# On automation, labor reallocation and welfare

Stéphane Auray, Aurélien Eyquem

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Stéphane Auray, Aurélien Eyquem. On automation, labor reallocation and welfare. Journal of Economic Dynamics and Control, 2025, 177, 10.1016/j.jedc.2025.105129. hal-05137753

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# On automation, labor reallocation and welfare ☆

Stéphane Auray<sup>a</sup>, Aurélien Eyquem<sup>b,*</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

JEL classification:

E24

E32

F41

Keywords:

Robots

Automation

Heterogeneous firms

Labor market

Open-economy

## ABSTRACT

We develop an open-economy model of endogenous automation with heterogeneous firms and labor-market reallocation to quantify the contribution of various trends to the adoption of robots in the U.S. economy. The decline in the relative price of robots is the major trend leading to automation, but interacts with other trends that either hinder (rising entry costs, rising markups) or slightly foster (rising labor productivity, declining trade costs) the adoption of robots. Taken alone, the decline in the relative price of robots produces moderate welfare gains in the long run, but less than labor productivity growth. We then exploit our model to show that a decline in the relative price of robots (i) generates small positive cross-country automation spillovers and (ii) produces inefficient labor-market reallocation since a small subsidy on robots combined with a training subsidy can generate small welfare gains. Our main conclusion is that automation can not be simply modeled as an exogenous decline in the price of robots, and must be analyzed in a broader framework taking into account trends affecting firms, such as the decline in business dynamism and the rise in markups.

## 1. Introduction

Since the 1980's, the relative price of robots falls very significantly, the number of robots per thousands of industry workers has been rising very fast (Acemoglu and Restrepo (2022)). However, this macroeconomic trend is entangled with other important and potentially interacting trends, such as the decline in business dynamism, the rise in markups, increasing trade openness or changes in labor productivity. All these changes in the environment are likely to both affect the extent of automation – how much robots have been adopted by firms – and automation is likely to affect the way these other trends have affected the economy. In this paper, we isolate the macroeconomic and welfare effects of the decline in the price of robots by means of a general equilibrium model fed with a collection of trends to match the data.

We build an open-economy model of endogenous automation with heterogeneous firms, endogenous business formation and tradability, and a costly labor-market reallocation mechanism. The model extends Ghironi and Melitz (2005) to include segmented labor markets – with imperfect insurance among households of different segments – and endogenous automation. It also separates workers from firms owners to identify the extent of labor reallocation across routine and non-routine sectors. Overall, the model allows

<footnote><sup>a</sup> CREST-Ensai and Rennes School of Business, ENSAI, Campus de Ker-Lann, Rue Blaise Pascal, BP 37203, 35172 BRUZ Cedex, France</footnote>

<footnote><sup>b</sup> Department of Economics, University of Lausanne, Internef, CH-1015 Lausanne, Switzerland</footnote>

<footnote>☆ We would like to thank the editor B. Ravikumar and two referees for their invaluable comments. We also thank Basile Grassi, Sergio Rebelo, Ariel Resheff and Farid Toubal as well as participants in various seminars and conferences for interesting feedbacks. We acknowledge financial support from the French National Research Agency (ANR-20-CE26-0018-02).</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: stephane.auray@ensai.fr (S. Auray), aurelien.eyquem@unil.ch (A. Eyquem).</footnote>
for a comprehensive account of the effects of automation on labor markets, business dynamism, productivity and trade. Further, as automation is endogenous, it does not only depend on the relative price of robots as in many models, but is more generally affected by changes in the environment of firms, namely the marginal production cost and the relative amounts of labor types used to produce, and both margins are affected by a variety of trends. Our main message is that one should take account of all the ongoing trends in the U.S. economy – especially those related to business dynamism and market power – together with the fall in the price of robots to analyze properly the macroeconomic and welfare effects of automation, as well as any potential welfare-maximizing fiscal policies.

We use the model to quantify the contribution to automation of various trends of the U.S. economy since the mid-80's: a fall in the price of robots, a rise in firms' markups (De Loecker et al. (2020)), a rise in entry costs to account for the observed decline in business dynamism (Akcigit and Ates (2023)), a fall in trade costs accounting for the greater observed trade openness of the U.S. economy, and a rise in labor productivity. Calibrated to match the data, these trends allow for a decomposition that singles out the pure effects of the observed fall in the price of robots. More generally, the procedure and model allow for a decomposition of the respective contributions of the various trends to GDP, automation, business dynamism, trade openness and welfare.

In the model, when robots become cheaper, they are used as perfect substitutes to labor in routine tasks and as mild complements to non-routine tasks. This gives rise to (i) a displacement effect that lowers the demand for routine labor and raises the demand for non-routine labor and a rise in the non-routine wage premium; (ii) a productivity effect that lowers the production cost and results in more output; (iii) a labor supply effect driven by the non-routine wage premium, as workers reallocate towards the non-routine sector, which further amplifies the displacement effect and (iv) a boost in business dynamism, which further deepens automation and magnifies its effects on output and labor markets. Our model also predicts large aggregate production and wage gains from automation but very unevenly distributed. As in Moll et al. (2022), routine workers loose by all possible measures – lower employment, lower wages and lower consumption – while non-routine workers gain. Firms owners gain the most, as suppliers of robots and through higher dividends allowed by lower marginal production costs. In the aggregate, the welfare gains from non-routine workers and firm owners more than compensate the losses from routine workers, and automation raises welfare. One key aspect to this is the reallocation of routine workers to the non-routine sector, which shrinks their weight in the working force.

Our decomposition of the observed dynamics of the U.S. economy also brings important results regarding the effects of other trends on automation. First, absent other trends, the fraction of automated tasks and labor-market reallocation would have been quantitatively more important. Negative trends such as the rise in markups or the decline in business dynamism had offsetting effects on these processes. Absent the rise in markups or entry costs, we find that the fraction of automated tasks would have risen to 60 percents instead of 41 percents in the baseline simulation. This would have led to stiffer wage growth for non-routine workers, a more massive reallocation of workers to the non-routine sector, and larger aggregate welfare gains. By the same logic, the positive trends of increasing labor productivity and reduced trade costs helped foster the automation process. Had labor productivity or trade costs remained constant, the fraction of automated tasks, labor reallocation and the non-routine wage gap would all have been slightly less. In our model, while the absence of changes in trade costs has relatively little additional consequences except for the degree of trade openness, the absence of labor productivity gains also significantly moderates wage growth for both types of workers, generating large welfare losses.

We then run counterfactual experiments. In the first one, we assume that the relative price of robots declines more or less abroad than in the Home economy. This experiment reveals that there are small positive cross-country spillovers from automation through trade and firm relocation. In the second experiment, we contemplate the possibility of a welfare-maximizing tax on robots, a training subsidy or both. When all trends are considered as driving forces, if the fiscal scheme is imposed at the world level, the optimal solution consists in imposing a large subsidy on robots around 15%, a large subsidy on training or a combination of both – the robot tax being a more powerful instrument. When the only driving force is the decline in the price of robots, we find a much smaller welfare-maximizing subsidy on robots, around 1.4%, and a much smaller subsidy on training. This experiment shows that, when all trends are considered together, the robot and training subsidies are used to correct for other inefficiencies, related to increasing markups and entry costs and associated distortions, rather than inefficiencies strictly related to the decline in robots' price. Finally, we find that the difference between an optimized tax/subsidy scheme implemented at the world level or unilaterally is quantitatively negligible, in line with the small spillovers identified in the first counterfactual experiment.

**Literature.** Recent contributions by Acemoglu and Restrepo (2020) and Acemoglu and Restrepo (2022) document the upward trend in robots' adoption and automation over the last 30 years around the world. Building on a task-model of automation, they also show theoretically and empirically that automation gives rises to a displacement effect, lowering the demand for routine labor and increasing the demand for non-routine labor, and to a positive productivity effect. Moll et al. (2022) offer a quantification of automation modeled as a rise in the share of capital in production. As a source of biased technical change, automation has also been considered a potentially important factor of job polarization in advanced economies (see Goos et al. (2009), Harrigan et al. (2021) among others). As such, it has been advocated to contribute to slow or jobless recoveries (see Jaimovich and Siu (2020) and Graetz and Michaels (2017) for challenging views). In addition, numerous questions regarding the inefficient nature of automation and its potential taxation have been raised and partly answered (see Guerreiro et al. (2021), Beraja and Zorzi (2025)).

However, most contributions on automation like Acemoglu and Restrepo (2020) or Eden and Gaggl (2018) focus on labor demand and production effects, but mostly abstract from labor supply effects and endogenous labor-market reallocation. A notable exception is Guerreiro et al. (2021), who build a closed-economy general equilibrium model of automation to analyze the optimal taxation of robots. However, we consider a more general CES production function that fits empirical evidence in a more compelling way, as it generates a substantial increase in the non-routine labor share, something a Cobb-Douglas production function can not do. Two important papers highlight the importance of using a CES production function. Humlum (2019) analyzes the labor reallocation
induced by robots in Denmark estimating the switching cost for workers. Rodrigo (2022) uses the occupation switching cost of the reallocation induced by robots to explain the slow short-term productivity growth after the new technology is adopted. Our paper is different from those however in considering an explicit and endogenous training decision to reallocate labor towards the non-routine sector. What is more, our paper considers the effect of automation in an environment with endogenous entry, which is, to the best of our knowledge, new. As such, the above papers may miss some of the effects of automation running through the relative supply of skills in general, but also in response to automation, or through changes in business dynamism or other trends of the U.S. economy.

