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Business cycles and government bond purchase by central banks in monetary union

Masato Nakao

Faculty of Commerce and Economics, Chiba University of Commerce

Toichiro Asada

Faculty of Economics, Chuo University 11th Nonlinear Economic Dynamics NED 2019, Kyiv, Ukraine Kyiv School of Economics September 5, 2019 1 / 39

Significance of Research

Purchasing government bond of a certain country can stabilize unsynchronized business cycles of two countries in a monetary union.

50 100 150 200 100 200 300 400 500 t Y1,Y2 Y1 Y2

Figure 1: Before purchase

50 100 150 200 250 200 250 300 350 t Y1,Y2 Y1 Y2

Figure 2: After purchase in country 1

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Outline

• 1. Introduction
• 2. Model
• 3. Numerical simulations
• 4. Conclusion

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Introduction

Background

The Euro area brought the euro crisis to an end by declaring an introduction of Outright Monetary Transactions (OMT). While the Euro area is not an optimum currency area (OCA), the Euro area exists today.

▶ Basis: Mundell (1961), McKinnon (1963), and Kenen (1969) ▶ Survey: Mongelli (2002), Baldwin and Wyplosz (2015), and De Grauwe (2018)

The theory of OCA focus on the synchronization of business cycles in a monetary union.

▶ Frankel and Rose (1996), G¨ achter et al. (2012), and De Grauwe and Ji (2016, 2017)

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Problems

Several studies have proved that the synchronization of business cycles is increased before introduction of the euro, while it is decreased after the introduction.

▶ Before: Altavilla (2004), Camacho et al. (2006), and Darvas and Szap´ ary (2008), ▶ After: Weyerstraß et al. (2011) and G¨ achter et al. (2012)

However, what seems to be lacking is an analysis on how the purcahase of government bonds by central banks in a monetary uninon affects the stability and the synchronization

• f business cycles.

It is important to analyze the combination of the theory of OCA and the stability and the synchronization of business cycles.

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Purpose and method

Purpose

▶ To discuss whether buying a government bond of a country in a monetary union stabilize the business cycles of several countries in a monetary union. ▶ To analyze whether purchasing a government bond of a coutnry stabilize the business cycles even if the business cycles are not synchronized.

Method

▶ Kaldorian two-country model with a monetary union and imperfect capital mobility ▶ Relevant research: Asada, Inaba and Misawa (2001), Asada, Chiarella, Flaschel and Franke (2003), and Asada (2004)

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Model

Assumption

Assumption 1 E = Ee = ¯ E = 1 (1) Assumption 2 pi = 1 (2)

The subscript i (i = 1, 2): the index number of a country E: exchange rate Ee: expected exchange rate of the near future pi: price level of country i

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Government bond

Bi = Bi

i + Bj i + θi,

(3) ˙ Bi = ˙ Bi

i + ˙

Bj

i + ˙

θi (4)

Bi: outstanding nominal government bonds of country i Bi

i: outstanding nominal government bonds of country i held by a private sector in

country i Bj

i : outstanding nominal government bonds of country i held by a private sector in

country j θi: outstanding government bonds of coutry i held by the supranational central bank system that is included central banks of each country

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Budget constraint

˙ Bi = Gi + Biri − Ti − ˙ θi (5)

Gi: real government expenditure ri: nominal rate of interest Ti: real income tax

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Net exports, capital account, and total balance

J1 + J2 = 0, (6) Q1 + Q2 = 0, (7) A1 + A2 = 0, (8) Ai = Ji + Qi (9) Ji = Ji(Yi, Yj) ; Ji

Yi = ∂Ji

∂Yi < 0, Ji

Yj = ∂Ji

∂Yj > 0, (10)

Ji: real net exports Qi: real capital account balance Ai: real total balance Yi: real net national income

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Money supply

M = M1 + M2, (11) ˙ M = ˙ θ1 + ˙ θ2, (12) ˙ Mi = Ai + ˙ θi, (13) Mi = Li(Yi, ri) ; ∂Li ∂Yi > 0, Li

ri = ∂Li

∂ri < 0, (14)

M: nominal money supply in the whole of monetary union Mi: nominal money supply Li: demand for money

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Capital account balance function

˙ B2

1 = β(r1 − r2) ; β > 0,

(15) ˙ B1

2 = β(r2 − r1),

(16) ˙ B2

1 = − ˙

B1

2 = β(r1 − r2),

(17) B2

1 + B1 2 = ¯

D, (18) Q1 = ˙ B2

1 − ˙

B1

2 + r2B1 2 − r1B2 1

= β(r1 − r2) − β(r2 − r1) + r2B1

2 − r1B2 1

= 2β(r1 − r2) + r2B1

2 − r1B2 1

(19)

