KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

March 13, 2024

Firm decisions about how much labour to hire.

Production function \(Y = F(L)\) where \(F^\prime > 0\) and \(F^{\prime\prime} < 0\)

\[ \max_{L} PF(L) - WL \]

FOC: \(F^\prime(L) = \frac{W}{P}\)

Downward-sloping labour demand \(\frac{\partial L}{\partial W} = \frac{1}{PF^{\prime\prime}(L)} < 0\)

Cost minimization problem

\(\min_{L, K} C(L, K) = WL + RK\) s.t. \(F(L, K) = \bar{Y}\)

Conditional demand functions \(\bar{K}(W, R, Y)\) and \(\bar{L}(W, R, Y)\)

\[ \frac{F_L(\bar{L}, \bar{K})}{F_K(\bar{L}, \bar{K})} = \frac{W}{R} \quad\text{and}\quad F(\bar{L}, \bar{K}) = \bar{Y} \]

Own-price elasticities: \(\eta_W^L = \frac{\partial \ln \bar{L}}{\partial \ln W} < 0\), \(\eta_R^K = \frac{\partial \ln \bar{K}}{\partial \ln R} < 0\)

Cross-price elasticities: \(\eta_R^L = \frac{\partial \ln \bar{L}}{\partial \ln R} > 0\) and \(\eta_W^K = \frac{\partial \ln \bar{K}}{\partial \ln W} > 0\)

Elasticity of substitution \(\sigma = \frac{\partial \ln\left(\frac{K}{L}\right)}{\partial \ln \left(\frac{W}{R}\right)} > 0\)

It is also possible to show that

\[ \eta_R^L = \sigma (1 - s) \quad \text{and} \quad \eta_W^L = -\sigma(1 - s) \]

where \(s = \frac{WL}{C}\) is labour share in total cost

\(\max_{Y} PY - C(W, R, Y)\)

Solution: \(P = C_Y(W, R, Y^*), L^* = \bar{L}(W, R, Y^*), K^* = \bar{K}(W, R, Y^*)\)

Total elasticities decomposed into **substitution** and **scale** effects:

\[ \varepsilon_W^L = \color{#8e2f1f}{\eta_W^L} + \color{#288393}{\eta_Y^L \varepsilon_W^Y} < 0 \]

\[ \varepsilon_R^L = \color{#8e2f1f}{\eta_R^L} + \color{#288393}{\eta_Y^L\varepsilon_R^Y} \lessgtr 0 \]

Shephard’s lemma: specify cost function and back out labour demand

Example: translog cost function with \(n\) inputs

\[ \ln C = a_0 + \sum_{i = 1}^n a_i \ln W^i + \frac{1}{2} \sum_{i = 1}^n \sum_{j = 1}^n a_{ij} \ln W^i \ln W^j + \frac{1}{\theta} \ln Y \]

\[ \Rightarrow s^i = a_i + \sum_{j = 1}^n a_{ij} \ln W^j \]

Estimate parameters \(a_{i}, a_{ij}\) and calculate implied elasticities.

Endogeneity

General equilibrium

Definitions of variables

Review by Hamermesh (1996) concludes that \(-\eta_W^L \in [0.15, 0.75]\).

If \(\eta_W^L = -0.30\) and given that \(s \approx 0.7\),

\[ \sigma = \frac{-\eta_W^L}{1 - s} \approx 1 \]

consistent with the Cobb-Douglas production function.

The review also suggests \(-\varepsilon_W^L \approx 1 \Rightarrow\) large scale effect.

Quadratic cost: \(C\left(\Delta L_t\right) = b\left(\Delta L_t - a\right)^2\)

Assymmetric convex costs: \(C\left(\Delta L_t\right) = -1 + e^{a\Delta L_t} - a\Delta L_t + \frac{b}{2}\left(\Delta L_t\right)^2\)

Linear cost: \(C\left(\Delta L_t\right) = \begin{cases}c_h \Delta L_t & \text{if }\Delta L_t \geq 0\\-c_f \Delta L_t & \text{if }\Delta L_t \leq 0\end{cases}\)

Fixed cost

Continuous time \(\Rightarrow \Delta L_t = \dot{L}_t = \frac{\text{d} L_t}{\text{d}t}\)

\[ \Pi_0 = \int_0^\infty \Pi_t dt = \int_0^\infty \left[F(L_t) - W_tL_t - \frac{b}{2}\dot{L}_t^2\right]e^{-rt}dt \]

