Effort and Performance in Public-Policy Contests

(10)	dPr*L	= ∂ Pr L ∂ x * L = ∂ x * L d I	∂Pr L ∂ x*H _l___
	dI		∂x*H	∂I
Note that -dPL- = - d x H	∂PrH *, ∂xH	∂ 2Prl _ * * ʌ * ∂x ∂x HL	- ∂2PrH ∂ x*H ∂ x	∂ Pri      1 — and ----= —rτ . Thus L         ∂ xl     nl(I )

may rewrite (10) as:

(11) ^dPr*L
dI

1	[      ∂2 Prh H
~d------ Bn n LH	nn Ч L H ∂ X, ∂ Xh

(η_H +η_L )

pPrH

Ч ∂x²H

η Ln² H

∂² Pr_L     2

₂ ^L ηHn²L
∂x l

we

Λ

JJ_J
J

This gives

Proposition 5:

d Pr* L
dI

> 0 if -∂LPTl.

<       ∂ x_L∂ x_H

(η_H +η_L )^>
<

p⅛

Ч ∂x²H

nH
η L —

∂² Pr_L

l    ∂x²L

nL
η_H

nh

By this proposition we get:

Corollary 5.1:

dPr^*L

(a) Under a type-(ι) reform with i=H, ------<

dI

if

∂² Pr

------L- ≤ 0 ;

∂x_L ∂x_H

d Pr^*L

(b) Under a type-(ιv) reform with i=L, ------

dI

> 0 if

∂² Pr_L

------L- ≥ 0 ;

∂x_L ∂x_H

© Under a type-(iv) reform,

dPr*L > 0 _if ∂²Prh
d I <      ∂x ² H

> ηH nL

2

< η_Ln_H

∂²PrL
and-----L

-0 ;

^∂xL ^∂xH ^≤

(d) Under a type-(v) reform,

^dP > 0 if
dI <

∂²PrH
∂x ² h

(^d2PrL VηH

_dx ²

Ч ^xLL J

2
n_L

^{> η}L ⁿH²

∂²PrL ≥
and--0.

∂x_H ∂x_L ≤

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