An Extended Lucas Critique of Inequality and Growth

Unemployment can be blamed on a deficiency in aggregate demand …; in some sense, the entire shortfall in aggregate demand … today can be blamed on extremes of inequality.

Stiglitz (2012)

Introduction

Economic theory provides a framework for public policy – without a foundation in theory, we have limited basis for forecasting future economic states, or for predicting the economic impact of proposed policy options. The ability to forecast macroeconomic conditions, with consideration to policy settings as a causal determinant, is therefore vital to policy making.

Macroeconomic models are able to reasonably reliably ‘predict’ the past and the present in so called ‘now-casts’ (Balcilar et al 2015). Their validity also carries forward a few quarters into the future (Balcilar et al 2015; Wieland 2013). But their predictive capacity reduces very rapidly beyond 2 years (Wieland & Wolters 2013, p.302 & 315). This is reminiscent of Niels Bohr’s quip “Prediction is very difficult, especially about the future”.

Perhaps as a result of the failure of models to reliably predict the future, persistent mysteries exist in macroeconomics – What is the cause of the long-run decline in economic growth since the 1960’s? (Darby n.d.; Gordon 2014; Martin n.d.) Why has ‘productivity’ steadily declined? (Jackson 2015) Why has inequality risen consistently over this period? (Piketty 2014)

One central question that dominates the study of the relationship between economic growth and inequality is whether the relationship is net positive or net negative (Cingano 2014) – does addressing rising inequality necessarily come at the expense of economic growth? (Okun 1975, Cingano 2014; Piketty 2014; Persson n.d.; Stiglitz 2015; Jackson 2015)

This paper will attempt to address these questions in a novel way, firstly by observing that three uncontentious statements lead inexorably to a radical conclusion –

Regression analysis is not functionally capable of accurately deriving parameters when there exists a feedback loop of mutual causation in the system being analysed
Macroeconomic analysis of the relationship between inequality and growth has been based almost exclusively on regression analysis (Cingano 2014). A similar claim could be made about macroeconomic empirical analysis more generally.
This paper will present a theoretical argument, based on an extension of the Lucas Critique (Lucas 1976), that the relationship between inequality and growth features a feedback loop that results in mutual causation.

I will therefore argued that the reason macroeconomics is unable to explain key trends in inequality and economic growth is that previous studies have relied on a methodology of analysis functionally incapable of embodying the operative causal relationship.

I will employ the following methodology:

An analytical framework is presented that lays out the correspondence between genres of mathematical models and the nature of the causal relationships each embodies.

This analytical framework suggests that relationships that involve mutual causation cannot be modelled through regression analysis.

An alternative form of model – cellular automata simulation (CAS) – is suggested as suitable for modelling relationships with feedback loops
A theoretical argument is presented that demonstrates that the relationship between inequality and economic growth involves mutual causation – a feedback loop
A ‘first-order’ CAS model of inequality and economic growth is developed
The forecasting accuracy of this CAS model is compared to ‘best-in-class’ current models, which indicates that the CAS model has superior forecasting accuracy.
Inferences are drawn regarding the nature of feedback loops and mutual causation inherent in the real-world economy based on the dynamics of the CAS model

The implications of this theory of causation in macroeconomic systems are profound, including that the goal of addressing economic inequality need not conflict with the goal of optimising economic growth.

Analytical Framework

The Trouble with Macroeconomics

‘My thesis in this lecture is that macroeconomics … has succeeded: Its central problem of depression prevention has been solved …’ (Lucas 2003)

This statement by Lucas, expressed just five years before the onset of the GFC, cannot now be read without seeing it as illustrative of unfounded hubris of macroeconomists. In the decade since the GFC the role of macroeconomic modelling in failing to anticipate the great recession has been the subject of concerted analysis and criticism:

Media and other commentators have criticized macroeconomists … for failing to predict the great recession … well-known academics … have published scathing commentaries on the current state of macroeconomic modeling. (Wieland et al 2012, p.1-2)

“Most work in macro-economics in the past 30 years has been useless at best and harmful at worst.” (Paul Krugman, June 2010, quoted in Wieland et al 2012, p.2)

Paul Romer’s (2016) scathing review of the state of macro-economic modelling ‘The Trouble with Macroeconomics’ satirises macroeconomics by drawing parallels with pre‑scientific thinking. He provides pseudoscientific epithets (shown below with Romer’s emphasis) for various concepts in macroeconomics:

‘While “demand” shocks such as the aether AKA risk premium, exogenous spending, and investment-specific phlogiston AKA technology shocks explain a significant fraction of the short-run forecast variance in output, both the troll’s wage mark-up (or caloric AKA labor supply) and, to a lesser extent, output-specific phlogiston AKA technology shocks explain most of its variation in the medium to long-run. … Third, inflation developments are mostly driven by the gremlin’s price mark-up shocks in the short run and the troll’s wage mark-up shocks in the long-run’ (Romer 2016, p.7)

The essence of Paul Romer’s (2016) critique is that current methods of macroeconomic modelling – where economic variables are asserted without proof, parameterised without significant critical analysis, and quantified using parameter-fitting – is just as likely to mislead as to elucidate.

I agree with the harsh judgment by Lucas and Sargent (1979) that the large Keynesian macro models … relied on identifying assumptions that were not credible. The situation now is worse. Macro models make assumptions that are no more credible and far more opaque. (Romer 2016, p.19)

The critical question is whether something fundamental needs to change for economic modelling to evolve from what Romer asserts is a pre-scientific state of blind-faith in parameterisation, into a science that can genuinely pursue understanding of causal relationships, such that it can develop genuine predictive capacity.

Current macro-economic modelling practice relies largely on methods that utilise parametric regression analysis (Wieland & Wolters 2013, Wieland et al 2012). Often these are applied somewhat interchangeably (read: ‘indiscriminately’) (Balcilar et al 2015, W&W 2011, Kisacikoglu 2013). However, these modelling methods have different functional properties, which map directly to appropriate (and inappropriate) uses of each model genre. While there are certainly some instances of overlap of functionality, there are also bad choices.

Bad choices include using a modelling method where it is functionally incapable of embodying the underlying relationship of causation. I will therefore summarise, before proceeding, a typology of macro-model genres linked to their functional capacity to embody causal relationships.

Typology of Macro-Models

Non-parametric Mathematical models

The simplest mathematical relationships are those that can be encapsulated in a non-parametric linear, polynomial or loglinear mathematical relationship. These tend to have greatest application where they are simply an axiomatic statement or a mathematical tautology:

Income is equal to consumption plus savings

We can illustrate this simple, one-to-one and one-way causal relationship between independent and dependent variables diagrammatically:

VAR Models

A Vector Autoregression (VAR) Modelling methodology permits multiple variables and parameters to be determined using statistical regression analysis. Again, the causation must be one‑way (not mutual). VAR Models are able to represent relationships where a system of independent variables (and control variables) is each related via linear, polynomial and loglinear relationships to a dependent variable.

This is a useful tool for solving what would otherwise be a complex system of mathematical relations, but it requires that causation is one-way – not mutual causation between independent and dependent variables.

Time-variable VAR Models

Time-variable VAR analysis supplements the VAR Model methodology with the capacity to introduce time-varying structural parameters (Granger & Terasvirta 1993). This permits the model to represent phases in time where different structural parameters are applicable, such as different conditions in growth and recessionary conditions (Granger 2008).

The requirement that the Model be ‘tractable’ prevents the model from explicitly embodying relationships of mutual causation with feedback loops (Evans 1997, p.12, Linde 2018, p.282).

Where a relationship of mutual causation exists, statistical analysis results in parameters that scramble the various drivers of change over time. The functional utility of such models is therefore limited because the derived time-varying parameters then confuse separate causes of parameter change. A time-varying VAR model cannot separately derive the portion of the change in structural parameters that is due to a quantifiable (and ongoing) relationship of mutual causation, as distinct from say, a short-term phase-change (growth phase versus recession).

A VAR Model can derive a separate parameter associated with a relationship of mutual causation only where an instrumental variable can be identified, but this is not possible in relation to macroeconomic variables. Instrumental variables are therefore not available to identify mutual causation in the relationship between inequality and economic growth, so time-varying VAR modelling is not suitable for analysing mutual causation in macroeconomic systems.

DSGE Models

Dynamic Stochastic General Equilibrium (DSGE) Models are widely utilised by institutions, sophisticated policy advisers and central banks in recent years (Wieland & Wolters 2011).

DSGE Models are able to represent complex interactions of a large number of parametric linear, polynomial and loglinear relationships that are either derived empirically or through theory, or a combination thereof. They can also be built-up iteratively and via components. This aids the process of iterative refinement of the Model (Granger 2008; Linde 2018).

However, the causal relationships that such models are capable of embodying are similar to Time-varying VAR models – they are limited by the assumption of general equilibrium and one-way causation (in the absence of instrumental variables to separately derive parameters related to relationships of mutual causation).

As a result, DSGE Models are a highly useful tool for quantifying parameters within already identified complex relationships, but they are much less useful for the process of unpicking the numerous causal relationships, networks and feedback loops that might be hypothesised to exist in a complex adaptive system (Hendry & Muellbauer 2018; Linde 2018).

Micro-founded Simulations and Cellular Automata Simulations

Cellular Automata Simulations (CAS) have been advocated since computing power first became sufficiently widely available to permit their development (Brock 1991; Holland 1995 & 2006; Puu 1991), although have arguably been underutilised.

