Cost minimizing automation measure ;

(1947,      0.1649)–(1952,     0.1649) node [pos=0.5,above,yshift=1.25em, font=]0.1677;

(1960, 0.2349)–(1965,0.2349) node [pos=0.5,above,yshift=0.25em, font=]0.235;

(1978,0.3098) –(1982, 0.3098) node [pos=0.5,above,yshift=0.25em, font=]0.309;

(1995, 0.2724)–(2001,0.2724)node [pos=0.5,above,yshift=0.5em, font=]0.2724;

(2006, 0.199)–(2012,  0.199)node [pos=0.5,above,yshift=0.15em,font=]0.199;

Automation measures. Notes— The steady-state measure of automation is obtained from (<ref>) by using the calibration in Table <ref> and the data of the  capital cost share and the rate of return of capital in Figure <ref>. The cost-minimizing automation measure follows from Lemma <ref>.

This strategy was backed by a faith in Keynesianism— crystallized in significant tax cuts for business and upper-income groups in 1964, the expansion of welfare throughout the 1960s and 70s, and the institutionalization of countercyclical economic policies—which ultimately  limited  the restoring effects that unemployment can have on profitability by setting a downward pressure on wage increases.[This conclusion  follows by increasing $Z_{t}$ and $\theta_{t}$ in equation  (<ref>). Particularly, if $Z_{t}$ was increased by the government's concern over social welfare and if $\theta$ was kept relatively large as a result of demand driven policies, the result is a decreasing rate of return of capital, which is what is shown in  Figure <ref>, Panel A. ]  From this perspective it is expected that in the face of  mounting labor costs, especially after 1964, firms  accelerated the adoption of labor replacing technologies. This is precisely what the data of mann2021benign and the steady-state measure of automation derived from equation  (<ref>) show in Figure <ref>. The increasing relative  costs of labor is picked up by the  rise in $1-\bar{m}_{t}$ from 1962 to to 1980,  which provides theoretical  support to the view that the rise of welfare and the expansion of the citizen wage created the condition for automation to become a powerful player in reducing labor costs and increasing unemployment.

By these standards, the computer revolution and similar technological improvements found  fertile grounds for creating potentially  disruptive effects on the labor market after the 1980s. According to the calibration of the model, by 1980 $1-\bar{m}_{t}$ was about twice as large as $1-m_{t}$, meaning that the system lied far above the boundary dividing  regions 1 and 2 in Figure <ref> and that even small changes in automation could have generated significantly negative effects on employment and real wages. The expansion of labor replacing technologies  throughout the two decades from the early 1980s to the late 1990s was, however, relatively small  according to all measures in Figure <ref>. One of the reasons for this may have been the conservative retrenchment that followed after the late 1970s, which reduced the expansion of labor costs relative to the costs of capital.

The 2000s saw a second revival of labor replacing technologies. Consistent with the findings of autor2018automation, Figure <ref> depicts a strong increase in the steady-state automation measure which moves along the data of Dechezlepretre2019 and mann2021benign.   A peculiar characteristic of this period is that it coincided with a fall in the labor share, suggesting that the rise in automation was probably not prompted by rising labor costs but rather by technology improvements or tax cuts for capital. acemoglu2020does present evidence suggesting that the adoption of labor replacing technologies may have resulted from changes in the US tax system favoring the adoption of capital over labor. If  acemoglu2020does are correct, the fall in the cost-minimizing automation measure was likely  lower than was is reported in Figure <ref>, meaning that the disruptive effects of automation probably  continued to be important even after 2010.[ It is likely that COVID-19 exacerbated  the effects of automation on the labor market  since one of the defining characteristics of the  pandemic was the unprecedented increase in unemployment benefits and tight labor markets following the lock-downs. In this scenario it can be expected  that labor replacing technologies will be potentially effective  in reducing production costs of firms, which may  help explain the growing concerns in the media over the  effects automation.]

§ CONCLUSIONS

This paper provides a framework that helps understand how institutions, automation, unemployment, and profitability  interact  in the dynamic setting of a general equilibrium analysis. At the center of the model is the notion that profits are  a surplus over costs of production, which brings back to light the importance of  political and institutional factors in the determination of income distribution.

Among  the attractive  features of the paper is that it creates a link between task-based models, the surplus approach of the Classical economists, and  the literature on equilibrium unemployment. The merger of these  approaches introduces three  important theoretical contributions to the literature. First, it  establishes an endogenous theory of profits based on the relations of power between capitalists and workers. Second, it formalizes how unemployment and wage  dynamics are directly affected by the assignment of tasks between capital and labor. Third, it establishes how institutions can directly affect the task allocation of factors by intervening in the relative prices of labor and capital.

The empirical strength of the model is captured using two complementary strategies. First it is shown that  the behavior of income shares, capital returns, the rate of unemployment, and the big ratios of macroeconomics  in the American economy can be  explained by specific changes in labor institutions and technology. The second part  supports this conclusion by showing that the implied changes in the equilibrium measures of worker power and automation are consistent  with the history of welfare and technical change in the  US. Ultimately, this strategy highlights that there is much to be gained by locating the abstract reasoning of  economic theory into  the  historical context of a specific society.

From a political economy perspective, the article presents a framework to understand how capitalism interacts with the institutions of liberal democracy. For instance, one of the conclusions drawn from the analysis  is that there may exist a conflict between the social and economic sphere of citizenship with the profit-making capacity of the capitalist system.  The empowerment of workers can increase the outside option to employment and lead to a reduction of the rate of return  which may ultimately threaten the reproduction of capital at an increasing scale. In this respect, one of the key policy implications of the paper is that  the search for more equitable societies must consider the constraint imposed by profitability,  since a steady-state growth path cannot be sustainable  with declining rates of profit.   It is equally important  to note that   governments  play an active in protecting workers from the disruptive effects of unregulated markets—some of which are portrayed by the negative consequences that automation can have on employment and wages. The bottom line in this respect is that the balance between  more progressive societies and a more capital friendly system is, to a large extent, a political choice resulting from the particular time and place in the process of history.

The paper  sets the stage for at least four avenues of future research.  The first  is a study of the dynamic behavior of the model by introducing stochastic variables and evaluating how the economy reacts to shocks at the business cycle time scale. A  second avenue is to introduce credit in order  to understand how interest payments are determined when—consistent with business accounting—interest rates are not counted as part of costs of production but are rather paid from the surplus extracted from the process of production. This could serve the purpose of showing how “capital costs" are separated from “pure profits,"  and could consequently help understand the changes of capital, labor and profit shares in the past  50 years in the US; see, e.g., karabarbounis2019accounting and barkai2020declining. The model can also benefit from the introduction  of bargaining models offering a more accurate representation of the conflicting interests between capitalists and workers in wage negotiation processes.   Lastly, it is important  to introduce the government sector as an active agent so that institutional and political factors are themselves an endogenous response to the economic outcomes of the system.

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§ APPENDIX

§ AUXILIARY DERIVATIONS

§.§ Proof of Equation (<ref>)

Given the property of Poisson processes and the assumption of a large number of projects <cit.>:

\begin{equation}\label{eq:poisson_prop}
\frac{\int_{i \in \mathcal{H}_{t}} \; x_{t}(i) di}{\pi_{I}}=\frac{\int_{i \notin \mathcal{H}_{t}} \; x_{t}(i) di}{1-\pi_{I}}=\int_{0}^{1} \; x_{t}(i)\; di \;\; \;\; \text{for }\;\; x_{t}(i)=\Big\{\mathcal{I}_{t}(i), \mathcal{I}^{\frac{\upsilon-1}{\upsilon}}_{t}(i) \Big\}.
\end{equation}

The objective of the investment   firms is to maximize $P^{I}_{t} I_{t} - P_{t} Y_{t}$ subject to (<ref>). Using (<ref>),  maximization problem yields  $P^{I}_{t}=P_{t} \Psi^{-1}_{t}$.  Similarly, using (<ref>),   investment expenditure can be expressed as

\begin{equation*}
\begin{split}
\mathcal{X}_{t} &= P^{I}_{t} \int_{0}^{1} \mathcal{I}_{t}(i) \; di =  P^{I}_{t}  \Bigg[ \int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}_{t}(i) \; di + \int_{i \notin \mathcal{H}_{t-1}}  \mathcal{I}_{t}(i) \; di \Bigg]\\
&=  P^{I}_{t}  \Bigg[  \int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}_{t}(i) \; di + (1-\pi_{I}) \Bigg\{\int_{i \notin \mathcal{H}_{t-1}}  \mathcal{I}_{t}(i) di \Bigg\}/(1-\pi_{I})  \Bigg]\\
&=  P^{I}_{t}  \Bigg[ \int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}_{t}(i) \; di + (1-\pi_{I}) \int_{0}^{1} \mathcal{I}_{t-1}  \Bigg]=  P^{I}_{t}  \Bigg[  \int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}_{t}(i) \; di + (1-\pi_{I}) \frac{\mathcal{X}_{t-1}}{P^{I}_{t-1}}\Bigg]
\end{split}
\end{equation*}

A similar argument can be extended to express the amount of investment. Using (<ref>) and (<ref>):

\begin{equation*}
\begin{split}
I_{t} &= \Bigg(\int_{i \in \mathcal{H}_{t}} \mathcal{I}_{t}(i)^{\frac{\upsilon-1}{\upsilon}} di\Bigg)^{\frac{\upsilon}{\upsilon-1}}=\Bigg( \pi_{I} \int_{0}^{1}  \mathcal{I}_{t}(i)^{\frac{\upsilon-1}{\upsilon}} di\Bigg)^{\frac{\upsilon}{\upsilon-1}}\\
&=\Bigg[ \pi_{I}  \Bigg( \int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}^{\frac{\upsilon-1}{\upsilon}}_{t}(i) \; di + \int_{i \notin \mathcal{H}_{t-1}}  \mathcal{I}^{\frac{\upsilon-1}{\upsilon}}_{t}(i) \; di \Bigg) \Bigg]^{\upsilon/(\upsilon-1)}\\
&= \Bigg[  \pi_{I}\Bigg(\int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}^{\frac{\upsilon-1}{\upsilon}}_{t}(i) \; di + (1-\pi_{I}) \Bigg\{\int_{i \notin \mathcal{H}_{t-1}}  \mathcal{I}^{\frac{\upsilon-1}{\upsilon}}_{t}(i) di \Bigg\}/(1-\pi_{I})  \Bigg)\Bigg]^{\upsilon/(\upsilon-1)}\\
&= \Bigg[  \pi_{I} \int_{i \in \mathcal{H}_{t-1}}  \mathcal{I}^{\frac{\upsilon-1}{\upsilon}}_{t}(i) \; di + (1-\pi_{I}) I_{t-1}^{\frac{\upsilon-1}{\upsilon}} \Bigg]^{\upsilon/(\upsilon-1)}.
\end{split}
\end{equation*}