β: degree of mobility of international capital flows

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Disequilibrium quantity adujustment process, consumption, and tax

˙ Yi = αi[Ci + Ii + Gi + Ji − Yi] ; αi > 0, (20) Ci = ci(Yi + riBi

i + rjBi j − Ti) + C0i ; 0 < ci < 1, C0i ≥ 0,

(21) Ti = τi(Yi + riBi

i + rjBi j) − T0i ; 0 < τi < 1, T0i ≥ 0,

(22)

Yi: real net national income Ci: real private consumption expenditure α: adjustment speed of the goods market ci: marginal propensity to consume C0i: basic consumption Ii: real net private investment expenditure Gi: real government expenditure τi: marginal tax rate T0i: negative income tax (or basic income)

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Investment, capital stock, and government expenditure

Ii = Ii(Yi, Ki, ri) ; Ii

Y i = ∂Ii

∂Yi > 0, Ii

Ki = ∂Ii

∂Ki < 0, Ii

ri = ∂Ii

∂ri < 0, (23) ˙ Ki = Ii, (24) Gi = G0i + γi( ¯ Yi − Yi) ; γi > 0, (25)

Ki: real capital stock G0i: basic government expenditure γi: degree of counter-cyclical fiscal policy ¯ Yi: the level of real national income that a government determine the counter-cyclical government expenditure (this is not necessarily natural output)

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Assumption

Assumption 3 ˙ θ1 = ˙ θ2 = 0 (26) ¯ M = M1 + M2 (27) Therefore, we transform Eqs. (5), (12) and (13) into the following equations. ˙ Bi = Gi + Biri − Ti, (28) ˙ M = 0 (29) ˙ Mi = Ai (30)

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Eight-dimensional dynamical system (i)

˙ Y1 =α1[{c1(1 − τ1) − 1}Y1 + c1(1 − τ1){(B1 − θ1 − B2

1)r1(Y1, M1) + ( ¯

D − B2

1)r2(Y2, ¯

M − M1)} + c1T01 + C01 + G01 + γ1( ¯ Y1 − Y1) + I1(Y1, K1, r1(Y1, M1)) + J1(Y1, Y2)] =F1(Y1, K1, B1, B2

1, Y2, M1; α1, γ1, θ1),

(31) ˙ K1 =I1(Y1, K1, r1(Y1, M1)) = F2(Y1, K1, M1), (32) ˙ B1 =G01 + γ1( ¯ Y1 − Y1) + B1r1(Y1, M1) − τ1{Y1 + (B1 − θ1 − B2

1)r1(Y1, M1) + ( ¯

D − B2

1)r2(Y2, ¯

M − M1)} + T01 =F3(Y1, B1, B2

1, Y2, M1; γ1, θ1),

(33) ˙ B2

1 =β{r1(Y1, M1) − r2(Y2, ¯

M − M1)} =F4(Y1, Y2, M1; β), (34)

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Eight-dimensional dynamical system (ii)

˙ Y2 =α2[{c2(1 − τ2) − 1}Y2 + c2(1 − τ2){(B2 − θ2 − ( ¯ D − B2

1))r2(Y2, ¯

M − M1) + B2

1r1(Y1, M1)}

+ c2T02 + C02 + G02 + γ2( ¯ Y2 − Y2) + I2(Y2, K2, r2(Y2, ¯ M − M1)) − J1(Y1, Y2)] =F5(Y1, B2

1, Y2, K2, B2, M1; α2, γ2, θ2),

(35) ˙ K2 =I2(Y2, K2, r2(Y2, M1)) = F6(Y2, K2, M1), (36) ˙ B2 =G02 + γ2( ¯ Y2 − Y2) + B2r2(Y2, M1) − τ2{Y2 + (B2 − θ2 − ( ¯ D − B2

1))r2(Y2, M1) + B2 1r1(Y1, M1)} + T02

=F7(Y1, B2

1, Y2, B2, M1; γ2, θ2),

(37) ˙ M1 =J1(Y1, Y2) + 2β{r1(Y1, M1) − r2(Y2, M1)} + ( ¯ D − B2

1)r2(Y2, M1)

− B2

1r1(Y1, M1)

=F8(Y1, B2

1, Y2, M1; β)

(38)

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Jacobian matrix of the system of Eqs. (31)–(38)

J =                F11 F12 F13 F14 F15 F18 F21 F22 F28 F31 F33 F34 F35 F38 F41 F45 F48 F51 F54 F55 F56 F57 F58 F65 F66 F68 F71 F74 F75 F77 F78 F81 F84 F85 F88                (39)

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Results from model analysis

Proposition 1 (i) Suppose that the parameter β is fixed at any level. Then, the equilibrium point of the system (31)–(38) is locally stable if at least

• ne of the parameters θ1, θ2, γ1 and γ2 is sufficiently large.