The work of Graham (2001); Dijk (2004), Holland (2006) and Yang (2018) each suggest that micro‑founded CAS models are able to represent the economy as a complex adaptive system. CAS involves building a computer simulation (Anderson et al 1998) as follows:

A system is analysed by breaking it down into interacting ‘cells’ – each cell represents a computational component of the whole
The ‘calculation’ within each cell is mathematically determinative, and based on the values of other cells.
Each time-period can be represented as an iteration of the model calculation, or typically as a column within a matrix of cell calculations
Typically, the output from one time-period forms the inputs to the next time-period
The pattern of interrelationships between cells is capable of embodying cyclical, iterative relationships and relationships of mutual or reciprocal causation.
The CAS simulation is ‘run’, and time-series outputs from the simulation can be statistically compared to empirical time-series data.

It is the reciprocal and/or networked dependencies that result in emergent behaviours of the system at the macro‑level – these complexities do not need to be ‘baked-into’ the relationships between cells at the micro-level. CAS simulations are therefore able to embody, demonstrate and derive dynamic feedback relationships that would render a relationship intractable for the purposes of VAR or DSGE modelling.

This aids in the process of theory development, as feedback loop dependencies remain logical and observable, rather than being lost in ‘scrambled’ statistically determined parameters.

CAS have been used successfully to model systems of economic geography, such as property prices or populations, as well as ecological systems such as predator/prey systems (Engelen et al 1995).

A crucial extension of the CAS methodology suggested in this report is that the Case Study extends the CAS methodology to model interacting cells where, instead of cells being representative of parts of a system that are physically proximate, the cells represent portions of an economic system that are causally proximate.

There are two types of causal relationships that are best suited to modelling via CAS:

Network Relationships

Iterative, mutual causation

CAS models are not limited to one-way causal relationships, and do not need to rely on the ubiquitous assumptions of equilibrium and (near perfect) rationality. CAS models can incorporate states of bounded rationality (where the information available within one cell is limited to only adjacent or causally linked cells, or only ‘upstream’ cells in a causal chain). They can also derive and demonstrate non‑equilibrium states such as emergent, dynamic macro-level effects.

A macro-level non‑equilibrium, emergent behaviour in an intractable system would be observed as a drift in the value of structural parameters if that system were modelled using a VAR or DSGE model.

One disadvantage of CAS is that the calculation is not expressible in the form of baseline regression formula, so it is not possible to use statistical techniques such as regression analysis to derive an optimised set of structural parameters. Instead, techniques such as search algorithms or Monte Carlo methods must be deployed to optimise any parameters within the model.

This collection of benefits and limitations has led some to conclude that the best use of micro‑founded simulations is for theory development, exploration of concepts where theory is undeveloped (or unreliable) and for heterodox theory development (Davis et al 2007).

Framework for selection of modelling method

As can be seen from the foregoing discussion, the choice of model should not be an indiscriminate choice, nor made on the basis of trial and error (as observed in Wieland & Wolters 2011; Balcilar et al 2015). Rather, there is a necessary correspondence between the nature of the underlying causal relationship of variables and the modelling technique capable of embodying that relationship.

The typology of macro-modelling techniques can therefore be matched to their appropriate application as summarised in Table 1 .

Typology of Economic Models

Empirical Analysis

Inequality and Growth – Unexplained

The most comprehensive research of the relationship between inequality and economic growth has been undertaken by Cingano (2014), which also provides a literature review and meta-analysis of relevant studies from the early 1960’s.

Cingano (2014) identified a statistically significant negative relationship between inequality and economic growth over the in-sample analysis period (1980 to 2010), however, this clearly represents an over‑simplification as it fails to recognise that alternating periods of positive and negative correlations are present, as can be observed in the following Figure 1 and Table 2 below.

Clearly a more nuanced explanation is required.

Growth and Inequality – Empirical Data USA, 1962-2016

Data Source: WID Database

Growth and Inequality Phase Transitions, USA 1962 – 2016

Period	Growth	Inequality	Correlation
1962 to 1975	Reducing growth rate	Reducing inequality	Positive
1975 to 2000	Steady or slightly increasing growth rate	Increasing inequality	Positive, or no correlation
2000 to 2016	Rapidly reducing growth rate	Increasing inequality	Negative

The causal nature of the growth and inequality relationship

Cingano (2014) summarises a range of opposing theorised independent causal factors identified in the literature hypothesised to contribute to the relationship between inequality and economic growth, as follows:

Negative contributors in the literature discussed by Cingano (2014, p.10-11) include:

“endogenous fiscal policy” theory (Persson and Tabellini 1994), whereby greater inequality becomes unacceptable to voters, who then insist on higher taxation and regulation, and policies generally that are discouraging of investment, risk taking and growth;
financial market imperfections create poverty traps where wealthier individuals have greater access to productive opportunities for investment, including investment in development of their own human capital through education (Galor and Zeira 1993 & 1998);
adoption of transformative technology depends on sufficient domestic demand (Krueger 2012; Bernstein 2013);

Positive contributors in the literature discussed by Cingano (2014, p.11-12) include:

High inequality provides incentives to work harder and invest, including productive risk taking (Mirrlees 1971);
Higher inequality fosters aggregate savings, and therefore capital accumulation because the rich have a lower propensity to consume (Kaldor 1956; Bourguignon 1981)

Cingano (2014), together with the previous studies in his literature survey, uses regression analysis of panel data from OECD nations over a 30-year period (1980 to 2010). This is equivalent to a VAR model (and Time-varying VAR analysis, where Cingano also includes a lagged dependent variable in the baseline regression formula). The causal relationship able to be explored using such analysis assumes there is no mutual causation or feedback loop inherent in the relationship:

However, it will be argued that the nature of the causal relationship of growth and inequality involves a feedback loop amongst interacting computational nodes. It is the presence of feedback loops amongst the interacting nodes that renders regression analysis incapable of accurately representing the underlying causal relationships.

This underlying incompatibility – a complex adaptive system incorporating a dynamic feedback loop of causation being represented as though it were a simpler system – could explain the poor performance of economic modelling with respect to growth and inequality, such as the large number of inconsistent findings over time in the literature reviewed by Cingano (2014, p.12) – “That survey highlights that there is no consensus on the sign and strength of the relationship …”

Using modelling techniques that fail to embody feedback loops and mutual causation can be expected to derive structural parameters that diverge from empirical data when used for out-of-sample forecasting, as the effects of the feedback loop iteratively change the value of the structural parameters.

The Lucas Critique, Extended

An example of a dynamic feedback loop can be illustrated clearly when we look at the relationship between growth and inequality, applying the mathematical formulation discussed by Lucas in his seminal essay that has since become known as the Lucas Critique (Lucas 1976).

For a more detailed background of Lucas’s mathematical formulation and argument, refer to Appendix D.

Lucas (1976) discussed how the VAR models prevalent at the time were founded on the contention that an economy over time can be represented by a formula of following form:

Where:

is aggregate income at time t+1

F is a function specified in advance, which may be theoretically founded

is aggregate income at time t

x_t is a vector of exogeneous forcing variables

is a vector of independent (through time) identically distributed random shocks

is a fixed parameter vector

Lucas states ‘For practical purposes, one usually thinks of [the challenge of economic forecasting being that of] estimating the values of … ’.

Under this construction the fixed parameter vector, includes the quantitative impact of the aggregate ‘decision rules’ of agents that make up the economy.

The key idea encapsulated in Lucas’s argument is that must change as a result of ‘adaption of the decision rules of agents’ following changes in policy. He argues that, where is derived empirically through regression analysis then future values of could not be determined based on previous values of , as the response to new policies cannot be judged based on the response to previous conditions.

This mathematical formulation continues to be the basis on which VAR and DSGE Modelling rests.

Based largely on Lucas’s argument, macroeconomists have inferred that where a drift in the value of the structural parameter vector, , is observed, this can be assumed to have been caused by the impact of policy changes on the aggregate decisions of agents in the economy.

However, there is an additional explanation for drift in the observed values of the structural parameter . This alternative explanation is an endogenously cause – not related to agents’ responses to policy changes – but rather is related to the emergent behaviour of a complex adaptive system that incorporates a dynamic feedback loop. These endogenous changes in the structural parameter can be quantitatively forecast within the context of a cellular automata model.

One such example is the case of increasing inequality:

Saving ratio varies systematically with wealth percentile (Schmidt-Hebbel 2000)
Both income and wealth inequality have increased consistently since 1975 (WID Database)
Consumption is axiomatically defined as income less savings
As inequality increases, aggregate consumption must be weighted more heavily by the savings propensity of the wealthy, who are in receipt of a larger share of aggregate economic activity (both in terms of labour income and return on capital)

Rising inequality therefore results in a systematic change in the structural parameters, θ, which occurs even in the absence of policy change or exogenous shocks.

In terms of Lucas’s formulation, this can be expressed as the observation that θis not a fixed parameter vector, but one that changes systematically in response to rising inequality.

I will argue that this change is able to be quantitatively forecast with accuracy based on simple, micro-founded decision rules. The CAS Simulation which forms the basis of the Case Study provides forecasts derived based on this contention.

It is also clear both empirically (Piketty 2014) and theoretically (Hagar 2018, p.18) that inequality rises as the result of economic activity because the wealthy are able to exploit economic activity to disproportionately benefit from return on capital of a larger share of aggregate wealth. The exception to this logical and empirical observation is where a highly progressive tax and transfer system is imposed (Piketty 2014 p.24, 316 & 499), and is sufficient to offset any unequal economic outcomes that result from economic activity.

θ then, is not fixed but a dependent variable – an outcome driven at least in part by differential savings propensity through the wealth spectrum, and progressivity of taxation.

Lucas (1976 p.41) states:

… given that the structure of all econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any change in policy will systematically alter the structure of econometric models.

We can now paraphrase Lucas as follows:

Given that the decision rules of economic agents (whether optimal or not) pertaining to consumption vary systematically with wealth and income, and the distribution of wealth and income changes as a result of economic activity, it follows that economic activity per se will systematically alter the structural parameters of econometric models. The structural parameters that define a model therefore cannot be taken to be constant, even in the absence of exogenous shocks or policy changes.

