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Endogenous growth in production networks

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We investigate the interplay between technological change and macro- economic dynamics in an agent-based model of the formation of production networks. On the one hand, production networks form the structure that determines economic dynamics in the short run. On the other hand, their evolution reflects the long-term impacts of competition and innovation on the economy. We account for process innovation via increasing variety in the input mix and hence increasing connectivity in the network. In turn, product innovation induces a direct growth of the firm’s productivity and the potential destruction of links. The interplay between both processes generates complex technological dynamics in which phases of process and product innovation successively dominate. The model reproduces a wealth of stylized facts about industrial dynamics and technological progress, in particular the persistence of heterogeneity among firms and Wright’s law for the growth of productivity within a technological paradigm. We illustrate the potential of the model for the analysis of industrial policy via a preliminary set of policy experiments in which we investigate the impact on innovators’ success of feed-in tariffs and of priority market access.

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  1. Here and in the following, we always consider implicitly that the vector (μinc,μim,μrad) is normalized so that μrad + μinc + μim = 1

  2. A cautionary note in this respect is that, as argued by Lamperti et al. (2015), market-based policies may not be sufficient to prevent environmental disasters while Command-and- Control policies are fully effective.

  3. In fact, everything goes as if the economy were receiving an external subsidy.


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This study was funded by the European Commission through the FP7 project IMPRESSIONS (603416) and the H2020 project Dolfins (640772), and through the Agence Nationale de la Recherche via Labex Louis Bachelier (ANR 11-LABX-0019) and Labex OSE (ANR-10-LABX-93-01).

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Correspondence to Antoine Mandel.

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Appendix A: General equilibrium and production networks

Appendix A: General equilibrium and production networks

1.1 A.1 A general equilibrium economy

One can associate to a production network A a general equilibrium economy as follows:

Definition 1

The general equilibrium economy \(\mathcal {E}(A)\)is defined by the given of:

  • A representative household supplying one unit of labor andhaving preferences represented by the Cobb-Douglas utility function\(u(x_{1},\cdots ,x_{m})=\prod _{i = 1}^{m} x_{i}^{\alpha _{0,i}}\)

  • A set of firms M with production functions of the form

    $$ f_{i}(x_{0},(x_{j})_{j = 1,\cdots,n_{i}})= x_{0}^{\alpha} \left( \sum\limits_{j = 1}^{n_{i}} x_{j}^{\theta}\right)^{(1-\alpha)/\theta} $$
  • A production network A consistent with Eq. 6in the sense that for alliM,\(\sum _{j = 1}^{m} a_{i,j}=n_{i}\).

1.2 A.2 Out-of-equilibrium dynamics

For a given network, the dynamics of prices and output follow from the application of simple behavioral rules. More precisely, given a production network A the dynamics of wealth \({w^{t}_{i}} \in {\mathbb {R}}_{+}\), output \({q^{t}_{i}} \in {\mathbb {R}}_{+}\) and prices \({p^{t}_{i}} \in {\mathbb {R}}_{+}\) within period t are determined as follows.

  1. 1.

    Each agent i receives the nominal demand \(\sum _{j \in N} \alpha ^{t}_{i,j} {w^{t}_{j}}\), which is implied by the current structure of the supply network

  2. 2.

    Agents adjust their prices frictionally toward their market-clearing values according to:

    $$ {p_{i}^{t}}=\tau_{p} \overline{p}_{i}^{t}+ (1-\tau_{p}) p^{t-1}_{i} $$

    where τp ∈ [0, 1] is a parameter measuring the speed of price adjustment and, given the nominal demand \(\sum _{j \in N} \alpha ^{t}_{i,j} {w^{t}_{j}}\) and the output stock \({q_{i}^{t}}\), \(\overline {p}_{i}^{t}\) is the market clearing price for firm i, that is:

    $$ \overline{p}_{i}^{t}=\frac{\sum_{j \in N} \alpha^{t}_{i,j} {w^{t}_{j}}}{{q_{i}^{t}}}. $$
  3. 3.

    Whenever τp < 1 markets do not clear (except if the system is at a stationary equilibrium). In case of excess demand, we assume that clients are rationed proportionally to their demand. In case of excess supply, we assume that the amount \(\overline {q}^{t}_{i}:=\sum _{j \in N} \alpha ^{t}_{i,j} {w^{t}_{j}}/p_{i}^{t}\) is actually sold and that the rest of the output is stored as inventory. Together with production occurring on the basis of purchased inputs, this yields the following evolution of the product stock:

    $$ q^{t + 1}_{i}={q^{t}_{i}}-\overline{q}^{t}_{i} + f_{i}\left( \frac{\alpha^{t}_{0,i}{w^{t}_{i}}}{{p_{0}^{t}}}, \left( \frac{\alpha^{t}_{j,i}{w^{t}_{i}}}{{p_{j}^{t}}}\right)_{j \in \Sigma_{i}(A^{t})}\right) $$

    Note that in the case where τp = 1, markets always clear (one has \(\overline {q}^{t}_{i} = {q^{t}_{i}}\)) and Eq. 9 reduces to

    $$ q^{t + 1}_{i}= f_{i}\left( \frac{\alpha^{t}_{0,i}{w^{t}_{i}}}{{p_{0}^{t}}},\left( \frac{\alpha^{t}_{j,i}{w^{t}_{i}}}{{p_{j}^{t}}}\right)_{j \in \Sigma_{i}(A^{t})}\right) $$
  4. 4.

