The Lindahl equilibrium in Schumpeterian growth models

Abstract

The main motivation of the paper is to determine the social value of innovations in a standard scale-invariant Schumpeterian growth model, which explicitly introduces knowledge diffusion over a Salop (Bell J Econ 10(1):141–156 1979) circle. The social value of an innovation is defined as the optimal value of the knowledge inherent in this innovation. We thus have to price optimally knowledge. For that purpose, contrary to what is done in standard growth theory, we complete the markets using Lindahl prices for knowledge. The Lindahl equilibrium, which provides the system of prices that sustains the first-best social optimum in an economy with non rival goods, appears as a benchmark. First, its comparison with the standard Schumpeterian equilibrium à la Aghion and Howitt (Econometrica (60)2:323–351 1992) enables us to shed a new light on the issue of non-optimality of the latter. Second, the Lindahl equilibrium also allows us to revisit the issue of R&D incentives in presence of cumulative innovations. Finally, this benchmark may be a first step to understand how knowledge is exchanged in new technology sectors.

Appendix

1.1 A.1 Law of knowledge accumulation

1.1.1 A.1.1 Proof of Lemma 1

Consider any given sector ω, ω ∈ Ω, and a time interval (t, t + Δt). The level of knowledge at date t in this sector is χ _{ω t} . Let k, $k\in \mathbb {N}$, be the number of innovations that occur during the interval (t, t + Δt).

Under Assumptions 1 and 2, the level of knowledge at date t + Δt, χ _{ω t + Δt} , is a random variable taking the values $\left \{ \chi _{\omega t}+k\sigma \mathcal {P}_{\omega t}\right \}_{k\in \mathbb {N}}$ with associated probabilities $\left \{ \frac {\left ({\int }^{t+{\Delta } t}_{t}\lambda l^{}_{\omega u}du \right )^{k}}{k!}e^{-{\int }^{t+{\Delta } t}_{t}\lambda l^{}_{\omega u} du} \right \}_{k\in \mathbb {N}}$. Accordingly, the expected level of knowledge at date t + Δt is

$$\begin{array}{@{}rcl@{}} &&\mathbb{E}\left[ \chi_{\omega\: t+{\Delta} t} \right]=\sum\limits^{\infty}_{k=0}\frac{\left( {\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du \right)^{k}}{k!}e^{-{\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du}\left[ \chi_{\omega t}+k\sigma\mathcal{P}_{\omega t}\right] \\ &&=\left[ \chi_{\omega t}\sum\limits^{\infty}_{k=0}\frac{\left( {\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du \right)^{k}}{k!}+\sigma \mathcal{P}_{\omega t} \left( {\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du \right)\right.\\ &&\kern15pt\left.\times\sum\limits^{\infty}_{k=1}\frac{\left( {\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u}du \right)^{k-1}}{(k-1)!} \right]e^{-{\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du} \end{array} $$

The MacLaurin series ${\sum }^{K}_{k=0}\frac {\left ({\int }^{t+{\Delta } t}_{t}\lambda l^{}_{\omega u} du \right )^{k}}{k!}$ converges to $e^{{\int }^{t+{\Delta } t}_{t}\lambda l^{}_{\omega u} du}$ as K → ∞. Thus, one gets

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[ \chi_{\omega\: t+{\Delta} t} \right]\!\!&=&\!\!\left[ \chi_{\omega t}e^{{\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du}\,+\,\sigma \mathcal{P}_{\omega t}\!\left( {\int}^{t+{\Delta} t}_{t}\!\lambda l^{}_{\omega u} du \right)e^{{\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du}\! \right]e^{-{\int}^{t+{\Delta} t}_{t}\lambda l^{}_{\omega u} du}\\ &&\Leftrightarrow\mathbb{E}\left[ \chi_{\omega\: t+{\Delta} t} \right]=\chi_{\omega t}+\lambda\sigma \left( {\int}^{t+{\Delta} t}_{t} l^{}_{\omega u} du \right) \mathcal{P}_{\omega t} \end{array} $$

Let Λ _{ω u} denote a primitive of $l^{}_{\omega u}$ with respect to the time variable u. Rewriting the previous expression, one exhibits Newton’s difference quotients of $\mathbb {E}\left [ \chi _{\omega t} \right ]$ and of Λ _{ω t} :

$$\frac{\mathbb{E}\left[ \chi_{\omega \, t+{\Delta} t} \right]-\chi_{\omega t}}{\Delta t}=\lambda\sigma\frac{{\Lambda}_{\omega t+{\Delta} t}-{\Lambda}_{\omega t}}{\Delta t}\mathcal{P}_{\omega t} $$

Finally, letting Δt tend to zero, one gets $\frac {\partial \mathbb {E}\left [ \chi _{\omega t} \right ]}{\partial t}\equiv \dot \chi _{\omega t}=\lambda \sigma l^{}_{\omega t}\mathcal {P}_{\omega t}$. This proves that the expected knowledge in any sector ω is a differentiable function of time. Its derivative gives the law of motion of the expected knowledge as given in Lemma 1, in which the expectation operator is dropped to simplify notations.

1.1.2 A.1.2 Particular cases

The law of motion derived in Lemma 1 is quite general. Indeed, choosing particular specifications of the pools $\mathcal {P}_{\omega t}$ enables us to obtain several laws of knowledge accumulation commonly used in the fully endogenous Schumpeterian growth theory. We propose to classify the various models proposed in this literature in four main ranges, according to the considered pools of knowledge (i.e. the considered types of knowledge spillovers).

No knowledge spillovers (neither inter nor intra-sectoral knowledge spillovers)

In Barro and Sala-i-Martin (2003 - Ch. 6) or in Peretto (2007), for instance, the knowledge production technology uses final good only. In this extreme case, there are neither inter-sectorial nor intra-sectoral knowledge spillovers. A similar framework in which new knowledge is produced only with private inputs can also be considered using our formalization. Assume that $\mathcal {P}_{\omega t}=1$; accordingly, one has $\dot \chi _{\omega t}=\lambda \sigma l^{}_{\omega t},\forall \omega \in {\Omega }$. In this case, the only input used in the production of knowledge is labor.

Only intra-sectorial knowledge spillovers (no inter-sectoral knowledge spillovers)

In the models proposed by Grossman and Helpman (1991), Segerstrom (1998), Peretto (1999), Acemoglu (2009 - Ch. 14), or Aghion and Howitt (2009 - Ch. 4), among others, it is implicitly assumed that spillovers are only intra-sectoral (there are no spillovers across sectors): the pool of knowledge used in each sector comprises only the knowledge previously accumulated within this sector. This type of model can be obtained assuming that $\mathcal {P}_{\omega t}=\chi _{\omega t},\forall \omega \in {\Omega }$. One gets the following knowledge production functions $\dot \chi _{\omega t}=\lambda \sigma l^{}_{\omega t}\chi _{\omega t}, \forall \omega \in {\Omega }$.

