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agents with income average human capital capital H 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 Jie endogenous growth model envelope theorem Equations finite number fraction of income future periods Glomm and Ravikumar growth and income growth Jie Zhang growth rate Hºiſ Hf 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 old agent's utility Optimal public investments optimal tax rate parents physical capital prefer subsidizing private provide public education pſo public education reduces public educational policy Ravikumar 1992 reduces income inequality subsidized private education Tamura tax effect UNESCO unless initial income utility function value function Zealand