Comparative statics

p^opt_{M U,t} = χ_{M U} γ²_{M U} + χ_{M U}

¡wt− θ^{M U,t}+ l^∗_{M U,t}¢

−γ_MU(1− γM U)

γ²_{M U} + χ_{M U} E_t^CCBp^{M U}_F,t , (13) where θM U,t is the union-wide productivity shock.

3.2 Comparative statics

The purpose of this model is to illustrate how changes in economic variables may affect the individual’s choice to vote for or against entrance in the monetary union.

This will be done here by comparing the individual’s loss under monetary inde-pendence to the monetary union case. Furthermore, the aim is to show how an individual’s decision to vote for or against membership in the EMU is affected by the country characteristics presented in Section 2 and represented here by the para-meters γ, ψ and λ.

3.2.1 Individual loss under monetary independence

Let the real wage for period t be given by Ωt= wt−p^t. Substituting in the expression for the nominal wage (6), the realized real wage becomes

Ωt=− (p^t− Et−1pt) + λθH,t. (14) From Equation (1), the difference between actual and expected price level is given by

pt− Et−1pt= γ (pH,t− Et−1pH,t) + (1− γ) (p^F,t− Et−1pF,t) . (15)

From the expression for optimal monetary policy, Equation (10), the difference (pH,t− Et−1pH,t) is

(pH,t− Et−1pH,t) =−χ (1− λ) γ² + χ θH,t

=−δ (1 − λ) θ^H,t. (16)

Note that if χ = 0, then pH,t = E_t−1pH,t and individuals can perfectly predict monetary policy, since then the central bank does not allow the productivity shock to show up in the domestic price level.

For simplicity, we will assume that Home individuals know with certainty the realized values of the MU and the RW productivity shocks, which in turn implies that ˜pM U,t = E_t−1p˜M U,t and ˜pRW,t = E_t−1p˜RW,t. The unexpected part in the price level on imported goods (pF,t− Et−1pF,t)is then fully due to the nominal exchange rate shocks,

(pF,t− Et−1pF,t) = φ_{M U,t}+ (1− ψ) φM R,t. (17) Substituting Equations (15), (16) and (17) into (14), we obtain the following expres-sion for the prediction error in real wages:

Ωt− Et−1Ωt= [γδ (1− λ) + λ] θ^H,t− (1 − γ)¡

φ_{M U,t}+ (1− ψ) φM R,t

¢. (18)

The first part of the first right-hand side term in Equation (18) arises as the central bank chooses to accommodate the productivity shock with the parameter δ (see Equation [16]). This affects the prices of a share γ of the overall consumption basket. The second part of the first term is present only if the nominal wages are indexed to productivity. Note that since γδ < 1, the larger the λ, the larger the real wage prediction error (for a given productivity shock). The second term comes from the fact that prices on imported goods (a fraction [1 − γ] of the overall consumption basket) are subject to shocks to the nominal exchange rates.

Taking expectations of the expression for employment, Equation (7), we find that the prediction error in employment can be written as (lH,t− Et−1lH,t) = (pH,t− Et−1pH,t) + (1− λ) θ^H,t. Then, again using Equation (16) above, this can be written simply

(lH,t− Et−1lH,t) = (1− λ) (1 − δ) θ^H,t. (19)

Therefore the uncertainty in employment is a function of the Home productivity shock, weighted by the rate of accommodation of the central bank, δ and the degree of wage flexibility in nominal wages, λ. By setting δ as close to 1 as possible (note from Equation [16] that δ can never be larger than 1) the central bank can choose to stabilize employment. Also, the larger the part of the wages indexed to productivity (larger λ), the smaller the prediction error in employment.

Substituting the Equations (18) and (19) into the individual’s loss function (12), we have:¹⁵

¡Λⁱ_t¢no−MU

= E_t−1£

(Ωt− Et−1Ωt)²+ µ (lH,t− Et−1lH,t)²¤

= E_t−1©

[γδ (1− λ) + λ] θ^H,t− (1 − γ)¡

φ_{M U,t}+ (1− ψ) φM R,t

¢ª2

+ +µE_t−1{(1 − λ) (1 − δ) θ^H,t}²

= Ψ^N₁ V ar (θH,t) + Ψ^N₂ V ar¡ φ_{M U,t}¢

+ Ψ^N₃ V ar¡ φ_{M R,t}¢ +Ψ^N₄ Cov¡

φ_{M U,t}, φ_{M R,t}¢

, (20)

where the parameters in (20) are defined as

Ψ^{M I}₁ = (γδ (1− λ) + λ)² + µ (1− λ)²(1− δ)², Ψ^{M I}₂ = (1− γ)²,

Ψ^{M I}₃ = (1− γ)²(1− ψ)², Ψ^{M I}₄ = 2 (1− γ)²(1− ψ) . 3.2.2 Individual loss in a monetary union

When the Home economy enters the monetary union, monetary policy is no longer conducted by the national monetary authority, but instead by a union-wide author-ity. The optimal policy of this common central bank was derived in Equation (13) where the union-wide productivity shock θM U,t is a weighted average of the country-specific productivity shocks of the individual member states. To be particular, after the Home economy has entered, the union-wide productivity shock can be written as (nθH,t+ (1− n) θ^{N H,t}) where n is the size of the Home country relative to all other countries in the monetary union, N H. The import price on goods produced

15Assuming that the covariance between the productivity shock and exchange rate shocks is zero.

outside the monetary union is now defined as p^{M U}_F,t = ˜pRW,t+ sM R,t

= ˜pRW,t+ s_{M R,t−1}+ φ_{M R,t}.

