Advanced (International) Macroeconomics

Transcription

1 Advanced (International) Macroeconomics Hartmut Egger University of Bayreuth Fall 2015 Hartmut Egger Advanced (International) Macroeconomics 1 of 114

2 Table of Contents 1 Intertemporal Trade and Current Account (O&R[1,2]) 2-Period Model Equilibrium in an endowment economy Equilibrium in a closed economy Equilibrium in a small open economy Intertemporal trade pattern, savings and international equilibrium Capital accumulation and production Closed economy with capital accumulation and production Small open economy with capital accumulation and production Adding government consumption Investment, savings and world interest rate in international equilibrium Hartmut Egger Advanced (International) Macroeconomics 2 of 114

3 Table of Contents 2 Balassa-Samuelson effect in economy with traded and non-traded goods (O&R[4]) Real exchange rate and purchasing power parity A small economy with tradeable and non-tradable goods Effects of productivity growth or interest rate shifts The Harrod-Balassa-Samuelson effect Production consumption and current account 3 Nominal interest rates and nominal exchange rate (R[1]) Structure of financial wealth and balance of payments Uncovered interest rate parity Foreign exchange market with imperfect capital mobility Forward market and covered interest rate parity Hartmut Egger Advanced (International) Macroeconomics 3 of 114

4 Table of Contents 4 Money and exchange rates (O&R[8,9]) Interaction of goods-, factor- and financial markets Money and exchange rates under full international arbitrage and fully flexible prices Nominal rigidities: Mundell-Fleming-Dornbusch Model Hartmut Egger Advanced (International) Macroeconomics 4 of 114

5 Intertemporal Trade and Current Account Hartmut Egger Advanced (International) Macroeconomics 5 of 114

6 2-Period Model Hartmut Egger Advanced (International) Macroeconomics 6 of 114

7 Within periods: Inter- and intraindustry trade (see trade theory from Ricardo, Heckscher/Ohlin to new trade theories with imperfect markets). Gains from international labor division (comparative advantages), exploitation of economies of scale or intensified competition. Across periods: Intertemporal trade. Gains from international borrowing and lending. Hartmut Egger Advanced (International) Macroeconomics 7 of 114

8 Intertemporal choice - preferences U = u(c 1 )+βu(c 2 ), 0 < β < 1 (1) u period (instantaneous) utility u > 0,u < 0,lim c 0 u (c) = β subjective discount factor (time-preference parameter) Marginal rate of substitution ( MRS dc ) 2 dc 1 U=const = u (c 1 ) βu (c 2 ) (2) Hartmut Egger Advanced (International) Macroeconomics 8 of 114

9 C 2 Intertemporal indifference curve Figure 1: Intertemporal indifference curve C 1 Hartmut Egger Advanced (International) Macroeconomics 9 of 114

10 Example utility functions Example 1: u(c t ) = lnc t Example 2: u(c t ) = c1 1/σ t 1 1/σ (Isoelastic utility functions), 0 < σ( 1). Hartmut Egger Advanced (International) Macroeconomics 10 of 114

11 Intertemporal budget constraint c 1 + c 2 1+r = y 1 + y 2 1+r (3) r interest rate (if there are variable interest rates, r t+1 denotes c 2 interest rate from t to t +1. Then etc.). 1+r 2 y t endowment (income) in period t. Hartmut Egger Advanced (International) Macroeconomics 11 of 114

12 C 2 (1+r)y 1 +y 2 y 2 A (Endowment point) 1+r y 1 y 1 + y 2 Figure 2: Intertemporal budget constraint 1+r C 1 Hartmut Egger Advanced (International) Macroeconomics 12 of 114

13 Optimal intertemporal choice Lagrange-Function: L = u(c 1 )+βu(c 2 )+λ max U s.t. (3) ( y 1 + y 2 1+r c 1 c ) 2 1+r Hartmut Egger Advanced (International) Macroeconomics 13 of 114

14 First-order conditions L = 0 = u (c 1 ) = λ c 1 L = 0 = βu (c 2 ) = λ c 2 1+r L = 0 = (3) λ combine to the so-called intertemporal Euler equation: u (c 1 ) = (1+r)βu (c 2 ) (4) Eq. (4) determines how consumption needs to be allocated intertemporally in order to maximize utility at a given interest rate r. Hartmut Egger Advanced (International) Macroeconomics 14 of 114

15 Interpretation of Euler Equation The intertemporal choice is optimal if there are no gains from intertemporal reallocation. Equation (4) is equivalent to where MRS is given by (2) MRS = 1+r (5) Hartmut Egger Advanced (International) Macroeconomics 15 of 114

