Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4. A two-countr extended Mundell-Fleming model = ² YR r + ² YQ + ² YG [g + ηg ], = ² YR r ² YQ + ² YG [g + ηg], m p = ² MY ² MR r, m p = ² MY ² MR r, = ω N ² NW [w p], = ω N ² NW [w p ], w = w + λp C, w = w + λ p C, p C = ω + p +( α), p C = ω + p ( α), (T3.) (T3.2) (T3.3) (T3.4) (T3.5) (T3.6) (T3.7) (T3.8) (T3.9) (T3.) Notes: All variables except the interest rate are in logarithms and starred variables refer to the foreign countr. Endogenous variables are the outputs (, ), the real exchange rate (), therateofinterest (r ), price levels (p, p ), nominal wages (w, w ), and consumer price indexes (p C,p C ). Exogenous are government spending (g, g ),themonestocks(m, m ), and the wage targets (w,w). Note that ω log Ω. We compute the effects of fiscal and monetar polic under the various regimes.
2 2 Nominal wage rigidit in both countries In this case we have λ = λ =so that the model reduces to: = ² YR r + ² YQ + ² YG [g + ηg ], () = ² YR r ² YQ + ² YG [g + ηg], (2) m w = δ N ² MR r, (3) m w = δ N ² MR r, (4) = ω N ² NW [p w ], (5) = ω N ² NW [p w], (6) where we have alread used (5)-(6) to simplif (3)-(4) and where the composite parameter δ N is defined as follows: δ N +² MY ω N ² NW ω N ² NW. (7) Writing the sstem ()-(4) in one matrix euation gives: g N r = Γ g N m, (8) m where N and Γ N are defined as follows: ² YR ² YQ N ² YR ² YQ δ N ² MR, (9) δ N ² MR and: Γ N ² YG η² YG η² YG ² YG. () The determinant and inverse of N are easil computed: N =2δ N ² YQ (² MR + δ N ² YR ) >, () We have used Maple V to compute the determinant and inverse of. Of course, one can also use Cramer s Rule.
3 N N (2) δ N ² YQ² MR δ N² YQ² MR ² YQ[² MR +2δ N² YR] ² YQ² MR δ N ² YQ² MR δ N² YQ² MR ² YQ² MR ² YQ[² MR +2δ N² YR] δ 2 N ² YQ δ 2 N ² YQ δ N ² YQ δ N ² YQ. δ N [² MR + δ N ² YR] δ N [² MR + δ N ² YR] ² MR + δ N ² YR [² MR + δ N ² YR] 2. Fiscal polic 2.. Domestic fiscal polic Using (2) and the first column of () we find: r N ² YG η² YG δ N ( + η) ² YQ ² MR g = δ N ( + η) ² YQ ² MR N δ 2 N ( + η) ² YQ δ N ( η)[² MR + δ N ² YR ] ² YGg (3) B using () we can simplif and obtain: r ( + η) ² MR = ( + η) ² MR δ N ( + η) ( η)[² MR + δ N ² YR ] /² YQ ² YG g 2[² MR + δ N ² YR ]. (4) 2..2 Foreign fiscal polic Using(2)andthesecondcolumnof()wefind: η² YG r ² YG N g ( + η) ² MR = ( + η) ² MR ² YG g δ N ( + η) 2[² MR + δ N ² YR ]. (5) ( η)[² MR + δ N ² YR ] /² YQ Apart from the effect on the real exchange rate (which changes sign), all effects in (4) and (5) coincide.
4 2.2 Monetar polic 2.2. Domestic monetar polic Using (2) and the third column of () we find: ² YQ [² MR +2δ N ² YR ] r N m = ² YQ ² MR N δ N ² YQ ² MR + δ N ² YR m (6) B using () we can simplif and obtain: ² MR +2δ N ² YR r = ² MR m δ N 2δ N [² MR + δ N ² YR ]. (7) [² MR + δ N ² YR ] /² YQ 2.2.2 Foreign monetar polic Finall, using (2) and the fourth column of () we find: ² YQ ² MR r N m = ² YQ [² MR +2δ N ² YR ] N δ N ² YQ m (8) [² MR + δ N ² YR ] B using () we can simplif and obtain: ² MR r = ² MR +2δ N ² YR m δ N 2δ N [² MR + δ N ² YR ]. (9) [² MR + δ N ² YR ] /² YQ
5 3 Real wage rigidit in both countries In this case we have λ = λ =so that the model reduces to: = ² YR r + ² YQ + ² YG [g + ηg ], (2) = ² YR r ² YQ + ² YG [g + ηg], (2) = ω N ² NW [ω + w +( α)], (22) = ω N ² NW [ω + w ( α)], (23) p = m ² MY + ² MR r, (24) p = m ² MY + ² MR r, (25) where (2)-(23) determine the real uantities (,, r, and ) and (24)-(25) residuall determine the nominal uantities. (The model features a Classical dichotom.) In matrix format the real sstem can be written as: R " # r = Γ g R g, (26) where R and Γ R are now given b: ² YR ² YQ R ² YR ² YQ δ R, (27) δ R and: Γ R ² YG η² YG η² YG ² YG. (28) The composite parameter δ R is defined as: δ R ( α) ω N ² NW. (29) The determinant and inverse of R are: R =2² YR (δ R + ² YQ ), (3) δ R ² YR δ R ² YR ² YR [δ R +2² YQ ] δ R ² YR R δ R ² YR δ R ² YR δ R ² YR ² YR [δ R +2² YQ ] R δ R + ² YQ δ R + ² YQ (δ R + ² YQ ) (δ R + ² YQ ). (3) ² YR ² YR ² YR ² YR
6 3. Fiscal polic 3.. Domestic fiscal polic Using (3) and the first column of (28) we find: δ R ( η) r η R ² YGg = δ R ( η) ( + η)(δ R + ² YQ ) /² YR ( η) ² YG g 2(δ R + ² YQ ). (32) 3..2 Foreign fiscal polic Using(3)andthesecondcolumnof(28)wefind: η δ R ( η) r R ² YGg = δ R ( η) ( + η)(δ R + ² YQ ) /² YR ( η) ² YG g 2(δ R + ² YQ ). (33) 3.2 Monetar polic For obvious reasons, monetar polic does not affect an of the real variables. It is left as an exercise for the reader to determine the effects of monetar polic on price levels (and thus on the nominal exchange rate).