Lucas’s further comment applies equally to this new contention:

For the question of the short-term forecasting, or tracking ability of econometric models, we have seen that this conclusion is of only occasional significance. For issues involving policy evaluation, in contrast, it is fundamental; for it implies that comparisons of the effects of alternative policy rules using current macro-econometric models are invalid regardless of the performance of these models over the sample period or in ex ante short-term forecasting.

Lucas 1976, p.41

If we want to take advantage of a theoretical basis for determining the rate of change of the structural parameters (such as may be possible in relation to the inequality / growth relationship), we must find a micro-theoretical construction for changing θ, rather than relying on parametric regression analysis.

Testing the Analytic Framework

The failure of current economic modelling to recognise and account for mutual causation and the resultant endogenous ‘drift’ of structural parameters can be expected to result in:

The ability to derive empirically (through regression analysis) structural parameters that accurately ‘now-cast’ historical data.
Reasonably good performance of models in forecasting steady conditions over short time periods – ie. where those empirically derived structural parameters continue to apply close to the ‘in-sample’ case for which the structural parameter was derived.
Poor performance of models around significant changes – models founded on fixed structural parameters will tend to forecast ‘status quo’ (such as continual growth at a steady pace) and will have low forecasting ability for changes in structural parameters.
Poor performance of models over the long-term because ‘drift’ of structural parameters is expected to occur even in the absence of policy changes due to the emergent property of increasing inequality changing the aggregate consumption function.

We can now review the performance of economic models to ascertain whether these characteristics are observed.

Forecasting Turning Points

Balcilar et al (2015) and Wieland & Wolters (2011) reviewed the performance of a range of macroeconomic model genres for forecasting turning points (such as recessions and recoveries from recessions).

Balcilar selected eight ‘best-in-class’ models for testing from a ‘rich set of model’ (2015, p.2986). Only one of the eight ‘best in class’ models successfully predicted a downturn coinciding with the Great Recession and ‘… the timing forecast for the downturn lagged by 4 quarters.’ (Balcilar 2015, p.3003).

Wieland & Wolters (2011) reviews the performance of six models over 5 recessions and finds that “Model forecasts compare particularly well to professional forecasts at a horizon of three to four quarters and during recoveries”.

Both Balcilar and Wieland & Wolters note that the accuracy of model forecasts decreases significantly with forecasting horizon – “the predictability of output growth quickly decreases with the forecast horizon” (Wieland & Wolters 2013, p.302). This is consistent with the prediction of the analytical framework that the error due to dynamic feedback loops will accumulate over time, resulting in an increasing divergence between forecast and observation.

Long-Run Growth Forecasts

Economic models perform particularly poorly in relation to long-run forecasting (10+ years). The academic and professional literature in relation to long-run growth projections is littered with the carcasses of discarded theories, delivered with confidence but promptly ignored by reality. Perhaps reality gives little heed to theories that preference ‘tractability’ of the model over genuinely representing economic behaviour in all its complexity, including embodying the dynamic feedback loops present in the real-world economy.

Some illustrations of poor performance of long-run growth forecasts include:

In a fully specified competitive equilibrium, per capita output can grow without bound, possibly at a rate that is monotonically increasing over time.” … “growth may be persistently slower in less developed countries and may even fail to take place at all.” (Romer, 1986)

From 1979 to 1989, growth in U.S. real gross domestic product (GDP) slowed to only 2.5 percent per year. … This growth is very slow compared to the trend growth of nearly 4 percent experienced in 1948-65 or even the 3.1 percent of 1965-79. (Darby, 1992)

Is the economy entering a new normal with a substantially lower growth rate? (Mester 2015, p.1) – a rhetorical question that Mester goes on to answer as follows:

“… growth was 4.3 percent over the 1982-1990 expansion and 3.6 percent over the 1991-2001 expansion. In contrast, over the current expansion, growth has been 2.2 percent. The Congressional Budget Office projects that real GDP will grow an average of 2.1 percent per year from 2020 to 2025, at the same pace as its estimate of maximum sustainable or potential growth. This is lower than the CBO’s pre-crisis potential growth estimate of 3.1 percent for the 1981-2007 period.” (Mester 2015, p.3)

Romer (2016, p.19-20), quoting Lucas and Sargent, is particularly unflattering when commenting on the ability of macroeconomists to predict long-term economic growth:

… these predictions were wildly incorrect, and that the doctrine on which they were based is fundamentally flawed, are now simple matters of fact …

… the task that faces contemporary students of the business cycle is that of sorting through the wreckage …(Lucas and Sargent, 1979, p. 49)

Focussing on the issue of declining trend growth in the USA since the 1960’s, it is common to point to declining productivity as the explanation:

“… over time, FOMC participants have lowered their projections for longer-run growth. The FOMC began releasing these longer-run projections in January 2009. At that time, the central tendency of the participants’ projections of longer-run GDP growth was 2.5 to 2.7 percent. In the projections released last month, the central tendency was down to 1.8 to 2.2 percent. … reflects the … downward revisions to past productivity growth.” (Mester 2015, p.2)

Relying on declining ‘productivity’ as a forecasting tool is a losing game of constantly adjusting structural parameters based on empirical data, without any theoretical basis for that adjustment. As indicated by the following series of CBO long-term economic forecasts (CBO n.d.), the absence of theoretical foundation results in the CBO forecasts since 2000 simply chasing declining growth to ever lower levels.

CBO Forecasts

Actual growth – WID Database; CBO forecasts – CBO n.d.

This is consistent with the expectations of our analytical framework – the use of historically derived structural parameters fails to forecast the impact of the feedback loop between inequality and growth on the changing value of structural parameters.

Declining growth – productivity or something else?

The long-run decline in economic growth between the 1960s and 2010s, which is often characterised as being ‘caused’ by a long-run decline in ‘productivity’ (Mester 2015, p.2) is, in Romer’s (2016, p.7) analysis, akin to invoking ‘phlogiston’^[1] as an explanatory mechanism.

The ‘declining productivity explanation’ is clearly founded on circular reasoning:

Why is there a long-run decline in the rate of GDP growth?
Because productivity is declining.
How do we know that productivity is declining?
We measure its decline as a reduction in output (GDP) per unit input.

This at best serves to reframe the original question about declining GDP growth rate in terms of output per unit input. At worst it merely serves to put a name to our ignorance, while making an unfounded, implicit claim that the causal agent is the ‘productivity’ of economic inputs.

It should not be controversial to assert that the level of production is a balance of forces of production (supply) and consumption (demand) – production decisions of firms are based on their assessment of likely future levels of consumption. Measuring output therefore does not indicate whether the causal agent is declining production productivity or declining expectations of consumption demand.

We could therefore speculate, with equal confidence to the ‘declining productivity explanation’, that the long-run decline in GDP is ‘caused’ by a long-run decline in the demand side of the market – the ability or desire of consumers to purchase goods and services. Of course, this would also demand an explanation founded on something more than mere circular reasoning.

I will show that increasing inequality causing reduced aggregate demand provides this explanation.

CAS Simulation of Growth and Inequality

Introduction

In order to investigate the potential of a CAS simulation to accurately model the feedback loop hypothesised above (refer The Lucas Critique, Extended), a ‘first order’ CAS model (the Simulation) has been constructed embodying the relationship between economic growth and inequality in the USA economy since 1962.

A Radically Different Approach

It is important, before proceeding, that I emphasise the radical difference in the theoretical foundation of this methodology, which goes far beyond merely adopting a different modelling technique.

Previous attempts to explain the relationship between inequality and growth have attempted to quantitatively explore explanatory variables by solving parameterised regression formulae (time-varying VAR models). That method implicitly assumes that any change in growth or inequality is driven by a change in independent variables.

That method ignores the potential for emergent forces (feedback loops) of the type described in The Lucas Critique, Extended. These emergent forces cause a gradual change in the equilibrium value in a system even in the absence of any change in independent variables.

The Simulation presented in this paper therefore takes the equilibrium state of exogenous variables as an empirical condition (implicitly defined via the 1962 data used as the observed initial conditions). The model then seeks to answer the following question:

If an emergent property (a dynamic feedback loop) exists, whereby economic growth causes increased inequality and increasing inequality reduces mean consumption propensity, how much of the observed change in inequality and growth can be explained by including quantification of that emergent property in the model?

Specifically, the Simulation incorporates the following hypotheses within its algorithm:

wealth inequality increases over time due to cumulative savings from labour and return on capital – wealth within each percentile band is initiated at the values taken from 1962 empirical data, and thereafter is calculated in the Simulation based on savings share of labour income and return on capital;
as wealth inequality increases, this produces a reduction in aggregate consumption as a result of a larger share of total income accruing to agents with a higher savings propensity;
this reduced aggregate consumption forms part of a production function, whereby reduced aggregate demand results in reduced production.

This is the type of emergent property or feedback loop where VAR and DSGE Models are incapable of providing forecasting accuracy due to their inability to forecast out-of-sample changes in the structural parameters derived through historical regression analysis (Lucas 1976).

Simulation Specification

Full details of the technical specification of the CAS Simulation are included in Appendix C.

This section provides an overview of:

Data sources
- Initial conditions
- Annual data inputs
- Assumptions
The iterative calculation embodied in the Simulation

Throughout the Simulation, the following percentile bands (with respect to both wealth and income distributions) have been adopted for the purpose of analysis:

Percentile Band Descriptor

P0-P50 Bottom Half

P50-P80 Middle 30%

P80-P90 80 to 90

P90-P95 90 to 95

P95-P99 95 to 99

P99-P99.9 Top 1%

P99.9-P100 Top 0.1%

These percentile bands were selected for modelling purposes as a balance between competing pressures, with consideration to the following constraints:

balance of accuracy versus simplicity
Best fit data availability across data sets and percentile bands (WID Database, CBO n.d.)
First-order estimation – the purpose of this study is to investigate whether incorporating the differential savings / consumption behaviour through the wealth distribution can improve macroeconomic forecasting. An initial simplified analysis is sufficient for investigative purposes.