Our second main contribution to the literature is the open-economy dimension. The latter received relatively little attention, to the exception of Artuc et al. (2023), who investigate the trade and wage effects of automation. Artuc et al. (2023) consider a Ricardian model of trade but disregard firms' entry in production and export markets, i.e. effects at the extensive margin. According to our results, both features – firm creation and entry in export markets – matter to fully assess the macroeconomic effects of automation, especially in light of the above-mentioned general equilibrium linkages through which automation proceeds. Mandelman and Zlate (2022) is more closely related to our work. They propose a model of offshoring, immigration, automation and trade in tasks to disentangle the respective contributions of these factors in explaining the polarization of U.S. labor markets.<fnref n="1" /> We adopt a somewhat similar approach in combining various trends to account for the observed effects of automation and grasp its welfare implications. However, rising markups and declining business dynamism appear to be salient trends of the U.S. economy that are not taken into account by Mandelman and Zlate (2022), while we highlight their critical importance in shaping the effects of automation. In addition, Mandelman and Zlate (2022) do not really discuss how trends other than declining robots' price affect the extent of automation while this is an important focus of our paper. From an empirical perspective, Fontagné et al. (2024) find that automation affects labor shares indirectly, by altering the structure of global value chains and the 'upstreamness' of industries that automate.

Our paper also draws on the literature dealing with rising markups and declining business dynamism. De Loecker et al. (2020) document a very large increase in market concentration and markups in the U.S. over the last 40 years. Although their finding is being discussed extensively in subsequent papers and potentially explained by changed in cost-price pass-through (Conlon et al. (2023)), by the rise of the service sector (Marto (2024)) or challenged on empirical grounds (Foster et al. (2022)), it is still considered a key ingredient to account for the dynamics of the U.S. economy, along with factors weighing on business dynamism (De Loecker et al. (2021)). The decline in U.S. business dynamism has been documented in details by Decker et al. (2016) and, according to Akcigit and Ates (2021), rationalized by interacting changes in market concentration and innovation, leading to decreasing knowledge diffusion (see also Akcigit and Ates (2023)). Recent contributions even connect the rise in markups and decline in business dynamism to automation or intangible inputs such as software (see Firooz et al. (2025) or De Ridder (2024) for instance). Our goal is not to explain the observed rise in markups or the decline in business dynamism but to highlight how these trends interact with automation. In particular, Bilbiie et al. (2014) or Bilbiie et al. (2019) shows that models with entry and markups can generate inefficiencies stemming from the various distortions: inefficient labor supply, inefficient supply of varieties given consumers' preferences, inefficient fluctuations of varieties over time. In particular, we find that combining trends affecting business dynamism and markups interacts strongly with the decline in the price of robots, altering macroeconomic and welfare consequences of automation.

The paper is structured as follows. Section 2 presents the model. Section 3 calibrates the model using U.S. data and assesses the fit of the model. Section 4 runs a decomposition of the drivers of automation. Section 5 runs counterfactual experiments to highlight positive cross-country spillovers and the potential welfare gains from subsidizing robots or training. Section 6 proposes concluding remarks.

## 2. Model

We build a two-country open-economy model with endogenous entry and export participation, endogenous automation and heterogeneous labor. In each country there are two families of workers supplying differentiated labor: routine workers and non-routine workers. The size of each family is endogenous and affected by the wage premium.

### 2.1. Families

We consider two countries of equal size, Home and Foreign, and denote Foreign variables with an asterisk. The Home economy represents the U.S. while the Foreign economy represents a pool of advanced economies. We focus on the Home economy (U.S.) and assume perfect symmetry in the structure and parameter values – except for the access to international financial markets.

**Home families.** In the Home economy, there are three families: non-routine workers (h), routine workers (u) and firm owners (e).<fnref n="2" /> Let $\lambda_t \in [0, 1]$ be the time-varying size of the family of non-routine workers. Workers can switch family: a fraction $s_t$ of routine workers can acquire non-routine skills by paying training costs and become skilled workers, and a fraction $\delta_{it}$ of non-routine workers face skill depreciation and become unskilled workers. As a result, the dynamics of the size of the family of non-routine workers is:

$$ \lambda_t = (1 - \delta_{it}) \lambda_{t-1} + s_{t-1} (1 - \lambda_{t-1}). \quad (1) $$

Both families of workers are hand-to-mouth. The family of non-routine workers does not make any decision while the family head of routine workers decides the training effort to maximize lifetime per-capita welfare:

<footnote n="1">¹ The paper adopts many of the assumptions featured in Mandelman (2016).</footnote>

<footnote n="2">² We separate the distribution of firm profits and access to international financial markets to isolate the effects of shocks on labor-market reallocation.</footnote>
$$V_t^u = \log(c_t^u) - \frac{\omega}{\zeta} s_t^\zeta + \beta \mathbb{E}_t \{ (1 - s_t) V_{t+1}^u + s_t V_{t+1}^h \}, \quad (2)$$

where $c_t^u$ denotes per-capita consumption at time $t$ in family $i$, $\omega s_t^\zeta / \zeta$ is a non-convex utility cost of training experienced by non-routine workers, and:

$$V_t^h = \log(c_t^h) + \beta \mathbb{E}_t \{ (1 - \delta_{tv}) V_{t+1}^h + \delta_{tv} V_{t+1}^u \}, \quad (3)$$

is the value function of the non-routine family. Firm owners have access to financial markets, invest in the creation of new firms and produce robot capital. They represent a fix proportion $\mu$ of the household sector and do not change types.

**Non-routine workers.** The family of non-routine workers has the following budget constraint:

$$c_t^h = w_t^h \chi_t^h, \quad (4)$$

where $\chi_t^h$ is an exogenous measure of hours worked by non-routine workers.

**Routine workers.** The family head of routine workers chooses consumption and training $s_t$ to maximize (2) subject to (1) and to the following budget constraint:

$$c_t^u = (1 - s_t) w_t^u \chi_t^u, \quad (5)$$

where $\chi_t^u$ is an exogenous measure of hours worked by routine workers. This equation highlights that the total wage earned by routine workers $(1 - s_t) w_t^u \chi_t^u$ is used to consume, so that training not only has a utility cost but also represents an opportunity cost. The optimal choice of training $s_t$ implies:

$$\frac{w_t^u \chi_t^u}{c_t^u} + \omega s_t^{\zeta-1} = \beta \mathbb{E}_t \{ V_{t+1}^h - V_{t+1}^u \}. \quad (6)$$

Optimal training equates the marginal utility cost of training, composed of the forgone labor income $w_t^u \chi_t^u$ and the marginal disutility $\omega s_t^{\zeta-1}$ from training, to the discounted expected marginal utility gain of becoming a non-routine worker next period $V_{t+1}^h - V_{t+1}^u$. These equations show that any force improving the prospective relative wage or consumption of non-routine workers results in labor reallocation through higher training $s_t$ by increasing the acceptable marginal cost of training, and thus trigger an increase in the relative size of the non-routine labor force $\lambda_t / (1 - \lambda_t)$. As we will start from a situation where $\lambda_0 = 1/2$, we define labor-market reallocation as $\Lambda_t = |\lambda_t / (1 - \lambda_t) - 1|$.

**Firm owners.** The last type of households represents a fraction $\mu$ of the household sector. They produce $x_t$ robots every period under perfect competition and sell them at price $\phi_t$, and own both intermediate firms which operate under perfect competition and generate zero profits. Last, they own final good producers – which generate profits – through a mutual fund and invest in the creation of new final good producers. They also have access to local bonds. Their budget constraint writes:

$$\mu c_t^e + s h_t (n_t + n_{et}) \bar{v}_t + b_t = r_{t-1} b_{t-1} + s h_{t-1} (\bar{k}_t + \bar{v}_t) n_t + \phi_t x_t \quad (7)$$

where $s h_t$ denotes shares of the mutual fund, $n_t = (1 - \delta)(n_{t-1} + n_{t-1}^e)$ the endogenous number of final goods producers, $n_{et}$ the number of new firms and $\bar{k}_t$ and $\bar{v}_t$ their average operating profits and market value, respectively.<fnref n="3" /> Variable $b_t$ denotes the amount of dollar-denominated bonds held by firm owners, returning $r_t$ between time $t$ and $t+1$. Assuming log preferences for firm owners, we also define their stochastic discount factor as $\beta_{t,t+1}^e = \beta c_t^e / c_{t+1}^e$ and derive the first-order conditions with respect to bonds and shares:

$$\mathbb{E}_t \{ \beta_{t,t+1}^e r_t \} = 1, \quad (8)$$

$$\mathbb{E}_t \{ \beta_{t,t+1}^e (1 - \delta) (\bar{k}_{t+1} + \bar{v}_{t+1}) \} = \bar{v}_t, \quad (9)$$

Assuming $s h_t = s h_{t-1} = 1$ as in Ghironi and Melitz (2005), their budget constraint simplifies to:

$$\mu c_t^e + n_t^e \bar{v}_t + b_t = r_{t-1} b_{t-1} + n_t \bar{k}_t + \phi_t x_t \quad (10)$$

**Consumption baskets.** For each family, the aggregate consumption basket is made of Home varieties $\omega \in \Omega$ and Foreign varieties $\omega^* \in \Omega$ where $\Omega$ denotes the space of possible varieties:

$$c_t^i = \left( (n_t)_{\frac{\zeta-1}{\theta_t}} \int_{\omega^* \in \Omega} c_{dt}^i (\omega)_{\frac{\theta_t-1}{\theta_t}} d\omega + (n_{xt}^*)_{\frac{\zeta-1}{\theta_t}} \int_{\omega^* \in \Omega} c_{xt}^i (\omega^*)_{\frac{\theta_t-1}{\theta_t}} d\omega^* \right)^{\frac{\theta_t}{\theta_t-1}}, \quad i = \{h, u, e\}. \quad (11)$$

Here $\theta_t > 1$ is the elasticity of substitution between goods, $n_t$ and $n_{xt}^*$ respectively denote the number of Home and imported varieties available for consumption in the Home country. Following Bénassy (1996), $\zeta$ captures the love for variety of consumers.