16 Equilibrium in an endowment economy Hartmut Egger Advanced (International) Macroeconomics 16 of 114

17 Endowment economies have no capital accumulation and no production. Aggregate supply (with symmetric agents): Y t = y t N t, t = 1,2 where N t is the population size in period t. In the following, N t is normalized to 1 so that Y t = y t. Aggregate demand: C t = c t, t = 1,2 Hartmut Egger Advanced (International) Macroeconomics 17 of 114

18 In closed economy (autarky): In open economy: C t = Y t, t = 1,2 (6) C t = Y t +r t B t CA t (7) where B t is the value of net foreign assets inherited from period t 1 and CA t is the current account balance (Ertragsbilanz, auch Leistungsbilanz). By definition, CA t = B t+1 B t (8) Hartmut Egger Advanced (International) Macroeconomics 18 of 114

19 Remarks on national accounting Gross domestic product (GDP) (Bruttoinlandsprodukt): Y t Gross national product (GNP) (Bruttosozialprodukt): Y t +r t B t i.e. GNP=GDP+net international factor payments net international factor payments here only includes interest and dividend earnings on net foreign assets but no workers remittances Trade balance (goods and services): Net exports NX t Hartmut Egger Advanced (International) Macroeconomics 19 of 114

20 Remarks on national accounting Capital account balance (Kapitalverkehrsbilanz, auch Kapitalbilanz; includes financial account balance): Net sales of foreign assets: (B t+1 B t ) Balance of payments (Zahlungsbilanz): NX t +r t B t = B t+1 B t Current account balance (Ertragsbilanz, auch Leistungsbilanz): within period perspective: CA t = NX t +r t B t (9) intertemporal perspective: CA t = B t+1 B t (10) Hartmut Egger Advanced (International) Macroeconomics 20 of 114

21 Remarks on national accounting 2013 current account balance in % of GDP Source: Worldbank database Hartmut Egger Advanced (International) Macroeconomics 21 of 114

22 Remarks on national accounting C32 Euro area b.o.p.: current account (seasonally adjusted; 12-month cumulated transactions as a percentage of GDP) current account balance Source: Euro Area statistics online Hartmut Egger Advanced (International) Macroeconomics 22 of 114

23 Remarks on national accounting C33 Euro area b.o.p.: direct and portfolio investment (12-month cumulated transactions as a percentage of GDP) net direct investment net portfolio investment Source: Euro Area Statistics online Hartmut Egger Advanced (International) Macroeconomics 23 of 114

24 Remarks on national accounting "%$ #$ %$ &%%#$ &%%'$ &%%($ &%%)$ &%%*$ &%"%$ &%""$ &%"&$ &%"+$ &%",$!#$!"%$!"#$ -./$ 012$ 023$ 42-$ 5/-$ 526$ 178$ 82/$ Current account figures for selected Euro member states Source: World Development Indicators Hartmut Egger Advanced (International) Macroeconomics 24 of 114

25 Remarks on national accounting Balance of payments of Germany Source: Deutsche Bundesbank: Monthly Report, March 2015 Hartmut Egger Advanced (International) Macroeconomics 25 of 114

26 billion Remarks on national accounting The Bundesbank's TARGET2 balance Year on year change, net capital exports + year end levels in billion Source: Deutsche Bundesbank: Monthly Reports, March 2015 Hartmut Egger Advanced (International) Macroeconomics 26 of 114

27 Remarks on national accounting Long-run impact of short-run imbalances: CA t = NX t +r t B t and CA t = B t+1 B t imply B t+1 = NX t +(1+r t )B t Repeating the argument for B t+2 we get B t+2 = NX t+1 +(1+r t+1 )NX t +(1+r t )(1+r t+1 )B t Hartmut Egger Advanced (International) Macroeconomics 27 of 114

28 Remarks on national accounting In a T +1-period world with r t = r (inheriting B t and leaving B t+t+1 ): B t+t+1 = (1+r) T+1 B t +(1+r) T NX t (1+r)NX t+t 1 +NX t+t ( ) 1 T t+t ( ) 1 s t B t+t+1 = (1+r)B t + NX s (11) 1+r 1+r s=t Hartmut Egger Advanced (International) Macroeconomics 28 of 114

29 Terminal condition B t+t+1 = 0 implies t+t s=t ( ) 1 s t NX s = (1+r)B t 1+r Hartmut Egger Advanced (International) Macroeconomics 29 of 114

30 For instance, for B 3 = B 1 = 0 (temporary imbalance 1 ): NX 1 + NX 2 = (1+r)B 1 = 0 1+r CA 1 +CA 2 = 0 and CA 1 = B 2 B 1 = B 2 CA 2 = B 3 B 2 = B 2 NX 1 = CA 1 rb 1 = CA 1 = B 2 NX 2 = CA 2 rb 2 = (1+r)B 2 1 if interested in more long-run dynamics, see Obstfeld/Rogoff Chapter 2 Hartmut Egger Advanced (International) Macroeconomics 30 of 114