Knowledge spillovers depending on average knowledge

The models of Aghion and Howitt (1998 - Ch. 12), Dinopoulos and Thompson (1998), Peretto (1998), Li (2003), among others, consider firm-specific knowledge production functions such that, as stated by Laincz and Peretto (2006), “spillovers depend on average knowledge”. Surveying this literature, these authors formalize this assumption in equation (9) of their paper. One can equivalently refer to equations (7) and (9) in Jones (1999), to equations (13) and (14) in Dinopoulos and Sener (2007), to equation (5) in Ha and Howitt (2007), or to the framework used in Aghion and Howitt (2009 - Ch. 4). Using our notations, this normalization assumption gives the following knowledge production function $\dot \chi _{\omega t}=\lambda \sigma l^{}_{\omega t}\mathcal {P}_{\omega t}$, where $\mathcal {P}_{\omega t}={\int }^{}_{\Omega }\frac {\chi _{h t}}{N}\:dh, \forall \omega \in {\Omega }$. Here, the new knowledge produced in any given sector depends on a knowledge aggregator, which is the average knowledge within the whole economy. This formalization has been introduced to remove the scale effect property while maintaining the endogenous ingredients of the seminal literature. Note, however, that it appears that the cases in which knowledge spillovers are only intra-sectoral and those in which they depend on average knowledge are closely related. Indeed, in both of these frameworks, there are no inter-sectoral knowledge spillovers: since one generally considers the symmetric case in which χ _{ω t} = χ _t ,∀ω ∈ Ω, one has $\mathcal {P}_{\omega t}=\frac {\chi _{t}}{N}{\int }^{}_{\Omega }dh=\chi _{t}, \forall \omega \in {\Omega }$. Scale effects are canceled by removing inter-sectoral knowledge diffusion.

Knowledge spillovers depending on the knowledge level of the frontier firms (“leading-edge technology”)

In the models of Aghion and Howitt (1992), Young (1998), Howitt (1999), Segerstrom (2000), or Garner (2010), among others, the increase in knowledge consecutive to the occurrence of an innovation in any given sector at date t depends on the level of knowledge reached in the most advanced sector. This type of framework can be directly obtained from our formalization. Indeed, assuming $\mathcal {P}_{\omega t}=\chi ^{max}_{t}$, where $\chi ^{max}_{t}\equiv \max \left \{\chi _{\omega t}, \omega \in {\Omega }\right \}$, one gets $\dot \chi _{\omega t}=\lambda \sigma l^{}_{\omega t} \chi ^{max}_{t}\:, \: \forall \omega \in {\Omega }$.

Global knowledge spillovers

A last range of models assumes that knowledge spillovers are global: each sector uses the whole disposable knowledge in the economy, that is $\mathcal {P}_{\omega t}={\int }^{}_{\Omega }\chi _{h t}\:dh=\mathcal {K}_{t}$. Accordingly, one gets the following knowledge production function:

$$ \dot \chi_{\omega t}=\lambda\sigma l^{}_{\omega t} \mathcal{K}_{t},\forall \omega\in{\Omega} $$

(22)

Comments on the law of knowledge accumulation (22) are given in Section 2 (see Section 2.1.2). In particular, we show how it relates to the ones originally introduced in the seminal papers of Romer (1990) and of Aghion and Howitt (1992). See the comments of the corollary to Proposition 1.

1.2 A.2 First-best social optimum - Proof of Proposition 2

The social planner maximizes the representative household’s discounted utility (3) subject to Eqs. 1, 2, 4, 5, 6 and 7. The maximization program can be written as follows:

$$\begin{array}{cc} \begin{array}{lllll} \text{Max} \mathcal{U}={\int}^{\infty}_{0}\ln(c_{t})e^{-\rho t}dt\, \text{subject to}\\ \{c_{t}\}_{t\in[0,\infty[}\\ \{{L^{Y}_{t}}\}_{t\in[0,\infty[}\\ \{l^{}_{\omega t}\}_{t\in[0,\infty[,\; \omega\in{\Omega}}\\ \{x_{\omega t}\}_{t\in[ 0,\infty[,\omega\in{\Omega}} \end{array} & \begin{array}{l} \left\{\begin{array}{lllll} \mathcal{K}_{t}={\int}_{\Omega}\chi_{\omega t}\:d\omega\\ \dot \chi_{\omega t}=\lambda\sigma l^{}_{\omega t}\mathcal{P}_{\omega t}\:, \: \omega\in{\Omega}\\ \mathcal{P}_{\omega t}={\int}^{}_{{\Omega}_{\omega}}\chi_{h t}\:dh \:, \: \forall \omega\in{\Omega}\\ L={L^{Y}_{t}}+{\int}_{\Omega}l^{}_{\omega t}\:d\omega \\ Y_{t}=({L^{Y}_{t}})^{1-\alpha}{\int}_{\Omega}\chi_{\omega t}(x_{\omega t})^{\alpha}d\omega \\ x_{\omega t}=\frac{y_{\omega t}}{\chi_{\omega t}},\; \omega\in{\Omega} \\ Y_{t}=Lc_{t}+{\int}_{\Omega}y_{\omega t}d\omega \end{array}\right. \end{array} \end{array} $$

where c _t , ${L^{Y}_{t}}$, $l^{}_{\omega t}$ and x _{ω t} , ω ∈ Ω, are the control variables, and χ _{ω t} , ω ∈ Ω, the continuum of state variables of the dynamic optimization problem.²³ We denote by ι _{ω t} , ω ∈ Ω, ν _t and μ _t , the co-state variables associated with the continuum of state variables, to the labor constraint, and to the final good resource constraint, respectively. After some rearrangement, one can write the Hamiltonian as:

$$\begin{array}{@{}rcl@{}} \mathcal{H}=\ln(c_{t})e^{-\rho t}&+&\mu_{t}\left[ ({L^{Y}_{t}})^{1-\alpha}{\int}_{\Omega}\chi_{\omega t}(x_{\omega t})^{\alpha}d\omega-Lc_{t}-{\int}_{\Omega}\chi_{\omega t}x_{\omega t}d\omega \right]\\ &+&\nu_{t}\left[L-{L^{Y}_{t}}-{\int}_{\Omega}l^{}_{\omega t}d\omega\right] + {\int}_{\Omega}\iota_{\omega t}\left[ \lambda\sigma l^{}_{\omega t}{\int}^{}_{{\Omega}_{\omega}}\chi_{h t}dh\right]d\omega \end{array} $$