If  firms minimize investment expenditure subject to (<ref>), they will all choose $\mathcal{I}_{t}(i)=\mathcal{I}_{t}$ for all $i \in \mathcal{H}_{t-1}$, so that

\begin{equation*}
\mathcal{I}_{t} = \pi_{I}^{\frac{2 \upsilon}{1-\upsilon}} \Big[I_{t}^{\frac{\upsilon-1}{\upsilon}} - (1-\pi_{I}) I_{t-1}^{\frac{\upsilon-1}{\upsilon}} \Big]^{\frac{\upsilon}{\upsilon-1}}.
\end{equation*}

Replacing the last equation on $\mathcal{X}_{t}$, it follows that

\begin{equation}
\begin{split}
\mathcal{X}_{t}&=  P^{I}_{t}  \Bigg[  \pi_{I}  \int_{0}^{1} \mathcal{I}_{t}(i) \; di + (1-\pi_{I}) \frac{\mathcal{X}_{t-1}}{P^{I}_{t-1}}\Bigg]\\
&= P^{I}_{t}  \Bigg[ \Omega(I_{t},I_{t-1}) + (1-\pi_{I}) \frac{\mathcal{X}_{t-1}}{P^{I}_{t-1}}\Bigg]
\end{split}
\end{equation}

where $\Omega(I_{t},I_{t-1}) \equiv \pi_{I}^{\frac{1+\upsilon}{1-\upsilon}} \Big[I_{t}^{\frac{\upsilon-1}{\upsilon}} - (1-\pi_{I}) I_{t-1}^{\frac{\upsilon-1}{\upsilon}} \Big]^{\frac{\upsilon}{\upsilon-1}}$.  This is precisely what is obtained from (<ref>).

§.§ First-order Conditions of Workers and Capitalists

The Lagrangian associated with the optimization problem of workers is

\begin{equation*}
\begin{split}
\mathcal{L}^{w}&=\sum_{t=0}^{\infty} \Big(\prod_{i=0}^{t} \beta^{w}_{i}\Big) \Bigg\{ \Big[  L_{t} \;\mathcal{U}^{we}(C^{we}_{t},h_{t}) +  U_{t}\; \mathcal{U}^{we}(C^{wu}_{t},0)\Big]+ \phi^{w}_{0,t}\Big(w_{t} N_{t} + U_{t} b_{t}- \\
& P_{t} C^{we}_{t} L_{t} -P_{t} C^{wu}_{t} U_{t} \Big)+ \phi^{w}_{1,t}\Big( (1-\lambda - U^{A}_{L_{t},t})L_{t} + f(\theta_{t})U_{t}-L_{t+1}\big) +\\
& \phi^{w}_{2,t}\Big(  U_{t} (1-f(\theta_{t}))   +  (\lambda + U^{A}_{L_{t},t} )L_{t} - U_{t+1} \big) \Bigg\}
\end{split}
\end{equation*}

Where $\phi^{w}_{i,t}$ are the Lagrange multipliers.  Using the  first order conditions it follows that:

\begin{equation}
\begin{split}
&C^{we}_{t}, C^{wu}_{t}:  \;\;\;   \mathcal{U}^{we}_{C^{we}_{t},t} - \phi^{w}_{0,t} P_{t} =0, \;\;   \;\;\;   \mathcal{U}^{wu}_{C^{wu}_{t},t} - \phi^{w}_{0,t} P_{t} =0\\
&L_{t}: \;  \beta^{w}_{t}\Big[ \mathcal{U}^{we}(C^{we}_{t},h_{t}) + \phi^{w}_{0,t} w_{t} h_{t}  -\phi^{w}_{0,t} P_{t} C^{we}_{t} + \phi^{w}_{1,t} (1-\hat{\lambda}_{t} ) + \phi^{w}_{2,t} \hat{\lambda}_{t}\Big] = \phi^{w}_{1,t-1}\\
&U_{t}: \;   \beta^{w}_{t}\Big[\mathcal{U}^{wu}(C^{wu}_{t},0) + \phi^{w}_{0,t} b_{t} -\phi^{w}_{0,t} P_{t} C^{wu}_{t} + \phi^{w}_{1,t} f(\theta_{t}) + \phi^{w}_{2,t}(1- f(\theta_{t}) )\Big]=\phi^{w}_{2,t-1}.
\end{split}
\end{equation}

Expressing all variables  in   consumption units by dividing by the marginal utility of consumption,  and expressing $\tilde{\beta}^{w}_{t} \equiv \beta^{w}_{t} \mathcal{U}^{w}_{C^{w}_{t},t}/ \mathcal{U}^{w}_{C^{w}_{t-1},t-1}$ and $\tilde{\phi}^{w}_{i,t}=\phi^{w}_{i,t}/\mathcal{U}^{w}_{C^{w}_{t},t}$,  the marginal value of an employed and unemployed worker satisfy

\begin{equation*}
\Phi_{L_{t+1},t+1} = \frac{\tilde{\phi}^{w}_{1,t}}{\tilde{\beta}^{w}_{t+1}} \;\; \text{and}\;\;\; \Phi_{U_{t+1},t+1} = \frac{\tilde{\phi}^{w}_{2,t}}{\tilde{\beta}^{w}_{t+1}},
\end{equation*}

which completes the result in (<ref>).

The Lagrangian associated with the capitalists optimization problem is

\begin{equation*}
\begin{split}
\mathcal{L}^{c}&= \sum_{t=0}^{\infty} {\beta^{c}}^{t}\;  \Bigg\{ \mathcal{U}^{c}_{t}(C^{c}_{t})+ \phi^{c}_{0,t}\Big( P_{t} Y_{t} - P^{k}_{t} I_{t} -w_{t} N_{t} - \kappa_{t} V_{t} -  P_{t} C^{c}_{t}-T_{t}\Big)\\
&+ \phi^{c}_{1,t}\Big( (1-\hat{\lambda}_{t} )L_{t} + q(\theta_{t})V_{t}-L_{t+1}\big)
\end{split}
\end{equation*}

As usual,  $\phi^{c}_{i,t}$ are the Lagrange multipliers. Expressing the first order conditions in consumption units using $\phi^{c}_{0,t} = \mathcal{U}^{c}_{C^{c}_{t},t}/P_{t}$, it follows that:

\begin{equation}\label{eq:foc_capitalist_app}
V_{t}: \; -\frac{\kappa_{t}}{P_{t}}  +\tilde{\phi}^{c}_{1,t} q(\theta_{t})=0; \;\;\;\;  L_{t} : \;  h_{t} Y_{N_{t},t} - \frac{w_{t}h_{t}}{P_{t}} + \tilde{\phi}^{c}_{1,t} (1-\hat{\lambda}_{t})=\frac{ \tilde{\phi}^{c}_{1,t-1}}{\tilde{\beta}^{c}_{t}}
\end{equation}

Where $\tilde{\phi}^{c}_{i,t} \equiv \phi^{c}_{i,t}/\mathcal{U}^{c}_{C^{c}_{t},t}$ and the value of an additional employed labor for the capitalist is $\Lambda_{L_{t+1},t+1}= \big(\tilde{\phi}^{c}_{1,t}/\tilde{\beta}^{c}_{t+1} \big)$.

The shadow cost of capital in consumption units can be obtained from the  maximization problem of capital good producers. The Lagrangian in this problem can be formulated as

\begin{equation*}
\mathcal{L}^{cc}=\sum_{t=0}^{\infty} \Big(\prod_{i=0}^{t} \tilde{\beta}^{c}_{i}\Big) \Bigg\{ P^{k}_{t} I_{t}- P^{I}_{t} X_{t} - \tilde{\phi}^{c}_{3,t} \Big(  \Omega(I_{t},I_{t-1}) +(1-\pi_{I})X_{t-1} -X_{t}\Big) \Bigg\}
\end{equation*}

Using the first order conditions:

\begin{equation}\label{eq:foc_capital_app}
\begin{split}
&I_{t}:\; P^{k}_{t}=\tilde{\phi}^{c}_{3,t} \Omega_{I_{t},t}  +  \tilde{\beta}^{c}_{t+1}  \tilde{\phi}^{c}_{3,t+1} \Omega_{I_{t},t+1} \\
& X_{t}: \;  P^{I}_{t}= \tilde{\phi}^{c}_{3,t} -   \tilde{\beta}^{c}_{t+1} \tilde{\phi}^{c}_{3,t+1} (1-\pi_{I})
\end{split}
\end{equation}

Using the marginal productivity equation of capital, it follows that the demand for capital satisfies

\begin{equation}\label{eq:demand_capita_euler}
Y_{K_{t}} = \frac{\delta P^{k}_{t}}{P^{c}_{t}} = \delta P^{I}_{t} \Omega_{I_{t},t} + \delta \Big[\Omega_{I_{t},t} + \frac{\Omega_{I_{t},t+1}}{1-\pi_{I}} \Big] \sum_{i=1}^{\infty} \Big(\prod_{j=1}^{i} \tilde{\beta}^{c}_{t+j}\Big) (1-\pi_{I})^{i} P^{I}_{t+i}.
\end{equation}

An important implication of (<ref>) is that, because $P^{I}_{t}=(1+\mu_{t})P^{c}_{t} \Psi_{t}^{-1}$, capital demand is a positive function of current and future rates of return.  If, for instance, capitalists receive news that in the future labor institutions will provide further support to workers, it can be expected that they will reduce the current demand for capital.