(ii) Suppose that the parameter θ1, θ2, γ1 and γ2 are relatively small and inequalities F11 > 0 and F55 > 0 hold. Then, the equilibrium point of the system (31)–(38) is locally unstable if the parameter β is sufficiently large.

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Numerical simulations

Simulations: assumption

Based on Asada (2004), assume the following parameter values. ci = 0.8, τi = 0.2, T0i = 10, C01 = 20, C02 = 40, G01 = 50, G02 = 60, ¯ M = 600, ¯ Yi = 500, ¯ D = 2, αi = 1, β = 5 ri = 15 √ Yi − Mi, (40) Ii = 20 √ Yi − 0.3Ki − ri, (41) J1 = −0.35Y1 + 0.25Y2 (42) Compute the trajectories by selecting several values of θ1, θ2, γ1 and γ2 and the following initial conditions of the variables: Y1(0) = 190, K1(0) = 1179, B1(0) = 1.6, B2

1(0) = 1,

Y2(0) = 280, K2(0) = 1351, B2(0) = 0.9, M1(0) = 280

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Business cycles under unstable economy

50 100 150 200 100 200 300 400 500 t Y1,Y2 Y1 Y2 Note: β = 5, θi = 0, γi = 0.35

Figure 3: High degree of capital movement and unsynchronized business cycles

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Purchasing government bond of country 1

50 100 150 200 250 200 250 300 350 t Y1,Y2 Y1 Y2 Note: β = 5, θ1 = 0.6, θ2 = 0, γi = 0.35

Figure 4: Convergence by purchasing government bond of country 1

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Purchasing government bond of country 2

200 400 600 800 1000 1200 200 250 300 350 400 t Y1,Y2 Y1 Y2 Note: β = 5, θ1 = 0, θ2 = 0.6, γi = 0.35

Figure 5: Convergence by purchasing government bond of country 2

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Purchasing government bond of both countries

50 100 150 200 240 260 280 300 320 340 360 t Y1,Y2 Y1 Y2 Note: β = 5, θ1 = 0.3, θ2 = 0.3, γi = 0.35

Figure 6: Convergence by purchasing government bond of country 2

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Counter-cyclical fiscal policy

20 40 60 80 100 120 140 240 260 280 300 320 340 360 t Y1,Y2 Y1 Y2 Note: β = 5, θ1 = 0, θ2 = 0, γi = 0.5

Figure 7: Counter-cyclical fiscal policy can stabilize business cycles fluctuations

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Purchasing government bond of both countries and counter- cyclical fiscal policy

20 40 60 80 100 250 300 350 t Y1,Y2 Y1 Y2

Note: β = 5, θ1 = 0.3, θ2 = 0.3, γi = 0.5

Figure 8: Counter-cyclical fiscal policy can stabilize business cycles fluctuations

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Results of simulations

Purchasing government bonds stabilize business cycles. Purchasing a government bonds stabilize business cycles in a monetary union even if business cycles are not synchronized. The combination of buying government bonds and a counter-cyclical fiscal policy stabilize business cycles more rapidly.

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Conclusion

Conclusion

A policy such as OMT that buy a bond of a country in a crisis contributes to the stabilization of business cycles. While the synchronization of business cycles is attracting attention in the theory of OCA, it is important that the purchase of government bonds by central banks can stabilize the cycle even if they are not synchronized. One element for the euro area to become the OCA was acquired through measures against the euro crisis.

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Future plan

Analyze the stability of the business cycle using a model with variable prices, in order to consider a policy aimed at influencing prices, such as quantitative easing.

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Thank you very much for your kind attention.

