Interactions Between Fiscal and Monetary Authorities in a Three-Country New-Keynesian Model of a Monetary Union

Abstract

In this paper we consider the effectiveness of various coordination arrangements between monetary and fiscal authorities within a monetary union if an economic shock has occurred. We address this problem using a multi-country New-Keynesian model of a monetary union cast in the framework of linear quadratic differential games. Using this model we study various coordination arrangements between fiscal and monetary players, including partial fiscal cooperation between only a subgroup of countries, which, to the best of our knowledge, has not been considered yet in the New-Keynesian literature. Using a simulation study we show that, in many cases and from the global point of view, partial fiscal cooperation between a subgroup of fiscal players is more efficient than non-coordination and that, in general, full cooperation without an appropriate transfer system is not a stable configuration. Furthermore, in case there is no full cooperation we show that the optimal configuration of the coordination structure depends on the type of shock that has occurred. We present a detailed analysis of the relationship between coordination structures and type of shock.

Appendix

1.1 A.1 Sensitivity Analysis with Respect to Structural Parameters

The assumed changes of one structural parameter at a time are presented in the first column of Table 5. The next columns show the social preference ordering over different regimes for four shocks.²⁷ The following notation is used: P—stands for regimes 4, 5 and 6; regime in bold means that players in a coalition are better off than in N; $\hat{C}/\hat{F}$—means that all fiscal players in grand/full fiscal coalition are better off than in N but the CB is not in this regime; $\check{C}/\check{F}$—vice versa; F—all players are better off than in N. In most of the cases the ordering based on $J_{U}^{\ast}$ was the same as based on J _U , with only few minor exception, therefore, only the preference ordering based on the former social loss is reported in Table 5.

Table 5 Sensitivity analysis of the benchmark model

For all combinations of parameters except for κ _i,y/π =0.5, from the social point of view, regime N is preferred over F; in other words full fiscal coordination in counter-productive. Thus, it can be said that for the large set of parameters our model confirms results of Beetsma et al. (2001).

The ordering CPNF prevails for symmetric inflation shock $v_{0S}^{\pi}$ and is robust to changes in parametrisation (except for lower value of κ _i,y/π ).²⁸ The same ordering is valid for country-specific inflation shock $v_{0A}^{\pi}$, however, due to asymmetry partial fiscal coalition 6 is (in general) characterised by different social loss than 4 and 5; thus, preference ordering over partial fiscal coalitions might vary. This leads to the conclusion that the grand coalition or partial fiscal coalitions should be sought as the socially efficient regimes in the case of inflation shock. The question whether such forms of coordination would be sustainable remains open. In Table 5 under $v_{0S}^{\pi}$ and $v_{0A}^{\pi}$ the grand coalition is preferred over non-cooperation by every player from an individual point of view. However, it does not tell us whether any player would like to deviate from this arrangement with hope that remaining fiscal players maintain cooperation and a partial fiscal regime emerges. In contrast, the fact that P-regimes are usually inferior to N for (fiscal) player(s) being in coalitions allows us to draw a conclusion that, certainly, these regimes are not sustainable in self-oriented (and myopic) environment.

The picture is less clear for output gap shocks as parameter changes have more influence on regimes’ social ordering here. Full fiscal cooperation is always least preferred (except for lower value of κ _i,y/π ) so the results of Beetsma hold also in this case. However, in contrast to inflation shocks, the grand coalition scores often worse than non-cooperation or P-regimes in the case of symmetric output gap shock.

To summarise, sensitivity analysis of the benchmark model confirms the result of Beetsma et al. (2001) as in all the cases but one full fiscal coordination is worse than non-cooperation. Furthermore, this result can be extended further, as it comes out that F is the worst of all regimes, including partial fiscal cooperation. For inflation shocks, the grand coalition is the socially optimal outcome, and this regime is better than non-cooperation from the individual point of view. However, whether C is sustainable remains still an open question. Partial fiscal cooperation is suboptimal w.r.t. the grand coalition but gives better results than non-cooperation from the social perspective. Unfortunately, in many cases these regimes are suboptimal from individual point of view, thus, possibly unsustainable in the non-cooperative environment, especially if players are myopic. The ineffectiveness results about full fiscal coordination hold also for output gap shocks, but it more difficult to draw some definite conclusions about the preference ordering over the other regimes as the differences in social loss between them are small and, therefore, sensitive, to changes in parametrisation.

It is apparent from Table 5 that the influence of forward-lookingness on our model calls for more attention, as lower values of this parameter may have an important impact on the result obtained above. Table 6 shows the optimal losses for symmetric inflation shock $v_{0S}^{\pi}$ and benchmark parametrisation but with κ _i,y/π =0.5. What is the reason for the improved effectiveness of F regime w.r.t. N when the economies in a monetary union are characterised by lower forward-lookingness?

Table 6 Optimal losses for ($v_{0S}^{\pi}$, κ _i,y =0.5)

Table 7 Symmetric inflation shock, the number of equilibria in LQDGs

Change of an important model parameter certainly influenced the reduced form matrices $\tilde{B}_{4}$ and $\tilde{B}_{5}$ which show the influence of control instruments on output gaps and inflations, respectively.

$$\begin{aligned} \tilde{B}_{4} =&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.7007 & 0.1468 & 0.1468 & -0.6631 \\ 0.1468 & 0.7007 & 0.1468 & -0.6631 \\ 0.1468 & 0.1468 & 0.7007 & -0.6631\end{array} \right ], \quad \mbox{and} \\ \tilde{B}_{5} =&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0.0134 & 0.0026 & 0.0026 & -0.0125 \\ 0.0026 & 0.0134 & 0.0026 & -0.0125 \\ 0.0026 & 0.0026 & 0.0134 & -0.0125\end{array} \right ]. \end{aligned}$$

The pattern is closely comparable to the benchmark (all the values are now more or less 15 % lower than before and all the signs are preserved). With a bigger backward-looking component, economies are more persistent and converge slower to the equilibrium. This is reflected by nearly two times as more losses in regimes N, C and P for κ _i,y/π =0.5 compared to benchmark with $\kappa_{i,y/\pi}=\frac{2}{3}$ in Table 3. As far as the cost of the control instruments is concerned, we observe the following pattern. For regimes where only fiscal players fully or partially cooperate (i.e. F and P), (total) cost of control of those players involved in the coalition gets lower when backward-looking component becomes more eminent. In the case of full fiscal cooperation. This reduction is drastic, from 7.2424 to 0.1974. In contrast, the loss of the fiscal players from the control effort (in P regimes) increases more than 10 times, from 0.0336 to 0.3682. The same holds for the non-cooperative regime, where the (non-cooperative) fiscal players have their control effort increased from 0.0285 to 0.6618. In spite of the somewhat increased volatility of inflation and output gap, the above decrease in control cost in the F regime accompanied by the increase in control cost in N regime (see Fig. 3) makes full fiscal cooperation more attractive than previously.

