- Why does the Schooling Gap Close when the Wage Gap Persists?
- Because the paid hours gap closes as the economy's service sector grows.

Ngai & Petrongolo (2017)
Restuccia & Vandenbroucke (2014)
Bils & Klenow (2000)
Bick, Fuchs-Schündeln, & Lagakos (2018)
Barro & Lee (2013)

- Ngai, L. R., & Petrongolo, B. (2017). Gender gaps and the rise of the service economy. American Economic Journal: Macroeconomics, 9(4), 1–44. https://doi.org/10.1257/mac.20150253
- Restuccia, D., & Vandenbroucke, G. (2014). Explaining educational attainment across countries and over time. Review of Economic Dynamics, 17(4), 824–841. https://doi.org/10.1016/j.red.2014.03.002
- Bils, M., & Klenow, P. J. (2000). Does schooling cause growth? American Economic Review, 90(5), 1160–1183. https://doi.org/10.1257/aer.90.5.1160
- Bick, A., Fuchs-Schündeln, N., & Lagakos, D. (2018). How do hours worked vary with income? Cross-country evidence and implications. American Economic Review, 108(1), 170–99. https://doi.org/10.1257/aer.20151720
- Barro, R. J., & Lee, J. W. (2013). A new data set of educational attainment in the world, 1950 2010. Journal of Development Economics, 104, 184–198. https://doi.org/10.1016/j.jdeveco.2012.10.001

\[ U(c,\ell, s^{f}, s^{m}) = \int_{t=0}^{T} \mathrm{e}^{-\rho t} \left( \log(c - \bar c) + \varphi \log(\ell) - \beta^{f}\unicode{x1D7D9}_{t\le s^{f}} - \beta^{m}\unicode{x1D7D9}_{t\le s^{m}}\right) \mathrm{d}t. \]

- Two individuals with distinct genders
- Traditional production throughout their lifespan
- Modern production after schooling finishes

- Cross-sector aggregator \[ c = \left( \sum_{i=A,M,S} \omega_{i} c_{i}^{\frac{\varepsilon-1}{\varepsilon}} \right)^{\frac{\varepsilon}{\varepsilon-1}} \]
- Within-sector aggregator \[ c_{i} = \left( \psi_{i} (c_{ir} )^{\frac{\sigma-1}{\sigma}} + (1-\psi_{i})(c_{ih} )^{\frac{\sigma-1}{\sigma}} \right)^{\frac{\sigma}{\sigma-1}} \]

- Cross-sector aggregator \[ c = \left( \sum_{i=A,M,S} \omega_{i} c_{i}^{\frac{\varepsilon-1}{\varepsilon}} \right)^{\frac{\varepsilon}{\varepsilon-1}} \]
- Within-sector aggregator \[ c_{i} = \left( \psi_{i} (c_{ir} )^{\frac{\sigma-1}{\sigma}} + (1-\psi_{i})(c_{ih} )^{\frac{\sigma-1}{\sigma}} \right)^{\frac{\sigma}{\sigma-1}} \]

- Cross-sector aggregator \[ c = \left( \sum_{i=A,M,S} \omega_{i} c_{i}^{\frac{\varepsilon-1}{\varepsilon}} \right)^{\frac{\varepsilon}{\varepsilon-1}} \]
- Within-sector aggregator \[ c_{i} = \left( \psi_{i} (c_{ir} )^{\frac{\sigma-1}{\sigma}} + (1-\psi_{i})(c_{ih} )^{\frac{\sigma-1}{\sigma}} \right)^{\frac{\sigma}{\sigma-1}} \]

- Leisure aggregator \[ \ell = \left( \xi^{f}_{l} (\ell^{f})^\frac{\eta_{l}-1}{\eta_{l}} + \xi^{m}_{l} (\ell^{m})^\frac{\eta_{l}-1}{\eta_{l}} \right)^\frac{\eta_{l}}{\eta_{l} - 1} \]
- Traditional production aggregator \[ c_{ih} = Z_{ih}\left( \xi^{f}_{ih} \left(L^{f}_{ih}\right)^{\frac{\eta-1}{\eta}} + \xi^{m}_{ih} \left(L^{m}_{ih}\right)^{\frac{\eta-1}{\eta}} \right)^{\frac{\eta}{\eta-1}} \]