Data Sources

The majority of data (for both inputs and empirical tests) has been taken from the World Inequality Database – which ‘aims to provide open and convenient access to the most extensive available database on the historical evolution of the world distribution of income and wealth, both within countries and between countries’ (Alvaredo et al 2018). Each WID data series is referenced by a code number (indicated: ref#).

The Simulation has adopted the WID definitions of data series, and utilises the following data:

Initial Conditions (based on both 1945 and 1962 values[2])

1962 wealth share by wealth percentile band (WID Database – ref# shweal992j)
1962 labour income share by income percentile band (WID Database – ref# spllin992j)
Gross National Income (GNI) (WID Database – ref# mnninc999i)

Capital Ratio (WID Database – ref# wwealn999i)

Annual Data Inputs:

Ratio of capital income to labour income (Piketty 2015)
Annual labour income share by income percentile band (WID Database – ref# spllin992j)
Average taxation by wealth percentile band (CBO n.d., Tax Policy Centre 2015, Bradford Tax Institute 2018)
Price Inflation (WID Database – ref# inyixx999i)
Economic multiplier, derived from NBER Recession Index (Federal Reserve Economic Data 2018) and Auerbach & Gorodnichenko 2010

Assumptions

The Simulation is based on the following assumptions:

Average savings / consumption ratio for wealth distribution bands – savings rises with wealth percentile, compatible with Schmidt-Hebbel 2000

Savings propensity increases with wealth percentile – the Simulation assumes an average savings percentile for each wealth percentile band, which are taken to be fixed over time.
There is a positive return to scale of return on capital – as concluded by Piketty (2014, p.430-467). In order to recognise both the changing mean cost of capital and the return to scale, the Simulation adopts a methodology whereby the calculated ‘capital clearing rate’ (or average return on capital) is multiplied by a ‘scale factor’ (applicable to each wealth band) that is derived through interpolation of WID wealth and return on capital data[3]. The scale factor is assumed to remain fixed for each wealth percentile band during the analysis period.
The taxation system incentivises holding assets in perpetuity or in “offshore” or other tax effective structures. Official data for capital income therefore represents only a portion of the total return on capital, as it is drawn from income tax records. Personal wealth multiplied by rate of return on capital is therefore taken to be a more accurate predictor of total capital income than reported (taxable) capital income.
A methodology has not yet been developed to quantitatively forecast labour income share within the Simulation. Labour income share is therefore input annually using WID Data for each income band. These data are combined with the calculated GNI derived by the Simulation, together with the Capital / Labour Income split to derive the aggregate labour income within each income band.
A range of other minor simplifications and assumptions are also built into the Simulation, as more fully defined in Appendix C.

Iterative Calculation

The iterative nature of the calculation on which the CAS Simulation is founded is best described diagrammatically (see Figure 3 – Iterative Calculation of CAS Simulation).

The following clarifications are provided in relation to this iterative methodology:

The iterative calculation for the CAS Model (Figure 3) is a cyclical calculation. It therefore must be initiated by entering empirical data. The Simulation is initiated with data for 1962 – the first year for which good quality distributional wealth and income data is available for the USA in the WID Database. The findings are not sensitive to this initiation date.
From 1962, the Simulation outputs from each year form the inputs for the following year. Other than the input of Annual Data Inputs (defined above), the basis of all calculations remains identical throughout the analysis period.
The calculation cannot be reduced to, or expressed in terms of, a single mathematical formula or regression equation.
The Simulation includes one key parameter that is neither assumed, nor calculable within the Simulation, so must be determined empirically – the Growth Scaling Factor. This acts as a multiplier on the value of business investment (aggregate business confidence in future income). For simplicity, the Simulation has taken this to be a fixed value over the analysis period, with the value being calculated using empirical data from 1962 to 1975[4]. The model requires a sufficient ‘tuning period’, but is not highly sensitive to the specific adopted tuning period.

The effect of this is that the 1962 to 1975 Simulation outputs should be considered to be an ‘in-sample’ forecast, whereas the Simulation outputs after 1975 constitute (predominantly) ‘out-of-sample’ forecasts[5] based in part on this empirically derived parameter.

Iterative Calculation of CAS Simulation

Outputs from CAS Simulation

Empirical Tests of Data

The following data sources are used to compare the Simulation outputs for the years 1963 to 2016 (where available) to provide an empirical test of the accuracy of the Simulation forecasts.

Wealth share by wealth percentile band (WID Database – ref# shweal992j)
Capital Ratio (WID Database ref# wwealn999i)
Gross National Income (GNI) (WID Database – ref# mnninc999i)
‘Missing wealth’ – additional off-shore wealth (as identified by Zucman 2013) has been proposed as a partial explanation for the difference between the Simulation forecast aggregate capital, and aggregate capital reported in the WID Database.

The long-run forecast generated by the CAS Simulation shows a high level of correlation with empirical data, indicating that the Simulation provides a reliable forecast over the long-term of both growth and inequality.

This suggests we can infer the accuracy of the logical decision rules embodied in the CAS Simulation, together with the dynamic behaviour of iterative cycle used to calculate return on capital, savings, and wealth accumulation.

Simulation Outputs of Key Metrics

The following outputs are produced by the Simulation. The correlation (r) with empirical data and Root Mean Square Error (RMSE) are reported:

Measures of Simulation Forecasting Accuracy (Tuning period 1962–1975, Forecast period 1976–2014)	Correlation (r)	RMSE
GNI Growth Rate (annual)*	40.7%	2.02%
GNI growth rate (5 year rolling ave)*	29.7%	1.28%
GNI ($)[6]	97.6%	5.61%
Capital Clearing Rate (5 year rolling ave)	87.2%	1.61%
Capital Ratio⁶	85.5%	12.39%
Capital Ratio (corrected)⁶	86.1%	11.52%
Income Gini (market)**	98.1%	17.69%
Top 1% Wealth Share	97.0%	13.05%

Further discussion and graphical outputs are provided in Appendix D, and a sensitivity analysis is provided in Appendix E.

* The correlation of GNI growth rate is clearly subject to significant ‘noise’, and is discussed in more detail in the following section. The appropriate measure of forecasting accuracy for GNI is therefore RMSE.

** Note that a part of the error in the Income Gini forecast is explained by the treatment of capital income – the Simulation treats all returns on capital as income, whereas the available WID Factor Income data includes only capital income, which doesn’t account for unrealised capital gains. The appropriate measure of forecasting accuracy for Income Gini is therefore the Correlation Coefficient.

Turning Point Forecasts

The Simulation accurately forecasts GNI growth rate for a period of approximately 20 years. The RMSE for the Simulation over the first 20 years is 1.01 %[7], equivalent to that of the published CBO forecasts over an average of 5 to 10 years.

The Simulation therefore extends the forecasting horizon for economic growth from an average of 7.6 years (CBO Forecasts), to 20 years without significant loss of forecasting accuracy:

Forecast Period	Simulation Forecast Accuracy		CBO Forecast Accuracy
Forecast Period	RMSE (annual rate)	RMSE (5 year ave)	RMSE *
5 year	2.06%	0.54%	0.65%
10 year	2.29%	0.75%	0.90%
15 year	2.41%	0.79%	Not published
20 year	2.21%	1.01%	Not published
30 year	2.36%	1.37%	Not published
39 year	2.38%	1.45%	Not published
* As the CBO Forecasts are generally flat over the forecast period, it is considered appropriate to compare their accuracy with that of the Simulation 5 year rolling average forecast

Simulation Forecast (20 years)

GNI Growth (Actual) from WID Database mnninc999i

‘Out-of-sample’ forecasting (beyond 1975) the Simulation accurately forecast the timing of turning points for 18 years – both the nadirs and zeniths in growth rates were forecast within 1 year of the observed turning point from 1975 to 1993.

This compares very favourably to the ‘best performing’ DSGE Models reviewed by Balcilar (2015) – one in eight ‘best-in-class’ Models was able to accurately forecast the GFC, given a forecasting horizon of only 3 years.

Best-in-class turning point forecasts

Balcilar 2015, p.3002

Long-Run Growth Forecasts

Current data availability permits comparison of the Simulation outputs over an ‘out-of-sample’ 39‑year forecast horizon (RMSE 1.45% for 5-year rolling average). By way of comparison, the CBO long range economic forecasts, published annually since 2000 provide a forecasting horizon of 10 years with RMSE of 0.9%.

Simulation Forecast (39 years)

CBO Long-term Growth Forecasts

Simulation Failure or Market Failure?

A superficial review suggests that the ability of the Simulation to forecast economic growth reduced significantly beyond the early 1990’s (RMSE of 5-year ave increased from 0.75% to 1.45%). The Simulation forecast of economic growth was consistently below actual growth data during the period 1993 to 2002 (Figure 6). However, this error reverses in the period 2005 to 2012.

This under-reporting, followed by correction, can be seen more clearly in the following graph of GNI showing the cumulative effect of annual growth.

GNI Growth – Forecast vs Actual

The gap between the Simulation forecast and empirical GNI may be explainable as follows:

The simulation continued to track underlying economic fundamentals
The economy was subject to a series of asset bubbles from the early 1990’s to 2008 driven by surplus capital, that resulted in a departure from economic fundamentals
The 2008 Great Recession (GFC) reflected a correction, with the economy returning to the Simulation’s forecast trend

The gap between the Simulation forecast and empirical GNI therefore may be interpreted as due to a market failure (borrowing against unsustainable asset prices and speculative investment of a capital glut), rather than necessarily a reduction in the forecasting accuracy of the Simulation. A modification and extension of the Simulation to include the impact of debt-funded consumption on aggregate demand may permit a future refinement of the model to eliminate this inconsistency.