Fig. 3

Benchmark vs. model with lower forward-lookingness, $v_{0S}^{\pi}$, regime N

The reason for these more moderate actions should be sought in the fact that more backward-lookingness in the model means that economies are more persistent and more slowly converging to equilibrium. This means that it is more costly to control them as the use of instruments is less efficient, which successfully diminishes the conflict between authorities, which choose more reasonable policies.

To improve our understanding of the meaning of forward-lookingness in the model similar sensitivity check to the previous one was conducted but now assuming that κ _i,y/π is set to 0.5 as the benchmark. The results are presented in Table 8 which was constructed in the same way as Table 5.²⁹

Table 8 Sensitivity analysis, benchmark model but with lower forward-lookingness

In most of the cases, $J_{U}^{\ast}(t_{0},F)$ is preferred over $J_{U}^{\ast}(t_{0},N)$ which confirms that forward-lookingness is decisive for the social profitability of full fiscal cooperation in our model. Similarly to Table 6 the social ordering is especially robust for both symmetric and asymmetric inflation shocks. In all these cases full fiscal coordination is the second best regime just after the grand coalition. Regimes P with smaller fiscal coalitions score worse, where as the worst result is obtained for non-cooperation. Consequently, in contrast to previous findings and Beetsma et al. (2001), the above results strongly advocate the need for coordination in the case of inflation shocks. Furthermore, any form of coordination is better than non-cooperation, not only from the social point of view, but also from the perspective of individual authorities.

The results are a little less clear for output gap shocks as, for a few parameter combinations, non-cooperation comes back to the position it took in Table 5, i.e. just after C but before N. It happens when the CB gets more powerful w.r.t. fiscal players in its influence on the output gap (either higher values of γ _i or lower value of η _i ).

1.2 A.2 Sensitivity Analysis w.r.t. Preference Parameters

Thus far, our sensitivity analysis was performed only w.r.t. structural parameters of the economies (excluding policy rules). It is interesting to take a closer look into preference parameters in players’ loss functions. In the benchmark, fiscal players cared five times more about development of output gaps than inflation whereas the CB cared five times more about inflation than output gap. We argued that such a preferences were in line with other parameters in the model as they guarantee that no variables are overrepresented in the total loss of players (as decompositions in previous sections confirm). In this section we vary relative preferences of fiscal and monetary authorities regarding output gap and inflation. In particular, let $r_{F}^{\pi/y}$ and $r_{CB}^{\pi/y}$ denote the ratio between α _i and β _i and between β _CB,i and α _CB,i , respectively, i.e. $r_{F}^{\pi/y}:=\frac{\alpha_{i}}{\beta_{i}}$ and $r_{CB}^{\pi/y}:=\frac{\beta_{CB,i}}{\alpha_{CB,i}}$. The sensitivity analysis will be performed for $r_{F}^{\pi/y}$ and $r_{CB}^{\pi/y}$ simultaneously changing from 0 to 1 with step 0.1. For $r_{F}^{\pi /y}=r_{CB}^{\pi/y}=0$ we have a situation where governments are concerned only about output gaps (i.e. are very liberal in stabilisation sense) while the CB is concerned only about inflation (i.e. is very conservative). In other words, this is the situation when preferences are totally opposite. In contrast, when $r_{F}^{\pi/y}=r_{CB}^{\pi/y}=1$ fiscal authorities as well as the monetary ones are equally interested in deviations of both variables, which, taking into account that the weight of the control instrument does not change and is equal between all the players, means that under symmetric shocks their objectives are the same in this extreme case. The results of the sensitivity check of the benchmark model with preferences amended in the above way are presented in Table 9. The ordering in this table, similarly to previous tables, is based on $J_{U}^{\ast}(t_{0})$.

Table 9 Sensitivity analysis, benchmark model with altered preference

It is evident from Table 9 that the grand coalition is the socially optimal regime for lower values of $r_{F/CB}^{\pi/y}$, when preferences of various authorities are opposite. The second best choice are partial fiscal cooperation regimes, where as fiscal cooperation scores worst, even worse than non-cooperative regime N. This pattern is observed in the neighbourhood of the benchmark for all shocks except for $v_{0S}^{y}$. In contrast, when $r_{F/CB}^{\pi/y}$ becomes larger, first partial fiscal cooperation becomes more socially profitable than C, then F more profitable than N and, finally, when preferences of governments and the CB coincide, F becomes the most profitable outcome of all. This last result is interesting as previously, in the majority of situations, C was the most socially desirable outcome. However, when $r_{F/CB}^{\pi /y}\approx r_{F/CB}^{\pi/y}$ this regime is not so efficient any more because under equal bargaining power assumption the loss of the CB is under-represented in the joint loss of the grand coalition. This leads to the situation in which the interest rate is less important in the joint loss than fiscal debts of individual countries and, therefore, is used more extensively than under F, where free-riding between fiscal coalition and the CB prevents both groups from an overuse of their control instruments. It is easily visible in Table 10 which shows the decomposed players losses for symmetric price shock and symmetric preferences, i.e. $r_{F}^{\pi /y}=r_{CB}^{\pi/y}=1$. Loss from $\chi_{F,C3}\widehat{\hat{f}}^{2}_{C3}$ under F is bigger than under C and, at the same time, $\chi_{CB}\widehat{\hat{\imath}}_{U}^{2}$ is lower under F than under C as fiscal players in F cannot rely so much as in C on interest rate to stabilise economies due to free-riding of the CB. Since, in social loss based on $J_{U}^{\ast}$ interest rate has relatively bigger share than in the joint loss of the grand coalition, full fiscal coordination scores better than the grand coalition where interest rate is relatively overused. Another important observation from Table 9 is that partial fiscal cooperation, as in most of the cases analysed before, is very often the second best choice.