<footnote n="3">³ More details regarding firm dynamics are given in the dedicated section below.</footnote>
Preferences boil down to CES preferences for $\zeta = 1$, $\zeta < 1$ ($\zeta > 1$) indicating a lower (higher) marginal welfare benefit from additional varieties. Defining $\rho_{dt}(\omega)$ and $\rho_{xt}^*(\omega^*)$ as the relative prices of each type of varieties, the demands for Home and imported goods are:

$$c_{dt}^i(\omega) = \rho_{dt}(\omega)^{-\theta_t} n_t^{\zeta-1} c_t^i, \text{ and } c_{xt}^i(\omega^*) = \rho_{xt}^*(\omega^*)^{-\theta_t} n_{xt}^{\zeta-1} c_t^i, \quad i = \{h, u, e\}. \quad (12)$$

**Foreign families.** The only difference in the problems solved by Foreign families is that firm owners have access to two types of bonds, the dollar-denominated bond and a local bond. In addition, changes in the amount of dollar-denominated bonds held require the payment of an adjustment cost<fnref n="4" />:

$$ac_t^{b*} = (\phi_b/2) (q_t^{-1} b_t^* - q_{t-1}^{-1} b_{t-1}^*)^2, \quad (13)$$

that depends on deviations of external assets from their past value, where $q_t$ is the consumption-based real exchange rate from the perspective of the Home economy. The resulting Euler equations of firm owners imply:

$$\mathbb{E}_t \left\{ \beta_{t,t+1}^{e*} r_t^* \right\} = 1, \quad (14)$$

$$\mathbb{E}_t \left\{ \beta_{t,t+1}^{e*} \left( r_t - r_t^* \frac{q_{t+1} (1 + \phi_b (q_t^{-1} b_t^* - q_{t-1}^{-1} b_{t-1}^*))}{q_t} \right) \right\} = 0, \quad (15)$$

where $\beta_{t,t+1}^{e*}$ is the counterpart of $\beta_{t,t+1}^e$ for Foreign firm owners. The first-one is a standard Euler equation on local bonds, and the second is a modified real interest rate parity condition that determines the dynamics of the real exchange rate.

## 2.2. Firms

**Intermediate goods.** In the intermediate sector, a representative firm uses a quantity of non-routine labor $\ell_t^h$ and a combination of effective routine labor $\psi_t \ell_t^u$ – where $\psi_t$ is labor productivity – and robots $x_t$ to produce $y_t$ units of intermediate goods. Among automatable tasks, only a time-varying fraction $a_t = \frac{x_t}{x_t + \psi_t \ell_t^u} \in [0, 1]$ of automatable tasks are effectively automated and performed by robots, and the remaining fraction $1 - a_t$ is performed by effective routine workers. As shown by Acemoglu and Restrepo (2022), the key underlying assumption is that *routine labor and robots are strong substitutes to perform automated tasks*.<fnref n="5" /> Guerreiro et al. (2021) build on this idea for automatable tasks and assume a Cobb-Douglas combination of automatable and non-automatable tasks. We adopt a more general CES representation, allowing both types of tasks to be complement, which results in rising demand for non-routine labor and a rising non-routine labor share in case of shocks fostering automation.<fnref n="6" /> Assuming *perfect substitutability among tasks performed by robots and non-routine workers*, our production function of intermediate goods writes:

$$y_t = \left( \alpha (\psi_t \xi \ell_t^h)^\nu + (1-\alpha) (x_t + \psi_t \ell_t^u)^\nu \right)^{\frac{1}{\nu}} \quad (16)$$

$$= \psi_t \left( \alpha (\xi \ell_t^h)^\nu + (1-\alpha) \left( \frac{\ell_t^u}{1-a_t} \right)^\nu \right)^{\frac{1}{\nu}}, \quad (17)$$

where $\xi$ is the relative productivity of non-routine workers, $\psi_t$ is the productivity of labor, $\nu$ determines the substitutability of routine and non-routine tasks. These equations show that automation plays the role of a routine-labor saving technology or equivalently an endogenous routine-biased technical change. Let $mc_t$ denote the relative price at which the intermediate good is sold to final goods producers, the representative intermediate firm maximizes its profits:

$$\Pi_{yt} = mc_t y_t - \phi_t x_t - w_t^u \ell_t^u - w_t^h \ell_t^h. \quad (18)$$

The first-order condition with respect to non-routine labor is independent from the type of task completed and yields:

$$\alpha m c_t \left( \frac{y_t}{\xi \ell_t^h} \right)^{1-\nu} = \frac{w_t^h}{\xi \psi_t}. \quad (19)$$

This equation shows that the demand for non-routine labor depends positively on the production of intermediate goods $y_t$ and negatively on the non-routine real wage relative to the price of the intermediate good. As discussed in Guerreiro et al. (2021), perfect substitutability among automatable tasks implies that anything else than $w_t^u/\psi_t = \phi_t$ either yields full automation or null automation. If the fraction of automated tasks $a_t$ is to be interior, a necessary condition is thus $w_t^u/\psi_t = \phi_t$ which is obvious from combining the first-order conditions on automatable tasks:

<footnote n="4">⁴ This assumption is purely technical and pins down the dynamics of net foreign assets as explained in Schmitt-Grohé and Uribe (2003) in small open economies or Ghironi and Melitz (2005) in two-country models.</footnote>

<footnote n="5">⁵ Acemoglu and Restrepo (2022) show clearly that this type of productive transformation is different from any alternative way of modeling automation (including skill-biased or capital-biased technical change).</footnote>

<footnote n="6">⁶ A Cobb-Douglas production function keeps the non-routine labor share constant, which does not fit the trend observed in the past decades.</footnote>
$$ (1 - \alpha) mc_t \left( \frac{y_t}{x_t + \psi_t \ell_t^u} \right)^{1-\nu} = \frac{w_t^u}{\psi_t} = \phi_t. \qquad (20) $$

Using the first-order condition on robots, we finally get:

$$ a_t = 1 - \left( \frac{1}{\alpha} \left( \frac{\phi_t}{(1-\alpha)mc_t} \right)^{\frac{\nu}{1-\nu}} - \frac{1-\alpha}{\alpha} \right)^{\frac{1}{\nu}} \frac{\ell_t^u}{\xi \ell_t^h}, \quad \text{with } 0 \le a_t \le 1. \qquad (21) $$

Equation (21) is key. It shows that the dynamics of automation critically depends on the relative price of robots $\phi_t$. As the latter goes down, a first direct effect stems from the direct substitution (displacement) of routine workers in the production process, pushing $m_t$ up. Further, as $\phi_t$ falls, so does the routine wage $w_t^u/\psi_t$ in equilibrium – unless $\psi_t$, the productivity of labor, falls more than $\phi_t$, which is not plausible. Since the demand for robots jumps, whether the demand for non-routine labor increases or not depends on whether automatable tasks and non-automatable tasks are substitutes or complements. In the case of complementarity, the demand for non-routine labor rises, which raises the non-routine wage premium. A positive response of the non-routine wage drives the marginal production cost $mc_t$ up, which amplifies automation endogenously. This second effect is completed by a third effect stemming from general equilibrium interactions. Since households reallocate among sectors depending on the non-routine wage premium, the resulting effects on $\ell_t^u / (\xi \ell_t^h)$ act as reinforcing or dampening automation. If reallocation leads to more labor in the non-routine sector in equilibrium, i.e. if $\ell_t^u / (\xi \ell_t^h)$ falls, then automation is reinforced. This equation thus shows that modeling automation as an endogenous process delivers much richer dynamics than just assuming exogenous robot capital deepening, as the latter also depends on general equilibrium effects. As a consequence, in our framework, any shock affecting the marginal production cost or the relative amounts of labor types used in production has implications for endogenous automation.

**Final goods.** In the final goods sector, a continuum of heterogeneous firms differentiates intermediate goods into varieties before selling them to consumers at home and abroad. The sector allows for endogenous entry and endogenous tradability. Over the entire space of potential varieties, only a subset will actually be created and commercialized. Each firm produces one variety. Firms have specific random productivity draws $z$, which remain fixed once firms have been created. Entry implies a once-and-for-all sunk cost $f_{et}$, paid in units of intermediate goods. At each period $t$, there are two types of firms: $n_t$ firms that are already productive at the beginning of the period and $n_{e,t}$ firms that are newly created but non-productive yet. At the end of period $t$, a fraction $\delta \in [0, 1]$ of existing firms is affected by an exogenous exit shock. The total number of varieties/firms thus evolves according to:

$$ n_t = (1 - \delta)(n_{t-1} + n_{et-1}). \qquad (22) $$

Among the firms created, only the most productive address the export market. Entry in the export market is subject to the repeated payment of a fixed cost $f_x$, also paid in units of intermediate goods, and incurs the payment of iceberg melting costs $\tau_t$.<fnref n="7" /> Firm-specific productivity $z$ has a Pareto distribution with lower bound $z_{min}$ and shape parameter $\epsilon > \theta_t - 1$. The probability density function of $z$ is $g(z) = \epsilon z_{min}^{\epsilon} / z^{\epsilon+1}$ and the cumulative density function is $G(z) = 1 - (z_{min}/z)^{\epsilon}$. Over the total number of potential firms, the number of existing firms will be determined by a free-entry condition. In addition, out of the total number of firms addressing the local market, the number of exporting firms $n_{xt}$ will be those that are productive enough to cover the additional fixed export costs and trade costs. Their number is:

$$ n_{xt} = (1 - G(z_{xt})) n_t = (z_{min}/z_{xt})^{\epsilon} n_t, \qquad (23) $$

where $z_{xt}$ is the individual productivity of the cut-off exporting plant. Firm $z$ produces a quantity $y_t(z)$ of variety $z$ using $y_{mt}(z)$ of the intermediate good and the following production function:

$$ y_t(z) = zy_{mt}(z). \qquad (24) $$

As such, the firm-specific marginal production cost is $mc_t(z) = mc_t/z$. Let $\kappa_t(z)$ denote the total current real profits of a firm with productivity $z$. Total current profits comprise domestic and export profits, $\kappa_{dt}(z)$ and $\kappa_{xt}(z)$, respectively defined as:

$$ \kappa_{dt}(z) = (\rho_{dt}(z) - mc_t/z) y_{dt}(z), \quad \text{and} \quad \kappa_{xt}(z) = (q_t \rho_{xt}(z) - (1 + \tau_t) mc_t/z) y_{xt}(z) - f_x mc_t, \qquad (25) $$

where $\rho_{dt}(z)$ is the relative price of good $z$ when sold in the Home market and $\rho_{xt}(z)$ its price expressed in terms of the Foreign currency when exported, with $q_t$ the real exchange rate. Variables $y_{dt}(z)$ and $y_{xt}(z)$ respectively denote the Home and Foreign total demand of domestic varieties, such that firm $z$ produces $y_t(z) = y_{dt}(z) + y_{xt}(z)$ and therefore demands $y_{mt}(z) = y_t(z) / z = (y_{dt}(z) + y_{xt}(z)) / z$ units of intermediate goods. The optimal pricing conditions are derived subject to the goods demand functions and optimal prices imply:

$$ \rho_{dt}(z) = \frac{\theta_t}{\theta_t - 1} \frac{mc_t}{z}, \quad \text{and} \quad \rho_{xt}(z) = (1 + \tau_t) \frac{\theta_t}{\theta_t - 1} \frac{mc_t}{q_t z}, \qquad (26) $$

Entry occurs one period before production starts and the productivity draw of the last entering firm remains fixed until the corresponding firm exits. Firms do not know their productivity draw prior entry. Hence, the entry condition equates the current entry

<footnote n="7">⁷ Trade costs are exogenous but time-varying. Out of a quantity $y$ produced and shipped, only $y / (1 + \tau_t)$ actually arrives. Firms need to produce $(1 + \tau_t) y$ to sell $y$.</footnote>
cost expressed in units of intermediate goods to the current value of the average firm: $\tilde{v}_t = f_{et}mc_t$, which combined with firm owners' Euler condition implies:

$$ \mathbb{E}_t \left\{ \beta_{t,f+1}^e (1 - \delta) \left( \tilde{\kappa}_{t+1} + f_{et} mc_{t+1} \right) \right\} = f_{et} mc_t. \quad (27) $$

where $\tilde{\kappa}_t$ denotes average profits. This equation shows the determinants of firms' entry. Given the definition of profits, entry is high when the current marginal cost is low, and when domestic and export markets are large. The entry condition also shows that entry is high when current entry costs are low or expected discounted entry costs higher than current entry costs. Among incumbents, only the most productive firms profitably enter the export market given that exporting requires the repeated payment of iceberg costs. The export productivity cut-off is thus $\kappa_{xt}(z_{xt}) = 0$ or, after using the optimal pricing and demand equations:

$$ z_{xt} = \frac{1 + \tau_t}{\theta_t - 1} \left( \frac{\theta_t mc_t}{q_t} \right)^{\frac{\theta_t}{\theta_t - 1}} \left( \frac{f_x}{n_{xt}^{\zeta-1} \Phi_t^*} \right)^{\frac{1}{\theta_t - 1}}, \quad (28) $$

where $\Phi_t^*$ denotes the aggregate Foreign demand. As in the case of firms' entry, this equation sheds light on the determinants of entry in the export market: low trade costs, a low marginal cost, a low relative price captured by a high value of $q_t$, low export costs, a large Foreign market and strong love for variety. The Foreign economy is characterized by symmetric conditions that are therefore not detailed.

## 2.3. Aggregation and data consistency

We define the average productivity of Home firms selling on the domestic market as $\tilde{z} = \nabla z_{min}$ where $\nabla = (\epsilon / (\epsilon - (\theta_t - 1)))^{\frac{1}{\theta_t - 1}}$ and the average productivity of Home firms addressing both markets as $\tilde{z}_{xt} = \nabla z_{xt}$ (see Ghironi and Melitz (2005) for a discussion and Melitz (2003) for proofs; see Hamano (2022) for comparable results with love for variety).

**Average prices.** Defining the average price of a Home good as $\tilde{\rho}_{dt} = \rho_{dt}(\tilde{z})$ and the average price of Home exported good as $\tilde{\rho}_{xt} = \rho_{xt}(\tilde{z}_{xt})$, we obtain real average prices:

$$ \tilde{\rho}_{dt} = \frac{\theta_t}{\theta_t - 1} \frac{mc_t}{\nabla z_{min}}, \quad \text{and} \quad \tilde{\rho}_{xt} = (1 + \tau_t) \frac{\theta_t}{\theta_t - 1} \frac{mc_t}{q_t \nabla z_{xt}}, \quad (29) $$

while conditions for Foreign goods are symmetrically defined and thus not reported.

**Average profits and variety effect.** Using the profit and pricing equations, average profits are given by:

$$ \tilde{\kappa}_{dt} = \frac{1}{\theta_t} \tilde{\rho}_{dt}^{1-\theta_t} n_t^{\zeta-1} \Phi_t, \quad \text{and} \quad \tilde{\kappa}_{xt} = \frac{(\theta_t - 1)}{\epsilon - (\theta_t - 1)} f_{xt} mc_t, \quad (30) $$

where $\Phi_t = (1 - \mu)(\lambda_t c_t^h + (1 - \lambda_t) c_t^u) + \mu c_t^e$. These equations can be used to obtain the dynamics of average total profits:

$$ \tilde{\kappa}_t = \tilde{\kappa}_{dt} + (n_{xt}/n_t) \tilde{\kappa}_{xt}. \quad (31) $$

Based on the expression of the CPI, we uncover the following variety effect:

$$ n_t^{\zeta} \tilde{\rho}_{dt}^{1-\theta_t} + n_{xt}^{*\zeta} \tilde{\rho}_{xt}^{*1-\theta_t} = 1. \quad (32) $$

**Aggregate productivity in final goods.** Endogenous changes in the number of producers and exporters affect aggregate productivity in the final goods sector. As more (less) varieties are created and/or more (less) varieties are exported, the productivity of the marginal producer (exporter) varies, even for a fixed and given distribution of productivity draws. Whether aggregate productivity in the sector is computed based on the density function $g(z)$ or based on weights reflecting relative output shares, as in Melitz (2003), entry and exit on domestic and export markets affect the aggregate productivity level through (i) changes in the average productivity level due to the marginal productivity of entering firms, and (ii) the reallocation of market shares among firms. We follow Melitz (2003) and define two measures of aggregate productivity, one for all domestic final goods producers and one for final goods producers that export:

$$ Z_t = n_t^{\frac{\zeta}{\theta_t - 1}} \nabla, \quad \text{and} \quad Z_{xt} = n_{xt}^{\frac{\zeta}{\theta_t - 1}} \nabla z_{xt}. \quad (33) $$

These equations show that aggregate productivity may increase either because the average productivity level rises, or because of a scale effect by which there are more producers on the markets. For the aggregate productivity of all domestic final goods producers, only the scale effect is present and any rise in the total number of firms will raise aggregate productivity. But as shown also in the above equations, the productivity gains are scaled by the love-for-variety parameter $\zeta$. Any mechanism that lowers the marginal production cost $mc_t$ or favors lower entry costs will have positive effects on aggregate productivity through the entry of new firms. The strength of the effect though depends on the substitutability of varieties and on love for variety.
**Market clearing, GDP and net foreign assets.** On the labor markets for routine and non-routine workers respectively, the equilibrium is:

$$ \ell_t^h = (1 - \mu) \lambda_t \chi^h, \text{ and } \ell_t^u = (1 - \mu) (1 - \lambda_t) (1 - s_t) \chi^u. \quad (34) $$

On goods markets, the clearing condition for Home intermediate goods reads:

$$ y_t = \tilde{\rho}_{dt}^{\theta_t} \nabla^{-1} n_t^{\zeta} \Phi_t + (1 + \tau_t) \tilde{\rho}_{xt}^{\theta_t} (\nabla z_{xt})^{-1} n_{xt}^{\zeta} \Phi_t^* + n_{et} f_{et} + n_{xt} f_{xt}, \quad (35) $$

where $\Phi_t^* = (1 - \mu) (\lambda_t^*(c_t^* + ac_t^{b*}) + (1 - \lambda_t^*) c_t^{u*}) + \mu c_t^{e*}$, and the market clearing condition for Home final goods is:

$$ y_t^c = n_t^\zeta \tilde{\rho}_{dt}^{1-\theta_t} \Phi_t + q_t n_{xt}^{\zeta} \tilde{\rho}_{xt}^{1-\theta_t} \Phi_t^*. \quad (36) $$

A data-consistent measure of GDP is obtained aggregating all sources of income from the different types of households:

$$ y_t^{gdp} = w_t^h \ell_t^h + w_t^u \ell_t^u + \phi_t x_t + n_t \tilde{\kappa}_t \quad (37) $$

On the market for dollar-denominated bonds, the international clearing condition gives:

$$ q_t b_t + b_t^* = 0, \quad (38) $$

while the Foreign local bond is assumed to be in zero net supply. The dynamics of Home net foreign assets are obtained by aggregating all budget constraints and combining with market clearing conditions:

$$ b_t - r_{t-1} b_{t-1} = q_t n_{xt}^{\zeta} \tilde{\rho}_{xt}^{1-\theta_t} \Phi_t^* - n_{xt}^{*\zeta} \tilde{\rho}_{xt}^{1-\theta_t} \Phi_t. \quad (39) $$

**Data consistency.** In principle, as explained by Ghironi and Melitz (2005), our model variables would have to be deflated by a price index capturing the aggregate variety effect. The presence of endogenous varieties with love for variety implies that welfare-based price indices may vary even though individual product prices remain fixed. However, given that we consider a long period of time and focus on structural change, we consider that statistical offices have enough time to update price indices to take account of new varieties. We thus focus on welfare-based macroeconomic aggregates and abstract from deflating key variables.

