3.1. Workers

There is a unit measure of workers with lifetime preferences

(1) \begin{equation} \int _{0}^{\infty }e^{-\gamma t}U\left (C_{w}(t),L(t)\right )dt, \end{equation}

where $U({\cdot})$ is strictly increasing and concave in both arguments, and where $C_{w}(t)$ is consumption and $L(t)$ is labor supply. Workers supply $L(t)\geq 0$ units of labor to firms, for which they receive an after-tax wage of $\left (1-\tau ^{\ell }(t)\right )w(t)$ . They also receive a transfer from the government, $T^{w}(t)$ . They consume all of their income, so their consumption isFootnote ¹⁰

(2) \begin{equation} C_{w}=\left (1-\tau ^{\ell }\right )wL+T^{w}. \end{equation}

We assume that workers are not able to own capital. Therefore they exhibit hand-to-mouth consumption behavior by assumption. Their only non-trivial decision is labor supply, which satisfies the condition

(3) \begin{equation} \left (1-\tau ^{\ell }\right )wU_{C}\left (C_{w},L\right )\leq -U_{L}\left (C_{w},L\right ), \end{equation}

which holds with equality when $L\gt 0$ .

A particularly useful utility function, which we use in our quantitative analysis, is log in consumption and leisure:

(4) \begin{equation} U\left (C,L\right )=\log \left (C\right )+\phi \log \left (1-L\right ), \end{equation}

in which case labor supply satisfies

(5) \begin{equation} L=\frac{1}{1+\phi }\cdot \max \left (0,1-\frac{\phi T^{w}}{\left (1-\tau ^{\ell }\right )w}\right ). \end{equation}

The condition for zero labor supply is

\begin{equation*} \frac {\phi T^{w}}{\left (1-\tau ^{\ell }\right )w}\geq 1. \end{equation*}

In other words, if the transfer is sufficiently large relative to the after-tax wage, the household does not supply any labor.

3.2. Entrepreneurs

There is also a measure $n_e\in (0,1)$ of entrepreneurs who have preferences given by

(6) \begin{equation} \int _{0}^{\infty }e^{-\rho t}\log \left (c_{e}\right ). \end{equation}

Entrepreneurs each own capital $k$ , which they rent to firms at rental rate $r$ . Capital depreciates at rate $\delta$ , and capital income is subject to a tax of $\tau ^{k}$ gross of depreciation. Therefore, an entrepreneur’s budget constraint is given by

\begin{equation*} \dot {{k}}+c_{e}=r^{k}k \end{equation*}

where

(7) \begin{equation} r^{k}\equiv \left (1-\tau ^{k}\right )r-\delta \end{equation}

is the after-tax net return on capital. Their consumption behavior satisfies the dynamic optimality condition

\begin{equation*} \dot {c}_e/c_e=r^{k}-\rho \end{equation*}

which, together with their budget constraint implies

\begin{equation*} c_e = \rho k. \end{equation*}

As there is a measure $n_e$ of entrepreneurs, per-capita consumption and capital holdings of entrepreneurs are related to aggregate variables by $C_e = n_e c_e$ and $K = n_e k$ . Then the expressions above imply the following aggregate expressions:

(8) \begin{equation} \dot{{K}} =r^{k}K - C_e \end{equation}

(9) \begin{equation} \dot{C}_e/C_e =r^{k}-\rho \end{equation}

(10) \begin{equation} C_e = \rho K \end{equation}

Equation (10) implies that entrepreneurs consume a constant flow share of their wealth. This result follows from the twin assumptions of log utility, which implies an intertemporal elasticity of substitution (IES) equal to $1$ , and zero non-capital income. Together, these assumptions imply that income and substitution effects cancel and the entrepreneur consumption function is not forward-looking. We make these assumptions to avoid issues arising from the manipulation of tax rates far in the future in the optimal policy problem we analyze in section 4, which would otherwise lead either to the non-existence of an interior steady state (for IES<1), or to slow convergence to zero capital tax rates in the long run (for IES>1). Since these results rely on perfect commitment to tax rates in the far future, and correspondingly forward-looking responses by entrepreneurs to these rates, they requires strong assumptions about commitment and rationality which are unlikely to hold in reality. Thus for our purposes log utility is not only more convenient, but also more realistic in its results.Footnote ¹¹

Note that we allow entrepreneurs’ discount rate $\rho$ to differ from that of workers $\gamma$ . We do this for two reasons. First, given the central importance of intertemporal substitution to any question of capital taxation, and especially to managing an episode of technical change occurring over a period of time, the discount rate is a critical parameter. Thus it is interesting to examine how our results depend on this rate, considered separately for workers and entrepreneurs. Second, we think that it is intuitive that entrepreneurs have a lower discount rate than workers, as we assume in our quantitative model in Section 6. This is consistent with workers holding no capital, and with a large body of literature supporting a link between heterogenous intertemporal discounting and wealth inequality.Footnote ¹²

3.3. Firms

The firms choose an optimal production plan for given wage $w$ and rental rate of capital $r$ . The production technology is constant-returns-to-scale (CRS) and is given by

(11) \begin{equation} Y=\left [\int _{0}^{1}\left (y\left (i\right )\right )^{1-\frac{1}{\sigma }}di\right ]^{\frac{\sigma }{\sigma -1}}, \end{equation}

where $y\left (i\right )$ is the contribution of task $i$ to overall output, $Y$ . Thus, $\sigma$ is the elasticity of substitution among tasks in production. As we will see later, in equilibrium it will also be the elasticity of substitution between capital and labor. There is a unit measure of tasks indexed by $i\in \left [0,1\right ]$ . Each task can be performed by labor or capital. The output of task $i$ satisfies:

(12) \begin{equation} y\left (i\right )=a\left (i\right )k\left (i\right )+b\left (i\right )\ell \left (i\right ) \end{equation}

where $a\left (i\right )$ denotes the productivity of capital in task $i$ , and $b\left (i\right )$ denotes the productivity of human labor in task $i$ . We assume, without loss of generality, that $a\left (i\right ),b\left (i\right )\geq 0$ for all $i$ , and that tasks are ordered such that $a\left (i\right )/b\left (i\right )$ is weakly decreasing in $i$ . We further require that $k\left (i\right ),\ell \left (i\right )\geq 0$ .

We adopt a task-based aggregation of the overall production process of the economy. This choice is more appropriate for our purposes as our focus is on automation and the process of automation is associated with substitution between capital and labor at the level of tasks, rather than goods. Therefore, tasks represent an intuitive disaggregated unit for the specification of the production technology.

3.3.1. Optimal production and factor demand

Next we characterize the optimal production plan of firms and derive expressions for factor demand.Footnote ¹³ We assume that capital and labor are both fully mobile across tasks. Then there is a single wage $w$ and rental rate of capital $r$ . Optimal production by firms yields an allocation of capital and labor across tasks $\{k(i),\ell (i)\}$ such that aggregate output $Y$ is the following function of aggregate capital $K=\int _ik(i)$ and labor $L=\int _i\ell (i)$ :

(13) \begin{equation} Y = F(K,L) \equiv \max _\alpha \left \{ \left [\alpha ^{\frac{1}{\sigma }}\left (A(\alpha ) \cdot K\right )^{1-\frac{1}{\sigma }}+\left (1-\alpha \right )^{\frac{1}{\sigma }}\left (B(\alpha ) \cdot L\right )^{1-\frac{1}{\sigma }}\right ]^{\frac{\sigma }{\sigma -1}} \right \} \end{equation}

where

(14) \begin{equation} A(\alpha) = \left [\frac{1}{\alpha }\int _{0}^{\alpha }\left (a\left (i\right )\right )^{\sigma -1}di\right ]^{\frac{1}{\sigma -1}}, \end{equation}

(15) \begin{equation} B(\alpha ) = \left [\frac{1}{1-\alpha }\int _{\alpha }^{1}\left (b\left (i\right )\right )^{\sigma -1}di\right ]^{\frac{1}{\sigma -1}} \end{equation}

are average capital and labor productivity respectively, and where $\alpha \in [0,1]$ is the share of tasks done by capital. The optimal choice of $\alpha$ satisfies:

(16) \begin{equation} \begin{cases} \dfrac{a\left (i\right )}{b\left (i\right )}\geq q(\alpha, K,L) & \text{for }i\lt \alpha \\[15pt] \dfrac{a\left (i\right )}{b\left (i\right )}\leq q(\alpha, K,L) & \text{for }i\gt \alpha . \end{cases} \end{equation}

where $q$ is given by

(17) \begin{equation} q(\alpha, K,L) = \left (\frac{\int _{0}^{\alpha }\left (a\left (i\right )\right )^{\sigma -1}di}{\int _{\alpha }^{1}\left (b\left (i\right )\right )^{\sigma -1}di}\cdot \frac{L}{K}\right )^{\frac{1}{\sigma }} \end{equation}

Note that, given our assumption that $a(i)/b(i)$ is weakly decreasing in $i$ , $q$ is strictly increasing in $\alpha$ for given $(K,L)$ , and therefore (16) defines a unique value of $\alpha$ . Inverse capital and labor demand satisfy:

(18) \begin{equation} w = F_{L}(K,L) = B^{1-1/\sigma }\left ( (1-\alpha ) Y/L \right )^{1/\sigma }, \end{equation}

(19) \begin{equation} r = F_{K}(K,L) = A^{1-1/\sigma }\left ( \alpha Y/K \right )^{1/\sigma }, \end{equation}

from which we see that $q \equiv r/w$ is the relative factor price.

Thus, the production function has a standard CES representation, with capital and labor productivity terms, for a given value of $\alpha$ . Moreover, as noted earlier, $\sigma$ , the elasticity of substitution among tasks, is also the elasticity of substitution between capital and labor. This representation closely resembles analogous expressions in other papers that use task-based models of automation, the closest being equation (7) in Acemoglu and Restrepo (Reference Acemoglu and Restrepo2018b). However, we also clearly highlight explicitly the share of tasks done by capital $\alpha$ as well as provide a succinct representation of the productivity of capital, $A(\alpha )$ , and labor, $B(\alpha )$ . This has an important advantage for our purposes as the effect of a marginal change in the capital efficiency schedule $a(i)$ can be decomposed into changes in $A$ and $\alpha$ . These two components quite intuitively capture the distinction between automation and traditional technical progress: technical change that produces an increase in $\alpha$ represents automation, since tasks previously done by labor are now being done by capital, whereas technical change that produces an increase in $A$ represents traditional technical progress.Footnote ¹⁴

3.4. Government

The government sets taxes on capital income and labor income. It uses this income to pay for transfers to workers and to finance expenditures, $G$ , which otherwise do not enter the model. The government budget constraint, therefore, is

(20) \begin{equation} \tau ^{\ell }wL+\tau ^{k}rK=T^{w}+G \end{equation}

Government expenditures are assumed to be a fixed fraction of GDP

(21) \begin{equation} G(t) = \omega Y(t) \end{equation}

and may be interpreted as the provision of core functions of government, such as defense, law enforcement, a legal system, basic infrastructure, and so on. Transfers may be considered all broad and untargeted government transfers, plus services that directly substitute for household consumption expenditures.