The first-order conditions $\frac {\partial \mathcal {H}}{\partial c_{t}}=0$ , $\frac {\partial \mathcal {H}}{\partial {L^{Y}_{t}}}=0$, $\frac {\partial \mathcal {H}}{\partial l^{}_{i t}}=0 \: (i\in {\Omega })$, $\frac {\partial \mathcal {H}}{\partial x_{it}}=0 \: (i\in {\Omega })$ and $\frac {\partial \mathcal {H}}{\partial \chi _{it}}=-\dot {\iota _{i t}} \: (i\in {\Omega })$ respectively yield:²⁴

$$ c_{t}^{-1}e^{-\rho t}=\mu_{t}L $$

(23)

$$ \mu_{t}(1-\alpha)\frac{Y_{t}}{{L^{Y}_{t}}}=\nu_{t} $$

(24)

$$ \iota_{i t}\lambda\sigma{\int}^{}_{{\Omega}_{\omega}}\chi_{h t}dh=\nu_{t}, \forall i\in{\Omega} $$

(25)

$$ \mu_{t}\left[\alpha ({L^{Y}_{t}})^{1-\alpha}\chi_{it}(x_{it})^{\alpha-1}-\chi_{it}\right]=0\;,\forall i\in{\Omega} $$

(26)

$$ \mu_{t}\left[ ({L^{Y}_{t}})^{1-\alpha}(x_{i t})^{\alpha}-x_{it} \right] +\lambda\sigma{\int}^{}_{{\Omega}_{i}}\iota_{ht}l^{}_{ht}dh= -\dot{\iota_{it}},\forall i\in{\Omega} $$

(27)

From Eq. 26, one gets:

$$ x_{it}=x_{t}=\alpha^{\frac{1}{1-\alpha}}{L^{Y}_{t}}\;,\;\forall i\in{\Omega} $$

(28)

Plugging (28) in Eq. 5, and using the definition of the whole disposable knowledge in the economy (given by Eq. 1), one gets:

$$ Y_{t}=\alpha^{\frac{\alpha}{1-\alpha}}{L^{Y}_{t}} \mathcal{K}_{t} \text{ and thus } g_{Y_{t}}=g_{{L^{Y}_{t}}}+g_{\mathcal{K}_{t}} $$

(29)

Moreover, plugging (28) in the final good resource constraint, Eq. 7 becomes $Y_{t}=Lc_{t}+\alpha ^{\frac {1}{1-\alpha }} {L^{Y}_{t}}\mathcal {K}_{t}$. Dividing both sides of this expression by Y _t and using the previous expressions of x _t and Y _t (respectively given in Eqs. 28 and 29), one obtains L c _t /Y _t = 1 − α; this gives

$$ g_{c_{t}}=g_{Y_{t}} $$

(30)

Finally, the first-order conditions (24) and (27) become, respectively

$$\begin{array}{@{}rcl@{}} &&\mu_{t}(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}} \mathcal{K}_{t}=\nu_{t} \\ &&\text{ and } \frac{\mu_{t}}{\iota_{it}}(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}} {L^{Y}_{t}}+\lambda\sigma{\int}^{}_{{\Omega}_{i}}\frac{\iota_{ht}}{\iota_{it}}l^{}_{ht}dh=-g_{\iota_{it}},\forall i\in{\Omega} \end{array} $$

(31)

We now consider the usual symmetric case in which l _{ω t} = l _t and χ _{ω t} = χ _t ∀ω ∈ Ω. Accordingly, one has $\mathcal {P}_{\omega t}=\mathcal {P}_{t}=\theta \chi _{t},\forall \omega \in {\Omega }$, and thus the following expression of the growth rate of the stocks of knowledge:

$$ g_{\mathcal{K}_{t}}=g_{\chi_{t}}=\frac{\dot \chi_{t}}{\chi_{t}}=\lambda\sigma\theta l_{t} $$

(32)

Moreover, Eq. 25 becomes $\iota _{i t}\lambda \sigma \theta \mathcal {K}_{t}/N=\nu _{t}, \forall i\in {\Omega }$. Hence, one has ι _{i t} = ι _t , ∀i ∈ Ω. Using Eq. 31 and the labor resource constraint, one gets $\frac {\mu _{t}}{\iota _{t}}=\frac {\lambda \sigma \theta }{(1-\alpha )\alpha ^{\frac {\alpha }{1-\alpha }} N}$ and thus

$$ g_{\mu_{t}}=g_{\iota_{t}}=-\left( \frac{\lambda\sigma\theta {L^{Y}_{t}}}{N}+\lambda\sigma\theta l_{t} \right)=-\frac{\lambda\sigma\theta L}{N}=-\frac{\lambda\sigma\theta}{\gamma} $$

(33)

Furthermore, the labor constraint (4) is now

$$ {L^{Y}_{t}}+Nl^{}_{t}=L\Leftrightarrow {L^{Y}_{t}}+\gamma Ll^{}_{t}=L\Leftrightarrow l_{t}=\frac{1}{\gamma}\left( 1-\frac{{L^{Y}_{t}}}{L} \right) $$

(34)

Finally, log-differentiating (23) gives $g_{c_{t}}+\rho =-g_{\mu _{t}}$; using Eq. 33 allows us to derive the optimal growth rate of per-capita consumption:

$$ {g^{o}_{c}}=\frac{\lambda\sigma\theta}{\gamma}-\rho $$

(35)

The social optimum is completely characterized by Eqs. 28, 29, 30, 32, 34 and 35; and therefore by the following system of equations (the superscript “ ^o” is used for “social optimum”):

$$ \left\{\begin{array}{lllll} {g^{o}_{c}}=\frac{\lambda\sigma\theta}{\gamma}-\rho &(l1)\\ g^{o}_{c_{t}}=g^{o}_{Y_{t}}=g^{o}_{{L^{Y}_{t}}}+g^{o}_{\mathcal{K}_{t}}=g^{o}_{{L^{Y}_{t}}}+\lambda\sigma\theta {l^{o}_{t}} &(l2)\\ {l^{o}_{t}}=\frac{1}{\gamma}\left( 1-L^{Yo}_{t}/L \right) &(l3)\\ {x^{o}_{t}}=\alpha^{\frac{1}{1-\alpha}} L^{Yo}_{t} &(l4) \end{array}\right. $$

(36)

From (l1), (l2) and (l3) one gets

$$\frac{\lambda\sigma\theta}{\gamma}-\rho =g^{o}_{{L^{Y}_{t}}}+\lambda\sigma\theta \,{l^{o}_{t}}\Leftrightarrow g^{o}_{{L^{Y}_{t}}}=\frac{\lambda\sigma\theta L^{Yo}_{t}}{\gamma L} -\rho $$

In order to solve for $L^{Yo}_{t}$, we use a variable substitution. Let $X_{t}=1/L^{Yo}_{t}$; one gets the first-order linear differential equation $\dot X_{t}-\rho X_{t}=-\lambda \sigma \theta /N$. Its solution is

$$X_{t}=\left( X_{0}-\frac{\lambda\sigma\theta}{\rho N} \right)e^{\rho t}+\frac{\lambda\sigma\theta}{\rho N}\Leftrightarrow L^{Yo}_{t}=\frac{1}{\left( \frac{1}{L^{Yo}_{0}}-\frac{\lambda\sigma\theta}{\rho N} \right)e^{\rho t}+\frac{\lambda\sigma\theta}{\rho N}} $$

Using the transversality condition, it can be shown that $L^{Yo}_{t}$ immediately jumps to its steady-state level $L^{Yo^{ss}}=\rho N/\lambda \sigma \theta$. The transversality condition is only satisfied when $L^{Yo}_{t}=L^{Yo^{ss}},\forall t$. Thus, one has $g^{o}_{{L^{Y}_{t}}}=0$.

