. As a result, the growth rate of the decentralized frictional economy (DE henceforth) is lower, as compared to a hypothetical economy in which firms can coordinate their efforts (CE henceforth).Footnote 7
The frictions also amplify the fraction of duplicative simultaneous innovation. In the CE firms know exactly which research avenues their competitors are working on and, as a consequence, all duplicative innovation is the result of intentional competition for patents. In the DE, on the other hand, firms cannot coordinate their efforts so some of the duplication which takes place is unintentional. Nonetheless, the higher fraction of duplicative simultaneous innovation in the DE is not a mechanical consequence of the frictions. Instead, it is the result of a general equilibrium effect. Intuitively, in both the DE and the CE firms enter the R&D sector until the cost of innovation equals the expected return which is given by the probability of receiving a patent times the net present value of monopoly profits. Since the probability of securing a patent is negatively related to the fraction of simultaneous innovation, this free entry condition determines the total fraction of wasteful simultaneous innovation whether some of it is intentional or not. Due to the lower growth rate, in the DE firms discount future profit streams at a lower rate. This increases the value of holding a patent as compared to the CE. Because of this, firms in the DE have a stronger incentive to engage in R&D which leads to higher congestion in equilibrium. As a result, the fraction of wasteful duplication of effort is amplified.
The coordination frictions also reduce the aggregate R&D intensity (as captured by the market tightness), even though firms have greater incentives to engage in innovation activities as compared to the CE. This is the case because for any market tightness, the coordination frictions reduce firms’ probability of securing a monopoly position. Given a tightness, the ratio of innovations to ideas is the same for both the DE and the CE. In the DE, however, there is a number of foregone innovations. Hence, a lower fraction of these innovations are distinct which leads to a lower number of patents to be distributed among firms. This reduced probability of securing a patent induces firms to decrease their entry into the R&D sector, and so the DE is left with a lower R&D intensity.
We also examine the implications of coordination frictions for the constrained planing allocation—the planner can choose the number of R&D entrants, but she cannot assign firms to projects. Aside from a monopoly pricing externality, the model also features the usual “stepping on toes” (ST henceforth) and “standing on shoulders” (SS henceforth) externalities. Formalizing the coordination frictions that take place in the economy uncovers a novel link between these two externalities—their relative sizes are endogenously determined by the tightness. The ST one arises due to the possibility of simultaneous innovation in our model. The planner finds the marginal R&D firm’s entry beneficial only if it is the sole inventor of its chosen project. The entrant, however, receives a positive payoff even if she faces competition for the patent as long as she wins the race. This creates incentives for firms to over-invest in R&D as compared to the socially optimal level. How large the incentives to over-invest are depended on the chance a firm secures a patent and the chance it is the sole inventor, both of which are endogenously determined by the market tightness. When it is either low or high, the incentives are small, whereas for intermediate values of the tightness they are large. The SS externality arises because firms cannot appropriate the benefits generated by any new ideas that come about from their innovation. As a result they do not value those new ideas. The planner, on the other hand, does because an increase in the pool of ideas permanently alleviates future coordination problems. Thus, the SS externality induces firms to under-invest in R&D as compared to the socially optimal level. When the market tightness is high, the chance the marginal entrant will be the sole inventor is low, hence, it is unlikely that new ideas will be generated from her innovation. This tends to decrease the size of the SS externality. At the same time, a high tightness implies a high level of congestion in the market. Thus, ideas are relatively scarce and, as a result, more valuable. This tends to increase the size of SS. It turns out this latter effect dominates so the SS externality increases in magnitude as the tightness goes up. Hence, the planner cares more about reducing congestion than foregone innovation: the size of the SS externality is primarily governed by how valuable ideas are to the planner, not by how many of them remain uninnovated. Consequently the planner chooses a relatively low market tightness because this delivers relatively low congestion. As a result, the ST externality is larger than the SS externality for all parameter values and implementing the planner’s allocation requires imposing a tax on R&D investment.
1.1. Relationship to the Literature
Our paper models firms’ choice of direction for their R&D efforts and the coordination problems inherent in this decision. As such, it is related to a recent literature on economic growth which emphasizes matching and other frictions in the innovation process [see, for example, Perla and Tonetti (Reference Perla and Tonetti2014), Lucas and Moll (Reference Lucas and Moll2014), Benhabib et al. (Reference Benhabib, Perla and Tonetti2014), Chiu et al. (Reference Chiu, Meh and Wright2017), Akcigit and Liu (Reference Akcigit and Liu2016), and Akcigit et al. (Reference Akcigit, Celik and Greenwood2016)]. For example, Akcigit and Liu (Reference Akcigit and Liu2016) emphasizes the importance of informational frictions for growth and welfare. In their model firms have access to two research avenues—a lucrative but risky one and a less profitable but safe one. In the process of R&D firms learn about the nature of the risky project (whether it is a dead-end or not) but have incentives to keep this information secret from their rivals. This creates a disclosure problem as firms may spend resources replicating the dead-end results of their rivals. The authors analyze firms’ behavior in a pure strategy equilibrium and investigate the welfare implications.
The work here complements the literature by examining a different source of friction. As Akcigit and Liu (Reference Akcigit and Liu2016) we analyze an information issue—firms keep their R&D activities secret. In contrast to that paper, we emphasize the coordination problems inherent with secrecy. In our economy, firms are unsure which research projects their rivals are working on, whereas in Akcigit and Liu (Reference Akcigit and Liu2016) firms are unsure what their rivals have learned in the process of R&D. Thus, we emphasize the importance of information even when learning is absent and research avenues have certain payoffs. In particular, to the best of my knowledge, this is the first growth paper to emphasize search frictions in the market for ideas which take the form of a coordination failure.Footnote 8
Our economy features the possibility of wasteful simultaneous innovation. In this regard the current paper relates to a literature which studies the implications of duplicative R&D investment for growth and welfare.Footnote 9 Papers within that literature include Segerstrom et al. (Reference Segerstrom, Anant and Dinopoulos1990), Corriveau (Reference Corriveau1994), Corriveau (Reference Corriveau1998), and Zeira (Reference Zeira2011) which focus on duplication of effort in patent races; Aghion et al. (Reference Aghion, Bloom, Blundell, Griffith and Howitt2005) and Acemoglu and Akcigit (Reference Acemoglu and Akcigit2012) which analyze economies where firms duplicate the previous successful innovations of their rivals; Akcigit and Liu (Reference Akcigit and Liu2016) which analyzes a model where firms may unknowingly spend R&D resources to duplicate the dead-end results of their rivals.Footnote 10 Jones and Williams (Reference Jones and Williams2000) quantify the size of the ST externality which stems from duplication of R&D effort and investigate whether or not the R&D spending in the U.S. economy is sub-optimally high/low; Strulik (Reference Strulik2007) tackles the same research question in a model with human capital and impure altruism; Chen et al. (Reference Chen, Chu, Chu and Lai2021) study the level of the optimal capital tax rate when the ST externality is present. In contrast to these studies, we focus on the link between the ST and the SS externalities that arises endogenously from the coordination frictions in the economy.