**Trends.** We consider five different trends capturing the following evolutions of the U.S. economy.

* First, we consider a steady decline in the relative price of robots $\phi_t$. In most contributions analyzing automation, this is the single driver of the increased use of robots in production. In our set-up however, because automation results from endogenous choices from intermediate goods producers, the relative price of robots interacts with other factors determining the marginal production cost of intermediate goods, and relative amounts of labor. Rising markups, higher entry costs, trade costs or labor productivity all contribute to the endogenous adoption of robots.

* We also include a decline in the elasticity of substitution between goods $\theta_t$, implying rising markups as documented by De Loecker et al. (2020). This trend captures the observed rise in market concentration. In our set-up with endogenous entry, markups play a dual role with ambiguous net effects. On the one hand, as in most macroeconomic models, they distort the divide of value-added into profits and factor shares in favor of profits, lower net factor payments and thus lead to less production. On the other hand, these effects on the intensive margin can be compensated by changes in the extensive margin of goods production, since higher profits imply higher expected gains for firms just below the entry cut-off, and higher markups might stimulate entry and business dynamism. Hence, to account for the observed decline in business dynamism, this trend is combined with a persistent rise in entry costs. Of course changes in markups also have implications for aggregate productivity through the extensive margin, as explained above and shown by Equation (33).

* A rise in the entry cost $f_{et}$ is introduced to account for the observed decline in business dynamism discussed by Akcigit and Ates (2023). While this trend is 'reduced-form', it aims at capturing the dynamics of the total number of producers which, as shown again by Equation (33), has implications for aggregate productivity and thus aggregate output.<fnref n="8" /> What's more, the decline in business dynamism has implications for the aggregate demand of both types of labor, and thus for the extent of automation.

* In our open-economy model, aggregate productivity and output are not only affected by domestic conditions but also by the dynamics of exports and the number of exported varieties. Equation (33) shows it clearly, and we also want to take into account the rise in trade openness of the U.S. economy over the sample, from 18% in the late 80's to 30% in the early 2010's.<fnref n="9" /> We thus consider a fall in trade costs $\tau_t$, because it also has implications for aggregate productivity, output and automation.

* Last, we consider a trend rise in labor productivity $\psi_t$. All the above trends, when combined, might result in positive or negative aggregate output dynamics. We thus introduce the possibility of a steady rise in labor productivity to account for any potential additional rise in aggregate GDP. Introducing this trend will also help disentangle the different productivity effects at work: those

<footnote n="8">⁸ De Ridder (2024) proposes a mechanism relying on intangible inputs to account for the observed slowdown of productivity growth, the decline in business dynamism and the rise of market power. Other mechanisms relating competitive, innovation and productivity growth tried to offer micro-foundations for these trends (see Aghion et al. (2023)).</footnote>

<footnote n="9">⁹ On measuring trade openness, see also Waugh and Ravikumar (2016).</footnote>
pertaining to the increased use of robots, those arising from changes in existing and traded varieties of goods, and those driven
by increasing labor productivity.

Contrary to the literature, since firms' entry and automation are both endogenous, all these trends will have implications for the number of existing and traded varieties as well as for the degree of automation, and thus effects on output, productivity and trade openness. We build on existing measurable evidence for the fall in the relative price of robots $\phi_t$, and for rising markups $\theta_t/(\theta_t - 1)$. The three other trends are set based on simulation methods to match the observed dynamics of various aggregates, as described in details below.

**Solution method.** Given the size of perturbations and adjustments, non-linearities are likely to be significant. We thus rule out linearization methods and solve the model using perfect foresight simulations with the five different growth trends.<fnref n="10" />

**3. Parameter values and data fit**

We use various data sources to calibrate our model, all pertaining to the U.S. economy. A subset of parameters is calibrated using empirical evidence about their value. The remaining subset of parameters is adjusted to fit time-series evidence.

**3.1. Calibrated parameters**

Both countries have similar size and are symmetric in the initial equilibrium, which implies $q = 1$ and $b = 0$. All trends will be symmetric (i.e. world-wide) in the baseline experiment.

*Households* The model is annual, and the discount factor is $\beta = 0.96$, which implies a 4% steady-state interest rate. We assume that firm owners represent $\mu = 10\%$ of the total number of households, the remaining $1 - \mu = 90\%$ are workers, which are initially equally split between the routine and non-routine sectors ($\lambda_0 = 1 - \lambda_1 = 0.5$). The depreciation rate of human capital is set to $\delta_w = 0.025$ following Dinerstein et al. (2022), which implies a training effort of $s = \delta_w = 0.025$ and, given the value of $\zeta$ given below, implies $\sigma = 1.2905 \times 10^5$.

*Automation and labor* Our baseline calibration is meant to capture the situation of the U.S. in the late 1980's, when the routine and non-routine labor shares were roughly equal. Eden and Gaggl (2018) report a share of ICT capital in 1987 in output of 2.95%, which implies assuming $m_0 = m_0^* = 0.058$ in our model, and adjusting the initial relative price of robots $\phi_0 = \phi_0^*$ accordingly. Further, we follow Guerreiro et al. (2021), and impose $\alpha = 0.5$. In addition, the productivity of labor is initialized to $\psi_0 = 1$. Matching the 1987 routine and non-routine shares of labor income given other parameters leads us to impose $\chi^h = 0.25$ and $\chi^u = 0.3$. The relative productivity parameter $\xi$ is then adjusted given other parameters to match the 1987 non-routine wage premium $w_0^h/w_0^u = 1.33$, implying $\xi = 1.2229$. As explained below, $v$ is set along with other parameters to match the dynamics key ratios and variables.

*Firms' entry and exit* In the final goods sector, without loss of generality and as usual in the models with endogenous entry, we impose $n=1$ which pins down the value of $f_{e0}$, the initial value of the entry cost. We also impose the initial share of exporting firms in the steady state at $n_{x0}/n_0 = 0.2$, and adjust the export cost $f_x$ accordingly. Based on U.S. Census Bureau Business Dynamics Statistics, the annual death rate of firms is set to $\delta = 0.125$ to match an observed gross entry rate in 1987 of $n_{e0}/n_0 = 0.143$ in the initial steady state. Regarding the initial elasticity of substitution between varieties $\theta_0$, we follow Ghironi and Melitz (2005) and set $\theta_0 = 3.8$, which delivers a 1.35 markup and compares well to the 1.32 estimate reported by De Loecker et al. (2020) for the U.S. in 1987. We also follow Ghironi and Melitz (2005) for the Pareto parameter determining the dispersion of firm's productivity and impose $\varepsilon = 3.4$.

*Trade and financial sector* For the calibration of the trade sector, we adjust the iceberg cost parameter to $\tau_0 = 2.25$ to reproduce the 18% U.S. openness ratio in 1987, measured as total trade divided by GDP using World Bank data. This number might seem high but is not far from the estimates reported by Anderson and van Wincoop (2004), and aligns well with the U.S. economy being one of the most 'closed' economies world-wide. Our calibration implies that final goods' exporters are 60.5% more productive than non-exporters, and that final goods' export prices are 102% higher than domestic prices. Finally, the international bond adjustment cost parameter in our annual set-up is $\phi_b = 0.01$ following Ghironi and Melitz (2005).

*Measured driving forces* Two of our driving forces can be measured directly from the existing empirical evidence. For the depreciation-adjusted relative price of ICT capital, we use data from Eden and Gaggl (2018).<fnref n="11" /> For markups, we rely on estimates for the U.S. economy from De Loecker et al. (2020). For both shocks, we consider a trend growth rate $g$ that equals the observed average growth rate between 1987 and 2013. Over the period, the relative price of ICT capital falls by $g_\phi = -0.0173$ every period, an overall 36% drop. Further, matching the growth rate of markups requires setting $g_\theta = -0.0102$, implying that the average markup increases by more than 17 percentage points (from 1.35 to 1.53) between 1987 and 2013.