For the moment we will take the path of tax rates and transfers $\{\tau ^k,\tau ^\ell, T^w\}$ as exogenous, subject to satisfying the government budget constraint (20). In section 4 we will analyze the optimal choice of taxes and transfers under the assumption of majority voting.

3.5. Existence of steady state with fixed tax rates

An interesting feature of models like this oneFootnote ¹⁵ is the possibility of continuous growth through capital accumulation. Thus, before turning to the analysis of optimal policy, we briefly discuss the question of the existence of a steady state for fixed technology and tax rates. The following proposition gives a condition under which the economy reaches a steady state:

Proposition 1. Suppose that $a(i)$ , $b(i)$ , $\tau ^\ell$ , and $\tau ^k$ are all constant. Let $A(1)$ be capital productivity $A(\alpha )$ , as defined in ( 14 ), as $\alpha \to 1$ . Then if

(22) \begin{equation} A\left (1\right ) \gt \frac{\rho +\delta }{1-\tau ^{k}}\equiv r^{\ast }\gt 0 \end{equation}

the growth rate of the capital stock, $\dot{K}/K$ , is bounded below by $A(1) - r^\ast \gt 0$ , and therefore the capital stock grows continually and the economy does not reach a steady state. By contrast, if $A(1) \lt r^\ast$ , the economy reaches a steady state with $r=r^\ast$ .

Proof. See Appendix A.2.

This result follows from the fact that, as $\alpha \to 1$ , the production function approaches $Y = A(1) \cdot K$ , where $A(1)$ is capital productivity (14) when $\alpha =1$ . Since $r$ is increasing in $L/K$ , and as $L/K\to 0$ the marginal product of capital is bounded below by $A(1)$ , it follows that $A(1)$ is also a lower bound on possible values of $r$ . Meanwhile, the critical interest rate $r^\ast$ is the steady state rental rate of capital, derived from the entrepreneur Euler equation. Then if $A(1)\gt r^\ast$ , the rental rate is always above the steady state level, and so no steady state is reached. Then, by the Euler equation, entrepreneur consumption $C_e$ always grows at a rate no less than $A(1)-r^\ast \gt 0$ , and since $C_e = \rho K$ the capital stock does as well.Footnote ¹⁶

In the following sections, we will focus on the case in which a steady state exists. This will allow us to focus on our main goal—to analyze the distributional consequences of an episode of automation, similar to what the developed world is likely to experience over the next few decades.Footnote ¹⁷

4.1. Statement and solution of the planning problem

We assume that policy is set by majority voting with full commitment. Since we assume that workers are in the majority, this amounts to setting policy to maximize the welfare of workers. Thus we can formulate the problem as a Ramsey Planning problem, with a single state variable ( $K$ ), and solve it as follows:

Proposition 2. The equilibrium under majority voting is equivalent to the solution to the following Ramsey Planning problem: For given $K(0)$ , choose $\{C_w,L,K\}$ to maximize

(24) \begin{equation} \max _{\left \{ C_{w},C_{e},K,L\right \} }\left \{ \int e^{-\gamma t}U\left (C_{w},L\right )\right \}, \end{equation}

subject to constraints

(25) \begin{equation} \dot{K} = (1-\omega ) F\left (K,L\right )-\delta K - \rho K - C_w \end{equation}

and $L\geq 0$ . The solution to this problem satisfies the conditions

(26) \begin{equation} U_{C}\left (C_{w},L\right ) =\lambda, \end{equation}

(27) \begin{equation} -U_{L}\left (C_{w},L\right ) \geq \lambda (1-\omega ) F_{L}\left (K,L\right ), \end{equation}

(28) \begin{equation} -\dot{\lambda }/\lambda =(1-\omega )F_{K}\left (K,L\right )-\delta -\rho -\gamma, \end{equation}

together with ( 25 ) holding with equality, and where ( 27 ) holds with equality when $L\gt 0$ .

Proof. See Appendix A.3.

Here equation (25) combines the resource constraint with the entrepreneur consumption rule (10). Notably, this embeds the government budget constraint, and therefore implies variation in transfers as well as tax rates, which would not be possible if the planner did not have access to a transfer to workers as a policy instrument.

Proposition2 gives conditions which define paths of $\{C_w,K,L,\lambda \}$ under majority voting, which likewise define $C_e = \rho K$ .Footnote ²⁰ To understand its implications for optimal policy, we first characterize the steady state, and then consider the transition.

4.2. Optimal policy in the long run

Consider the equilibrium under optimal policy, as given by Proposition2. Given that technology is stationary and $A(1)$ satisfies condition (23), in the long run the economy will converge to a steady state. Corollary (1) characterizes equilibrium quantities and policy variables that prevail in this steady state.