Our paper is also related to the literature on coordination frictions. We study the coordination problems inherent in firms’ choice of R&D projects and in this way contribute to the existing literature which has traditionally examined coordination failure in other contexts, such as the labor and goods markets: Julien et al. (Reference Julien, Kennes and King2000), Burdett et al. (Reference Burdett, Shi and Wright2001), and Shimer (Reference Shimer2005). In these directed search studies firms/sellers post wages/prices and workers/buyers choose which firm to send an application to/which seller to visit.Footnote 11 In equilibrium it is possible that, due to coordination frictions, many workers apply for the same position/many buyers queue at the same seller. This generates congestion in the market. Similarly, in our setting firms choose towards which idea to direct their innovative activity and congestion arises because of the possibility that several firms innovate the same idea simultaneously. An important difference between our model and these aforementioned papers is that in ours congestion has an intertemporal aspect. The R&D firms of today make innovations which decrease the congestion faced by the R&D firms of tomorrow, an effect which the R&D firms of today do not internalize. Thus, the R&D firms of tomorrow play an analogous role to that of workers in the traditional search and matching framework. Furthermore, as we discuss in Section 4, the literature typically finds the decentralized equilibrium to be efficient whereas, in our study, we do not partially because of this dynamic aspect of congestion. Within the macroeconomic literature the closest paper is Gabrovski (Reference Gabrovski2020). In the Gabrovski (Reference Gabrovski2020) economy firms are randomly matched with ideas according to the same urn-ball matching function that arises endogenously in our model. In contrast to our current paper, Gabrovski (Reference Gabrovski2020) does not focus on the coordination problems in the economy and the resulting implications for growth and welfare. Instead, the focus there is on the cyclical properties of R&D investment when innovation quality is endogenous. Within the literature on industrial organization the two closest papers are Kultti et al. (Reference Kultti, Takalo and Toikka2007) and Kultti and Takalo (Reference Kultti and Takalo2008) which also feature search frictions in the market for ideas. In these papers there is the possibility of simultaneous innovation due to a matching technology which is the same as the equilibrium one in our paper. Kultti et al. (Reference Kultti, Takalo and Toikka2007) and Kultti and Takalo (Reference Kultti and Takalo2008) focus on intellectual property rights in a partial equilibrium framework with a fixed number of ideas and without free entry into the innovation sector. In contrast, our model focuses on a general equilibrium framework with growth, an endogenously determined number of ideas, and an endogenously determined market tightness through free entry in the R&D sector.
We begin by introducing the environment. We then examine the decentralized equilibrium and highlight the impact of coordination frictions. Next, we study the implications of the frictions for the planner’s (constrained) efficient allocation. The last section concludes.
2. The economy
The environment is an augmented version of the textbook model in Barro and Sala-i-Martin (Reference Barro and Sala-i-Martin2003) Chapter 6 (BSM henceforth).Footnote 12 We set the model in discrete time, so as to allow for the possibility of simultaneous innovation. There are three types of agents—a final good producer, R&D firms, and a unit measure of households. The only point of departure from BSM is in the R&D sector, so as to emphasize the novel features of the model. In particular, R&D projects are scarce and R&D entrants can direct their efforts towards a specific project, but they cannot coordinate their research activities.
2.1. Final Good Sector
The final good is produced by a single price taker, using the following technology
\begin{equation} Y_t=AL^{1-\lambda }\sum _{n=0}^{N_t}X^{\lambda }_{t,n}, \quad 0\lt \lambda \lt 1, \end{equation}
where
$Y_t$
is output,
$L$
is the fixed labor supply of households,
$N_t$
is the number of intermediate varieties, and
$X_{t,n}$
is the amount of a particular variety
$n$
employed in production. The price of the final good is normalized to unity. The final good firm faces a competitive market for labor, which is hired at the wage
$w_t$
, and a monopolistically competitive market for varieties, where a unit of each variety
$n$
is bought at the price
$P_{t,n}$
.
2.2. R&D Sector
The novel features of our model are contained within the R&D sector of the economy. The innovation process has three stages and makes a distinction between potential innovations (ideas) and actual innovations (new varieties). At stage one, firms enter the R&D sector at a cost
$\eta \gt 0$
units of the final good. The number of R&D entrants is denoted by
$\mu _t$
and is to be determined in equilibrium. At stage two firms direct their innovative effort towards a particular R&D project from a pool of ideas. The number of ideas in the pool is denoted by
$\nu _t$
. The choice is private knowledge and firms cannot coordinate their efforts. To capture this coordination failure, we follow the existing literature and focus on a symmetric equilibrium where firms use identical mixed strategies.Footnote
13
Ideas are identical and if a firm innovates an idea, it transforms into exactly one new variety. Innovation takes one period—a firm which enters at time
$t$
innovates the chosen project at time
$t+1$
. Thus, the only source of uncertainty in our model is the random realization of firms’ equilibrium mixed strategies—some ideas may be innovated by many firms simultaneously, while others may not be innovated at all. We should note that in an economy with a small number of ideas this idiosyncratic uncertainty translates into aggregate uncertainty. However, due to positive long-run growth, as we will see later, the pool of ideas eventually becomes large, i.e.
$\nu _t\rightarrow \infty$
, regardless of the initial number of ideas,
$\nu _0$
. As a result, there is no aggregate uncertainty along the BGP. Since our analysis focuses precisely on the BGP, we restrict our attention to the limit economy, where the number of participants in the coordination game is large. This is a standard approach in the literature on coordination frictions—see, for example, Shi (Reference Shi2001), Shi (Reference Shi2005), Julien et al. (Reference Julien, Kennes and King2000), Julien et al. (Reference Julien, Kennes and Ritter2018), Gomis-Porqueras et al. (Reference Gomis-Porqueras, Julien and Wang2017), Shimer (Reference Shimer2005), Julien et al. (Reference Julien, Kennes and Ritter2014). Innovators apply for a patent which grants perpetual monopoly rights over the variety. Each innovation is protected by exactly one patent—if several firms simultaneously apply for the same patent, then each has an equal chance of receiving it. Stage three is as in BSM. Patent holders supply their variety in a monopolistically competitive market. Both the average and marginal costs of production are normalized to unity so profits are given by
$\pi _{t,n}=(P_{t,n}-1)X_{t,n}$
. Furthermore, the value of holding a monopoly over a variety
$n$
at time
$t$
is given by
$V_{t,n} \equiv \sum _{i=t+1}^{\infty }d_{it}\pi _i(n)$
, where
$d_{it}$
is the factor with which firms discount future profits.
A necessary condition for positive long term growth in our model is that the number of ideas,
$\nu _t$
, grows at a positive rate. We follow Kortum (Reference Kortum1997) and Romer (Reference Romer1990), among others, and assume that knowledge is cumulative. Patenting an idea at time
$t$
allows firms to “stand on the shoulders of giants” and gain access to
$M\gt 1$
new research avenues at
$t+1$
.Footnote
14
Once an idea is innovated, it is no longer a potential R&D project and so it is removed from the pool.Footnote
15
Thus, the net increase in the pool of ideas from innovating one new variety is
$M-1$
. Due to the frictions in our model, there is a chance that an idea is not innovated, i.e. no firm directs its research efforts towards the idea in question. Let us denote the fraction of uninnovated ideas by
$\phi _t$
, then the law of motion for ideas is given by
\begin{equation} \nu _{t+1}=\nu _t +(1-\phi _t)(M-1)\nu _t. \end{equation}
As each innovated idea is transformed into a new variety, it follows that
\begin{equation} N_{t+1}=N_t+(1-\phi _t)\nu _t. \end{equation}
2.3. Households
Households are endowed with a discount factor
$\beta$
and a per-period utility function
$U(C_t)=\mathrm{ln}C_t$
. They save by accumulating assets, which in this economy are claims on intermediate firms’ profits. In particular, households have access to a mutual fund that covers all intermediate good firms. Let
$a_t$
denote the amount of shares held by the representative household at the beginning of period
$t$
. Each period all profits are redistributed as dividends, thus the total assets of the household entering period
$t$
are
$a_t\!\sum _{n=0}^{N_t}\!(\pi _{t,n}+V_{t,n})$
. At time
$t$
households decide on the shares they would like to hold at
$t+1$
,
$a_{t+1}$
. The mutual fund at that time covers all firms which exist at time
$t+1$
,
$N_{t+1}$
. Hence, the household’s budget constraint is given by
\begin{align} a_{t+1}\sum _{n=0}^{N_{t+1}}V_{t,n}=a_t\sum _{n=0}^{N_t}(\pi _{t,n}+V_{t,n})+w_tL-C_t. \end{align}
The household’s first order conditions imply the Euler equation below
\begin{align} \frac{1}{C_t}=\frac{\beta }{C_{t+1}}\left (\sum _{n=0}^{N_{t+1}}(\pi _{t+1}(n)+V_{t+1}(n))\right )\left (\sum _{n=0}^{N_{t+1}}V_{t,n}\right )^{-1}. \end{align}
The intuition is standard—consumers equate the marginal utility at time
$t$
with the discounted marginal utility at time
$t+1$
, times the gross rate of return on their assets.
