6. Technological shift and labour markets

KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

April 3, 2024

Labour market of educated workers

Source: Figure 2 from Acemoglu and Autor (2011)

Source: Figure 1 from Acemoglu and Autor (2011)

Technological change and the labour market

Canonical model

Canonical model


  • Two types of labour: high- and low-skill
    Typically, high edu and low edu (can be relaxed)

  • Skill-biased technological change (SBTC)
    New technology disproportionately \(\uparrow\) high-skill labour productivity

  • High- and low-skill are imperfectly substitutable
    Typically, CES production function with elasticity of substitution \(\sigma\)

  • Competitive labour market

Canonical model

Production function

\[ Y = \left[\left(A_L L\right)^\frac{\sigma - 1}{\sigma} + \left(A_H H\right)^\frac{\sigma - 1}{\sigma}\right]^\frac{\sigma}{\sigma - 1} \]

  • \(A_L\) and \(A_H\) are factor-augmenting technology terms

  • \(\sigma \in [0, \infty)\) is the elasticity of substitution

    • \(\sigma > 1\) gross substitutes
    • \(\sigma < 1\) gross complements
    • \(\sigma = 0\) perfect complements (Leontieff production)
    • \(\sigma \rightarrow \infty\) perfect substitutes
    • \(\sigma = 1\) Cobb-Douglas production

Canonical model


  1. There is only one good and \(H\) and \(L\) are imperfect substitutes in production
  2. There are two goods \(Y_H = A_H H\) and \(Y_L = A_L L\) and consumers have CES utility \(\left[Y_L^\frac{\sigma - 1}{\sigma} + Y_H^\frac{\sigma - 1}{\sigma}\right]^\frac{\sigma}{\sigma - 1}\)
  3. Combination of the two: different sectors produce goods that are imperfect substitutes and \(H\) and \(L\) are employed in both sectors.


Supply of \(H\) and \(L\) is assumed inelastic \(\Rightarrow\) firm problem is sufficient.

Canonical model

Equilibrium wages

\[ \begin{align} w_L &= A_L^\frac{\sigma - 1}{\sigma} \left[A_L^\frac{\sigma - 1}{\sigma} + A_H^\frac{\sigma - 1}{\sigma}\left(\frac{H}{L}\right)^\frac{\sigma - 1}{\sigma}\right]^\frac{1}{\sigma - 1}\\ w_H &= A_H^\frac{\sigma - 1}{\sigma} \left[A_L^\frac{\sigma - 1}{\sigma}\left(\frac{H}{L}\right)^{-\frac{\sigma - 1}{\sigma}} + A_H^\frac{\sigma - 1}{\sigma}\right]^\frac{1}{\sigma - 1} \end{align} \]

Comparative statics:

  • \(\frac{\partial w_L}{\partial H/L} > 0\) low-skill wage rises with \(\frac{H}{L}\)

  • \(\frac{\partial w_H}{\partial H/L} < 0\) high-skill wage falls with \(\frac{H}{L}\)

  • \(\frac{\partial w_i}{\partial A_L} > 0\) and \(\frac{\partial w_i}{\partial A_H} > 0, ~\forall i \in \{L, H\}\) both wages rise with \(A_L\) and \(A_H\)

Canonical model

Skill premium

\[ \frac{w_H}{w_L} = \left(\frac{A_H}{A_L}\right)^\frac{\sigma - 1}{\sigma} \left(\frac{H}{L}\right)^{-\frac{1}{\sigma}} \]

\(\Delta\) relative supply

\[ \frac{\partial \ln \frac{w_H}{w_L}}{\partial \ln \frac{H}{L}} = -\frac{1}{\sigma} < 0 \]

Relative demand curve is downward-sloping

\(\Delta\) technology

\[ \frac{\partial \ln\frac{w_H}{w_L}}{\partial \ln\frac{A_H}{A_L}} = \frac{\sigma - 1}{\sigma} \lessgtr 0 \]

  • Gross substitutes: \(\sigma > 1 \Rightarrow \frac{\partial \ln w_H/ w_L}{\partial \ln A_H / A_L} > 0\)
  • Gross complements: \(\sigma < 1 \Rightarrow \frac{\partial \ln w_H/ w_L}{\partial \ln A_H / A_L} < 0\)
  • Cobb-Douglas: \(\sigma = 1 \Rightarrow \frac{\partial \ln w_H/ w_L}{\partial \ln A_H / A_L} = 0\)

Tinbergen’s race in the data

Katz and Murphy (1992)

The log-equation of skill premium is extremely attractive for empirical analysis

\[ \ln\frac{w_{H, t}}{w_{L, t}} = \frac{\sigma - 1}{\sigma} \ln\left(\frac{A_{H, t}}{A_{L, t}}\right) -\frac{1}{\sigma} \ln \left(\frac{H_t}{L_t}\right) \]

