# The share of the advantaged group that gets the investment is, therefore, F (Λ_{A }

v) and the share

of the disadvantaged group is F (Λ_{D }

v), where F (.) is the normal distribution. The inequality in

the investment is measured by the difference in these shares: F (Λ_{A }

v)

F (Λ_{D }

v). The analysis

focuses on the change in this quantity as v changes: as the average cost decreases, how does the inequality change?

# Denote the difference in the share with the investment in each group as

. Under the

assumption of normally distributed costs, this difference is simply

=

Z Λ Λ D

v

v

√

1 2πσ^{2 }

exp

(x) 2σ^{2 }

_{2}!

dx

Integrating out, and differentiating with respect to v implies that

d

dv

=

√

1

2πσ^{2 }

exp

(Λ_{A }

2σ^{2 }

v)^{2 }

!

+

exp

(Λ_{D }

2σ^{2 }

v)

^{2 }!!

The sign of this differential changes based on v. The result is summarized in Proposition 1.

Proposition 1. When investment costs are high on average, decreases in the cost result in increased bias towards the advantaged group. As average investment costs decrease, the sign of this e ect switches and further decreases result in decreases in inequality.

Proof. We begin by restating the proposition as follows: There exists v^{∗ }such that when v^{∗ }< v < ∞,

d dv

< 0 and when v = v^{∗},

d dv

= 0 and when 0 ≤ v < v^{∗},

d dv

>

0. Further, this v

^{∗ }=

Λ +Λ 2

^{D }. Given

this, the proof is straightforward.

Note that the sign of

d dv

is the same as the sign of (Λ_{A }

v)^{2 }

(Λ_{D }

v)^{2}.

negative, the differential is negative, and if it is positive, the differential is positive.

If this is We will

therefore focus on conditions to sign that object.

# In particular, we will focus on signing it in three

cases: when v >

Λ +Λ 2

^{D }, when v =

Λ +Λ 2

## D

and when v <

Λ +Λ_{D }2

1. v > (Λ_{D }

Λ +Λ

v)

2 2

> (Λ_{A }

v)^{2},

^{D }. Rearranging, this holds when

which

implies

that

2. v < (Λ_{D }

Λ +Λ

v)

2 2

< (Λ_{A }

v)^{2},

^{D }. Rearranging, this holds when

which

implies

that

3. v = (Λ_{D }

Λ +Λ

v)

2 2

= (Λ_{A }

v)^{2},

^{D }. Rearranging, this holds when

which

implies

that

(Λ_{D }

(Λ_{A }

v)^{2 }v) > Λ (Λ A D

(Λ_{D }

(Λ_{A }

v)^{2 }v) < Λ (Λ A D

(Λ_{D }

(Λ_{A }

v)^{2 }v) = Λ (Λ A D

v. v)^{2 }

v. v)^{2 }

v. v)^{2 }

# Squaring both sides, we find

< 0, so

d dv

< 0.

# Squaring both sides, we find

>

0, so

d dv

>

0.

# Squaring both sides, we find

= 0, so

d dv

= 0.

The proposition suggests that, beginning in a situation with very high investment costs (hence, limited investments), increases in access will make the disadvantaged group relatively worse off. Further increases, however, are predicted to decrease inequality.

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