31 Equilibrium in a closed economy Hartmut Egger Advanced (International) Macroeconomics 31 of 114

32 Goods market equilibrium C t = Y t, t = 1,2 and optimal consumption choice (cf. intertemporal Euler equation) u (C 1 ) = (1+r)βu (C 2 ) give us the autarky real interest rate 1+r A = u (Y 1 ) βu (Y 2 ) (12) Hartmut Egger Advanced (International) Macroeconomics 32 of 114

33 (Budget constraint (3) is obviously fulfilled for C t = Y t.) 1+r A is the willingness to pay for present consumption. Virtual price in closed economy without investment possibilities. Relevant when opening up. Hartmut Egger Advanced (International) Macroeconomics 33 of 114

34 C 2 Y 2 A (Autarky) U[A] 1+r A (Absolute value of slope) Y 1 Figure 3: Autarky equilibrium C 1 Hartmut Egger Advanced (International) Macroeconomics 34 of 114

35 Effect of time preference on r A β > β implies MRS(Y 1,Y 2 ) < MRS(Y 1,Y 2 ) where MRS(Y 1,Y 2 ) u (Y 1 ) βu (Y 2 ) Hartmut Egger Advanced (International) Macroeconomics 35 of 114

36 C 2 Y 2 U[A] 1+r A 1+ r A Ũ[A] Y 1 C 1 Figure 4: r A rises with impatience. Hartmut Egger Advanced (International) Macroeconomics 36 of 114

37 Effect of output changes on r A Assume linear consumption expansion path (MRS(λY 1,λY 2 ) = MRS(Y 1,Y 2 )) r A rises if positive output shock is expected. No change of r A if present and future output rise pari passu. Hartmut Egger Advanced (International) Macroeconomics 37 of 114

38 C 2 1+ r A Expansion path à A 2 Y 2 A 1 1+r A 1+r A Y 1 C 1 Figure 5: Effect of output changes on r A Hartmut Egger Advanced (International) Macroeconomics 38 of 114

39 Equilibrium in a small open economy Hartmut Egger Advanced (International) Macroeconomics 39 of 114

40 2 periods: B 1 = B 3 = 0 i.e. NX 1 + NX 2 1+r = 0 r exogenously given by the world market Intertemporal equilibrium allocation C 1,C 2 determined by: u (C 1 ) = (1+r)βu (C 2 ) (13) C 1 + C 2 1+r = Y 1 + Y 2 1+r (14) Hartmut Egger Advanced (International) Macroeconomics 40 of 114

41 Special case If subjective discount factor is equal to market discount factor β = 1 1+r, the solution of (13) & (14) is given by C 1 = C 2 C (15) C = (1+r)Y 1 +Y 2 2+r (16) For β < 1 1+r the allocation is biased in favor of C 1. Hartmut Egger Advanced (International) Macroeconomics 41 of 114

42 Open vs. closed economy equilibrium C 2 1+r A A (Endowment point; Autarky equilibrium) (1+r)Y 1 +Y 2 Y 2 C 2 NX 2 C (Open economy equilibrium) NX 1 U[A] 1+r U[C] Y 1 C 1 Y 1 + Y 2 1+r C 1 Figure 6: Comparing open and closed equilibrium if r A > r Hartmut Egger Advanced (International) Macroeconomics 42 of 114

43 Access to international capital markets allows intertemporal income shifting. Here from future to present (borrowing) since r < r A. Gains from intertemporal trade U[C] > U[A] Debt from current net imports NX 1 = Y 1 C 1 < 0 must be paid back by future net exports NX 2 = (1+r)NX 1 = Y 2 C 2 > 0 Hartmut Egger Advanced (International) Macroeconomics 43 of 114

44 Implications for trade flows and capital account According to (7), (9) and (10): C 1 = Y 1 +rb 1 CA 1 = Y 1 +(1+r)B 1 B 2 = Y 1 NX 1 C 2 = Y 2 +rb 2 CA 2 = Y 2 +(1+r)B 2 B 3 = Y 2 NX 2 Hartmut Egger Advanced (International) Macroeconomics 44 of 114

45 In 2-period world: B 1 = B 3 = 0 Trade flows C 1 Y 1 = NX 1 Y 2 C 2 = (1+r)(C 1 Y 1 ) = NX 2 Net foreign assets B 2 = (C 1 Y 1 ) Y 2 C 2 = (1+r)B 2 = (1+r)(C 1 Y 1 ) Hartmut Egger Advanced (International) Macroeconomics 45 of 114