Table 10 Optimal losses for $v_{0S}^{\pi}$, $\alpha_{F,i}=\alpha _{CB}=\frac{1}{2}\alpha_{CB}=\frac{1}{2}\alpha_{F,i}$

Next to every column with social preference ordering we show average social loss obtained for different levels of $r_{F/CB}^{\pi/y}$. Obviously, the less conflicting preferences are the lower average common loss suffered by the union is. Thus, the percentage difference between the average losses for $r_{F/CB}^{\pi/y}=0$ and $r_{F/CB}^{\pi/y}=1$ is the highest for symmetric and asymmetric inflation shocks (61.5 % and 59 %, respectively), and much more moderate for both output shock (13.5 % and around 0), which confirms are previous results that the biggest gains from choosing an appropriate regime is to be expected in the former case.

A number of interesting conclusions can be drawn the above analysis. First of all, the relative antagonism between the CB and governments in the monetary union is an important factor which strongly determines the profitability of full fiscal coordination. In contrast to other various findings from the literature, in our model, strongly independent bank of a monetary union is not so profitable from the common perspective and more intermediate arrangements are advisable. Secondly, if bargaining power in the grand coalition do not coincide with socially optimal preferences, this regime might be counter-productive w.r.t. full-fiscal coalition, which in turn, can turn out to be optimal due to free-riding.

The analyses in Table 9 is made under the assumption that both types of authorities simultaneously change their preferences from the most conflicting to the same ones. This was rather theoretical simulation as having little chances to be realised in (the European) practice as the ECB independence is strongly safeguarded by relevant treaties. It is also interesting to study the more realistic situation in which the strong CB’s focus on inflation remains unchanged while governments, at the beginning fixed only at inflation, become gradually interested in inflation. More formally, we consider the case where $r_{CB}^{\pi/y}$ is kept constant at 0 where as $r_{F}^{\pi/y}$ changes from 0 to 1. One possible interpretation of such simulations in Table 11, which one can think of, are more and more stringent provisions of the SGP, which additionally to fiscal debt issues regulates also inflation in the EMU Member States.

Table 11 Sensitivity analysis, benchmark model with altered preference

In general, the outcomes in Table 11 are reasonably similar to those from Table 9 as far as main trends are considered, i.e. the less intensive conflict between authorities makes partial fiscal cooperation regimes as well as full fiscal cooperation more interesting from the social point of view. Of course, always restrictive CB makes it impossible to reach the same outcome as in the previous case. For $r_{CB}^{\pi/y}=0$ and $r_{F}^{\pi/y}=1$ (i.e. the last row of Table 11) the social orderings are similar $r_{F/CB}^{\pi /y}=0.5$ previously (the middle of the Table 9). Accordingly, minimal average loss obtained for the last case is now higher than when authorities’ preferences were more alike. However, what is important, social losses at the end of both tables are not much different which shows that similar low social welfare can be obtained either by making preferences of fiscal and monetary authorities more parallel, or by safeguarding the CB independence and making government to be more equally oriented about inflations and output gaps. The first proposition seems to be rather unacceptable by the modern economic school, but the second one seems not only to be acceptable from this point of view, but actually implemented in the current European practice (in the form of the strongly independent ECB and the SGP, which makes governments more “inflation-aware”).

The issue related to the SGP will be discussed further in this paper but first we will consider (nearly) optimal policy rules.

1.3 A.3 Nearly Optimal Policy Rules

In the LQDG framework it is not possible to analytically optimise certain parameters of the model, however, an approximate analyses can be performed numerically. We will use this method to study how various combinations of policy rules’ parameters influence output gap and inflation volatility and which of them are likely to bring (nearly) optimal outcome from the social point of view. Due to the space constraints we will focus mainly on the symmetric inflation shock. Tables 12, 13, 14 show (optimal) losses together with their decomposition in the non-cooperative regime for different values of $\theta_{y}^{i}$ and $\theta _{\pi}^{U}$. More specifically, Table 12 shows cases in which $\theta_{\pi}^{U}=1.25$ and $\theta_{y}^{i}$ changes from 0 to −1; Table 13 cases in which $\theta_{\pi }^{U}=1.5$ and $\theta_{y}^{i}$ as before; and, finally, Table 14 shows cases in which $\theta_{\pi}^{U}=1.75$ and as before. Such an analysis, albeit approximate, may give us an important insight into efficiency of different policy rules’ combinations.

Table 12 Optimal losses for $\theta^{M}_{\pi}=1.25$ and $\theta^{F}_{y}=0, 0.1,0.3, 0.5, 0.8, 1.0$

Table 13 Optimal losses for $\theta^{M}_{\pi}=1.50$ and $\theta^{F}_{y}=0, 0.1, 0.3, 0.5, 0.8, 1.0$

Table 14 Optimal losses for $\theta^{M}_{\pi}=1.75$ and $\theta^{F}_{y}=0, 0.1, 0.3, 0.5, 0.8, 1.0$

Figure 4 compares $J_{U}^{\ast}$ losses for different combinations of $\theta_{\pi}^{U}$ and $\theta_{y}^{i}$. In general, from the monetary authority perspective, a rule less focused on inflation (i.e. $\theta_{\pi}^{U}=1.25$, Table 12) results in higher losses than for the benchmark value (i.e. $\theta _{\pi }^{U}=1.50$, Table 13), whereas a rule more focused on inflation (i.e. $\theta_{\pi}^{U}=1.75$, Table 14) generates lower losses. The only exception from this pattern is a combination of coefficients $\theta_{\pi}^{U}=1.50$ and $\theta_{y}^{i}=0$ which produces the lowest social loss, i.e. is an optimal Taylor rules parameters’ combination (ceteris paribus) for the non-cooperative regime. From the fiscal authority perspective the stronger reaction to output, the higher loss and vice versa. Finally, it should be mentioned that for combinations $(\theta_{\pi}^{U}=1.25, \theta_{y}^{i}=-0.3)$ and $(\theta_{\pi}^{U}=1.25, \theta _{y}^{i}=-0.5)$ strong irregularities emerge, explanation of which should be sought in mathematical properties of the model.³⁰