Assume a log-linear trend in relative productivities

\[ \ln \left(\frac{A_{H, t}}{A_{L, t}}\right) = \alpha_0 + \alpha_1 t \]

and plug it into the log skill premium equation:

\[ \ln\frac{w_{H, t}}{w_{L, t}} = \frac{\sigma - 1}{\sigma}\alpha_0 + \frac{\sigma - 1}{\sigma} \alpha_1 t -\frac{1}{\sigma} \ln\left(\frac{H_t}{L_t}\right) \]

Tinbergen’s race in the data

Katz and Murphy (1992)

Estimated the skill premium equation using the US data in 1963-87

Implies elasticity of substitution \(\sigma \approx \frac{1}{0.612} =\) 1.63

Agrees with other estimates that place \(\sigma\) between 1.4 and 2 (Acemoglu and Autor 2011)

Tinbergen’s race in the data

Source: Figure 19 from Acemoglu and Autor (2011)

Very close fit up to mid-1990s, diverge later

Fit up to 2008 implies \(\sigma \approx\) 2.95

Accounting for divergence:

  • non-linear time trend in \(\ln\frac{A_H}{A_L}\)
    brings \(\sigma\) back to 1.8, but implies \(\frac{A_H}{A_L}\) slowed down

  • differentiate labour by age/experience as well

Canonical model


  1. Simple link between wage structure and technological change
  2. Attractive explanation for college/no college wage inequality1
  3. Average wages \(\uparrow\) (follows from \(\partial w_i / \partial A_H\) and \(\partial w_i/ \partial A_L\))

However, the model cannot explain other trends observed in the data:

  1. Falling \(w_L\)
  2. Earnings polarization
  3. Job polarization

Also silent about endogeneous adoption or labour-replacing technology.

Unexplained trend: falling real wages

Source: Figure 1 in Acemoglu and Restrepo (2022)

Unexplained trend: earnings polarization

Source: Figure 8 in Acemoglu and Autor (2011)

Unexplained trend: job polarization

Source: Figure 10 in Acemoglu and Autor (2011)

Task-based model

Task-based model


Task is a unit of work activity that produces output

Skill is a worker’s endowment of capabilities for performing tasks

Key features:

  1. Tasks can be performed by various inputs (skills, machines)
  2. Comparative advantage over tasks among workers
  3. Multiple skill groups
  4. Consistent with canonical model predictions

Task-based model

Production function

Unique final good \(Y\) produced by continuum of tasks \(i \in [0, 1]\)

\[ Y = \exp \left[\int_0^1 \ln y(i) \text{d}i\right] \]

Three types of labour: \(H\), \(M\) and \(L\) supplied inelastically.

\[ y(i) = A_L \alpha_L(i) l(i) + A_M \alpha_M(i) m(i) + A_H \alpha_H(i) h(i) + A_K \alpha_K(i) k(i) \]

  • \(A_L, A_M, A_H\) are factor-augmenting technologies

  • \(\alpha_L(i), \alpha_M(i), \alpha_H(i)\) are task productivity schedules

  • \(l(i), m(i), h(i)\) are number of workers by types allocated to task \(i\)

Task-based model

Comparative advantage assumption

\(\alpha_L(i)/\alpha_M(i)\) and \(\alpha_M(i)/\alpha_H(i)\) are continuously differentiable and strictly decreasing.

Market clearing conditions

\[ \int_0^1 l(i) \text{d}i \leq L \qquad \int_0^1 m(i) \text{d}i \leq M \qquad \int_0^1 h(i) \text{d}i \leq H \]

Task-based model

Equilibrium without machines

Lemma 1

Given comparative advantage assumption, there exist \(I_L\) and \(I_H\) such that

Note that boundaries \(I_L\) and \(I_H\) are endogenous

This gives rise to the substitution of skills across tasks

Task-based model

Law of one wage

Output price is normalised to 1 \(\Rightarrow \exp\left[\int_0^1 \ln p(i) \text{d}i\right] = 1\)

All tasks employing a given skill pay corresponding wage

\[ \begin{align} w_L &= \frac{A_L \alpha_L(i)}{p(i)} &\Rightarrow& \qquad l(i) &=& \frac{L}{I_L} &\forall& i < I_L \\ w_M &= \frac{A_M \alpha_M(i)}{p(i)} &\Rightarrow& \qquad m(i) &=& \frac{M}{I_H - I_L} &\forall& I_L < i < I_H \\ w_H &= \frac{A_H \alpha_H(i)}{p(i)} &\Rightarrow& \qquad h(i) &=& \frac{H}{1 - I_H} &\forall& i > I_H \end{align} \]