46 Long-run effects of short-run trade deficit If endowment expectations are wrong, the associated short run trade deficit may have long-run implications (3 or more periods, B 1 = 0). CA 1 = NX 1 = B 2 CA 2 = rb 2 = rnx 1 B 3 = CA 2 +B 2 = (1+r)NX 1 B t+1 = B t, t 3 requires CA t = NX t +rb t = 0 = For t 3: NX t = r(1+r)nx 1 = C t( 3) < Y t B 4 = 0 requires CA 3 = B 3 = NX 3 = (1+r) 2 NX 1 = C 3 < C t( 3) < Y 3 Hartmut Egger Advanced (International) Macroeconomics 46 of 114

47 C t+1 Ã = (A 1,Ã2)ex ante expected B 3 = (1+r)NX 1 A = (A 1,A 2 )ex post realised (C 2 = A 2 ) NX 2 = 0 C 1 C t( 3) C 3 1+r NX 1 Figure 7: Long-run effects of trade deficit (3 per., B 1 = 0) C t Hartmut Egger Advanced (International) Macroeconomics 47 of 114

48 Intertemporal trade pattern, savings and international equilibrium Hartmut Egger Advanced (International) Macroeconomics 48 of 114

49 Case r < r A (see figure 6): Implies NX 1 < 0, NX 2 > 0, given B 1 = B 3 = 0. The country is a net importer in period 1, net exporter in period 2. 1+r is the price of present consumption (here the import good) in terms of future consumption (export good). Hartmut Egger Advanced (International) Macroeconomics 49 of 114

50 Terms of trade Terms of trade = price of exports price of imports = 1 1+r decline in r i) terms of trade improve positive income and wealth effect on C 1 ii) substitution effects also favors C 1 Hartmut Egger Advanced (International) Macroeconomics 50 of 114

51 r A NX 1 (r) 0 NX 1 Figure 8: Net exports and interest rate if r < r A Hartmut Egger Advanced (International) Macroeconomics 51 of 114

52 Case r > r A (see figure 9): Implies NX 1 > 0, NX 2 < 0, given B 1 = B 3 = 0. Country is net exporter of present output, terms of trade 1+r rise in r i) positive terms of trade effect on C 1 in sum, the NX 1 -reaction is ambiguous. ii) negative substitution effect on C 1 Hartmut Egger Advanced (International) Macroeconomics 52 of 114

53 Case r > r A C 2 C A 1+r 1+r A C 1 NX 1 Figure 9: Intertemporal trade with r > r A Hartmut Egger Advanced (International) Macroeconomics 53 of 114

54 Interest rate NX 1 (r) r A (β,y 2 /Y 1 ) (-) (+) 0 NX 1 (= S1) Figure 10: Net exports and interest rate if r > r A Hartmut Egger Advanced (International) Macroeconomics 54 of 114

55 The position of the curve is fixed by r A. Remember: r A declines with β and (for linear expansion path of consumption) rises with Y 2 /Y 1. Moreover: For B 1 = 0, CA 1 = NX 1 and thus S 1 Y 1 +rb 1 C 1 = NX 1 Hartmut Egger Advanced (International) Macroeconomics 55 of 114

56 International equilibrium Two country world: Home: r A (β,y 2 /Y 1 ) Foreign: r A (β,y 2/Y 1) integrated world is like closed economy with goods market equilibrium condition C t +C t = Y t +Y t Hartmut Egger Advanced (International) Macroeconomics 56 of 114

57 Using C t +NX t = Y t, Ct +NX t = Yt, we get This determines world interest rate r. NX t +NX t = 0 (17) Hartmut Egger Advanced (International) Macroeconomics 57 of 114

58 Interest rate NX 1 (r) r A r NX 1 NX 1 (r) NX 1 r A 0 NX(r) Figure 11: Equilibrium world interest rate if r A > r A Hartmut Egger Advanced (International) Macroeconomics 58 of 114

59 If increasing impatience (β or β ) or rising future output (Y 2 /Y 1 or Y 2 /Y 1 ) raise ra or r A the equilibrium world interest rate rises, ceteris paribus. This worsens terms of trade for net importer Foreign improves terms of trade for net exporter Home Hartmut Egger Advanced (International) Macroeconomics 59 of 114

60 Capital accumulation and production Hartmut Egger Advanced (International) Macroeconomics 60 of 114

61 Basic model assumptions Production function: Y t = F (K t ) Properties: F(0) = 0 F > 0 F < 0 Inada conditions: lim K 0 F (K) = lim K F (K) = 0 Since N t = 1, level of capital stock K t and capital intensity k t = K t /N t coincide. Hartmut Egger Advanced (International) Macroeconomics 61 of 114

