First-time buyers: how do they finance their purchases and what’s changed? – Technical Appendix

Proof

In the main post, we decompose an individual’s income, Y_i, into the differential w.r.t population average income, Y_pop, and average income of their cohort, Y_Ci (thanks to Fahmida Rahman and the Resolution Foundation for helping us with cohort income data from their Intergenerational Audit):

$Y_i=Y_{pop}+(Y_{Ci}-Y_{POP})+(Y_i-Y_{Ci})$

Averaging this expression over all first time buyers, we just sum it up over all individual FTBs and divide by the number of them (N):

$\overline{Y}_{FTB}=\frac{\Sigma_{i=1}^NY_i}{N} =\frac{NY_{POP}}{N} + \frac{\Sigma_{i=1}^N (Y_{Ci}-Y_{POP})}{N} + \frac{\Sigma_{i=1}^N (Y_{i}-Y_{Ci})}{N}$

Notice that because, the cohort average income, is the same for each member of a given cohort, we can write the sum of Ci’s across all individuals as a weighted average of cohort average incomes. αCj denotes the share of FTBs from a given cohort and Cj denotes the average income of cohort j, and note here that the sigma operator is now adding up over G cohort groups, not N individuals.

$\overline{Y}_{FTB}=Y_{POP}+\Sigma_{j=1}^{G}(\alpha_{Cj}Y_{Cj})-Y_{POP}+\overline{Y_{i}-Y_{Ci}}$

We then apply the first difference operator:

$\overline{\Delta Y}_{FTB}=\Delta Y_{POP}+\Sigma_{j=1}^{G}(\Delta\alpha_{Cj}\, \textup{L.}Y_{Cj})+\Sigma_{j=1}^{G}(\textup{L.}\alpha_{Cj}\Delta Y_{Cj})+\Sigma_{j=1}^{G}(\Delta \alpha_{Cj} \Delta Y_{Cj})-\Delta Y_{POP}+\Delta (\overline{Y_{i}-Y_{Ci}})$

Re-ordering these terms gives us the four effects:

Where for notational simplicity we use the simple sigma to denote summing over cohort groups.

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