2.4. Balanced Growth Path Equilibrium
The usual profit maximization of intermediate and final good firms implies that
$P_{t,n}=1/\lambda$
and
$X\,{:\!=}\,X_{t,n}=(\lambda ^2A)^{1/(1-\lambda )}L$
. Thus, every intermediate good firm receives the same per period profits of
$\pi \,{:\!=}\,\pi _{t,n}=X(1-\lambda )/\lambda$
. This implies that
$V_t\,{:\!=}\,V_{t,n}=\sum _{i=t+1}^{\infty }d_{it}\pi$
—every firm is equally valuable. Since each variety carries the same amount of profits, the stage two equilibrium strategy of firms is to direct their R&D effort towards each idea with equal probability. This implies the following equilibrium outcome.
Proposition 1.
The number of firms which direct their R&D effort towards a particular idea follows a Poisson distribution with mean
$\theta _t$
, where
$\theta _t \equiv \mu _t/\nu _t$
.
A proof is in the appendix. The random realization of firms’ equilibrium strategies gives rise to the standard urn-ball matching technology.Footnote
16
In equilibrium some firms find themselves competing with many rivals for the same patent while others may not face any competition at all. The average number of firms that compete for the same patent is given by the ratio of firms to ideas,
$\theta _t$
. This is the tightness in the market for ideas and is analogous to the queue length in models of directed search in the labor market.Footnote
17
In the labor market, a high
$\theta _t$
implies each firm receives on average more applications from workers and so the market is more congested. In the current context, a higher
$\theta _t$
implies more firms, on average, find themselves competing for the same innovation and so the market is more congested. In particular, an R&D firm becomes a monopolist with probability
$\sum _{m=0}^{\infty }Pr($
exactly
$m$
rival firms direct their research effort towards the particular idea
$)/(m+1)=$
$\sum _{m=0}^{\infty }e^{-\theta _t}\theta _t^m/(m+1)!=\left(1-e^{-\theta _t}\right)/\theta _t$
. Since only one firm gets the patent, the simultaneous innovations made by all other rivals represent wasteful duplication of effort. Thus, a fraction
$\omega \equiv 1- (1-e^{-\theta _t})/\theta _t$
of all firms make a wasteful simultaneous innovation.Footnote
18
This duplication of effort is analogous to the congestion in the labor market where the firm can hire only one worker and the applications of all other workers who apply for the same job are turned down. The market tightness also dictates the fraction of foregone innovation
$\phi _t=Pr(\mathrm{no firm directs its research effort towards a particular idea})=e^{-\theta _t}$
. As
$\theta _t$
increases the chance that no firm chooses to work on any particular project is lower and so the fraction of foregone innovation decreases as well.Footnote
19
Hence, the laws of motion for ideas and varieties transform to
\begin{equation} \nu _{t+1}=\nu _t +\left(1-e^{-\theta _t}\right)\!(M-1)\nu _t, \end{equation}
\begin{equation} N_{t+1}=N_t+\left(1-e^{-\theta _t}\right)\nu _t. \end{equation}
Given free entry, it follows thatFootnote 20
\begin{equation} \eta =\frac{1-e^{-\theta _t}}{\theta _t}V_t. \end{equation}
The equilibrium tightness is pinned down by the net present value of profits and the entry cost. As a consequence the fractions of wasteful duplication of effort,
$\omega _t$
, and foregone innovation,
$\phi _t$
, are determined by free entry as well. Higher profits (or lower costs) induce firms to tolerate a lower chance of securing a monopoly position which results in a higher equilibrium tightness. This implies more duplication of effort and lower foregone innovation. It is worth noting that due to the usual “over-grazing” problem firms have an incentive to compete for patents so some of the duplication which takes place in the economy is intentional. At the same time, due to the coordination problems firms may end up in a patent race with too many rivals. Hence, some of the simultaneous innovation which takes place is unintentional. R&D entrants foresee this possibility and so in equilibrium the free entry condition pins down the aggregate level of wasteful innovation whether it is intentional or not.
It is straightforward to establish that output, varieties, consumption, entry into R&D, and the stock of ideas all grow at the same rate,
$g$
, along the BGP. Namely,
$g=(1-e^{-\theta })(M-1)$
, where
$\theta$
is the value of the market tightness along the BGP.Footnote
21
As in BSM
$Y_t$
,
$C_t$
,
$N_t$
, and
$\mu _t$
all grow at the same rate. In our model, the number of ideas,
$\nu _t$
, also grows at this rate. In fact, the expansion of
$\nu _t$
is the ultimate source of growth in the economy. Because knowledge is cumulative, innovation today increases the number of ideas in the future. This permanently reduces the severity of the coordination problems and subsequently the cost of securing a monopoly position.Footnote
22
This lower cost in turn induces higher entry into R&D up to the point where congestion and, as a consequence, the tightness reach their BGP levels.
It is convenient to solve the model by looking at the stable ratios
$\theta$
,
$\frac{\nu }{N}$
and
$\frac{C}{N}$
. From the law of motion of ideas and varieties, and from
$g_N=g_{\nu }$
, it follows that
$\frac{\nu }{N}=M-1$
. Next, combining the household budget constraint with the free entry condition and the law of motion for varieties yields
$C/N=(1+\lambda )\pi/\lambda -\eta \theta (M-1)$
. Lastly, we can use the fact that
$g_C=g_{\nu }$
, the Euler equation, the law of motion for
$\nu _t$
, and the free entry condition to find an implicit solution for the market tightness.
\begin{equation} \eta =\left (\frac{1-e^{-\theta }}{\theta }\right )\frac{\beta \pi }{1+(1-e^{-\theta })(M-1)-\beta }. \end{equation}
Even though we cannot explicitly solve for
$\theta$
, it is straightforward to establish that the solution is unique. Intuitively, as
$\theta$
increases the market for ideas gets more congested and each firm’s chance of becoming a monopolist decreases. At the same time, higher market tightness implies a higher growth rate. This, in turn, increases the rate with which firms discount future profit streams and, as a consequence, decreases the value of holding a patent. Both of these effects decrease the incentives to enter the R&D sector when the market tightness is high and vice versa.
Proposition 2.
The equilibrium market tightness,
$\theta$
, the growth rate,
$g$
, and the fraction of wasteful innovation,
$\omega$
, are:
-
increasing in
$A$
and
$\beta$
-
decreasing in
$\eta$
and
$M$
The proof, derived by taking the total derivative of (9), appears in the appendix. Intuitively, an increase in productivity leads to higher profits. High profits, in turn, raise the value of being a monopolist,
$V_t$
. This increases firms’ incentives to innovate, which leads to a higher number of R&D entrants and subsequently to a higher market tightness. This higher tightness implies the market is more congested and so a higher fraction of firms make a duplicative simultaneous innovation, i.e.
$\omega$
increases. At the same time a higher
$\theta$
reduces the fraction of foregone innovation, which ultimately increases the growth rate. Similarly, a higher entry cost,
$\eta$
, discourages entry into R&D, which decreases
$\theta$
,
$\omega$
, and
$g$
. An increase in
$\beta$
or a decrease in
$M$
both lead to an increase in the effective discount factor,
$\beta C_t/C_{t+1}$
, along the BGP. Thus, firms value future profits more, which increases the value of a patent,
$V_t$
, and ultimately the market tightness, the fraction of wasteful duplication of effort, and the growth rate.
