Automation and unemployment: help is on the way

Abstract

This paper examines the evolution of unemployment in a task-based model that allows for two types of technical change. One is automation, which turns labor tasks into mechanized ones. The second is addition of new labor tasks, which increases specialization, as in the expanding variety literature. The paper shows that in equilibrium the unemployment caused by automation converges to zero over time.

Appendix

1.1 A. Proofs

Proof of Proposition 1

We prove it by negation, assuming that $\varphi (M_{t} ) > R_{t}^{{\frac{\theta }{1 - \theta }}}$. The definition of φ implies:

$$ \int\limits_{0}^{{M_{t} }} {R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} k(j)^{{\frac{ - \theta }{{1 - \theta }}}} dj} = R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} \varphi (M_{t} ) > 1 \ge \int\limits_{0}^{{M_{t} }} {p_{t} (j)^{{\frac{ - \theta }{{1 - \theta }}}} dj} . $$

Hence, there is a set of positive measure of automated tasks for which $p_{t} (j) > R_{t} k(j)$. This violates perfect competition and free entry in the markets for automated goods.□

Proof of Proposition 2

Note first that since $\theta^{\theta /(1 - \theta )} < 1$ we get:

$$ T_{t - 1} - M_{t - 1} \ge T_{t - 2} - M_{t - 1} + \left( {T_{t - 1} - T_{t - 2} } \right)\theta^{{\frac{\theta }{1 - \theta }}} . $$

Applying Eq. (10) we get:

$$ T_{t - 1} - M_{t - 1} \ge \frac{{R_{t - 1}^{{\frac{\theta }{1 - \theta }}} - \varphi \left( {M_{t - 1} } \right)}}{{\varphi^{\prime}\left( {M_{t - 1} } \right)}}. $$

Applying this inequality to the number of unemployed due to automation in Eq. (21), we get the result of the proposition.□

Proof of Proposition 3

We prove the proposition by negation. Assume that automation is bounded. We first show that the number of tasks T is bounded as well. If not, Eq. (10) shows that if T is unbounded, while M is bounded, the gross interest rate R_t goes to infinity. We next show that this is impossible. If that is the case, the equilibrium in the capital market implies that from some time on $K_{t + 1} = w_{t}$. Multiplying with employment $E_{t}$ on both sides of this equality and using Eqs. (15) and (16) on shares of labor and capital, implies:

$$ \frac{{Y_{t + 1} }}{{Y_{t} }} = \frac{{1 - R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} \varphi \left( {M_{t} } \right)}}{{E_{t} R_{t + 1}^{{\frac{ - 1}{{1 - \theta }}}} \varphi \left( {M_{t + 1} } \right)}}. $$

(32)

If the gross interest rate R goes to infinity, while M is bounded, the RHS of this equation goes to infinity. This means that output growth rises to infinity. Clearly this is impossible, since according to Eq. (13) the numerator of output grows at a rate which is a multiple of the rate of growth of T, which is bounded, while the denominator of (13) converges to 1 and does not grow in the long run. Hence, R cannot grow to infinity and therefore T is bounded, just as M is.If T and M are bounded, they converge, as increasing sequences, to finite numbers, $M_{\infty }$ and $T_{\infty }$ correspondingly. As a result, the rate of growth of tasks converges to zero. Note that it is impossible that $M_{\infty } = T_{\infty }$, since then Eq. (12) implies that employment E converges to zero, while it should always be larger than 1. If $M_{\infty } < T_{\infty }$, Proposition 5 implies that the rate of automation unemployment converges to zero, as the changes in M converge to zero. Equation (20) implies that structural unemployment also converges to zero in this case. Hence, the LHS of (12) converges to $1 + e$. However, the RHS of (12) converges to a number smaller than $1 + e$, due to Assumption 1. Hence, bounded automation leads to a contradiction.□

Proof of Proposition 4

Assume by negation that interest rates are not bounded from above. Then:

$$ R_{t}^{{\frac{\theta }{1 - \theta }}} \mathop{\longrightarrow}\limits_{{t \to \infty }}^{}\infty . $$

(33)

As the rate of interest rises and is above $1 + \rho$, the capital market equilibrium condition (18) implies that $K_{t + 1} = w_{t}$ from some period on. Multiplying this equality by employment $E_{t}$ on both sides and using the shares of capital and labor in Eqs. (15) and (16), leads to the equality (32) as in the proof of Proposition 3. Applying (33) to (32) implies that the rate of growth of output converges to infinity. However, the numerator in Eq. (13) grows at a rate proportional to the rate of growth of T, which is bounded, while its denominator converges to 1. Hence, the rate of growth cannot rise to infinity.

