The ultimate determinants of central bank independence

10

function.

Hence, optimal commitment in monetary policy involves trading off the credibility gains
associated with lower average inflation versus loss of flexibility due to a distorted response
to output shocks.

III.3. The Ultimate Determinants of Central Bank Independence

Proposition (3.1) is Rogoff’s theorem. Rogoff is unable to write down a closed-form
solution for ε*. Therefore, he is also unable to derive propositions concerning the
comparative static properties of this equilibrium. The following section can be seen as an
extension of the Rogoff theorem.

Using a graphical method, we develop an alternative way of determining the optimal
degree of central bank independence. Next, we show how this result is conditioned on the
natural rate of unemployment (U), society’s preferences for unemployment stabilization
(χ), the variance of productivity shocks (σ_μ²) and the slope of the Phillips curve ((1-β)^¹).

By setting (3.3) equal to zero we obtain the first-order condition for a minimum of E_t-1

_LI_t

0 = -^dnI + -^drI                                                                      (3.6)

∂ε     ∂ε

Substituting (3.4) and (3.5) into (3.6), yields

-χ²u²      ₊     ^χ2(^{1 β )2e σμ}    ₌ 0                                            (3.7)

⁽¹ -^β )²(^{1 + ε})³     [(¹ + ^ε)(¹ -^β )2+χ]³

Equation (3.7) determines ε* as an implicit function of χ, u, σ_μ² and β. A solution for ε*
always exists and is unique.

To show this we adapt a graphical method used by Cukierman (1992, pp. 170-172) in
the context of a dynamic game.

Rewrite (3.7) as

ɛ ₌ [(¹+εX1-^β)² ₊ χ]³u² ≡ F(_g)                                                      (3.8)

⅛¹ -^{β )4(1} +ɛ)³

The function F(ε) on the right-hand side of equation (3.8) is monotonically decreasing in

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