3. The impact of coordination frictions
The goal of our analysis is to study the impact of coordination frictions in the economy. To this end we compare the DE’s BGP to the BGP of a hypothetical CE. The only difference between the latter economy and the DE is that firms can coordinate their research efforts at stage two of the innovation process. Let superscript
$c$
denote the value of any variable in the CE along the BGP.Footnote
23
Since firms can coordinate their R&D investments all projects are innovated by the exact same number of firms simultaneously. Moreover, all research avenues are undertaken, and subsequently, all ideas are innovated. Given free entry, however, the CE features a positive fraction of wasteful duplication of effort due to the usual “over-grazing” problem, i.e.
$\mu ^c_t \gt \nu _t^c$
. Firms intentionally choose to compete for patents and so their chance of securing a monopoly position is
$1/\theta ^c$
, which is less than unity in equilibrium. Nonetheless, the resulting waste,
$\omega ^c = 1- 1/\theta ^c$
, is smaller than the one in the DE.
Proposition 3.
In the coordination economy all ideas are innovated and the growth rate equals
$M-1$
. Furthermore,
\begin{equation} \omega -\omega ^c=\frac{e^{-\theta }\!(M-1)\eta }{\beta \pi }\gt 0. \end{equation}
A proof is in the appendix. The coordination frictions amplify the fraction of wasteful simultaneous innovation. This, however, is not a mechanical consequence of the frictions, but rather the result of an endogenous general equilibrium effect—free entry dictates the level of congestion in both economies, and so the fraction of wasteful innovation is an endogenous object, whether some of the duplication that takes place is unintentional or not. As a result the relative sizes of
$\omega$
and
$\omega ^c$
are pinned down by how lucrative it is to hold a monopoly position in each of the two economies. In the CE there is no forgone innovation,
$\phi ^c=0$
, so the growth rate is higher as compared to the one in the DE. This higher growth rate increases the effective discount factor, which in turn reduces the value of holding a monopoly position. Hence,
$V_t\gt V_t^c$
so firms find holding a patent more lucrative in the DE as compared to the CE. Analytically, the level of amplification,
$\omega -\omega ^c$
, equals the difference in the growth rates,
$g^c-g=e^{-\theta }(M-1)$
, divided by the discounted normalized profits,
$\beta \pi/\eta$
.
Proposition 4.
The market tightness in the decentralized economy is lower than that in the coordination economy, i.e.
$\theta \lt \theta ^c$
.
A proof is in the appendix. The coordination frictions reduce the economy-wide R&D intensity (as captured by the market tightness), even though the DE features a higher fraction of wasteful simultaneous innovation. This is the case because, for a given market tightness, the frictions reduce an entrant’s chance of securing a monopoly position. In particular, the probability of securing a patent in the DE for a given tightness
$\tilde{\theta }$
,
$Pr(\mathrm{patent}|\tilde{\theta })=(1-e^{-\tilde{\theta }})/\tilde{\theta }$
, is only a fraction
$1-e^{-\tilde{\theta }}$
of the one in the CE,
$Pr(\mathrm{patent}|\tilde{\theta })^c=1/\tilde{\theta }$
. As firms cannot coordinate their efforts, in the DE only a fraction
$1-e^{-\tilde{\theta }}$
of ideas are patented. Thus, even though the number of patent applications per idea,
$\tilde{\theta }$
, is the same in both economies, in the DE there are relatively less patents to be distributed among innovators. This decreases each entrant’s chance of securing a monopoly position and subsequently reduces the incentives to enter the R&D sector. This is true even though the DE features a higher value of holding a patent. In other words, the effect of a lower probability of securing a patent dominates the effect of an increase in the net present value of profits and ultimately reduces the incentives to enter the R&D sector and decreases the market tightness. Furthermore, the decrease in the tightness provides an indirect channel through which the presence of foregone innovation reduces the growth rate in the DE—a lower tightness decreases each idea’s chance of being innovated which results in a lower aggregate number of innovations.
The economic impact of coordination frictions is higher when profits are low, households are more inpatient, and the entry cost is higher. The next proposition states the result.
Proposition 5.
The level of amplification of wasteful innovation,
$\omega -\omega ^c$
, and the amount by which the tightness is reduced,
$\theta ^c-\theta$
are
-
decreasing in
$A$
and
$\beta$
-
increasing in
$\eta$
A proof is in the appendix. The fraction of foregone innovation depends only on the market tightness in the DE. When the tightness is low, the probability that an idea is not matched with any firm is high, which leads to a high fraction of foregone innovation and vice versa. The level of the amplification of wasteful innovation moves in the same direction as
$\phi$
. This is because firms in the DE are willing to tolerate lower probability of securing a monopoly position only due to the higher value of holding a patent induced by
$g\lt g^c$
. As the fraction of foregone innovation decreases, the difference in the growth rates decreases as well. This reduces the incentives for firms to tolerate extra congestion, which decreases the amplification.
When
$\phi$
is low, the incentives for firms in the DE to over-invest (as compared to the CE) induced by the difference in the growth rates is low as well. This generates an upward pressure on the difference in research intensities,
$\theta ^c-\theta$
. At the same time, a smaller fraction of forgone innovation implies that, for a given market tightness, there are relatively more patents to be distributed among firms in the DE. Hence,
$Pr(\mathrm{patent}|\tilde{\theta })^c-Pr(\mathrm{patent}|\tilde{\theta })$
decreases, which in turn increases the incentives for firms in the DE (relative to CE) to enter the R&D sector. Consequently, this generates a downward pressure on
$\theta ^c-\theta$
. For a decrease in the fraction of foregone innovation induced by changes in
$A$
,
$\beta$
, or
$\eta$
the latter effect dominates the former.
4. The constrained planning problem
Next we turn to the planner’s constrained efficient allocation—she chooses the optimal BGP allocation subject to the coordination frictions in the market for ideas. Without loss of generality, we impose symmetry in the intermediate varieties, i.e.