Proof of Proposition 5

Divide the proof to three cases, $A < (1 + \rho )^{\theta /(1 - \theta )}$, $A > (1 + \rho )^{\theta /(1 - \theta )}$ and $A = (1 + \rho )^{\theta /(1 - \theta )}$. In the first case the proof is straightforward, since $R_{t} \ge 1 + \rho$ for all t.

In the second case, we prove the proposition by negation. If the proposition does not hold, we have:

$$ R_{t}^{{\frac{\theta }{1 - \theta }}} \mathop{\longrightarrow}\limits_{{t \to \infty }}^{}A. $$

(34)

In the second case Proposition 1 and Assumption 2 imply that from some period on, $R_{t} > 1 + \rho$ and hence $K_{t + 1} = w_{t}$. We next multiply this equality by employment $E_{t}$ and derive an equality similar to (32) in the proof of Proposition 3:

$$ \frac{{Y_{t + 1} }}{{Y_{t} }} = \frac{{1 - R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} \varphi \left( {M_{t} } \right)}}{{E_{t} R_{t + 1}^{{\frac{ - 1}{{1 - \theta }}}} \varphi \left( {M_{t + 1} } \right)}}. $$

(35)

Note that condition (34) implies that the numerator in the RHS of (35) converges to zero. The denominator is bounded below by a positive number according to Proposition 4. Hence, the rate of growth of output converges to zero. However, Eq. (13) implies that output grows both because T rises continuously and increases the numerator, and because the denominator converges to zero due to (34). Hence, the rate of growth of output must be positive and $Y_{t + 1} /Y_{t}$ should be greater or equal to 1. This contradiction proves the proposition in this case.

In the third case we prove the proposition by negation as well. Assume that (34) holds. If the interest rates are above $1 + \rho$, then the analysis of the second case holds and we reach the same contradiction. If not, from some period on $R_{t} = 1 + \rho$. Hence, capital satisfies: $K_{t + 1} \le w_{t}$. We can derive, similar to the derivation of (32), the following inequality:

$$ \frac{{Y_{t + 1} }}{{Y_{t} }} \le \frac{{1 - A^{ - 1} \varphi \left( {M_{t} } \right)}}{{E_{t} A^{{\frac{ - 1}{\theta }}} \varphi \left( {M_{t + 1} } \right)}}. $$

(36)

Since this ratio converges to zero, it contradicts Eq. (13).□

Proof of Proposition 6

Note first that Eq. (10) implies that:

$$ \frac{{M_{t} }}{{T_{t} - M_{t} }} < \frac{{M_{t} \varphi^{\prime}\left( {M_{t} } \right)}}{{R_{t}^{{\frac{\theta }{1 - \theta }}} - \varphi \left( {M_{t} } \right)}}. $$

(37)

The denominator of the RHS of (37) converges to $R_{\infty }^{\theta /(1 - \theta )} - A$, which is positive, due to Proposition 5. As for the numerator of the RHS of (37) we get from Fig. 

Fig. 3

Estimation of $M\varphi^{\prime} (M)$

3, where the function ψ is determined by the intersection between the tangent to φ and the vertical axis, that:

$$ M\varphi^{\prime}\left( M \right) = \varphi \left( M \right) - \psi \left( M \right). $$

However, both $\psi (M)$ and $\varphi (M)$ converge to $A$. Hence, the numerator of the RHS of (37) converges to zero.□