Fig. 4

Social loss for various combinations of policy rules parameters, regime N

Let us now focus on the individual players’ perspective. Due to the irregularities mentioned above we will exclude from our analysis Table 12. For $\theta_{\pi}^{U}=1.50$, the loss of the fiscal players is not monotonic and reaches its minimum for the benchmark parametrisation (i.e. $\theta_{y}^{i}=0.50$). However, for $\theta _{\pi}^{U}=1.75$ the results are more clear-cut as the stronger reaction of governments to output always leads to the lower loss. This is totally at odds with the CB losses which behave in the exactly opposite way. The reasons of this difference can be found in the decomposition of losses. Stronger reaction of fiscal debt to output gap leads to its lower volatility (in Table 14 $\beta_{F,C1}\hat{y}_{C1}^{2}$, and, consequently, $\beta_{CB}\hat{y}_{CB}^{2}$ decrease with $\theta _{y}^{i}$), however, this positive effect is reached at the very expense of inflation volatility which grows accordingly. This is detrimental for the CB as this authority is mainly concerned about inflation under benchmark parametrisation. Overall, the conflict of interest between both types of authorities is clearly visible here. Highest value of $\theta_{\pi}^{U}$ without a counter-response in fiscal rule is damaging to loss of governments as the CB’s strong reaction to inflation makes output gap very volatile. Thus, governments use the most of their control effort to improve the situation, however, only higher (absolute) values of $\theta_{y}^{i}$ make them increasingly better off. This pattern is robust even for $\theta _{y}^{i}$ reaching minus one. In contrast, as mentioned above, for the more moderate CB’s policy rule (i.e. $\theta_{\pi}^{U}=1.5$) and for $\theta_{y}^{i}$ higher than half, fiscal loss start to increase. This means that, if fiscal rule responses to tempered monetary rule too strongly, there is an effect of overshooting the fiscal policy rule. As a result, governments are pushed to deviate from such a rule much stronger than before because the (overshot) rule must be discretionary corrected. Consequently, $\chi_{F,C1}\widehat{\hat{f}}^{2}_{C1}$ grows from 0.0269 in benchmark to 0.6282 for $\theta_{y}^{i}=-1$. Comparing the CB’s losses between Tables 12, 13, 14 it is evident that more reactionary stance is in the interest of the CB as its loss decreases with increasing $\theta_{\pi}^{U}$ due to the lower inflation deviation in the union.

To sum up, from the individual point of view, we have a situation where CB has incentives to increase $\theta_{\pi}^{U}$ and, at the same time, for high values of $\theta_{\pi}^{U}$, governments have incentives to increase $\theta_{y}^{i}$. When both types of authorities do it at the same time, the economies end up in a position which is not only suboptimal from the social point of view (right-down corner in Fig. 4), but also from individual ones. For $\theta_{\pi}^{U}=1.75$ and $\theta _{y}^{i}=-1.0$ governments obtain loss of 2.4264 and the CB of 6.5807, which is the outcome Pareto-dominated by other combinations, e.g. the benchmark. Unfortunately, the socially optimal combination $(\theta _{\pi}^{U}=1.5, \theta_{y}^{i}=0)$ does not Pareto-dominate combination $(\theta_{\pi}^{U}=1.75, \theta_{y}^{i}=-1.0)$ so the mutual agreement to move toward the optimum seems unlikely to be obtained.

1.4 A.4 SGP Analysis

Within our model we can also investigate the effects of the major policy-surveillance institution of the EMU, namely the SGP. The SGP imposes a framework of fiscal stringency and coordination measures that aim at securing the implementation of the BEPGs. In our model the effects of various levels of the SGP stringency can be studied by considering (i) different levels of the countercyclical parameter $\theta_{y}^{i}$ in the fiscal rule; and (ii) different weights associated with the domestic fiscal deficit, χ _i , in the objective functions of the fiscal players. We compare the following three cases, each characterised by stricter SGP provisions than the other:³¹

• I. In the fiscal rule the coefficient measuring a countercyclical reaction of fiscal debt to deviation of output gap is two times smaller than in the benchmark, i.e.: $\theta_{y}^{i,new}=-0.25$;

• II. As above, but, additionally, deviations from the rule are more costly (i.e. are more severely punished by the SGP provisions), $\chi_{i}^{new,II}=1.5\chi_{i}$;

• III. As in point I, but, additionally, $\chi_{i}^{new,III}=3\chi _{i}$.

It is expected that smaller countercyclical reaction of the fiscal rule is going to force fiscal authorities to deviate stronger from the rule than in the benchmark. On the other hand, more costly deviations from the rule in cases II and III are likely to diminish the use of fiscal instrument w.r.t. case I. As far as individual losses of players are concerned, it is possible to directly compare new cases to the benchmark, however, it is not exactly obvious whether we can do so with the social loss. Whereas governments, as public authorities, might be bound by tougher SGP provisions, it does not have to lead to an automatic increase of the social loss. In the benchmark we assumed that χ _U =χ _i . Now, we are going to compute “adjusted” social loss of the entire union $J_{U}^{\ast}(t_{0})$, denoted $J_{U}^{\ast A}(t_{0})$, by assuming that cost of the deviation of the fiscal instrument from the rule is unchanged, i.e. equals to χ _U =χ _i as in the benchmark, instead of $\chi_{U}=\chi _{i}^{new,II}$ or $\chi_{U}=\chi_{i}^{new,III}$. By doing so we will see what is the contribution of a change in use of a fiscal instrument in the total change of the loss from stabilisation effort.³²

