Solution Methods

RunGTAP offers a variety of different solution methods:

•The SINGLE-STEP SOLUTION, or Johansen method, treats the model as a linear system, linearized around the initial solution. This approach is the simplest and quickest computationally. However, since GTAP is actually a non-linear system, Johansen results are not quite accurate, except for small shocks. The errors are super-proportional to the size of the shock: double the size of the shock, and the size of the errors will more than double.

•MULTI-STEP SOLUTION procedures are used to reduce linearization errors which arise from the default one-step or Johansen solution method. Briefly, the Euler multi-step procedure automatically divides the exogenous shock into a (user-specified) number of equal components. For example, a 10% increase in labour supply might be computed as two successive increases of 4.88% (1.0488 x 1.0488 = 1.1). Results for the first 4.88% instalment are calculated and the database is updated accordingly. Using the new database, results are calculated for the second 4.88% instalment. Since errors are super-proportional to the size of the shock, halving the shock leads to errors at each step which are less than half the size of the error produced by a single, full-size, step. Thus, the results from the two steps may be combined to produce a solution that is more accurate than that obtained by a single step. The more steps, the more accuracy.

The Gragg and Midpoint methods are variations on the Euler method -- they can sometimes produce more accurate results for a given number of steps.

•SEQUENCE OF SOLUTIONS WITH EXTRAPOLATION -- Early experiments in solving models by the Euler method led to the following observations. The differences between a 8-step and a 16-step solution are often about half those between a 4-step and an 8-step solution. The differences between a 16-step and a 32-step solution are about half those between a 8-step and a 16-step solution, and so on. This rule enables us to predict what results would be generated by a solution with an infinite number of steps -- that is, the exact solution.

For example, we might choose to run 3 Euler solutions with respectively 4, 8 and 12 steps. GEMPACK automatically uses these results to extrapolate to a solution that is more accurate than any of the 3 individual solutions. For Gragg and Midpoint methods the numbers of steps in a sequence of solutions must be either all odd or all even.