RDP 9007: Operating Objectives for Monetary Policy Appendix

DERIVATION OF RESULTS IN SECTION 3

The full model consists of equations (I) to (VI), combined with one of the following policy rules:

For each choice of policy rule, the model has solutions for prices and output which are of the form

Assuming the stochastic terms are all independent, comparative results for price and output variables under the three rules can be obtained by comparing the absolute values of corresponding coefficients in the above solution equations. For example, π₁ measures the effect of domestic supply shocks on the variance of output, and we aim to rank the values of π₁ obtained using each of the policy rules. The procedure is then repeated for the remaining coefficients in equaztions (A1) and (A2).

To solve the model, we begin by collapsing equations (I) to (VI) down to the following two-equation system:^[12]

The interest rate rules can then be expressed in terms of the exogenous shocks as follows:

1. R = 0.
2. 
3. .

This provides sufficient information to determine the coefficients in equations (A1) and (A2). It will be useful to note that Δ₁ and Δ₂ are both less than one, and that Δ₁ < (1+β)Δ₂ < 1.

Domestic supply shock

 

1. 

   

2. π₂ = −(1−λ)λβΔ₀ + λ + βλ(1−λ)Δ₀Δ₁

   = −λβΔ₀(1−λ)(1−Δ₁) + λ, which is positive and greater than π₁.

   λ₂ = −(1−λ)λΔ₀ + λ(1−λ)Δ₁Δ₀

   = −λ(1−λ)Δ₀(1−Δ₁), which is negative and less than λ₁ in absolute value.

3. π₃ = −(1−λ)λβΔ₀ + λ + βλ(1+β)(1−λ)Δ₀Δ₂ − βλΔ₂

   = −(1−λ)λβΔ₀(1 − (1+β)Δ₂) + λ(1−βΔ₂).

   This expression is positive because 1−(1+β)Δ₂ < 1−βΔ₂ , and is less than π₁, since (1+β)(1−λ)Δ₀ < 1.

   λ₃ = −λ(1−λ)Δ₀ + λ(1+β)(1−λ)Δ₂Δ₀ − λΔ₂.

   Here the sum of the second and third terms is negative, so λ₃ is greater than λ₁ in absolute value.

Export supply shock

1. π₁ = λβ(1−λ)Δ₀ + (1−λ) > 0

   λ₁ = λ(1−λ)Δ₀ > 0.

2. π₂ = λβ(1−λ)Δ₀ + (1−λ) − βλΔ₁Δ₀(1−λ)

   = λβ(1−λ)Δ₀ (1−Δ₁) + (1−λ) < π₁.

   λ₂ = λ(1−λ)Δ₀ − λ(1−λ)Δ₁Δ₀

   = λ(1−λ)Δ₀(1−Δ₁) < λ₁.

3. π₃ = λβ(1−λ)Δ₀ + (1−λ) − βλ(1+β)(1−λ)Δ₂Δ₀ − β(1−λ)Δ₂

   = λβ(1−λ)Δ₀(1−(1+β)Δ₂) + (1−λ)(1−βΔ₂) , which is positive and less than π₂.

   λ₃ = λ(1−λ)Δ₀ − λ(1−λ)(1+β)Δ₂Δ₀ − (1−λ)Δ₂.

This expression is ambiguous in sign, and it is also ambiguous whether or not it is smaller in absolute value than λ₁ or λ₂. It should be noted, however, that there exist values of the policy parameter γ for which λ₃ < λ₁ in absolute value: γ (and hence Δ₂) can always be chosen to be sufficiently small that the second and third terms in the above expression do not over-compensate for the absolute size of the first.

Domestic demand shock

1. π₁ = λβΔ₀ > 0

   λ₁ = λΔ₀ > 0.

2. π₂ = λβΔ₀ − λβΔ₀Δ₁ < π₁

   λ₂ = λΔ₀ − λΔ₀Δ₁ < λ₁.

3. π₃ = λβΔ₀ − λβΔ₀(1+β)Δ₂ < π₂ (since (1+β)Δ₂ > Δ₁)

   λ₃ = λΔ₀ − λ(1+β)Δ₀Δ₂ < λ₂ (since (1+β)Δ₂ > Δ₁).

Terms of trade shock

1. π₁ = λβΔ₀(1−λ)(1+β) + β(1−λ) > 0

   λ₁ = λΔ₀(1−λ)(1+β) > 0.

2. π₂ = λβΔ₀(1−λ)(1+β) + β(1−λ) − βλΔ₀(1−λ)(1+β)Δ₁ < π₁

   λ₂ = λΔ₀(1−λ)(1+β) − λΔ₀(1−λ)(1+β)Δ₁ < λ₁.

3. π₃ = λβΔ₀(1−λ)(1+β) + β(1−λ) − βΔ₀λ(1−λ)(1+β)Δ₂ − β(1−λ)(1+β)Δ₂ < π₂ (since (1+β)Δ₂ < 1).

   λ₃ = λΔ₀(1−λ)(1+β) − λ(1+β)(1−λ)(1+β)Δ₀Δ₂ − Δ₂(1−λ)(1+β).

This expression has the same ambiguity of relative size as was remarked upon in the case of the export supply shock above.

Footnote

For simplicity of exposition, it is assumed that the demand elasticities g and h are equal to one, but this does not materially affect the qualitative results. [12]