$X_{t,n}=X_t(n')$
for any varieties
$n$
and
$n'$
. Thus, the planner faces the problem of choosing production of varieties, consumption, a number of varieties, a number of ideas, and a market tightness in order to maximize welfare subject to the resource constraint, the laws of motion for ideas and varieties, and the coordination frictions. The planner’s problem, using a recursive formulation, is summarized below:
\begin{align} V^P\!(N_t,\nu _t) = \max _{C_t, X_t, \theta _t, N_{t+1}, \nu _{t+1}}\mathrm{ln}C_t + \beta V^P(N_{t+1},\nu _{t+1})\nonumber \\[3pt] \mathrm{s.t.} \quad AL^{1-\lambda }N_tX_t^{\lambda }=N_tX_t + C_t +\eta \theta _t \nu _t, \end{align}
\begin{align} N_{t+1}=N_t+\left(1-e^{-\theta _t}\right)\nu _t, \end{align}
\begin{align} \nu _{t+1}=\nu _t+(1-e^{-\theta _t})(M-1)\nu _t. \end{align}
Maximizing with respect to
$X_t$
yields the usual solution for varieties
$X^*\,{:\!=}\,X_t=(\lambda A)^{1/(1-\lambda )}L$
. As in BSM the difference between the planner’s solution and the decentralized outcome comes from the monopoly pricing of intermediate goods. Let
$\pi ^*=X^{*}(1-\lambda )/\lambda$
denote the implied per period monopoly profits at the efficient level of intermediate varieties. Then, it is straightforward to establish that the rest of the first order conditions can be reduced to the following system of equations:
\begin{align} \frac{h_t}{\phi _t}=\beta \frac{C_t}{C_{t+1}}\left (\pi ^*+\frac{h_{t+1}}{\phi _{t+1}}\right ), \end{align}
\begin{align} \frac{\psi _t}{\phi _t}=\beta \frac{C_t}{C_{t+1}}\left (-\eta \theta _{t+1}+\left(1-e^{-\theta _{t+1}}\right)\!\left (\frac{h_{t+1}}{\phi _{t+1}}+\frac{\psi _{t+1}}{\phi _{t+1}}(M-1)\right )+\frac{\psi _{t+1}}{\phi _{t+1}}\right ), \end{align}
\begin{align} \eta = e^{-\theta _t}\!\left (\frac{h_t}{\phi _t} + \frac{\psi _t}{\phi _t }(M-1)\right ), \end{align}
where
$\phi _t$
,
$h_t$
,
$\psi _t$
are the multipliers associated with (11), (12), and (13), respectively. The first equation above, (14), characterizes the planner’s valuation of varieties: the value of a variety equals the discounted sum of per period profits,
$\pi ^*$
, and the continuation value
$h_{t+1}/\phi _{t+1}$
. There are only two differences as compared to the DE [equation (5)]—the level of profits is higher and the planner chooses a different tightness. The second equation, (15), characterizes how the planner values an idea: it is the discounted sum of several terms. The first term is the dividend,
$-\eta \theta _{t+1}$
, which represents the average R&D effort spent per idea. It captures the intuition that unlike other assets, which carry positive returns, an idea is only valuable if it is innovated. Hence, the planner finds it costly to keep a stock of ideas because it diverts resources away from consumption and into R&D. The second term represents the capital gain from innovation—the probability an idea is innovated,
$(1-e^{-\theta _{t+1}})$
, times the social benefit from innovating. This benefit is the value of the extra variety,
$h_{t+1}/\phi _{t+1}$
, plus the value of the extra ideas that would be added to the pool because of innovation,
$\psi _{t+1}/\phi _{t+1}(M-1)$
. Lastly, the idea carries its continuation value
$\psi _{t+1}/\phi _{t+1}$
. The last equation, (16), depicts the entry decision of the planner. She finds it optimal for firms to enter up until the expected social benefit from the marginal entrant is equal to the cost of innovating,
$\eta$
. The benefit that the entrant generates is the value of the extra variety and the extra ideas which come about from her innovation, times the probability she is the sole inventor,
$e^{-\theta _t}$
.Footnote
24
Apart from the monopoly pricing of intermediate goods, there are two additional externalities in the model which are illustrated in equation (16). First, the ST externality manifests through the difference in the fraction of socially and privately beneficial innovations. The planner finds the marginal entry beneficial only if the firm is the sole inventor. Firms, on the other hand, value entry even if they duplicate an innovation, as long as they receive the patent for it. In particular, due to this “stepping-on-toes” effect, the probability of a privately beneficial innovation is
$(1-e^{-\theta _t})/\theta _t\gt e^{-\theta _t}$
. Hence, the ST externality induces firms to over-invest in R&D as compared to the efficient allocation. Second, there is the SS externality—firms cannot appropriate the benefit of any ideas that come about from their innovations, so they do not value them. The planner, on the other hand, does because they permanently alleviate future coordination problems. Specifically, more innovation today increases the amount of future research avenues, which allows the economy to innovate more varieties without increasing the congestion problems. Thus, the extra ideas permanently reduce the cost of discovering new varieties.Footnote
25
As a result, the SS externality creates incentives for firms to under-invest as compared to the efficient allocation.
The ST and SS externalities in our model are tightly linked to the coordination frictions in the economy. Thus, it is instructive to compare our externalities to those studied in the labor search literature. To begin with, we can draw an analogy between the ST externality in our model and the congestion externality present in the baseline random search model of the labor market [Pissarides (Reference Pissarides2000), Chapter 8].Footnote 26 In both models, firms do not take into account that their entry increases congestion in the market, which makes it harder for other firms to get matched. In the labor market, the congestion externality induces entry to be either too high or too low depending on the way the firm and the worker share the surplus of the match. If the firm extracts too much of the surplus—there is too much entry in equilibrium, if it extracts too little—entry is too low.Footnote 27 When firms (or workers) direct their search, however, competition drives surplus shares for the firm and the worker to their efficient levels, so the congestion externality disappears [Julien et al. (Reference Julien, Kennes and King2000); Shimer (Reference Shimer2005); Shi (Reference Shi2001)]. In our model, on the other hand, the ST externality is present even though search is directed. This is because, in our setting, ideas are not active economic actors, so firms always extract the full amount of the surplus. In addition, because firms extract the full surplus of the firm-idea match, the ST externality always induces over-entry in equilibrium. Next, the SS externality in our model can be viewed as analogous to the thick-market externality studied in the labor search literature [see, for example, Diamond (Reference Diamond1982) and Howitt and McAfee (Reference Howitt and McAfee1987)]. Both externalities have a similar flavor—if more agents participate in the market congestion is lower and therefore it is easier for matches to occur.Footnote 28 This thick-market externality operates through a matching function that has increasing returns to scale, so a larger market today leads to lower congestion today. In contrast, the SS externality is intertemporal—more R&D entrants today leads to higher levels of innovation, which reduces congestion tomorrow.
To study the magnitude of the externalities, we can decompose the difference between the planner’s valuation of the benefit of entry at the efficient allocation,
$\eta$
, and the firm’s valuation of this benefit,
$V$
times the probability of securing a patent. Using
$\mathcal{M}$
,
$\mathcal{SS}$
, and
$\mathcal{ST}$
to denote the magnitude of the monopoly pricing, SS, and ST externalities, respectively, this decomposition is given by
\begin{equation} \eta - \left (\frac{1-e^{-\theta }}{\theta }\right )V = \underbrace{\left (\frac{h}{\phi }-V\right )\left (\frac{1-e^{-\theta }}{\theta }\right )}_{\mathcal{M}}+\underbrace{e^{-\theta }(M-1)\frac{\psi }{\phi }}_{\mathcal{SS}}-\underbrace{\Bigg (\frac{1-e^{-\theta }}{\theta }-e^{-\theta }\Bigg )\frac{h}{\phi }}_{\mathcal{ST}}. \end{equation}
Conceptually, each of
$\mathcal{M}/\mathcal{SS}/\mathcal{ST}$
measures how much more would the planner value the marginal entrant if monopoly pricing/SS/ST was the only externality in the model. In the case of
$\mathcal{M}$
, this quantity is given by the difference of the planner’s valuation of the extra variety minus the firm’s valuation, multiplied by the chance the entrant secures a monopoly position; in the case of
$\mathcal{SS}$
, it is the chance the entrant is the sole inventor times the number of extra ideas that are generated by this sole inventor times the value of each idea; in the case of
$\mathcal{ST}$
, it is given by the planner’s valuation of the extra variety times the difference between the chance an entrant receives a patent and the chance she is the sole inventor.
The probability an entrant is the sole inventor and the probability she secures a patent are both endogenous objects determined by the coordination frictions. Consequently, since these probabilities determine
$\mathcal{ST}$
and
$\mathcal{SS}$
, the frictions in our economy link the ST and SS externalities. To shed light on this novel link, which operates through the market tightness, use equations (16) and (14) to derive expressions for
$\mathcal{SS}$
and
$\mathcal{ST}$
that are explicit in
$\theta$
:
\begin{equation} \mathcal{SS} = \eta - \frac{e^{-\theta }\beta \pi ^*}{1+(1-e^{-\theta })(M-1)-\beta }, \end{equation}
\begin{equation} \mathcal{ST} = \Bigg (\frac{1-e^{-\theta }}{\theta }-e^{-\theta }\Bigg )\frac{\beta \pi ^*}{1+(1-e^{-\theta })(M-1)-\beta }. \end{equation}
The ST externality is relatively small when the tightness is either small or large. For intermediate values of
$\theta$
, it is relatively large. Intuitively, when the tightness is low, there is a small chance that the marginal entrant would compete with another firm for the patent. Thus, the probability of securing a monopoly position is essentially the probability of being the sole inventor, both of which are close to
$1$
. When the tightness is large, there is little chance the marginal entrant will be the sole inventor. At the same time, the firm also expects to win the race with a small probability because the market is congested and she is likely to face many rivals. Thus, both probabilities are close to
$0$
. When the tightness is in an intermediate range, on the other hand, the entrant expects to see some competition for the patent but the expected number of rivals is not too high. This leaves her with a high chance of securing a patent even when she is not the sole inventor. This drives a wedge between the two probabilities and consequently increases
$\mathcal{ST}$
.
