Appendix 4.E Proof of Proposition 1
To obtain the optimal allocation of the qualitative easing to each member, we differentiate Eq.
(4.58) with respect to υH:
By Eq. (4.E1) and the condition for asset purchase in each country, we have two conditions, as shown below.
2
υ
H − = and 1 0υ
H +υ
F = . (4.E2) 1Using the above conditions, we prove the Proposition 1:
1 2
opt opt
H F
υ =υ = . (4.E3)
Appendix 4.F Proof of eigenvalue
12μN
Substituting { }
φ
1,t t T∞= +c 1 =0 into Eqs. (4.68a) and (4.68b), we have:2, 2, 1
(
U Y
cλ
π)Π +tφ
t−φ
t− =0, (4.F1)(U Yc
λ
x)Xt−κφ
2,t =0. (4.F2)Substituting Eq. (4.F2) into Eq. (4.68f), the NKPC can be rewritten as:
1
1 ( ) 2,
t
β
t+κ U Y
cλ κφ
x − tΠ = Π + . (4.F3)
By Eqs. (4.F1) and (4.F3), we have the system difference equations as:
( )
‧
μN is the two eigenvalues of the system difference equations, Eq.
(4.F4), and thus we have:
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Chapter 5
Conclusions
With rising concerns over great differences in the economic performance of member states in the Euro area, the linkage between monetary policy and economic stability of an individual member and the whole union is important for the study of the monetary union regime. In this dissertation, we focus on two key features, namely, trade openness and fiscal fundamentals, of member states to analyze the issue and provide some reasons to explain the differences in the economic performance of member states. The main findings in each chapter can be summarized as follows.
In Chapter 2, we compare the impacts of the productivity and fiscal shocks on countries with different degrees of openness under both a monetary union regime and a flexible exchange rate regime. It is shown that the decline in output is greater for a country with low openness under both regimes. However, the exchange rate flexibility helps dampen the output declines, and thus the output of a country with lower openness declines more in a monetary union where the exchange rate flexibility is absent. In Chapter 3, we use a two-country DSGE model with one common currency and both short-term and long-term government bonds. Based on the differences in fiscal fundamentals among members, we deduce that the ECB’s the public sector purchase programme could cause asymmetric macroeconomic effects on member states. In Chapter 4, we set up a log-linearized model of optimal monetary policy bases on a two-country DSGE model of a monetary union with both conventional and unconventional monetary policies. It is shown that the differences in the economic performance of member states influence the union-wide welfare in the monetary union. In addition, we find that there is a lower union-wide welfare loss by adding the qualitative easing under either discretion or commitment.
We know that if various member states in the monetary union are not at the same point in the economic performance, enforcing a suitable monetary policy becomes a difficult assignment.
Nevertheless, in line with De Grauwe (2012), besides the two key features above, there are other
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important features that may result in different economic performance of member states, for example, unit labor costs, flexibility of wages and prices among member states. In general, in the Euro area, the divergences of economic trends among member states are caused by the interactions of important economic features. As we consider the divergence problem among member states, it is important to keep in mind the results caused by these key features.
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References
De Grauwe, P. (2012), The Economics of Monetary Union, 9th Edition, Oxford: Oxford University Press.