§ ACCOUNTING STRUCTURE OF THE MODEL

The analysis in the text is carried out on the basis of an accounting structure where capitalists finance all capital investments and labor costs using retained earnings. This  assumption is made in order to highlight the nature of profits as a surplus before complicating the model with the introduction of interest payments and rents. Essentially, given that in the Classical tradition  interests and rents are paid from the aggregate surplus of society in the circulation process of capital, the first logical step is to understand how  profits are reproduced  before introducing exogenous sources of liquidity to  the model.

The underlying accounting structure of the model is well represented in Figure <ref> using Marx's circuits of capital as formalized by foley1986understanding. At stage (a), the chain of intermediate and final good producers use a share of their money funds to hire labor services from workers and pay for new units of capital to capital good producers.

The aggregate flow of capital outlays is committed to a production process which results in a flow value of unsold finished output ($P^{c}_{t} Y_{t}$).  As noted by numerous authors (e.g., haavelmo1960),   not all components of capital outlays are transferred as   finished output after the same length of time in the production process, so the relation between $\mathcal{C}_{t}=w_{t}N_{t} + P^{k}_{t} I_{t} $ and $P^{c}_{t} Y_{t}$ can be accounted for by a convolution <cit.>:

\begin{equation}\label{eq:conserv_p1}
\sum_{t'=-\infty}^{t} \mathcal{C}_{t'} \mathcal{A}(t-t';t')=P^{c}_{t} Y_{t},
\end{equation}

where $\mathcal{A}(t-t';t') \geq 0$ for $ t-t' \geq 0$ represents the distributed shares in the value of capital outlays and $\sum_{t'=-\infty}^{t} \mathcal{A}(t-t';t')=1$. Intuitively, (<ref>) says that finished goods   valued at costs (since it is unsold output) is equal to the weighted sum of the value of all previous capital outlays committed to the process of production.

To simplify the mathematical analysis, I  assume that each  unit of capital  stays in the production process for a time period $T_{P_{t}}$ and then emerges all at  once as a finished good. This is represented, similar to [p. 70]foley1986understanding, as

\begin{equation*}
P^{c}_{t} Y_{t} = \mathcal{C}_{t-T_{P_{t}}}.
\end{equation*}

That is, the value of the final good  measured at cost prices must be equal to the value of capital outlays entering the production process in period $t-T_{P_{t}}$. Given Assumption <ref>, the current value of capital outlays is discounted exponentially with a discount rate  $g_{t}$, denoting  the growth rate of the value of the capital stock,  such that

\begin{equation}\label{eq:prod_time}
P^{c}_{t} Y_{t} = e^{-g_{t} T_{P_{t}}} \mathcal{C}_{t}= \mathcal{C}_{t-T_{P_{t}}}.
\end{equation}

The value of the final good valued at costs $(P^{c}_{t} Y_{t})$  subtracts from the value of the stock of productive capital in the transition from (a) to (b). The value of sales results from the sum of transition flows (c) and (d), such that   $P_{t}Y_{t}=P^{c}_{t} Y_{t} +\Pi_{t}$. In the transition from (d) to  (f), capitalists pay  taxes and vacancy expenses out of their realized profits. From this, final good producers decide   how much to retain in the circuit of capital with a value of $s_{t} \bar{\Pi}_{t}$, $s_{t}$ being the rate of savings, and how much to consume, with a value equal to $(1-s_{t})\bar{\Pi}_{t}$.

[node distance = 7.25 cm, on grid]
(q0) [gray,text width=2.4cm] Financial capital;
(q1) [above right = of q0,gray, text width=6.5 cm] Productive capital $(\mathcal{K}_{t}=P^{k}_{t}K_{t})$;
(q2) [below right = of q1,gray,text width=2.4cm] Commerical capital;
(q5) [below  = of q0,gray] ;
(q6) [below  = of q2,gray] ;
(q0) – (q2) node[pos=0.3,gray] (q4) ;
(q4) – (q5) node[pos=0.29,black] (q3) Final good capitalist consumption;
(q4) – (q2) node[pos=0.5,gray] (q9) ;
(q9) – (q6) node[pos=0.285,black] (q8) Expenses;
(q0) edge node [left,pos=0.67,text width=3.4cm] $\mathcal{C}_{t}=w_{t} N_{t}+P^{k}_{t} I_{t}$ (q1)
(q1) edge  node[above right,pos=0.5, text width=4.3cm] $P^{c}_{t}Y_{t}=\int_{M_{t-1}}^{M_{t}}p_{t}(j) y_{t}(j) dj=\delta P^{k}_{t} K_{t}+w_{t} N_{t}$ (q2)
(q1) edge  node[right,pos=0.08,blue] (b)  (q2)
(q2) edge   [bend right] node[below,pos=0.2,blue] (c)  (q0)
(q2) edge  node[above,pos=0.5,black] $\Pi_{t}$  (q9)
(q9) edge  node[below,pos=0.2,blue]  (q4)
(q9) edge  node[above,pos=0.6] $\bar{\Pi}_{t}$  (q4)
(q0) edge  node[left,pos=0.2,blue] (a)  (q1)
(q0) edge  node[right,pos=0.95,black]   $ T_{P_{t}} $  (q1)
(q2) edge  node[below,pos=0.5,blue] (d)  (q9)
(q4) edge  node[below,pos=0.75,blue] (h)  (q0)
(q9) edge  node[below,pos=0.75,blue] (f)  (q4)
(q4) edge  node[above] $s_{t} \bar{\Pi}_{t}$  (q0)
(q2) edge [bend right] node[above] $P^{c}_{t}Y_{t}$ (q0);
[->,snake=snake,segment amplitude=.5mm,
segment length=2mm,
line after snake=2mm,black] (q4) – (q3)
node[ right, midway,black]$(1-s_{t}) \bar{\Pi}_{t}$
(q4) – (q3)
node[ left, pos=0.85,blue](g);
[->,snake=snake,segment amplitude=.5mm,
segment length=2mm,
line after snake=2mm,black] (q9) – (q8)
node[ right, midway,black]$T_{t} + \kappa_{t} V_{t}$
(q9) – (q8)
node[ left, pos=0.8,blue](e);

Reproduction process of capital. Notes— The  transitions of the flows of capital are described with  blue letters in parenthesis and the different forms of capital are represented in gray.

The two main assumption in the description of the circuit of capital in Figure <ref> are that: (i)  commodities are sold immediately after they are produced; and (ii) that there is a zero time-lag between the moment the final good is sold and the moment firms demand new capital outlays. Formally, assumption (i) implies that there is no variation in commercial capital (i.e., there are no variations in final good inventories). Similarly, assumption (ii) states that the variations of financial capital ($\mathcal{K}^{F}_{t}$) are equal to zero, that is

\begin{equation}\label{eq:equil_finc_capital}
\frac{d \mathcal{K}^{F}_{t}}{dt} = P^{c}_{t} Y_{t} + s_{t} \tilde{\Pi}_{t} -  \mathcal{C}_{t} =0.
\end{equation}

This last equation states that savings $(s_{t} \tilde{\Pi}_{t})$ are equal to net investment $(P^{k}_{t} (I_{t}-\delta K_{t}))$. This relation is also used to derive the financial constraint of capitalist households (<ref>). Using flows (f), (g) and (h) in Figure <ref>, it follows that

\begin{equation*}
\begin{split}
(1-s_{t}) \bar{\Pi}_{t} &= P_{t} C^{c}_{t} = \tilde{\Pi}_{t}-s_{t}\tilde{\Pi}_{t}\\
&= \tilde{\Pi}_{t}- \mathcal{C}_{t}  +  P^{c}_{t} Y_{t} \\
&= \Pi_{t} - T_{t} - \kappa_{t} V_{t} - \mathcal{C}_{t}  +  P^{c}_{t} Y_{t} \\
&= P_{t} Y_{t} - P^{c}_{t} Y_{t}  - T_{t} - \kappa_{t} V_{t} - \mathcal{C}_{t}  +  P^{c}_{t} Y_{t} \\
&= P_{t} Y_{t}- T_{t} - \kappa_{t} V_{t}- w_{t} N_{t} - P^{k}_{t} I_{t}.
\end{split}
\end{equation*}

Using the financial constraint of workers and the definition of profits of capital good producers in (<ref>), we can derive aggregate consumption in (<ref>) and the aggregate resource constraint as

\begin{equation*}
P_{t} C_{t} = P_{t} C^{w}_{t}  + P_{t} C^{c}_{t}  + \Pi^{c}_{t}=P_{t}Y_{t} - \kappa_{t} V_{t} -P^{I}_{t} X_{t}.
\end{equation*}

Ultimately, though I preferred to omit all discussions about the circuit of capital in the main text, it is clear that all relevant accounting relations and financial constraints are obtained from Figure <ref>.