Finally, replacing $L^{Yo}_{t}$ in the system of Eq. 36 and using the assumption N = γ L, one obtains the characterization of the social optimum as exhibited in Proposition 2.

1.3 A.3 Schumpeterian equilibrium à la Aghion and Howitt (1992) and private value of innovations - Proof of Propositions 3 and 4

In this section, we provide the detailed analysis of a decentralized economy à laAghion and Howitt (1992), we fully characterize the set of equilibria as functions of the public tools vector (ψ, φ) and we compute the private value of innovations. As stated in Definition 2, at each vector (ψ, φ) is associated a particular Schumpeterian equilibrium, which consists of time paths of set of prices

$$\left\{ \left( w_{t}\left( \psi,\varphi\right),r_{t}\left( \psi,\varphi\right),\left\{q_{\omega t}\left( \psi,\varphi\right)\right\}_{\omega \in{\Omega}}\right) \right\}^{\infty}_{t=0} $$

and of quantities

$$\left\{ \!\left( c_{t}\!\left( \psi,\!\varphi\right)\!,\!Y_{t}\!\left( \psi,\!\varphi\right),\!\left\{x_{\omega t}\!\left( \psi,\!\varphi\right)\right\}_{\omega \in{\Omega}}\!,\!{L^{Y}_{t}}\!\!\left( \psi,\!\varphi\right),\!\left\{l_{\omega t}\!\left( \psi,\!\varphi\right)\right\}_{\omega\in{\Omega}}\!,\!\left\{\chi_{\omega t}\!\left( \psi,\varphi\right)\right\}_{\omega \in{\Omega}}\right)\! \right\}^{\infty}_{t=0} $$

such that: the representative household maximizes his utility; firms maximize their profits; the final good market, the financial market and the labor market are perfectly competitive and clear; on each intermediate good market, the innovator is granted an infinitely-lived patent and monopolizes the production and sale until replaced by the next innovator; and there is free entry on each R&D activity. For all the computations, in order to simplify notations, we drop the (ψ, φ) arguments for all variables.

The representative household maximizes his intertemporal utility given by Eq. 3subject to his budget constraint, $\dot b_{t}=w_{t}+r_{t}b_{t}-c_{t}-T_{t}/L$, where b _t denotes the per capita financial asset and T _t is a lump-sum tax charged by the government in order to finance public policies. This yields the usual Keynes-Ramsey condition:

$$ r_{t}=g_{c_{t}}+\rho $$

(37)

In the final sector, the competitive firm maximizes its profit given by ${\pi ^{Y}_{t}}=({L^{Y}_{t}})^{1-\alpha }{\int }_{\Omega }\chi _{\omega t}(x_{\omega t})^{\alpha }d\omega -w_{t}{L^{Y}_{t}}-{\int }_{\Omega }(1-\psi )q_{\omega t}x_{\omega t}d\omega$. The first-order conditions yield

$$ w_{t}=(1-\alpha)\frac{Y_{t}}{{L^{Y}_{t}}} \quad \text{and} \quad q_{\omega t}=\frac{\alpha({L^{Y}_{t}})^{1-\alpha}\chi_{\omega t}(x_{\omega t})^{\alpha-1}}{1-\psi} \;,\; \forall \omega \in{\Omega} $$

(38)

Consider any sector ω, ω ∈ Ω. Given the governmental intervention on behalf of R&D activities, the incumbent innovator, having successfully innovated at date t, receives at any date τ > t the net profit $\pi ^{x_{\omega }}_{\tau }=(1+\varphi )\left (q_{\omega t}x_{\omega t}-y_{\omega t}\right )$ with probability $e^{-{\int }^{\tau }_{t}\lambda l^{}_{\omega u} du}$. Differentiating (9) with respect to time gives the standard arbitrage condition in each R&D activity ω:

$$ r_{t}+\lambda l^{}_{\omega t}=\frac{\dot{\Pi}^{x}_{\omega t}}{{\Pi}^{x}_{\omega t}}+\frac{\pi^{x_{\omega}}_{t}}{{\Pi}^{x}_{\omega t}} \;,\; \forall \omega\in{\Omega} $$

(39)

In what follows, we focus on the case of unconstrained monopoly, that is, the case of drastic innovations. As shown below in the remark (see Eq. 57), innovations are drastic if and only if $\sigma \theta >1/\alpha ^{\frac {\alpha }{1-\alpha }}-1$. In each intermediate good sector ω, ω ∈ Ω, the incumbent monopoly maximizes the instantaneous net profit $\pi ^{x_{\omega }}_{t}$, where the demand for intermediate good ω, x _{ω t} , is given by Eq. 38 and the quantity of final good used in the production is y _{ω t} = x _{ω t} χ _{ω t} (see Eq. 6). Maximization gives the usual symmetric use of intermediate goods in the final good production and the usual mark-up on the price of intermediate goods:

$$ x_{\omega t}=x_{t}=\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{1}{1-\alpha}}{L^{Y}_{t}} \text{ and } q_{\omega t}=\frac{\chi_{\omega t}}{\alpha},\; \forall \omega \in{\Omega} $$

(40)

The monopoly profits now write:

$$\begin{array}{@{}rcl@{}} \pi^{x_{\omega}}_{\tau}&=&(1+\varphi)\left( q_{\omega t}x_{\omega t}-y_{\omega t}\right)=(1+\varphi)\left( \frac{1-\alpha}{\alpha}\right)\chi_{\omega t}x_{\omega t} \\ &=&(1+\varphi)\left( \frac{1-\alpha}{\alpha}\right)\chi_{\omega t}\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{1}{1-\alpha}}{L^{Y}_{t}},\forall \omega\in{\Omega} \end{array} $$

(41)