Corollary 1. Under majority voting and stationary technology that satisfies ( 23 ), a steady state is reached in which $\{K,L,C_w\}$ satisfy:

(29) \begin{equation} -\frac{U_{L}\left (C_{w},L\right )}{U_{C}\left (C_{w},L\right )} =(1-\omega )F_{L}\left (K,L\right ), \end{equation}

(30) \begin{equation} C_{w} =(1-\omega )F\left (K,L\right )-\left (\rho +\delta \right )K, \end{equation}

(31) \begin{equation} F_{K}\left (K,L\right ) =\frac{\rho +\gamma +\delta }{1-\omega } = r^\ast . \end{equation}

Steady state tax rates and transfers satisfy:

(32) \begin{equation} \tau ^{\ell } =\omega, \end{equation}

(33) \begin{equation} \tau ^{k} =\omega +\frac{\left (1-\omega \right )\gamma }{\rho +\delta +\gamma } \equiv \omega +\frac{\gamma }{r^{\ast }}, \end{equation}

(34) \begin{equation} T^w = \gamma K. \end{equation}

Proof. See Appendix A.3.

Equations (29) – (31) jointly define $\{L,C_w,K\}$ , which imply a long-run marginal product of capital equal to $r^\ast$ . The long-run taxes and transfers that implement this equilibrium are given by equations (32) – (34). Notably, long-run tax rates are independent of technology, and only depend on the government consumption share, depreciation, and the discount rates of entrepreneurs and workers. The steady state capital tax rate is always higher than $\omega$ , and therefore higher than the labor tax rate.Footnote ²¹

To gain intuition for these results, it is helpful to first consider the case of a representative agent economy. In this case, the optimal transfer would be zero, and optimal taxes on labor and capital would both equal the share of government spending $\omega$ (and in particular would be zero when $\omega =0$ ). Intuitively, in the absence of government spending linked directly to GDP, households would not choose to distort the labor or capital markets at all, since at the margin the only purpose of taxing would be to transfer funds to themselves. By contrast, when government spending is a share of GDP, a fraction of any increase in production goes to increased government spending, which has no benefit to households, and thus is perceived by them as a tax on production. Therefore, they optimally choose to internalize this cost by setting the labor and capital taxes equal to the share of marginal GDP consumed by the government.

Compare this with the present model with workers and entrepreneurs. Here there is an additional purpose of taxes—to transfer funds from entrepreneurs to workers. Since entrepreneurs earn no labor income, this motive provides no incentive to raise labor taxes, and therefore $\tau ^\ell$ remains unchanged. By contrast, capital taxes allow the net transfer of funds from entrepreneurs to workers, which has two effects. First there is the direct effect of higher consumption due to the transfer. This is counterbalanced by an indirect pecuniary effect, since higher capital taxes lead to lower capital, which lowers $w/r$ and reduces the relative income of workers.Footnote ²² Thus optimal policy involves an intertemporal tradeoff between current and future consumption of workers. Nevertheless, the net effect of this tradeoff is that the capital tax rate will rise above $\omega$ , with the proceeds from the excess portion transferred to workers. This occurs for two reasons. First, entrepreneurs consume flow consumption $C_e = \rho K$ , which from the perspective of workers acts as a tax on capital, effectively lowering the return on investment. Thus the planner raises capital taxes by $\rho /r^\ast$ , so that entrepreneurs internalize this “tax” and reduce investment accordingly. Second, since workers are more impatient than entrepreneurs, they prefer a lower rate of investment. To bring this about, the planner raises the capital tax rate by $(\gamma -\rho )/r^\ast$ . Summing these two motives yields an increase in the tax rate of $\gamma /r^\ast$ relative to the representative agent economy. The proceeds from these higher taxes are transferred to workers, and thus workers receive a flow transfer equal to a fixed fraction $\gamma$ of the economy’s capital stock.

4.3. Optimal policy over the transition

It is also instructive to analyze how the path of taxes and transfers relates to other macroeconomic variables during transition to develop intuition for the character of the optimal policy. For this purpose, it is not necessary to calculate explicit expressions for these policy variables in terms of exogenous parameters as done above in Corollary1. Instead, it is enough to relate them to the paths of the other macroeconomic variables as in:

Corollary 2. During transition the optimal policy under majority voting is related to other macroeconomic variables as follows:

(35) \begin{equation} \tau ^\ell = \omega, \end{equation}

(36) \begin{equation} \tau ^{k} = \omega +\frac{\gamma -\dot{\lambda }/\lambda -\dot{K}/K}{F_K}, \end{equation}

(37) \begin{equation} T^w = \left (\gamma -\dot{\lambda }/\lambda -\dot{K}/K\right )K, \end{equation}

where $\{C_w,L,K,\lambda \}$ satisfy ( 25 ) – ( 28 ).

Proof. See Appendix A.3.

Corollary2 shows that, in line with our discussion of optimal policy above, the labor tax rate remains fixed at the government expenditure share $\omega$ throughout. However, the capital tax rate and transfers will differ from their long-run level over the course of the transition. This occurs for two reasons. The first relates to the tradeoff between the direct consumption effect of the transfer and its indirect pecuniary effect. When capital is below its long-run level, there is both a smaller direct consumption effect (since there is less capital income to tax), and a greater pecuniary effect (since the marginal effect of more capital on wages is larger when there is less capital). Thus the optimal tax rate and transfer are both increasing in capital, as can be seen by their respective dependence on $F_K$ and $K$ . However, their dynamics differ somewhat in that the optimal tax rate depends on the marginal product of capital, which varies directly with technology, while the optimal transfer depends only on the capital stock.