<footnote n="10">¹⁰ As an alternative, previous versions of the paper used the extended path algorithm (see Fair and Taylor (1983)) but this required shocks to be treated as unexpected every period, and persistent – instead of permanent – which is debatable. In any case both algorithms yield relatively close results.</footnote>

<footnote n="11">¹¹ The relative price of robots is adjusted for the observed rise in ICT capital depreciation based on a user-cost approach using the depreciation rate provided by the same authors.</footnote>
<table><caption>**Table 1**
Parameter values.</caption><thead><tr><th>Parameter</th><th>Target / Source</th><th>Value</th></tr></thead><tbody><tr><td>Discount factor</td><td>Annual interest rate of 4%</td><td>β = 0.96</td></tr><tr><td>Share of firm owners</td><td>Imposed</td><td>μ = 0.1</td></tr><tr><td>Convexity of training cost</td><td>Optimized</td><td>ζ = 4.4372</td></tr><tr><td>Depreciation of non-routine skills</td><td>Imposed</td><td>δ<sub>ω</sub> = 0.025</td></tr><tr><td>Non-routine hours worked</td><td>Non-routine labor share</td><td>χ<sup>h</sup> = 0.27</td></tr><tr><td>Routine hours worked</td><td>Routine labor share</td><td>χ<sup>u</sup> = 0.30</td></tr><tr><td>Non-routine labor rel. pdty</td><td>Wage premium u<sup>h</sup><sub>0</sub>/u<sup>u</sup><sub>0</sub> = 1.33 (data)</td><td>ξ = 1.2229</td></tr><tr><td>Initial cost of robots</td><td>Share of ICT capital (m<sub>0</sub> = m<sub>0</sub><sup>*</sup> = 0.06)</td><td>φ<sub>0</sub> adjusted</td></tr><tr><td>Non-routine labor share</td><td>Guerreiro et al. (2021)</td><td>α = 0.5</td></tr><tr><td>Factor elast. of subs.</td><td>Optimized</td><td>1/(1 - ν) = 0.1839</td></tr><tr><td>Initial entry cost</td><td>Normalization (n<sub>0</sub> = 1)</td><td>f<sub>e0</sub> adjusted</td></tr><tr><td>Export cost</td><td>Normalization (n<sub>x0</sub>/n<sub>0</sub> = 0.2)</td><td>f<sub>x</sub> adjusted</td></tr><tr><td>Exogenous exit rate</td><td>n<sub>ω0</sub>/n<sub>0</sub> = 0.143 (U.S. Census Bureau)</td><td>δ = 0.125</td></tr><tr><td>Initial elast. of subs.</td><td>Ghironi and Melitz (2005)</td><td>θ<sub>0</sub> = 3.8</td></tr><tr><td>Pareto curvature</td><td>Ghironi and Melitz (2005)</td><td>ε = 3.4</td></tr><tr><td>Love for variety</td><td>Optimized</td><td>ζ = 0.1863</td></tr><tr><td>Initial trade costs</td><td>Trade/GDP ratio of 18%</td><td>τ<sub>0</sub> = 2.25</td></tr><tr><td>Portfolio adjustment cost</td><td>Ghironi and Melitz (2005)</td><td>φ<sup>b</sup> = 0.01</td></tr><tr><td>Pre-2000 labor prod. growth</td><td>Optimized</td><td>g<sub>ω,1</sub> = 0.0219</td></tr><tr><td>Post-2000 labor prod. growth</td><td>Optimized</td><td>g<sub>ω,2</sub> = 0.0104</td></tr><tr><td>Trade costs growth</td><td>Optimized</td><td>g<sub>τ</sub> = -0.033</td></tr><tr><td>Entry costs growth</td><td>Optimized</td><td>g<sub>f<sub>e</sub></sub> = 0.065</td></tr></tbody></table>

### 3.2. Adjusted parameters and data fit

We are left with a subset of free parameters, which do not have clear data counterpart and/or are unobservable: the training cost parameter $\zeta$, the elasticity of substitution between factors $v/(v-1)$, and the parameter governing households' love for variety $\zeta$, and the trend growth rates of labor productivity $\psi_t$, trade costs $\tau_t$ and entry costs $f_{et}$. Regarding trends, we consider two sub-periods for labor productivity to match the apparent break in GDP per capita after 2000. We set the value of the 7 remaining unknown parameters: $\{\zeta, v, \zeta, g_{\psi,1}, g_{\psi,2}, g_{\tau}, g_{f_e}\}$ to matches the following set of observed trends: (i) the fall in the share of routine labor in output, (ii) the rise in labor-market reallocation $\Lambda_t$, (iii) the rise in the non-routine wage premium, (iv) the decline in business dynamism, measured by the fall in the entry rate $(n_{et}/n_t)$, (v) the rise in trade openness and (vi) the rise in GDP per capita.<fnref n="12" />

We find $\zeta = 4.4372$, $v = -4.4384$, $\zeta = 0.1863$, $g_{\psi,1} = 0.0219$, $g_{\psi,2} = 0.0104$, $g_{\tau} = -0.0330$, $g_{f_e} = 0.065$. The production parameter $v$ implies a strong complementarity between routine and non-routine tasks, as the elasticity of substitution is $1/(1 - v) = 0.1839$. In a framework where robots and routine labor are perfect substitutes, complementarity is needed to account for a substantial rise in the non-routine labor share, consistent with what is observed – although not targeted by our procedure – in the data. The parameter governing the strength of love for variety implies that households value varieties – roughly four times – less than in the typical CES case since $\zeta = 0.2669 < 1$, in line with previous findings (see Lewis and Poilly (2012) for instance). Labor productivity is growing fast (2.2% per year) before 2000 and at a slower pace after 2000 (1.04% per year), trade cost decrease significantly (by 58% overall) and entry costs surge (by 400%). The calibration is summarized in Table 1 and Fig. 1 reports our baseline simulation.

In Fig. 1, the two first rows are our targets and their dynamics are all well matched. The third row and the two first panels of the last row report the dynamics of exogenous processes, showing the large drop in the relative price of robots, the rise in markups, the rise in entry costs, the rise in labor productivity – with a kink around 2000 due to productivity slowdown – and the fall in trade costs. Finally, the last panel of the last row reports the dynamics of the non-routine labor share, which is not targeted by our procedure, showing that the model does a good job in replicating its observed dynamics.<fnref n="13" />

### 4. Trends and mechanisms

Let us now dig further in the effects of the fall in the relative price of robots. As already mentioned, our model of endogenous automation implies that the share of automated tasks is not only affected by the declining trend in the relative price of robots. Fig. 2 offers a decomposition of the contribution of all the trends to key variables. We could introduce trends one by one and look at their contributions. However, this approach would result in large non-linearities and discrepancies, especially because large negative trends – that decrease automation – hit the zero lower bound of the fraction of automated tasks, which is -5.8 percentage points, very fast. But when interacted with positive trends – that increase automation, the effects of negative trends might be much larger through

<footnote n="12">¹² The share of routine labor, the relative non-routine employment used to compute labor reallocation, and the non-routine wage premium are taken from Eden and Gaggl (2018). The entry rate comes from the Census Bureau Business Dynamics Statistics. GDP per capita and the trade openness ratio are taken from World Bank data.</footnote>

<footnote n="13">¹³ A Cobb-Douglas production function subjected to automation shocks would simply reallocate the income shares between the automated and non-automated routine tasks, that is, between robot capital and routine labor, and leave the non-routine labor share constant and equal to its steady-state level.</footnote>
<figure><img src="image_3.png" /></figure>

Note: Data taken from De Loecker et al. (2020) (markups), Eden and Gaggl (2018) (routine labor share, non-routine to routine employment and adjusted price of ICT capital), Census Bureau Business Dynamics Statistics (entry rate) and World Bank (GDP per capita, trade openness).

<caption>Fig. 1. Data vs. simulated time series.</caption>

lower induced automation. So we proceed as follows to make sure that the zero lower bound of automated tasks is never hit. First, we consider the marginal contribution of the trend on $\phi_t$ simulating the model with this trend only. For other trends, we run model simulations interacting each trend with the trend on $\phi_t$, and compute their marginal contribution of as the difference between the resulting trajectory and the simulation with the trend on $\phi_t$ only. Finally, we impute the difference between the baseline simulation and the sum of the marginal contributions – the residual – to the interaction between trends.<fnref n="14" />

## 4.1. The declining price of robots

Let us start with the marginal effects of the fall in the price of robots, the blue areas in Fig. 2. Taken alone, the fall in the relative price of robots is mirrored by an equivalent fall in the wage of routine workers, which implies an equivalent negative response of routine workers' consumption $c_t^u$ (panel (d)). Equation (21) also shows that a fall in $\phi_t$ fosters automation and leads the fraction of routine tasks performed by robots to surge from 5.8% in 1987 to more than 53%. Given the mild complementarity between automatable and non-automatable tasks, the more intensive use of robots is also met with a reallocation of labor to the non-routine sector through higher training (panel (c) and (h)). The higher demand of non-routine labor is illustrated by the large rise in the non-routine wage premium (+108%, panel (g)). Labor-market reallocation and the large increase in the wage premium induce the consumption of non-routine workers to rise (+36%). Clearly, routine workers loose while non-routine workers gain from the fall in robots' prices. However, since the proportion of non-routine workers increases, this trend generates aggregate welfare gains (+11.3%, panel (i)), further strengthened by the huge increase in firm owners' consumption (+280%). All else equal, the stronger use of robots

<footnote n="14">¹⁴ In all cases, from now on and to allow for welfare evaluations, given that welfare is forward-looking, we solve the model for an additional 50 periods assuming the economy stays in its 2013 steady state driven by the labor productivity trend only.</footnote>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

Note: Welfare is expressed as the weighted sum of Hicksian equivalent changes in consumption of routine workers, non-routine workers and firm owners. Variables are reported in percentage or percentage point difference from the initial equilibrium.

<caption>Fig. 2. Trends decomposition. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</caption>
and the stimulation of entry made possible by the lower marginal cost both increase firm profits much more than it raises the average consumption of workers, concentrating income across the distribution.

Indeed, beyond the displacement effect, the fall in the relative price of robots also lowers the production cost of intermediate goods $mc_t$. Hence, on top of the direct increase in production made possible by the greater use of robots, the fall in the relative price of robots generates indirect output gains at the extensive margin by inducing more firms to enter the market – although not immediately, as shown by the rising entry rate (up to +2.8 pp, starting from 14.2% in 1987). Since aggregate demand expands, expected profits rise and new firms are created, which further fuels aggregate demand, and even more final goods producers enter. Regarding the open-economy dimension, the fall in the relative price of robots does not appear to play a key role for trade openness or the export threshold per se, and its contribution even appears to be slightly negative.

