Dynamic Macroeconomics (Concepts, Techniques, Applications)

Dynamic Macroeconomics (Concepts, Techniques, Applications) this version: June 2013 Thomas Steger University of Leipzig Institute for Theoretical Economics Macroeconomics e-mail: steger@wifa.uni-leipzig.de 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. (). 5.1.1 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

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

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 = 0 +. 2

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 12.2.6), who applies the method of variation of parameters. Consider the following DE 0 + 1 = () 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

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

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 := + 1 2 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

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 ) + + 0 + (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

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

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

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 0 1 2... 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

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 4 5.4.1 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

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= 0. 5.4.2 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 = 0 2 0 3 0 In more general notation, the solution to a linear, homogenous, first-order difference equation is =. 11

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

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 ( ) + 1+ 5.4.3 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). 5.4.4 Second-order difference equations A second-order difference equation is one that contains the second difference 2 = ( )= ( +1 ) = ( +2 1 ) ( +1 ) = +2 2 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

linear, non-homogenous, second-order difference equations with constant coefficients, like the following gives +2 + 1 +1 + 2 = (17) Particular solution. We try a solution of the form =. Plugging this into (17) + 1 + 2 = = = 1+ 1 + 2 Assuming that 1 + 2 6= 1 this gives = 1+ 1 + 2. If, on the other hand, 1 + 2 = 1, the trial solution should be =, which yields ( +2)+ 1 ( +1)+ 2 = = = 1 +2 and thus = 1 ("a moving equilibrium"). +2 Complementary solution. Thisisthesolutionto +2 + 1 +1 + 2 =0 (18) Motivated by experience (see first-order difference equations) one can try a solution of the form =. The preceding equation then becomes +2 + 1 +1 + 2 =0 = 2 + 1 + 2 =0. (19) Hence, assuming that the trial solution is non-trivial ( 6=0)implies 2 + 1 + 2 =0. This quadratic equation in has two roots 12 = 1 ± p 2 1 4 2 2 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 2 1 4 2, the square root is a real number, and 1 and 2 are real and 14

distinct. The terms 1 and 2 are linearly independent, and can be written as = 1 1 + 2 2. 2. When 2 1 =4 2, the square root vanishes, and the characteristic roots are repeated 1 = 2 = 1 2. In this case, we have = 3 + 4 with 3 = 1 + 2. 3. When 2 1 4 2, the characteristic roots are conjugate complex: 12 = ± with = 42 1 = 2 1.Inthiscase,weget 2 2 = 1 1 + 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

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, 737-753. Judd, K., Numerical Methods in Economics, MIT Press, Cambridge (Massachusetts), 1999. Shone, R., Economic Dynamics, Cambridge University Press, Cambridge UK, 2002. 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, 1998. Turnovsky, Stephen J., Methods of Macroeconomic Dynamics, MIT Press, Cam- 16

bridge Mass, Part V, 1996. Jitka Dupacova, Jan Hurt, Josef Stepan, Stochastic Modeling in Economics and Finance (Applied Optimization, 75), Kluwer Academic Publishers; 2002. 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 (http://www.waelde.com/aio.html) 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. 190-214. [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, 1996. [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. 586-604. [6] Varian, Hal, Microeconomic Analysis, W. W. Norton & Company, 1992. [7] Applied Intertemporal Optimization, 2012 (http://www.waelde.com/aio.html) 17