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agents with income average human capital child education compulsory schooling consumption discount factor distributional effect Eckstein and Zilcha education and public education reduces income education to public educational spending educational subsidies endogenous growth model envelope theorem Equations F₁ finite number fraction of income future periods Glomm and Ravikumar growth and income growth Jie Zhang growth rate H₁ H₁+1 imply income below average income distribution inequality is high inequality is sufficiently initial income inequality invested in child investments in education investments in human Lewis Evans majority voting median agent median agent's welfare number of periods old agent maximizing Optimal public investments optimal tax rate parents physical capital prefer public education prefer subsidizing private provide public education public education reduces public educational policy Ravikumar 1992 reduces income inequality ẞ is large subsidized private education subsidy rate Tamura tax effect UNESCO unless initial income utility function Zealand