Players’ losses in the first three regimes in the case of a symmetric inflation shock and benchmark parametrisation are shown in Tables 15 and 16. In spite of vast differences between cases a few general conclusions can be drawn. First of all, lower counter-cyclical reaction of fiscal debt (case I) always makes the losses of fiscal players higher than in the benchmark, which means that the assumed value $\theta _{y}^{i}=-0.5$ was chosen relatively well for the initial simulations and which confirms our findings from the previous section.³³ Secondly, in all the new cases higher SGP stringency leads to increasing losses of fiscal players from output gap volatility. This is natural as fiscal authorities refrain from using the more expensive control instruments. Interestingly, in different regimes we obtain different relationships between SGP stringency and the amount of the control instrument used. Whereas in all regimes with any form of cooperation (i.e. C, F, and P-regimes) the higher χ _i , the less control instrument is used (compare cases II to I and III to II), then under non-cooperation this relationship is highly non-linear. For $\chi _{i}^{new,II}$ governments decide to use $\frac{\chi_{C1}\widehat{\hat {f}}^{2}_{i}}{\chi_{C1}}=\widehat{\hat{f}}^{2}_{i}=1190$ which is nearly 40 % more (not less) than for χ _i in case I, however when χ _i increases to $\chi_{i}^{new,III}$ they contract substantially control action to $\frac{\chi_{C1}\widehat{\hat{f}}^{2}_{i}}{\chi_{C1}}= \widehat{\hat{f}}^{2}_{i}=300$. The SGP regulating the use of control instrument influences also the use of interest rate by the CB of the union. In many cases when control action of fiscal authorities is diminished the response of the CB also fades out, i.e. conflict between both types of authorities is hampered. In relative terms, the biggest reductions in the control effort of the monetary authority is obtained under F ^III where cost of the control effort is lowered from 25.83 to 3.00. On the other hand, under N ^III we also witness quite a reduction in the fiscal control effort w.r.t. N ^I, but the main driving force in this case is a free-riding of fiscal players, which forces the CB to increase its engagement in the union economy not stabilised enough by national governments. As the loss from the CB’s control instrument is an important part of $J_{U}^{\ast}(t_{0})$ and $J_{U}^{A\ast}(t_{0})$, this leads to higher social loss under N ^III than under F ^III.

Table 15 Regimes N, C and F for different levels of the SGP stringency

Table 16 Regimes N, C and F for different levels of the SGP stringency

To summarise, we established the third factor (next to the degree of backward-lookingness and loss functions’ preferences) which heavily determined the results obtained for the benchmark parametrisation of the model. The increased SGP stringency reduces incentives of fiscal players to use control instruments, therefore, in situations where high social losses where driven by the conflict between authorities (notably regime F), such a firmer stance is beneficial to the union-wide economic interest. However, in situations in which free-riding is present (notably regime N under benchmark) increased SGP stringency may lead to more extensive free-riding of governments as undertaking any actions become more costly. This, in turn, makes the CB to intervene and increases social loss of the union. In other words, the stringent SGP has both positive and negative effects in the context of this paper and is able to make unprofitable regime to become profitable.

Similar analysis has been performed for 3 other shocks. Since the conflict under $v_{0A}^{\pi}$ is less eminent also the social gains from higher SGP stringency are lower. As before, both output shocks are characterised by the lower variability of losses between different regimes, however, still SGP stringency is able to make non-cooperation inferior to fiscal cooperation, at least, in the case of the symmetric shock.

1.5 A.5 Model Derivations

1.5.1 A.5.1 Reduced Form of the Model

Defining

$$\begin{aligned} K_{y} :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \kappa_{1,y} & 0 & \cdots& 0 \\ 0 & \kappa_{2,y} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \kappa_{n,y}\end{array} \right ] , \qquad K_{\pi}:=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \kappa_{1,\pi} & 0 & \cdots& 0 \\ 0 & \kappa_{2,\pi} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \kappa_{n,\pi}\end{array} \right ] , \\ G :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \gamma_{1} & 0 & \cdots& 0 \\ 0 & \gamma_{2} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \gamma_{n}\end{array} \right ] , \qquad E:=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \eta_{1} & 0 & \cdots& 0 \\ 0 & \eta_{2} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \eta_{n}\end{array} \right ] , \\ \varXi :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \xi_{1} & 0 & \cdots& 0 \\ 0 & \xi_{2} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \xi_{n}\end{array} \right ] , \qquad B:=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \beta_{1} & 0 & \cdots& 0 \\ 0 & \beta_{2} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \beta_{n}\end{array} \right ] , \\ R :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0 & \rho_{12} & \cdots& \rho_{1n} \\ \rho_{21} & 0 & \cdots& \rho_{2n} \\ \cdots& \cdots& \cdots& \cdots\\ \rho_{n1} & \rho_{n2} & \cdots& 0\end{array} \right ] , \\ D :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \delta_{12} & \delta_{13} & \cdots& \delta_{1n} \\ -\sum_{j\in F/2}\delta_{2j} & \delta_{23} & \cdots& \delta_{2n} \\ \cdots& \cdots& \cdots& \cdots\\ \delta_{n2} & \delta_{n3} & \cdots& -\sum_{j\in F/n}\delta_{nj}\end{array} \right ] , \\ \varPsi_{y} :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \psi_{1,y} & 0 & \cdots& 0 \\ 0 & \psi_{2,y} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \psi_{n,y}\end{array} \right ] \\ V :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \varsigma_{12} & \varsigma_{13} & \cdots& \varsigma_{1n} \\ -\sum_{j\in F/2}\varsigma_{2j} & \varsigma_{23} & \cdots& \varsigma _{2n} \\ \cdots& \cdots& \cdots& \cdots\\ \varsigma_{n2} & \varsigma_{n3} & \cdots& -\sum_{j\in F/n}\varsigma _{nj}\end{array} \right ] , \\ \varPsi_{\pi} :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \psi_{1,y} & 0 & \cdots& 0 \\ 0 & \psi_{2,y} & \cdots& 0 \\ \cdots& \cdots& \cdots& \cdots\\ 0 & 0 & \cdots& \psi_{n,y}\end{array} \right ] \\ \varPsi :=&\left [ \begin{array}{@{}c@{\quad}c@{}} \varPsi_{y} & 0 \\ 0 & \varPsi_{\pi} \end{array} \right ] , \quad \mbox{and} \quad v_{t}:=\left [ \begin{array}{@{}c@{}} v_{t}^{y} \\ v_{t}^{\pi}\end{array} \right ] , \\ \iota_{n} :=&\left [ \begin{array}{@{}c@{}} 1 \\ 1 \\ 1 \\ \cdots\\ 1\end{array} \right ] _{n}, \qquad S:=\left [ \begin{array}{@{}c@{\quad}c@{}} -\iota_{ ( n-1 ) } & I_{n-1} \end{array} \right ] , \\ \varTheta_{\pi}^{F} :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \theta_{\pi}^{1} & \theta_{\pi}^{2} & \cdots& \theta_{\pi}^{n}\end{array} \right ] ^{T} \quad\mbox{and}\quad \varTheta_{y}^{F}:= \left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \theta_{y}^{1} & \theta_{y}^{2} & \cdots& \theta_{y}^{n}\end{array} \right ] ^{T} \end{aligned}$$