Together with the definition of the whole disposable knowledge in the economy (1), (40) allows us to rewrite the final good production function (5) and the wage expression given in Eq. 38, respectively, as

$$ Y_{t}=\left( \frac{\alpha^{2}}{1-\psi}\right)^{\frac{\alpha}{1-\alpha}}{L^{Y}_{t}}\mathcal{K}_{t} \text{ and } w_{t}=(1-\alpha)\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{\alpha}{1-\alpha}}\mathcal{K}_{t} $$

(42)

The free entry condition condition in any R&D activity ω is $w_{t}=\lambda {\Pi }^{x}_{\omega t}$, where $\lambda {\Pi }^{x}_{\omega t}$ is the expected revenue when one unit of labor is invested in R&D (from Assumption 1), and w _t is the cost of one unit of labor (given in Eq. 42). This condition gives the private value of an innovation in sector ω at date t, as defined in Eq. 9:

$$ {\Pi}^{x}_{\omega t}={{\Pi}^{x}_{t}}=\frac{(1-\alpha)}{\lambda}\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{\alpha}{1-\alpha}}\mathcal{K}_{t},\forall \omega\in{\Omega} $$

(43)

Consequently, one has $\dot {\Pi }^{x}_{\omega t}/{\Pi }^{x}_{\omega t}=g_{\mathcal {K}_{t}}$. Moreover, using Eqs. 41 and 43, one gets

$$ \frac{\pi^{x_{\omega}}_{t}}{{\Pi}^{x}_{\omega t}}=\frac{(1+\varphi)\left( \frac{1-\alpha}{\alpha}\right)\chi_{\omega t}\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{1}{1-\alpha}}{L^{Y}_{t}}}{\frac{(1-\alpha)}{\lambda}\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{\alpha}{1-\alpha}}\mathcal{K}_{t}}=\frac{(1+\varphi)\lambda\alpha\chi_{\omega t}{L^{Y}_{t}}}{(1-\psi)\mathcal{K}_{t}}, \forall \omega\in {\Omega} $$

(44)

The arbitrage condition (39) now writes

$$ r_{t}+\lambda l^{}_{\omega t}=g_{\mathcal{K}_{t}}+\frac{(1+\varphi)\lambda\alpha {L^{Y}_{t}}\chi_{\omega t}}{(1-\psi)\mathcal{K}_{t}}\;,\; \forall\omega\in{\Omega} $$

(45)

Log-differentiating with respect to time the expression of the final good production function given in Eq. 42 gives

$$ g_{Y_{t}}=g_{{L^{Y}_{t}}}+g_{\mathcal{K}_{t}} $$

(46)

Furthermore, using Eq. 1, 6 and 40, the final good resource constraint (7) can be rewritten as $Y_{t}=Lc_{t}+\left [\alpha ^{2}/(1-\psi )\right ]^{\frac {1}{1-\alpha }}{L^{Y}_{t}}\mathcal {K}_{t}$. Dividing both sides by Y _t and using the expression of Y _t given in Eq. 42, one gets L c _t /Y _t = 1 − α ²/(1 − ψ). Log-differentiating this expression gives

$$ g_{Y_{t}}=g_{c_{t}} $$

(47)

As usually in the standard literature, we focus on the symmetric equilibrium, in which l _{ω t} = l _t and χ _{ω t} = χ _t , ∀ω ∈ Ω. Consequently, one has $\mathcal {K}_{t}=N\chi _{t}$. Hence, the growth rate of the whole disposable knowledge is $g_{\mathcal {K}_{t}}=g_{\chi _{t}}$. Moreover, the pools of knowledge and the laws of accumulation of knowledge in each sector ω are respectively given by $\mathcal {P}_{\omega t}=\mathcal {P}_{t}=\theta \chi _{t}$ and $\dot \chi _{\omega t}=\dot \chi _{t}=\lambda \sigma \theta \, l_{t}\chi _{t}$, ∀ω ∈ Ω. Therefore, one has

$$ g_{\chi_{\omega t}}=g_{\chi_{t}}=g_{\mathcal{K}_{t}}=\lambda\sigma\theta \,l_{t}\;,\;\forall\omega\in{\Omega} $$

(48)

Finally, one can rewrite (45) - the arbitrage condition in any R&D activity ω, ω ∈ Ω - as follows:

$$ r_{t}+\lambda l^{}_{t}=\lambda\sigma\theta l_{t}+\frac{(1+\varphi)\lambda\alpha {L^{Y}_{t}}}{(1-\psi)N} $$

(49)

The equilibrium quantities, growth rates and prices are characterized by Eqs. 4, 37, 40, 42, 46, 47, 48 and 49:

$$ \left\{\begin{array}{lllll} L_{t}={L^{Y}_{t}}+Nl_{t}\\ r_{t}=g_{c_{t}}+\rho\\ x_{\omega t}=x_{t}=\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{1}{1-\alpha}}{L^{Y}_{t}} \quad \text{ and } \quad q_{\omega t}=\frac{\chi_{\omega t}}{\alpha}\quad,\; \forall \omega \in{\Omega}_{t}\\ Y_{t}=\left( \frac{\alpha^{2}}{1-\psi}\right)^{\frac{\alpha}{1-\alpha}}{L^{Y}_{t}}\mathcal{K}_{t}\text{ and } w_{t}=(1-\alpha)\left( \frac{\alpha^{2}}{1-\psi} \right)^{\frac{\alpha}{1-\alpha}}\mathcal{K}_{t}\\ g_{Y_{t}}=g_{{L^{Y}_{t}}}+g_{\mathcal{K}_{t}}\\ g_{Y_{t}}=g_{c_{t}}\\ g_{\chi_{\omega t}}=g_{\chi_{t}}=g_{\mathcal{K}_{t}}=\lambda\sigma\theta l_{t}\;,\;\forall\omega\in{\Omega}\\ r_{t}+\lambda l^{}_{ t}=\lambda\sigma\theta l_{t}+\frac{(1+\varphi)\lambda\alpha {L^{Y}_{t}}}{(1-\psi)N} \end{array}\right. $$

(50)

From Eqs. 37 and 49, one gets $g_{c_{t}}+\rho +\lambda l^{}_{t}=\lambda \sigma \theta l_{t}+\frac {(1+\varphi )\lambda \alpha {L^{Y}_{t}}}{(1-\psi )N}$. From Eqs. 46, 47 and 48, one gets $g_{c_{t}}=g_{Y_{t}}=g_{{L^{Y}_{t}}}+g_{\chi _{t}}=g_{{L^{Y}_{t}}}+\lambda \sigma \theta \, l_{t}$. Using the labor constraint (4) and the assumption N = γ L together with these two expressions, one gets the following differential equation in ${L^{Y}_{t}}$.