1.2 B. Adding capital to production by labor

This appendix extends the benchmark model, by assuming that production by labor of each intermediate good $j \in (M_{t} ,T_{t} ]$ requires not only a worker, but also a structure of size s, which is capital in labor production. Assume further that $k(j) \ge s,\,\,{\text{for}}\;{\text{all}}\,\,j \ge 0$, as automation requires a structure as well and we include it in the function k. Assume further that structures fully depreciate within one period, so that all capital depreciates at a rate of 1. In this extension the price of an old labor task unit is:

$$ p_{t} \left( j \right) = w_{t} + R_{t} s. $$

The price of a new labor task unit is:

$$ p_{t} \left( j \right) = \frac{{w_{t} + R_{t} s}}{\theta }. $$

Substituting these prices in the demand condition (4) in the paper we get that the wage in this extension is described by:

$$ w_{t} = \left[ {\frac{{T_{t - 1} - M_{t} + \left( {T_{t} - T_{t - 1} } \right)\theta^{{\frac{\theta }{1 - \theta }}} }}{{1 - R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} \varphi \left( {M_{t} } \right)}}} \right]^{{\frac{1 - \theta }{\theta }}} - R_{t} s. $$

(38)

Equating income in the R&D sector and in labor in production leads to equilibrium condition similar to (11):

$$ Y_{t} = \left( {w_{t} + R_{t} s} \right)^{{\frac{1}{1 - \theta }}} \frac{{\theta^{{\frac{ - \theta }{{1 - \theta }}}} }}{{bT_{t - 1} \left( {1 - \theta } \right)}}\frac{{w_{t} }}{{w_{t} + R_{t} s}}. $$

(39)

We can use this equilibrium condition and the FOC to calculate the amount of labor for each task and in the R&D sector and get a labor market equilibrium condition similar to (12):

$$ E_{t} = \left( {T_{t - 1} - M_{t} } \right)\frac{{\theta^{{\frac{ - \theta }{{1 - \theta }}}} }}{{bT_{t - 1} \left( {1 - \theta } \right)}}\frac{{w_{t} }}{{w_{t} + R_{t} s}} + \left( {T_{t} - T_{t - 1} } \right)\frac{\theta }{{bT_{t - 1} \left( {1 - \theta } \right)}}\frac{{w_{t} }}{{w_{t} + R_{t} s}} + \frac{{T_{t} - T_{t - 1} }}{{bT_{t - 1} }}. $$

(40)

From (39) and (40) we get that output per worker in the economy is:

$$ y_{t} = \frac{{\left( {w_{t} + R_{t} s} \right)^{{\frac{1}{1 - \theta }}} }}{{T_{t - 1} - M_{t} + \left( {T_{t} - T_{t - 1} } \right)\theta^{{\frac{\theta }{1 - \theta }}} + \left( {T_{t} - T_{t - 1} } \right)\theta^{{\frac{\theta }{1 - \theta }}} \left( {1 - \theta } \right)\frac{{w_{t} }}{{w_{t} + R_{t} s}}}}. $$

(41)

Note that the share of labor is equal to:

$$ S_{t}^{L} = \frac{{w_{t} }}{{y_{t} }} = \frac{{w_{t} + R_{t} s}}{{y_{t} }} - \frac{{R_{t} s}}{{y_{t} }}. $$

(42)

Using Eqs. (38) and (41) we get that:

$$ \frac{{w_{t} + R_{t} s}}{{y_{t} }} = 1 - R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} \varphi \left( {M_{t} } \right) + \left[ {1 - R_{t}^{{\frac{ - \theta }{{1 - \theta }}}} \varphi \left( {M_{t} } \right)} \right]\frac{{\left( {T_{t} - T_{t - 1} } \right)\left( {1 - \theta } \right)\theta^{{\frac{\theta }{1 - \theta }}} \frac{{R_{t} s}}{{w_{t} }}}}{{T_{t - 1} - M_{t} + \left( {T_{t} - T_{t - 1} } \right)\theta^{{\frac{\theta }{1 - \theta }}} }}. $$

Clearly, this ratio converges in the long-run to $1 - R_{\infty }^{ - \theta /(1 - \theta )} A$. As the ratio $R_{t} s/y_{t}$ converges to 0 as well, we get from (42) that in the long-run the share of labor in output converges to $1 - R_{\infty }^{ - \theta /(1 - \theta )} A$, just as in the benchmark model.