7 4 Mixed case Inthemixedcasewehaveλ =and λ =. The model can be written as: = ² YR r + ² YQ + ² YG [g + ηg ], (34) = ² YR r ² YQ + ² YG [g + ηg], (35) = δ R ω N ² NW [ω + w ], (36) m w = δ N ² MR r, (37) p = m ² MY + ² MR r, (38) p = + w ω N ². NW (39) Onl domestic mone is neutral. The real sstem, consisting of (34)-(37), can be written as follows: M g r = Γ M g, m (4) where M and Γ M are given b: ² YR ² YQ M ² YR ² YQ δ R δ N ² MR, (4) and: Γ M ² YG η² YG η² YG ² YG, (42) The determinant and inverse of M are: M = δ R (² MR + δ N ² YR )+² YQ (² MR +2δ N ² YR ), (43) M M δ R (² MR + δ N ² YR ) δ N δ R ² YR ² YQ (² MR +2δ N ² YR ) δ R ² YR ² YQ ² MR (δ R + ² YQ ) ² MR ² YQ ² MR ² YR (δ R +2² YQ ) ² YQ δ N (δ R + ² YQ ) δ N δ N ² YQ (δ R + ² YQ ) (² MR + δ N ² YR ) δ N ² YR ² MR + δ N ² YR ² YR (44)
8 4. Fiscal polic 4.. Domestic fiscal polic Using (44) and the first column of (42) we find: r η M ² YGg δ R [² MR +( η) δ N ² YR ] = ² MR [( + η) ² YQ + ηδ R ] δ N [( + η) ² YQ + ηδ R ] [² MR +( η) δ N ² YR ] ² YG g M. (45) 4..2 Foreign fiscal polic Using(44)andthesecondcolumnof(42)wefind: r η M ² YGg δ R [η² MR ( η) δ N ² YR ] = ² MR [( + η) ² YQ + δ R ] δ N [( + η) ² YQ + δ R ] ² YG g M. (46) ² MR +( η) δ N ² YR 4.2 Monetar polic 4.2. Domestic monetar polic Domestic monetar polic does not affect an of the real variables. It is left as an exercise to the reader to see how it affects the nominal price level and the nominal exchange rate.
9 4.2.2 Foreign monetar polic Using (44) and the third column of (42) we find: r M m δ R ² YR = ² YR (δ R +2² YQ ) m (δ R + ² YQ ) M. (47) ² YR
5 Fiscal polic multipliers In the text we represent the reduced-form expressions for the output levels b: = g + ζg = g + ζ g (48) (49) Implicitl we hold constant m, m, w, w,andω and normalize them to zero. We can then represent the various regimes in terms of ζ and ζ. 5. Nominal wage rigidit in both countries From (4) and (5) we find that: = [g + g ] ( + η) ² MR² YG 2[² MR + δ N ² YR ], (5) = [g + g ] ( + η) ² MR² YG 2[² MR + δ N ² YR ]. (5) Normalizing the common multiplier b unit, we can represent this case b ζ = ζ =. 5.2 Real wage rigidit in both countries From (32) and (33) we find that: = [g g ] ( η) δ R² YG 2(δ R + ² YQ ), (52) = [ g + g ] ( η) δ R² YG 2(δ R + ² YQ ). (53) Normalizing the common multiplier b unit, we can represent this case b ζ = ζ =. 5.3 Mixed case From (45) and (46) we find: = + ( η) δ N² YR ² MR ( + η) ²YQ = + η g + δ R g + η ( η) δ N² YR ² MR + ( + η) ² YQ δ R g δr ² MR ² YG, (54) M g δr ² MR ² YG. (55) M Inthetextweimplicitlassumethatforeignfiscal polic constitutes a beggar-th-neighbour polic, i.e. we assume that η² MR < ( η) δ N ² YR in (54). In the firststepwenormalizethe common multiplier appearing in (54)-(55), i.e. we set: δ R ² MR ² YG M =. (56)
In the second step, we transform our measure of g and g b writing: ḡ ḡ g + ( η)δ, (57) N ² YR ² MR g + (+η)² YQ δ R. (58) Using (56)-(58) we can then write (54)-(55) in the form of (48)-(49), though with ḡ and ḡ appearing, where ζ and ζ are defined as follows: ζ = η ( η)δ N ² YR ² MR + (+η)² <, (59) YQ δ R < ζ = (+η)² YQ δ R + η + ( η)δ N ² YR ² MR <. (6) Note that in the text we ignore this transformation and work directl with (59)-(6).