$$ g_{{L^{Y}_{t}}}-\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right]L^{Y}_{t}=-\left( \frac{\lambda}{\gamma}+\rho \right) $$

(51)

We use the variable substitution $X_{t}=1/{L^{Y}_{t}}$. Log-differentiation with respect to time writes $g_{X_{t}}=-g_{{L^{Y}_{t}}}$. Substituting into (51) gives

$$\begin{array}{@{}rcl@{}} -g_{X_{t}}-\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right]\frac{1}{X_{t}}&=&-\left( \frac{\lambda}{\gamma}+\rho\right)\\ &\Leftrightarrow& \dot X_{t}-\left( \frac{\lambda}{\gamma}+\rho \right)X_{t}=-\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right] \end{array} $$

The solution of this first-order linear differential equation is

$$X_{t}=e^{\left( \frac{\lambda}{\gamma}+\rho \right) t}\left( X_{0}-\frac{ 1}{\frac{\lambda}{\gamma}+\rho}\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right] \right)+\frac{ 1}{\frac{\lambda}{\gamma}+\rho}\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right] $$

Accordingly, one obtains

$${L^{Y}_{t}}=\frac{1}{e^{\left( \frac{\lambda}{\gamma}+\rho \right) t}\left( \frac{1}{{L^{Y}_{0}}}-\frac{ 1}{\frac{\lambda}{\gamma}+\rho}\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right]\right)+\frac{1}{\frac{\lambda}{\gamma}+\rho}\frac{\lambda}{\gamma L}\left[ 1+\frac{1+\varphi}{1-\psi}\alpha \right]} $$

Using the transversality condition in the program of the representative household, we can show that ${L^{Y}_{t}}$ immediately jumps to its steady-state level $L^{Y^{ss}}=\left (\frac {\lambda }{\gamma }+\rho \right )/\left (\frac {\lambda }{\gamma L}\left [ 1+\frac {1+\varphi }{1-\psi }\alpha \right ]\right )$. The transversality condition is only satisfied when ${L^{Y}_{t}}={L^{Y}_{0}}=\gamma L\left (\frac {\lambda }{\gamma }+\rho \right )/\left (\lambda \left [ 1+\frac {1+\varphi }{1-\psi }\alpha \right ]\right ), \forall t$. Thus, one has $g_{{L^{Y}_{t}}}=0$.

Substituting into the system (50), one proves Propositions 3 and 4, in which we provide the complete characterization of the decentralized Schumpeterian equilibrium and the private value of an innovation, respectively.

1.4 A.4 Remark: drastic versus nondrastic innovations

In the final sector, the competitive firm cost minimization program is

$$\begin{array}{lllll} \text{Min} \mathcal{C}=w_{t}{L^{Y}_{t}}+{\int}_{\Omega}(1-\psi)q_{\omega t}x_{\omega t}d\omega\,\,\text{subject to}\,\, Y_{t}=({L^{Y}_{t}})^{1-\alpha}{\int}_{\Omega}\chi_{\omega t}(x_{\omega t})^{\alpha}d\omega\\ \{{L^{Y}_{t}}\},\{x_{\omega t}\}_{\omega\in{\Omega}} \end{array} $$

The first-order conditions with respect to ${L^{Y}_{t}}$ and x _{ω t} give respectively:

$$ {L^{Y}_{t}}=\frac{\eta(1-\alpha)Y_{t}}{w_{t}} $$

(52)

$$ \text{and }\left( 1-\psi \right)q_{\omega t}=\eta\alpha({L^{Y}_{t}})^{1-\alpha}\chi_{\omega t}(x_{\omega t})^{\alpha-1},\forall\omega\in{\Omega} $$

(53)

Furthermore, in the symmetric case, one has

$$ Y_{t}=({L^{Y}_{t}})^{1-\alpha}N\chi_{t}(x_{t})^{\alpha} $$

(54)

Then, Eq. 53 writes

$$ x_{t}=\frac{\eta\alpha({L^{Y}_{t}})^{1-\alpha}N\chi_{t}(x_{t})^{\alpha}}{N(1-\psi) q_{t}}=\frac{\eta\alpha Y_{t}}{N(1-\psi) q_{t}} $$

(55)

From Eqs. 52 and 55, one gets ${L^{Y}_{t}}/x_{t}=(1-\alpha )N(1-\psi )q_{t}/\alpha w_{t}$; together with Eq. 54 one gets the following expressions of ${L^{Y}_{t}}$ and x _t :

$${L^{Y}_{t}}=\frac{Y_{t}}{N\chi_{t}}\left( \frac{(1-\alpha)N(1-\psi)q_{t}}{\alpha w_{t}}\right)^{\alpha} \text{ and } x_{t}=\frac{Y_{t}}{N\chi_{t}}\left( \frac{(1-\alpha)N(1-\psi)q_{t}}{\alpha w_{t}}\right)^{\alpha-1} $$

Plugging in the total cost, $\mathcal {C}=w_{t}{L^{Y}_{t}}+{\int }_{\Omega }(1-\psi )q_{\omega t}x_{\omega t}d\omega$, one gets $\mathcal {C}=\frac {Y_{t}}{N\chi _{t}}\left [\frac {(1-\alpha )N}{\alpha }\right ]^{\alpha }\left [(1-\psi )q_{t}\right ]^{\alpha }\left (w_{t} \right )^{1-\alpha }+N(1-\psi )q_{t}\frac {Y_{t}}{N\chi _{t}}\left [\frac {(1-\alpha )N(1-\psi )q_{t}}{\alpha w_{t}}\right ]^{\alpha -1}$. Finally, the cost function of the final sector is

$$ \mathcal{C}\left( Y_{t},\chi_{t},q_{t},w_{t} \right)=\frac{N^{\alpha-1}}{1-\alpha}\left( \frac{1-\alpha}{\alpha }\right)^{\alpha}\frac{Y_{t}}{\chi_{t}}\left[(1-\psi)q_{t}\right]^{\alpha}\left( w_{t} \right)^{1-\alpha} $$

(56)