The long-run gross rate of interest $R_{\infty }$ should also be the same as in the benchmark model. The reason is that the demand for capital is determined by the share of capital in output and that is equal in the long-run to the share of capital in the benchmark model. Hence, the share of labor in this paper should not be affected by the narrow definition of capital, which includes only equipment without structures.

1.3 C. Skill, wages and technical change

This appendix introduces differences in skill between workers. We add it mainly to examine the effect of skill-biased technical change on the share of labor in output, as explained in Sect. 5 in the paper. For simplicity assume that each generation lives only one period. Assume that there are two skill levels, high and low. To become high skilled a person needs to acquire education at the beginning of the period. We denote the amount of high skilled in the population by H and of low skilled L, with $H + L = 1$. Utility of each person is equal to $c/z$, where z is the effort of acquiring education. This effort is distributed randomly across the population in the following way. If H people acquire skill, denote the effort of the marginal person by $z(H)$, where z is an increasing function. Clearly the others who acquire education have a lower effort. Assume also that $z(0) > 0$. If the high skill wage in period t is $w_{H,t}$ and the low skill wage is $w_{L,t}$, maximization of utility implies:

$$ \frac{{w_{H,t} }}{{w_{L,t} }} = z\left( H \right). $$

(43)

Hence, the wage ratio, or the skill premium, depends positively on the amount of high skilled in the economy.

We abstract in this section from unemployment, to focus mainly on the share of labor. We also assume that the economy is open and small, so that there is no sector in this economy that invents new tasks, and these are created abroad. This assumption also implies that the interest rate is exogenous, and determined in the global capital market, so we assume that it is constant at R.

Next, we assume that labor tasks can be either high or low skilled. There is an indicator function on tasks h, that is equal to 1 if a task requires high skill labor and to 0, if it requires low skill labor. Denote:

$$ d\left( {x,y} \right) = \frac{1}{y - x}\int\limits_{x}^{y} {h(j)dj} . $$

Hence, d is the density of skilled tasks in the interval [x, y]. Actually, this density measures the skill bias of technical change, as it measures the demand for skill across new tasks. To simplify the analysis, assume that the distribution of demand for skill is uniform across tasks, namely that d is constant over all intervals of tasks.

The rest of the economy is similar to the benchmark model, but we allow for the cost of machines to differ between high skill and low skill tasks. Hence, there are two cost of machines functions, $k_{H} \,\,{\text{and}}\,\,k_{L}$. Each cost function defines its own φ function:

$$ \begin{gathered} \varphi_{H} \left( M \right) = \int\limits_{0}^{M} {k_{H} \left( j \right)^{{\frac{ - \theta }{{1 - \theta }}}} dj} , \hfill \\ \varphi_{L} \left( M \right) = \int\limits_{0}^{M} {k_{L} \left( j \right)^{{\frac{ - \theta }{{1 - \theta }}}} dj} . \hfill \\ \end{gathered} $$

Each φ function converge to its own limit: $A_{H} = \mathop {\lim }\limits_{M \to \infty } \varphi_{H} (M)\,\,{\text{and}}\,\,A_{L} = \mathop {\lim }\limits_{M \to \infty } \varphi_{L} (M)$.

Since the production function of the final good is the same as in the benchmark model, the equilibrium condition (4) holds as well. The supply prices of the intermediate goods, or tasks, are equal to:

$$ p_{t} \left( j \right) = \left\{ \begin{gathered} Rk_{H} (j),\,\,{\text{if}}\,\,0 \le j \le M_{H,t} ,\,\,h(j) = 1 \hfill \\ Rk_{L} (j),\,\,{\text{if}}\,\,0 \le j \le M_{L,t} ,\,\,h(j) = 0 \hfill \\ w_{H,t} ,\,\,{\text{if}}\,\,M_{H,t} \le j \le T_{t} ,\,\,h(j) = 1 \hfill \\ w_{L,t} ,\,\,{\text{if}}\,\,M_{L,t} \le j \le T_{t} ,\,\,h(j) = 0. \hfill \\ \end{gathered} \right. $$

(44)