§ AGGREGATE PRODUCTION FUNCTION AND THE ACCOUNTING IDENTITY

The aggregate production function in the main text can be written as

\begin{equation*}
Y_{t} = B\Big[(1-m_{t})^{1/\sigma} \Big(\Gamma^{K}_{t} K_{t}\Big)^{\frac{\sigma-1}{\sigma}} + \Big( \int_{0}^{m_{t}}{\Gamma^{N}_{t}}^{\sigma-1}(j) dj\Big)^{1/\sigma}\;  \Big(\Gamma^{N}_{t}(J^*_{t}) N_{t}\Big)^{\frac{\sigma-1}{\sigma}} \Big]^{\frac{\sigma}{\sigma-1}}
\end{equation*}

To simplify the mathematics I will assume that $\Gamma^{K}_{t}=\Psi_{t}=1$ and work with the fact that $ \int_{0}^{m_{t}}{\Gamma^{N}_{t}}^{\sigma-1}(j) dj \approx m_{t}$, so given that $\Gamma^{N}_{t}(j)=e^{\alpha j}$, then:

\begin{equation}\label{eq:ces_app}
y_{t} = B \Big[(1-m_{t})^{1/\sigma}  k_{t}^{\frac{\sigma-1}{\sigma}} + {m_{t}}^{1/\sigma}\; N_{t}^{\frac{\sigma-1}{\sigma}} \Big]^{\frac{\sigma}{\sigma-1}}
\end{equation}

Where $k_{t} \equiv K_{t} e^{-\alpha J^*_{t}}$ and $y_{t} \equiv Y_{t}  e^{-\alpha J^*_{t}}$. Using the properties of the aggregate production function, we know that $Y_{t} = \delta P^{k}_{t} K_{t} + w_{t} N_{t}=$ and $y_{t} = \delta P^{k}_{t} k_{t} + \hat{w}_{t} N_{t}$, where $\hat{w}_{t} = w_{t} e^{-\alpha J^{*}_{t}}$.  Working in continuous time (for simplicity), denoting $\dot{x} \equiv dx/dt$ and $g_{x} \equiv \dot{x}/x$, then:

\begin{equation*}
g_{y} =(1-\Omega^{c}_{t})g_{P^{k}} + \Omega^{c}_{t} g_{\hat{w}} + (1-\Omega^{c}_{t})g_{k} + \Omega^{c}_{t} g_{N} = \psi_{t} +  (1-\Omega^{c}_{t})g_{k} + \Omega^{c}_{t} g_{N}.
\end{equation*}

Here $\Omega^{c} = wN/Y$ is the labor share on costs and $(1-\Omega^{c})=\delta P^{k}K/Y$. We are interested in finding some functional form that will track the behavior of $\Omega^{c}_{t}$ such that it will always provide a good prediction of $y_{t}$ at any moment in time. Our purpose is to show that the CES production function (<ref>) solves this problem.

Start with the partial derivative of $y_{t}$ in (<ref>) with respect to time, from which it follows that

\begin{equation}\label{eq:ces_growth_app}
g_{y} = \frac{\dot{m}_{t}}{(1-\sigma)D_{t}} \Big[(1-m_{t})^{\frac{1-\sigma}{\sigma}}k_{t}^{\frac{\sigma-1}{\sigma}} - m_{t}^{\frac{1-\sigma}{\sigma}}N_{t}^{\frac{\sigma-1}{\sigma}}\Big] + \frac{(1-m_{t})^{1/\sigma} k_{t}^{\frac{\sigma-1}{\sigma}}}{D_{t}} g_{k} + \frac{m_{t}^{1/\sigma} N_{t}^{\frac{\sigma-1}{\sigma}}}{D_{t}} g_{N}.
\end{equation}

Where $D_{t} \equiv \Big[(1-m_{t})^{1/\sigma}  k_{t}^{\frac{\sigma-1}{\sigma}} + {m_{t}}^{1/\sigma}\; N_{t}^{\frac{\sigma-1}{\sigma}} \Big]$. Using the marginal productivity conditions in (<ref>) (and the assumption that $\Gamma^{K}=1$):

\begin{equation*}
\begin{split}
\delta P^{k}_{t} &= B^{\frac{\sigma-1}{\sigma}} (1-m_{t})^{1/\sigma} (Y_{t}/K_{t})^{1/\sigma}\\
w_{t}  & \approx m_{t}^{1/\sigma} B^{\frac{\sigma-1}{\sigma}} e^{\alpha(\sigma-1)J^*_{t}/\sigma} (Y_{t}/N_{t})^{1/\sigma},
\end{split}
\end{equation*}

it implies that $\Omega^{c}_{t}  \equiv w_{t}N_{t}/Y_{t} \approx  m_{t}^{1/\sigma} N^{\frac{\sigma-1}{\sigma}}_{t}/D_{t}$ and $1-\Omega^{c}_{t}  \equiv \delta P^{k}_{t}K_{t}/Y_{t} = (1-m_{t})^{1/\sigma} k^{\frac{\sigma-1}{\sigma}}_{t}/D_{t}$. Replacing these equations in (<ref>), we get:

\begin{equation}\label{eq:ces_growth_app}
g_{y} = (1-\Omega^{c}_{t})g_{k} + \Omega^{c}_{t} g_{N}
\end{equation}

That is, the aggregate production function will hold exactly if—after correcting for the time trend of labor productivity—$\psi_{t} \approx 0$, which is what is expected on average out of the stationary rates of growth of wages and capital prices.

This close connection with the accounting identity can be used to reveal  the possible illusion of  forming causal interpretations from  marginal productivity equations.  For example, one may hope to test the validity of marginal productivity equations by using $\Omega^{c}_{t}  \approx  m_{t}^{1/\sigma} N^{\frac{\sigma-1}{\sigma}}_{t}/D_{t} $ and estimating

\begin{equation*}
g_{\Omega^{c}} = \frac{1}{\sigma} g_{m} - \frac{\sigma-1}{\sigma}(1-\Omega^{c}_{t})\Big[g_{k} - g_{N}\Big],
\end{equation*}

not knowing that this equation must necessarily hold if $\psi_{t} \approx 0$. The equation above presents two alternatives to interpret a fall in the wage share.  First, as noted by [p. 85]felipe2013aggregate, it may decline if capital and labor are gross substitutes and $K_{t}$ grows faster than $N_{t}$. This is the approach used by  piketty2014capital and others to “explain" the fall in the wage share. Second, the share of labor may fall if $m_{t}$ declines—regardless of the value of $\sigma$. This is one of the innovations of the production function with automation, since it presents an alternative that can account for the falling wage share without assuming gross substitution between labor and capital.

One of the problems here is that $g_{\Omega^{c}}$ may decrease as a result of, say, a lower bargaining power of workers, but by interpreting this fall using marginal productivity equations one may wrongly associate it with, say, increasing automation. Furthermore, one may confuse the direction of causality by thinking that changes in the distribution of income are caused by the components of the production function when these may go the other way: the fall in the wage share may explain the changes in the parameters of the production function.

The bottom line here is that it is generally questionable to assign claims of causality to aggregate production functions when these may just be representing an accounting relation. The model in the paper partly addresses this issue in two ways. First, by creating a relation between the aggregate production function and the aggregate costs of production, rather than aggregate income,  the model is capable of introducing institutional and political factors as determinants of the labor share. Secondly, by restricting $\sigma<1$, the model can identify changes in technology with automation if these are also associated with increasing unemployment, lower steady-state real wages, and a higher capital-output ratio.

§  PROOFS

§.§ Proof of Theorem <ref>

Part A. Let the stationary values be defined   $\hat{Y}_{t}=Y_{t} e^{-\alpha J^{*}_{t}}$,  $\hat{C}^{c}_{t}=C^{c}_{t} e^{-\alpha J^{*}_{t}}$, $\hat{C}^{w}_{t}=C^{w}_{t} e^{-\alpha J^{*}_{t}}$, $\hat{w}_{t}=w_{t} e^{-\alpha J^{*}_{t}}$, $\hat{K}_{t} = K_{t}  e^{-\alpha J^{*}_{t}} \Psi^{-1}_{t}$, $\hat{I}_{t} = I_{t}  e^{-\alpha J^{*}_{t}} \Psi^{-1}_{t}$, $\hat{P}^{I}_{t}=P^{I}_{t}  \Psi_{t}$, $\hat{P}^{k}_{t}=P^{k}_{t}  \Psi_{t}$, $\hat{\mathcal{X}}_{t}=\mathcal{X}_{t} e^{-\alpha J^{*}_{t}}$,  and $\hat{\mathcal{K}}_{t}=\mathcal{K}_{t} e^{-\alpha J^{*}_{t}}$. Given the accounting structure defined in online Appendix <ref>, the evolution of capital can be described as

\begin{equation*}
\mathcal{K}_{t+1} = \mathcal{K}_{t} + s_{t}(r_{t}-\tau_{t}-\zeta_{t})\mathcal{K}_{t}.
\end{equation*}

Where $r_{t} \equiv  \Pi_{t}/\mathcal{K}_{t}$ is the rate of profit,  $\tau_{t} \equiv T_{t}/\mathcal{K}_{t}$ is the  share of taxes on capital value,  $\zeta_{t} = \kappa_{t} V_{t}/\mathcal{K}_{t}$ is the share of vacancy costs to capital, and $\mathcal{K}_{t} = P^{k}_{t} K_{t}$ is the current value of the capital stock.  Defining $\mathcal{K}_{t}=\hat{\mathcal{K}} e^{\alpha J^{*}_{t}}$  and denoting the current rate of growth of productive capital as $g_{t} = (\mathcal{K}_{t+1}-\mathcal{K}_{t})/\mathcal{K}_{t}$, then

\begin{equation*}
e^{\alpha (J^{*}_{t+1}-J^{*}_{t})}= 1 + g_{t}.
\end{equation*}

Given Assumption <ref> (ii), it follows that $g_{t} \approx g$. Now, the existence of an equilibrium where $g_{t} \rightarrow g$ depends on the existence and stability of the equilibrium in the labor market.  This equilibrium is guaranteed if $b_{1}$ is sufficiently small. To verify this claim we can evaluate the partial derivatives of $(1+\mu^{S})$ and $\mu^{D}$ near the equilibrium and check the conditions by which $\partial \mu^{D}/\partial \theta >0$ and $\partial (1+\mu^{S})/\partial \theta <0$.