Task-based model

Endogenous thresholds: no arbitrage

Threshold task \(I_H\): equally profitable to produce with either \(H\) or \(M\) skills

\[ \frac{A_M \alpha_M(I_H) M}{I_H - I_L} = \frac{A_H \alpha_H(I_H) H}{1 - I_H} \]

Similarly, for \(I_L\):

\[ \frac{A_L \alpha_L(I_L) L}{I_L} = \frac{A_M \alpha_M(I_L) M}{I_H - I_L} \]

Task-based model

Endogenous thresholds: no arbitrage

Task-based model

Comparative statics: wage elasticities

\[ \begin{matrix} \frac{\text{d} \ln w_H / w_L}{\text{d}\ln A_H} > 0 & \frac{\text{d} \ln w_M / w_L}{\text{d}\ln A_H} < 0 & \frac{\text{d} \ln w_H / w_M}{\text{d}\ln A_H} > 0 \\ \frac{\text{d} \ln w_H / w_L}{\text{d}\ln A_M} \lesseqqgtr 0 & \frac{\text{d} \ln w_M / w_L}{\text{d}\ln A_M} > 0 & \frac{\text{d} \ln w_H / w_M}{\text{d}\ln A_M} < 0 \\ \frac{\text{d} \ln w_H / w_L}{\text{d}\ln A_L} < 0 & \frac{\text{d} \ln w_M / w_L}{\text{d}\ln A_L} < 0 & \frac{\text{d} \ln w_H / w_M}{\text{d}\ln A_L} > 0 \end{matrix} \]

Consider \(\uparrow A_M\):

  1. \(M\) more productive \(\Rightarrow ~ \uparrow\) set of medium-skill tasks \(\uparrow I_H\) and/or \(\downarrow I_L\)
  2. Since supply is constant, \(\uparrow w_M\) and \(\downarrow w_H, w_L\)
  3. Depending on comparative advantages:
    1. \(\left|\beta^\prime_L(I_L) I_L\right| > \left|\beta^\prime_H(I_H)(1 - I_H)\right| \Rightarrow \frac{w_H}{w_L} \downarrow\)
    2. \(\left|\beta^\prime_L(I_L) I_L\right| < \left|\beta^\prime_H(I_H)(1 - I_H)\right| \Rightarrow \frac{w_H}{w_L} \uparrow\)

Task-based model

Task replacing technologies

Assume in \([\underline{I}, \bar{I}] \subset [I_L, I_H]\) machines outperform \(M\)1. Otherwise, \(\alpha_K(i) = 0\).

In this case,

  1. \(w_H / w_M\) increases
  2. \(w_M / w_L\) decreases
  3. \(w_H / w_L \uparrow \color{#9a2515}{\left(\downarrow\right)}\) if \(\left|\beta^\prime_L(I_L) I_L\right| \stackrel{<}{\color{#9a2515}{>}} \left|\beta^\prime_H(I_H)(1 - I_H)\right|\)

Task-based model

Endogenous supply of skills

Each worker \(j\) is endowed with some amount of each skill \(l^j, m^j, h^j\)

Workers allocate time to each skill given

\[ \begin{align} &t_l^j + t_m^j + t_h^j \leq 1 \\ &w_L t_l^j l^j + w_M t_m^j m^j + w_H t_h^j h^j \end{align} \]

Assuming that \(\frac{h^j}{m^j}\) and \(\frac{m^j}{l^j}\) are decreasing in \(j\), there exist \(J^\star\left(\frac{w_H}{w_M}\right)\) and \(J^{\star\star}\left(\frac{w_M}{w_L}\right)\)

Task-based model

Illustration in the data

Suppose \(A_H \uparrow\). The model predicts that \(\frac{w_H}{w_M} \uparrow\) and \(\frac{w_M}{w_L} \downarrow\).

Use occupational specialization at some \(t = 0\) as comparative advantage.

  • \(\gamma_{sejk}^i\) share of 1959 population employed in \(i\) occupations, \(\forall i \in \{H, M, L\}\)

\[ \Delta w_{sejk\tau} = \sum_t \left[\beta_t^H \gamma_{sejk}^H + \beta_t^L \gamma_{sejk}^L\right] 1\{\tau = t\} + \delta_\tau + \phi_e + \lambda_j + \pi_k + e_{sejk\tau} \]

Descriptive regression informed by the model!

