Finding The Solution In A Simple SFC Model

Stock-Flow Consistent models are the mathematical workhorse of much of modern post-Keynesian thought (including Modern Monetary Theory). These models are fairly easy to work with, but there are a lot of details in how the model solution is determined. In this article, I discuss the solution for the simplest model with government money in the textbook Monetary Economics by Godley and Lavoie -- model SIM -- although I have simplified things even further (model SIMplest?).

The model I discuss is taken from Chapter 3 of Monetary Economics. Although the explanation by Godley and Lavoie is the best discussion of mathematical economic models I have seen, it is still not completely obvious to a non-economist. For example, their "simple" model SIM has 11 variables to be determined, and the exact solution method was not obvious to first inspection. Some of this complexity is necessary -- even model SIM tracks the interactions of three sectors of the economy.

It took me time to get used to their style of working with the 11 variables in 11 equations; in control systems, all of these variables would be collapsed into vectors.However, it is advantageous as their exposition is similar to the rest of the academic economic literature.

It appears that academic economists like developing their models in the same way that circuit theory was taught when I took introductory electrical engineering courses in the early 1990s: you write down lots of equations, and then solve them via algebraic manipulations.  (Given the inertia in undergraduate courses, I imagine that the same is still true.) Meanwhile, in the real world, electrical engineers simulate circuits by drawing them computer-aided design packages, and the circuit equations are solved via systematic use of matrix algebra.

Within this article, I am simplifying the discussion by only looking at how the solution is determined within the first period. The process I discuss would have to be repeated in later steps, but this would make things more complex. (I assume that the household sector starts with no money; that is no longer true after the first period.) Even this one-period calculation has hidden complexities.

UPDATE: The article "Techniques for Finding SFC Model Solutions" introduces techniques for solving these models without the simplifying assumptions made here. However, the discussion in this article helps provide an intuition with regards to the numerical solution techniques described there.

The extreme simplicity of this model cannot be emphasised enough: one cannot conclude anything about SFC models in general by looking at what this model alone can do. Within Godley and Lavoie, simplifications are eliminated in almost every direction.

Chart: Generated from Chapter 3 model of code by Raza and Kinsella

In order to see what the full solutions look like, there is code available to solve the models in Godley and Lavoie  at the "sfc-models" website: sfc-models.net. They now have R code, by Hamid Raza and Stephen Kinsella (which I used to generate the chart above; I have not had a chance to look at the code). That site has a good deal of Eviews code for various SFC models. Eviews is a commercial econometric package; I currently use R in my work here as it is open source.

The Pre-History Of The Model (t <0)

The model works in discrete time steps, labelled t=0,1,2,... These represent accounting periods, such as months, quarters, or even days. We have no notion of any data that is not at the model frequency. For example, if the model represents quarterly data, there is no way of representing monthly time series within the model. We follow mathematical convention and are starting the simulation at t=0.

We assume that there is very little happening before t=0. There is no government money, and no economic activity, although we somehow have a labour force and capital in place at t=0. This is obviously unrealistic, but to be expected in a mathematical model.

If we look at how we solve at t=0, a solution with no activity is exactly what is predicted. Even though real resources existed, money (or more generally, financial assets) drives model economic activity. (In the real world, one would expect the private sector to create private money, but such an asset is precluded in model SIM.)

The Government Sector

The government starts issuing money at t=0. It kicks off economic activity by ordering $20 worth of services (produced by the business sector, as described below.)

I am simplifying model SIM by assuming that government taxation is zero during the first period. (This means that the results I derive do not match the example values in the textbook.)

Since we are fixing government spending to be $20 for this period, and taxes are zero, we know that the government sector must run a fiscal deficit of $20 in the first period. The only financial asset within this economy is government money, and so we know that the private sector will hold $20 in money at the end of the period. (In model SIM, the amount of taxes levied depends upon economic activity. By eliminating taxation, solving the equations is much simpler.)