Before deriving expressions for the uncertainty in real wages and employment once the Home economy is a member of the monetary union, we need to find an expression for the Home consumer price index within the monetary union. Starting from the original expression (1) and using the fact that now, with a common currency, pH,t=

˜

pM U,t, it is possible to derive the following expression for the Home consumer price index within the monetary union:

(pt)^MU = (γ + (1− γ) ψ) ˜pM U,t+ (1− γ) (1 − ψ)¡

˜

pRW,t+ s_MR,t−1+ φ_{M R,t}¢ , since (1 − γ) ψ is the share of all goods consumed in Home that is produced within the currency union. Thus, the expectational error in the consumer price index is given by

(pt− Et−1pt)^{M U}= (γ + ψ (1− γ)) (˜pM U,t− Et−1p˜M U,t) + + (1− γ) (1 − ψ) φM R,t,

assuming, as above, that ˜pRW,t = E_t−1p˜RW,t. Note also that the simplifying assump-tion that the only shock to productivity unknown to Home individuals is the country specific part, θH,t, implies that θN H,t− Et−1θN H,t = 0. Therefore,

˜

pM U,t− Et−1p˜M U,t=−(1− λ) χM U

γ²_{M U} + χ_{M U}nθH,t

=−δ^{M U}nθH,t.

Substituting in the derived expression for (pt− Et−1pt)into the real wage expression (14) gives

(Ωt− Et−1Ωt)^{M U} = (γ + ψ (1− γ)) δ^{M U}nθH,t+ λθH,t− (1 − γ) (1 − ψ) φM R,t. Employment in the Home economy, once it is a member of the monetary union, is given by (lH,t)^{M U} =−Et−1pt+ pM U,t+ (1− λ) θ^H,t. The difference between realized

and expected employment when the Home economy is a member in the monetary union is therefore given as,

(lH,t− Et−1lH,t)^{M U}= (pM U,t− Et−1pM U,t) + (1− λ) θ^H,t

= [(1− λ) − δ^{M U}n] θH,t.

Substituting the derived expressions for (Ωt− Et−1Ωt)^{M U} and (lH,t− Et−1lH,t)^{M U} into the individual loss function, it is now possible to derive the following expression for individual loss in a monetary union:

¡Λⁱ_t¢M U

= E_t−1·³

(Ωt− Et−1Ωt)^{M U}´2

+ µ³

(lH,t− Et−1lH,t)^{M U}´2¸

= E_t−1©

(γ + ψ (1− γ)) δ^{M U}nθH,t+ λθH,t− (1 − γ) (1 − ψ) φM R,t

ª2

+µE_t−1{((1 − λ) − δ^{M U}n) θH,t}²

= Ψ^{M U}₁ V ar (θH,t) + Ψ^{M U}₂ V ar¡

φ_{M R,t}¢

, (21)

where the parameters in (21) are defined as

Ψ^MU₁ = [((γ + ψ (1− γ)) δ^{M U}n) + λ]² + µ [(1− λ) − δ^{M U}n]², Ψ^MU₂ = (1− γ)²(1− ψ)².

3.2.3 Comparing the two cases

Therefore, theoretically, when voting for or against membership in a monetary union, an individual compares the respective loss functions, and a condition for voting “yes”

is that the loss from entering the union is smaller than the loss from staying monetary independent: Λ^{M I} − Λ^{M U} > 0. Using the expressions derived above, this condition for voting for entrance in the union becomes

0 <¡ Λⁱ_t¢M I

−¡ Λⁱ_t¢M U

⇔

0 < Ψ1V ar (θH,t) + Ψ2V ar¡ φ_{M U,t}¢

+ Ψ3Cov¡

φ_{M U,t}, φ_{M R,t}¢

, (22)

where Ψ1 = Ψ^{M I}₁ − Ψ^{M U}1 , Ψ2 = Ψ^{M I}₂ and Ψ3 = Ψ^{M I}₄ . Note that the parameters in front of the term V ar¡

φ_{M R,t}¢

are cancelled out. In the next part we will see how this

decision rule is affected by differences in the key parameters: share of domestically produced goods in total consumption (γ), nominal wage indexation (λ) and country size (n).