Consider any given sector ω, ω ∈ Ω. If an innovation occurs at date t, the quality reached by its producer (i.e. the new level of knowledge in this sector attained by the latest innovator) is $\chi _{\omega t}+{\Delta }\chi _{\omega t}=\chi _{\omega t}+\sigma \mathcal {P}_{\omega t}$, where χ _{ω t} is the quality reached by the previous innovator (i.e. the level of knowledge achieved through the previous innovations in this sector). In the symmetric case, one has χ _{ω t} = χ _t , and thus $\mathcal {P}_{\omega t}=\theta \chi _{t},\forall \omega \in {\Omega }$; then, the quality attained by the latest innovator writes (1 + σ θ)χ _t . The final sector can buy the intermediate good from the latest innovator at the marked-up price $q_{t}=\frac {\left (1+\sigma \theta \right )\chi _{t}}{\alpha }$, or from the previous innovator at the competitive price q _t = χ _t . In the former case, the cost function is $\mathcal {C}\left (Y_{t},\left (1+\sigma \theta \right )\chi _{t},q_{t}=\frac {\left (1+\sigma \theta \right )\chi _{t}}{\alpha },w_{t} \right )$; in the latter case, it is $\mathcal {C}\left (Y_{t},\chi _{t},q_{t}=\chi _{t},w_{t} \right )$. The innovation is drastic if and only if

$$\begin{array}{@{}rcl@{}} &&\mathcal{C}\left( Y_{t},\left( 1+\sigma\theta\right)\chi_{t},q_{t}=\frac{\left( 1+\sigma\theta\right)\chi_{t}}{\alpha},w_{t} \right)<\mathcal{C}\left( Y_{t},\chi_{t},q_{t}=\chi_{t},w_{t} \right) \\ &&\Leftrightarrow \frac{N^{\alpha-1}}{1-\alpha}\left( \frac{1-\alpha}{\alpha }\right)^{\alpha}\frac{Y_{t}}{\left( 1+\sigma\theta\right)\chi_{t}}\left[(1-\psi)\frac{\left( 1+\sigma\theta\right)\chi_{t}}{\alpha}\right]^{\alpha}\left( w_{t} \right)^{1-\alpha}\\ &&<\frac{N^{\alpha-1}}{1-\alpha}\left( \frac{1-\alpha}{\alpha }\right)^{\alpha}\frac{Y_{t}}{\chi_{t}}\left[(1-\psi)\chi_{t}\right]^{\alpha}\left( w_{t} \right)^{1-\alpha} \\ &&\Leftrightarrow\sigma\theta> \frac{1}{\alpha^{\frac{\alpha}{1-\alpha}}}-1 \end{array} $$

(57)

1.5 A.5 Lindahl equilibrium and social value of innovations - Proof of Propositions 5 and 6

The representative household maximizes his intertemporal utility given by Eq. 3 subject to his budget constraint, $\dot b_{t}=w_{t}+r_{t}b_{t}-c_{t}-T_{t}/L$, where b _t denotes the per capita financial asset and T _t is a lump-sum tax charged by the government in order to finance public policies and fund R&D expenditures. This yields the usual Keynes-Ramsey condition:

$$ r_{t}=g_{c_{t}}+\rho $$

(58)

Differentiating (11) with respect to time gives an arbitrage condition stating that the rate of return is the same on the financial market and on any R&D investment:

$$ r_{t}=\frac{v_{\omega t}}{V_{\omega t}}+\frac{\dot V_{\omega t}}{V_{\omega t}},\forall \omega\in{\Omega} $$

(59)

As usual in the standard literature, we focus on a symmetric equilibrium in which l _{ω t} = l _t and χ _{ω t} = χ _t , ∀ω ∈ Ω. Consequently, since N = γ L, one has $\mathcal {K}_{t}=N\chi _{t}=\gamma L\chi _{t}$ and $L={L^{Y}_{t}}+\gamma Ll_{t}$. The pools of knowledge and the laws of accumulation of knowledge in each sector ω are now respectively given by $\mathcal {P}_{\omega t}=\mathcal {P}_{t}=\theta \chi _{t}$ and $\dot \chi _{\omega t}=\dot \chi _{t}=\lambda \sigma \theta l_{t}\chi _{t}$, ∀ω ∈ Ω. Therefore, one has

$$ g_{\chi_{\omega t}}=g_{\chi_{t}}=g_{\mathcal{K}_{t}}=\lambda\sigma\theta l_{t},\forall\omega\in{\Omega} $$

(60)

Using Eqs. 1, 6, and 14, the final good production function (5), the wage (12), and the final good resource constraint (7) can be rewritten respectively as

$$ Y_{t}=\alpha^{\frac{\alpha}{1-\alpha}}{L^{Y}_{t}}\mathcal{K}_{t},\;w_{t}=(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}}\mathcal{K}_{t} \text{ and }Y_{t}=Lc_{t}+\alpha^{\frac{1}{1-\alpha}}{L^{Y}_{t}}\mathcal{K}_{t} $$

(61)

Dividing both sides of the expression of the final good resource constraint given in Eq. 61 by Y _t , one gets L c _t /Y _t = 1 − α. Log-differentiating with respect to time this expression as well as the final good production function given in Eq. 61, one gets

$$ g_{c_{t}}=g_{Y_{t}}=g_{{L^{Y}_{t}}}+g_{\mathcal{K}_{t}} $$

(62)

From Eqs. 16 and refsocialvalue.ProductionFinalGoodCOMPACT, one gets the following social value of one unit of knowledge χ _{h t} :

$$ V_{ht}=V_{t}=\frac{(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}}\gamma L}{\lambda\sigma \theta},\forall h\in{\Omega} $$

(63)

Using Eqs. 14, 63, the marginal profitabilities of knowledge given in Lemma 3 can be rewritten as follows:

$$v^{Y}_{\omega t}=\alpha^{\frac{\alpha}{1-\alpha}}{L^{Y}_{t}}\text{, }v^{x}_{\omega t}\,=\,-\alpha^{\frac{1}{1-\alpha}}{L^{Y}_{t}}\text{ and } v^{\chi_{h}}_{\omega t}=\left\{\begin{array}{lllll}\frac{(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}}\left( L-{L^{Y}_{t}} \right)}{\theta} \text{ if } h\in\mathcal{D}_{\omega}\\ 0 \text{ if } h\notin\mathcal{D}_{\omega} \end{array}\right.,\forall\omega\in{\Omega} $$

Accordingly, the instantaneous social value of one unit of knowledge χ _{ω t} at date t is

$$ v_{\omega t}=v^{Y}_{\omega t}+v^{x}_{\omega t}+{\int}_{\Omega}v^{\chi_{h}}_{\omega t}dh=(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}}L,\forall\omega\in{\Omega} $$

(64)

From Eqs. 63 and 64, one has $\dot V_{\omega t}/V_{\omega t}=0$ and v _{ω t} /V _{ω t} = λ σ θ/γ, ∀ω ∈ Ω. Thus, from the arbitrage condition (59), one obtains the equilibrium interest rate:

$$ r_{t}=\frac{\lambda\sigma \theta}{\gamma} $$

(65)

