# Rise of the Machines: The Effects of Labor-Saving Innovations on Jobs and Wages

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13 level of an optimally designed machine. For simplicity, we omit the task-neutral productivity term A K here, as it does not affect optimal machine design. Then we have α N (s, σ, τ) sup z F (z; σ) [ 1 τ s z], [ ] α K (s K, σ) sup z F (z; σ) 1 1 s K z, A unique interior solution to the worker training problem exists provided τ > 0, while the machine design problem always admits a unique interior solution. 18 levels z N (s, σ, τ) and z K (s K, σ) are pinned down by the first-order conditions The optimal knowledge f(z(s, σ, τ); σ) [ 1 τ s z(s, σ, τ)] = τ s F (z(s, σ, τ); σ), [ ] f(z(s K, σ); σ) 1 1 s K z(s K, σ) = 1 s K F (z(s K, σ); σ). (1) Optimality requires that the benefit of learning the solution to an additional problem the probability that the problem occurs times the number of efficiency units left for production, be equal to the cost of doing so the number of efficiency units lost times the fraction of problems these efficiency units would have solved. We will formalize the concept of innateness by assuming that some tasks feature τ = 0. It is immediate that in such innate ability tasks, α N (s, σ, 0) = 1. Thus, optimal worker and machine productivities are given by and F (z(s, σ, τ); σ) [1 τ ] s z(s, σ, τ) if τ > 0 α N (s, σ, τ) = 1 if τ = 0 [ α K (s K, σ) = F (z(s K, σ); σ) 1 1 ] z(s K, σ). s K We impose Assumption 1 for the remainder of the paper. Let the set of worker skills be given by S = [s, s] and let s be an element in set S = s K S. By the above equations, we have that α N ( s, σ, 1) α K ( s, σ). Thus, workers and machines face the same productivity schedule in training-intensive tasks. We drop superscripts and define the function α( s, σ) = F (z( s, σ); σ) [1 1 s ] z( s, σ) s S = s K [s, s], (2) where z( s, σ) is implicitly given by (1) when τ = 1. We now turn to the properties of the productivity schedule α( s, σ). First notice that α 18 A unique interior solution to the worker training problem exists if τ > 0 because first, the problem is strictly concave as f is strictly decreasing; second, the derivative of the objective at z = 0 is strictly positive; finally, the value of the objective function becomes negative for a sufficiently large z. The same arguments also establish the result for the machine design problem. 13

18 1 * 0 * 0 1 Figure 1: Assignment of labor and capital to tasks. Knowledge intensity σ is plotted on the vertical axis, while training intensity τ is plotted on the horizontal axis. Dotted and dashed lines indicate tasks performed by machines and workers, respectively. 18

19 and that the wage schedule is given by d log w(s) ds = log α(s, M(s)). (11) s The last equation is due to the fact that in equilibrium, a firm producing training-intensive task σ chooses worker skill s to minimize marginal cost w(s)/α(s, σ). Once differentiability of the matching function has been established, (10) can easily be derived from the market clearing condition (8) given Lemma 2, and using (6) and (7). In particular, Lemma 2 and (8) imply s s v(s )ds = σ σ1 n 1 (M 1 (σ ), σ )dσ. Changing variables on the RHS of the last expression and differentiating with respect to s yields v(s) = n 1 (s, M(s))M (s), and substituting (5) we obtain M (s) = α(s, M(s))v(s). (12) y(m(s)) After eliminating task output and price using (6) and (7), (10) follows. Figure 2 illustrates how the matching function assigns workers to training-intensive tasks. In order to characterize the equilibrium more fully, and for comparative statics exercises, it is necessary to derive equations pinning down the endogenous variables σ 0, σ 1, and s. These equations are due to a set of no-arbitrage conditions. In particular, firms producing the marginal tasks are indifferent between hiring labor or capital, and the marginal worker is indifferent between performing innate ability tasks or the marginal training-intensive tasks. Formally, the price and wage functions must be continuous, otherwise the zero-profit condition (7) could not hold. This is a well-known result in the literature on comparative-advantage-based assignment models. Hence, the no-arbitrage conditions for the marginal tasks are and r A K α(s K, σ 0 ) = w(s) for all s s (13) r A K α(s K, σ1 ) = w(s ) α(s, σ1 (14) ), and the no-arbitrage condition for the marginal worker is w(s) = w(s ) for all s s. (15) The last result implies that there is a mass point at the lower end of the wage distribution. The mass point is a result of normalizing A, the amount of problems drawn, to one for all workers. Relaxing this assumption would complicate the analysis, although the main results would go 19

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