the SNKM model can be rewritten as:

$$\begin{aligned} y_{t} = {}& K_{y}E_{t}y_{t+1}+ ( I_{n}-K_{y} ) y_{t-1}-G ( \mathbf{\iota}_{n}i_{t}-E\mathbf{\pi}_{t+1} ) +Ef_{t} \\ &{}- K_{y}RE_{t}y_{t+1}+Ry_{t}- ( I_{n}-K_{y} ) Ry_{t-1}-K_{y}DE_{t}s_{t+1} \\ &{}+ Ds_{t}- ( I_{n}-K_{y} ) Ds_{t-1}+v_{t}^{y}, \end{aligned}$$

(12)

$$\begin{aligned} \pi_{t}={} & K_{\pi}BE_{t} \pi_{t+1}+ ( I_{n}-K_{\pi} ) B\pi _{t-1}+ \varXi y_{t}+\varXi Vs_{t}+v_{t}^{\pi}, \end{aligned}$$

(13)

$$\begin{aligned} s_{t+1}={} & s_{t}+S\pi_{t+1}, \end{aligned}$$

(14)

$$\begin{aligned} v_{t+1}={} & \varPsi v_{t}+\varepsilon _{t+1}, \end{aligned}$$

(15)

where I _m is m×m identity matrix (m=n−1,n), s _t :=[s _12,t ⋯ s _1n,t ] and y _t , π _t , $v_{t}^{y}$ and $v_{t}^{\pi}$ are appropriately defined vectors of size n each. In particular, it can be shown that every s _ij,t :=p _j,t −p _i,t (j≠i) can be expressed in terms of s _12,t ,…,s _1n,t . For example, in a three-country monetary union we have six bilateral real exchange rates: s _12,t =p _2,t −p _1,t , s _13,t =p _3,t −p _1,t , s _21,t =p _1,t −p _2,t , s _23,t =p _3,t −p _2,t , s _31,t =p _1,t −p _3,t , and s _32,t =p _2,t −p _3,t . Clearly, the last four variables can be expressed as a combination of the first two, i.e. s _21,t =−s _12,t , s _23,t =s _13,t −s _12,t , s _31,t =−s _13,t and s _32,t =s _12,t −s _13,t .

Defining fiscal and monetary policy rule vectors as:

$$\begin{aligned} f_{t} &:=\varTheta_{\pi}^{F} \pi_{t}+\varTheta_{y}^{F}y_{t},\quad \mbox{ and} \end{aligned}$$

(16)

$$\begin{aligned} i_{t} &:=\theta_{\pi}^{U} \omega^{T}\pi_{t}+\theta_{y}^{U}\omega ^{T}y_{t}, \end{aligned}$$

(17)

substituting them into system (12)–(15) and rearranging we get:

$$\begin{aligned} &{-}K_{y} ( I_{n}-R ) E_{t}y_{t+1}-GE_{t} \pi_{t+1}+K_{y}DE_{t}s_{t+1} \\ &\quad=- \bigl( I_{n}-E\varTheta_{y}^{F}+G \iota_{n}\theta_{y}^{U}\omega ^{T}-R \bigr) y_{t}- \bigl( G\iota_{n}\theta_{\pi}^{U} \omega ^{T}-E\varTheta_{\pi}^{F} \bigr) \pi_{t} \\ &\qquad{}+ ( I_{n}-K_{y} ) ( I_{n}-R ) y_{t-1}+Ds_{t}- ( I_{n}-K_{y} ) Ds_{t-1}+I_{n}v_{t}^{y}, \end{aligned}$$

(18)

$$\begin{aligned} &{-}K_{\pi}BE_{t}\pi_{t+1}=\varXi y_{t}-\pi_{t}+ ( I_{n}-K_{\pi} ) B \pi_{t-1}+\varXi Vs_{t}+I_{n}v_{t}^{\pi}, \end{aligned}$$

(19)

$$\begin{aligned} &s_{t+1}-SE_{t}\pi_{t+1}=s_{t}, \end{aligned}$$

(20)

$$\begin{aligned} &v_{t+1}=\varPsi v_{t}+\varepsilon _{t+1}. \end{aligned}$$

(21)

Introducing three additional vectors of variables a _t+1:=y _t , b _t+1:=π _t and c _t+1:=s _t we may rewrite system (18)–(21) as:

$$\begin{aligned} &{-}K_{y} ( I_{n}-R ) E_{t}y_{t+1}-GE \pi_{t+1}+K_{y}Ds_{t+1} \\ &\quad=- \bigl( I_{n}-E\varTheta_{y}^{F}+G \iota_{n}\theta_{y}^{U}\omega ^{T}-R \bigr) y_{t}- \bigl( G\iota_{n}\theta_{\pi}^{U} \omega ^{T}-E\varTheta _{\pi}^{F} \bigr) \pi_{t} \\ &\qquad{}+Ds_{t}+ ( I_{n}-K_{y} ) ( I_{n}-R ) a_{t}- ( I_{n}-K_{y} ) Dc_{t}+I_{n}v_{t}^{y} \end{aligned}$$

(22)

$$\begin{aligned} &{-}K_{\pi}BE\pi_{t+1}=\varXi y_{t}- \pi_{t}+\varXi Vs_{t}+ ( I_{n}-K_{\pi } ) Bb_{t}+I_{n}v_{t}^{\pi}, \end{aligned}$$