A higher market tightness, on the other hand, always leads to an increase in the magnitude of the SS externality. As
$\theta$
increases two opposing effects impact
$\mathcal{SS}$
. First, the chance that an entrant would be the sole inventor decreases. Thus, the probability that any new ideas would be added to the pool as a consequence of the firm’s entry decreases. This effect tends to reduce
$\mathcal{SS}$
. Second, a higher tightness implies that ideas are relatively more scarce and so the value of the new ideas generated by the entrant is larger. This effect tends to increase
$\mathcal{SS}$
. Given the equilibrium urn-ball matching technology, it turns out that the latter effect dominates the former. Hence, the magnitude of the SS externality is increasing in the market tightness.
Furthermore, we find that the magnitudes of the two externalities are such that
$\mathcal{ST}\gt \mathcal{SS}$
holds for all parameter values. To see this, first observe that along the BGP the planner’s allocation is characterized by
\begin{equation} \left (\frac{\nu }{N}\right )^{*}=M-1, \end{equation}
\begin{equation} \left (\frac{C}{N}\right )^{*}=\pi ^{*}-\eta \theta ^{*}(M-1), \end{equation}
\begin{equation} 1+\left(1-e^{-\theta ^{*}}\right)(M-1)=\beta \left (1+\frac{\pi ^{*}}{\eta }e^{-\theta ^{*}}+(1-e^{-\theta ^{*}}-\theta ^{*}e^{-\theta ^{*}})(M-1)\right ). \end{equation}
Then, from equations (14) and (22), it follows that
$\mathcal{ST}\gt \mathcal{SS}$
if and only if
$\pi ^{*}\!-\eta \theta ^{*} (M-1){\gt} 0$
. But this has to hold, from equation (21), as the efficient allocation must feature
$C_t\gt 0$
.
Proposition 6. The absolute value of the ST externality is larger than that of the SS externality.
The proof is in the preceding text. Intuitively, congestion in the economy is undesirable because it diverts resources away from consumption and into duplicative simultaneous innovation. Foregone innovation, on the other hand, is undesirable because it leaves some ideas uninnovated and, as a consequence, congestion tomorrow is higher than it could have been otherwise. Thus, reducing congestion is of first-order importance to the planner, whereas reducing foregone innovation only comes second. This intuition is also demonstrated by the behavior of the SS externality as we vary the market tightness. A higher tightness leads to a higher
$\mathcal{SS}$
because congestion is high (which makes ideas more valuable), even though foregone innovation (the chance the firm is the sole inventor) is low. Thus, when the planner picks an optimal market tightness she does so by striking a balance between congestion and foregone innovation in favor of low congestion. As a result, the ST externality dominates the SS one. Since the planner chooses a market tightness that delivers low congestion for any set of parameter values,
$\mathcal{ST}\gt \mathcal{SS}$
holds always.
We should note, however, that the results in Proposition 6 rely on our assumptions regarding the nature of competition between firms for innovations. Under alternative assumptions the size of the ST externality may be lower (or even zero), in which case it may be possible for the SS externality to dominate the ST one for some parameter values. First, our analysis focuses exclusively on the rivalrous nature of innovation. Thus, we have not considered any potential positive spill-over effects that might be present when several firms simultaneously innovate the same idea. For example, Gabrovski (Reference Gabrovski2020) considers a framework where ideas invented by more firms simultaneously contribute, on average, more new ideas to the future pool of ideas. In that context congestion decreases the cost of future innovation (in the same spirit as the SS externality) and, as a consequence, the size of the ST externality is lower. Second, we have assumed that firms receive patents for their innovations. In reality many firms rely instead on secrecy to protect their intellectual property (see, for example, Hall et al. (Reference Hall, Helmers, Rogers and Sena2014) and the references therein). Under the alternative assumption that firms protect their innovations through secrecy, the size of the ST externality would depend on the nature of competition between firms. For example, if we assume competition is such that firms which innovate simultaneously share the monopoly profits from the new variety, then the size of the ST externality would be the same as in our baseline setting with patents.Footnote
29
On the other hand, if we assume Bertrand competition firms only expect to receive positive profits if they are the sole inventor, i.e. with probability
$e^{-\theta }$
. Thus, competition among firms reduces the surplus each firm can extract from the innovation, so the ST externality disappears completely.Footnote
30
Since the ST externality dominates the SS one, implementing the efficient allocation in the DE requires the government to impose a tax on R&D investment.Footnote
31
In particular, suppose that the government imposes a subsidy on the purchases of intermediate varieties at a rate
$s$
and a tax on R&D activities at a rate
$\tau$
. Furthermore, if the government keeps a balanced budget through the means of lump-sum transfers, then the optimal policy is summarized below.
Proposition 7.
The optimal subsidy on the purchase of intermediate varieties is given by
$s^{*}=1-\lambda$
. The optimal tax rate on R&D entry is given by
\begin{align} \tau ^{*}=\frac{\beta \pi ^{*}(1-e^{-\theta ^{*}})}{\eta \theta ^{*}[1+(1-e^{-\theta ^{*}})(M-1)-\beta ]}-1\gt 0. \end{align}
A proof is in the appendix.
5. Conclusion
In practice firms have technological access to a scarce number of research avenues. They choose an avenue towards which to direct their R&D efforts but because of informational issues cannot coordinate their choices. Motivated by these observations, we develop an expanding-variety endogenous growth model to study the implications of coordination frictions for growth and welfare. Our analysis shows the frictions generate a number of foregone innovations and reduce the economy-wide research intensity. Both of these effects reduce the economy’s growth rate. This leads to a general equilibrium effect that endogenously amplifies the fraction of wasteful simultaneous innovation as compared to a frictional economy where firms can coordinate their R&D activities. When we turn our attention to welfare we find that formalizing the coordination frictions uncovers a novel link between the “stepping on toes” and “standing on shoulders” externalities. This link, which operates through the market tightness, determines the relative sizes of the two externalities. Either one may dominate the other in size, depending on the level of congestion in the economy. However, we show that at the efficient allocation the ST externality is larger for all parameter values. Thus, implementing the planner’s allocation requires imposing a tax on R&D investment.
APPENDIX: Proofs omitted from the text
Proof of Proposition 1
Proof. We follow the literature on coordination frictions in the labor market and treat our economy with a large number of ideas of ideas as the limiting case of a finite-idea economy [see, for example, Shi (Reference Shi2001), Shi (Reference Shi2005), Julien et al. (Reference Julien, Kennes and King2000), Julien et al. (Reference Julien, Kennes and Ritter2018), Gomis-Porqueras et al. (Reference Gomis-Porqueras, Julien and Wang2017), Shimer (Reference Shimer2005), Julien et al. (Reference Julien, Kennes and Ritter2014)]. In particular [as in Julien et al. (Reference Julien, Kennes and King2000) and Burdett et al. (Reference Burdett, Shi and Wright2001)], we characterize the equilibrium when the number of ideas and firms is finite and then evaluate the resulting equilibrium outcome in the limit as the number of projects tends to infinity (keeping the market tightness constant).