Second, as is typical in intertemporal optimization problems, the intertemporal marginal rate of substitution affects the optimal investment rate. For the planner, the relevant rate is the growth rate of workers’ marginal utility of consumption $\dot{\lambda }/\lambda$ , whereas entrepreneurs’ marginal utility grows at the rate $-\dot{C}_e/C_e = -\dot{K}/K$ (given log utility, and since $C_e = \rho K$ ). Thus the capital tax is increased by $({-}\dot{\lambda }/\lambda ) - \dot{K}/K$ , so that entrepreneurs internalize this difference in the desired rate of intertemporal substitution. Intuitively, if this term is positive it implies that workers have a greater desire to smooth consumption over time than entrepreneurs, and thus choose to raise capital taxes to transfer funds to themselves for current consumption.

5.1. Uniform labor productivity

We now turn to characterizing the long-term effects of an episode of technical change. But first we need to make an assumption about labor productivity. Here we assume that labor productivity is uniform across tasks, and for convenience is normalized to $1$ , that is, $b(i)=1$ for $i \in [0,1]$ . We can then derive simple expressions for major steady state quantities:

Proposition 3. Given that $a(i)$ satisfies ( 38 ), $b(i)=1$ , and that conditions ( 23 ) and ( 39 ) hold, in steady state we have $\alpha = \bar{\alpha }$ , and

(40) \begin{equation} K/L = \bar{\alpha }\left (1-\bar{\alpha }\right )^{\frac{1}{\sigma -1}}\left ((r^{*}/a)^{\sigma -1}-\bar{\alpha }\right )^{\frac{\sigma }{1-\sigma }}/a \gt 0,\end{equation}

(41) \begin{equation} Y = \left (\left (\bar{\alpha }\right )^{\frac{1}{\sigma }}\left (aK\right )^{\frac{\sigma -1}{\sigma }}+\left (1-\bar{\alpha }\right )^{\frac{1}{\sigma }}L^{\frac{\sigma -1}{\sigma }}\right )^{\frac{\sigma }{\sigma -1}}, \end{equation}

(42) \begin{equation} s_{L} =1-\bar{\alpha }\left (r^{\ast }/a\right )^{1-\sigma } \in (0,1), \end{equation}

(43) \begin{equation} w = \left (\frac{1-\bar{\alpha }}{s_{L}}\right )^{\frac{1}{\sigma -1}}, \end{equation}

where $s_L = wL/Y$ is the labor share.

Proof. See Appendix A.4.

Note that, while steady state capital and output are defined only up to steady state labor supply, which depends on equations (2) and (3) as well, the steady state wage and labor share do not depend on $L$ . This is convenient, as we are chiefly interested in the distributional implications of automation, which we summarize using the labor share and the wage. These depend on the share of tasks done by capital $\bar{\alpha }$ , and on the ratio between capital task productivity $a$ and the long-run rental rate of capital, which satisfies $r=r^\ast$ .Footnote ²⁵

The expressions in Proposition3 immediately yield the following comparative statics for traditional technical progress and automation for the wage and labor share:

Corollary 3. Traditional technical progress, that is, an increase in capital task productivity $a$ , increases the steady state wage, whereas the steady state labor share of income (i) increases for $\sigma \lt 1$ ; (ii) remains constant for $\sigma =1$ (the Cobb-Douglas case); and (iii) decreases for $\sigma \gt 1$ . In particular,

(44) \begin{equation} \frac{dw}{da}\frac{a}{w}=\frac{1-s_{L}}{s_{L}} \gt 0 \end{equation}

and

(45) \begin{equation} \frac{ds_{L}}{da}\frac{a}{s_{L}}=\left (1-\sigma \right )\left (\frac{1-s_{L}}{s_{L}}\right )\gtreqqless 0\qquad as\qquad \sigma \lesseqqgtr 1. \end{equation}

where $s_L$ satisfies ( 51 ).

Proof. See Appendix A.4.

Corollary 4. Automation, that is, an increase in the automated share of tasks $\bar{\alpha }$ , increases the steady state wage but decreases the steady state labor share. In particular,

(46) \begin{equation} \frac{dw}{d\bar{\alpha }}\frac{\bar{\alpha }}{w}=\frac{1}{\sigma -1}\left (\frac{1}{s_{L}}-\frac{1}{1-\bar{\alpha }}\right )\gt 0 \end{equation}

and

(47) \begin{equation} \frac{ds_{L}}{d\bar{\alpha }}\frac{\bar{\alpha }}{s_L}=-\left (\frac{1-s_{L}}{s_L}\right )\lt 0 \end{equation}

where $s_L$ satisfies ( 42 ).

Proof. See Appendix A.4.

Intuitively, traditional technical progress increases the wage, since this corresponds to increasing the effective capital supply, and capital is a complement of labor. It is likewise natural that changes in the labor share depend on $\sigma$ , i.e., the degree of substitutability between capital and labor. As is well known, when $\sigma =1$ the opposing substitution effect and the Edgeworth complementarity exactly cancel each other and the labor share is constant. When $\sigma \gt 1$ , the substitution effect dominates causing the labor share to decline despite the increase in wages, whereas the reverse is true for $\sigma \lt 1$ .

