Visualizing Multi-Dimensional Functions In Economics

This paper shows how a forward-shooting algorithm can be easily implemented using the Matlab programming language. In the paper we develop and implement a forward-shooting numerical algorithm for computing the dynamics of a small representative agent macroeconomic model when subjected to an exogenous shock. The model has a number of important characteristics that effect its dynamic solution. These characteristics are common to a wide range of economic models. One characteristic is that the model is highly non-linear so that numeric solutions to the dynamics are necessary. Another characteristic is that the solution must lie on a stable manifold. This requires the model to contain jumping variables and makes the numeric solution of the dynamics difficult as the solution will easily 'fall off' the stable manifold. The model terminal solution is particularly sensitive to initial conditions and to computational errors introduced in the solution.A forward-shooting algorithm can be used to find the necessary jumps to ensure the model lies on the stable manifold. In the paper we use computer visualisation techniques to show the algorithm searching for the necessary jumps to determine the dynamic solution. The algorithm searches for the initial conditions of the jumping variables that result in the terminal solution of the model being close to the known values.The paper shows how the algorithm, and its visualisation, can be implemented in the Matlab programming language using standard Matlab routines. The program implements a Nelder-Mead simplex search to find the particular initial conditions of the jumping variables that minimises the 2-norm between known terminal solution and that generated by the candidate initial conditions. For each candidate initial conditions a Runge-Kutta solver is used to generate the dynamics of the model, and plotting routines are used to visualise these dynamics.