(23)

$$\begin{aligned} &{-}S\pi_{t+1}+s_{t+1}=s_{t}, \end{aligned}$$

(24)

$$\begin{aligned} &a_{t+1}=y_{t}, \end{aligned}$$

(25)

$$\begin{aligned} &b_{t+1}=\pi_{t}, \end{aligned}$$

(26)

$$\begin{aligned} &c_{t+1}=s_{t}, \end{aligned}$$

(27)

$$\begin{aligned} &v_{t+1}=\varPsi v_{t}+\varepsilon _{t+1}. \end{aligned}$$

(28)

Defining: A ₁₁=−K _y (I−R), A ₁₂=−G, A ₁₃=K _y D, $B_{11}=- ( I-E\varTheta _{y}^{F}+G\iota _{n}\theta_{y}^{U}\omega^{T}-R )$, $B_{12}=- ( G\iota _{n}\theta _{\pi}^{U}\omega^{T}-E\varTheta_{\pi}^{F} )$, B ₁₃=D, B ₁₄=(I _n −K _y )(I _n −R), B ₁₆=−(I _n −K _y )D, B ₁₇=[0 _n×n I _n ], A ₂₂=−K _π B, B ₂₁=Ξ, B ₂₂=−I _n , B ₂₃=ΞV, B ₂₅=(I _n −K _π )B, B ₂₇=[I _n 0 _n×n ], A ₃₂=−S, B ₇₇=Ψ, the system (22)–(28) in state-space form as:

$$ E_{t}z_{t+1}=A^{-1}Bz_{t}+F \upsilon_{t}, $$

(29)

where $z_{t}:= [ y_{t}^{T} \ \pi_{t}^{T} \ s_{t}^{T} \ a_{t}^{T} \ b_{t}^{T} \ c_{t}^{T} \ v_{t}^{T}] ^{T}$ or $z_{t}:= [ z_{1,t}^{T}\ z_{2,t}^{T}\ v_{t}^{T} ] ^{T}$ with $z_{1,t}:= [ a_{t}^{T}\ b_{t}^{T}\ c_{t}^{T} ] ^{T}$, $z_{2,t}:= [ y_{t}^{T}\ \pi_{t}^{T}\ s_{t}^{T} ] ^{T}$,

$$\begin{aligned} \upsilon_{t} :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} 0_{1\times n} & 0_{1\times n} & 0_{1\times ( n-1 ) } & 0_{1\times n} & 0_{1\times n} & 0_{1\times ( n-1 ) } & \varepsilon _{t}^{T}\end{array} \right ] ^{T}, \\ A :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} A_{11} & A_{12} & A_{13} & 0_{1} & 0_{1} & 0_{2} & 0_{3} \\ 0_{1} & A_{22} & 0_{2} & 0_{1} & 0_{1} & 0_{2} & 0_{3} \\ 0_{4} & A_{32} & I_{ ( n-1 ) } & 0_{4} & 0_{4} & 0_{5} & 0_{6} \\ 0_{1} & 0_{1} & 0_{1} & I_{n} & 0_{1} & 0_{2} & 0_{3} \\ 0_{1} & 0_{1} & 0_{1} & 0_{1} & I_{n} & 0_{2} & 0_{3} \\ 0_{4} & 0_{4} & 0_{5} & 0_{4} & 0_{4} & I_{ ( n-1 ) } & 0_{6} \\ 0_{7} & 0_{7} & 0_{8} & 0_{7} & 0_{7} & 0_{8} & I_{2n}\end{array} \right ], \\ B :=&\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} B_{11} & B_{12} & B_{13} & B_{14} & 0_{1} & B_{16} & B_{17} \\ B_{21} & B_{22} & B_{23} & 0_{1} & B_{25} & 0_{2} & B_{27} \\ 0_{4} & 0_{4} & I_{ ( n-1 ) } & 0_{4} & 0_{4} & 0_{5} & 0_{6} \\ 0_{1} & 0_{1} & 0_{1} & I_{n} & 0_{1} & 0_{2} & 0_{3} \\ 0_{1} & 0_{1} & 0_{1} & 0_{1} & I_{n} & 0_{2} & 0_{3} \\ 0_{4} & 0_{4} & 0_{5} & 0_{4} & 0_{4} & I_{ ( n-1 ) } & 0_{6} \\ 0_{7} & 0_{7} & 0_{8} & 0_{7} & 0_{7} & 0_{8} & \varPsi \end{array} \right ], \\ F :=&\left [ \begin{array}{@{}c@{\quad}c@{}} 0_{1} & 0_{1} \\ 0_{1} & 0_{1} \\ 0_{4} & 0_{4} \\ 0_{1} & 0_{1} \\ 0_{1} & 0_{1} \\ 0_{4} & 0_{4} \\ I_{n} & 0_{1} \\ 0_{1} & I_{n}\end{array} \right ], \end{aligned}$$

where 0₁, 0₂, 0₃, 0₄, 0₅, 0₆, 0₇, 0₈, 0₉ are zero matrices of dimensions n×n, n×(n−1), n×2n, (n−1)×n, (n−1)×(n−1), n×2n, 2n×n, 2n×(n−1) and 2n×1.

In order to obtain LQDG NKM, we assume that E _t z _1,t+1=z _1,t+1, i.e. that economic agents in the deterministic NKM make neither systematic nor random errors when predicting the future. Furthermore, substituting monetary and fiscal rules (6)–(7) in which deviation is possible into system (12)–(15) in the way presented above and performing similar transformations we obtain the system:

$$\begin{aligned} &{-}K_{y} ( I_{n}-R ) E_{t}y_{t+1}-GE \pi_{t+1}+K_{y}Ds_{t+1} \\ &\quad=- \bigl( I_{n}-E\varTheta_{y}^{F}+G \iota_{n}\theta_{y}^{U}\omega ^{T}-R \bigr) y_{t}- \bigl( G\iota_{n}\theta_{\pi}^{U} \omega ^{T}-E\varTheta _{\pi}^{F} \bigr) \pi_{t} \\ &\qquad{}+ Ds_{t}+ ( I_{n}-K_{y} ) ( I_{n}-R ) a_{t}- ( I_{n}-K_{y} ) Dc_{t}+I_{n}v_{t}^{y}+E \hat{f}_{t}-G\hat{\imath}_{t} \end{aligned}$$

