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. 500 350 400 300 300 Y 1 , Y 2 Y 1 , Y 2 Y 1 Y 1 200 Y 2 Y 2 250 100 0 200 0 50 100 150 200 0 50 100 150 200 250 t t Figure 1 : Before purchase Figure 2 : After purchase in country 1 2 / 39
Outline 1. Introduction 2. Model 3. Numerical simulations 4. Conclusion 3 / 39
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) 4 / 39
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 of business cycles. It is important to analyze the combination of the theory of OCA and the stability and the synchronization of business cycles. 5 / 39
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) 6 / 39
Model
Assumption Assumption 1 E = E e = ¯ E = 1 (1) Assumption 2 p i = 1 (2) The subscript i ( i = 1 , 2 ): the index number of a country E : exchange rate E e : expected exchange rate of the near future p i : price level of country i 7 / 39
Government bond B i = B i i + B j i + θ i , (3) B i = ˙ ˙ i + ˙ B j i + ˙ B i θ i (4) B i : outstanding nominal government bonds of country i B i i : outstanding nominal government bonds of country i held by a private sector in country i B j i : outstanding nominal government bonds of country i held by a private sector in country j θ i : outstanding government bonds of coutry i held by the supranational central bank system that is included central banks of each country 8 / 39
Budget constraint B i = G i + B i r i − T i − ˙ ˙ θ i (5) G i : real government expenditure r i : nominal rate of interest T i : real income tax 9 / 39
Net exports, capital account, and total balance J 1 + J 2 = 0 , (6) Q 1 + Q 2 = 0 , (7) A 1 + A 2 = 0 , (8) A i = J i + Q i (9) Y i = ∂J i Y j = ∂J i J i = J i ( Y i , Y j ) ; J i < 0 , J i > 0 , (10) ∂Y i ∂Y j J i : real net exports Q i : real capital account balance A i : real total balance Y i : real net national income 10 / 39
Money supply M = M 1 + M 2 , (11) ˙ M = ˙ θ 1 + ˙ (12) θ 2 , ˙ M i = A i + ˙ θ i , (13) M i = L i ( Y i , r i ) ; ∂L i r i = ∂L i > 0 , L i < 0 , (14) ∂Y i ∂r i M : nominal money supply in the whole of monetary union M i : nominal money supply L i : demand for money 11 / 39
Capital account balance function ˙ B 2 1 = β ( r 1 − r 2 ) ; β > 0 , (15) ˙ B 1 2 = β ( r 2 − r 1 ) , (16) ˙ 1 = − ˙ B 2 B 1 2 = β ( r 1 − r 2 ) , (17) 2 = ¯ B 2 1 + B 1 (18) D, Q 1 = ˙ 1 − ˙ B 2 B 1 2 + r 2 B 1 2 − r 1 B 2 1 = β ( r 1 − r 2 ) − β ( r 2 − r 1 ) + r 2 B 1 2 − r 1 B 2 1 = 2 β ( r 1 − r 2 ) + r 2 B 1 2 − r 1 B 2 (19) 1 β : degree of mobility of international capital flows 12 / 39
Disequilibrium quantity adujustment process, consumption, and tax ˙ Y i = α i [ C i + I i + G i + J i − Y i ] ; α i > 0 , (20) C i = c i ( Y i + r i B i i + r j B i j − T i ) + C 0 i ; 0 < c i < 1 , C 0 i ≥ 0 , (21) T i = τ i ( Y i + r i B i i + r j B i j ) − T 0 i ; 0 < τ i < 1 , T 0 i ≥ 0 , (22) Y i : real net national income C i : real private consumption expenditure α : adjustment speed of the goods market c i : marginal propensity to consume C 0 i : basic consumption I i : real net private investment expenditure G i : real government expenditure τ i : marginal tax rate T 0 i : negative income tax (or basic income) 13 / 39
Investment, capital stock, and government expenditure Y i = ∂I i K i = ∂I i r i = ∂I i I i = I i ( Y i , K i , r i ) ; I i > 0 , I i < 0 , I i < 0 , (23) ∂Y i ∂K i ∂r i ˙ K i = I i , (24) G i = G 0 i + γ i ( ¯ Y i − Y i ) ; γ i > 0 , (25) K i : real capital stock G 0 i : basic government expenditure γ i : degree of counter-cyclical fiscal policy ¯ Y i : the level of real national income that a government determine the counter-cyclical government expenditure (this is not necessarily natural output) 14 / 39
