International macroeconomics (advanced level) Lecture notes

1 International macroeconomics (advanced level). Lecture notes Nikolas A. M uller-Plantenberg*. 2017 2018. * Address: Departamento de An alisis Econ omico - Teor a Econ omica e Historia Econ omica, Universidad Aut onoma de Madrid, 28049 Madrid, Spain. International macroeconomics OUTLINE. Outline I Aims of the course 10. II Basic models 11. 1 Balassa-Samuelson effect 11. III Difference equations 17. 2 Introduction to difference equations 17. 3 Modelling currency flows using difference equations 29. IV Differential equations 36.

2 4 Introduction to differential equations 36. 5 First-order ordinary differential equations 36. 6 Currency crises 39. 7 Systems of differential equations 45. 8 Laplace transforms 48. 9 The model of section revisited 53. 10 A model of currency flows in continuous time 54. V Intertemporal optimization 56. 11 Methods of intertemporal optimization 56. 21 December 2017 2. International macroeconomics OUTLINE. 12 Intertemporal approach to the current account 56. 13 Ordinary maximization by taking derivatives 58. VI Dynamic optimization in continuous time 70.

3 14 Optimal control theory 70. 15 Continuous-time stochastic processes 74. 16 Continuous-time dynamic programming 78. 17 Examples of continuous-time dynamic programming 84. 18 Portfolio diversification and a new rule for the current account 86. 19 Investment based on real option theory 90. 20 Capital accumulation under uncertainty 98. 21 December 2017 3. International macroeconomics CONTENTS. Contents I Aims of the course 10. II Basic models 11. 1 Balassa-Samuelson effect 11. Growth accounting .. 11. Example 1 .. 11. Example 2.

4 11. Example 3 .. 11. The price of non-traded goods with mobile capital .. 12. Balassa-Samuelson effect .. 14. Accounting for real exchange rate changes .. 14. Theory versus empirics .. 15. Real appreciation of the yen .. 15. Conclusions .. 16. III Difference equations 17. 2 Introduction to difference equations 17. Definition .. 17. Examples .. 17. Difference equation with trend, seasonal and irregular .. 17. Random walk .. 17. Reduced-form and structural equations .. 18. Error correction .. 19. General form of difference equation.

5 19. Solution to a difference equation .. 20. Lag operator .. 20. 21 December 2017 4. International macroeconomics CONTENTS. Solving difference equations by iteration .. 20. Sums of geometric series .. 20. Iteration with initial condition - case where |a1 | < 1 .. 21. Iteration with initial condition - case where |a1 | = 1 .. 21. Iteration without initial condition - case where |a1 | < 1 .. 22. Iteration without initial condition - case where |a1 | > 1 .. 22. The exchange rate as an asset price in the monetary model .. 23. Alternative solution methodology.

6 23. Example: Second-order difference equation .. 25. Solving second-order homogeneous difference equations .. 26. Roots of the general quadratic equation .. 26. Homogeneous solutions .. 26. Particular solutions .. 28. 3 Modelling currency flows using difference equations 29. A benchmark model .. 30. A model with International debt .. 31. IV Differential equations 36. 4 Introduction to differential equations 36. 5 First-order ordinary differential equations 36. Deriving the solution to a differential equation .. 37. Applications.

7 37. Inflation .. 37. Price of dividend-paying asset .. 38. Monetary model of exchange rate .. 38. 21 December 2017 5. International macroeconomics CONTENTS. 6 Currency crises 39. Domestic credit and reserves .. 39. A model of currency crises .. 40. Exchange rate dynamics before and after the crisis .. 41. Exhaustion of reserves in the absence of an attack .. 42. Anticipated speculative attack .. 43. Fundamental causes of currency crises .. 43. 7 Systems of differential equations 45. Uncoupling of differential equations .. 45.

8 Dornbusch model .. 46. The model's equations .. 46. Long-run characteristics .. 46. Short-run dynamics .. 47. 8 Laplace transforms 48. Definition of Laplace transforms .. 48. Standard Laplace transforms .. 49. Properties of Laplace transforms .. 49. Linearity of the Laplace transform .. 49. First shift theorem .. 49. Multiplying and dividing by t .. 49. Laplace transforms of the derivatives of f (t) .. 50. Second shift theorem .. 50. Solution of differential equations .. 50. Solving differential equations using Laplace transforms.

9 50. First-order differential equations .. 51. Second-order differential equations .. 51. Systems of differential equations .. 52. 9 The model of section revisited 53. 21 December 2017 6. International macroeconomics CONTENTS. 10 A model of currency flows in continuous time 54. The model's equations .. 54. Solving the model as a system of differential equations .. 54. Solving the model as a second-order differential equation .. 55. V Intertemporal optimization 56. 11 Methods of intertemporal optimization 56. 12 Intertemporal approach to the current account 56.

10 Current account .. 57. A one-good model with representative national residents .. 57. 13 Ordinary maximization by taking derivatives 58. Two-period model of International borrowing and lending .. 58. Digression on utility functions .. 59. Logarithmic utility.. 59. Isoelastic utility.. 60. Linear-quadratic utility.. 60. Exponential utility.. 61. The HARA class of utility functions .. 61. Two-period model with investment .. 62. An infinite-horizon model .. 63. Dynamics of the current account .. 64. A model with consumer durables.