(30)

$$\begin{aligned} &{-}K_{\pi}BE\pi_{t+1}=\varXi y_{t}- \pi_{t}+\varXi Vs_{t}+ ( I_{n}-K_{\pi } ) Bb_{t}+I_{n}v_{t}^{\pi} \end{aligned}$$

(31)

$$\begin{aligned} &{-}S\pi_{t+1}+s_{t+1}=s_{t} \end{aligned}$$

(32)

$$\begin{aligned} &a_{t+1}=y_{t} \end{aligned}$$

(33)

$$\begin{aligned} &b_{t+1}=\pi_{t} \end{aligned}$$

(34)

$$\begin{aligned} &c_{t+1}=s_{t} \end{aligned}$$

(35)

$$\begin{aligned} &v_{t+1}=\varPsi v_{t}, \end{aligned}$$

(36)

which, compared to the system (22)–(28), has two additional vectors of control variables $\hat{f}_{t}$ and $\hat {\imath}_{t}$. System (30)–(36) in state-space form can we written as:

$$ z_{t+1}=A^{-1}Bz_{t}+A^{-1}Cu_{t}, $$

(37)

where $u_{t}:= [ \hat{f}_{t}^{T}\ \hat{\imath}_{t} ] ^{T}$ and

$$ C:=\left [ \begin{array}{@{}c@{\quad}c@{}} E & -G\iota_{n} \\ 0_{1} & 0_{1} \\ 0_{1} & 0_{1} \\ 0_{1} & 0_{1} \\ 0_{4} & 0_{4} \\ 0_{7} & 0_{9}\end{array} \right ]. $$

1.5.2 A.5.2 Initial Condition Derivation

Initial condition z ₀ should position the system on the saddle-path so that the model would converge to the equilibrium. We propose two alternative ways of deriving this initial condition:

• 1. One way to obtain z ₀ which positions the system on the saddle-path is to solve the RE version of the model and then use the initial state obtained. This initial state, by definition (if RE-model is stable), meets the required condition because it positions the system on the saddle path. In particular, at t=0 vector of endogenous non-predetermined variables z _1,t will “jump” to a saddle path whereas vector of endogenous state (predetermined) variables z _2,t will have a value of 0. The initial value of shock vector v _t should follow the same assumptions made while solving the RE SNKM, i.e. its initial value should equal to standard deviation of ε _t . A number of freeware applications is available to solve RE model with DYNARE by Juillard (1996) being probably the most famous.³⁴

• 2. Another method to position the system on the saddle-path is to calculate the orthogonal projection of the shock v _t onto the stable subspace at time t=1. This method will be described below in more details.

Let

$$ z_{t+1}=\bar{A}z_{t},\quad z(0)=z_{0}, $$

(38)

be the deterministic NKM, where $\bar{A} :=A^{-1}B$.

Now, let $\bar{A}=SJS^{-1}$ be a Jordan decomposition of $\bar{A}$ such that $J=\operatorname{diag} ( \varLambda_{S}, \varLambda_{U} )$ and S=[S _S S _U ], where Λ _S contains all stable eigenvalues of $\bar{A}$ and Λ _U all unstable eigenvalues of $\bar{A}$ and S _S (S _U ) is the with Λ _S (Λ _U ) corresponding stable (unstable) subspace of $\bar{A}$. Then, if z ₀ belongs to S _S we have z(0)=S _S y for some y ($y= ( S_{S}^{T}S_{S} ) ^{-1}S_{S}^{T}z_{0}$).³⁵ In that case we may write:

$$\begin{aligned} z(t)& =e^{\bar{A}t}z(0)=Se^{Jt}S^{-1}S_{\bar{S}}y \\ &=Se^{Jt} \left [ \begin{array}{c} I \\ 0\end{array} \right ] y=S\left [ \begin{array}{c} e^{\varLambda_{S}t} \\ 0\end{array} \right ] y =S\left [ \begin{array}{c} e^{\varLambda_{S}t} ( S_{S}^{T}S_{S} ) ^{-1}S_{S}^{T}z_{0} \\ 0\end{array} \right ], \end{aligned}$$

which is the solution for t≥0. In our simulations we always consider such orthogonal projection of z onto the stable subspace at time t=1 as the initial condition $\tilde{z}_{0}$.

1.5.3 A.5.3 Change from a Discrete- to a Continuous-Time Model

Following Kwakernaak (1976) let the continuous time system be:

$$\begin{aligned} \dot{x}& =Ax+Bu, \\ y& =Cx+Du. \end{aligned}$$

Under the assumptions that u(t)=u(t _i ), t _i ≤t≤t _i+1 and Δ=t _i+1−t _i the equivalent discrete-time system is:

$$\begin{aligned} x(i+1)& =A_{Cl}x(i)+B_{d}u(i) \end{aligned}$$

(39)

$$\begin{aligned} y(i)& =C_{d}x(i)+D_{d}u(i), \end{aligned}$$

(40)

where

$$\begin{aligned} A_{Cl} =&e^{A\varDelta }, \\ B_{d} =& \biggl( \int_{0}^{\varDelta }e^{A\tau}d\tau \biggr) B, \\ C_{d} =&Ce^{A\varDelta } \quad\mbox{and} \\ D_{d} =&C \biggl( \int_{0}^{\varDelta }e^{A\tau}d\tau \biggr) B+D. \end{aligned}$$

Assuming Δ=1 we may rewrite the continuous time system in terms of discrete time system matrices as:

$$\begin{aligned} A& =\log A_{Cl}=\log ( I+A_{Cl}-I ) \approx A_{Cl}-I, \end{aligned}$$

(41)

$$\begin{aligned} B& = \biggl[ \int_{0}^{1}e^{A\tau}d \tau \biggr] ^{-1}B_{d}= \bigl( e^{A}-I \bigr) ^{-1}AB_{d}, \end{aligned}$$

(42)

$$\begin{aligned} C& =C_{d}e^{-A}, \end{aligned}$$

(43)

$$\begin{aligned} D& =D_{d}-C \biggl( \int_{0}^{\varDelta }e^{A\tau}d \tau \biggr) B \\ & =D_{d}-C \bigl[ A^{-1} \bigl( e^{A}-I \bigr) \bigr] B. \end{aligned}$$

(44)