1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

When Will Incumbents Avoid a Primary Challenge? 245

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IfπRI∈[π, 1 ),Sis constant for all values ofπRI(and equal to zero), so an

increase inπRIwill not affect it.

A.6 Proof of Lemma 4

I calculate the differential ofSwith respect toqand check its sign, remembering

that the values ofπandπareπ= (^1 −q)

2

1 − 2 q+ 2 q^2 andπ=

q^2

1 − 2 q+ 2 q^2. According to the

values ofSin Theorem1,ifπ∈( 0 ,π),∂S∂q=0; similarly ifπ∈(π, 1 ),∂S∂q=0. So

in those intervals,Sis unresponsive to marginal changes inq.

However, ifπ∈(π,^12 ),∂S∂q= 2 πq−π+ 1 −qwhich is strictly positive; ifπ=^12 ,

∂S

∂q=

1

2 which is strictly positive; ifπ∈(

1

2 ,π),

∂S

∂q=−^2 πq+π+qwhich is strictly

positive. So in those intervals,Sis strictly increasing with marginal increases inq.

To analyze the cases whereπ=πandπ=π, note that∂q∂( (^1 −q)

2

1 − 2 q+ 2 q^2 )<0, so

with a marginal increase inq,πremains in the interval[ (^1 −q)

2

1 − 2 q+ 2 q^2 ,

1

2 ]where I just

proved thatSis increasing withq. Similarly note that∂q∂( q

2

1 − 2 q+ 2 q^2 )>0, so with a

marginal increase inq,πremains in the interval[^12 , q

2

1 − 2 q+ 2 q^2 ]where I just proved

thatSis increasing withq.

To summarize,Sis unresponsive to marginal changes inqforπ∈( 0 ,π)∪(π, 1 ),

and is strictly increasing withqforπ∈{π}∪(π,^12 )∪{^12 }∪(^12 ,π)∪{π}.

A.7 Proof of Lemma 5

See the proof of Lemma 1 in Serra ( 2011 ).

A.8 Proof of Theorem 3

See the proof of Theorem 2 in Serra (2011).

A.9 Proof of Theorem 4

For points 1, 2, 5, 6 and 7, see the proof of points 1, 2, 6, 7 and 8 of Theorem 4 in

Serra (2011), respectively.

To study the effect ofq(point 3 in the theorem), we note that it only has an

indirect effect onTthrough its effect onS. I proved in Lemma5 thatqhas a strictly

positive effect onSwhenever forπRI∈[π,π]. And I have proved (in point 2 of

the theorem) thatShas a strictly positive effect onT. Therefore, combining both