Starting with $\partial (1+\mu^{S})/\partial \theta <0$, it is useful to note that $\hat{Z} = \hat{b} + (\epsilon_{1}/(1+\epsilon_{1}) h \hat{Y}_{N}$ and $\hat{b} = b_{0}^{\beta} + b_{1}^{\beta} (1-L)$. Thus, the Nash solution in steady-state can be written as

\begin{equation*}
(1+\mu^{S})= \frac{h \hat{Y}_{N}}{ (1-\eta_{w})\big( b_{0}^{\beta} + b_{1}^{\beta} (1-L) \big) + (\epsilon_{1} + \eta_{w})/(1+\epsilon_{1}) \;  h \hat{Y}_{N} + \eta_{w} \hat{\kappa} \Big[ (1-\tilde{\beta})(1-\hat{\lambda})q(\theta)^{-1} + \tilde{\beta} \theta\Big] }
\end{equation*}

where $\tilde{\beta} \equiv \beta^{w}/\beta^{c}$. Expressing $\bar{\omega} = (1-\eta_{w})\big( b_{0}^{\beta} + b_{1}^{\beta} (1-L) \big) + (\epsilon_{1} + \eta_{w})/(1+\epsilon_{1}) \;  h \hat{Y}_{N} + \eta_{w} \hat{\kappa} \Big[ (1-\tilde{\beta})(1-\hat{\lambda})q(\theta)^{-1} + \tilde{\beta} \theta\Big] $, it follows that

\begin{equation}\label{eq:derivative_nash}
\frac{\partial (1+\mu^{S})}{\partial \theta} = \frac{h_{\theta} \hat{Y}_{N}}{\bar{\omega}} + \frac{h \hat{Y}_{N \theta}}{\bar{\omega}} - \frac{h \hat{Y}_{N}}{\bar{\omega}^2} \Big[  \hat{\kappa} \eta_{w} \big(\tilde{\beta} - (1-\tilde{\beta})(1-\hat{\lambda}) \frac{ q'(\theta)}{q(\theta)^{2} } \big)+ \frac{\epsilon_{1}+\eta_{w}}{1+\epsilon_{1}} \big( h_{\theta} \hat{Y}_{N} + h \hat{Y}_{N \theta}\big) - (1-\eta_{w}) b_{1}^{\beta} L_{\theta} \Big]
\end{equation}

If $b^{\beta}_{1} =0$, we would get an equation close to the usual Nash solution, which is known to have a negative slope. However, if  $b^{\beta}_{1}  >0$, we can have $\frac{\partial (1+\mu^{S})}{\partial \theta}  >0$, specially for small values of $\theta$ given that  $L_{\theta}$ increases as $\theta \rightarrow 0$.  This explains why the slope of $\mu^{S}$ is initially  positive in Figure <ref>.

Correspondingly, the partial derivative of $\mu^{D}$ with respect to $\theta$ is

\begin{equation*}
\frac{\partial \mu^{D}}{\partial \theta} = -{\mu^{D}}^{2} \Big[ \frac{h_{\theta} \hat{Y}_{N} q(\theta) }{\hat{\lambda} \hat{\kappa}} + \frac{h\hat{Y}_{N \theta}}{\hat{\lambda} \hat{\kappa}} + \frac{h \hat{Y}_{N}q'(\theta)}{\hat{\lambda} \hat{\kappa}} \Big],
\end{equation*}

which is  positive given the assumption of decreasing marginal returns and $q'(\theta) <0$. Thus, a stable equilibrium in the labor market can be obtained if $b^{\beta}_{1}$ is sufficiently low.

§.§ Part B

Marginal productivity of capital.  Using (<ref>), we know that the marginal productivity of capital satisfies $Y_{K_{t}} = \delta P^{k}_{t}/P^{c}_{t}$. Expressing this equation in its stationary form it follows that $\hat{Y}_{K_{t}} = Y_{K_{t}} \Psi_{t} = \delta P^{k}_{t} \Psi_{t}/P^{c}_{t}$. Now, given the first order conditions of capitalists in  Appendix <ref>:

\begin{equation*}
\hat{Y}_{K_{t}} = \frac{\delta \hat{P}^{k}_{t}}{P^{c}_{t}} = \delta (1+\mu_{t}) \Omega_{I_{t},t} + \delta \Big[\Omega_{I_{t},t} + \frac{\Omega_{I_{t},t+1}}{1-\pi_{I}} \Big] \sum_{i=1}^{\infty}  (1-\pi_{I})^{i} e^{-i\bar{z}}(1+ \mu_{t+i}).
\end{equation*}

where I assume (without any loss of generality) that $\beta^{c}=1$. In the limit, since

\begin{equation*}
\begin{split}
\Omega_{I_{t},t} \rightarrow \Omega_{I} = \pi^{\frac{1+\upsilon}{1-\upsilon}}_{I} \Big[1 -(1-\pi_{I}) e^{-\frac{\bar{z}(\upsilon-1)}{\upsilon}}\Big]^{\frac{1}{\upsilon-1}}\\
\Omega_{I_{t},t+1} \rightarrow \Omega^{+}_{I} = -(1-\pi_{I}) \pi^{\frac{1+\upsilon}{1-\upsilon}}_{I} \Big[1 -(1-\pi_{I}) e^{-\frac{\bar{z}(\upsilon-1)}{\upsilon}}\Big]^{\frac{1}{\upsilon-1}}e^{\bar{z}/\upsilon},
\end{split}
\end{equation*}

it follows that

\begin{equation}
\hat{Y}_{K_{t}} = \frac{\delta \Omega_{I} (1+\mu)}{1-(1-\pi_{I})e^{-\bar{z}}} \Big[1-(1-\pi_{I})e^{-\bar{z}(\upsilon-1))/\upsilon}\Big]
\end{equation}

which is the result in (<ref>).

Rate of profit. The rate of profit, unlike the rate of return of capital, is the ratio of a flow over a stock. By definition, $r_{t} \equiv \Pi_{t}/(P^{k}_{t}K_{t})$. Making use of equation (<ref>) in Appendix <ref>:

\begin{equation*}
\begin{split}
g_{t} &= \frac{(w_{t} N_{t} + P^{k}_{t} I_{t})\big(1- e^{g_{t} T_{P_{t}}}\big)}{\mathcal{K}_{t} }= \frac{\Big[s_{t}\big(	\Pi_{t} - T_{t} - \kappa_{t} V_{t}\big) + P^{c}_{t} Y_{t} \Big] \big(1- e^{g_{t} T_{P_{t}}}\big)}{\mathcal{K}_{t} }\\
g_{t}&=\Big[s_{t}(r_{t}-\tau_{t}-\zeta_{t}) + \frac{P^{c}_{t}Y_{t}}{\mathcal{K}_{t}}\Big](1-e^{-g_{t}T_{P_{t}}})\\
g_{t}& = \Big[s_{t}(r_{t}-\tau_{t}-\zeta_{t} ) + \frac{r_{t}}{\mu_{t}}\Big](1-e^{-g_{t}T_{P_{t}}}).
\end{split}
\end{equation*}

Where $\tau_{t} \equiv T_{t}/\mathcal{K}_{t}$ is the  share of taxes on capital value,   $\zeta_{t} = \kappa_{t} V_{t}/\mathcal{K}_{t}$ is the share of vacancy costs to capital, and $T_{P_{t}}$ is the average production time of the final good defined in Appendix <ref>. Given that in equilibrium $s_{t}(r_{t}-\tau_{t}-\zeta_{t})=g_{t}$, it follows that

\begin{equation}\label{eq:prof_rate_approx}
\begin{split}
e^{g_{t}T_{P_{t}}} &= 1+ \frac{g_{t}\mu_{t}}{r_{t}}\\
\Rightarrow r_{t} &\approx \frac{\mu_{t}}{T_{P_{t}}}.
\end{split}
\end{equation}

Given Assumption <ref> (ii) and the assumption that the final good is produced after an average time lag $T_{P_{t}}$, it must hold that $P^{k}_{t-T_{P_{t}}} I_{t-T_{P_{t}}}  + w_{t-T_{P_{t}}}  N_{t-T_{P_{t}}}  = e^{-g_{t} T_{P_{t}}} \big[ P^{k}_{t}I_{t} +w_{t} N_{t}\big] = P^{c}_{t} Y_{t} = \delta P^{k}_{t} K_{t} + w_{t} N_{t}$, such that

\begin{equation*}
T_{P_{t}} = \frac{1}{g_{t}} \mathrm{log} \Bigg(\frac{P^{k}_{t}I_{t} +w_{t} N_{t}}{\delta P^{k}_{t} K_{t} + w_{t} N_{t}}\Bigg) = \frac{1}{g_{t}} \mathrm{log} \Bigg(\frac{\hat{P}^{k}_{t}\hat{I}_{t} +\hat{w}_{t} N_{t}}{\delta \hat{P}^{k}_{t} \hat{K}_{t} + \hat{w}_{t} N_{t}}\Bigg)
\end{equation*}

Now, since $\hat{I}_{t} \approx (\delta + g_{t}) \hat{K}_{t}$, the last equation becomes

\begin{equation*}
T_{P_{t}} = \frac{1}{g_{t}} \; \mathrm{log}\Big(1 + g_{t} \frac{1}{\delta + (\hat{w}_{t} N_{t}/\hat{P}^{k}_{t}\hat{K}_{t})}\Big).
\end{equation*}

Using equation (<ref>), $(\hat{w}_{t}N_{t}/\hat{P}^{k}_{t}\hat{K}_{t})= (\hat{w}_{t}N_{t}/P_{t}\hat{Y}_{t}) \times (P_{t}\hat{Y}_{t}/P^{k}_{t}\hat{K}_{t})$ and  $(P
_{t}\hat{Y}_{t}/P^{k}_{t}\hat{K}_{t})=(1+\mu_{t})\delta/(1-(\hat{w}_{t}N_{t}/P^{c}_{t}\hat{Y}_{t}))$, such that $(\hat{w}_{t}N_{t}/\hat{P}^{k}_{t}\hat{K}_{t})=\delta \Omega^{c}_{t}/(1-\Omega^{c}_{t})$ and

\begin{equation}\label{eq:stat_prod_time_app}
T_{P_{t}} \rightarrow  \frac{1}{g} \; \mathrm{log} \Big(1+ g \frac{1-\Omega^{c}}{\delta}\Big) \approx \frac{1-\Omega^{c}}{\delta}.
\end{equation}

Replacing (<ref>) in (<ref>), it follows that

\begin{equation}
r_{t} \rightarrow \frac{\delta \mu}{1-\Omega^{c}}
\end{equation}

which is the result in (<ref>).