The partition of labor and the growth rates are characterized by Eqs. 4, 58, 60, 62 and 65. From Eq. 58, 62 and 65, one gets $g_{c_{t}}=g_{Y_{t}}=g_{{L^{Y}_{t}}}+g_{\mathcal {K}_{t}}=\lambda \sigma \theta /\gamma -\rho$; and from Eqs. 4 and 60, one gets $g_{\mathcal {K}_{t}}=g_{\chi _{t}}=\lambda \sigma \theta \left (1/\gamma -{L^{Y}_{t}}/\gamma L\right )$. From these two expressions, one obtains $g_{{L^{Y}_{t}}}-\frac {\lambda \sigma \theta }{\gamma L}{L^{Y}_{t}}=-\rho$. Using the variable substitution $X_{t}=1/{L^{Y}_{t}}$, one gets a first-order linear differential equation:

$$g_{X_{t}}+\frac{\lambda\sigma\theta}{\gamma L}\frac{1}{X_{t}}=\rho\Leftrightarrow\dot X_{t}-\rho X_{t}=-\frac{\lambda\sigma\theta}{\gamma L} $$

Its solution is $X_{t}=e^{\rho t}\left (X_{0}-\frac {\lambda \sigma \theta }{\rho \gamma L}\right )+\frac {\lambda \sigma \theta }{\rho \gamma L}$. Hence, one obtains

$${L^{Y}_{t}}=\frac{1}{e^{\rho t}\left( \frac{1}{{L^{Y}_{0}}}-\frac{\lambda\sigma\theta}{\rho\gamma L}\right)+\frac{\lambda\sigma\theta}{\rho\gamma L}} $$

Using the transversality condition of the program of the representative household, it can be shown that ${L^{Y}_{t}}$ immediately jumps to its steady-state level $L^{Y^{ss}}=\rho \gamma L/\lambda \sigma \theta$. The transversality condition is only satisfied when ${L^{Y}_{t}}={L^{Y}_{0}},\forall t$. Hence, one has ${L^{Y}_{t}}=\rho \gamma L/\lambda \sigma \theta =L^{Yo},\forall t$, and thus $g^{o}_{{L^{Y}_{t}}}=0$.

Thus, the partition of labor, the quantities of intermediate good, the quantities of knowledge and the growth rates are

$$\begin{array}{@{}rcl@{}} {L^{Y}_{t}}=L^{Yo}=\frac{\rho\gamma L }{\lambda\sigma\theta};\ l_{\omega t}=l^{o}=\frac{1}{\gamma}-\frac{\rho }{\lambda\sigma\theta},\forall\omega\in{\Omega}; \\x^{}_{\omega t}=x^{o}=\alpha^{\frac{1}{1-\alpha}}\frac{\rho\gamma L }{\lambda\sigma\theta},\forall\omega\in{\Omega}; \\ \chi_{\omega t}={\chi^{o}_{t}}=\frac{\mathcal{K}^{o}_{t}}{\gamma L},\forall\omega\in{\Omega};\ \mathcal{K}^{o}_{t}=e^{g^{o}t}; \\\text{and } g_{c_{t}}=g_{Y_{t}}=g_{\mathcal{K}_{t}}=g_{\chi_{\omega t}}=g^{o}=\frac{\lambda\sigma\theta}{\gamma}-\rho,\forall\omega\in{\Omega} \end{array} $$

(66)

This proves that the quantities and growth rates computed in the Lindahl equilibrium are indeed those of the first-best social optimum. The system of prices is as follows.

• The prices of rival goods are ${w^{o}_{t}}=(1-\alpha )\alpha ^{\frac {\alpha }{1-\alpha }}\mathcal {K}^{o}_{t}$; ${r^{o}_{t}}=\frac {\lambda \sigma \theta }{\gamma }$; $q^{o}_{\omega t}={q^{o}_{t}}={\chi ^{o}_{t}}=\frac {\mathcal {K}^{o}_{t}}{\gamma L},\forall \omega \in {\Omega }$.

• Regarding the pricing of knowledge, one has the following results.

  • The personalized prices (Lindahl prices) of one unit of knowledge χ _{ω t} for the final good sector, the intermediate sector ω, and R&D sector h, h ∈ Ω, are $v^{Yo}_{\omega t}=\alpha ^{\frac {\alpha }{1-\alpha }}\frac {\rho \gamma L }{\lambda \sigma \theta },\forall \omega \in {\Omega }$; $v^{xo}_{\omega t}=-\alpha ^{\frac {1}{1-\alpha }}\frac {\rho \gamma L }{\lambda \sigma \theta },\forall \omega \in {\Omega }$; and $v^{\chi _{h}o}_{\omega t}=\left \{\begin {array}{lllll}\frac {(1-\alpha )\alpha ^{\frac {\alpha }{1-\alpha }}}{\theta }\left (L-\frac {\rho \gamma L }{\lambda \sigma \theta }\right ), \text {if } h\in \mathcal {D}_{\omega }\\ 0, \text {if } h\notin \mathcal {D}_{\omega } \end {array}\right .,\forall \omega \in {\Omega }$.

  • The instantaneous income received by the producer of one unit of knowledge χ _{ω t} is $v^{o}_{\omega t}=v^{Yo}_{\omega t}+v^{xo}_{\omega t}+v^{R\&Do}_{\omega t}=v^{o}=(1-\alpha )\alpha ^{\frac {\alpha }{1-\alpha }}L,\forall \omega \in {\Omega }$, where $v^{R\&Do}_{\omega t}={\int }_{\Omega }v^{\chi _{h}o}_{\omega t}dh=(1-\alpha )\alpha ^{\frac {\alpha }{1-\alpha }}\left (L-\frac {\rho \gamma L }{\lambda \sigma \theta }\right ),\forall \omega \in {\Omega }$.

This proves Proposition 5. Finally, as seen in Eq. 11, an innovation consists in an increase in knowledge of ${\Delta }\chi _{\omega t}=\sigma \mathcal {P}_{\omega t}$ new units; moreover, from Eq. 63, the social value of one unit of knowledge χ _{ω t} at date t is $V^{o}_{\omega t}={\int }^{\infty }_{t}v^{o}_{\omega t}e^{-{{\int }^{s}_{t}}r_{u}du}ds=(1-\alpha )\alpha ^{\frac {\alpha }{1-\alpha }}\gamma L/\lambda \sigma \theta$. The social value of an innovation in any sector ω, ω ∈ Ω, is thus $\mathcal {V}^{o}_{\omega t}=\sigma \mathcal {P}^{o}_{\omega t}V^{o}_{\omega t}$, where $\mathcal {P}^{o}_{\omega t}=\theta \chi ^{o}_{\omega t}=\theta \mathcal {K}^{o}_{t}/\gamma L$. Finally, one gets

$$ \mathcal{V}^{o}_{\omega t}=\frac{(1-\alpha)\alpha^{\frac{\alpha}{1-\alpha}} }{\lambda}\mathcal{K}^{o}_{t},\forall\omega\in{\Omega} $$

(67)

This proves Proposition 6.