KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

April 3, 2024

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 productivityHigh- and low-skill are imperfectly substitutable

Typically, CES production function with elasticity of substitution \(\sigma\)Competitive labour market

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

\(A_L\) 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

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

**Note**

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

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

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 \(A_H\)

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

**\(\Delta\) 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: \(\sigma = 1 \Rightarrow \frac{\partial \ln w_H/ w_L}{\partial \ln A_H / A_L} = 0\)

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:

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

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)

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 downdifferentiate labour by age/experience as well

- Simple link between wage structure and technological change
- Attractive explanation for college/no college wage inequality
^{1} - 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:

- Falling \(w_L\)
- Earnings polarization
- Job polarization

Also silent about endogeneous adoption or labour-replacing technology.

**Task** is a unit of work activity that produces output

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

Key features:

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

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 \(i\)

**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

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

**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**

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

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

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 \(I_L\):

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

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

Consider \(\uparrow A_M\):

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

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

In this case,

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

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 \(J^{\star\star}\left(\frac{w_M}{w_L}\right)\)

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!

- A rich model that can accommodate numerous scenarios
- Outsourcing tasks to lower-cost countries
- Endogenous technological change
- Creation of new tasks

- Useful tool to study effect on inequality and job polarization

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

attribute 100% of the decline to automation

predict given industry adoption of automation technology

Source: Figure 7

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

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 \(w_H\).

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.