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References i

Altavilla, C. (2004) “Do EMU Members Share the Same Business Cycle?” Journal of Common Market Studies, Vol. 42, No. 5, pp. 869-896. Asada, T. (2004) “A Two-Regional Model of Business Cycles with Fixed Exchange Rates: A Kaldorian Approach,” Studies in Regional Science, Vol. 34, No. 2, pp. 19-38. Asada, T., T. Inaba, and T. Misawa (2001) “An Interregional Dynamic Model: The Case of Fixed Exchange Rates,” Studies in Regional Science, Vol. 31, No. 2, pp. 29-41. Asada, T., C. Chiarella, P . Flaschel, and R. Franke (2003) Open Economy Macrodynamics: An Integrated Disequilibrium Approach, Berlin: Springer-Verlag Berlin Heidelberg. Baldwin, R E. and C Wyplosz (2015) The Economics of European Integration: McGraw Hill, 4th edition. Camacho, M., G. Perez-Quiros, and L. Saiz (2006) “Are European Business Cycles Close Enough to Be Just One?” Journal of Economic Dynamics and Control, Vol. 30, No. 9-10, pp. 1687-1706. Darvas, Z. and G. Szap´ ary (2008) “Business Cycle Synchronization in the Enlarged EU,” Open Economies Review, Vol. 19, No. 1, pp. 1-19. De Grauwe, P . (2018) Economics of Monetary Union: Oxford University Press, 12th edition. 31 / 39

References ii

De Grauwe, P . and Y. Ji (2016) “Flexibility Versus Stability: A Difficult Tradeoff in the Eurozone,” Credit and Capital Markets – Kredit und Kapital, Vol. 49, No. 3, pp. 375-413. (2017) “The International Synchronisation of Business Cycles: The Role of Animal Spirits,” Open Economies Review, Vol. 28, No. 3, pp. 383-412. Frankel, J. and A. Rose (1996) “The Endogeneity of the Optimum Currency Area Criteria,” The Economic Journal, Vol. 108, No. July, pp. 1009-1025, DOI: http://dx.doi.org/10.3386/w5700. G¨ achter, M., A. Riedl, and D. Ritzberger-Gr¨ unwald (2012) “Business Cycle Synchronization in the Euro Area and the Impact of the Financial Crisis,” Monetary Policy & the Economy, Vol. 2, pp. 33-60. Kenen, P B. (1969) “The Theory of Optimum Currency Areas: An Eclectic View,” in Monetary Problems of International Economy: The University of Chicago Press, pp. 41-60. McKinnon, R I. (1963) “Optimum Currency Areas,” The American Economic Review, Vol. 53, No. 4, pp. 717-725. Mongelli, F P . (2002) “”New” Views on the Optimum Currency Area Theory: What Is EMU Telling Us?” European Central Bank Workig Paper Series, No. 138. Mundell, R. (1961) “A Theory of Optimum Currency Areas,” American Economic Review, Vol. 51,

• pp. 657-665.

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References iii

Weyerstraß, K., B. van Aarle, M. Kappler, and A. Seymen (2011) “Business Cycle Synchronisation with(in) the Euro Area: In Search of a ‘Euro Effect’,” Open Economies Review, Vol. 22, No. 3,

• pp. 427-446.

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Business cycles of Unstable economy

50 100 150 200

• 2
• 1

1 2 3 4 t B1,B2,B1

2

B1 B2 B1

2

Note: β = 5, θi = 0, γi = 0

Figure 9: Not synchronized business cycles

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Purchasing government bond of country 1

50 100 150 200 250 1.0 1.5 2.0 2.5 3.0 3.5 t B1,B2,B1

2

B1 B2 B1

2

Note: β = 5, θ1 = 0.6, θ2 = 0, γi = 0.35

Figure 10: Convergence by purchasing government bond of country 1

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Purchasing government bond of country 2

200 400 600 800 1000 1200 1400 1 2 3 4 t B1,B2,B1

2

B1 B2 B1

2

Note: β = 5, θ1 = 0, θ2 = 0.6, γi = 0.35

Figure 11: Convergence by purchasing government bond of country 2

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Purchasing government bond of both countries

50 100 150 200 1.0 1.5 2.0 2.5 3.0 t B1,B2,B1

2

B1 B2 B1

2

Note: β = 5, θ1 = 0.3, θ2 = 0.3, γi = 0.35

Figure 12: Convergence by purchasing government bond of country 2

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Counter-cyclical fiscal policy

20 40 60 80 100 120 140 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t B1,B2,B1

2

B1 B2 B1

2

Note: β = 5, θ1 = 0, θ2 = 0, γi = 0.5

Figure 13: Counter-cyclical fiscal policy can stabilize business cycles fluctuations

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Purchasing government bond of both countries and counter- cyclical fiscal policy

20 40 60 80 100 120 140 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t B1,B2,B1

2

B1 B2 B1

2

Note: β = 5, θ1 = 0.3, θ2 = 0.3, γi = 0.5

Figure 14: Counter-cyclical fiscal policy can stabilize business cycles fluctuations

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