62 Capital accumulation: K t+1 = K t +I t (18) (Depreciation ignored, K can be eaten up, i.e. I t = K t.) Capital demand under perfect competition: r t = F (K t ) (19) Wages (labor demand) under perfect competition: W t = F (K t ) r t K t (20) Hartmut Egger Advanced (International) Macroeconomics 62 of 114

63 Closed economy with capital accumulation and production Hartmut Egger Advanced (International) Macroeconomics 63 of 114

64 Intertemporal production and investment Goods market equilibrium C t +I t = Y t (21) C 1 Y 1 = F(K 1 ) Y 2 = F(K 2 ) I 1 C 2 I 2 K 1 K 1 +I 1 = K 2 K 3 t 1 Period t = 1 Period t = 2 t 2 Hartmut Egger Advanced (International) Macroeconomics 64 of 114

65 Intertemporal transformation curve ( Production possibilities Frontier PPF) C 2 +K 3 = F(K 2 )+K 2 Intertemporal PPF: = F K 1 +F(K 1 ) C }{{ 1 +K } 1 +F(K 1 ) C 1 (22) I 1 C 2 + C 2 +K 3 C 2 + = F K 1 +F(K 1 ) C }{{} 1 +K 1 +F(K 1 ) C }{{} 1 K 2 K 2 Hartmut Egger Advanced (International) Macroeconomics 65 of 114

66 dc + 2 dc 1 d 2 C + 2 dc 2 1 = 1+F K 1 +F(K 1 ) C }{{} 1 < 0 K 2 = F (K 2 ) < 0 Hartmut Egger Advanced (International) Macroeconomics 66 of 114

67 C + 2 F(K 1 +F(K 1 ))+K 1 +F(K 1 ) X 2 = F(K 2 )+K 2 X K 1 +I 1 = K 2 1+F (K 2 ) X 1 F(K 1 )+K 1 C 1 Figure 12: Intertemporal production possibilities frontier Hartmut Egger Advanced (International) Macroeconomics 67 of 114

68 Intertemporal budget constraint of representative household Period 1 Inherited wealth K 1 S 1 wealth saved for t = 2 Income Y 1 C 1 Period 2 Capital wealth (K 1 +S 1 )(1+r) C 2 Labor income W 2 Bequest K 3 Hartmut Egger Advanced (International) Macroeconomics 68 of 114

69 Intertemporal consumption possibility line (CPL) Intertemporal budget constraint: C 2 +K 3 = W 2 +(K 1 +Y 1 C 1 )(1+r) (23) Since W 2 = F(K 2 ) rk 2 and K 1 +Y 1 C 1 = K 2 (23) is consistent with (22). That means: Savings behavior of households leads to a point on the economy s PPF. The question is: which point? Hartmut Egger Advanced (International) Macroeconomics 69 of 114

70 Optimal intertemporal choice K 3 - choice depends on the bequest motive. Can be captured by u(c 1 )+βu(c + 2 ) C + 2 C + 2 = C 2 +K 3... bequest motive = C 2... no bequest motive max u(c 1 )+βu(c + 2 ) s.t. C+ 2 = W 2 +(K 1 +Y 1 C 1 )(1+r) Hartmut Egger Advanced (International) Macroeconomics 70 of 114

71 Optimal intertemporal choice yields first-order condition MRS u (C 1 ) βu (C + 2 ) = 1+r where K 3 = 0 without bequest motive. (K 3 = 0 implies K 2 +I 2 = 0 and thus S 2 = I 2 = K 2.) In dubio, assume K 3 = 0, i.e. C 2 = C 2 + in the following. Hartmut Egger Advanced (International) Macroeconomics 71 of 114

72 C + 2 CPL F(K 2 )+K 2 A 1+r A PPF U[A] K 1 +I 1 = K 2 Figure 13: CPL and PPF F(K 1 )+K 1 C 1 Hartmut Egger Advanced (International) Macroeconomics 72 of 114

73 Small open economy with capital accumulation and production Hartmut Egger Advanced (International) Macroeconomics 73 of 114

74 Goods market equilibrium and intertemporal foreign account (see (11)) C t +I t +NX t = Y t (24) imply NX 2 +(1+r)NX 1 = B 3 (1+r) 2 B 1 }{{} D C 2 +(1+r)C 1 = Y 2 I 2 +(1+r)(Y 1 I 1 ) D = F(K 2 ) [K 3 K 2 ]+(1+r)[F(K 1 ) (K 2 K 1 )] D Hartmut Egger Advanced (International) Macroeconomics 74 of 114

