Dynamic Macroeconomics (Concepts, Techniques, Applications)

Size: px
Start display at page:

Download "Dynamic Macroeconomics (Concepts, Techniques, Applications)"

Transcription

1 Dynamic Macroeconomics (Concepts, Techniques, Applications) this version: June 2013 Thomas Steger University of Leipzig Institute for Theoretical Economics Macroeconomics 1 Introduction 2 Production and Technology 3 Utility and Welfare 4 Dynamic Optimization 5 Dynamic Systems 5.1 Differential Equations 1 Differential equations (DE) are functional equations. That is, the solution of a DE () = [()] is itself a function of time, i.e. () First-order Linear Differential Equations A first-order differential equation has the following form + () = (), (1) 1 This section closely follows Chiang and Wainwright (2005, Chapters 15/16) 0

2 where R denotes the independent variable (often the time index) and :=. This is a first-order DE since only the first derivative of with respect to time occurs. Since both and appear only in firstdegreeandthereisnosuchterm, thisisa linear DE. The homogenous case (constant coefficient). Assuming () = (constant coefficient) and () =0 (homogenous DE) we have + =0 A solution is readily found as follows. At first, we rewrite the DE as Integrating both sides w.r.t. time gives 1 = Z Z 1 = Considering the LHS, we know that (chain rule) 1 = ln and hence Z 1 = Z ln =ln + 1 Turning to the RHS we have Z = + 2 and equating both sides gives ln = + with := 2 1 1

3 Finally, taking the antilog on both sides and defining := yields () = (general solution) () = 0 (definite solution), where the definite solution results from the determination of the arbitrary constant by exploiting a border condition like (0) = 0. The non-homogenous case (constant coefficient). Assuming that () = (constant coefficient) and () = in (1) one gets + = (2) The solution of this DE consists of the sum of two terms, namely the complementary function (denoted )andtheparticular integral (denoted ). The component is the general solution of the homogenous DE, whereas is simply any particular solution of the non-homogenous equation. From above we know that = As regards, we can try the simplest possible type of solution, namely =. In this case (2) becomes = (since =0) and the solution reads = (assuming 6= 0). The general solution of the non-homogenous equation, i.e. complete equation, then is = + = + The definite solution is found once the arbitrary constant of integration has been determined by exploiting a border condition like (0) = 0 to read =

4 5.1.2 Variable coefficients and variable term The homogenous case. Consider (1) with () =0, i.e. + () =0,whichmay be expressed as 1 = () Integrating both sides, we get LHS = Z 1 = Z RHS = Z ln =ln + Z () = () Equating the LHS to the RHS gives the general solution Z ln = () = () = () Digression: non-homogenous DE with constant coefficient. We follow Gandolfo (1997, Chapter ), who applies the method of variation of parameters. Consider the following DE = () with 0 6=0or + = 1 0 () with = 1 0.As a trial solution we use which implies () =() (3) = () () Plugging the RHS into + = 1 0 () yields () () {z } + () = 1 () {z } 0 () = 1 0 () = () = 1 0 () 3

5 Integrating both sides gives () = 1 0 Z () + where is an arbitrary constant of integration. Replacing () in (3) according to the RHS of the preceding equation yields µ Z 1 () = 0 () + (4) The non-homogenous case (variable coefficient). Consider + () = (). We employ the same procedure as before. The trial solution implies = () () = = () () ()() () Plugging the RHS into + () = () gives () () ()() () {z } + ()() () = () {z } () () = () = () =() () Formingtheantilogonbothsidesgives Z () = () () + and plugging the RHS into the trial solution = () () gives (see also Chiang, 1984, pp. 487/488) µz = () () + () where is an arbitrary constant. Notice that this solution, again, consists of two additive components. 4

6 5.2 Two applications Application #1: The time path of wealth in a small open economy. TheDE that describes the evolution of per capita wealth () is given by () ( )() = () (where and are constant) and () =(0) ( ). How does the solution () look like? follows Noting that + = () µz has solution () = () + we can conclude that the solution to () ( )() = () should read as µ Z () = ( ) + ( ()) ( ) (5) Noting ( ) = ( ) for and constant and = (0) ( ) one gets Z () = µ ( ) + Z ( ) (0) ( ) ( ) Moreover, rewrite ( ) = (1 ) + () = ( ) µ + () = ( ) µ + and use R = 1 + to get (0) ( ) ( ) + 1 ( ) ( ) 2 (1 ) + (0) ( ) ( ) ( ) ( ) (1 ) + This equation contains two unknowns, namely := and (0). To determine these two unknowns we exploit two (border) conditions, namely () = 0 and lim () ( ) =0(NPC together with the TVC). The latter, assuming 0 ( ) 0, gives and (6) lim () ( ) = =0 5

7 Turning to (0), noticethat =0together with () = 0 implies 0 = ( ) (0) (1 ) + µ (1 ) + (0) = ( 1) 0 + (7) (8) Remark #1: Notice that the PDV of consumption may be expressed as Z 0 (0) ( ) ( ) = (1 ) + (1 ) + (1 ) + (0) (0) (1 ) + 0 = (1 ) + (0) and hence (7) requires that R (0) ( ) ( ) = 0 0 +, i.e. consumption must equal total wealth. Remark #2: Equ. (6) together with =0and (8) gives () = ( ) µ (1 ) (1 ) + {z } (0) () = µ 0 + ( ) This is the solution stated in Barro and Sala-i-Martin (1995, p. 99). the PDV of ( ) Application #2: The price of an asset and the no-arbitrage condition. The price of an asset, at time, which entitles the holder to a cash flow () [ ] is given by () = Z () (), (9) where R () represents the cumulative discount factor between and (in the simpler case where () =, the discount factor simply is ). We now differentiate this equation w.r.t. time to receive the no-arbitrage condition in the capital market (there are two assets: bonds paying interest rate () and an asset described above). 6

