The ultimate determinants of central bank independence

APPENDIX B. DERIVATION OF THE PROPERTIES OF THE FUNCTION F(ε) IN
THE FIRST-ORDER CONDITION.

∂F

(1) Demonstrationthat _ < 0.

∂ε

The first derivative of F with respect to ε is given by

∂F ₌ -3U² χ[(1 +g)(1 -β )² + χ]²

(B.1)

^d≈ σμ(1 -β )⁴(1+e)4

which is negative.

∂²F

(2) Demonstration that ^υ > 0.

∂ε²

The second derivative fo F with respect to ε is given by

∂²F ₌ 6U²χΓ[Γ-χ]

(B.2)

^dε² (1 -β )⁴(1₊ε)5 σ²_μ

where Γ ≡ (1+ε)(1-β)² + 2χ, (B.2) is positive.

(3) Demonstration that F(0) = ^[(1 ~ ^β) ^+χ] ^u .

(1 -β )⁴

This can be shown by direct examination of the right-hand side of equation (3.8) at ε = 0.

(4) Demonstration that ^{u (1}^β ) < F(ε) < ^[(1^β ) ^+χ] ^u^σμ ⁽¹ -^β )⁴^σμ

Since F(0) = ^[(1^β) ^{+ χ}] ^u ,
σ²_μ(1 -β )⁴

lim = -u⁽¹^β ) _and ^dF < 0, F(_ε) must be bounded between
e-∞ _σ² ∂ε

^{u (1}^β ) and F(0).
σ²μ

∂F

(5) Demonstration that > 0.

∂U

The first derivative of F with respect to U is given by