The ultimate determinants of central bank independence

11

th that⁸)

∣ ,.       [(1   β)2 ⁺ χ]3u² г l./ ∖     (1   β)2uu²  and

F(0) = jʌ___—__—__, lim F(ε) = 2___an__ ^and

σ⅛1 -β )^{4     i}                σj

(1 -β)²u² < F(_ε) < [(1 -β)²÷χ]³u²
σ2                   σ²μ(1 -β )⁴

We are now ready to prove:

PROPOSITION 3.2: ⁽¹ ~^β)2^u2 < ɛ* < ^[(1 ~^β)2 ^{+ χ}]3^u2
^σ2                    ^σ2μ(¹ -^β )4

Proof: The left-hand side of (3.8) is a 45-degree straight line through the origin. Since

F(0) = ^{[(1 β}) ^{+ χ}] ^u and ^dF < 0, these two functions must intersect at one and only
σ²μ(1 -β )⁴          əɛ

one point. Moreover, since

^{u (1} ~^β) < F(_ε) < ^[(1 ~^β) ^+χ] ^u , the intersection occurs at a

σ²μ                    σ²2(1 -β )⁴

value of ɛ that is bounded between ^{(1 β} ) ^u and ^{[(1 β}) ^+χ] ^u
σ2            ^σμ(¹ -^β )4

Figure 3.1 illustrates the argument graphically. Clearly, a solution for ɛ exists and is
unique.

⁸⁾ These statements are demonstrated in Appendix B to this paper.

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