2. Labour Supply

KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

March 7, 2024

Labour supply

How people choose whether and how much they work?

Static model

Static labour supply


  • 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

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

Static labour supply


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

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

Static labour supply

Comparative statics

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:

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

Static labour supply

Comparative statics

Source: Wikipedia

Static labour supply

Labour supply curve

Source: Wikipedia

Household model

Intrahousehold labour supply

Unitary model

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

Intrahousehold labour supply

Collective model

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.

Intertemporal model

Intertemporal labour supply


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 \(B_t\)

Intertemporal labour supply


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

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

Intertemporal labour supply

Wage elasticities of labour supply

  • 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 \(\psi_{hw} \geq \eta_{hw} \geq \varepsilon_{hw}\)


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

Intertemporal labour supply


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

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

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


Empirical specifications

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 \(\mathcal{R}_{it}\)!

Empirical specifications

Two-stage budgeting

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

  1. Solve static labour supply model given \(C_t = \mathcal{R}_t + w_t H_t\)
  2. 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: \(\alpha_w - \alpha_R wH\)

Empirical specifications

Frisch elasticity

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: \(\alpha_w\)

Empirical specifications

Practical issues

  • 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


Observational data

Source: Pencavel (1986)


Experimental data: drop in tax rates in the UK 1978-92

Source: Blundell, Duncan, and Meghir (1998)


Intensive vs extensive margin

Source: Chetty et al. (2012)

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


Measurement errors

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


Measurement of consumption

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)


Micro vs macro elasticities

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!


Discrete choice dynamic programming

Incorporate discrete choices into model of labour supply

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

Uncompensated dynamic elasticity
Eckstein and Wolpin (1989) 5.0
Van Der Klaauw (1996) 3.6
Francesconi (2002) 5.6
M. P. Keane and Wolpin (2010) 2.8

Source: M. P. Keane (2011)


  • 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.