Investment-output ratio The function $\Omega_{t}$ can be written as $\Omega_{t} = \pi^{\frac{1+\upsilon}{1-\upsilon}}_{I} I_{t} \Big[ 1- (1-\pi_{I})(\frac{\hat{I}_{t-1}}{\hat{I}_{t}})^{\frac{\upsilon -1}{\upsilon}} e^{-\bar{z}(\upsilon-1)/\upsilon}\Big]^{\frac{\upsilon}{\upsilon-1}}$. Given that $X_{t}=\hat{X}_{t} \Psi_{t} e^{\alpha J^{*}_{t}}$, in equilibrium it holds that

\begin{equation*}
\hat{X} = \frac{\hat{I} \pi^{\frac{1+\upsilon}{1-\upsilon}}_{I}  \Big[ 1- (1-\pi_{I})e^{-\bar{z}(\upsilon-1)/\upsilon}\Big]^{\frac{\upsilon}{\upsilon-1}}}{1-(1-\pi_{I})e^{-\bar{z}}}
\end{equation*}

Using $\hat{I} \approx (\delta +g) \hat{K}$ and $\hat{P}^{I}_{t}=P^{I}_{t} \Psi_{t}$:

\begin{equation*}
\frac{\hat{X}}{\hat{Y}} = \frac{(\delta +g) \hat{K}\pi^{\frac{1+\upsilon}{1-\upsilon}}_{I}  \Big[ 1- (1-\pi_{I})e^{-\bar{z}(\upsilon-1)/\upsilon}\Big]^{\frac{\upsilon}{\upsilon-1}}}{\hat{Y}\big(1-(1-\pi_{I})e^{-\bar{z}}\big)}.
\end{equation*}

Solving $\hat{K}/\hat{Y}$ from (<ref>) we obtain (<ref>).

§.§ Proof of  Lemma <ref>

This part of the proof follows from Lemma A2 of acemoglu2018race. The main difference here is that the automation measure function depends on the rate of return of capital and is bounded in regions strictly inside $(0,1)$.

We can begin by noting that  at the boundary of region 2 in Figure <ref>, it must be true that $\delta \hat{P}^{k}_{t}/\gamma_{k} = w_{t} e^{-\alpha J^{*}_{t}} = \hat{w}_{t}$. Using this condition in the ideal price index, it follows that:

\begin{equation*}
1 = \Bigg(\frac{\delta \hat{P}^{k}_{t}}{\gamma_{k}}\Bigg)^{1-\sigma} \; \Bigg[1 -\bar{m}(\mu_{t}) + \frac{e^{\alpha(\sigma-1)\bar{m}(\mu_{t})}-1}{\alpha(\sigma-1)}\Bigg]
\end{equation*}

The sign of $\bar{m}(\mu_{t})$ can be deduced using a Taylor expansion on $\mathrm{exp}(x) = 1 + x +x^2/2 + \mathcal{O}(x^3)$, with $x = \alpha(\sigma-1)\bar{m}(\mu_{t})$, such that[Using the calibration in Table <ref>, the errors of this approximation are at most 5$\%$.]

\begin{equation*}
\bar{m}(\mu_{t})   \approx \sqrt{\frac{2}{\alpha(1-\sigma)} \Bigg( 1 - \Big(\frac{\delta \hat{P}^{k}_{t}}{\gamma_{k}}\Big)^{\sigma-1}\Bigg)} = \sqrt{\frac{2}{\alpha(1-\sigma)} \Bigg( 1 - \Big(\frac{\hat{Y}_{k_{t}}}{\gamma_{k}}\Big)^{\sigma-1}\Bigg)}
\end{equation*}

The assumption that $\gamma_{k} = \delta P^{k}_{t}(0)$ guarantees that $\bar{m}(\mu_{t}) >0$ since $\gamma_{k} < \delta P^{k}_{t}(\mu_{t})$ for all $\mu_{t} >0$. Furthermore, given that $\hat{Y}_{k}$ is an increasing function of $\mu$, then $\bar{m}(\mu)$ is also an increasing function of the rate of return.

§.§ Proof of Proposition <ref>

§.§.§ Automation

Wages. The first part is analogous to  the proof of Lemma A2 of acemoglu2018race. Defining $\hat{w}_{t} = w_{t} e^{-\alpha J^{*}_{t}}$ and using the ideal price index condition

\begin{equation*}
\hat{w}^{1-\sigma}_{t} = \frac{B^{1-\sigma} -(1-m^*_{t})\Big(\delta P^{k}_{t}/\Gamma^{k}_{t}\Big)^{1-\sigma}}{\int_{0}^{m^{*}_{t}} \Gamma^{N}_{t}(j) dj}.
\end{equation*}

By implicit differentiation, it follows that

\begin{equation*}
\hat{w}'_{t}/\hat{w}_{t} = \frac{1}{1-\sigma} \Bigg( \frac{\Big(\delta P^{k}_{t}/\Gamma^{k}_{t}\Big)^{1-\sigma}}{\Big(\int_{0}^{m^{*}_{t}} \Gamma^{N}_{t}(j) dj\Big)\hat{w}^{1-\sigma}_{t}} - \frac{e^{\alpha(\sigma-1)m^{*}_{t}}}{\int_{0}^{m^{*}_{t}} \Gamma^{N}_{t}(j) dj} \Bigg)
\end{equation*}

Using  $\tilde{w}_{t} = w_{t} e^{-\alpha M_{t}}$ and  $\int_{0}^{m^{*}_{t}}  \tilde{w}^{1-\sigma}_{t}  {\Gamma^{N}_{t}}^{(1-\sigma)} (j) dj = \hat{w}^{1-\sigma}_{t} \int_{0}^{m^{*}_{t}}  \hat{w}^{1-\sigma}_{t}  {\Gamma^{N}_{t}}^{(\sigma-1)} (j) dj$

\begin{equation*}
\hat{w}'_{t}/\hat{w}_{t} = \frac{1}{1-\sigma} \Bigg( \frac{\Big(\delta P^{k}_{t}/\Gamma^{k}_{t}\Big)^{1-\sigma} - \tilde{w}^{1-\sigma}_{t}}{ \tilde{w}^{1-\sigma}_{t} \; \int_{0}^{m^{*}_{t}} \Gamma^{N}_{t} (j) dj } \Bigg)
\end{equation*}

Meaning that $\hat{w}'_{t} >0$ if $\tilde{w}_{t} < \delta \hat{P}^{k}_{t}/\gamma_{k}$, which is the condition to be in region 1 in Figure <ref>.

Hours. Using the Nash solution with SEP preferences, it follows that

\begin{equation*}
\epsilon_{0} h^{1/\epsilon_{1}}_{t} =\frac{\hat{Y}_{N_{t}}}{\hat{C}^{we}_{t} P_{t}} = \frac{1}{L_{t}h_{t} + (1-L_{t})\hat{b}_{t}/\hat{w}_{t}},
\end{equation*}

since $C^{we}_{t}=C^{wu}_{t}=C^{w}_{t}$ and $w_{t}=Y_{N_{t}}$. By implicit differentiation:

\begin{equation*}
\Big[\epsilon_{0}/\epsilon_{1} h^{1/\epsilon_{1} -1}_{t}  + \frac{L_{t}w^2_{t}}{(P_{t}C^{w}_{t})^2}\Big] dh =  -\frac{dm}{(P_{t}C^{w}_{t}/w_{t})^2} \Bigg[ L_{\theta} \theta_{m} \big(h_{t} - \hat{b}_{t}/\hat{w}_{t}\big) - (1-L_{t})\frac{\hat{w}'_{t}}{\hat{w}_{t}} \frac{\hat{b}_{t}}{\hat{w}_{t}} \Bigg]
\end{equation*}

The product $(1-L_{t})\frac{\hat{w}'_{t}}{\hat{w}_{t}} \frac{\hat{b}_{t}}{\hat{w}_{t}} \approx 0$, so

\begin{equation*}
\frac{dh}{dm} \propto - L_{\theta} \theta_{m} \big(h_{t} - \hat{b}_{t}/\hat{w}_{t}\big)
\end{equation*}

where $L_{\theta} \equiv \partial L_{t}/\partial \theta_{t} >0$ and $\theta_{m} \equiv \partial \theta_{t}/ \partial m_{t}$. Since $\hat{w}_{t} > \hat{b}_{t}$ and $h_{t} \approx 1$ by assumption, the sign of $dh/dm$ is determined by $\theta_{m}$.

Rate of return and vacancy/unemployment ratio. This is the main part of the proof of Proposition <ref>. Using the equations in (<ref>), we can explore the changes in the labor market to a variation in $m_{t}$ near the equilibrium $\theta=\theta^*$. Starting with the Nash solution for wages:

\begin{equation*}
\left.{\frac{\partial (1+\mu^{S})}{\partial m}}\right\vert_{\theta=\theta^*}= \frac{h_{m} \hat{w}}{\bar{\omega}} + \frac{\hat{w}' h}{\bar{\omega}} - \frac{h \hat{w}}{\bar{\omega}^2} \Big[ (1-\eta_{w}) \hat{Z}_{m} +\eta_{w} h_{m} \hat{w} + \eta_{w} h \hat{w}'\Big].
\end{equation*}

Where $ \bar{\omega} \equiv (1-\eta_{w})\hat{Z} + \eta_{w}\Big( h \hat{w} + \hat{\kappa}\Big( (1-\beta^{w}/\beta^{c})(1-\hat{\lambda})/q(\theta) + (\beta^{w}/\beta^{c}) \theta \Big)$. Given SEP preferences, the steady-state value of the opportunity cost of employment is $\hat{Z}_{t} = \hat{b}_{t} + \epsilon_{1}/(1+\epsilon_{1}) h_{t} \hat{w}_{t}$. From this it follows that $\hat{Z}_{m} = \epsilon_{1}/(1+\epsilon_{1}) h_{m} \hat{w} + \epsilon_{1}/(1+\epsilon_{1}) h \hat{w}'$. Now, given that we are fixing $\theta=\theta^*$, then $h_{m} \approx 0$, such that

\begin{equation*}
\begin{split}
\left.{\frac{\partial (1+\mu^{S})}{\partial m}}\right\vert_{\theta=\theta^*} \approx  &  \frac{\hat{w}' h}{\bar{\omega}} - \frac{h \hat{w}}{\bar{\omega}^2} \Big[ (1-\eta_{w}) \frac{\epsilon_{1}}{1+\epsilon_{1}} h \hat{w}' + \eta_{w} h \hat{w}' \Big]\\
=&  \frac{\hat{w}'h}{\bar{\omega}} \Bigg[1 - \frac{\hat{w}h}{\bar{\omega}} \Big(\eta_{w} + (1-\eta_{w})\frac{\epsilon_{1}}{1+\epsilon_{1}} \Big) \Bigg].
\end{split}
\end{equation*}

Thus, $\left.{\frac{\partial (1+\mu^{S})}{\partial m}}\right\vert_{\theta=\theta^*} >0$ if $\hat{w}'_{t} >0$, which occurs in region 1 of Figure <ref>. In this case,  the Nash solution moves to the right with an increase of $m$ (or to the left when $m$ declines, as shown in Figure <ref>).