The above results suggest that the falling relative price of robots is a key driver of automation. It generates displacement effects leading to a rising non-routine wage premium and pushes workers to reallocate from the routine to the non-routine sector. Further, it brings positive productivity effects that lower the production cost and boost output at the intensive margin. A decline in the price of robots also adds positive effects at the extensive margin and potentially raises business dynamism. The resulting welfare gains fall partly on non-routine workers, routine workers experience welfare losses, but they mainly benefit firm owners and robot producers.

## 4.2. The role of other trends

Fig. 2 also shows that automation is not only driven by the decline in the price of robots. In particular, panel (b) shows that negative trends – rising markups (red areas) and entry costs (yellow areas) – dampened the adoption of robots in the production process. Both trends contribute to lower the fraction of automated tasks by 16pp (when taken together), and decrease the non-routine wage premium (by roughly 100% when taken together, markups contributing much more than entry costs). These relative wage dynamics hurt the consumption of non-routine workers and discourages routine workers from training ($s_t$ falls by 1.3 percentage points). Both trends thus push the labor reallocation process in the opposite direction, from the non-routine to the routine sector. The rise in markups acts mostly through the intensive margin and lowers aggregate demand but leaves the entry rate almost constant. The rise in entry costs acts more clearly through the extensive margin and lowers the consumption of non-routine workers and firm owners in the long run. Taken together, the two trends lower output roughly by 47% compared to its 1987 level. In terms of welfare, the rising trend in markups taken alone generates welfare losses around 15% while the rise in entry cost taken alone generates welfare losses around 6.5%.

Fig. 2 also shows that positive trends such as the rise in labor productivity (purple areas) or the decline in trade costs (green areas) had effects on the process of automation, although quantitatively less massive than the negative effects of markups and entry costs. The decline in trade costs slightly increases automation while the rise in labor productivity lowers it, all else equal. Labor productivity mostly redistributes income from firm owners to workers, routine workers benefiting much more than non-routine workers. Both trends unambiguously produce aggregate lifetime welfare gains of 2.7% (trade costs) and 34% (labor productivity) respectively.

## 4.3. Alternative assumptions

We now contrast the results of the baseline model with those produced by alternative models/assumptions. First, we consider a closed-economy version to fully assess the contribution of the open economy – both in terms of trade and financial markets – to the results.<fnref n="15" /> Second, we consider a version with a constant number of produced and traded varieties to evaluate the contribution of the extensive margin of output.<fnref n="16" /> Third, we look at a version where all household types share risks to achieve the same level of consumption within a single large family.<fnref n="17" /> Fig. 3 represents the transition under these alternatives when considering all trends and compares the results with the baseline case.

In a closed economy (dotted black line), the transition is characterized by a slightly less intense automation process, leading to dampened labor reallocation towards the non-routine sector and an attenuated rise in the wage premium. The consumption of routine workers remains unaffected, since it is entirely driven by the price of robots and labor productivity. The consumption of non-routine workers and firm owners both increase substantially less in the baseline, resulting in lower welfare gains. These alternative dynamics align very closely with the marginal contribution of the trend in trade costs highlighted in the above decomposition.

The model with constant produced and traded varieties displays radically different dynamics: output increases more than two times more, driven by much larger automation and subsequent labor reallocation towards the non-routine sector. The consumption gains for non-routine workers and firm owners are both more than doubled. As a result aggregate welfare gains are much larger than in the baseline case. These dynamics are the result of (i) the neutralization of the negative trend on entry costs and (ii) the translation of larger quantities produced at the intensive margin to trade flows, leading to much more trade openness. Overall these effects are

<footnote n="15">¹⁵ We generate a closed economy by imposing very large steady-state trade costs $τ = 200$ instead of $τ = 2.25$ and by neutralizing the decline in trade costs.</footnote>

<footnote n="16">¹⁶ In this case we replace the entry conditions and the export thresholds of the baseline model by a relation that keeps produced and traded varieties constant and equal to their initial steady-state values. The entry cost trend has no traction anymore.</footnote>

<footnote n="17">¹⁷ With perfect insurance across households, we pose $c_t = (1 - \mu)\lambda_t c_t^h (1 - \lambda_t) c_t^e + \mu c_t^e$ and log preference in aggregate consumption, and use the associated stochastic discount factor in the Euler equation on the mutual fund shares (owning firms) and the Home and Foreign bonds.</footnote>
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

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<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<caption>Fig. 3. Alternative model assumptions.</caption>

<footnote>Note: Welfare is expressed in percentage change of initial lifetime welfare. Variables are reported in percentage or percentage point difference from the initial equilibrium.</footnote>
also very much consistent with our previous analysis of the marginal (negative) effects of the trend in entry costs, which are muted
in this experiment by definition.<fnref n="18" />

Our last experiment with perfect insurance across households of the same country highlights that aggregate dynamics are only marginally affected, except for the short-run dynamics of the entry rate. As such, perfect insurance mostly alters the dynamics of welfare/consumption gains across different types of households: with perfect insurance winners have to share their gains with losers, which in our case means that firm owners experience dramatically lower gains, non-routine workers slightly larger gains and routine workers now gain overall instead of losing.<fnref n="19" />

4.4. Summary

The above decomposition stems from an open-economy model with endogenous automation, business formation and labor-market reallocation that replicates most trends of the U.S. economy over the last 40 years: the rise of automation, the labor-market reallocation from the routine to the non-routine sector, the rise in markups, the decline in business dynamism, the rise in trade openness and the observed rise in aggregate output. The decomposition suggests that automation was mostly driven by the fall in the price of robots, but also resulted from opposing forces induced by other trends. The observed rise in markups and entry costs dampened automation and labor-market reallocation while the decline in labor productivity and trade costs slightly reinforced them. Labor productivity mostly acts to offset the redistributive effects of trend on robots' price and brings welfare gains to all types of households. If the sole driver of the U.S. economy over this period had been the fall in the price of robots, automation and labor-market reallocation would have been quantitatively much more important, bringing a much more unequal distribution of welfare gains to the benefit of firm owners. However, other forces came into play to generate a different structure of welfare gains: rising markups and the decline in business dynamism lowering aggregate welfare and labor productivity increase aggregate welfare.

5. Counterfactual analyzes

Let us now use our model to consider two counterfactual exercises. In the first one, we assume that the rest of the world faces a larger or smaller decline in the relative price of robots. In the second one, we look at the potential effects of an optimized tax/subsidy on robots, a subsidy on training and a combination of both. While a complete analysis of an optimal policy is beyond the scope of this paper, it still sheds light on the potential inefficiencies induced by the decline in robots' price, those induced by other trends, and can inform about international fiscal interactions.

5.1. Asymmetric decline in the relative price of robots

Let us first assume that the rest of the world faces a larger or smaller decline in the growth rate of robots price $g_{\phi}^{*}$ than the U.S. $g_{\phi}$. Fig. 4 reports the marginal effects – compared to the symmetric baseline – of the relative price of robots falling 10% faster or slower abroad.

Fig. 4 shows that when the rest of the world faces a faster decline in the relative price of robots, the Home (U.S.) economy grows faster, automates more, and faces a decline in business dynamism. As the rest of the world automates faster and experiences more productivity gains, more firms locate there and less firms are created in the Home economy, as shown by the decline in the Home entry rate. However, the Home economy benefits from the faster decline in robots prices through cheaper imports and stronger demand for its goods produced locally. The initial relative real exchange rate depreciation further stimulates Home exports and generates a short-run trade surplus.

These movements lower the marginal production cost $mc_t$, which triggers additional automation in the Home economy. As firms adopt more robots to perform automatable tasks, the non-routine wage premium rises and labor reallocates towards the non-routine sector. So the Home economy gets an extra kick of automation through trade linkages when the rest of the world faces a faster decline in the price of robots, and this trend generates positive cross-country spillovers in our framework.

Quantitatively speaking, these positive spillovers from a faster decline in robots price are very small, just an fraction of percentage points. One can also notice that, when the Foreign economy experiences a slower growth of robots prices, the opposite effects are somewhat larger, especially on the entry rate, trade balance and real exchange rate. This reflects the non-linearity of the model and more particularly the asymmetry of the Pareto distribution of firms' productivity and entry/export cut-offs.

5.2. Optimized tax on robots or training subsidy

For the moment we only raised positive questions regarding the ability of our model to account for the observed dynamics of the U.S. economy. However, we used our model to quantify the welfare effects of the various trends we considered. We now ask whether the introduction of a tax on the robots $\tau_{rt}$ or a training subsidy $\tau_{st}$ can deliver transition paths with higher aggregate welfare for the

<footnote n="18">¹⁸ Note also that the kink around 2000 due to productivity slowdown is much attenuated since its effects running through the entry rate and firm dynamics are muted.</footnote>

<footnote n="19">¹⁹ The one-family case also features larger aggregate welfare gains, mostly as a result of aggregating consumption before taking the log in utility rather than summing the logs of consumption levels.</footnote>
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<figure><img src="image_37.png" /></figure>

Note: Welfare is expressed in percentage change of initial lifetime welfare. Variables are reported in percentage or percentage point difference from the baseline case. The real exchange rate is expressed so that an increase represents a real depreciation.

<caption>Fig. 4. Asymmetric fall in the relative price of robots.</caption>
household sector. In both cases, we consider that tax proceeds (expenditures in the case of subsidies) are rebated to (paid for by) households in a lump-sum way.

