Time Representation in Economics

In this paper, we study general polynomial discretizations in backward and forward looking, and the preservation of stability properties. We apply these results to the Ramsey model [1]. Its discrete-time version is a hybrid discretizations of a backward-looking budget constraint and a forward-looking Euler equation. Saddle-path stability is a robust property under discretization.


Discretizations
Continuous-time systems can be approximated by discretetime systems.In the spirit of Krivine, Lesne and Treiner [2], we bridge continuous and discrete-time dynamics through general polynomial discretizations.
Discretizations can differ according to the step, the order and the direction of discretization.The step gives the length of the period in discrete time.The order is that of the Taylor expansion of a continuous-time model.The direction depends on the backward or forward-looking nature of this Taylor expansion.A hybrid discretization mixes backward and forward-looking approximations.
We want to show that the steady state is invariant to the step, the order and the direction of discretization and its continuous-time stability properties (sink, saddle, source) are preserved under a sufficiently small discretization step in any case (backward, forward or hybrid).
Instead of considering a continuous variable t and the corresponding position   x t determined by an m-dimensional system of ordinary differential equations: where 0 , f C  jointly with the initial condition let us pick up a regular sequence of time values: where h is a (possibly small) positive constant (discretization step), and the associated values: The path from n x to 1 n x  can be reconstructed component by component through an appropriate integration of . Focusing on the ith component of the vector , we can integrate the time derivative on the right or on the left to obtain, respectively, ).The Euler-Taylor discretization is a polynomial approximation.Assuming that and considering the qth order polynomial, we obtain a backward or a forward-looking discretization: . A discretization is said to be hybrid if (1) holds for some components of the vector x and (2) holds for the others.

Setting
, we obtain from (1) and (2) a first-order discretization: where the subscript i denotes the ith component of the vector.

  
x nh , exact solution to system   = x f x  : the smaller h, the more accurate the representation.
Higher-order discretizations are also possible.Let us discretize the continuous-time dynamical system by second-order Taylor polynomials, that is approximate the ith component of n 1 x  ith a quadratic form.Using (1) and (2), we obtain in backward and forward-looking, respectively: where the subscript i denotes the ith component of the vector.
If f is an analytic function, infinite-order backward or forward discretizations converge exactly to 1 n n x x   and (1) and (2) now hold with equality: In this case, the Taylor polynomials become a convergent series and the discretized dynamics represent exactly the continuous-time system whatever the step h.
In general, a discretization is a closer approximation of a continuous-time system when the step h is smaller or the order of discretization q higher.The dynamic proper-ties of a continuous-time system can be preserved lowering h or increasing q.

Dynamic Equivalence
To compare continuous-time and discrete-time system, we study approximations in a neighborhood of the steady state and focus on the persistence of local stability properties.
Focus first on the steady state.The system have the same steady state.Indeed, in both the cases, we require   = 0 f x (respectively, and = 0 We further notice that the system of m equations   = 0 f x neither depends on the discretization degree h nor on the discretization method (forward or backward-looking).Therefore, the steady state is invariant to discretization.
Focus now on the stability properties.Are they preserved under discretization in a neighborhood of the steady state?
Without loss of generality, we consider two-dimensional dynamics.In the spirit of Samuelson [3], we can represent the stability properties in the plane of trace T and determinant D of the Jacobian matrix J of the system evaluated at the steady state.
In the following, the subscripts and 1 will denote variables in continuous or discrete time respectively.0 1) In continuous time, stability depends on the real part of these eigenvalues.If both the real parts are negative (positive), the steady state is a sink (source) (in this case, the trace of 0 J is negative (positive) and the determinant of 0 J is positive (positive)).If the signs of the real parts are different, the eigenvalues are real and the steady state is a saddle point (in this case, the determinant is negative).
2) In discrete time, the modulus of an eigenvalue a ib  matters.When ( > 1) the eigenvalue is inside (outside) the unit circle.If both the eigenvalues are inside (outside) the unit circle, the steady state is a sink (source).If one is inside and the other outside the unit circle, the steady state is a saddle point.We can evaluate the characteristic polynomial Along the line , one eigenvalue is equal to -1 because , the two eigenvalues are nonreal and conjugate with unit modulus.Consider first the points that neither belong to these lines nor to the segment.Inside the triangle defined by and , T D lies on the left sides of both the lines Copyright © 2012 SciRes.
TEL < T ).It is a source otherwise.
At l 1 D  a two-dimensional system is required to study east, the three cases (sink, saddle and source) together and to consider hybrid discretizations.Without loss of generality, we linearize the following system of ordinary differential equations dynamics around the steady state are Local by represented the Jacobian matrix 0 J evaluated at the steady state ). isc 0 er d We focus len
We linearize the backward-looking discretization m (5) around the c of the syste where I a ommon steady state and we obtain   J are the two-d and Jacobian atrix of system (6).We observe that 0 imensional identity matrix m J depends on the steady state x which, in turn, does depend on h.Then tization step etermine the intervals of equivalence between the continuous and the discrete-time dynamics:  2). 3)If the steady state is a source in continuous time, th generically undergoes a Hopf bifurcation at en the source property is preserved whatever > 0 h (Figure 3).
The system 1 H h and flip bifurcations at Fi h , = 1, 2 i .Proof From ( 7) and (8), it is ss pl po ible to ot a curve for each one of these different cases:   Copyright We linearize now the forward-looking discre Differently from the previous case, the Jacobian matri 0 x of system (9) is no longer linear in h.The trace and th of 1 e determinant J are now given by   As above, we set three critical values:  3) Let the steady state be a source in continuous time.< < and a sink if . 4 The system generically undergoes a Hopf bifurcation at and flip bifurcations at Fi Proof The proof is similar to that of Proposition 1. See Bosi and Ragot [4] for more details.h , .= 3, 4 i Corollary 4 (topological equivalence in forward looking) In every case of Proposition 3, there exists a nonempty interval   0, h  for the discretization step h where the stability properties of the continuous-time system are preserved.
Proof Straightforward.Simply observe that, in cases (1) and (2.1), = h   .The same happens in the case