8 To do this, one needs to apply the Leibniz rule 2 = ( ) () () {z} =0 =0 z } { Z () + {z} =1 Z () () The integrand of the last expression on the RHS can be expressed as 3 () () = () () = () () µ Z () To evaluate the last expression on the RHS we apply the Leibniz rule again Z () = () () Z () + = () {z} {z} {z } =0 =1 =0 Putting everything together gives Z = () () () () = ()() () (10) This is the well-known no-arbitrage condition (sometimes called "Fisher equation"), which must hold at each instant of time under capital market equilibrium (absence of profit opportunities). Buying one unit of the asset at price () under consideration (say a "share") provides a gain of ()+ (instantaneous profit plus change in asset price). This must equal the reward that can be earned by investing the amount () into bonds which pays ()(). Remark. Differentiating equ. (9) w.r.t. time yields the no-arbitrage condition (10). R 2 () Leipniz rule: () ( ) = ( ())0 () ( ()) 0 () + R () () ( ) (Sydsaeter et al., 2005, p. 60). 3 Notice that () = () 0 () 7

9 However, solving equ. (10), which represents a first-order, linear, non-homogenous DE with variable coefficient, yields equ. (9) provided that the following boundary condition (No-Ponzi-Game condition) holds (see also Blanchard/Fischer, 1989, pp. 48/51) lim () () =0 The other direction. Multiplying both sides of ()() = () by 0 () and integrate forward to get Z 0 () [ ()()] = Z 0 () () (11) 0 0 h i 0 () () 0 = Z 0 0 () () (12) The integration on the LHS can be understood by noting that (Leibniz rule of differentiation) h i 0 () () = 0 () ()+ 0 () () = () 0 () ()+ 0 () () Hence, we have lim 0 () () (0) = Z 0 0 () () (13) The No-Ponzi-Game condition (together with the TVC) requires that lim 0 () () = 0 such that (0) = Z 0 0 () () (14) 8

10 5.3 Digression: The No-Ponzi-Game condition Consider an agent who lives for a finite time period and has access to the capital market. Let () denote financial wealth at time (time may be continuous or discrete). The No-Ponzi-Game condition (NPGC) then reads as follows ( ) 0 i.e. ( ) 0 is excluded. The economic significanceisasfollows. TheNo-Ponzi-Game condition (NPGC) represents an equilibrium constraint that is imposed on every agent. Everyone must repay his/her debt, i.e. leave the scene without debt at terminal point in time. continuous): In case of an infinitely lived agent, the NPGC reads as follows (time is lim () 0 i.e. lim () 0 is excluded. To see the economic significance, assume that Mr. Ponzi (and his dynasty) wishes to increase consumption today by (with being measured in monetary units). Consumption expenditures are being financed by borrowing money. Debt repayment as well as interest payments are being financed by increasing indebtedness further. Debt then evolves according to debt (1 + ) (1 + ) 2... Debt at time evolves evolves according to = (1 + ) if N () = if R Noting that () = () the above NPGC may be expressed as lim () =lim() 0 9

11 i.e. lim () 0 is excluded. If Mr. Ponzi increases consumption by, financed by employing his innovative financing scheme, debt evolves according to () = such that the present value of debt would remain positive, which is excluded since lim = 0 Charles Ponzi became known 1920s as a swindler in for his money making scheme. He promised clients huge profits by buying discounted postal reply coupons in other countries and redeeming them at face value in the US as a form of arbitrage. In reality, Ponzi was paying early investors using the investments of later investors. This type of scheme is now known as a "Ponzi scheme". (Wikipedia, June 3rd 2013) 5.4 Difference Equations Discrete time and difference equations We now assume that the time index can take only integer values ( N); often it is said that time is discrete. The difference quotient can then be stated simply as (since =1). A first-order difference equation is one that contains the first difference := +1 4 This section closely follows Chiang and Wainwright (2005, Chapter 17,18) 10

12 Thus a first-order difference of is transformable into a sum of terms involving a one-period time lag. Hence, one can define a first-order difference equation as one that contains a one-period lag in the dependent variable. In what follows, we consider the solution to linear, non-homogenous, first-order difference equations with constant coefficients, as the following +1 =0, (15) where 6= Solving a first-order difference equation Iterative method. Equation (15) may be expressed as +1 = Now if 0 is given we can develop the solution simply as follows 1 = 0 2 = ³ 1 = 3 = ³ 2 = Hence, the general solution appears to be ³ = In more general notation, the solution to a linear, homogenous, first-order difference equation is =. 11

13 General method. Suppose we are seeking the solution to +1 + = (16) where and are constants. The general solution comprises two parts: a particular solution (any solution of the complete non-homogenous equation (16)) and a complementary solution, which is the general solution of +1 + =0. The part represents the intertemporal equilibrium, while gives the deviation from this stationary solution. We consider first. As a solution for +1 + =0we try = (with 6= 0). This implies +1 = +1 such that +1 + =0becomes +1 + =0 = ( + ) =0 = = Hence, for the initial trial solution = to work we must set = implying = ( ). Next we turn to, which is any solution to the complete equation. As a trial solution we chose the simplest example, i.e. = (implying +1 = ). Hence we have + = = = 1+ Since this particular value satisfies (16), the particular integral can be written as = 6= 1 1+ If it happens that = 1, the particular solution = 1+ case, one should try a solution of the form =, whichgives ( +1)+ = = = +1+ = is not defined. In this 12

14 and hence = In summary, the general solution of (16) is given by (assuming that 6= 1) = + = ( ) + 1+ To eliminate (an arbitrary constant) we assume that 0 is given, which yields 0 = + 1+ = = 0 1+ and the definite solution to (16) then reads ( 6= 1) = µ 0 ( ) The dynamic stability of equilibrium 1+ Consider the general solution of the form = +. The stability of obviously depends on the value of. If 1 ( 1), then the equilibrium is stable (unstable). In addition, provided that 0 ( 0), the solution is monotonic (exhibits oscillations) Second-order difference equations A second-order difference equation is one that contains the second difference 2 = ( )= ( +1 ) = ( +2 1 ) ( +1 ) = Thus a second-order difference of is transformable into a sum of terms involving a two-period time lag. Hence, we define a second-order difference equation as one that contains a two-period lag in the variable. In what follows we consider the solution to 13

