Find taxes that arise at a median voter, Business Economics

In a model where a continuum of individuals have preferences for consumption and leisure as follows

wⁱ = cⁱ + ln(xⁱ)

and individuals differ in their labour endowments ei, suppose two candidates run for office and offer a platform of imposing a proportional tax rate τ to finance lump sum transfers uniformly to all the population.

Find taxes that arise at a median voter eqm
Check whether eqm taxes maximize f.
Compare eqm with the "planner's solution"

Where the planner maximizes the average individual's welfare. Are taxes higher or lower at the median voter solution? Why?

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