As shown in Corollary4, automation also results in an increase in the steady state wage, mainly due to the increase in steady state capital. However, the effect of automation on the labor share, unlike that of traditional technical progress, does not depend on the elasticity of substitution $\sigma$ . This is a vital difference that speaks to the current debate centered around automation and associated policy proposals like universal basic income. The key to this difference lies intuitively in the fact that the effect of automation manifests as substitution within tasks (or at the task level), whereas the effect of traditional technical progress ( $a$ ) operates between tasks (or across tasks at the aggregate level). Thus, the distributional effects of automation are mainly driven by perfect substitutability of capital and labor within a task whereas the effects of traditional technical progress depend on the degree of substitutability of capital and labor between tasks (i.e. $\sigma$ ). Moreover, since this difference between the degree of substitutability (between capital and labor) within and between tasks is larger when $\sigma \lt 1$ , the qualitative effects of automation compared to traditional technical progress are worse in this case compared to when $\sigma \gt 1$ . This is substantiated by the fact that when $\sigma \lt 1$ an increase in $a$ increases the labor share of income whereas it falls for an increase in $\bar{\alpha }$ , but when $\sigma \gt 1$ the labor share of income falls in both scenarios.Footnote ²⁶

5.2. Variable labor productivity and wage declines

The result in Corollary4 that automation necessarily raises wages in the long run may be taken as evidence against concerns about automation harming human labor. However, this result depended on our assumption of constant labor productivity across tasks. By contrast, if labor productivity differs across tasks, and relatively high productivity tasks are automated, automation may lower wages.

To illustrate this possibility, suppose that capital productivity is piecewise across tasks as given by (38), and that labor productivity $b(i)$ is now

(48) \begin{equation} b\left (i\right )=\begin{cases} b_{m} & i\in [0,\bar{\alpha }_{1}]\\[5pt] b_{1} & i\in (\bar{\alpha }_{1},1] \end{cases}. \end{equation}

Then we consider an episode of automation that shifts $\bar{\alpha }$ from some level below $\bar{\alpha }_1$ to $\bar{\alpha }_1$ . That is, $b_m$ is the productivity of the marginal tasks automated in this episode of automation, whereas $b_1$ is the productivity of the remaining tasks at its end. Then we can derive a simple condition under which the steady state wage declines:

Proposition 4. Suppose that capital productivity is given by ( 38 ) and labor productivity by ( 48 ). Suppose that initially the economy is in steady state with $\bar{\alpha } = \bar{\alpha }_0 \lt \bar{\alpha }_1$ , and then there is a one-time shift in the capital productivity curve to $\bar{\alpha } = \bar{\alpha }_1$ . Then the steady state wage declines if and only if

(49) \begin{equation} \frac{b_{m}}{b_{1}}\gt \left [\frac{\left (a/r^{\ast }\right )^{1-\sigma }-\bar{\alpha }_{1}}{1-\bar{\alpha }_{1}}\right ]^{\frac{1}{1-\sigma }}. \end{equation}

Proof. See Appendix A.5.

As we previously assumed $a\gt r^{\ast }$ , condition (49) will never be satisfied for $b_{m}\leq b_{1}$ , that is, when the productivity of tasks automated is equal to or lower than the productivity of remaining tasks. However, when $b_m \gt b_1$ , this condition may be satisfied, and thus it is possible for the wage to decline in the long run. Whether this condition holds depends on the relative productivity of the automated tasks to the remaining tasks $b_m/b_1$ , the degree of substitutability between capital and labor $\sigma$ , and the ratio of capital task productivity to the steady state rental rate of capital $a/r^\ast$ . The last matters because this determines the level of steady state capital.

6.1. Calibration

For the quantitative exercise, we choose parameter values that are very standard in the macro/real business cycle literature, and/or based on the data. We calibrate the model so that the initial steady state is typical in terms of tax rates, the ratio of consumption to output, the labor income share, and the ratio of capital to output, among other variables. The initial steady state of the calibrated model is shown in Table 1.

Table 1. Initial steady states with calibrated policy vs. majority voting: $\sigma =0.8$

For $\sigma$ , the elasticity of substitution between capital and labor, the neoclassical literature typically assumes a value of 1, which corresponds to the Cobb-Douglas case. However, as Acemoglu and Restrepo (Reference Acemoglu and Restrepo2018b) note, there is considerable debate on its value and, in particular, on whether it is less than or greater than 1. While recent estimates in Karabarbounis and Neiman (Reference Karabarbounis and Neiman2014) put it to be greater than one, the bulk of the earlier evidence in the literature points to a value between 0.5 and 1 (for further discussion, see Oberfield and Raval (Reference Oberfield and Raval2021) and references therein). We follow Acemoglu and Restrepo (Reference Acemoglu and Restrepo2018b) in viewing the consensus in the literature as more supportive of a value of $\sigma$ below $1$ , and thus set $\sigma =0.8$ as our baseline calibration. However, given the debate around its value, we also consider the value of $\sigma =1.2$ , but since the main result of the paper (and also overall results) are very similar, as explained later the $\sigma =1.2$ case is relegated to an appendix.Footnote ²⁷

We assume that workers have log utility, as given in equation (4). Log utility is a standard and tractable assumption, and implies a moderate intertemporal elasticity of substitution. We set the entrepreneurs’ discount rate $\rho$ to .04 which implies an annual real interest rate of 4%, a standard value in quantitative macro models (e.g. as in Atolia et al. (Reference Atolia, Chatterjee and Turnovsky2010)). As it is common to assume that workers are more impatient than entrepreneurs, we set their discount rate $\gamma$ to .06. $\delta =.05$ corresponds to a 5% depreciation rate for capital, which is a standard value in the literature.Footnote ²⁸ It is also very close to the average US depreciation rate for 1960–2022 of 4.96%, and allows our calibrated model to closely match the US capital-output ratio over the same period (2.90 vs. 2.93).Footnote ²⁹ In a similar vein, we set $\phi =1.4$ , which implies a labor supply of $L=.339$ in our calibrated model.Footnote ³⁰