15 linear, non-homogenous, second-order difference equations with constant coefficients, like the following gives = (17) Particular solution. We try a solution of the form =. Plugging this into (17) = = = Assuming that = 1 this gives = If, on the other hand, = 1, the trial solution should be =, which yields ( +2)+ 1 ( +1)+ 2 = = = 1 +2 and thus = 1 ("a moving equilibrium"). +2 Complementary solution. Thisisthesolutionto =0 (18) Motivated by experience (see first-order difference equations) one can try a solution of the form =. The preceding equation then becomes =0 = =0. (19) Hence, assuming that the trial solution is non-trivial ( 6=0)implies =0. This quadratic equation in has two roots 12 = 1 ± p each of which is acceptable in the solution.infactboth 1 and 2 should appear in the solution, because the general solution to (18) must indeed consist of two linearly independent parts. Three cases must be distinguished: 1. When , the square root is a real number, and 1 and 2 are real and 14

16 distinct. The terms 1 and 2 are linearly independent, and can be written as = When 2 1 =4 2, the square root vanishes, and the characteristic roots are repeated 1 = 2 = 1 2. In this case, we have = with 3 = When , the characteristic roots are conjugate complex: 12 = ± with = 42 1 = 2 1.Inthiscase,weget 2 2 = = 1 ( + ) + 2 ( ). Using De Moivre s theorem, the solution can be transformed into a trigonometric form to read = ( 5 cos + 6 sin ) with = 2 (assumed positive), 5 = 1 + 2,and 6 =( 1 2 ). 5 The general solution of (17) is again = +. Stability requires that the real part of all roots ( 12 ) is less than one. 5.5 Some textbooks Dynamic optimization Chiang, A. C. (1992), Elements of Dynamic Optimization, McGraw Hill, New York. Dockner, E.J. et al., Differential Games in Economics and Management Science, 2000, Cambridge University Press, Cambridge. Feichtinger, Gustav and Richard F. Hartl (1986), Optimale Kontrolle ökonomischer Prozesse, Anwendung des Maximumprinzips in den Wirtschaftswissenschaften, de Gruyter, Berlin. (a valuable book, probably there is an english version?) Kamien, Morton I. and Nancy L. Schwartz (1981), Dynamic Optimization, The Calculus of Variations and Optimal Control in Economics and Management, North- Holland, New York. Silberberg, Eugene (1990), The Structure of Economics: A Mathematical Analysis, Mc Graw-Hill, New York. (the respective chapter contains a nice summary on control theory) 5 The parameter is determined by cos = and sin =. 15

17 Intriligator, Michael D. (1971), Mathematical Optimization and Economic Theory, Prentice-Hall, Inc., Englewood Cliffs, N.J. ("desipte its age" an excellent book on optimisation methods) Seierstad, Atle and K. Sydsaeter (1987) Optimal Control Theory with Economic Applications, North Holland. Dynamic systems Tu, Pierre N.V. (1994), Dynamical Systems, An Introduction with Applications in Economics and Biology, Springer-Verlag, Berlin. Lorenz, Hans-Walter (1989), Nonlinear Dynamical Economics and Chaotic Motion, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin. Gandolfo, Giancarlo (1996), Economic Dynamics, Springer-Verlag, Berlin, Heidelberg, New York. Chiang, A. C., Fundamental Methods of Mathematical Economics, McGraw-Hill, 1984, PART 5. Numerical solution: simulation of transitional dynamics Abell, M. L. andj. P. Braselton, Differential Equations with Mathematica, 1997, Academic Press, San Diego. Brunner, M. and H. Strulik, Solution of perfect foresight saddlepoint problems: a simple method and applications, Journal of Economic Dynamics and Control, 2002, 26, Judd, K., Numerical Methods in Economics, MIT Press, Cambridge (Massachusetts), Shone, R., Economic Dynamics, Cambridge University Press, Cambridge UK, Stochastic growth Dixit A. K. and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, New Jersey. Mikosch, T., Elementary Stochastic Calculus with Finance in View, World Scientific, Singapore, Turnovsky, Stephen J., Methods of Macroeconomic Dynamics, MIT Press, Cam- 16

18 bridge Mass, Part V, Jitka Dupacova, Jan Hurt, Josef Stepan, Stochastic Modeling in Economics and Finance (Applied Optimization, 75), Kluwer Academic Publishers; Kamien, Morton I. and Nancy L. Schwartz (1981), Dynamic Optimization, The Calculus of Variations and Optimal Control in Economics and Management, North- Holland, New York. SECTION 22 Wälde, Klaus, Applied Intertemporal Optimization, 2012 ( References [1] Chiang, A. C. and K. Wainwright (2005), Elements of Dynamic Optimization, McGraw Hill, New York. [2] Diewert, W.E., Index Numbers, in: Durlauf, Steven N. and Lawrence E. Blume (eds.), The New Palgrave Dictionary of Economics, Second Edition, Volume 4, Palgrave Macmillan, 2008, pp [3] Intriligator, Michael D. (1971), Mathematical Optimization and Economic Theory, Prentice-Hall, Inc., Englewood Cliffs, N.J. [4] Varian, Hal, Mathematica for Economists, Handbook of Computational Economics, Chapter 11, Vol. I, H.M. Amman, D.A. Kendrick, and J. Rust (eds.), Elsevier Science, [5] McKenzie, Lionel W., General Equilibrium, in: Durlauf, Steven N. and Lawrence E. Blume (eds.), The New Palgrave Dictionary of Economics, Second Edition, Volume 3, Palgrave Macmillan, 2008, pp [6] Varian, Hal, Microeconomic Analysis, W. W. Norton & Company, [7] Applied Intertemporal Optimization, 2012 ( 17

The Ramsey Model. (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 2013)

The Ramsey Model. (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 2013) The Ramsey Model (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 213) 1 Introduction The Ramsey model (or neoclassical growth model) is one of the prototype models in dynamic macroeconomics.