75 Hence, C 2 +K }{{} 3 +(1+r)C 1 = F(K 2 )+K 2 +(1+r)[F(K }{{} 1 )+K 1 K 2 ] D (25) }{{} C + X 2 2 X 1 where X = (X 1,X 2 ) is a point at the PPF. Hartmut Egger Advanced (International) Macroeconomics 75 of 114

76 C + 2 PPF X 2 X CPL(r) given by (25) C 2 C 1+r X 1 C 1 Figure 14: Consumption possibilities line (CPL) under world interest and D = 0. Hartmut Egger Advanced (International) Macroeconomics 76 of 114 C 1

77 C + 2 D > 0 D < 0 D = 0 X 1+r C 1 Figure 15: CPL under long-run imbalances (D 0) In the following D = 0 (e.g. B 1 = B 3 = 0). Hartmut Egger Advanced (International) Macroeconomics 77 of 114

78 C + 2 PPF 1+r NX 2 X U[C] C NX 1 Figure 16: Equilibrium production (X) equilibrium consumption (C), and trade balances (NX) C 1 Hartmut Egger Advanced (International) Macroeconomics 78 of 114

79 From autarky to open economy equilibrium Case r < r A : C r 1+r A X A C I 1 C 1 C 1 Figure 17: Double gains from intertemporal trade Hartmut Egger Advanced (International) Macroeconomics 79 of 114

80 In addition to the picture for the endowment economy: Production structure shifts from A to X by higher investments I 1. Increase in current consumption by C 1. Current account deficit NX 1 = I 1 + C 1 paid back by increased future production (+ possibly lower consumption). Hartmut Egger Advanced (International) Macroeconomics 80 of 114

81 Case r > r A : C + 2 C A A X NX 1 1+r Figure 18: A net exporting country C 1 Hartmut Egger Advanced (International) Macroeconomics 81 of 114

82 Production shifts in favor of current output by decreasing investment I 1 < 0. Additional output allows net exports. Net exports today allow higher future consumption C 2 > 0 by future imports (NX 2 = (1+r)NX 1 ). Present consumption C 1 may shrink (A ) or increase (A) depending on the relative strength of income and substitution effect (plus output shift). Hartmut Egger Advanced (International) Macroeconomics 82 of 114

83 Adding government consumption Hartmut Egger Advanced (International) Macroeconomics 83 of 114

84 With government consumption, period utility has the following additive form: u(c)+v(g). The budget constraint in the two period model is C 1 + C 2 1+r = Y 1 T 1 I 1 + Y 2 T 2 I 2, 1+r where T t denotes taxes and Y t T t is disposable income of the private sector in period t. Goods market equilibrium in period t: C t +I t +G t +NX t = Y t (26) Hartmut Egger Advanced (International) Macroeconomics 84 of 114

85 Current account balance (recall (9),(10)): CA t = NX t +rb t = Y t +rb t C t G t I t = Y t +rb t C t T }{{} t + T t G }{{} t St P private savings public savings I t With a balanced budget T t = G t of the public sector, private savings are equal to total savings (S P t = S t ) and CA t = St P +T t G }{{} t I t (27) S t total savings B t+1 = B t +S t I t (28) Hartmut Egger Advanced (International) Macroeconomics 85 of 114

86 Impact of G in small open economy Increase in G 1 (G 2 ) shifts transformation curve (PPF) for private sector leftward (downward). In the following illustration (with a balanced budget of the government: T t = G t ): Initial situation: G 1 = G 2 = 0 and NX 1 = NX 2 = 0 Shock 1: Shock 2: G 1 G 2 Hartmut Egger Advanced (International) Macroeconomics 86 of 114

87 C r B C A NX 1 C 1 Figure 19: Impact of G 1 Hartmut Egger Advanced (International) Macroeconomics 87 of 114

88 Impact of G 1 : Private feasible output shifts from A to B. Would decrease C 1 by the full amount of G 1 = BA leaving C 2 unaffected Individuals prefer C by borrowing from abroad. Hartmut Egger Advanced (International) Macroeconomics 88 of 114

89 C + 2 C A B NX 1 Figure 20: Impact of anticipated G 2 increase Individuals hedge against tax G 2 by lending to Foreign in Period 1. Hartmut Egger Advanced (International) Macroeconomics 89 of 114 C 1

90 Investment, savings and world interest rate in international equilibrium Hartmut Egger Advanced (International) Macroeconomics 90 of 114

91 The investment function Production function: Y t = A t F(K t ) A t : Productivity parameter Accumulation equation: K 2 = K 1 +I 1 In the following, we consider a 2-period model with B 1 = B 3 = 0, K 3 = 0 and G t = T t = 0. Hartmut Egger Advanced (International) Macroeconomics 91 of 114