Household Sector

For simplicity, we assume that there is an unlimited number of workers available within the household sector. As a result, there is no concept of an "unemployment rate"; there is effectively an infinite number of unemployed potential workers, and a certain number of employed workers. We would need to model the labour market in order for the unemployment rate to have a realistic effect on the model solution.

The implication is that workers are either employed full time or are unemployed. This is different than the convention in representative agent DSGE models, where all households are assumed to adjust their working hours based on the labour market situation. That is, an unemployment rate of 7% within one of these DSGE models is really households deciding to work hours that are 93% of "maximum employment" hours. The message is that if you are a household in a mathematical model, you want to live in a DSGE model, and not a post-Keynesian one.

The household sector works at the business sector, receiving wages. (In model SIM, it would also pay taxes based on wages received.) Not all wages are spent; some are saved and the household ends the period holding money. How much is saved is discussed below.

The Business Sector And Production

The business sector is assumed to hold capital (that does not depreciate) with infinite productive capacity; output is solely dependent upon labour inputs. Production is linear: for every worker hired, there will be an output of 1 unit of services. Services cannot be held in inventory, so the model is simplified as there is no need to track the stock (level) of inventories.

As is typical in a post-Keynesian model, prices are administered. This is done in an exceedingly simple fashion here, once again, for simplicity. (The model would be nonlinear if we violate this assumption.) The business firm has taken the notion of "non-profit" to an extreme, and pays workers $1 over the period, and charges $1 for one unit of output. (Since the worker produces 1 unit, the firm has exactly zero profit on every unit sold.)

The cash flows during the first period is illustrated below.

Flows in the simplified SIM model

A key simplification of model SIM is that the only financial asset holdings within the economy are by the household sector, This means that the business sector cannot hold money at the end of the period. As a result, the total revenue (government spending of $20 plus household consumption) has to equal the wage bill by assumption.

This implies a form of "model-consistent expectations," which is related to the concept of "rational expectations." The business sector has to accurately forecast its total revenue in order to determine the number of workers hired. If it did not, it could potentially incur a loss, which is assumed to be impossible.

There is a lot of arguments online about the use of "rational expectations" within models. Stock-Flow Consistent models can incorporate model-consistent expectations (as is assumed here), or expectation errors can be incorporated. As a result, SFC models are more flexible than mainstream models which assume that model-consistent expectations must always hold. Starting from this model, we could allow for business sector forecast errors if we added in business sector money holdings. At which point we also would need to introduce dividends (and possibly split the household sector into workers and capitalists).

"Accounting-Only" Model Solution

Since we assume that wage expense equals business sector revenue, we can inspect the diagram to see that the solution implies that the household sector saves $20. Beyond that, the model is still under-determined. That is, if all we do is insist that accounting relationships hold, there are an infinite number of solutions.

• The business sector could hire 20 workers to produce the services demanded by the government, and those workers could save 100% of their wages. Total business revenue is $20.
• The business sector could hire 1000 workers, and those workers spend most of their wages ($980 out of $1000). Total business revenue is $1000.

Needless to say, this is somewhat unrealistic. Part of the lack of realism reflects the assumption about an infinite labour supply. Nevertheless, even if we modelled capital and labour constraints, there would be a range of solutions.

In other words, accounting relationships are not enough. We need to add a description of entity behaviour within the model.

Household Savings

The model needs a mechanism to pin down how much the household sector spends versus saves. (Looking at the previous diagram, we see that the household use of money is the only decision point within the model: all other flows are determined.) Stock-Flow Consistent models opt for fairly straightforward consumption functions. The standard functions used in Godley and Lavoie consist of fixed parameters multiplying disposable income (wages less taxes) and initial wealth (money) holdings.

For period t=0 in my simplified model, initial money holdings are 0, and so the consumption function is given by:

Consumption = a (Wages), 

where 0 < a < 1. 

For example, if a=0.6, we can back out that wages have to be $50, since the household savings of $20 is 40% of $50. (Full formula: Total Wages = $20/(1-a).) The lower the personal saving rate (higher a)  the greater the level of wages and output. (Total output equals government and household consumption.)