For tax rates, we use the calculations in Fernández-Villaverde et al., (Reference Fernández-Villaverde, Guerrón-Quintana, Kuester and Rubio-Ramírez2015) updated to more recent data, and we use OECD data for government transfers.Footnote ³¹ Taxes and transfers are averaged over the period 2007 to 2019, which excludes the COVID-19 pandemic and post-pandemic years. The tax rate on labor $\tau ^\ell$ comes to 23.1 percent and the capital tax rate $\tau ^k$ amounts to 31.9 percent. The transfer as a percentage of GDP is 17.4. Using these values, the share of government consumption, $G$ , as a share of GDP, $\omega$ , is computed residually to satisfy the government’s budget constraint and amounts to 9.07 percent, as noted in Table 1. We do not include consumption taxes. Under our calibration, including consumption taxes would increase taxes collected but would go entirely to funding a larger $G$ , as transfers are directly calibrated from data while $G$ is calculated residually.

Next we discuss the technology. We normalize labor task productivity to $b(i)=1$ . As mentioned above, capital task productivity takes the piecewise form of equation (38). Given the values of $\rho$ , $\delta$ , $\sigma$ , and $\tau ^k$ , the values of $a$ and $\bar{\alpha }_o$ together determine the capital share of income, for which the mid-point estimate over 1972–2012 is slightly below 38 percent (see Figure II in Karabarbounis and Neiman (Reference Karabarbounis and Neiman2014)). Also, the value of $a$ must be sufficiently large to satisfy (16) so that tasks with $i\lt \bar{\alpha }$ are done by capital in steady state. With these considerations in view, we set $\bar{\alpha }_0=.5$ and $a=0.5$ , which yield a capital income share of 38.3 percent.

To calibrate the new higher value of $\bar{\alpha }=\bar{\alpha }_1$ after the episode of automation, we discipline our choice with the past experience of a declining labor share documented by Karabarbounis and Neiman (Reference Karabarbounis and Neiman2014). In their Figure II, we see the share of labor declining by about 5 percentage points from about 65 to 60 percent over the 37-year period of 1975–2012. Since our focus is on the new wave of automation arising from recent major breakthroughs in robotics and AI, which based on our theoretical results in Corollaries3 and 4 is likely to lead to a larger decline in the labor share than past episodes of technical progress, we choose a value of $\bar{\alpha }_1$ to target a 10 percentage point decline in the labor share. This requires setting $\bar{\alpha }_1 = 0.6249$ , which implies output increases by 106% in the long run.

As argued in Atolia et al. (Reference Atolia, Chatterjee and Turnovsky2012), it is more natural to assume that an increase in productivity like this will happen gradually over time. Therefore, to have a realistic assessment of the dynamics, the increase in $\bar{\alpha }$ is allowed to occur gradually. In particular, $\bar{\alpha }$ follows the following exogenous process:

(50) \begin{equation} \dot{\bar{\alpha }} = \theta (\bar{\alpha }_1-\bar{\alpha }) \end{equation}

with $\bar{\alpha } = \bar{\alpha }_0$ initially and $\theta =0.1$ . This implies that the impact of automation lasts for about three to four decades—a typical duration for a general purpose technology.Footnote ³²

6.2. Initial policy and steady state under majority voting

Before analyzing the episode of automation, we first ask how the steady state tax rates from our majority voting model differ from the corresponding rates in US data, and how these rates affect the initial steady state. As discussed above, capital and labor tax rates in the USA have recently averaged 31.9% and 23.1% respectively. By contrast, given our other parameters, the steady state tax rates under majority voting will be 45.4% and 9.7% respectively. That is, capital tax rates will be significantly higher, and labor taxes significantly lower.

The steady states under the two sets of tax rates are shown in Table 1. Unsurprisingly, given the higher capital tax rate, steady state capital is lower, which likewise reduces steady state output. The higher capital taxes fund a larger net transfer to workers, equal to 14.6% of GDP, compared with 8.8% using historical tax rates.Footnote ³³ Thus, workers consume a larger share of output, so that although total output is about 6.3% lower, worker consumption is only 0.2% lower. Conversely, entrepreneur consumption is significantly lower, in both absolute and relative terms. The wage is lower due to the lower capital stock, but the after-tax wage is higher, and so labor supply is higher.Footnote ³⁴

6.3. Quantitative analysis of an episode of automation

We now consider the effect of the episode of automation. We assume the economy begins in the steady state under majority voting with $\bar{\alpha } = \bar{\alpha }_o$ . At time $t=0$ , the economy is hit by a one-time automation shock, which steadily increases $\bar{\alpha }$ from $\bar{\alpha }_{o}$ to $\bar{\alpha }_{1}$ as described in section 6.1. In order to isolate the effect of automation (from the confounding effects of transitioning to optimal policy), we consider the impact of automation under optimal government policy. Thus, the economy is assumed to be in the optimal initial steady state when hit by the shock, and then transitions under optimal policy to the new steady state.

