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J. Lagerlöf (U of Copenhagen) 12
2q for q 2 [0, θ ] . Microeconomics (MikØk) 3: L2-II Spring ‘
11 25 / 29 Fully Non-Linear Tari¤: Serving both types? (2/3)
With this utility function, whether both types or only the high
type are served depends on how ν relates to the cuto¤ value
ν z and z For ν > b both types are served:
qSB = θ = qFB and θ /θ . (1 υ ) ( θ θ )
υ qSB = θ < qFB = θ and
V SB =
For ν 1
2 h υ qSB 2 + (1 2 υ) qSB b only the high type is served:
qSB = θ J. Lagerlöf (U of Copenhagen) and i and E qSB = qFB . qSB = 0. Microeconomics (MikØk) 3: L2-II Spring ‘
11 26 / 29 Fully Non-Linear Tari¤: Serving both types? (3/3)
Plotting b and ν (the corresponding cuto¤ value when using a
two-part tari¤) against z (= θ /θ ) yields: v*, v^ 1.0 0.8 0.6 0.4 0.2 0.0
J. Lagerlöf (U of Copenhagen) 0.1 0.2 0.3 0.4 0.5 0.6 Microeconomics (MikØk) 3: L2-II 0.7 0.8 0.9 1.0 z Spring ‘
11 27 / 29 Appendix (1/2)
Here we show a bit more carefully that, at Step 3 of the
analytical solution, both constraints must bind.
Imagine that each of constraints may or may not hold, and write
the constraints as
θ u (q ) t = ∆IR t = θu q where ∆IR L 0 and ∆IC
a constraint is slack.
I Thus ∆IR J. Lagerlöf (U of Copenhagen) L H (IR-L-new) L, t + ∆IC H, (IC-H-new) 0 are the “gaps” that are left if = 0 i¤ IR-L binds. (And similarly with ∆IC Microeconomics (MikØk) 3: L2-II H .) Spring ‘
11 28 / 29 Appendix (2/2)
By rewriting IR-L-new and IC-H-new, we have
∆IR t = θu q (6) L and
t = θ u (q ) θu q ∆IC H +t = θ u (q ) θu q ∆IC H + θu q = θ u (q ) θ ∆IC θuq H ∆IR
∆IR L (7) L. Now plug (6) and (7) into the objective function (1):
V q, q = ν θu q
θ u (q ) ∆IR
θ L c q + (1 θuq ∆IC ν)
H ∆IR L cq . This expression is decreasing in both gaps.
Hence, it is optimal to make both gaps equal to zero, that is, to
let both constraints bind.
J. Lagerlöf (U of Copenhagen) Microeconomics (MikØk) 3: L2-II Spring ‘
11 29 / 29...
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