KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

March 7, 2024

How people choose whether and how much they work?

- Utility from consumption of goods (\(C\)) and leisure (\(L\)): \(U(C, L)\).
- Total time endowment \(L_0\)
- Agent chooses \(h\) how much time to work such that \(L = L_0 - h\).
- Budget constraint is \(C \leq wh + Y \Rightarrow C + w L \leq w L_0 + Y\)
- \(w\) is real hourly wage
- \(Y\) is non-labour income

\[ \max_{C, h} U(C, L_0 - h) \quad\text{subject to} \quad C \leq wh + Y \]

First-order conditions of the Lagrangian are

\[ U_C(C, L) = \lambda \qquad U_L(C, L) = \lambda w \]

Solution pair \(C^*(w, Y)\) and \(h^*(w, Y)\) satisfies

\[ \frac{U_L(C^*, L^*)}{U_C(C^*, L^*)} = w \quad \text{and} \quad C^* = wh^* + Y \]

How does optimal labour supply change with \(w\)?

Marshallian (uncompensated) wage elasticity: \(\varepsilon_{hw} = \frac{\partial \ln h^*}{\partial \ln w}\)

Hicksian (compensated) wage elasticity: \(\eta_{hw} = \frac{\partial \ln \hat{h}}{\partial \ln w}\)

Decomposition into **substitution** and **income** effects:

\[ \varepsilon_{hw} = \color{#8e2f1f}{\eta_{hw}} + \color{#288393}{\frac{wh}{Y} \varepsilon_{hY}} \]

Household represented by single utility function \(U(C, L_1, L_2)\)

Budget constraint \(C + w_1 L_1 + w_2 L_2 \leq Y_1 + Y_2 + (w_1 + w_2) L_0\)

Simple extension of static model

Not consistent with observed data

Individual utility functions \(U_1(C_1, L_1), U_2(C_2, L_2)\)

Budget constraint \(C_1 + C_2 + w_1 L_1 + w_2 L_2 \leq R_1 + R_2 + (w_1 + w_2) L_0\)

Partner utility constraint (Pareto efficiency) \(U_2(C_2, L_2) \geq \bar{U}_2\)

In this case, individual program can be represented by

\[ \max_{C_i, L_i} U_i(C_i, L_i) ~ \text{s.t.} C_i + w_i L_i \leq w_i L_0 + \Phi_i \]

where \(\Phi_i\) describes how resources \(R_1 + R_2\) are shared in the household.

General utility function \(U(C_0, \ldots, C_T; L_0, \ldots, L_T)\) (intractable)

Separable utility function \(\sum_{t = 0}^T U(C_t, L_t, t)\)

Budget constraint \(A_t = (1 + r_t) A_{t - 1} + B_t + w_t(1 - L_t) - C_t\)

- savings rate \(r_t\)
- total time normalized to one: \(h_t + L_t = 1\)
- assets \(A_t\)
- non-labour income \(B_t\)

\[ \mathcal{L} = \sum_t U(C_t, L_t, t) - \sum_t \nu_t \left[A_t - (1 + r_t) A_{t - 1} - B_t - w_t(1 - L_t) + C_t\right] \]

First-order conditions:

\[ \begin{align*}\frac{U_L(C_t, L_t, t)}{U_C(C_t, L_t, t)} = &w_t\\\nu_t = &(1 + r_{t + 1})\nu_{t + 1} \end{align*} \qquad \forall t \in [0, T] \]

Iterating over all periods: \(\ln \nu_t = - \sum_{\tau = 1}^t \ln\left(1 + r_\tau\right) + \ln\nu_0\)

Frisch elasticity \(\psi_{hw}\) (holding \(\nu_t\) constant)

Marshallian elasticity \(\varepsilon_{hw}\) (takes into account \(\nu_t\))

Hicksian elasticity \(\eta_{hw}\) (holding lifetime utility constant)

It is possible to show that \(\psi_{hw} \geq \eta_{hw} \geq \varepsilon_{hw}\)

**Interpretation**

*Transitory* changes in wages affect labour supply more than permanent changes.

