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More about the Theory of Subtotals Suppose a model was written: (1) Z= F(X,Y) Here Z is an endogenous variable and there are two exogenous variables X and Y. In a real model, Z would be a vector and there would be many more exogenous variables. However, this simple model is enough to demonstrate the principle of subtotals. For small changes around an initial solution we can write the linearized form: (2) delZ= [Fx]delX + [Fy]delY where Fx and Fy are the derivatives of F with respect to X and Y. It seems natural to say that [Fx]delX is that part of delZ due to the change in X, and similarly that the change in Z due to Y is [Fy]delY. GEMPACK solves non-linear equation systems by repeatedly solving systems such as (2). The changes (delZ) computed at each step are accumulated to give an exact change solution. Similarly, we can cumulate the terms [Fx]delX and so on to obtain contributions Cx and Cy from each exogenous variable. The contributions will add up to the total change in Z. During the computation, a succession of tiny shocks to delX will cause X to move from its initial value X0 to its final value X1. Similarly Y will move from Y0 to Y1. We can think of the movement of these 2 exogenous variables as a path over a plane, from [X0,Y0] to [X1,Y1]. Clearly, there are many possible paths from [X0,Y0] to [X1,Y1]. Luckily, whichever path we take, as long as individual steps delX and delY are small enough, we will arrive at the correct total change in Z. It turns out however, for non-linear models, that the values of the individual cumulated contributions Cx and Cy do depend [usually rather weakly] on the path followed by X and Y. Of course, the sum of these contributions does not so vary. The path that GEMPACK chooses for exogenous variables is a straight line between initial and final values of exogenous variables. There are some theoretical reasons to suppose that this path gives the least biased estimates of Cx and Cy. Obviously, the decomposition of total changes into those parts attributable to changes in individual exogenous variables must depend on which variables we think to be exogenous. Consider, for example, a simulation with GTAP where tariffs on two commodities, cloth and wheat, were shocked, giving rise to endogenous changes in the two corresponding import volumes. We could decompose the total effect on national welfare between the two tariff shocks. Now imagine a companion simulation where this time the two tariffs are endogenous, and the corresponding import volumes (quotas) are exogenous and shocked so that they change by just the same amounts as in the previous simulation. We should find that all variables changed by the same amount in both simulations. In this sense, the changes to the tariffs and import quotas are equivalent. If however we decomposed the total effect on national welfare in the second simulation between the two import volume shocks, we would normally obtain a different division of the same change in national welfare. The part of the welfare change due to the change in the cloth tariff in simulation 1 would not be the same as the part of the welfare change due to the change in the cloth quota in simulation 2. The reason for the difference is simply that the exogenous variables are different in the two simulations -- it does not require any non-linearities in the model or differences between the two simulations in the paths followed by variables. The theory of subtotals is explained more fully in: CoPS/IMPACT Working Paper Number IP-73 Title: Decomposing Simulation Results with Respect to Exogenous Shocks Authors: W. Jill Harrison, J. Mark Horridge and K.R. Pearson Download from: http://www.copsmodels.com/elecpapr/ip-73.htm URL of this topic: www.copsmodels.com/webhelp/rungtap/hc_theorysubtot.htm Link to full GEMPACK Manual Link to GEMPACK homepage |