More information

Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path

Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path Ryoji Ohdoi Dept. of Industrial Engineering and Economics, Tokyo Tech This lecture note is mainly based on Ch. 8 of Acemoglu

More information

Lecture 6: Competitive Equilibrium in the Growth Model (II)

Lecture 6: Competitive Equilibrium in the Growth Model (II) Lecture 6: Competitive Equilibrium in the Growth Model (II) ECO 503: Macroeconomic Theory I Benjamin Moll Princeton University Fall 204 /6 Plan of Lecture Sequence of markets CE 2 The growth model and

More information

Advanced Macroeconomics

Advanced Macroeconomics Advanced Macroeconomics The Ramsey Model Marcin Kolasa Warsaw School of Economics Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 30 Introduction Authors: Frank Ramsey (1928), David Cass (1965) and Tjalling

More information

PRESENTATION OF MATHEMATICAL ECONOMICS SYLLABUS FOR ECONOMICS HONOURS UNDER CBCS, UNIVERSITY OF CALCUTTA

PRESENTATION OF MATHEMATICAL ECONOMICS SYLLABUS FOR ECONOMICS HONOURS UNDER CBCS, UNIVERSITY OF CALCUTTA PRESENTATION OF MATHEMATICAL ECONOMICS SYLLABUS FOR ECONOMICS HONOURS UNDER CBCS, UNIVERSITY OF CALCUTTA Kausik Gupta Professor of Economics, University of Calcutta Introductory Remarks The paper/course

More information

DYNAMIC LECTURE 5: DISCRETE TIME INTERTEMPORAL OPTIMIZATION

DYNAMIC LECTURE 5: DISCRETE TIME INTERTEMPORAL OPTIMIZATION DYNAMIC LECTURE 5: DISCRETE TIME INTERTEMPORAL OPTIMIZATION UNIVERSITY OF MARYLAND: ECON 600. Alternative Methods of Discrete Time Intertemporal Optimization We will start by solving a discrete time intertemporal

More information

Monetary Economics: Solutions Problem Set 1

Monetary Economics: Solutions Problem Set 1 Monetary Economics: Solutions Problem Set 1 December 14, 2006 Exercise 1 A Households Households maximise their intertemporal utility function by optimally choosing consumption, savings, and the mix of

More information

Dynamic Optimization Problem. April 2, Graduate School of Economics, University of Tokyo. Math Camp Day 4. Daiki Kishishita.

Dynamic Optimization Problem. April 2, Graduate School of Economics, University of Tokyo. Math Camp Day 4. Daiki Kishishita. Discrete Math Camp Optimization Problem Graduate School of Economics, University of Tokyo April 2, 2016 Goal of day 4 Discrete We discuss methods both in discrete and continuous : Discrete : condition

More information

Dynamic Optimization: An Introduction

Dynamic Optimization: An Introduction Dynamic Optimization An Introduction M. C. Sunny Wong University of San Francisco University of Houston, June 20, 2014 Outline 1 Background What is Optimization? EITM: The Importance of Optimization 2

More information

A suggested solution to the problem set at the re-exam in Advanced Macroeconomics. February 15, 2016

A suggested solution to the problem set at the re-exam in Advanced Macroeconomics. February 15, 2016 Christian Groth A suggested solution to the problem set at the re-exam in Advanced Macroeconomics February 15, 216 (3-hours closed book exam) 1 As formulated in the course description, a score of 12 is

More information

Study Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981).

Study Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981). Study Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981). (Copies are available from the Shell Centre for Mathematical

More information

Foundations of Modern Macroeconomics Second Edition

Foundations of Modern Macroeconomics Second Edition Foundations of Modern Macroeconomics Second Edition Chapter 5: The government budget deficit Ben J. Heijdra Department of Economics & Econometrics University of Groningen 1 September 2009 Foundations of

More information

Topic 5: The Difference Equation

Topic 5: The Difference Equation Topic 5: The Difference Equation Yulei Luo Economics, HKU October 30, 2017 Luo, Y. (Economics, HKU) ME October 30, 2017 1 / 42 Discrete-time, Differences, and Difference Equations When time is taken to

More information

Lecture 6: Discrete-Time Dynamic Optimization

Lecture 6: Discrete-Time Dynamic Optimization Lecture 6: Discrete-Time Dynamic Optimization Yulei Luo Economics, HKU November 13, 2017 Luo, Y. (Economics, HKU) ECON0703: ME November 13, 2017 1 / 43 The Nature of Optimal Control In static optimization,

More information

Session 4: Money. Jean Imbs. November 2010

Session 4: Money. Jean Imbs. November 2010 Session 4: Jean November 2010 I So far, focused on real economy. Real quantities consumed, produced, invested. No money, no nominal in uences. I Now, introduce nominal dimension in the economy. First and

More information

slides chapter 3 an open economy with capital

slides chapter 3 an open economy with capital slides chapter 3 an open economy with capital Princeton University Press, 2017 Motivation In this chaper we introduce production and physical capital accumulation. Doing so will allow us to address two

More information

Volume 30, Issue 3. A note on Kalman filter approach to solution of rational expectations models

Volume 30, Issue 3. A note on Kalman filter approach to solution of rational expectations models Volume 30, Issue 3 A note on Kalman filter approach to solution of rational expectations models Marco Maria Sorge BGSE, University of Bonn Abstract In this note, a class of nonlinear dynamic models under

More information

Chapter 4. Applications/Variations

Chapter 4. Applications/Variations Chapter 4 Applications/Variations 149 4.1 Consumption Smoothing 4.1.1 The Intertemporal Budget Economic Growth: Lecture Notes For any given sequence of interest rates {R t } t=0, pick an arbitrary q 0

More information

Lecture 3: Dynamics of small open economies

Lecture 3: Dynamics of small open economies Lecture 3: Dynamics of small open economies Open economy macroeconomics, Fall 2006 Ida Wolden Bache September 5, 2006 Dynamics of small open economies Required readings: OR chapter 2. 2.3 Supplementary

More information

Foundations of Modern Macroeconomics B. J. Heijdra & F. van der Ploeg Chapter 6: The Government Budget Deficit

Foundations of Modern Macroeconomics B. J. Heijdra & F. van der Ploeg Chapter 6: The Government Budget Deficit Foundations of Modern Macroeconomics: Chapter 6 1 Foundations of Modern Macroeconomics B. J. Heijdra & F. van der Ploeg Chapter 6: The Government Budget Deficit Foundations of Modern Macroeconomics: Chapter

More information

Dynamic (Stochastic) General Equilibrium and Growth

Dynamic (Stochastic) General Equilibrium and Growth Dynamic (Stochastic) General Equilibrium and Growth Martin Ellison Nuffi eld College Michaelmas Term 2018 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term 2018 1 / 43 Macroeconomics is Dynamic

More information

Solving the intertemporal consumption/saving problem in discrete and continuous time