Hybrid Discretizations
In economics, many higher-dimensional models require a hybrid discretization to recover the equivalence between discrete and continuous time, that is a mix of discretization in backward and forward looking.Without loss of generality, we consider a system where the first equation is discretized backward and the second one forward.Thus, the system of differential Equations (5) becomes: The steady state is invariant to the choice of time and to the type of discretization (backward/forward).The trace and the determinant of the Jacobian matrix 1 J of the hybrid system (10)-(11) become Notice that, in the particular case 2 2 = 0 f x   , (12) and (13) write , then the steady state is a saddle point.

Ramsey Model
In the seminal Ramsey [1], the planner maximizes the undiscounted dynastic utility: , un- where t and t denote the individual capital and consumption.The initial endowment is given.
The intensive production function f k is strictly increasing and strictly concave in the capital intensity and satisfies the Inada conditions.The felicity   u c is also strictly increasing and strictly concave in the consumption level.c denotes the bliss point, that is the steady state value of consumption: The planner maximizes the Hamiltonian: to find the first-order conditions: where . The strict concavity of u ensures that is a well-defined function of the multiplier  .
In discrete time, the planner maximizes under a sequence of resource constraints: , to obtain the firstorder conditions: We want to prove that the discrete-time system ( 16)-( 17) is a discretization of the continuous-time system ( 14 we recover exactly the discrete-time resource constraint (16) with a unit discretization step ( ).However, the intertemporal arbitrage requires a forward-looking discretization.Focus on (15) and apply (4): which gives exactly the discrete-time Euler Equation (17) under a unit discretization step .= 1 h The forward-looking discretization of (15) is more suitable to capture saving decisions.Indeed, the expected productivity affects the arbitrage between consumption today and consumption tomorrow.
Let us consider the steady state.For all the three dynamical systems (14)-( 15), ( 16)-( 17) and ( 18)-(19) the steady state is defined by:   = Focus now on the stability properties.The Jacobian matrix 0 J of the continuous time system (14)-( 15) is given by: Pro odel is a saddle point (as in the continuous-time case) whatever the discretization step Proof The Jacobian matrix 1 J of the hybrid Euler discretization (18)-( 19) is: where A and B are defined above.The trace and determinant become B and .We obt 1 r th ath stability r the discreti h room for bifur scr

REFERENCES
[1] F. Ramsey, " of Saving," Economic Journal , pp. 543-559.property, whateve zation step .There is no cations, as in the continuous-time case.Therefore, the saddle-path stability is a robust property of the Ramsey model because it holds whatever the di 1 < D T   etization step.
on the past value n x (future value 1 n x  ) on the right-hand side.Equation (3) is the classical Euler discretization.In economics, forward-looking discretizations are of interest because agents behave according to their expectations.The sequences   n x us denote the trace and determinant of J and 1

4
technology and preferences ensure its existence and uniqueness).
the continuous-time case.
If the steady state is a sink in continuous time, then the sink property is preserved in discrete time whatever If the steady state is a sink in continuous time, then the steady state in discrete time is a sink if Let the steady state be a saddle in continuous time.
, then the steady state is a saddle if 0 <