Assumption Assumption 3 θ 1 = ˙ ˙ θ 2 = 0 (26) ¯ M = M 1 + M 2 (27) Therefore, we transform Eqs. (5), (12) and (13) into the following equations. ˙ B i = G i + B i r i − T i , (28) ˙ M = 0 (29) ˙ M i = A i (30) 15 / 39
Eight-dimensional dynamical system (i) ˙ Y 1 = α 1 [ { c 1 (1 − τ 1 ) − 1 } Y 1 1 ) r 1 ( Y 1 , M 1 ) + ( ¯ 1 ) r 2 ( Y 2 , ¯ + c 1 (1 − τ 1 ) { ( B 1 − θ 1 − B 2 D − B 2 M − M 1 ) } + c 1 T 01 + C 01 + G 01 + γ 1 ( ¯ Y 1 − Y 1 ) + I 1 ( Y 1 , K 1 , r 1 ( Y 1 , M 1 )) + J 1 ( Y 1 , Y 2 )] = F 1 ( Y 1 , K 1 , B 1 , B 2 1 , Y 2 , M 1 ; α 1 , γ 1 , θ 1 ) , (31) ˙ K 1 = I 1 ( Y 1 , K 1 , r 1 ( Y 1 , M 1 )) = F 2 ( Y 1 , K 1 , M 1 ) , (32) B 1 = G 01 + γ 1 ( ¯ ˙ Y 1 − Y 1 ) + B 1 r 1 ( Y 1 , M 1 ) 1 ) r 1 ( Y 1 , M 1 ) + ( ¯ 1 ) r 2 ( Y 2 , ¯ − τ 1 { Y 1 + ( B 1 − θ 1 − B 2 D − B 2 M − M 1 ) } + T 01 = F 3 ( Y 1 , B 1 , B 2 1 , Y 2 , M 1 ; γ 1 , θ 1 ) , (33) ˙ 1 = β { r 1 ( Y 1 , M 1 ) − r 2 ( Y 2 , ¯ B 2 M − M 1 ) } = F 4 ( Y 1 , Y 2 , M 1 ; β ) , (34) 16 / 39
Eight-dimensional dynamical system (ii) ˙ Y 2 = α 2 [ { c 2 (1 − τ 2 ) − 1 } Y 2 + c 2 (1 − τ 2 ) { ( B 2 − θ 2 − ( ¯ 1 )) r 2 ( Y 2 , ¯ D − B 2 M − M 1 ) + B 2 1 r 1 ( Y 1 , M 1 ) } + c 2 T 02 + C 02 + G 02 + γ 2 ( ¯ Y 2 − Y 2 ) + I 2 ( Y 2 , K 2 , r 2 ( Y 2 , ¯ M − M 1 )) − J 1 ( Y 1 , Y 2 )] = F 5 ( Y 1 , B 2 1 , Y 2 , K 2 , B 2 , M 1 ; α 2 , γ 2 , θ 2 ) , (35) ˙ K 2 = I 2 ( Y 2 , K 2 , r 2 ( Y 2 , M 1 )) = F 6 ( Y 2 , K 2 , M 1 ) , (36) B 2 = G 02 + γ 2 ( ¯ ˙ Y 2 − Y 2 ) + B 2 r 2 ( Y 2 , M 1 ) − τ 2 { Y 2 + ( B 2 − θ 2 − ( ¯ D − B 2 1 )) r 2 ( Y 2 , M 1 ) + B 2 1 r 1 ( Y 1 , M 1 ) } + T 02 = F 7 ( Y 1 , B 2 1 , Y 2 , B 2 , M 1 ; γ 2 , θ 2 ) , (37) M 1 = J 1 ( Y 1 , Y 2 ) + 2 β { r 1 ( Y 1 , M 1 ) − r 2 ( Y 2 , M 1 ) } + ( ¯ ˙ D − B 2 1 ) r 2 ( Y 2 , M 1 ) − B 2 1 r 1 ( Y 1 , M 1 ) = F 8 ( Y 1 , B 2 1 , Y 2 , M 1 ; β ) (38) 17 / 39
Jacobian matrix of the system of Eqs. (31)–(38) F 11 F 12 F 13 F 14 F 15 0 0 F 18 0 0 0 0 0 F 21 F 22 F 28 F 31 0 F 33 F 34 F 35 0 0 F 38 F 41 0 0 0 F 45 0 0 F 48 J = (39) F 51 0 0 F 54 F 55 F 56 F 57 F 58 0 0 0 0 F 65 F 66 0 F 68 0 0 0 F 71 F 74 F 75 F 77 F 78 F 81 0 0 F 84 F 85 0 0 F 88 18 / 39
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 one of the parameters θ 1 , θ 2 , γ 1 and γ 2 is sufficiently large . (ii) Suppose that the parameter θ 1 , θ 2 , γ 1 and γ 2 are relatively small and inequalities F 11 > 0 and F 55 > 0 hold. Then, the equilibrium point of the system (31)–(38) is locally unstable if the parameter β is sufficiently large . 19 / 39
Numerical simulations
Simulations: assumption Based on Asada (2004), assume the following parameter values. c i = 0 . 8 , τ i = 0 . 2 , T 0 i = 10 , C 01 = 20 , C 02 = 40 , M = 600 , ¯ ¯ Y i = 500 , ¯ G 01 = 50 , G 02 = 60 , D = 2 , α i = 1 , β = 5 √ r i = 15 Y i − M i , (40) √ I i = 20 Y i − 0 . 3 K i − r i , (41) J 1 = − 0 . 35 Y 1 + 0 . 25 Y 2 (42) Compute the trajectories by selecting several values of θ 1 , θ 2 , γ 1 and γ 2 and the following initial conditions of the variables: Y 1 (0) = 190 , K 1 (0) = 1179 , B 1 (0) = 1 . 6 , B 2 1 (0) = 1 , Y 2 (0) = 280 , K 2 (0) = 1351 , B 2 (0) = 0 . 9 , M 1 (0) = 280 20 / 39
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