Solving the intertemporal consumption/saving problem in discrete and continuous time Chapter 9 Solving the intertemporal consumption/saving problem in discrete and continuous time In the next two chapters we shall discuss the continuous-time version of the basic representative agent model,

More information

Ergodicity and Non-Ergodicity in Economics

Ergodicity and Non-Ergodicity in Economics Abstract An stochastic system is called ergodic if it tends in probability to a limiting form that is independent of the initial conditions. Breakdown of ergodicity gives rise to path dependence. We illustrate

More information

Advanced Macroeconomics

Advanced Macroeconomics Advanced Macroeconomics The Ramsey Model Micha l Brzoza-Brzezina/Marcin Kolasa Warsaw School of Economics Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 47 Introduction Authors:

More information

ECOM 009 Macroeconomics B. Lecture 2

ECOM 009 Macroeconomics B. Lecture 2 ECOM 009 Macroeconomics B Lecture 2 Giulio Fella c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 2 40/197 Aim of consumption theory Consumption theory aims at explaining consumption/saving decisions

More information

1 The Basic RBC Model

1 The Basic RBC Model IHS 2016, Macroeconomics III Michael Reiter Ch. 1: Notes on RBC Model 1 1 The Basic RBC Model 1.1 Description of Model Variables y z k L c I w r output level of technology (exogenous) capital at end of

More information

We shall finally briefly discuss the generalization of the solution methods to a system of n first order differential equations.

We shall finally briefly discuss the generalization of the solution methods to a system of n first order differential equations. George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 1 Ordinary Differential Equations In this mathematical annex, we define and analyze the solution of first and second order linear

More information

In the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now

In the Ramsey model we maximized the utility U = u[c(t)]e nt e t dt. Now PERMANENT INCOME AND OPTIMAL CONSUMPTION On the previous notes we saw how permanent income hypothesis can solve the Consumption Puzzle. Now we use this hypothesis, together with assumption of rational

More information

Lecture 4 The Centralized Economy: Extensions

Lecture 4 The Centralized Economy: Extensions Lecture 4 The Centralized Economy: Extensions Leopold von Thadden University of Mainz and ECB (on leave) Advanced Macroeconomics, Winter Term 2013 1 / 36 I Motivation This Lecture considers some applications

More information

Eco504 Spring 2009 C. Sims MID-TERM EXAM

Eco504 Spring 2009 C. Sims MID-TERM EXAM Eco504 Spring 2009 C. Sims MID-TERM EXAM This is a 90-minute exam. Answer all three questions, each of which is worth 30 points. You can get partial credit for partial answers. Do not spend disproportionate

More information

ECON4510 Finance Theory Lecture 1

ECON4510 Finance Theory Lecture 1 ECON4510 Finance Theory Lecture 1 Diderik Lund Department of Economics University of Oslo 18 January 2016 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 1 18 January 2016 1 / 38 Administrative

More information

Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time

Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:

More information

The Real Business Cycle Model

The Real Business Cycle Model The Real Business Cycle Model Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations of aggregate economic variables around their trend.

More information

Economic Growth: Theory and Policy (Timo Boppart s part)

Economic Growth: Theory and Policy (Timo Boppart s part) Economic Growth: Theory and Policy (Timo Boppart s part) Lecturer: Timo Boppart Stockholm University, PhD Program in Economics Q4, March/April, 2015. 1 General Information This part of the course consists

More information

Beiträge zur angewandten Wirtschaftsforschung

Beiträge zur angewandten Wirtschaftsforschung Institut für Volkswirtschaftslehre und Statistik No. 68- Stable Solutions to Homogeneous Difference- Differential Equations with Constant Coefficients Manfred Krtscha and Ulf von Kalckreuth Beiträge zur

More information

Rational Asset Price Bubbles and the Real Economy

Rational Asset Price Bubbles and the Real Economy Rational Asset Price Bubbles and the Real Economy Lecture in Advanced Macroeconomics II Benjamin Larin Leipzig University Institute for Theoretical Economics Macroeconomics July 2 and 6, 2015 (updated:

More information

MA Advanced Macroeconomics: Solving Models with Rational Expectations

MA Advanced Macroeconomics: Solving Models with Rational Expectations MA Advanced Macroeconomics: Solving Models with Rational Expectations Karl Whelan School of Economics, UCD February 6, 2009 Karl Whelan (UCD) Models with Rational Expectations February 6, 2009 1 / 32 Moving

More information

Course 16:198:520: Introduction To Artificial Intelligence Lecture 13. Decision Making. Abdeslam Boularias. Wednesday, December 7, 2016

Course 16:198:520: Introduction To Artificial Intelligence Lecture 13. Decision Making. Abdeslam Boularias. Wednesday, December 7, 2016 Course 16:198:520: Introduction To Artificial Intelligence Lecture 13 Decision Making Abdeslam Boularias Wednesday, December 7, 2016 1 / 45 Overview We consider probabilistic temporal models where the

More information

Introduction Optimality and Asset Pricing

Introduction Optimality and Asset Pricing Introduction Optimality and Asset Pricing Andrea Buraschi Imperial College Business School October 2010 The Euler Equation Take an economy where price is given with respect to the numéraire, which is our

More information

Toulouse School of Economics, Macroeconomics II Franck Portier. Homework 1. Problem I An AD-AS Model

Toulouse School of Economics, Macroeconomics II Franck Portier. Homework 1. Problem I An AD-AS Model Toulouse School of Economics, 2009-2010 Macroeconomics II Franck Portier Homework 1 Problem I An AD-AS Model Let us consider an economy with three agents (a firm, a household and a government) and four

More information

ES10006: Core skills for economists: Mathematics 2

ES10006: Core skills for economists: Mathematics 2 ES10006: Core skills for economists: Mathematics 2 Seminar 10 Problem set no. 7: Questions 6 and 7 Simultaneous difference equations The direct method The following system of n = 2 linear difference equations

More information

ES10006: Core skills for economists: Mathematics 2

ES10006: Core skills for economists: Mathematics 2 ES10006: Core skills for economists: Mathematics 2 Seminar 6 Problem set no. 4: Questions 1c, 1d, 2, 4, 7, 8b First-order linear differential equations From Chiang and Wainwright (2005, Section 15.1),

More information

Linear Riccati Dynamics, Constant Feedback, and Controllability in Linear Quadratic Control Problems

