Rt q v q u0 q or u0 q or 1 c 1 u 0 qsb c

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Unformatted text preview: w problem: maximize this expression for V w.r.t. q and q (without any constraints). J. Lagerlöf (U of Copenhagen) Microeconomics (MikØk) 3: L2-II Spring ‘ 11 15 / 29 Fully Non-Linear Tari¤: Analytical solution (7/9) Step 4 (Solving the Unconstrained Problem) The FOC w.r.t. q: ∂V ∂q ν θu0 q or θu0 q or = (1 c 1 ν ν " θ u 0 qSB = c 1 I | θ u0 q = 0 ν) θ θ u0 q = c θ 1 νθ ν {z >1 θ θ # 1 (4) . } Conclusion: The low-valuation consumers buy less than the socially optimal quantity (marginal utility > marginal cost). J. Lagerlöf (U of Copenhagen) Microeconomics (MikØk) 3: L2-II Spring ‘ 11 16 / 29 Fully Non-Linear Tari¤: Analytical solution (8/9) The FOC w.r.t. q: ∂V = (1 ∂q ν ) θ u 0 (q ) c =0, θ u 0 qSB = c . I (5) Conclusion: The high-valuation consumers buy the socially optimal quantity (marginal utility equals marginal cost). It also follows from (4) and (5) that qSB < qSB : 0— — qSB — — qFB — — — — — qSB — — — — — > q =q FB J. Lagerlöf (U of Copenhagen) Microeconomics (MikØk) 3: L2-II Spring ‘ 11 17 / 29 Fully Non-Linear Tari¤: Analytical solution (9/9) Step 5 (Checking that IC-L is satis…ed at the optimum) We must verify that t SB θ u qSB θ u qSB by IR-L t SB [= Since IR-L is binding, this amounts to θ u qSB Recall the expression for t in (3) above: t SB = θ u qSB 0] . t SB 0. θ u qSB . θ Using this expression for t SB in our inequality we get θ u qSB θ u qSB + θ or θ θ h u qSB θ u qSB u qSB i 0 0. This inequality holds if qSB > qSB , which we know is the case. J. Lagerlöf (U of Copenhagen) Microeconomics (MikØk) 3: L2-II Spring ‘ 11 18 / 29 Fully Non-Linear Tari¤: Conclusions (1/3) 1 The high-valuatio...
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