The role of speculative investment as a potential underlying driver of the observed ‘bubble economy’ is revealed in the in the following graphs, which shows that aggregate accumulated capital began to depart in the early 1990’s from the NPV of forward 20 year aggregate capital income (after having closely tracked for at least 45 years).

It may be this capital glut that has been searching for a financial return from the early 1990’s, and finding that return cannot be delivered within the income-constrained economy (Hoffman & Stewen 2016).

Accumulated Capital vs NPV of Future Capital Income

Simulation outputs, closely correlated with WID Data

Growth of Capital Surplus 1970 to 2015

Deconstruction of Simulation

As discussed earlier, one feature of CAS Simulations is that this form of model cannot be expressed in the form of a mathematical formula or baseline regression equation.

The methods for analysing CAS Simulation include:

Comparison of time-series forecasts with observed historical data (refer previous section); or
Deconstruction of the dynamics of the model, to reveal emergent behaviours, such as dynamic feedback loops

The comments within this section are of the second type – they relate to analysis of emergent properties embodied by the Simulation. The high correlation of the Simulation outputs with empirical data suggests that the behaviour of the Simulation may provide a good analogue for the corresponding behaviour of real‑world macroeconomic systems.

Virtuous Cycles and Vicious Cycles

The Simulation dynamics reveal that the behaviour of economic growth over the period 1962 to 2016 is predominantly determined by three ‘self‑amplifying’ iterative cycles that arise within the Simulation as emergent properties. These cycles are not ‘coded for’ within the Simulation – they arise as macro-level behaviours that result from the cumulative and iterative effect of micro‑founded decision rules and dependencies between calculation cells.

These self-amplifying cycles cause predictable endogenous changes in structural parameters.

The ‘Virtuous Investment Cycle’

Growth begets capital accumulation that enables investment, which drives more growth.

Analysis of the Simulation outputs indicates that this cycle dominated from 1975 to 1995.

Vicious Cycle of Reduced Consumption

As the economy grows, the benefits of growth accrues disproportionately to those at the top of the wealth distribution, who also have a higher savings ratio. A larger share of economic activity is therefore saved, and does not contribute to aggregate demand in the following year. The reduced aggregate demand produces a corresponding reduction in aggregate income and aggregate production.

Analysis of the Simulation outputs suggests that this cycle began to dominate from 1995.

Vicious Cycle of Return on Capital

Return on capital tends to be larger than economic growth (Piketty 2014, p.25 – “r>g”) in the absence of a highly progressive tax and transfer system.

Accumulated capital therefore grows at a faster rate than GNI, increasing the capital ratio (WID Database, Figure 14). Market return on capital therefore trends downward because the increasing supply of capital competes for a return available from the proportionately smaller aggregate capital income. Refer steadily reducing interest rates and bond rates since the 1970’s.

Owners of capital ‘push back’ against this declining rate of return by exerting political influence to maintain the rate of return on capital through policies that result in an increasing share of overall economic activity going to return on capital at the expense of return on labour (Piketty 2015). Such strategies include adjustments that reduce real wages (such as diminishing the bargaining power of unions) or reducing the progressivity of the taxation system. These trends have been noted in sociological research (Hager 2018). As capital owners disproportionately have a higher savings ratio, this tends to decrease aggregate demand and therefore production.

Analysis of the Simulation outputs suggests that this cycle has been dominant (over the Virtuous Cycle) since 1995.

Centrality of Capital Ratio

Each of the three observed virtuous and vicious cycles creates a causal loop – each iteration of the loop is therefore self-amplifying, while also competing against the other cycles.

We can observe that the magnitude of the virtuous investment cycle is governed by the size of the economy (GNI), whereas the two vicious cycles are governed by the degree of income and wealth inequality respectively. The differential determinants of the power of each cycle ensures that their relative importance changes over time as an emergent property of the macroeconomic system.

It is worth reviewing briefly the post-war economic period in terms of these dominant dynamic feedback loops with reference to the empirical data in Figure 14:

Dynamic Feedback in Key Growth Drivers

1945 to 1962 – Highly Progressive Taxation

During the period immediately following WWII (from 1945), a highly progressive taxation system existed which substantially prevented the accumulation of private capital, directing surplus earnings instead into aggregate demand through direct fiscal action. This maintained the opposing cycles almost in balance (albeit capital ratio declined slowly over this period).

1962 to 1975 – Capital Constrained Economy

In the period 1962 to 1975 (while the taxation system remained extremely progressive) the capital ratio had declined to an historical minimum, and growth began to be constrained by inadequate access to capital through the mechanism of extremely high interest rates. After approximately 40 years of extremely progressive taxation, the capital ratio reached a nadir in around 1975 which was inadequate to support growth.

1970s, 80s and 90s – Releasing the Virtuous Cycle

During this period capital ratio and inequality were low and the ‘virtuous investment cycle’ dominated (compared to ‘vicious cycles’). The progressivity of the taxation system was reduced throughout the 1980s, releasing the ability to accumulate investment capital, which resulted in the return to gradual increase in the trend rate of economic growth.

1995 to 2010s – Income Constrained Economy

However, as the benefits of growth accumulation snowballed in the form of increasing capital ratio and increasing inequality, the marginal benefit of additional stored capital was reduced. From 1993, the growth in accumulated capital was no longer matched by an increase in the NPV of future capital income (refer Figure 9), which indicates that additional stored capital had begun to become unproductive. The economy had entered a phase where it was ‘income constrained’ rather than ‘capital constrained’. As a result, the two vicious cycles began to dominate.

This explains the reversion to falling growth rates following 1995.

Economic growth can therefore be retarded either by capital ratio being inadequate (1970) or capital ratio being excessive combined with high inequality (post-1995). It follows that there must be a ‘happy medium’ where the ‘Virtuous Cycle’ and the ‘Vicious Cycles’ are in balance, and economic growth is optimised.

One challenge for public policy is therefore to identify the growth-optimising level of capital ratio and inequality, and to identify the progressivity of taxation that is able to stabilise the balance between opposing self-amplifying cycles at this optimum level.

Looking Forward

The Simulation does not rely on parameters derived through regression analysis of historical data, but rather includes within its function the decision rules that adjust structural parameters continuously. It is therefore possible to extrapolate the Simulation forward with a reasonable degree of confidence, to examine the impact of the identified dynamic feedback loops on long-term growth trends.

The following assumptions are necessary to enable long-run out-of-sample forecasts:

Labour / Capital Share – we have now identified that the downward trend in the labour share of GNI is driven by the ‘vicious cycle of return on capital’. This is not inevitable, but a politically-driven policy feedback loop. We can therefore investigate alternative policy scenarios using the Simulation.
Distribution of Labour income – although beyond the scope of this paper, there are good reasons to believe that the trend of increasing labour income inequality will continue, albeit at a reducing rate.
Inflation – A full consideration of inflation is beyond the scope of this report, but there are good reasons to believe that inflation will remain low, particularly in scenarios where economic growth remains low.
NBER Recession Index – the function of the recession index in the Simulation (together with interest rates) is that of synchronising the Simulation forecasts with the short-term fluctuations in the business cycle. As it is the mean value that impacts long-term trend growth, out‑of‑sample forecasting can adopt the mean level of the recession index values over the period 1962-2016 as a basis for estimating future conditions.

When the Simulation is extrapolated forward on this basis, it predicts (in the absence of return to a highly progressive taxation system) a long-run increase in inequality, a growing capital glut, and a long-run decline in real GNI growth (with the rate trending toward zero). This is because the two ‘vicious cycles’ continue to grow at a faster rate than the ‘virtuous cycle’.

Long-Run Growth Forecast

The action of the three feedback loops creates a tendency toward economic stagnation whenever the following assumptions are satisfied:

There is a positive return to scale with respect to return on capital
There is increasing savings propensity with increasing wealth share
The tax and transfer system is not so highly progressive as to prevent capital accumulation at a rate faster than economic growth

This is because, while a less progressive tax system initially results in a higher equilibrium growth rate (Cingano 2014), over time this also reinforces the two vicious cycles that drive the growth rate lower.

Equilibrium Growth Rate by Progressivity of Taxation

This conclusion may be seen as providing an alternative derivation of the phenomenon of Malthusian Stagnation.

Malthusian Stagnation therefore need not rely on the absence of technology improvement (declining marginal productivity of capital) as has previously been asserted (Ashraf & Galor 2011). In fact, the Simulation suggests that the causation may well run the other way – stagnation in the Malthusian era may have been caused by high wealth inequality, and the resultant absence of aggregate demand led to the absence of consumption markets for technological development.

This accords with historical empirical evidence, whereby most historical societies and modern societies without a significant welfare state have tended toward extreme inequality paired with near zero growth. The Simulation suggests that the path from boom to stagnation takes several decades to play out. However, is likely to impact even high growth developing economies if inequality in those nations is not addressed through introduction of more progressive taxation.

Simulation Findings

Capital Ratio

The Simulation suggests that there exists an optimum capital ratio (and corresponding level of inequality), sufficient to provide the capital required for economic growth (Virtuous Investment Cycle), but not so high that it acts as a suppressor of aggregate demand (Vicious Cycles 1 & 2).

Reduced aggregate demand, not declining productivity

It may be inferred from the Simulation that the phenomenon that has generally been characterised as ‘declining productivity’ can be explained in terms of rising inequality causing a reduction in aggregate demand. This suggests a significantly different set of policy responses are required.

Goals of growth and equality need not be in conflict

Addressing rising inequality has traditionally been asserted to come at a cost to the rate of economic growth (Okun 1975). The Simulation implies that when the capital ratio is sufficient to fund growth there is no trade-off between goals of greater equality and optimising growth.