Linear Riccati Dynamics, Constant Feedback, and Controllability in Linear Quadratic Control Problems Linear Riccati Dynamics, Constant Feedback, and Controllability in Linear Quadratic Control Problems July 2001 Revised: December 2005 Ronald J. Balvers Douglas W. Mitchell Department of Economics Department

More information

Financial Factors in Economic Fluctuations. Lawrence Christiano Roberto Motto Massimo Rostagno

Financial Factors in Economic Fluctuations. Lawrence Christiano Roberto Motto Massimo Rostagno Financial Factors in Economic Fluctuations Lawrence Christiano Roberto Motto Massimo Rostagno Background Much progress made on constructing and estimating models that fit quarterly data well (Smets-Wouters,

More information

Macroeconomics: A Dynamic General Equilibrium Approach

Macroeconomics: A Dynamic General Equilibrium Approach Macroeconomics: A Dynamic General Equilibrium Approach Mausumi Das Lecture Notes, DSE Jan 23-Feb 23, 2018 Das (Lecture Notes, DSE) DGE Approach Jan 23-Feb 23, 2018 1 / 135 Modern Macroeconomics: the Dynamic

More information

ECON 5118 Macroeconomic Theory

ECON 5118 Macroeconomic Theory ECON 5118 Macroeconomic Theory Winter 013 Test 1 February 1, 013 Answer ALL Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages

More information

ECON4510 Finance Theory Lecture 2

ECON4510 Finance Theory Lecture 2 ECON4510 Finance Theory Lecture 2 Diderik Lund Department of Economics University of Oslo 26 August 2013 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 2 26 August 2013 1 / 31 Risk aversion and

More information

Macroeconomics Qualifying Examination

Macroeconomics Qualifying Examination Macroeconomics Qualifying Examination January 2016 Department of Economics UNC Chapel Hill Instructions: This examination consists of 3 questions. Answer all questions. If you believe a question is ambiguously

More information

KIER DISCUSSION PAPER SERIES

KIER DISCUSSION PAPER SERIES KIER DISCUSSION PAPER SERIES KYOTO INSTITUTE OF ECONOMIC RESEARCH Discussion Paper No.992 Intertemporal efficiency does not imply a common price forecast: a leading example Shurojit Chatterji, Atsushi

More information

1 Overlapping Generations

1 Overlapping Generations 1 Overlapping Generations 1.1 Motivation So far: infinitely-lived consumer. Now, assume that people live finite lives. Purpose of lecture: Analyze a model which is of interest in its own right (and which

More information

Optimal Control. Macroeconomics II SMU. Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112

Optimal Control. Macroeconomics II SMU. Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112 Optimal Control Ömer Özak SMU Macroeconomics II Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112 Review of the Theory of Optimal Control Section 1 Review of the Theory of Optimal Control Ömer

More information

Local Determinacy of Prices in an Overlapping Generations Model with Continuous Trading

Local Determinacy of Prices in an Overlapping Generations Model with Continuous Trading MPRA Munich Personal RePEc Archive Local Determinacy of Prices in an Overlapping Generations Model with Continuous Trading Hippolyte d Albis and Emmanuelle Augeraud-Véron and Herman Jan Hupkes 7. October

More information

The basic representative agent model: Ramsey

The basic representative agent model: Ramsey Chapter 10 The basic representative agent model: Ramsey As early as 1928 a sophisticated model of a society s optimal saving was published by the British mathematician and economist Frank Ramsey (1903-1930).

More information

Economics 232c Spring 2003 International Macroeconomics. Problem Set 3. May 15, 2003

Economics 232c Spring 2003 International Macroeconomics. Problem Set 3. May 15, 2003 Economics 232c Spring 2003 International Macroeconomics Problem Set 3 May 15, 2003 Due: Thu, June 5, 2003 Instructor: Marc-Andreas Muendler E-mail: muendler@ucsd.edu 1 Trending Fundamentals in a Target

More information

A Variant of Uzawa's Theorem. Abstract

A Variant of Uzawa's Theorem. Abstract A Variant of Uzawa's Theorem Ekkehart Schlicht Department of Economics, University of Munich Abstract Uzawa (1961) has shown that balanced growth requires technological progress to be strictly Harrod neutral

More information

A simple macro dynamic model with endogenous saving rate: the representative agent model

A simple macro dynamic model with endogenous saving rate: the representative agent model A simple macro dynamic model with endogenous saving rate: the representative agent model Virginia Sánchez-Marcos Macroeconomics, MIE-UNICAN Macroeconomics (MIE-UNICAN) A simple macro dynamic model with

More information

1. Money in the utility function (start)

1. Money in the utility function (start) Monetary Economics: Macro Aspects, 1/3 2012 Henrik Jensen Department of Economics University of Copenhagen 1. Money in the utility function (start) a. The basic money-in-the-utility function model b. Optimal

More information

Last Update: March 1 2, 201 0

Last Update: March 1 2, 201 0 M ath 2 0 1 E S 1 W inter 2 0 1 0 Last Update: March 1 2, 201 0 S eries S olutions of Differential Equations Disclaimer: This lecture note tries to provide an alternative approach to the material in Sections

More information

Suggested Solutions to Homework #3 Econ 511b (Part I), Spring 2004

Suggested Solutions to Homework #3 Econ 511b (Part I), Spring 2004 Suggested Solutions to Homework #3 Econ 5b (Part I), Spring 2004. Consider an exchange economy with two (types of) consumers. Type-A consumers comprise fraction λ of the economy s population and type-b

More information

"0". Doing the stuff on SVARs from the February 28 slides

0. Doing the stuff on SVARs from the February 28 slides Monetary Policy, 7/3 2018 Henrik Jensen Department of Economics University of Copenhagen "0". Doing the stuff on SVARs from the February 28 slides 1. Money in the utility function (start) a. The basic

More information

The representative agent model

The representative agent model Chapter 3 The representative agent model 3.1 Optimal growth In this course we re looking at three types of model: 1. Descriptive growth model (Solow model): mechanical, shows the implications of a given

More information

ECON FINANCIAL ECONOMICS

ECON FINANCIAL ECONOMICS ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Spring 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International

