Abstract
This paper predicts that local wage distributions will contract with an improvement in amenities by imposing structure on the standard model of amenity capitalization with multiple worker types (Roback in Econ Inq 26(1):23–41, 1988). The added structure is Ellickson’s (Am Econ Rev 61(2):334–339, 1971) single-crossing condition, which assumes that the willingness to accept higher housing prices for better amenities increases with income. The model predicts that under empirically plausible conditions, workers at the upper end of the wage distribution will earn less in locations with better amenities while those at the lower end will actually earn more.
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Notes
When using the “traditional hedonic model”, Bayer et al. (2009) estimate a positive relationship between amenities (lack of air pollution) and wages using an OLS regression and an insignificant positive relationship using instrumental variables. Gyourko and Tracy (1991) estimate a negative wage differential for the disamenity student/teacher ratio.
However, given the nature of the specific productive amenity in Moretti (2004), it is most likely that the correct separation for that study is in fact education group.
It is assumed that the utility function of the workers is such that a positive amount of land l is always purchased.
A utility function that satsfies this condition is provided in Ellickson (1971) as the nested CES function \(U(x,l;\varPhi )=\{[a_{1}\varPhi ]^{\frac{1}{\rho }}+[(a_{3}X)^{\frac{1}{\omega }}+(a_{2}l)^{\frac{1}{\omega }}]^{\frac{\omega }{\rho }}\}^{\rho }\) where the a’s, \(\rho \), and \(\omega \) are constants, \(\rho =\frac{\sigma }{\sigma -1}\), and \(\sigma <1\).
Roy’s identity is used here to convert preference ratios into land consumption \(-V_{l}^{k}/V_{w}^{k}=l_{k}\).
Recall the unit cost of x is equal to one.
This is because the firm pays \(\theta _{r}\) of each unit of revenue to land and \(\theta _{w}^{k}\) to worker k, who spends \(s_{l}^{k}\) of \(\theta _{w}^{k}\) on land.
Too see this, apply the implicit function theorem to \(V(r_j,w_{j}^{k};\varPhi _{j})={\overline{V}}^{k}\) and note that \(V_{\varPhi }>0\) and \(V_{r}<0\).
See appendix for proof.
Note that the magnitude of this difference is decreasing in \(\varDelta \), the percentage of revenue from x production that accrues to land. A similar theoretical prediction in Black et al. (2009) states that the more amenities are capitalized into rents, the less the return to education will be.
See Kerr (2011) for a theoretical treatment of the remaining two cases.
The value of \({\tilde{\eta }}_{w}^{B}=0\) is arbitrarily chosen for convenience of the graph’s readability. Under single-crossing, \({\tilde{\eta }}_{w}^{B}\gtrless 0\) is also possible. All other thresholds (\({\tilde{\eta }}_{w}^{A}\), \({\tilde{\eta }}_{r}\), and \({\tilde{\eta }}\)) are guaranteed to be the sign they are shown to be in Fig. 2 and \({\tilde{\eta }}_{w}^{A}<{\tilde{\eta }}_{w}^{B}<{\tilde{\eta }}_{r}\) always holds.
See appendix for proof.
See Mayo (1981) for a review of early studies and Hansen et al. (1998) for evidence from Lorenz curve approach as well as the traditional approach. The general conclusion from this line of research is that income elasticities of housing is less than unity, even when elasticities are allowed to vary by income.
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Appendix
Appendix
Proof that \({\tilde{\eta }}_{w}^{A}<{\tilde{\eta }}_{w}^{B}<{\tilde{\eta }}_{r}\)
when
The single-crossing assumption guarantees that the right hand side is positive. Likewise,
when
where all terms on the ride hand side are positive. By transitivity, the proof is complete.
Proof that \(({\hat{w}}^{A}/{\hat{\varPhi }})-({\hat{w}}^{B}/{\hat{\varPhi }})<0\) when \(s_{l}^{A}<s_{l}^{B}\), \(s_{\varPhi }^{B}<s_{\varPhi }^{A}\) and \(\eta <{\tilde{\eta }}_{r}\)
Signing Equation 14 requires solving for
or
Therefore, \({\tilde{\eta }}\) is the value of \(\eta \) for which \(({\hat{w}}^{A}/{\hat{\varPhi }})=({\hat{w}}^{B}/{\hat{\varPhi }})\). What remains is to show is that this value is greater than \(\eta _{r}\) as shown in Figs. 2 and 3. In this case, we will have \(\eta<{\tilde{\eta }}_{r}<{\tilde{\eta }}\) rendering Eq. 19 negative.
The assumptions \(s_{\varPhi }^{A}>s_{\varPhi }^{B}\) and \(s_{l}^{A}<s_{l}^{B}\) imply that \({\tilde{\eta }}_{r}<{\tilde{\eta }}\). To see this, note that the latter inequality holds when
which can be rearranged as
where \(\varDelta \equiv \theta _{r}+s_{l}^{A}\theta _{w}^{A}+s_{l}^{B}\theta _{w}^{B}>0\) as before. Clearly, this holds when \(s_{\varPhi }^{A}>s_{\varPhi }^{B}\). The assumption \(s_{l}^{A}<s_{l}^{B}\) dictates the direction of the inequality since simplification of Eq. 21– 22 requires multiplying both sides by \((s_{l}^{B}-s_{l}^{A})\). Thus, under the assumptions \(s_{l}^{A}<s_{l}^{B}\) and \(s_{\varPhi }^{B}<s_{\varPhi }^{A}\), \(({\hat{w}}^{A}/{\hat{\varPhi }})-({\hat{w}}^{B}/{\hat{\varPhi }})<0\) whenever \(\eta <{\tilde{\eta }}_{r}\).
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Kerr, C. The effect of amenities on local wage distributions. Lett Spat Resour Sci 10, 215–228 (2017). https://doi.org/10.1007/s12076-017-0183-0
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DOI: https://doi.org/10.1007/s12076-017-0183-0