First, by assumption, the firm’s probability of securing a monopoly position given that there are exactly
$n$
rivals,
$Pr(\mathrm{monopoly}|n)=1/(n+1)$
. In a symmetric equilibrium all firms place the same probability
$s_i$
of directing their effort towards a particular idea
$i$
. Then, the chance that a firm would face exactly
$n$
rivals is
\begin{align*} Pr(n)=\binom{\mu _t-1}{n}s_i^n(1-s_i)^{\mu _t-1-n}. \end{align*}
Hence, the probability of securing a monopoly position is given by
\begin{align} Pr(\mathrm{monopoly})&=\sum _{n=0}^{\mu _t-1}Pr(\mathrm{monopoly}|n)P(n)=\sum _{n=0}^{\mu _t-1}\binom{\mu _t-1}{n}s_i^n(1-s_i)^{\mu _t-1-n}\frac{1}{n+1}\nonumber \\[3pt] &=\frac{1}{\mu _t}\sum _{n=0}^{\mu _t-1}\binom{\mu _t}{n+1}s_i^n(1-s_i)^{\mu _t-1-n}=\frac{1}{\mu _ts_i}\left (\sum _{n=0}^{\mu _t}\binom{\mu _t}{n}s_i^n(1-s_i)^{\mu _t-n}-(1-s_i)^{\mu _t}\right )\nonumber \\[3pt] &=\frac{1-(1-s_i)^{\mu _t}}{\mu _ts_i}. \end{align}
Next, we show that
$s_k=s_j$
for all
$k,j \in \nu _t$
. Suppose not. Then, there exists some
$k,j$
such that
$s_k\gt s_j$
. But for any
$i \in \nu _t$
, we have that
\begin{align} \frac{\partial Pr(\mathrm{monopoly})}{\partial s_i}=\frac{\mu _t^2s_i(1-s_i)^{\mu _t-1}-\mu _t[1-(1-s_i)^{\mu _t}]}{(\mu _ts_i)^2}. \end{align}
For any
$s_i \in (0,1)$
, it follows that
$Pr(\mathrm{monopoly})$
is decreasing in
$s_i$
if and only if
$(1-s_i)^{\mu _t-1}\lt Pr(\mathrm{monopoly})$
which clearly holds since
$\mu _t\geq 2$
. Now, for
$s_i=1$
, we have that
$\partial Pr(\mathrm{monopoly})/\partial s_i=-1/\mu _t\lt 0$
. Furthermore, it is easy to see that
$\lim _{s_i\rightarrow 0}\partial Pr(\mathrm{monopoly})/\partial s_i=-(\mu _t-1)/2\lt 0$
. Hence,
$Pr(\mathrm{monopoly})$
is decreasing in
$s_i$
everywhere in its domain. Then,
$s_k\gt s_j$
implies that
$Pr_k(\mathrm{monopoly})\lt Pr(\mathrm{monopoly})_j$
, which then implies that
$Pr_k(\mathrm{monopoly})V_{k,t}\lt Pr_j(\mathrm{monopoly})V_{j,t}$
since all varieties are equally profitable. Thus,
$s_k\gt s_j$
cannot be an equilibrium. Hence, we must have
$s_i=s_j=1/\nu _t$
for all
$i,j \in \nu _t$
. Finally, it is evident that no firm has an incentive to deviate from that strategy since the expected payoff from innovating an idea is the same for all research avenues.
Then, it follows that
\begin{align} Pr(i\ \mathrm{is\ matched\ with\ exactly}\ n\ \mathrm{firms})=\binom{\mu _t}{n}\left (\frac{1}{\nu _t}\right )^n\left (1-\frac{1}{\nu _t}\right )^{\mu _t-n}. \end{align}
Taking the limit as
$\mu _t, \nu _t \rightarrow \infty$
(keeping the ratio
$\theta _t$
constant) we get that
\begin{align} Pr(i\ \mathrm{is\ matched\ with\ exactly}\ n\ \mathrm{firms})\rightarrow \frac{\theta _t^n e^{-\theta _t}}{n!}. \end{align}
Proof of Proposition 2
Proof.
Totally differentiating both sides of (9) with respect to
$\pi$
yields
\begin{equation} \frac{\theta }{\pi }=\frac{\beta }{\eta }\left (\frac{1-e^{-\theta }}{\theta }\right )\left [e^{-\theta }(M-1)+\frac{\beta \pi }{\eta }\!\left (\frac{1-e^{-\theta }-\theta e^{-\theta }}{\theta ^2}\right )\right ]^{-1}\gt 0, \end{equation}
which is positive since
$1-e^{-\theta }-\theta e^{-\theta }\gt 0$
. As
$\frac{\theta }{A}=\frac{\theta }{\pi }\times \frac{\pi }{A}$
and profits,
$\pi$
, are increasing in productivity,
$A$
, it follows that
$\frac{\theta }{A}\gt 0$
. Similarly, totally differentiating (9) with respect to
$\beta$
,
$\eta$
, and
$M$
yields
\begin{align} \frac{\theta }{\beta }=\left [1+\frac{\pi }{\eta }\left (\frac{1-e^{-\theta }}{\theta }\right )\right ]\left [e^{-\theta }(M-1)+\frac{\beta \pi }{\eta }\!\left (\frac{1-e^{-\theta }-\theta e^{-\theta }}{\theta ^2}\right )\right ]^{-1}\gt 0, \end{align}
\begin{align}\frac{\theta }{\eta }=-\frac{\beta \pi }{\eta ^2}\left (\frac{1-e^{-\theta }}{\theta }\right )\Bigg]\left [e^{-\theta }(M-1)+\frac{\beta \pi }{\eta }\!\left (\frac{1-e^{-\theta }-\theta e^{-\theta }}{\theta ^2}\right )\right ]^{-1}\lt 0, \end{align}
\begin{align} \frac{\theta }{M}=-\left(1-e^{-\theta }\right)\left [e^{-\theta }(M-1)+\frac{\beta \pi }{\eta }\!\left (\frac{1-e^{-\theta }-\theta e^{-\theta }}{\theta ^2}\right )\right ]^{-1}\lt 0. \end{align}
Proof of Proposition 3
Proof.
Let us first explicitly characterize the BGP of the CE. The assumption we have placed on the parameter vales ensures that firms find all research avenues profitable. Hence, in equilibrium, all ideas are innovated, i.e.
$\mu _t\geq \nu _t$
, and each firm secures a patent with probability
$Pr(\mathrm{monopoly})=1/\theta _t$
. Hence, the laws of motion for ideas and varieties are given by
\begin{align} \nu _{t+1}=M\nu _t, \end{align}
\begin{align} N_{t+1}=N_t+\nu _t. \end{align}
Since the final good sector and the intermediate varieties production technology are as in the DE, it follows that in equilibrium it is still the case that
$P_{t,n}=1/\lambda$
,
$X=(\lambda ^2A)^{1/(1-\lambda )}L$
,
$Y_t=(\lambda ^{2\lambda }A)^{1/(1-\lambda )}LN_t$
,
$\pi =X(1-\lambda )/\lambda$
,
$V_t^c=\sum _{i=t+1}^{\infty }d_{it}\pi$
. As all ideas are equally productive, the free entry condition is now given by
$\eta =V_t^c/\theta _t$
. Moreover, consumers face the same problem as in the DE. Then, it is easy to see that
$g_Y=g_C=g_N=g_{\mu }=g_{\nu }$
. However, now from the law of motion for ideas, it follows that
$g_{\nu }=M-1$
. Next, using the laws of motion for ideas and varieties, it follows that along the BGP we still have,
$\nu/N=M-1$
. Furthermore, from the resource constraint, it follows that
$C/N=(1+\lambda )\pi/\lambda -\eta \theta ^c(M-1)$
. Lastly, using the free entry condition and the Euler equation, it follows that the market tightness is given by
$\theta ^c=\beta \pi/[\eta (M-\beta )]$
.