More information

Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6

Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6 1 Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6 1 A Two-Period Example Suppose the economy lasts only two periods, t =0, 1. The uncertainty arises in the income (wage) of period 1. Not that

More information

An Introduction to Noncooperative Games

An Introduction to Noncooperative Games An Introduction to Noncooperative Games Alberto Bressan Department of Mathematics, Penn State University Alberto Bressan (Penn State) Noncooperative Games 1 / 29 introduction to non-cooperative games:

More information

Permanent Income Hypothesis Intro to the Ramsey Model

Permanent Income Hypothesis Intro to the Ramsey Model Consumption and Savings Permanent Income Hypothesis Intro to the Ramsey Model Lecture 10 Topics in Macroeconomics November 6, 2007 Lecture 10 1/18 Topics in Macroeconomics Consumption and Savings Outline

More information

Simple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) Jonathan Heathcote. updated, March The household s problem X

Simple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) Jonathan Heathcote. updated, March The household s problem X Simple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) subject to for all t Jonathan Heathcote updated, March 2006 1. The household s problem max E β t u (c t ) t=0 c t + a t+1

More information

Interest Calculation and Dimensional Analysis. Estola, Matti. ISBN ISSN no 27

Interest Calculation and Dimensional Analysis. Estola, Matti. ISBN ISSN no 27 Interest Calculation and Dimensional Analysis Estola, Matti ISBN 952-458-710-6 ISSN 1795-7885 no 27 Interest Calculation and Dimensional Analysis Matti Estola University of Joensuu, Dept of Business and

More information

Handout 4: Some Applications of Linear Programming

Handout 4: Some Applications of Linear Programming ENGG 5501: Foundations of Optimization 2018 19 First Term Handout 4: Some Applications of Linear Programming Instructor: Anthony Man Cho So October 15, 2018 1 Introduction The theory of LP has found many

More information

Economics 210B Due: September 16, Problem Set 10. s.t. k t+1 = R(k t c t ) for all t 0, and k 0 given, lim. and

Economics 210B Due: September 16, Problem Set 10. s.t. k t+1 = R(k t c t ) for all t 0, and k 0 given, lim. and Economics 210B Due: September 16, 2010 Problem 1: Constant returns to saving Consider the following problem. c0,k1,c1,k2,... β t Problem Set 10 1 α c1 α t s.t. k t+1 = R(k t c t ) for all t 0, and k 0

More information

Introduction to Real Business Cycles: The Solow Model and Dynamic Optimization

Introduction to Real Business Cycles: The Solow Model and Dynamic Optimization Introduction to Real Business Cycles: The Solow Model and Dynamic Optimization Vivaldo Mendes a ISCTE IUL Department of Economics 24 September 2017 (Vivaldo M. Mendes ) Macroeconomics (M8674) 24 September

More information

The Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over,

The Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over, The Derivative of a Function Measuring Rates of Change of a function y f(x) f(x 0 ) P Q Secant line x 0 x x Average rate of change of with respect to over, " " " " - Slope of secant line through, and,

More information

Online Appendix I: Wealth Inequality in the Standard Neoclassical Growth Model

Online Appendix I: Wealth Inequality in the Standard Neoclassical Growth Model Online Appendix I: Wealth Inequality in the Standard Neoclassical Growth Model Dan Cao Georgetown University Wenlan Luo Georgetown University July 2016 The textbook Ramsey-Cass-Koopman neoclassical growth

More information

STRUCTURE Of ECONOMICS A MATHEMATICAL ANALYSIS

STRUCTURE Of ECONOMICS A MATHEMATICAL ANALYSIS THIRD EDITION STRUCTURE Of ECONOMICS A MATHEMATICAL ANALYSIS Eugene Silberberg University of Washington Wing Suen University of Hong Kong I Us Irwin McGraw-Hill Boston Burr Ridge, IL Dubuque, IA Madison,

More information

Graduate Macroeconomics 2 Problem set Solutions

Graduate Macroeconomics 2 Problem set Solutions Graduate Macroeconomics 2 Problem set 10. - Solutions Question 1 1. AUTARKY Autarky implies that the agents do not have access to credit or insurance markets. This implies that you cannot trade across

More information

THE SOLOW-SWAN MODEL WITH A NEGATIVE LABOR GROWTH RATE

THE SOLOW-SWAN MODEL WITH A NEGATIVE LABOR GROWTH RATE Journal of Mathematical Sciences: Advances and Applications Volume 9, Number /,, Pages 9-38 THE SOLOW-SWAN MODEL WITH A NEGATIVE LABOR GROWTH RATE School of Economic Mathematics Southwestern University

More information

Chaos in GDP. Abstract

Chaos in GDP. Abstract Chaos in GDP R. Kříž Abstract This paper presents an analysis of GDP and finds chaos in GDP. I tried to find a nonlinear lower-dimensional discrete dynamic macroeconomic model that would characterize GDP.

More information

Mathematical models in economy. Short descriptions

Mathematical models in economy. Short descriptions Chapter 1 Mathematical models in economy. Short descriptions 1.1 Arrow-Debreu model of an economy via Walras equilibrium problem. Let us consider first the so-called Arrow-Debreu model. The presentation

More information

University of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming

University of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming University of Warwick, EC9A0 Maths for Economists 1 of 63 University of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming Peter J. Hammond Autumn 2013, revised 2014 University of

More information

Structural Dynamics Prof. P. Banerji Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 1 Introduction

Structural Dynamics Prof. P. Banerji Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 1 Introduction Structural Dynamics Prof. P. Banerji Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 1 Introduction Hello, I am Pradipta Banerji from the department of civil engineering,

More information

Lecture 1: The Classical Optimal Growth Model

Lecture 1: The Classical Optimal Growth Model Lecture 1: The Classical Optimal Growth Model This lecture introduces the classical optimal economic growth problem. Solving the problem will require a dynamic optimisation technique: a simple calculus

More information

1 Bewley Economies with Aggregate Uncertainty

1 Bewley Economies with Aggregate Uncertainty 1 Bewley Economies with Aggregate Uncertainty Sofarwehaveassumedawayaggregatefluctuations (i.e., business cycles) in our description of the incomplete-markets economies with uninsurable idiosyncratic risk