Growth and equality are opposing goals only when the capital ratio is inadequate, but not when economic growth is ‘income constrained’. When capital ratio is high, policies that enhance capital accumulation are likely to add to the stockpile of idle capital, push down the capital clearing rate, reduce the rate of return to capital owners, and drive down economic growth.

Quantifying feedback loops is the next frontier

Greater attention should be given to studying, and ultimately quantifying, dynamic feedback loops in macroeconomic systems. These go beyond the lagged dependent variable correlations currently incorporated within regression analyses.

Theory and modelling of complex nonlinear behaviours such as the balance of virtuous and vicious feedback loops, together with empirical research that focuses on measuring these self-amplifying cycles may enable the identification of a reliable barometer of whether the economy is ‘income constrained’ or ‘capital constrained’. This will inform whether growth will be enhanced or inhibited by increasing the progressivity of taxation, paired with increased fiscal action to recycle wealth as consumption.

Stagnation is inevitable without progressive taxation

The Simulation suggests that an economy that features both positive return to scale of return on capital and increasing savings rate with wealth percentile, in the absence of a highly progressive taxation system, will tend toward extremely high inequality and a sustained low growth rate.

‘Malthusian Stagnation’ can therefore be explained by high capital ratio combined with high inequality; and does not require an assumption of declining marginal productivity of capital.

Conclusions and Recommendations

CAS Simulations can improve macroeconomic modelling, especially for relationships where a mutually causal relationships or dynamic feedback loop exists
CAS Simulations have a valuable role to play in experimental exploration and theory development
The Lucas Critique should be extended to include an understanding that structural parameters can change dynamically as an emergent property, driven by structural changes such as increasing inequality.
The CAS Simulation presented is merely a ‘first-order’ partial model, but two significant findings are suggested for further cross-country research:
1. The phenomenon of ‘Malthusian Stagnation’ may be derived without the need to assume a declining rate of return on capital, but instead results from the combination of increasing rate of return on capital with wealth; and increasing rate of savings with wealth (in the absence of highly progressive taxation).
1. The economic truism of a trade-off between goals of equality and growth is not necessarily inviolate. The Simulation suggests that these goals are opposing when the stock of capital is inadequate to support future economic growth, but when the capital ratio is high economic growth is likely to be enhanced by pursing policies that reduce economic inequality.

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Wieland V & Wolters M 2013, Chapter 5 – Forecasting and Policy Making, in: Elliott, G., Timmermann, A. (Eds.), Handbook of Economic Forecasting, Handbook of Economic Forecasting. Elsevier, pp. 239–325. https://doi.org/10.1016/B978-0-444-53683-9.00005-0

Wieland V & Wolters MH 2011, The diversity of forecasts from macroeconomic models of the US economy. Economic Theory; Heidelberg 47, 247–292. http://dx.doi.org.virtual.anu.edu.au/10.1007/s00199-010-0549-7

Yang A 2018, Causal decomposition in the mutual causation system. Nature Communications.

Zucman G 2013, The Missing Wealth of Nations: Are Europe and the U.S. net Debtors or net Creditors?. The Quarterly Journal of Economics 128, 1321–1364. https://doi.org/10.1093/qje/qjt012

Zucman G 2013b The Missing Wealth of Nations: Are Europe and the U.S. net Debtors or net Creditors? – Data Appendix

Glossary and Abbreviations

Capital – capital in the context of this paper is used in the same sense as it is used by Piketty (2014). This is not limited to ‘capital assets’ in the sense of durable but ultimately depreciable assets. Capital in this paper is used to refer to the ‘stock’ of asset values, as distinct from a ‘flow’ of income. Aggregate capital therefore includes the total value of all assets in an economy.

CAS (Cellular Automata Simulation) – A Cellular Automata Simulation is a simulation constructed of interacting ‘cells’, where the relationship between cells is typically a simple, mathematically determinative or logical decision rule. The CAS model is defined by a complex pattern of such interactions between cells, where the pattern of interactions is able to represent cells that are causally dependent. The relationships may be cyclical, and may embody relationships of mutual dependency, or cycles of causal dependency involving multiple cells.

This complex web of simple cellular interactions permits the model to display emergent behaviours of the system – it is therefore hypothesised that a CAS is able to model the behaviour of a Complex Adaptive System.

Complex Adaptive System – Holland (2006) describes a Complex Adaptive System as – systems that involve many components that adapt or learn as they interact. This ‘learning’ process need not be that of conscious agents – just as a market can ‘learn’ a new equilibrium price following an exogenous shock, a complex adaptive system can involve several interacting causally dependent elements that respond to both exogenous shocks and endogenous processes.

Dynamic Feedback Loop – is a self-reinforcing or self-amplifying causal loop, where two or more variables are causally dependent via a cycle of causation. The cyclical nature of this causation can result in iterative amplification, or iterative dampening.

DSGE Models – A Dynamic Stochastic General Equilibrium Model is a system of mathematical relationships that typically combines both applied general equilibrium theory and microeconomic foundations to provide a system that is able to forecast macroeconomic conditions, typically on the assumption that the system exists in or tends toward a ‘general equilibrium’.

Emergent property – In a complex adaptive system, it is possible for emergent properties (or emergent behaviours) to exist only at the macro-level – they are properties that emerge in the interaction of the components of the system, only when the system exists a whole. They do not exist as behaviours of the component parts of the system.

For instance, an ocean could be thought of as comprising billions of litres of water. Any litre of water, in isolation, exhibits forces and properties of thermal equilibrium – conduction and convection of heat, and exchange of radiant heat with its surrounding environment. But it is only when the ocean exists as complete system that the emergent property of an ocean current can arise. An ocean current doesn’t exist in any of the individual litres of water that make up the ocean – only in the macro-level system.

GNI – Gross National Income – see WID Database ref# mnninc999i

In-sample forecast – Where the modelling methodology employs parameterisation of independent variables, an ‘in-sample’ forecast relates to a calculation derived through application of the model, during the period to which the data relates that was used to derive the parameters.

Intractable – Intractable relationships are those that do not follow any of the rules of mathematical conventions, and are therefore not able to be represented using the mathematical formulas or notations so far devised. There need not be anything inherently different about a ‘tractable’ relationship, other than the fact that a mathematical convention and widely known notation exists to refer to the relationship.

Now-cast – is a ‘prediction’ of the ‘present situation, based on theory or a model output and applying data for the current value of independent variables. The term is used to distinguish ‘predictions’ about the present (or ex post ‘forecasts’ of past events), from true predictions about future events (relative to the time at which the forecast is made).

Out-of-sample forecast – Where a modelling methodology employs parameterisation of independent variables, an ‘out-of-sample’ forecast involves using the values of parameters determined through historical data to a different time period. The set of parameters (the structural parameters) are extended to a period beyond the range of the sample data which was used to determine the structural parameters on the assumption that the structural parameters remain constant through time.

VAR – Vector Autoregression – a form of mathematical model (and notation) whereby the relative importance of several independent variables with respect to a single dependent variable can be deduced through statistical analysis.

WID Database – World Inequality Database – see references: Alvaredo et al 2018

Summary of the Lucas Critique

Lucas’s mathematical formulation (Lucas 1976) and argument is summarised as follows:

One describes the economy in a time period t by a vector Y_t of state variables,

a vector x_t of exogeneous forcing variables, and a vector of independent (through time) identically distributed random shocks. The motion of the economy is determined by a difference equation

the distribution of , and a description of the temporal behavior of the forcing variables, x_t. The function f is taken to be fixed but not directly known; the task of empiricists is then to estimate f.

For practical purposes, one usually thinks of estimating the values of a fixed parameter vector , with

and F being specified in advance.

Lucas goes on to say, in the context of discussing modelling of the impact of policy changes or choices:

For such comparisons to have any meaning, it is essential that the structure (F, ) not vary systematically with the choice of { }.

Everything we know about dynamic economic theory indicates that this presumption is unjustified.

He asserts:

It is likely that the “drift” in which the adaptive models describe stochastically

reflects, in part, the adaptation of the decision rules of agents to the changing character of the series they are trying to forecast.

…

For longer term forecasting and policy simulations, however, ignoring the systematic sources of drift will lead to large, unpredictable errors.

…

… there are compelling empirical and theoretical reasons for believing that a structure of the form

(F known, fixed, “arbitrary”) will not be of use for forecasting and policy

evaluation in actual economies. For short-term forecasting, these arguments have

long been anticipated in practice, and models with good (and improvable) tracking

properties have been obtained by permitting and measuring “drift” in the parameter

vector .

Specification of Simulation

Key Features of Simulation

A CAS (Simulation) has been constructed that is representative of the complex relationships inherent in the dynamics of economic growth and inequality.

Cellular automata (Holland 2006) are separate computational components – ‘cells’ – within a complex system that interact via deterministic rules. In the case of the Simulation, the economy has been analysed in terms of cells representative of ‘computational components’ of the US economy during the period from 1962. For example, the Simulation includes cells representative of the following computational elements:

the consumption behaviour of the P0-50 segment of the wealth distribution in 1970;
the savings from labour income of the P90-95 segment of the wealth distribution in the year 1985;
taxation receipts collected by government in 1999; and
the net aggregate investment decisions of businesses in the year 2000.

The Simulation is built in Microsoft Excel, and is structured such that each column represents the key computational components of the economy in any one year.

The outputs of the calculations within any one year form the inputs (together with exogenous variables) for the next year.

The decision rules that define the behaviour of any one cell are simple, and remain consistent from one year to the next. While the decision rules remain consistent, the inputs change based on the iterative calculations from year to year, thereby giving the Simulation its ability to represent nonlinear calculations.

Crucially, the approach of using a cellular automata enables the Simulation to reflect relationships that cannot be expressed in the form of a ‘linear’ mathematical relationships.