6.3.1. Steady states

We begin by considering the long-run effects of the automation episode. Table 2 shows key steady state values before and after the episode. The automation episode causes the labor share to decline by 10 percentage points as calibrated. This decrease in the labor share is consistent with the results we derived in section 5. So is the increase in the wage, which rises by 68.9 percent. Output increases to 2.06 times its original value, as stated earlier, driven by a large increase in capital. The consumption of workers increases by 74.9%, less than the 106% increase in output. By contrast, the consumption of entrepreneurs increases by 137%, reflecting the increase in the capital share. Finally, it is important to note here once again that the optimal tax rates in the long run do not depend on technology and, therefore, remain unchanged across steady states, when $G/Y$ is held constant. However, the optimal transfer as a share of GDP increases, since the optimal steady state transfer is a fixed share of capital, and the capital-output ratio increases as a result of the episode of automation.

Table 2. Steady states before and after an episode of automation under optimal policy: $\sigma =0.8$

Figure 1. Dynamics for gradual increase with $\sigma =.8$ under optimal policy.

6.3.3. Fixed tax counterfactual

The previous analysis considered the impact of an episode of automation under optimal policy set by majority voting. However, a true assessment of the impact of these policies requires comparison to an appropriate counterfactual benchmark. Accordingly, we consider a counterfactual in which tax rates remain fixed over the course of the transition. To isolate the effects of the optimal dynamic adjustment of taxes from changes in their levels, we assume that tax rates are fixed at their optimal steady state levels under majority voting throughout.Footnote ⁴⁰ Thus the initial and final steady states are the same as shown in Table 2, and all that changes are the transition paths.

Figure 2. Dynamics for gradual increase with $\sigma =.8$ with fixed taxes.

The transition paths under the fixed tax counterfactual are shown in Figure 2, which depicts the same variables as Figure 1, except that the time path of automation ( $\bar{\alpha }$ ) is shown in place of the capital tax rate (which is now fixed).

In terms of the dynamics, two things stand out. First, the transition itself is much slower, as can be seen by directly comparing Figures 1 and 2. This is reflected in the time scales of the figures, which is 100 years in the fixed tax case, where previously we only showed 75. This follows directly from the fixed capital tax rate. As discussed above, under optimal policy the tax rate is lowered over the transition to boost investment and accelerate the accumulation of capital. In the fixed tax case, this does not take place, and thus it takes significantly longer for capital and output to reach their new higher steady state levels.

Secondly, there is significant overshooting of three variables: the labor share, labor supply, and transfers to workers as a share of GDP. These dynamics all relate to the labor share, so we focus on that. The labor share satisfies:

(51) \begin{equation} s_{L}=\left [1+\left (\bar{\alpha }/\left (1-\bar{\alpha }\right )\right )^{1/\sigma }\left (aK/L\right )^{1-1/\sigma }\right ]^{-1}. \end{equation}

Thus the labor share is decreasing in the share of tasks done by capital $\bar{\alpha }$ , but increasing in $K/L$ (since $\sigma \lt 1$ ).Footnote ⁴¹ To explain the overshooting, we return to our observation above that, in the absence of lower capital taxes during the transition, capital takes significantly longer to reach its steady state value. By contrast, the exogenous variable $\bar{\alpha }$ is unaffected, and therefore converges to its steady state value significantly faster than $K$ (as can be seen by comparing the first and third panels of Figure 2). Early in the transition the value of $\bar{\alpha }$ is rises rapidly, and the labor share declines. As $\bar{\alpha }$ nears its steady state value, $K$ (and therefore $K/L$ ) remain well below its steady state value, and thus the labor share falls below its steady state value, that is, overshoots. In the last phase, $K/L$ increases further, causing the labor share to rise. The dynamics of labor supply and the transfer to workers as a share of GDP exhibit the same dynamics, because their values are closely tied to the labor share.Footnote ⁴²

The slower pace of the transition under fixed capital taxes turns out to have very important implications for welfare between the two scenarios. Recall that, under optimal policy, the welfare gain for workers was 26.18 percent (in consumption-equivalent terms), versus 65.81 percent for entrepreneurs. By contrast, in the fixed tax counterfactual the workers’ welfare gain is 25.50 percent, whereas the entrepreneurs’ gain is 44.21 percent (again, both in terms of consumption equivalents). Two things are notable here. First, optimally varying capital taxes over the transition substantially increases the welfare gains of entrepreneurs from the episode of automation. Indeed, this comparison allows a decomposition of the welfare effect—67% of the welfare gain to entrepreneurs under optimal taxes is due to the technological change, and 33% is due to optimal policy. Second, in contrast to entrepreneurs, workers benefit very little from optimal policy—optimally varying tax rates over the transition increases workers’ welfare by only 0.68 percentage points, whereas entrepreneurs’ gain by a whopping 21.60 percentage points, more than an order of magnitude higher! This difference is particularly striking given that policy was set to maximize worker welfare, placing zero weight on entrepreneurs.

What explains this result? While it is better from the workers’ perspective to cut capital taxes over the transition in order to accelerate the accumulation of capital, this produces only a small, second-order gain in welfare. The reason is that the resulting benefit to workers in terms of a faster wage increase is offset by the steep cut in transfers that is required to implement the initial capital tax reduction. Workers are trading off higher consumption in the long run against lower consumption in the short run. By contrast, the lower capital taxes in the short run results in higher consumption for entrepreneurs in the short run, and the faster accumulation of capital results in higher entrepreneur consumption in the long run. Thus entrepreneurs do not experience a tradeoff—they simply enjoy higher consumption in both the short run and the long run, and therefore experience a much larger, first-order gain in welfare. Moreover, as the preceding explanation indicates, the logic of the underlying mechanism does not rely on the value of $\sigma$ and whether it is above or below 1. This is attested to by the results for $\sigma =1.2$ in Appendix B.