92 Condition for optimal capital input under perfect competition: r = A 2 F (K 1 +I 1 ) (29) (29) defines investment curve I 1 = I(r/A 2 ), I < 0 The negative slope follows from F < 0. Hartmut Egger Advanced (International) Macroeconomics 92 of 114

93 r r A I for A 2 Ĩ for Ã2 > A 2 I 1 Figure 21: Investment curve and productivity shifts Shifts in A 1 have no effect on investment since K 1 is already fixed from past decisions. Hartmut Egger Advanced (International) Macroeconomics 93 of 114

94 The saving function Reconsidering the endowment economy: From the endowment economy we know that B 1 = B 3 = 0 implies S 1 = Y 1 C 1 = NX 1 (r). Furthermore, we can note that, ds 1 /dr = dnx 1 /dr = dc 1 /dr. To determine the impact of interest rate r on savings (or, equivalently, NX 1 ), we can first look at the intertemporal Euler equation u (C 1 ) = (1+r)βu (C 2 ). Substituting the budget constraint C 2 = (1+r)(Y 1 C 1 )+Y 2 gives u (C 1 ) = (1+r)βu ((1+r)(Y 1 C 1 )+Y 2 ). (30) Hartmut Egger Advanced (International) Macroeconomics 94 of 114

95 Implicitly differentiating (30) with respect to r gives dc 1 dr = βu (C 2 )+β(1+r)u (C 2 )(Y 1 C 1 ) u (C 1 )+β(1+r) 2 u. (31) (C 2 ) Noting u > 0, u < 0, it is immediate that dnx 1 (r)/dr = dc 1 /dr > 0 if C 1 > Y 1 (or, equivalently, NX 1 < 0). However, dnx 1 (r)/dr = dc 1 /dr < 0 cannot be ruled out if Y 1 > C 1 (or, equivalently, NX 1 > 0) see Figure 10. Hartmut Egger Advanced (International) Macroeconomics 95 of 114

96 Consumption in a model with capital accumulation and production Substituting the budget constraint C 2 = (1+r)[A 1 F(K 1 ) C 1 I 1 ]+A 2 F(K 1 +I 1 )+K 1 +I 1 for C 2 in the Euler equation u (C 1 ) = (1+r)βu (C 2 ), gives u (C 1 ) = (1+r)βu {(1+r)[A 1 F(K 1 ) C 1 I 1 ] Implicitly differentiating with respect to r yields +A 2 F(K 1 +I 1 )+K 1 +I 1 }. (32) dc 1 dr = βu (C 2 )+β(1+r)u (C 2 )[A 1 F(K 1 ) C 1 I 1 ] u (C 1 )+β(1+r) 2 u (C 2 ) + β(1+r)u (C 2 ){A 2 F (K 1 +I 1 ) r} I/ r u (C 1 )+β(1+r) 2 u (C 2 ). Hartmut Egger Advanced (International) Macroeconomics 96 of 114

97 Accounting for A 2 F (K 1 +I 1 ) = r further implies dc 1 dr = βu (C 2 )+β(1+r)u (C 2 )[A 1 F(K 1 ) C 1 I 1 ] u (C 1 )+β(1+r) 2 u. (33) (C 2 ) Hence, the derivative in (33) is precisely the same as the derivative in (31), but with Y 1 C 1 replaced by the date 1 current account for an investment economy with B 1 = 0: A 1 F(K 1 ) C 1 I 1. That means that, given current account balances, the slope of the saving schedule is the same as for the endowment economy! Hartmut Egger Advanced (International) Macroeconomics 97 of 114

98 An intuition for this result The symmetry in the reaction of savings to interest rate adjustments in the endowment and the investment economy is a consequence of the envelope theorem. The first-order condition for profit-maximizing investment ensures that a small deviation from optimum investment does not alter the present value of national output, evaluated at the world interest rate. Consequently, at the margin, the investment adjustment I 1 / r has no effect on net lifetime resources, and hence no effect on consumption response. Hartmut Egger Advanced (International) Macroeconomics 98 of 114

99 From consumption to saving As noted above, savings in period 1 are given by S 1 = Y 1 C 1 or, equivalently, S 1 = A 1 F(K 1 ) C 1. Hence, we can write savings as function of r, A 1, A 2 and β: S 1 = S(r,A 1,A 2,β), with S 1 / r > 0 in the regular (non-perverse) case. Saving curve and productivity shift An increase of A t has analogous effects to an increase of Y t in endowment economy. According to slide 54 a rise in Y 1 shifts the S-curve to the right. A rise in Y 2 shifts the S-curve to the left. Rising impatience (a fall in β) also shifts the saving curve to the left. Hartmut Egger Advanced (International) Macroeconomics 99 of 114

