# 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, Yi, into the differential w.r.t population average income, Ypop, and average income of their cohort, YCi (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.