- Leisure aggregator \[ \ell = \left( \xi^{f}_{l} (\ell^{f})^\frac{\eta_{l}-1}{\eta_{l}} + \xi^{m}_{l} (\ell^{m})^\frac{\eta_{l}-1}{\eta_{l}} \right)^\frac{\eta_{l}}{\eta_{l} - 1} \]
- Traditional production aggregator \[ c_{ih} = Z_{ih}\left( \xi^{f}_{ih} \left(L^{f}_{ih}\right)^{\frac{\eta-1}{\eta}} + \xi^{m}_{ih} \left(L^{m}_{ih}\right)^{\frac{\eta-1}{\eta}} \right)^{\frac{\eta}{\eta-1}} \]

Time constraint

\begin{align} M^{g} = L^{g} - \ell^{g} - \sum_{i=A,M,S} L^{g}_{ih}. \label{eq:total-labor-allocation} \end{align}Budget constraint

\begin{align} \int_{t=0}^{T} \mathrm{e}^{-\rho t} \sum_{i=A,M,S} p_{ir}c_{ir} \mathrm{d}t & = \sum_{g = m,f} \int_{t=s^{g}}^{T} \mathrm{e}^{-\rho t} M^{g} w^{g} H(s^{g}) \mathrm{d}t. \label{eq:budget-constraint} \end{align}

- Modern technology production function \[ y_{ir} = Z_{ir} l_{ir}, \label{eq:modern-production} \] \[ l_{ir} = \left( \xi^{f}_{ir} \left( \delta(s^{f}) H(s^{f}) l^{f}_{ir}\right)^{\frac{\eta-1}{\eta}} + \xi^{m}_{ir} \left(\delta(s^{m})H(s^{m}) l^{m}_{ir}\right)^{\frac{\eta-1}{\eta}} \right)^{\frac{\eta}{\eta-1}} \]
- Human capital \[ H(s^{g}) = \exp\left(\frac{\zeta}{1-\nu}\left(s^{g}\right)^{1-\nu}\right) \]

A **competitive equilibrium** is a collection of market prices \(\{p_{ir}\}_{i = A,M,S}\), wages \(\{w^{f}, w^{m}\}\), consumption and labor supply allocations \(\{c_{ir}, L^{f}_{ih}, L^{m}_{ih}, L^{f}_{ir}, L^{m}_{ir}\}_{i = A,M,S}\), leisure choices \(\{\ell^{f}, \ell^{m}\}\), years of schooling \(\{s^{f}, s^{m}\}\), and labor demand and output choices \(\{l^{f}_{ih}, l^{m}_{ih}, y_{ir}\}_{i = A,M,S}\) such that

- households maximize their preferences subject to their budget and time constraints, the human capital, and the traditional production technology;
- firms maximize their profits subject to modern production technology;
- commodity and services markets clear (\(c_{ir} = y_{ir}\) for \(i = A,M,S\));
- traditional production is in autarky (\(c_{ih} = Z_{ih} L_{ih}\) for \(i = A,M,S\)); and
- labor markets clear (\(l^{g}_{ir} = L^{g}_{ir}\) for \(i = A,M,S\) and \(g=f,m\)).

Proposition:

Suppose that \(\zeta < ((1-\nu) / \rho)^{\nu-1}\). For any equilibrium allocation, the elasticity of paid hours ratio with respect to the ratio of schooling years is positive along paths for which wages are constant.

\[ \left. \frac{\partial \log \tilde M}{\partial \log \tilde s} \right|_{\tilde w} > 0 \]

Proposition:

Suppose that \(\zeta < ((1-\nu) / \rho)^{\nu-1}\). For any equilibrium allocation, the elasticity of paid hours ratio with respect to the ratio of schooling years is positive along paths for which wages are constant.

\[ \left. \frac{\partial \log \tilde M}{\partial \log \tilde s} \right|_{\tilde w} > 0 \]

Proposition:

Suppose that \(\zeta < ((1-\nu) / \rho)^{\nu-1}\). For any equilibrium allocation, the elasticity of paid hours ratio with respect to the ratio of schooling years is positive along paths for which wages are constant.

\[ \left. \frac{\partial \log \tilde M}{\partial \log \tilde s} \right|_{\tilde w} > 0 \]

- \(\zeta = 0.32\), \(\nu = 0.58\) (Bils & Klenow, 2000)
- \(\rho = 0.04\) (Restuccia & Vandenbroucke, 2014)
- \(((1-\nu) / \rho)^{\nu-1} = 0.37 > \zeta\)

\[
\log(w_{ig}) = c_g + \beta^g_1 age_i + \beta^g_2 age_i^2 + \rho^g_M Ind^M_i + \rho^g_S Ind^S_i + \theta^g_m Educ^m_i + \theta^g_h Educ^h_i + \epsilon_{ig}
\]

\[ \frac{L^{f}_{ih}}{L^{m}_{ih}} = \left(\frac{\xi^{f}_{ir}}{\xi^{m}_{ir}}\right)^{\eta} \left(\frac{\xi^{f}_{ir}d(s^{f})H(s^{f})}{\xi^{m}_{ir}d(s^{m})H(s^{m})}\right)^{-\eta} = \tilde\xi_{ih}^{\eta} \left(\tilde{w} \tilde{d} \tilde{H}\right)^{-\eta}. \]

\[ \frac{L^{f}_{ih}}{L^{m}_{ih}} = \left(\frac{\xi^{f}_{ir}}{\xi^{m}_{ir}}\right)^{\eta} \left(\frac{\xi^{f}_{ir}d(s^{f})H(s^{f})}{\xi^{m}_{ir}d(s^{m})H(s^{m})}\right)^{-\eta} = \tilde\xi_{ih}^{\eta} \left(\tilde{w} \tilde{d} \tilde{H}\right)^{-\eta}. \]

\[
\tilde \xi_{is} = \frac{\xi^{f}_{is}}{\xi^{m}_{is}} > \frac{\xi^{f}_{jq}}{\xi^{m}_{jq}} = \tilde \xi_{jq}
\]

\[
\tilde \xi_{is} = \frac{\xi^{f}_{is}}{\xi^{m}_{is}} > \frac{\xi^{f}_{jq}}{\xi^{m}_{jq}} = \tilde \xi_{jq}
\]

- Data:
- household survey data from (Bick et al., 2018),
- schooling data from (Barro & Lee, 2013) gathered from censuses,
- wage gap persists in cross-country income comparisons,
- but paid hours and schooling gaps are positively correlated.

- Model:
- General equilibrium, multi-sector, -gender, and -production technology model,
- qualitatively explains the positive elasticity between paid hours and schooling for fixed wages, and
- makes accurate quantitative predictions for schooling choices across country income groups.

- Communique:
- As the service sector of the economy grows, the paid hours gap closes,
- leads to a schooling gap decline,
- even when the wage gap stays constant.

Barro, R. J., & Lee, J. W. (2013). A new data set of educational attainment in the world, 1950 2010. *Journal of Development Economics*, *104*, 184–198. https://doi.org/10.1016/j.jdeveco.2012.10.001

Bick, A., Fuchs-Schündeln, N., & Lagakos, D. (2018). How do hours worked vary with income? Cross-country evidence and implications. *American Economic Review*, *108*(1), 170–199. https://doi.org/10.1257/aer.20151720

Bils, M., & Klenow, P. J. (2000). Does schooling cause growth? *American Economic Review*, *90*(5), 1160–1183. https://doi.org/10.1257/aer.90.5.1160

Ngai, L. R., & Petrongolo, B. (2017). Gender gaps and the rise of the service economy. *American Economic Journal: Macroeconomics*, *9*(4), 1–44. https://doi.org/10.1257/mac.20150253

Restuccia, D., & Vandenbroucke, G. (2014). Explaining educational attainment across countries and over time. *Review of Economic Dynamics*, *17*(4), 824–841. https://doi.org/10.1016/j.red.2014.03.002