100 r S for A 1,A 2,β S for à 1 > A 1 à 2 < A 2 β > β r A Figure 22: Saving curve S 1 Hartmut Egger Advanced (International) Macroeconomics 100 of 114

101 Investment, savings and current account r S CA 1 = S 1 I 1 r A CA 1 = I 1 S 1 I Figure 23: Investment, savings and current account S 1,I 1 Hartmut Egger Advanced (International) Macroeconomics 101 of 114

102 International equilibrium in a two-region world ( Metzler Diagrams ) World equilibrium requires CA 1 +CA 1 = 0 i.e. S 1 I 1 = (S 1 I 1) Hartmut Egger Advanced (International) Macroeconomics 102 of 114

103 r r I S CA 1 r A B r A r A B CA 1 A S I Home Foreign Figure 24: World equilibrium interest rate r A < r < r A Hartmut Egger Advanced (International) Macroeconomics 103 of 114

104 r r I S S r r CA 1 CA 1 CA 1 CA 1 S I Home Foreign Figure 25: Impact of rising impatience in Foreign (β ) World interest rate rises and current account CA 1 from Home to Foreign increases. Investment decreases in both regions. Hartmut Egger Advanced (International) Macroeconomics 104 of 114

105 Impact of positive productivity shock in Foreign Consider a productivity shock of the form A 2 World interest rate rises In Home Investment falls Saving and CA 1 -surplus rise In Foreign Investment reaction ambiguous CA 1-deficit rises Hartmut Egger Advanced (International) Macroeconomics 105 of 114

106 r r I Ĩ S S r r CA 1 CA 1 CA 1 CA 1 S I Home Foreign Figure 26: Impact of positive productivity shock in Foreign Hartmut Egger Advanced (International) Macroeconomics 106 of 114

107 Stability of international equilibrium and Marshall-Lerner condition The world equilibrium condition CA t +CA t = 0 is (Use CA t = S t I t ) S t (r)+s t(r) = I t (r)+i t (r) (34) As addressed in Figure 10, the saving curve may be backward bending, so that multiple equilibria and unstable equilibria cannot be excluded. Hartmut Egger Advanced (International) Macroeconomics 107 of 114

108 (Walrasian) stability condition: A market is stable in the Walrasian sense if a small increase in the price of the good traded there causes excess supply, while a small decrease causes excess demand. The stability condition defining Walrasian stability in the market for world savings is that a small rise in r should lead to an excess supply of savings: d [S 1 (r)+s 1 (r)] dr > d [I 1(r)+I 1 (r)] dr (35) Stability guarantees that market forces tend to eliminate imbalances resulting from small disturbances of international equilibrium. Hartmut Egger Advanced (International) Macroeconomics 108 of 114

109 r I +I S +S r Figure 27: Savings and Investment World saving and investment Hartmut Egger Advanced (International) Macroeconomics 109 of 114

110 For B 1 = B 3 = 0 national accounting identities imply NX 1 = CA 1 = S 1 I 1 NX 1 = CA 1 = S 1 I 1 Moreover (see (11)), NX 1 + NX 2 1+r = 0 Hartmut Egger Advanced (International) Macroeconomics 110 of 114

111 Using this in international equilibrium condition (34), we get S 1 I 1 +S 1 I 1 = NX 1 NX 2 1+r Thus (35) is equivalent to [ d NX 1 (r) NX 2 (r) ] 1+r dr > 0 (36) Hartmut Egger Advanced (International) Macroeconomics 111 of 114

112 [ d NX 1 (r) NX 2 (r) ] 1+r dr = NX 1 NX 2 (1+r) NX 2 (1+r) 2 = NX2 [ (1+r)NX 1 (1+r) 2 NX 1 ] NX 2 (1+r) NX2 +1 NX 1 (1+r) NX 2 Hartmut Egger Advanced (International) Macroeconomics 112 of 114

113 In equilibrium NX 1 (1+r) = NX 2 = NX2. Thus the square bracket is negative (positive) if (1+r)NX 1 NX 1 } {{ } η + (1+r)NX 2 NX2 > (<)1 (37) }{{} η If Home is net importer today (NX 1 < 0), then NX2 condition (35) is equivalent to < 0 and stability η +η > 1. (38) Hartmut Egger Advanced (International) Macroeconomics 113 of 114

114 Interpretation (NX 1 < 0) η is the (absolute value of) negative import elasticity of Home with respect to price 1+r of current consumption. η is the (positive) elasticity of Foreign s future imports. (38) is the intertemporal analogue to the so-called Marshall-Lerner condition. Remark When Home happens to be the exporter in period 1, rather than the importer, (38) still characterizes the Walras-stable case, but with import elasticities defined so that Home s and Foreign s role are interchanged. Hartmut Egger Advanced (International) Macroeconomics 114 of 114