Optimal Income Taxation with Endogenous Human Capital: Appendix

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1 Optimal Income Taxation with Endogenous Human Capital: Appendix Bas Jacobs January, 2003 Household optimization Consolidating the household budget constraint yields - omitting indices : c = tw x py + twlφ. + 2g. Substitution of the household budget constraint in the utility function yields and unconstrained maximization problem: max u = ln tw x py + twlφ. + 2g l+/ε. {l,x,y} + /ε This is equivalent to maximization of the transformed problem: max {l,x,y} u = tw x py + twlφ. + 2g l+/ε + /ε. First-order conditions are: = twφ. l /ε = 0, l = twlφ x. tw = 0, y = twlφ y. p = 0. University of Amsterdam, Tinbergen Institute, and CPB Netherlands Bureau for Economic Policy Analysis. Address: Faculty of Economics and Econometrics, University of Amsterdam, Roetersstraat, 08 WB Amsterdam, The Netherlands. Phone: Fax: b.jacobs@uva.nl.

2 Rewriting yields: l = [ twφ.] ε, φ x = γy tw =, φ y υx p lφ x =. We can solve for the optimal values of l, x, and y. First, use the marginal rate of technical substitution for goods and time invested in education to get: υw t y = x. Substitute the last result in the equation for labor supply then we get l as a function of x only: υε υw t l = [ tw] ε h ε x γε y υε = [ tw] ε h ε x x γε. Simplifying yields: l = ευ υ h ε w t ε+υ x εγ+υ. pγ Second, we can rewrite the first-order condition for leisure: And we have: lφ. = [ tw] ε φ. +ε. γlφ. = x, which follows from the arbitrage condition for learning. Substitution of the rewritten first-order condition for leisure in the last expression gives: γ[ tw] ε φ. +ε = x, Third, substitute the production function of human capital to obtain: υ+ε υw t γ[ tw] ε h +ε x x γ+ε = x. Simplifying results in the expression for the optimal x: x = γ h +ε ε+υ+ε υ w υ+ε ε+υ+ε t, where + εγ + υ > 0. y and l follow from plugging the value for x into the equations for y and l. 2

3 Second-order conditions To check the second-order conditions we first derive the utility function as a function of x only. Then, we evaluate the second derivative of the utility function at the optimum. If this second derivative is negative we know that utility reaches a maximum in x, y, l space, since optimum values of y and l are positive transformations of x. Substitution of the optimal values of y and l yields indirect utility as a function of x only. First use: + /ε l+/ε = Second, note that: + /ε [ twφ.]ε +/ε = + /ε [ twφ.]+ε. twφ.l = twφ. [ twφ.] ε = [ twφ.] +ε. Indirect utility is given by: v = tw x pωx + twlφ. + 2g + /ε l+/ε. Now substitute the expressions for +/ε l+/ε and twφ.l to obtain: v = tw x pωx + + ε [ twφx]+ε + 2g. tw Where we substituted y = υ x Ωx, and Φx φx, Ωx. γ p The second derivative of the utility function at the optimum values for y and l is: 2 v = 2 tw+ε Φx ε ε Φ2 x Φ + Φ xx. For utility to reach a maximum the term in brackets must be negative, since all other terms are positive. Next use the properties of Φ: Φ x = hγ + υω υ x γ+υ, and Φ xx = hγ + υγ + υ Ω υ x γ+υ 2. Upon substitution in the term in brackets we derive: + εγ + υ <. 3

4 Derivation optimum tax rate The Lagrangian for maximization of social welfare is given by: L = ΨV + η twlφ. + tw x G Λ df, where η is the Lagrange multiplier associated with the government budget constraint. First, we have: G = Ψ λ η + ηt H df = 0. G Note that there are no income effects on both labor supply and investment in human capital, so that H = 0. This equation can be rewritten using the G definition of b: so that: Second, we have: b df = 0, bdf = b =. = ηtwφ. l + η Ψ wlφ. + w x + η wlφ. + w x + twl φ x + φ y y tw df = 0. If goods invested in education are deductible we would have an additional in the last term in brackets. Rewriting yields: p y = Ψ twφ. l + twl η + wlφ. + w x + φ x + φ y y tw df = 0. 4

5 This formula can be simplified in four steps. First, use the definition of ξ to rewrite the first term: Ψ η + wlφ. + w x df = ξ HdF. Second, rewrite the second term: twφ. l df = t t wlφ. t l l df = t t ε lt Third, note that from the first order condition for x we have: twlφ x tw = 0. wlφ.df. And, fourth, rewrite the last term: twl φ x + φ y y tw df = y twlφ y df = t wlφ. φ yy t y t φ. y df = t t υε yt wlφ.df. If goods invested in education are deductible this term is zero. In that case the first line would contain the extra term tp y. Similarly as above one y y may then substitute the first order condition for y: twlφ y tp = 0. Substituting all terms in the first-order condition for t gives after simplifying: t t = ξ ωε lt + υε yt, where ω wlφ.df/ wlφ. + w xdf is the average ratio of second period income in total income. 5