Repeating the same exercise with the labor demand equation:

\begin{equation*}
\left.{\frac{\partial \mu^{D}}{\partial m}}\right\vert_{\theta=\theta^*}  =- \frac{{\mu^{D}}^{2} \beta^{c} \hat{w}'q(\theta)}{(1-\beta^{c}(1-\hat{\lambda}))\hat{\kappa}}.
\end{equation*}

That is, $\left.{\frac{\partial \mu^{D}}{\partial m}}\right\vert_{\theta=\theta^*}  <0$ if $\hat{w}'_{t} >0$. Given that $\mu^{S}$ is a decreasing function and $\mu^{D}$ is increasing close to the initial equilibrium, it follows that the labor market reaches a new equilibrium $\theta^{**}>\theta^{*}$ following an increase in $m$ if $m>\bar{m}(\mu)$. The increase in the vacancy/unemployment ratio it follows that $L_{t}$ rises with a higher $m$.

The final effect on $\mu_{t}$ depends on the model parameters and cannot be determined  a priori. It is most likely, however,  that $\partial \mu/\partial m \approx 0$ given that the labor supply and demand equations move in opposite directions. In what follows I will work with this assumption.

Labor Share on Costs of Production. Using <ref>, the share of labor on costs of production satisfies $\Omega^{c} = 1- (B\gamma_{k})^{\sigma-1} (1-m^*) \Big(\frac{\delta (1+\mu)\Omega_{I}\big(1-(1-\pi_{i})e^{-\bar{z}(\upsilon-1)/\upsilon}\big) }{1-(1-\pi_{I})e^{-\bar{z}}}\Big)^{1-\sigma}$, such that

\begin{equation*}
\left.{\frac{\partial \Omega^{c}}{\partial m}}\right\vert_{\mu=\mu^*}  =  (B\gamma_{k})^{\sigma-1}  \Bigg[\frac{   \delta (1+\mu)\Omega_{I}\big(1-(1-\pi_{I})e^{-\bar{z}(\upsilon-1)/\upsilon}\big)}{1-(1-\pi_{I})e^{-\bar{z}}}\Bigg]^{1-\sigma}
\end{equation*}

The sign of $\left.{\frac{\partial \Omega^{c}}{\partial m}}\right\vert_{\mu=\mu^*} $ is positive if $m^*=m$, which is the case when $m > \bar{m}(\mu)$.

Investment Expenditure to Output ratio.  Starting with the steady-state investment-output ratio in (<ref>), it readily follows that $\left.{\frac{\partial \hat{X}_{t}/\hat{Y}_{t}}{\partial m}}\right\vert_{\mu=\mu^*} $ is negative if $m^*=m$.

§.§.§ Unemployment Benefits
This part of the proof is equivalent for unemployment benefits and the discount factor of workers.[This is because we are working with SEP preferences. Other preference functions might lead to qualitatively different results for changes in $\hat{b}$ and $\beta^w$. ]  Similar to the previous part of the proof, it is convenient to start with the steady-state  equation of wages.

Wages. Using the ideal price condition, we can express the steady-state value of wages as

\begin{equation*}
\hat{w}^{1-\sigma} = \frac{B^{1-\sigma} - (1-m^*)\Big(\delta\hat{P}^{k}/\gamma_{k}\Big)^{1-\sigma} }{\int_{0}^{m^*} {\Gamma^{N}}^{\sigma-1}_{t}(j) dj} = \frac{B^{1-\sigma} - (1-m^*)\Big(\hat{Y}_{k}\gamma_{k}\Big)^{1-\sigma} }{\int_{0}^{m^*} {\Gamma^{N}}^{\sigma-1}_{t}(j) dj}
\end{equation*}

The partial derivative of $\hat{w}$ with respect to is then determined by $\partial \mu/\partial \hat{b}$ given that $\partial \hat{Y}_{k}/\partial \mu >0$.

Hours. Using again the Nash solution with SEP preferences:

\begin{equation*}
\Bigg[\frac{\epsilon_{0}}{\epsilon_{1}} h^{1/\epsilon_{1} - 1} + \frac{L \hat{w}^2}{(P \hat{C}^{w})^2}\Bigg] dh = \frac{d \hat{b}}{(P \hat{C}^{w}/\hat{w})^2} \Bigg[ L_{\theta} \theta_{\hat{b}} \big[ h - \hat{b}/\hat{w} \big] -(1-L) \frac{\hat{b}}{\hat{w}} \frac{\hat{w}_{\hat{b}}}{\hat{w}} + (1-L)/\hat{w}\Bigg].
\end{equation*}

As before, given that $(1-L)\hat{b}/\hat{w} \approx 0$, then

\begin{equation*}
\frac{dh}{d\hat{b}} \approx - \text{contant} \; \times \; L_{\theta} \theta_{\hat{b}} \big[ h - \hat{b}/\hat{w} \big].
\end{equation*}

The sign of $\frac{dh}{d\hat{b}} $ is consequently determined, for the most part, by the sign of $\theta_{\hat{b}}$.

Rate of return and vacancy/unemployment ratio. Given that the sign of $\hat{w}_{\hat{b}}$ and $\frac{dh}{d\hat{b}} $ are determined by $\partial \mu/\partial \hat{b}$ and $\theta_{\hat{b}}$, we can work this part with the initial  assumption that $\frac{dh}{d\hat{b}} = \frac{dh}{d\hat{b}}  = 0$. Starting with the Nash solution near the initial equilibrium $\theta=\theta^*$:

\begin{equation*}
\left.{\frac{\partial (1+\mu^{S})}{\partial \hat{b}}}\right\vert_{\theta=\theta^*} = -(1-\eta_{w}) \frac{h \hat{w}}{\bar{\omega}^2} < 0
\end{equation*}

Similarly, given the same assumptions:

\begin{equation*}
\left.{\frac{\partial \mu^{D}}{\partial \hat{b}}}\right\vert_{\theta=\theta^*} = 0.
\end{equation*}

Joining the effects on the labor supply and labor demand equations, it follows that $\mu \downarrow$, $\theta \downarrow$, $L \downarrow$, $\hat{w} \uparrow $ and $h \uparrow$.

Labor share on costs of production. Using (<ref>) it is straightforward to show that

\begin{equation*}
\frac{\partial \Omega^{c}}{\partial \hat{b}} = -(1-\sigma) (1-m^*)(B\gamma_{k})^{\sigma-1} {\hat{Y}_{k}}^{-\sigma} \frac{\partial \hat{Y}_{k}}{\partial \mu} \frac{\partial \mu}{\partial \hat{b}} > 0
\end{equation*}

if $\frac{\partial \mu}{\partial \hat{b}} < 0$ and $\sigma<1$.

Investment-Output and Capital-Output ratio. Given that $\partial \hat{Y}_{k}/\partial \hat{b} > 0$, it is obvious from (<ref>) and (<ref>) that $\partial (\hat{K}/\hat{Y})/\partial \hat{b} >0$ and $\partial (\hat{X}/\hat{Y})/\partial \hat{b} >0$ when $\partial \mu/\partial \hat{b}<0$.

§ DATA DESCRIPTION AND ANALYSIS

§.§ Data Description
I use data from the BEA-BLS integrated industry-level production account from 1947 to 2016 with the intention of creating a  mapping between the model in the paper and the sectors of the economy that is meant to represent. Particularly, given that the paper focuses on the profit-making capacity of the economy, it makes sense that it concentrates on the specific sectors that are  contributing to the direct creation of aggregate profits. Though this is a controversial topic that was largely submerged with the rise of neoclassical economics—which interpreted all potentially marketable activities to be production activities— it is instrumental for understanding the conditions allowing the reproduction of capital at an increasing scale. Here I take on the view that new wealth results from the creation of aggregate profits, and that these are the outcome of a social relation joining capitalists and workers in the production process of tangible output (goods or services). Taking a practical approach to this problem I concentrate exclusively on what basu2013dynamics denoted as “value-adding sectors", which consist on the following industries:[Sectors not included in the list,  like the financial industry ( Finance, Insurance,, and Real Estate—FIRE), Education and Health Services, and Professional and Business Services, share the characteristic that national accounts impute value added onto them to make it equal to the incomes generated. The BEA, for example, calculates the value added of the banking sector from interest rate spreads between lending and deposits rates, which has direct relation with the production of goods and services in the economy.]

* Utilities

* Construction

* Manufacturing

* Wholesale trade

* Retail trade

* Transportation and warehousing

* Information

* Administrative and waste management

* Arts, entertainment, and recreation

* Accommodation and food services

* Other services, expect government

Basing the analysis on these sectors and the theoretical argument of the model,  the rate of return of capital is measured

\begin{equation*}
\mu_{t} = \frac{P_{t}Y_{t}-P^{c}Y_{t}}{P^{c}Y_{t}}= \frac{P_{t}Y_{t} - \delta P^{k}_{t} K_{t}-w_{t} N_{t}}{\delta P^{k}_{t} K_{t}+w_{t} N_{t}},
\end{equation*}

where $P_{t}Y_{t}$ is nominal gross output minus nominal intermediate input. To obtain the value of depreciation I use Table 3.4ESI from the Fixed Assets Tables. The value of labor costs is the sum of nominal college labor input and nominal non-college labor input in the BEA-BLS data.

The labor share is estimated as nominal college labor input plus nominal non-college labor input over net output (nominal gross output minus nominal intermediate input). The capital cost share is the nominal value of depreciation over nominal net output. The investment-output ratio, in turn, is the nominal value of fixed investment obtained from Table 3.7ESI of the Fixed Assets Tables over nominal net output.