More information

Department of Economics, UCSB UC Santa Barbara

Department of Economics, UCSB UC Santa Barbara Department of Economics, UCSB UC Santa Barbara Title: Past trend versus future expectation: test of exchange rate volatility Author: Sengupta, Jati K., University of California, Santa Barbara Sfeir, Raymond,

More information

The Pasinetti-Solow Growth Model With Optimal Saving Behaviour: A Local Bifurcation Analysis

The Pasinetti-Solow Growth Model With Optimal Saving Behaviour: A Local Bifurcation Analysis The Pasinetti-Solow Growth Model With Optimal Saving Behaviour: A Local Bifurcation Analysis Pasquale Commendatore 1 and Cesare Palmisani 2 1 Dipartimento di Teoria Economica e Applicazioni Università

More information

Getting to page 31 in Galí (2008)

Getting to page 31 in Galí (2008) Getting to page 31 in Galí 2008) H J Department of Economics University of Copenhagen December 4 2012 Abstract This note shows in detail how to compute the solutions for output inflation and the nominal

More information

A Simple No-Bubble Theorem for Deterministic Sequential Economies

A Simple No-Bubble Theorem for Deterministic Sequential Economies A Simple No-Bubble Theorem for Deterministic Sequential Economies Takashi Kamihigashi March 26, 2016 Abstract We show a simple no-bubble theorem that applies to a wide range of deterministic sequential

More information

Linear Riccati Dynamics, Constant Feedback, and Controllability in Linear Quadratic Control Problems

Linear Riccati Dynamics, Constant Feedback, and Controllability in Linear Quadratic Control Problems Linear Riccati Dynamics, Constant Feedback, and Controllability in Linear Quadratic Control Problems July 2001 Ronald J. Balvers Douglas W. Mitchell Department of Economics Department of Economics P.O.

More information

Economic Growth: Lecture 9, Neoclassical Endogenous Growth

Economic Growth: Lecture 9, Neoclassical Endogenous Growth 14.452 Economic Growth: Lecture 9, Neoclassical Endogenous Growth Daron Acemoglu MIT November 28, 2017. Daron Acemoglu (MIT) Economic Growth Lecture 9 November 28, 2017. 1 / 41 First-Generation Models

More information

Comprehensive Exam. Macro Spring 2014 Retake. August 22, 2014

Comprehensive Exam. Macro Spring 2014 Retake. August 22, 2014 Comprehensive Exam Macro Spring 2014 Retake August 22, 2014 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question.

More information

American Journal of Economics and Business Administration Vol. 9. No. 3. P DOI: /ajebasp

American Journal of Economics and Business Administration Vol. 9. No. 3. P DOI: /ajebasp American Journal of Economics and Business Administration. 2017. Vol. 9. No. 3. P. 47-55. DOI: 10.3844/ajebasp.2017.47.55 ACCELERATORS IN MACROECONOMICS: COMPARISON OF DISCRETE AND CONTINUOUS APPROACHES

More information

Topic 6: Projected Dynamical Systems

Topic 6: Projected Dynamical Systems Topic 6: Projected Dynamical Systems John F. Smith Memorial Professor and Director Virtual Center for Supernetworks Isenberg School of Management University of Massachusetts Amherst, Massachusetts 01003

More information

Macroeconomics Theory II

Macroeconomics Theory II Macroeconomics Theory II Francesco Franco FEUNL February 2011 Francesco Franco Macroeconomics Theory II 1/34 The log-linear plain vanilla RBC and ν(σ n )= ĉ t = Y C ẑt +(1 α) Y C ˆn t + K βc ˆk t 1 + K

More information

Information Choice in Macroeconomics and Finance.

Information Choice in Macroeconomics and Finance. Information Choice in Macroeconomics and Finance. Laura Veldkamp New York University, Stern School of Business, CEPR and NBER Spring 2009 1 Veldkamp What information consumes is rather obvious: It consumes

More information

The Australian National University: Research School of Economics. An Introduction to Mathematical Techniques for Economic Analysis

The Australian National University: Research School of Economics. An Introduction to Mathematical Techniques for Economic Analysis The Australian National University: Research School of Economics An Introduction to Mathematical Techniques for Economic Analysis January and February 2016 Syllabus Instructor s Details Instructor: Dr

More information

Foundations of Modern Macroeconomics Second Edition

Foundations of Modern Macroeconomics Second Edition Foundations of Modern Macroeconomics Second Edition Chapter 4: Anticipation effects and economic policy BJ Heijdra Department of Economics, Econometrics & Finance University of Groningen 1 September 2009

More information

ECON2285: Mathematical Economics

ECON2285: Mathematical Economics ECON2285: Mathematical Economics Yulei Luo FBE, HKU September 2, 2018 Luo, Y. (FBE, HKU) ME September 2, 2018 1 / 35 Course Outline Economics: The study of the choices people (consumers, firm managers,

More information

A Summary of Economic Methodology

A Summary of Economic Methodology A Summary of Economic Methodology I. The Methodology of Theoretical Economics All economic analysis begins with theory, based in part on intuitive insights that naturally spring from certain stylized facts,

More information

Impatience vs. Incentives

Impatience vs. Incentives Impatience vs. Incentives Marcus Opp John Zhu University of California, Berkeley (Haas) & University of Pennsylvania, Wharton January 2015 Opp, Zhu (UC, Wharton) Impatience vs. Incentives January 2015

More information

Endogenous Growth: AK Model

Endogenous Growth: AK Model Endogenous Growth: AK Model Prof. Lutz Hendricks Econ720 October 24, 2017 1 / 35 Endogenous Growth Why do countries grow? A question with large welfare consequences. We need models where growth is endogenous.

More information

Mathematics Camp for Economists. Rice University Summer 2016

Mathematics Camp for Economists. Rice University Summer 2016 Mathematics Camp for Economists Rice University Summer 2016 Logistics Instructor: TA: Schedule: Time: Location: Office Hours: Metin Uyanık, muyanik1@jhu.edu Atara Oliver, sao5@rice.edu July 1 - July 29,

More information

One Variable Calculus: Foundations and Applications

One Variable Calculus: Foundations and Applications One Variable Calculus: Foundations and Applications Prof. Manuela Pedio 20550 Quantitative Methods for Finance August 2018 About myself I graduated from Bocconi s MSc in Finance in 2013 (yes, I remember

More information