Solving and analysing SFC models

Once the model’s accounting and behavioural equations have been specified, it is possible to solve the model. In practice, most SFC models are solved via numerical simulation. However, it is also possible to derive an analytical solution to simpler SFC models.26 _{The rest of this section discusses}

the numerical (Section 3.1.5.1) and analytical (Section 3.1.5.2) simulation methods, along with the methods of analysis that correspond with each simulation technique.

3.1.5.1 The numerical simulation method

The numerical simulation method can be summarised as being a solution method that first specifies the value of the model’s parameters, starting stocks and lagged variables, before using a computer program to solve the model. Thus, the results of a numerically simulated model are only true for the chosen values of these parameters/stocks/flows.

In general, there are three possible approaches for choosing the parameter/stock/flow values. The first of these is to base the values on empirical estimates. This means estimating parameter values econometrically, and basing stock, lagged and exogenous variables on estimates from the national accounts. The second approach is to base the values on ‘stylised facts’, while the third is to choose these values for their convenience. A mixture of these approaches is also possible. In general, the approach taken will depend on the aims of the modeller.

26 _{Because the model presented in Chapter 4 is relatively complex, it is solved using the numerical simulation} method.

Once values have been obtained for the parameters, stocks, and lagged and exogenous flow variables, the model can be solved via numerical simulation, and the results analysed. Several outcomes of a numerical simulation are possible. For most of the theoretical SFC literature,

numerical simulation leads to a steady state. It is also possible that the numerical simulation will lead to a recurring cycle, cycles within cycles (such as in Ryoo [2010]), or explosive behaviour (such as accelerating towards positive or negative infinity or collapsing to zero). What happens will depend upon the structure of the model and the values chosen for the parameters, starting stocks, and lagged and exogenous flow variables.

There are two methods for analysing numerically simulated SFC models (Caverzasi and Godin, 2015). In the first type, the model starts from a steady state (or recurring cycle). The model is then shocked by altering the value of a parameter or exogenous variable. The outcome of this shock can then be analysed. For example, if the model finds a new steady state (or recurring cycle) then the two steady states (or cycles) can be compared with each other, along with the transition between them.

In the second type of analysis, the model does not start from a steady state (or recurring cycle). Instead, the model is numerically simulated, and the result of the simulation is then analysed. The model may also be simulated again under a different set of assumptions (e.g. regarding parameter values, starting stocks, etc.) and the differences between the outcomes analysed (Caverzasi and Godin, 2015). This type of analysis is suited to models that might not find a steady state (or cycle). It is also the type of analysis performed on the empirical SFC models.

The first method of analysis (shocking the model) is used extensively in Chapters 6 and 7 (the simulation chapters) to analyse the effects of different types of transitions and changes in market conditions. Each simulation is also performed under a number of different parameter regimes, so as to better understand how the effect of each shock depends on particular assumptions (see Chapter 5).

An advantage of using the numerical simulation method over the analytical method is that it allows models of greater complexity to be solved (Caverzasi and Godin, 2015). However, the method also has some drawbacks. For example, the steady state is usually dependent on the values taken by the model parameters. Different parameter values may lead to different steady states, and there is no way of knowing how many other steady states exist or how stable these steady states are (Lavoie and Godley, 2001). The results of simulations may also depend on the model’s parameter values. Another issue with the numerical simulation method is that the causal structures in complex models may be difficult to determine. While this is not a direct result of the simulation method itself, the

ability to solve numerically facilitates the construction of more complex models. Indeed, Dos Santos and Macedo e Silva (2009, p.8) point out that this:

“inner complexity—and, in some cases, arbitrariness—may have been acting as a major obstacle to the diffusion of the approach and to the fluidity of the conversation among the many heterodox currents. Happily enough, we think there is an alternative: a simpler SFC model which can

simultaneously satisfy some of the heterodox demands for realism and convey at least some of the most important intuitions of the approach, while allowing for analytical solutions for some

interesting conceptual experiments. There is room, thus, for recasting (old and new) heterodox issues both in SFC analytical models and in computer simulations.”

3.1.5.2 The analytical solution method

Along with the numerical simulation method, it is also possible to derive analytical solutions for some of the simpler SFC models. Examples of analytical solutions can be found in Godley and Lavoie (2007a), Dos Santos and Zezza (2008), Dos Santos and Macedo e Silva (2008), and Ryoo and Skott (2013). The analytical solution method (as used in Dos Santos and Macedo e Silva, 2008) relies on the fact that in a steady state, all stocks and flows will grow at the same rate (by definition), so that the ratios between all the variables are fixed. Thus, in the steady state, the determinants of these ratios will also be equal to each other. It is, therefore, possible to set up a series of equalities that can be rearranged to obtain equilibrium conditions for various parameters. These solutions can be presented graphically, as in Godley and Lavoie (2007a) and Dos Santos and Macedo e Silva (2009) (who present the algebraic solutions found in Dos Santos and Zezza, 2008).

A major difference between the two approaches to solving SFC models is that the analytical method does not require the specification of the model’s parameter values, starting stocks or lagged or exogenous flow variables. Instead, the model’s solution is presented as an equilibrium condition. This is a major benefit of the analytical method: the solution is able to illuminate linkages and dependencies that may not be obvious from solving the model numerically or looking at the model’s equations. The drawback of the analytical approach is that it may not be possible to solve complex models. Thus, a desire to employ the analytical method may result in simpler models and a loss of realism.