To estimate the equilibrium capitalist savings rate I used Table 5.1. Saving and Investment by Sector and estimated        Undistributed corporate profits -      Capital consumption adjustment, corporate as the measure of retained profits for the demand of capital outlays. Net aggregate profits are measured by Corporate Profits After Tax with Inventory Valuation Adjustment (IVA) and Capital Consumption Adjustment (CCAdj). The rate of savings is ( Undistributed corporate profits -      Capital consumption adjustment) divided by net aggregate profits. Lastly, to avoid double y-axis I added 0.25 to the savings rate.

§.§ Additional Data and Figures

This subsection compares the labor shares by sectors in the US economy using the BEA-BLS integrated industry-level production account <cit.>. In the first row of Figure <ref> I compare the labor shares of the total non-farm economy with the measure proposed of productive sectors.  As noted above, the sectors excluded from the analysis are those with questionable assignments of value added. By employing this modification we find two important changes: the labor share is generally higher than the total non-farm sector; and the fall in the labor share is much clearer in the productive sectors right after the early 1980s. This is an important difference given that a variety of measures of the share of labor only exhibit a clear fall after the 2000s.

The data in Figures <ref> and <ref> offer a way of interpreting the behavior of the labor shares. First, it quite clear that—with the exception of unproductive service sectors (defined below)—the share of wages of non-college decreased significantly in the decade between 1970 and 1980.  That is, dividing the share of labor between college and non-college workers, it is clear that the entire fall of the aggregate wage share can be accounted for the declining participation of non-college workers on aggregate income. This, of course, is not necessarily a problem if the share of college workers increases by the same amount that the share of non-college labor is falling. The  data reveals an heterogeneous behavior between sectors: some show a considerable decline in the total share of labor in spite of the rising participation of college workers, while others show a general rise explained by the surge of the wages of college graduates.

By and large, the sectors excluded from the analysis because of their questionable assignments of value added show a steady or even rising share of wages. For instance, the labor share in unproductive labor services—which is the sum of Professional, scientific, and technical services, Management of companies and enterprises, Educational services, and Health care and social assistance—show a remarkable rise from about 40 percent in 1950 to about 90 percent since the 1990s. The FIRE sector also shows a slight increase in the labor share since the  1970s, which contrasts with the behavior of all other major sectors in  Figures <ref> and <ref>. Even without counting manufacturing, which  shows the steepest decline in the wage share since the late 1970s, we see that sectors like productive services, utilities and information, retail, transportation and warehousing, and wholesale trade all exhibit a considerable decline in the wage share since 1980s.

These differences in the sectors of the economy are of great importance for understanding the changes in income distribution. Particularly, one of the conclusions that can be drawn from this data is that different forces may be shaping the behavior of income distribution in each sector  depending on the role they play in the production of wealth in the economy.

Labor shares. Notes— The productive service sector in the sum of Administrative and waste management, arts, Entertainment, and recreation, Accommodation and food services, and Other services, expect government. All the data is from the BEA-BLS integrated industry-level production account.

Labor shares. Notes— The unproductive service sector is the sum of Professional, scientific, and technical services, Management of companies and enterprises, Educational services, and Health care and social assistance . All the data is from the BEA-BLS integrated industry-level production account.

§.§ Additional Empirical results
Figure <ref> shows some results which complement the empirical results in the main text. Panel A corroborates the results of Panel E in Figure <ref>, showing that the model does capture the main movements of the labor market through time. The results of net investment-output ratio, rate of profit, and the capital-output show a similar pattern as that of the investment-output ratio in Figure <ref>. Essentially, these results indicate that the calibration of the depreciation rate was probably too high, given that with a lower $\delta$, the fit of the model in Panel C of Figure <ref> would have required a higher capital-output ratio, and this would have implied  higher values of $P^{k} K/ PY$, $(P^{I} X-\delta P^{k} K)/P Y$ (panel B en Figure <ref>), on one hand, and a lower value for the rate of profit, on the other

A lower $\delta$, however, would have increased the steady-state values of the savings rate. The reason for this is that—according to Theorem <ref>—in the steady-state equilibrium: $s \approx g(1-\Omega^c)/(\delta \mu -(1-\Omega^c)(\tau + \zeta))$. In the paper I made the decision to underpredict the value of $P^{k} K/ PY$ to maintain relatively low rates of savings, even though I am aware that the model should overpredict the value of $s$ given the assumption that all investments are financed from retained earnings.

Having cleared out why the model underpredicts   the capital(investment)-output ratio, we may turn our attention to Panel C, which displays the rate of profit (measured as profits over the capital stock), the rate of return (measured as profits over costs of production), and the implied equilibrium rate of profit from equation (<ref>). One of the key results here is that, unlike the rate of return, the rate of profit was kept relatively constant after the 1980s. This is explained by the increase in the rate of automation which, in accordance to Proposition <ref> and  Figure <ref>, reduces the rate of profit because it increases the value of capital relative to the final good. This negative effect on $r$ is balanced with the increase in $\mu$ caused by the deterioration of the bargaining power of labor.

The data of the marginal productivity of capital is obtained using (<ref>) and estimating

\begin{equation*}
\hat{Y}_{K_{t}} = \frac{\delta (1+\mu_{t}) \Omega_{I} (1-(1-\pi_{I})e^{-\bar{z}(\upsilon-1)/\upsilon}}{1-(1-\pi_{I})e^{-\bar{z}}}
\end{equation*}

From this perspective, the marginal productivity of capital is directly obtained by the rate of return of capital, which—in turn—is determined as a social outcome resulting from the bargaining process of wages between capitalists and workers. Stated differently, the  marginal productivity of capital is here a meaningless concept if it is detached from the social elements determining the return of capital.  Note, in addition, that the marginal productivity of capital—unlike the rate of profit—increases considerably after the 1980s, which is precisely what is expected from  Proposition <ref> and  Figure <ref>.

Finally, Panel F shows a clear positive correlation between the wage premium and the rate of return of capital. Though the wage premium is not a topic which is directly treated in the paper, the data in Figure <ref> provides some evidence to the hypothesis that the  increase of the marginal productivity of skilled workers is responding to external factors like rising sales and  profits, all of which may be the result of political changes favoring top income earners  (see, e.g.,piketty2013capital) . The role that, e.g., managers and CEOs play in the creation of profits is a topic which is well worth studying in future research.

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Additional equilibria. Notes—  Panels  B,C,D,E, and F   are  based on  the BLS-BEA integrated data set  <cit.>. Panel A uses the non-farming  vacancy data of  petrosky2021unemployment. The wage-premium is normalized to the 1981 value of the rate of return.   As in Figure <ref>, the green diamonds are the result of adjusting $m$ and $\beta^w$ to the time averages of the capital cost share and the labor share.

§.§ Equation (<ref>)

The public policy equation in (<ref>)  is estimated using a Kalman filter and the following priors on the model:

\begin{equation*}
\beta^{b} \sim N(\beta^{b}_{0}, M_{0}), \; \;\; \varphi^{2}_{b} \sim IG(\nu_{0}, s_{0}),\;\; \varphi^{2}_{\beta^{b}_{i}} \sim IG(\nu^{\beta}_{0},s^{\beta}_{0}) \; \text{for} \; i=0,1,...,p+1.
\end{equation*}

Time-varying models often require information priors to narrow down the uncertainty of the parameters. Here I set $\beta^{b}_{0} =0_{p+1 \times 1}$ to avoid including any bias in the sign of the parameters and set $M_{0}=0.5 \times I_{p+1 \times p+1}$ to convey the message that the initial values are probably close to zero. Working with the assumption that the time-varying parameters are not too far from the fixed parameter solution, I set $s^{\beta}_{0}=(s^{\beta}_{0}-1) \times [0.1^2, 0.01^2, 0.01^2, 0.05^2]$ and $\nu^{\beta}=5$. That is, the expected of the variance of the parameters is equal to two times the vector on the right-hand side of $s^{\beta}_{0}$. Similarly, using the sample variance of UI extensions over labor productivity is about 0.00018, I set $s_{0}=(\nu_{0}/2-1)\times 0.01^2$ with $\nu_{0}=5$.

Given these priors it is quite simple to estimate the posterior distribution of the parameters using a Kalman filter and a Gibbs sampler. Further computational details can be found in prado2010time.

§.§.§ Analysis of vacancy costs
It is important to note that the data of rates of return and of  labor market tightness require vacancy costs which are significantly higher than those commonly reported in the literature. Using the calibration in Table <ref> and the results  in Figure <ref>, the proportional costs of hiring are on average 4.7 times greater than the average productivity of labor. Though this may be  an  implausibly large value in light of the empirical studies of silva2009labor, who show that recruiting costs are about 14 percent of quarterly pay per hire, and the estimates of merz2007labor, showing that the marginal costs of hiring are about 1.5 times the average productivity of labor, it is necessary if the model is to satisfy the conditions for steady-state growth paths in Theorem <ref>.

Implied rates of return in the literature

Vacancy costs ($\kappa/Y_{N}$)     $\mu$     $\mu^{\min}$      Source

0.21       0.015      0.046     shimer2005cyclical

0.585     0.0124       0.0684      hagedorn2008cyclical

0.433      0.0154      0.0242      hall2008limited

0.356      0.017      0.0272        pissarides2009

0.7      0.0926      0.1192      petrosky2018endogenous

0.531      0.022      0.0311        petrosky2021unemployment

*Notes— The minimum rate of return is obtained by setting aggregate profits equal to the sum of vacancy costs plus taxes as in equation (<ref>).

Table <ref> reports some   commonly used calibrations for vacancy costs relative to average labor productivity and shows that under no circumstance can any of the models of the equilibrium unemployment literature satisfy the condition where capitalists are capable of paying  taxes,  vacancy costs and have a remnant for financing their own consumption.  Search and matching models generally avoid this problem by working under the assumption that all households share the ownership of capital, so it makes  no  difference  whether consumption is financed from  wages or profits.  This, however, merely hides the problem since it does not address the key issue  of showing that—given the assumptions of the model—the economy can expand at an increasing scale through time.