Cellular automata relationships may be distinguished from other branches of mathematics in that they are able to express mutually interdependent relationships, whereby (for instance) aggregate net investment decisions of firms depends on anticipated future aggregate consumption, while consumption is also stimulated by investment. Such ‘circular’, self-reinforcing and self-amplifying relationships cannot be expressed in simple linear mathematical formulae, but can be represented by deterministic decisions rules and relationships operating between discrete computational components within a complex system.

Growth

Economic Growth is not treated within the Simulation as being representable by a linear formula. Instead, the growth rate is represented as the balance between forces of demand and supply, which are calculated within a complex system of interdependent computational components (cellular automata).

The demand side of this relationship is calculated based on the hypothesised consumption behaviour of private individuals (analysed in discrete bands within the wealth spectrum) and firms, together with the investment behaviour of firms who are assumed to invest on the basis of anticipated future consumption. Fiscal demand is currently calculated on the assumption that government spends all tax receipts in the year received (a simplifying assumption).

The supply side of this relationship is seen to be a result of resource constraints calculable, at least in principle, in the form of a Natural Growth Rate. The current iteration of the Simulation uses interest rates and the NBER recession index as proxies for resource constraints (Natural Growth Rate), which either stimulates or inhibits the investment response of firms, together with the consumption response to net investment (via the economic multiplier).

The formula detailed in Table 3 – Formal Definition of Simulation (below) is used to calculate growth rate in any one year, but the inputs to the formula also depend on the outputs from the corresponding calculation in the previous year. This iterative calculation permits the growth model to capture the nonlinear, mutual causation of growth on inequality, and inequality on growth.

Savings

The Simulation does not assume that savings is a constant ratio. Instead, the savings rate is defined implicitly through the adoption of a differential average consumption rate and marginal consumption rate for each band within the wealth distribution (Schmidt-Hebbel, 2000). The Simulation specifies savings rates for capital income and labour income (Schmidt-Hebbel 2000). The Simulation assumes that these average savings / consumption rates and marginal savings / consumptions rates remain constant within each wealth band over time.

The aggregate savings rate is therefore calculated separately for each year, and is dependent on both the relative mix of capital and labour income, and the levels of wealth and income inequality.

Bounded Rationality

The Simulation assumes actors behave rationally, but the relationships and decision rules within the Simulation are based on bounded rationality – actors are not assumed to have access to unlimited information, but instead to base their decisions on information available at the time of their decision (each column in the Simulation represents the state of the economy in a particular year, so rational actors do not have the ability to ‘look forward’, but are assumed to make decisions based on predictions about the future based on the information available at the time of their decision).

Role of Capital

Aggregate Capital is not assumed to exhibit a diminishing return to scale. Instead, the Capital Clearing Rate is derived at each point in time as a market response to the interaction of aggregate capital income and aggregate capital. The rate of return on capital may increase or decrease within the Simulation, dependent on the complex relationships and interactions of cellular automata.

The Capital Clearing Rate is calculated as the Aggregate Capital Income divided by Aggregate Accumulated Capital.

Inequality / Growth Relationship

The Simulation does not use statistical tools to investigate data correlations to ‘look for’ a particular relationship between inequality and growth. The Simulation does not assume a set of ‘independent variables’ are responsible for either growth or inequality.

The approach adopted by the Simulation is therefore not forced to assume that that the relationship between Inequality and Growth is linear or quadratic in nature, in order to explore the potential interactions and dependencies.

Technical Specification of Simulation

The following table summarises the key inputs (exogenous variables), calculations and outputs (dependent variables), sources of researched data, and assumptions.

Formal Definition of Simulation

Category	Variables	Source	Source / Description / Discussion
Initial conditions at commencement of model
Starting Conditions (1962 USA)	GNI	WID – GNI (mnninc999i)	GNI for 1962 is input as a starting condition, and is forecast by the model thereafter
	Capital Ratio	WID – Capital Ratio Private Wealth (wwealn999i)	Capital Ratio in 1962 is input as a starting condition, and is forecast by the model thereafter.
	Wealth Distribution	WID – Private Wealth Distribution (shweal992j)	Throughout the model the following wealth bands have been adopted for analysis: P0-P50, P50-P80, P80-P90, P90-P95, P95-P99, P99-P99.9, P99.9-P100
	Labour Income Distribution	WID – Pre-Tax Labour Income Distribution (spllin992j)	Throughout the model the following income bands have been adopted for analysis: P0-P50, P50-P80, P80-P90, P90-P95, P95-P99, P99-P99.9, P99.9-P100 A simplification has been adopted whereby income distribution bands are assumed to correspond to equivalent wealth distribution bands.
	Labour (vs. Capital) share of GNI	OECD / Piketty	Piketty and Zucman (2014) as cited in OECD – ‘The Labour Share in G20 Economies)’


Exogenous Policy Settings and annual inputs
Annual inputs	Effective Taxation Rate on Labour and Capital income, by income percentile band	CBO Analysis	Congressional Budget Office Analysis, as cited in https://en.wikipedia.org/wiki/Progressive_tax
	Labour income distribution	WID (spllin992j)	P0-P50, P50-P80, P80-P90, P90-P95, P95-P99, P99-P99.9, P99.9-P100
	Economic Multiplier Table, reflective of various scenarios	Research Paper – Auerbach A et al 2010	The NBER Recession Index is used, together with Economic Multiplier Table (from Auerbach A et al 2010, ‘Measuring the Output Responses to Fiscal Policy’) to calculate the annual economic multiplier on additional business investment. The combination of Inflation rate, together with the Economic Multiplier is used within the Simulation as a proxy for the moderating action of the Natural Growth Rate (which is not explicitly calculated in the model) on the calculated rate of aggregate demand, which together forms the GNI growth calculation. A future development of the Simulation may seek to make endogenous the calculation of Inflation Rate and Economic Multiplier by explicitly calculating the Natural Growth Rate, in lieu of importing these exogenous data inputs.
	Recession Index	NBER Recession Index
	Inflation Rate	WID Price Index (inyixx999i)
	Labour (vs. Capital) share of GNI	OECD Research	Piketty and Zucman (2014) as cited in OECD – ‘The Labour Share in G20 Economies)’
Global Settings (assumptions)
Distributional data, by wealth percentile band	Multiplier of Market Rate of Return on Capital by Wealth Percentile	Analysis of WID data (shweal992j, spkkin992j)	Analysis of WID data for Personal Capital Income and Personal Wealth has been used to calculate an average ‘return to scale’ of capital, in terms of increasing rate of Return on Capital with increasing wealth percentile band. The empirical basis of this data also incorporates any impact on differential Return on Capital of any financial market failure with respect to inequality and/or poverty. It is assumed that each wealth percentile band achieves a fixed multiplier of the ‘base’ market Capital Clearing Rate.
	Average Consumption Rate (Labour Income)	Assumed value, pending identification of high-quality empirical data or future research	Literature suggests that consumption ratio decreases as wealth percentile increases (Schmidt-Hebbel 2000), and that the consumption rate is lower (ie. higher savings rate) for capital income compared to labour income. Information sources are currently qualitative, so the quantitative values require further research and refinement.
	Average Consumption Rate (from RoC)
	Marginal Consumption Rate (Labour Income)
	Marginal Consumption Rate (from RoC)

Endogenous Variables and Simulation Outputs		Discussion	Empirical Test Data
	Capital Ratio	Calculated, by wealth percentile band, on an annual basis – aggregate cumulative capital is the original 1962 wealth share plus savings from labour income plus total Return on Capital less consumption from Return on Capital. The aggregate cumulative capital is then divided by forecast GNI.	WID – Capital Ratio Private Wealth (wwealp999i)
	Capital Clearing Rate	The base level for Return on Capital has been calculated as the Capital Clearing Rate, which is calculated as the aggregate capital income divided by accumulated capital.	10 Year Treasury Bond Interest Rate°
	Annual Increase in Aggregate Demand – from Net Business Investment	The increase in aggregate demand is calculated each year based on the preceding years’ macroeconomic conditions, income distribution data, and wealth distribution data, and aggregate consumption.	The value of ‘Net Business Investment’ is moderated by proxies for the Natural Growth Rate, in the form of the Inflation Rate (currently input as an exogenous variable) and the Economic Multiplier, which is subsequently applied as a multiplier of Net Business Investment.
	Economic Multiplier	Calculated on an annual basis for a trailing period of 5 years. The Economic multiplier is calculated by using the Recession Index to drive interpolation between different scenarios for Economic Multiplier values (based on research as per Auerbach A et al 2010)	No applicable empirical test data
	GNI	GNI is calculated based on the preceding years’ GNI plus the Annual Increase in Aggregate Demand from Net Business Investment moderated by the applicable Economic Multiplier.	WID – GNI (mnninc999i)
	Capital Glut	Calculated as the surplus / (deficit) of accumulated capital compared to the NPV of the 20 year forward annual Aggregate Capital Income	Close to nil in the period 1962 to 1993. Capital Glut grows to a significant value from 1995 to 2016.
Measures of Inequality	Top 1% Wealth Share	Top 1% Wealth Share is calculated as the Accumulated Capital (from labour income savings and retained Return on Capital) of the Top 1%	WID – Private Wealth (shweal992j) adjusted to reflect ‘Missing Wealth’**
	Top 1% Income Share	Top 1% Income Share is calculated as the forecast earnings from Labour and forecast earnings from Return on Capital of the Top 1% of the wealth distribution.	WID – Pre-Tax Income (spllin992j)
	Income Gini (market)	Income Gini is calculated annually based on forecast labour and capital earnings within each wealth band.	The Chartbook of Economic Inequality
	Wealth Gini	Calculated based on the forecast cumulative wealth held by each wealth band, based on savings from labour income and retained RoC.	No empirical test data identified
° T-Bond Rates sourced from ”Useful Data’ Blog published by Professor of Finance at Stern School of Business, New York, Available at – http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datacurrent.html °° Income Gini Data available at – https://www.chartbookofeconomicinequality.com/inequality-by-country/usa/ * A ‘Growth Scaling Factor’ has been applied within the model, which applies a fixed adjustment over the term from 1962 to 2014 to each annual forecast of increase in annual aggregate demand, such that the GNI at 1975 is ‘scaled’ to match empirical data for that year. ** Corrected data refers to adjustment of aggregate capital to correct for capital held in off-shore tax effective holding structures – based on research by Zucman G – ‘The Missing Wealth of Nations’