Period utility \(U(C_t, L_t, t) = \frac{C_t^{1 + \rho}}{1 + \rho} - \beta_t \frac{H_t^{1 + \gamma}}{\gamma}\)

FOC: \(H_t^\gamma = \frac{1}{\beta_t} \nu_t w_t \Rightarrow \ln H_t = \frac{1}{\gamma}\left(-\ln \beta_t + \ln \nu_t + \ln w_t\right)\)

- Evolutionary changes along anticipated wage profile \(\frac{\partial \ln H_t}{\partial \ln w_t} = \frac{1}{\gamma} > 0\)
- Transitory changes \(\frac{\partial \ln H_t}{\partial \ln w_t} = \frac{1}{\gamma}\left(1 + \underbrace{\frac{\partial \ln \nu_0}{\partial \ln w_t}}_{<\approx 0}\right) > 0\)
- Permanent changes \(\frac{\partial \ln H_t}{\partial \ln w_t} = \frac{1}{\gamma}\left(1 + \frac{\partial \ln \nu_0}{\partial \ln w_t}\right) \lessgtr 0\)
- Lottery win \(\frac{\partial \ln H_t}{\partial \ln B_t} = \frac{1}{\gamma} \frac{\partial \ln \nu_0}{\partial \ln B_t} < 0\)

Basic regression equation

\[ \ln H_{it} = \alpha_w \ln w_{it} + \alpha_R \mathcal{R}_{it} + \theta X_{it} + v_{it} \]

Interpretation of \(\alpha_w\): Frisch, Marshallian or Hicksian? Depends on \(\mathcal{R}_{it}\)!

**Two-stage budgeting**

Solution method of lifecycle labour supply models (Blundell and Macurdy 1999)

- Solve static labour supply model given \(C_t = \mathcal{R}_t + w_t H_t\)
- Solve for series \(\mathcal{R}_1, \ldots, \mathcal{R}_T\) to maximize lifetime utility

\[ \ln H_{it} = \alpha_w \ln w_{it} + \alpha_R \left(C_{it} - w_{it} H_{it}\right) + \theta X_{it} + v_{it} \]

Marshallian wage elasticity: \(\alpha_w\)

Income effect: \(\alpha_R w H\)

Hicksian wage elasticity: \(\alpha_w - \alpha_R wH\)

Recall that \(\ln \nu_t = -\sum_{\tau = 1}^t \ln (1 + r_\tau) + \ln \nu_0 \equiv -\ln(1 + r) t + \ln \nu_0\) (if \(r_\tau = r ~ \forall \tau\))

Substitute \(\alpha_R\mathcal{R}_{it} = \rho t + \alpha_R\ln \nu_{0, i}\) into basic equation:

\[ \begin{align*} \ln H_{it} &= \rho t + \alpha_w \ln w_{it} + \alpha_R \ln \nu_{0, i} + \theta X_{it} + v_{it} \\ \Delta \ln H_{it} &= \rho + \alpha_w \Delta \ln w_{it} + \theta \Delta X_{it} + \Delta v_{it} \end{align*} \]

Frisch wage elasticity: \(\alpha_w\)

Wages and hours worked are endogeneous

Hours (\(H | H > 0\)) and participation (\(H > 0\))

Measurement errors

Measures of \(C_{it}\)

Individual vs aggregate labour supply

Also some research on work effort for given hours of work (Dickinson 1999)

Classical measurement error in \(w_{it}\) attenuates the estimate of \(\alpha_w\)

“Denominator bias” \(\downarrow \alpha_w\) if wages are computed as ratio of earning and hours with measurement errors. M. P. Keane (2011) computes average Hicksian elasticity

among all papers: 0.31

among papers with direct measure of \(w_{it}\): 0.43

PSID (US) dataset only includes food consumption data

Consumption measure | Marshall | Hicks | Income | Frisch |
---|---|---|---|---|

PSID unadjusted | -0.442 | 0.094 | -0.536 | 0.148 |

Food + imputed (food prices, demographics) | -0.468 | 0.328 | -0.796 | 0.535 |

Food + imputed (house value, rent) | -0.313 | 0.220 | -0.533 | 0.246 |

Source: (M. P. Keane 2011, Table 5)

Macro elasticities of labour supply typically higher than micro estimates

M. Keane and Rogerson (2012) highlight:

- extensive vs intensive margin
- model misspecification due to human capital accumulation
- aggregation is not straightforward

Many ways to reconcile imply different mechanisms!