Euler equation: \(\frac{\partial \Pi_t}{\partial L} = \frac{\text{d}}{\text{d}t}\left(\frac{\partial \Pi_t}{\partial \dot{L}_t}\right) \Rightarrow b\ddot{L}_t - rb\dot{L}_t + F'(L_t) - W_t = 0\)

Optimal path: \(\dot{L}_t = \gamma \left[L^* - L_t\right]\) where \(\gamma\) is decreasing in \(b\).

\[ \Pi_0 = \int_0^\infty \left[F(L_t) - W_tL_t - C(\dot{L}_t)\right]e^{-rt}dt \]

where \(C\left(\dot{L}_t\right) = \begin{cases}c_h \dot{L}_t & \text{if }\dot{L}_t \geq 0\\-c_f \dot{L}_t & \text{if }\dot{L}_t \leq 0\end{cases}\)

Optimal labour demand path is derived from

\[ \begin{cases}F'(L_t) = W_t + r c_h & \text{if }\dot{L}_t \geq 0 \\ F'(L_t) = W_t - r c_f & \text{if }\dot{L}_t < 0\end{cases} \]

**Quadratic adjustment cost**

Assume linear quadratic production function

Estimate \(L_{it} = \lambda L_{i, t - 1} + X_{it} \beta + \mu_i + \varepsilon_{it}\)

- accounting for correlation between \(L_{i, t - 1}\) and \(\mu_i + \varepsilon_{it}\)

**Other adjustment costs and production functions**

Estimate Euler equation directly

Current employment \(L_t\) depends on past and future variables

Appropriate econometric methods (Hamilton 1994 book)

Adjustments happen fast (1-2 quarters) (Hamermesh 1996, chap. 7)

Dynamic substitutes: utilization of capital increases with \(L_t - L^*\)

Hours of work are adjusted faster than number of workers

What do the models we have considered so far predict?

lower labour demand (both compensated and uncompensated)

(maybe) higher labour supply

Any “problems” with these conclusions?

**Typically not supported by empirical evidence!**

On April 1, 1992 minimum wage in New Jersey \(\uparrow\) from $4.25 to $5.05.

It stayed at $4.25 in Pennsylvania.

It stayed at $4.25 in Pennsylvania.

- Compare before and after:

\(E_{t1}^{NJ} - E_{t0}^{NJ}\) = 0.59 (se = 0.54)

- Compare before and after:

\(E_{t1}^{NJ} - E_{t0}^{NJ}\) = 0.59 (se = 0.54) - Compare NJ and PA:

\(E_{t}^{NJ} - E_{t}^{PA}\) = -2.89 (se = 1.44)

- Compare before and after:

\(E_{t1}^{NJ} - E_{t0}^{NJ}\) = 0.59 (se = 0.54) - Compare NJ and PA:

\(E_{t}^{NJ} - E_{t}^{PA}\) = -2.89 (se = 1.44) - Diff-in-diff:

\(\left(E_{t1}^{NJ} - E_{t0}^{NJ}\right) - \left(E_{t1}^{PA} - E_{t0}^{PA}\right)\) = 2.75 (se = 1.34)

Seattle \(\uparrow\) min wage from $9.47 up to

- $11 in April 2015
- $13 in January 2016

Causal design:

**synthetic control**: weighted average of other counties that match pre-Seattle**nearest neighbour matching**: find “closest” worker outside of Seattle matching treated worker in Seattle

- Negative effect on hours worked stronger than on employment
- Experienced workers are better off

However,

- Potentially cascading effect
- Excluded large low-wage employers (like McDonald’s) (monopsony)

same policy + synthetic control = no change in employment

Review in Clemens (2021)

- Price pass-through (Leung 2021; Renkin, Montialoux, and Siegenthaler 2022)
- Non-wage labour cost (Clemens, Kahn, and Meer 2018)
- Flexibility (theoretical Clemens and Strain 2020)
- Effort (Ku 2022; Coviello, Deserranno, and Persico 2022)
- Firm profit (Draca, Machin, and Van Reenen 2011; Bell and Machin 2018)
- Firm exit (Luca and Luca 2019; Dustmann et al. 2022)

Basic static and dynamic models of labour demand

Application to minimum wage policy

- Ongoing research (little consensus)
- Clear that basic models are insufficient
- Typical frameworks: heterogeneous labour, monopsony
- Non-wage margins important and can interact with labour supply

Next: Human Capital

Bell, Brian, and Stephen Machin. 2018. “Minimum Wages and Firm Value.” *Journal of Labor Economics* 36 (1): 159–95. https://doi.org/10.1086/693870.

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