    As for the evolution of agents’ wealth, it is determined on the one hand by their purchases of inputs and their sales of output. On the other hand, we assume that the firm sets its expenses for next period at (1 − λ) times its current revenues and distributes the rest as dividends to the representative household. That is one has:

    $$ \forall i \in M,\ w^{t + 1}_{i}=(1-\lambda)\overline{q}^{t}_{i} {p_{i}^{t}} $$
    $$ w^{t + 1}_{0}= {q}_{0}^{t} {p_{0}^{t}} +\lambda \sum\limits_{i \in M}\overline{q}^{t}_{i} {p_{i}^{t}} $$

    Note that Eq. 12 can be interpreted as assuming that firms have myopic expectations about their nominal demand (i.e they assume they will face the same nominal demand next period) and target a fixed profit/dividend share λ ∈ (0, 1).

  5. 5.

    As for the evolution of input shares, agents adjust frictionally their input combinations toward the cost-minimizing value according to:

    $$ \alpha_{i}^{t + 1}=\tau_{w} \overline{\alpha}_{i}^{t} +(1-\tau_{w}) {\alpha_{i}^{t}} $$

    where τw ∈ [0, 1] measures the speed of technological adjustment and \(\overline {\alpha }_{i}^{t} \in {\mathbb {R}}^{M}\) denotes the optimal input weights for firm i given prevailing prices. Those weights are defined as the solution to the following optimization problem:

    $$ \left\{\begin{array}{cc} \max & f_{i}\left( \frac{\alpha_{0},i}{{p_{0}^{t}}},\left( \frac{\alpha_{j},i}{{p_{j}^{t}}}\right)_{j \in \Sigma_{i}(A^{t})}\right) \\ \text{s.t} & \sum_{j \in \Sigma_{i}(A^{t})} \alpha_{j,i}= 1 \end{array}\right. $$

1.3 A.3 Convergence

These behavioral rules in fact define out-of-equilibrium dynamics in the economy \(\mathcal {E}(A)\). Their asymptotic properties are extensively studied in Gualdi and Mandel (2015). In particular, for a fixed network and over a vast region of the parameter space, one observes convergence to a general equilibrium of the underlying economy with a fixed mark-up rate λ. It corresponds to general equilibrium in the common-sense if λ = 0 and is defined as follows:

Definition 2

A λ-mark-up equilibriumof the economy \(\mathcal {E}(A)\)is acollection of prices \((p^{*}_{0},\cdots ,p^{*}_{n}) \in {\mathbb {R}}^{M}_{+}\),production levels \((q^{*}_{0},\cdots ,q^{*}_{n}) \in {\mathbb {R}}^{M}_{+}\)andcommodity flows \((x^{*}_{i,j})_{i,j = 0\cdots n} \in {\mathbb {R}}^{M\times M}_{+}\)such that:

  • Markets clear. That is for all iM,one has

    $$q^{*}_{i} = \sum\limits_{j = 1}^{M} x^{*}_{i,j}. $$
  • The representative consumer maximizes his utility. That is\((q^{*}_{0},(x^{*}_{0,j})_{j = 1,\cdots ,n})\)isa solution to

    $$\left\{\begin{array}{c} \max u_{i}((x_{0,j})_{j = 1,\cdots,n}) \\ \\ \text{s.t } \sum_{j = 1}^{n} p^{*}_{j} x^{*}_{0,j} \leq 1 \end{array}\right. $$

    (with the price of labor normalized to 1)

  • Production costs are minimized. That is for alliM,(xi,j∗)j= 0⋯nis the solutionto

    $$\left\{\begin{array}{cc} \min & \sum_{j \in \Sigma_{i}(A)} p^{*}_{j} x_{j} \\ \text{s.t} & f_{i}(x_{j})\geq q^{*}_{i} \end{array}\right. $$
  • Prices are set as a mark-up over production costs at rate\(\frac {\lambda }{1-\lambda }\). That is one has for alliN :

    $$p^{*}_{i} =\left( 1+ \frac{\lambda}{1-\lambda}\right)\frac{\sum_{j \in \Sigma_{i}(A)} p^{*}_{j} x^{*}_{i,j}}{q^{*}_{i}} $$

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Gualdi, S., Mandel, A. Endogenous growth in production networks. J Evol Econ 29, 91–117 (2019).

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