Task-based model

Illustration in the data

Source: Table 10 in Acemoglu and Autor (2011)

Task-based model


  1. A rich model that can accommodate numerous scenarios
    1. Outsourcing tasks to lower-cost countries
    2. Endogenous technological change
    3. Creation of new tasks
  2. Useful tool to study effect on inequality and job polarization

Empirical results

Acemoglu and Restrepo (2022)


Multi-sector model with imperfect substitution between production inputs

\[ \text{Task displacement}_g^\text{direct} = \sum_{i \in \mathcal{I}} \omega_g^i \frac{\omega_{gi}^R}{\omega_i^R} \left(-d \ln s_i^{L, \text{auto}}\right) \]

  • \(\omega_g^i\) - share of wages earned by worker group \(g\) in industry \(i\)
    (exposure to industry \(i\))

  • \(\frac{\omega_{gi}^R}{\omega_i^R}\) - specialization of group \(g\) in routine tasks \(R\) within industry \(i\)

  • \(-d \ln s_i^{L, \text{auto}}\) - % decline in industry \(i\)’s labour share due to automation

    1. attribute 100% of the decline to automation

    2. predict given industry adoption of automation technology

Source: Figure 4 in Acemoglu and Restrepo (2022)

Acemoglu and Restrepo (2022)

Task displacement

Source: Figure 5

Acemoglu and Restrepo (2022)

Task displacement and changes in real wages

Source: Figure 6

Acemoglu and Restrepo (2022)

General equilibrium results

Source: Figure 7

Acemoglu and Restrepo (2022)

Model fit

Source: Figure 8

Source: Table VIII


Two theories linking technological advancements and labour markets

  • Canonical model (SBTC)

    • Simple application of two-factor labour demand theory
    • Empirically attractive characterization of between-group inequality
    • Fails to account for within-group inequality, polarization, and displacement
  • Task-based model (automation)

    • Rich model linking skills to tasks to output
    • Explains large share of changes in the wage structure since 1980s

Next: Labour market discrimination

Appendix: derivation of wage equations

The firm problem is to choose entire schedules \(\left(l(i), m(i), h(i)\right)_{i=0}^1\) to

\[ \max PY - w_L L - w_M M - w_H H \]

Consider FOC wrt \(l(j)\):

\[ PY \int_0^1 \frac{1}{y(j)} A_L \alpha_L(j) \text{d}i - w_L \int_0^1 \text{d}i = 0 \]

We normalised \(P = 1\); therefore, the FOC can be rewritten as

\[ \frac{Y}{y(j)} A_L \alpha_L(j) = w_L \]

On the next slide, we see that \(\frac{Y}{y(j)} = \frac{1}{p(j)}\).

Similar argument for \(w_M\) and \(w_H\).

Appendix: derivation of skill allocations

Numeraire price (\(P=1\)) is chosen such that \(\frac{y(i)}{Y} = \frac{p(i)}{P} = p(i), \forall i \in [0, 1]\)

Therefore, productivity of \(L\) is \(\frac{Y}{L} = \frac{y(i)}{p(i)L} = \frac{y(j)}{p(j)L}, \forall i \neq j < I_L\)

If we plug in definition of \(y(i)\) and notice that only \(l(i)\) matters when \(i < I_L\), then

\[ \frac{A_L \alpha_L(i) l(i)}{p(i) L} = \frac{A_L \alpha_L(j)l(j)}{p(j)L} \]

Given the wage equations in Law of one wage, we know \(\frac{A_L\alpha_L(i)}{p(i)} = \frac{A_L \alpha_L(j)}{p(j)}\). Therefore, \(l(i) = l(j), \quad \forall i\neq j < I_L\).

Plug it into the market clearing condition for \(L\)

\[ L = \int_0^{I_L} l(i) \text{d}i = l I_L \quad \Longrightarrow \quad l(i) = l = \frac{L}{I_L}, \forall i < I_L \]

Similar argument for \(m(i)\) and \(h(i)\).


Acemoglu, Daron, and David Autor. 2011. “Chapter 12 - Skills, Tasks and Technologies: Implications for Employment and Earnings.” In Handbook of Labor Economics, edited by David Card and Orley Ashenfelter, 4:1043–1171. Elsevier. https://doi.org/10.1016/S0169-7218(11)02410-5.
Acemoglu, Daron, and Pascual Restrepo. 2018. “The Race Between Man and Machine: Implications of Technology for Growth, Factor Shares, and Employment.” American Economic Review 108 (6): 1488–1542. https://doi.org/10.1257/aer.20160696.
———. 2022. “Tasks, Automation, and the Rise in U.S. Wage Inequality.” Econometrica 90 (5): 1973–2016. https://doi.org/10.3982/ECTA19815.
Autor, David H., Frank Levy, and Richard J. Murnane. 2003. “The Skill Content of Recent Technological Change: An Empirical Exploration*.” The Quarterly Journal of Economics 118 (4): 1279–1333. https://doi.org/10.1162/003355303322552801.
Katz, Lawrence F., and Kevin M. Murphy. 1992. “Changes in Relative Wages, 1963-1987: Supply and Demand Factors.” The Quarterly Journal of Economics 107 (1): 35–78. https://doi.org/10.2307/2118323.