Basic Tools for Dynamic Economic Analysis

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Basic Tools for Dynamic Economic Analysis

• Difference Equations
• Dynamic Programming

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• Dynamic macroeconomics involves the behavior of
  variables over time.
• We will briefly describe some useful tools for
  studying economic dynamics
• (1) Dynamic Programming ? Algorithm for solving
  dynamic optimization problems (alternative to
  Lagrangian)
• (2) Difference Equations ? Characterizing the
  solution to a dynamic equation or system of
  equations.

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Difference Equations

• An n-th order ordinary difference equation with
  endogenous variable x
• A first order linear difference equation
• where are constants.
• If z 0 this is called a homogenous difference
  equation.

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• The general solution is given by the sum of two
  parts
• (1) Homogenous (or time-varying) solution found
  by setting z 0 ?
• (2) A steady state (or constant) solution where
  xt xt1 x ?
• (3) General Solution ?

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• Example 1 Homogenous equation
• ?
• where x0 is given.
• (1) Homogenous solution
• (2) Steady State Solution
• (3) General Solution

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• CASES
• (i) a 0 ? xt 0 for all t
• (ii) a 1 ? xt x0 for all t
• (iii) 0 lt a lt 1 ? Convergence.
• (iv) a gt 1 ? Divergence.

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• Example 2 Non-Homogenous equation
• where x0 is given, z constant.
• Solution
• (1) Homogeneous Set z 0 ?
• where A is a constant.
• (2) Steady State Set xt xt1 x

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• (3) General Solution
• Given xt x0 when t 0 ?
• ?

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• CASES
• (i) 0 lt a lt 1 ? Convergence
• (ii) a gt 1 ? Divergence
• (iii) -1 lt a lt 0 ? Damped Oscillations
  
• The homogenous solution can be used to study the
  stability of economic variables over time (e.g.
  value of stock market, inflation).
• It is often times convenient to rule out time
  varying solutions and focus on the steady state
  solution.