As this equation shows, there are two frontiers of automation in this extension of the model, $M_{H,t} \,\,{\text{and}}\,\,M_{L,t}$. These are determined by the following condition of automation adoption, similar to the benchmark model:

$$ \begin{gathered} Rk_{H} \left( {M_{H,t} } \right) = w_{H,t} ,\,\,{\text{and}} \hfill \\ Rk_{L} \left( {M_{L,t} } \right) = w_{L,t} . \hfill \\ \end{gathered} $$

(45)

Substituting the supply prices (44) in condition (4) leads to the following equilibrium condition:

$$ \begin{gathered} d\left[ {1 - R^{{\frac{ - \theta }{{1 - \theta }}}} \varphi_{H} \left( {M_{H,t} } \right)} \right] + \left( {1 - d} \right)\left[ {1 - R^{{\frac{ - \theta }{{1 - \theta }}}} \varphi_{L} \left( {M_{L,t} } \right)} \right] \\ = w_{H,t}^{{\frac{ - \theta }{{1 - \theta }}}} d\left( {T_{t} - M_{H,t} } \right) + w_{L,t}^{{\frac{ - \theta }{{1 - \theta }}}} \left( {1 - d} \right)\left( {T_{t} - M_{L,t} } \right). \\ \end{gathered} $$

(46)

This equation describes a negative relation between the wages of high and of low skilled. To determine the two wage levels, we examine what determines the wage ratio, or the skill premium. First note that due to the FOC (3) in the paper for the labor tasks we get:

$$ \begin{gathered} Y_{t} = x_{t} \left( j \right)w_{H,t}^{{\frac{1}{1 - \theta }}} = \frac{H}{{d\left( {T_{t} - M_{H,t} } \right)}}w_{H,t}^{{\frac{1}{1 - \theta }}} \,\,{\text{if}}\,h\left( j \right)\,{ = 1} \hfill \\ Y_{t} = x_{t} \left( j \right)w_{L,t}^{{\frac{1}{1 - \theta }}} = \frac{L}{{d\left( {T_{t} - M_{L,t} } \right)}}w_{L,t}^{{\frac{1}{1 - \theta }}} \,\,{\text{if}}\,\,h\left( j \right) = 0. \hfill \\ \end{gathered} $$

(47)

From equating the two equations in (47) and using (43) we get:

$$ z\left( H \right) = \frac{{w_{H,t} }}{{w_{L,t} }} = \left[ {\frac{{T_{t} - M_{H,t} }}{{T_{t} - M_{L,t} }}\frac{1 - H}{H}\frac{d}{1 - d}} \right]. $$

Note that in the long-run $(T_{t} - M_{H,t} )/(T_{t} - M_{L,t} )$ converges to 1, since tasks grow faster than automation. Hence in the long-run H depends only on d and it depends positively. Thus, the long-run wage gap depends positively on the degree of skill bias, as expected.

We next turn to calculate the share of labor in output. Using (47) we get:

$$ S_{t}^{L} = \frac{{Hw_{H,t} + Lw_{L,t} }}{{Y_{t} }} = d\left( {T_{t} - M_{H,t} } \right)w_{H,t}^{{\frac{ - \theta }{{1 - \theta }}}} + \left( {1 - d} \right)\left( {T_{t} - M_{L,t} } \right)w_{L,t}^{{\frac{ - \theta }{{1 - \theta }}}} . $$

Applying (46) we get:

$$ S_{t}^{L} = d\left[ {1 - R^{{\frac{ - \theta }{{1 - \theta }}}} \varphi_{H} \left( {M_{H,t} } \right)} \right] + \left( {1 - d} \right)\left[ {1 - R^{{\frac{ - \theta }{{1 - \theta }}}} \varphi_{L} \left( {M_{L,t} } \right)} \right]. $$

As a result, the share of labor converges to:

$$ d\left[ {1 - R^{{\frac{ - \theta }{{1 - \theta }}}} A_{H} } \right] + \left( {1 - d} \right)\left[ {1 - R^{{\frac{ - \theta }{{1 - \theta }}}} A_{L} } \right]. $$

Hence, the skill bias reduces the share of labor only if $A_{H} > A_{L}$. This happens only if the machine cost functions satisfy: $k_{H} < k_{L}$. This is rather unlikely, as it means that machines that replace high skill labor are less costly than machines that replace low skill labor.