The model is thus modified in the following dimensions, focusing on the Home economy and considering that symmetric relations potentially hold for the Foreign economy. First, a potential tax on robots affects the production cost of intermediate goods producers, and their profits now write:

$$ \Pi_{yt} = mc_t y_t - (1 + \tau_{rt}) \phi_t x_t - w_t^u \ell_t^u - w_t^h \ell_t^h. \quad (40) $$

Optimal conditions for automation are modified $w_t^u/\psi_t = (1 + \tau_{rt}) \phi_t$ and Equation (21) becomes:

$$ m_t = 1 - \left( \frac{1}{\alpha} \left( \frac{(1 + \tau_{rt}) \phi_t}{(1 - \alpha) mc_t} \right)^{\frac{\nu}{1-\nu}} - \frac{1 - \alpha}{\alpha} \right)^{\frac{1}{\nu}} \frac{\ell_t^u}{\xi \ell_t^h}, \quad \text{with } 0 \le m_t \le 1. \quad (41) $$

This equation shows very transparently that the taxation of robots can slow down automation and the associated effects: labor-market reallocation, output growth and rise in business dynamism.

Further, since we assume that the proceeds from the tax on robots are redistributed lump-sum, and that a potential training subsidy affects the training decision, the budget constraints of non-routine and routine workers become:

$$ c_t^h = w_t^h \chi_t^h - T_t \quad (42) $$

$$ c_t^u = (1 - s_t) w_t^h \chi_t^h + \tau_{st} s_t - T_t \quad (43) $$

where $T_t$ is a lump-sum tax/transfer expressed in per-capita terms. The first-order conditions of non-routine workers and firm owners – whose budget constraint is modified but not reported here – are not affected by the potential taxes and subsidies, but the optimal training decision becomes:

$$ \frac{w_t^u \chi_t^u - \tau_{st}}{c_t^u} + \omega s_t^{\zeta-1} = \beta \mathbb{E}_t \{ V_{t+1}^h - V_{t+1}^u \}. \quad (44) $$

Finally, the budget constraint of the government writes:

$$ \tau_{rt} \phi_t x_t + T_t = \tau_{st} s_t. \quad (45) $$

We assume that the tax rates and subsidies respond to the growth rate of robots' price:

$$ \tau_{rt} = d_r g_{\phi}, \quad \text{and/or} \quad \tau_{st} = d_s g_{\phi}. \quad (46) $$

Given the sequence of trends imposed, we choose $d_r$ and $d_s$ so as to maximize aggregate lifetime welfare, which with a constant growth rate of $\phi_t$ will result in constant tax or subsidy rates. In the baseline case, the tax scheme is implemented unilaterally in the U.S.

**All trends.** Let us start with the optimized robot tax path when all trends are considered together. Fig. 5 below tracks the dynamics of key variables with an optimized world tax on robots, an optimized training subsidy or a combination of both, and compares these cases to the baseline (tax-free) path.

Fig. 5 shows that the welfare maximizing tax rate on robots is negative, around 15%; meaning that maximizing welfare would actually imply subsidizing robots. This would boost output, labor-market reallocation, training and business dynamism. While routine workers would clearly loose in terms of consumption and welfare, aggregate welfare gains would materialize, since (a) routine workers massively become non-routine workers over time, and (b) non-routine workers experience substantial consumption and welfare gains through amplified wage premium dynamics. An optimal 28% training subsidy achieves a similar outcome qualitatively, although its quantitative effects are much smaller than those of an optimal robot subsidy. The weaker potency of the training subsidy is further illustrated by the fact that the optimal mix of both instruments is an almost unchanged 14% robot subsidy along with a much smaller 8% training subsidy.

**Trend on $\phi_t$ only.** The above experiment is biased because it may not be driven by automation only. Indeed the tax on robots or the training subsidy may be used to undo other distortions, for instance induced by the (partly inefficient) rise of entry costs or markups. To answer this concern, we now run the exact same experiment but with the trend on $\phi_t$ as the only driving force. We report the results in Fig. 6.

First, Fig. 6 shows that, absent other trends and associated distortions, the robot subsidy is much smaller, around 1.4% – against 14-15% with all trends. The training subsidy also declines significantly to 0.6% in the case of an optimal mix – against 8% with all trends. These results show that other trends and implied distortions partly hindered the automation process and the associated labor-market reallocation, overturned the automation-induced rise in business dynamism and deeply affected the dynamics of welfare gains and losses over time, leading to an inefficiently low degree of automation.

Second, it suggests that optimal tax rate/subsidies with all trends in Fig. 5 were used to partly offset the negative welfare effects of rising markups and entry costs. Once these distortions are muted, the need to correct the path of robots' adoption is diminished.
<figure><img src="image_38.png" /></figure>

Note: Welfare is expressed in percentage change of initial lifetime welfare. Variables are reported in percentage or percentage point difference from the initial equilibrium.

<caption>Fig. 5. Dynamics with welfare-maximizing taxation in the US – all trends.</caption>
<figure><img src="image_39.png" /></figure>

Note: Welfare is expressed in percentage change of initial lifetime welfare. Variables are reported in percentage or percentage point difference from the initial equilibrium.

<caption>Fig. 6. Dynamics with taxation – trend on $\phi_t$ only.</caption>
Through the lens of our model, automation is close-to-efficient when the price of robots is the only driving force. The need to subsidize robots' adoption and training stems from the inefficient choice of routine workers regarding the speed at which they reallocate towards the non-routine sector.

These results further strengthen our main point: analyzing automation in relation to contemporaneous trends affecting business dynamism and market power is of prime importance. Not only to properly account for the drivers of automation, but also for the potential correction of induced inefficiencies.

**World-wide implementation.** In the above experiments, optimized tax rates and subsidies are implemented unilaterally in the U.S. However, what is the optimized tax scheme is implemented world-wide? Fig. 7 in Appendix B presents the results obtained from taxing/subsidizing robots and training globally. We focus on the jointly optimized tax/subsidy on robots and training with $\phi_t$ as the only driving force, and compare the world-wide and unilateral schemes, along with the baseline case without taxes or subsidies. The main result is that the global subsidies on robots and training are both slightly larger than the unilateral. Unilateral implementation thus leads to inefficiently low automation and labor-market reallocation, and a better outcome could be achieved by coordinating policies world-wide. Quantitatively speaking though, the global implementation does not make a large difference, in line with our result regarding the small size of international spillovers from an asymmetric decline in robots' price.

## 6. Conclusion

Since the 1980's, the number of robots per thousands of industry workers has been rising very fast and it should intuitively result in large productivity and output gains. We match key trends in the U.S. economy, among which the rise in automation, with an open-economy model featuring endogenous automation, heterogeneous firms and labor-market reallocation.

We show that the decline in the relative price of robots is a key factor leading to automation, and that it affects the economy in expected ways. However, other factors such as rising markups, rising entry costs, rising labor productivity or declining trade costs interact significantly. In particular, the joint decline in business dynamism and rise in markups strongly hindered automation. Singling out the effects of a decline in robots prices, we find that it generates welfare gains but less than labor productivity growth.

Last, counterfactual experiments suggest that declining robots' price generate small positive cross-country spillovers, and inefficient labor-market reallocation since a simple tax experiment shows that a constant subsidy on robots combined with a training subsidy can improve aggregate welfare.

## Appendix A. Details of the worker problems

The value function of non-routine workers writes:

$$ 
\begin{aligned}
V_t^h = \log(c_t^h) & - \mu_{1t}^h [c_t^h - w_t^h \chi_t^h] \\
& - \mu_{2t}^h [\lambda_t - (1 - \delta_w) \lambda_{t-1} - s_{t-1} (1 - \lambda_{t-1})] \\
& + \beta \mathbb{E}_t \{(1 - \delta_w) V_{t+1}^h + \delta_w V_{t+1}^u \},
\end{aligned} 
\quad (47) 
$$

while that of routine workers is given by:

$$ 
\begin{aligned}
V_t^u = \log(c_t^u) & - \frac{\bar{\omega}}{\zeta} s_t^\zeta - \mu_{1t}^u [c_t^u - (1 - s_t) w_t^u \chi_t^u] \\
& - \mu_{2t}^u [\lambda_t - (1 - \delta_w) \lambda_{t-1} - s_{t-1} (1 - \lambda_{t-1})] \\
& + \beta \mathbb{E}_t \{(1 - s_t) V_{t+1}^u + s_t V_{t+1}^h \}.
\end{aligned} 
\quad (48) 
$$

First-order conditions of the routine family are:

$$ c_t^u : \frac{1}{c_t^u} = \mu_{1t}^u, \quad (49) $$

$$ s_t : \beta \mathbb{E}_t \{(V_{t+1}^h - V_{t+1}^u)\} = \mu_{1t}^u w_t^u \chi_t^u + \bar{\omega} s_t^{\zeta-1}, \quad (50) $$

and imply

$$ \frac{w_t^u \chi_t^u}{c_t^u} + \bar{\omega} s_t^{\zeta-1} = \beta \mathbb{E}_t \{V_{t+1}^h - V_{t+1}^u\}, \quad (51) $$

with

$$ V_t^h - V_t^u = \log(c_t^h) - \left( \log(c_t^u) - \frac{\bar{\omega}}{\zeta} s_t^\zeta \right) + \beta \mathbb{E}_t \{(1 - \delta_w - s_t) (V_{t+1}^h - V_{t+1}^u)\}, \quad (52) $$

and implicitly defines the optimal training choice $s_t$ as a choice equalizing the expected utility gain from training to its current marginal utility cost.
Appendix B. World-wide implementation of optimized taxes and subsidies

<figure><img src="image_40.png" /></figure>

Note: Welfare is expressed in percentage change of initial lifetime welfare, non-routine workers and firm owners. Variables are reported in percentage or percentage point difference from the initial equilibrium.

<caption>Fig. 7. Dynamics with taxation – trend on $\phi_t$ only – Unilateral vs global implementation.</caption>
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