In Simple Words
Think of a team relay race where swapping a fast runner onto Team A makes Team A faster but makes Team B equally slower, leaving the combined average time of all runners completely unchanged. Unless the swap somehow inspires individual runners to run faster on their own regardless of who runs next to them, overall race performance remains identical.
Research question
Determining whether school vouchers increase net social welfare or merely redistribute existing educational outcomes requires a formal model separating peer-sorting externalities from direct productive gains.
Participants
Theoretical general equilibrium model applied to secondary education populations in urban school districts.
Study Design
Mathematical nesting model specifying student outcomes as $Y_i = \beta_0 X_i + \beta_1 \bar{X}_s + \beta_2 P + \epsilon_i$, where $\beta_1$ isolates redistributive peer effects and $\beta_2$ isolates direct productive effects.
Results
The theoretical framework demonstrates that under linear-in-means peer assumptions, student re-sorting does not alter aggregate population test scores ($\bar{Y} = \beta_0 \bar{X} + \beta_1 \bar{X}$); individual gains are strictly offset by losses to non-participants unless $\beta_2 > 0$.
Takeaway
Incorporate a direct productive coefficient alongside baseline peer averages when evaluating the aggregate social welfare impact of school choice policies.