Next, we can compare the percent of wasteful innovations in the two economies. In the CE there are
$\mu _t$
innovations and
$\nu _t$
of those are beneficial. Hence,
$\omega ^c=1-1/\theta ^c$
. Then, it follows that
\begin{equation} \omega -\omega ^c=\frac{\eta (M-\beta )}{\beta \pi }-\frac{\eta (1+g-\beta )}{\beta \pi }=\frac{e^{-\theta }(M-1)\eta }{\beta \pi }\gt 0. \end{equation}
Proof of Proposition 4
Proof.
From
$\theta ^c=\beta \pi/[\eta (M-\beta )]$
it follows that
\begin{align} \frac{\theta ^c}{1-e^{-\theta }}=\frac{\beta \pi }{\eta (M-\beta )(1-e^{-\theta })}\gt \frac{\beta \pi }{\eta (1+(1-e^{-\theta })(M-1)-\beta )}=\frac{\theta }{1-e^{-\theta }}, \end{align}
where the inequality follows because
$\beta \lt 1 \Rightarrow 1+(1-e^{-\theta })(M-1)-\beta \gt (M-\beta )(1-e^{-\theta })$
. Hence,
$\theta ^c\gt \theta$
.
Proof of Proposition 5
Proof.
The results for the fraction of foregone innovation are immediate from Proposition 2. Next, let us look at the difference in the fraction of wasteful simultaneous innovation. Totally differentiating equation (10) with respect to
$\pi$
,
$\beta$
,
$\eta$
, and
$M$
yields
\begin{align} \frac{(\omega -\omega ^c)}{\pi }=-\frac{\omega -\omega ^c}{\pi }-(\omega -\omega ^c)\frac{\theta }{\pi }\lt 0, \end{align}
\begin{align} \frac{(\omega -\omega ^c)}{\beta }=-\frac{\omega -\omega ^c}{\beta }-(\omega -\omega ^c)\frac{\theta }{\beta }\lt 0, \end{align}
\begin{align} \frac{(\omega -\omega ^c)}{\eta }=\frac{\omega -\omega ^c}{\eta }-(\omega -\omega ^c)\frac{\theta }{\eta }\gt 0, \end{align}
\begin{align} \frac{(\omega -\omega ^c)}{M}=\frac{\eta e^{-\theta }}{\beta \pi }-(\omega -\omega ^c)\frac{\theta }{M}\gt 0. \end{align}
Furthermore,
$\frac{(\omega -\omega ^c)}{A}=\frac{(\omega -\omega ^c)}{\pi }\times \frac{\pi }{A}\gt 0$
.
Next, let us look in the difference in the market tightness. Given
$\theta ^c=\beta \pi/(\eta (M-\beta ))$
, it follows that
\begin{align} \frac{\theta ^c}{\pi }=\frac{\beta }{\eta (M-\beta )}, \end{align}
\begin{align} \frac{\theta ^c}{\eta }=-\frac{\beta \pi }{\eta ^2(M-\beta )}=-\left (\frac{\pi }{\eta }\right )\!\frac{\theta ^c}{\pi }, \end{align}
\begin{align} \frac{\theta ^c}{\beta }=\frac{M\pi }{\eta (M-\beta )^2}=\left (\frac{M\pi }{\beta (M-\beta )}\right )\!\frac{\theta ^c}{\pi }. \end{align}
Then, using equations (27) and (39) and straightforward algebra, it follows that
\begin{align} \frac{(\theta ^c-\theta )}{\pi }\lt 0. \end{align}
Hence,
$\frac{(\theta ^c-\theta )}{A}=\frac{(\theta ^c-\theta )}{\pi }\times \frac{\pi }{A}\lt 0$
.
Similarly, equations (29) and (40) imply that
\begin{align} \frac{(\theta ^c-\theta )}{\eta }=-\left (\frac{\pi }{\eta }\right )\frac{(\theta ^c-\theta )}{\pi }\gt 0. \end{align}
Lastly, equation (28) implies that
\begin{align} \frac{\theta }{\beta } &= \left (\frac{\eta \theta }{\beta (1-e^{-\theta })}\right )\left (1+\frac{\pi }{\eta }\left (\frac{1-e^{-\theta }}{\theta }\right )\right )\frac{\theta }{\pi },\nonumber\\[3pt]
\frac{\theta }{\beta } &= \left (\frac{\pi }{\beta }+\frac{\pi }{1-\beta +g}\right )\frac{\theta }{\pi }, \nonumber\\[3pt]
\frac{\theta }{\beta } &= \left (\frac{(1+g)\pi }{\beta (1-\beta +g)}\right )\frac{\theta }{\pi }. \end{align}
Since
$(1+g)\pi/(\beta (1-\beta +g))\gt M\pi/(\beta (M-\beta ))$
, it follows that
\begin{align} \frac{(\theta ^c-\theta )}{\beta }\lt \left (\frac{M\pi }{\beta (M-\beta )}\right )\frac{(\theta ^c-\theta )}{\pi }\lt 0. \end{align}
Proof of Proposition 7
Proof.
The government imposes a tax on R&D activities at a rate
$\tau$
and subsidizes the purchase of intermediate varieties at a rate
$s$
. Furthermore, it keeps a balanced budget through the means of lump-sum transfers to households in the amount
$T_t$
. Thus, the government’s budget constraint is given by
$T_t=\sum _{n=0}^{N_t}sP_{t,n}X_{t,n} - \tau \eta \mu _t$
.
The final good firm chooses labor and intermediate inputs to maximize profits, now given by
$Y_t-w_tL-\sum _{n=0}^{N_t}(1-s)P_{t,n}X_{t,n}$
. The first order conditions yield the same labor demand equation as in the DE.
At stage three of the innovation process, the monopolist faces an analogous problem as in the DE. The only difference now is in the inverse demand function. Hence, in equilibrium,
$P=1/\lambda$
,
$X=[A\lambda ^2/(1-s)]^{1/(1-\lambda )}L$
,
$\pi =(1-\lambda )X/\lambda$
,
$Y_t=[A(\lambda ^2/(1-s))^{\lambda }]^{1/(1-\lambda )}LN_t$
.
As in the economy without government intervention, all ideas are equally profitable, so the matching technology is as in the DE. The free entry condition is now given by
\begin{align} \eta (1+\tau )=\frac{1-e^{-\theta _t}}{\theta _t}V_t, \end{align}
where the value of the monopoly position,
$V_t$
, is defined as in the DE.
The laws of motion for ideas and varieties, and the Euler equation are as in the DE. Hence, along the BGP, we still have that
$\nu _t/N_t=M-1$
. Then it is easy to see that the solution for
$C/N$
and the tightness is given by
\begin{align} \frac{C}{N}=\frac{1-s-\lambda ^2}{(1-\lambda )\lambda }\pi -\eta \theta (M-1), \end{align}
\begin{align} 1+(1-e^{-\theta })(M-1)=\beta \left (1+\frac{\pi }{\eta (1+\tau )}\left (\frac{1-e^{-\theta }}{\theta }\right )\right ). \end{align}
Then, setting
$s=s^{*}$
implies that
$\pi =\pi ^{*}$
and setting
$\tau =\tau ^{*}$
implies that
$\theta =\theta ^{*}$
. Thus,
$C/N=(C/N)^{*}$
.
To see that the optimal tax rate is positive, observe that
$\eta \tau ^{*} = \mathcal{ST}^*-\mathcal{SS}^*$
, where the star subscript indicates the magnitude of the externalities is evaluated at the socially optimal level of the tightness. Then, by Proposition 6,
$\tau ^*\gt 0$
.