Incorporate discrete choices into model of labour supply

- labour force participation (Eckstein and Wolpin 1989)
- marriage (Van Der Klaauw 1996)
- fertility (Francesconi 2002)

M. P. Keane and Wolpin (2010) combine all + school and welfare participation choices

Looked at standard models of labour supply

- Important intertemporal considerations

Mostly covered seminal papers, but many ongoing works

- Tax and benefit policies
- Cross-wage elasticities

Next: Labour Demand

Blundell, Richard, Alan Duncan, and Costas Meghir. 1998. “Estimating Labor Supply Responses Using Tax Reforms.” *Econometrica* 66 (4): 827–61. https://doi.org/10.2307/2999575.

Blundell, Richard, and Thomas Macurdy. 1999. “Chapter 27 - Labor Supply: A Review of Alternative Approaches.” In *Handbook of Labor Economics*, edited by Orley C. Ashenfelter and David Card, 3:1559–1695. Elsevier. https://doi.org/10.1016/S1573-4463(99)03008-4.

Cahuc, Pierre. 2004. *Labor Economics*. Cambridge (Mass.): MIT Press.

Chetty, Raj. 2012. “Bounds on Elasticities with Optimization Frictions: A Synthesis of Micro and Macro Evidence on Labor Supply.” *Econometrica* 80 (3): 969–1018. https://www.jstor.org/stable/41493842.

Chetty, Raj, Adam Guren, Day Manoli, and Andrea Weber. 2012. “Does Indivisible Labor Explain the Difference Between Micro and Macro Elasticities? A Meta-Analysis of Extensive Margin Elasticities.” *NBER Macroeconomics Annual* 27: 1–56. https://doi.org/10.1086/669170.

Chiappori, Pierre-André. 1992. “Collective Labor Supply and Welfare.” *Journal of Political Economy* 100 (3): 437–67. https://www.jstor.org/stable/2138727.

Dickinson, David L. 1999. “An Experimental Examination of Labor Supply and Work Intensities.” *Journal of Labor Economics* 17 (4): 638–70. https://doi.org/10.1086/209934.

Eckstein, Zvi, and Kenneth I. Wolpin. 1989. “Dynamic Labour Force Participation of Married Women and Endogenous Work Experience.” *The Review of Economic Studies* 56 (3): 375–90. https://doi.org/10.2307/2297553.

Francesconi, Marco. 2002. “A Joint Dynamic Model of Fertility and Work of Married Women.” *Journal of Labor Economics* 20 (2): 336–80. https://doi.org/10.1086/338220.

Keane, Michael P. 2011. “Labor Supply and Taxes: A Survey.” *Journal of Economic Literature* 49 (4): 961–1075. https://doi.org/10.1257/jel.49.4.961.

Keane, Michael P., and Kenneth I. Wolpin. 2010. “The Role of Labor and Marriage Markets, Preference Heterogeneity, and the Welfare System in the Life Cycle Decisions of Black, Hispanic, and White Women.” *International Economic Review* 51 (3): 851–92. https://www.jstor.org/stable/40784808.

Keane, Michael, and Richard Rogerson. 2012. “Micro and Macro Labor Supply Elasticities: A Reassessment of Conventional Wisdom.” *Journal of Economic Literature* 50 (2): 464–76. https://doi.org/10.1257/jel.50.2.464.

Pencavel, John. 1986. “Chapter 1 Labor Supply of Men: A Survey.” In *Handbook of Labor Economics*, 1:3–102. Elsevier. https://doi.org/10.1016/S1573-4463(86)01004-0.

Van Der Klaauw, Wilbert. 1996. “Female Labour Supply and Marital Status Decisions: A Life-Cycle Model.” *The Review of Economic Studies* 63 (2): 199–235. https://doi.org/10.2307/2297850.