The reason for my various simplifications of the model SIM is that it is possible to easily "read off" the solution for the system based on the fact that we know household savings equals $20. Otherwise, we need to solve the many equations of the system simultaneously to determine the solution. Given my aversion to masses of algebra, I would inevitably end up redeveloping the notation from scratch. This means that my version would no longer resemble the original solution method in Monetary Economics.

From a multiplier perspective, $20 in government spending generates $50 in GDP. Given the linearity of various relationships within this model, GDP will scale up if government spending is scaled up. As Godley and Lavoie discuss, the model will converge towards a steady state -- if the government imposes a non-zero tax rate for t>0 (as shown below) -- and the steady state GDP will have an even higher multiplier. In other words, the "long run" multiplier for constant government spending is higher than the "short-run" (one period) multiplier.

Excel Simulation

Portion of Excel simulation by Javier López Bernardo (sf-models.net). (Solution continues to right.)

[Update] There is an Excel file for simulating model SIM, by Javier López Bernardo (link to sfc-models page). As you can see, the spreadsheet is unhappy with my choice of a 0% tax rate (there is no steady state solution with a constantly growing deficit), but it shows that I did the math correctly for the first period (yay!). The screen shot above also shows the solution for the next accounting period as well.

Time series with no taxes

For a longer view, the graph above depicts a couple of time series from the Excel model. The solution is blowing up, which is telling us that eliminating taxes from a model economy gives fairly nonsensical results. Returning the tax rate to the 20% used in the text gives more sensible looking results (shown below).

Time series with a tax rate of 20%

Expectations And The Time Period

The model developed was in discrete time, and there some unsettling implications about the multiplier process. (This discussion harks back to my article Discrete Time Models Are Not Frequency Invariant.) Can expectations adjust this quickly?

Imagine that the time period is daily. The business sector was previously mired in a state of no activity whatsoever. It suddenly receives a government contract for $20. Certainly, the recipient of the contract would quickly hire at least 20 workers. But would it hire an additional 30 workers that same day, on the basis of the expected circular flows of revenue within the economy? Moreover, in the real world, there are multiple firms, and revenue would be split up amongst them. Until revenue starts to come in, why would they start hiring?

This implausibility suggests that we should replace "perfect foresight" expectations with a version of expectations that allows for errors. (This is discussed in depth in Godley and Lavoie.) In this case, the multiplier effect would work with more of a lag. This would be particularly true if demand is met from inventories; firms can wait for inventories to be unusually depleted before ramping up production. 

Conversely, if the time period within the model is annual, businesses would notice increased activity during the year. One would expect hiring and output to ramp up within the model time step, and reach a steady state before the end of the accounting period. In such a case, model-consistent expectations may be a good approximation of reality.

These considerations mean that we cannot expect model dynamics to be similar across different model frequencies. That said, the difficulties faced by fitting these models to data are so large, that this is hardly our greatest worry.

Concluding Remarks

It is an understatement to say that there are a lot of online arguments about the merits of differing approaches to macroeconomics. Some arguments are inherently political, but others revolve around seemingly esoteric model assumptions. (Although one could argue that there issues from political economy buried under those assumptions.)

If we compare SFC modelling to DSGE modelling, the clear advantage is held by SFC models (at least for those of us who have no interest in publishing in economic journals).

One can replicate almost all of the alleged advantages of DSGE models -- such as model-consistent expectations -- within a mathematical framework that is robust. Moreover, it allows for model behaviour that better matches empirical evidence from microeconomics (such as how prices are set by firms). Even if you hold the strong ideological belief that model behaviour must reflect optimising decisions by economic entities, you probably have a better chance of approximating that behaviour in a computationally tractable SFC model than in a DSGE model.

See Also:

• sfc-models.net has code to solve the models in Godley and Lavoie.
• Techniques for Finding SFC Model Solutions
• My recent post - Modern Monetary Theory Not Modern - Sigh, which touches on the use of mathematics in Modern Monetary Theory. Modern Monetary Theory research uses SFC models to do mathematical analysis.
• My SFC/MMT resource page.

(c) Brian Romanchuk 2015