Adopted simplifications and potential future refinements

The Simulation currently adopts a range of simplifications, which may warrant further refinement and iterative model development:

Fiscal policy is currently excluded – the Simulation currently assumes that all government taxation receipts are spent in the year of receipt, and that there is neither an excess nor deficit of government spending.
Fiscal debt does not currently form part of the Simulation. It is anticipated that refining the Simulation to include consideration of public debt would result in improved correlation between the forecast rate of Return on Capital and relevant empirical data.
The Wealth and Income Distributions are currently assumed to be corresponding distributions. This is not strictly true as an individual may in principle have very high income without also having very high wealth. It may be possible to refine the Simulation in the future to eliminate this simplification. Otherwise, empirical research may be warranted to explore the impact of this assumption on the Simulation validity.
The Data currently identified (CBO) for effective taxation rate by income band is based on US Federal Taxes. It is not clear what, if any, assumptions are included in this data regarding State and Local taxes. Further research is required to clarify, if possible, rates for effective taxation inclusive of all taxes.
The Simulation does not consider public and private consumption funded via debt – all public and private consumption is currently assumed to be funded from tax collections and income respectively. The Simulation may be refined in the future to account for increases in public and private debt, including debt funded consumption (and net debt retirement).
The Simulation currently treats the economy as a ‘closed’ system. The Simulation may be refined in future to account for net foreign income and net foreign capital flows.
The Simulation is currently expressed in nominal dollars and inflation is input based on empirical data. Further refinement is required to permit calculation of inflation as an endogenous variable, which it is anticipated would be achieved through calculating a Natural Growth Rate based on exogenous inputs reflective of non-financial resources.
Changes to income distribution by percentile are currently input based on empirical data. Further work is required to permit the derivation of distribution of labour income within the Simulation.

Simulation Output Graphs

Comments in relation to Simulation output graphs follow:

Economic growth is accurately forecast by the Simulation (r = 99.2%) based on Simulation inputs for 1962, and the Growth Scaling Factor using the data for 1962 to 1975.

Capital Ratio empirical data has been presented based on the WID Data set (Capital Ratio Private Wealth – wwealp999i), and secondly the WID adjusted to reflect the ‘missing wealth’ held is offshore tax havens based on the research by Zucman G – ‘The Missing Wealth of Nations’. The midrange forecast of Zucman for the USA has been adopted here, and where data does not exist (prior to 1990) the value of ‘additional private wealth’ has been assumed to be linear from 1970 to 1990.

The ‘missing wealth’ (Zucman G) has been assumed to be held in the following proportions:

P95-99: 5%

P99-99.9: 45%

P99.9-100: 50%

The consistently higher forecast of Top 1% income compared to the WID Database empirical data is explained, as the Simulation currently uses total Return on Capital to calculate capital income, whereas WID wealth data is based on taxable capital income (which does not account for unrealised capital gains). The important measure of forecasting accuracy is the correlation between the Simulation forecast and the empirical data.

The explanation provided above also applies to calculation of Gini Coefficient – as total Return on Capital is calculated within the Simulation, rather than reportable (taxable) capital income, the income of those with a larger capital share (at the top of the wealth distribution) is systematically higher in the Simulation, resulting in a higher calculated Income Gini. The important measure of forecasting accuracy is the correlation between the Simulation forecast and the empirical data.

Return on Capital is calculated as the ‘Capital Clearing Rate’ for the purpose of providing a base within the Simulation for forecasting return on capital. The relatively low level of correlation here is expected as:

The Simulation currently does not incorporate all determinants of return on capital (such as the total value of public debt and private debt)
The 10-Year treasury bond rate used here as a point of comparison does not necessarily reflect the average cost of capital in the market, but is provided here merely as a readily available benchmark that has some level of expected relationship to the capital clearing rate.

A future refinement of the Simulation should incorporate more comprehensive modelling of the Capital Clearing Rate and expected bond rates (or other appropriate empirical test metric).

Simulation Sensitivity Analysis

Forecasting accuracy of Simulation – variance with duration of tuning period

Tuning period	5 years		10 years		12 years		15 years		20 years
Forecast Period	1968 to 2014		1973 to 2014		1975 to 2014		1978 to 2014		1982 to 2014
Variable	Correlation (r)	RMSE	Correlation (r)	RMSE	Correlation (r)	RMSE	Correlation (r)	RMSE	Correlation (r)	RMSE
GNI Growth Rate – annual	35.9%	2.4%	34.9%	2.5%	40.7%	2.0%	38.4%	2.0%	35.8%	1.9%
GNI growth rate (5 year rolling ave)	25.5%	1.4%	31.4%	1.5%	29.7%	1.3%	29.3%	1.3%	26.2%	1.2%
GNI ($)*	98.3%	5.8%	98.0%	18.2%	97.6%	5.6%	97.3%	6.4%	96.6%	7.1%
Capital Clearing Rate (5 year rolling ave)	85.6%	1.6%	83.3%	1.7%	87.2%	1.6%	88.7%	1.6%	94.9%	1.3%
Capital Ratio*	88.4%	13.8%	86.1%	11.1%	85.5%	12.4%	83.9%	11.9%	79.8%	13.7%
Capital Ratio (with ‘missing wealth’)*	89.2%	n/a	86.8%	12.8%	86.1%	11.5%	84.5%	11.1%	80.7%	11.4%
Income Gini (market)	98.8%	19.0%	98.4%	18.3%	98.1%	17.7%	97.9%	17.3%	96.8%	16.5%
Top 1% Wealth Share	89.2%	13.1%	96.1%	13.9%	97.0%	13.1%	96.9%	13.1%	96.9%	12.4%
Top 1% Wealth Share (with ‘missing wealth’)	97.8%	6.3%	97.5%	2.0%	97.0%	2.1%	96.6%	2.0%	95.5%	2.2%
Top 1% Income Share	97.8%	11.2%	98.5%	11.3%	98.4%	10.9%	98.2%	10.8%	97.7%	10.4%
Top 1% Income Share (with ‘missing wealth’)	98.8%	9.8%	98.7%	9.1%	98.6%	8.8%	98.4%	8.8%	97.8%	8.5%

* The RMSE for GNI ($) and Capital Ratio is assessed in terms of the % error of each annual forecast (compared to the WID observed value) to prevent the exponential nature of data skewing the error measurement over time

[1] Named after the long discredited ‘substance’ hypothesised by 18^th century chemists then thought to exist in all combustible materials and to be responsible for the process of combustion

[2] The Simulation commences with a simplified methodology in 1945 (based on only a capital / labour income split). The full model commences in 1962 when WID wealth and income distribution data is available.

[3] Average pre-tax capital income (apkkin_992_j_US), population (ntaxma_992_t_US) and net personal wealth (ahweal_992_j_US)

[4] 1975 is selected somewhat arbitrarily, on the basis that 10 to 12 years of data should be sufficient to empirically determine (or ‘tune’) the value of this parameter. A sensitivity analysis is provided (Appendix E) for alternate ‘tuning periods’ – 5, 10, 12, 15 and 20 years.

[5] The Simulation forecasts are a combination of ex post / ex ante forecasts as during the ‘forecasting period’ the Simulation continues to rely on inputs for labour income distribution, capital / labour income ratio, exogenous tax rate changes, price escalation, and NBER recession index.

[6] The RMSE for GNI ($) and Capital Ratio is assessed in terms of the % error of each annual forecast (compared to the WID observed value) to prevent the exponential nature of data skewing the error measurement over time

[7] Based on comparison of 5 year rolling averages for forecast and actual

Appropriate Modelling Technique	Underlying causal relationship of variables	Independent / Dependent Variable relationship	Diagrammatic representation of causal relationship	Qualitative Characteristics of Data
Single variable linear, polynomial and loglinear mathematical model	Polynomial or loglinear one-way causal relationship	One-to-one causal relationship		Highly ordered, predictable data
VAR Models	Polynomial or loglinear one-way causal relationship	Many-to-one causal relationship		‘Noisy’ data, explainable in terms of interaction(s) of multiple independent variables
Time-variable VAR Models	Relationships with nonlinear (lagged) relationships involving one-way causation	One-to-one or many-to-one variables, with non‑linear lagged dependent variable causality		‘Noisy’ data, involving interaction(s) of multiple independent variables
DSGE Models	Relationships that combine multiple groups of one-way many-to-one casual dependencies within a one-way network of causation	One-to-one or many-to-one variables, with non‑linear lagged dependent variable causality		‘Noisy’ data, involving interaction(s) of multiple independent variables
Cellular Automata Simulations	Nonlinear ‘intractable’ relationships that may include mutual causation and feedback loops	Networked nodes interacting via complex webs of causal dependencies		Data displaying emergent properties or patterns, cycles or chaotic ‘noise’ that is not explained by multiple explanatory variables
Cellular Automata Simulations		Iterative, mutual dependency in causal relationships

Why Macroeconomic Models Fail

An Extended Lucas Critique of Inequality and Growth

Introduction

Analytical Framework

The Trouble with Macroeconomics

Typology of Macro-Models

Non-parametric Mathematical models

VAR Models

Time-variable VAR Models

DSGE Models

Micro-founded Simulations and Cellular Automata Simulations

Framework for selection of modelling method

Empirical Analysis

Inequality and Growth – Unexplained

The causal nature of the growth and inequality relationship

The Lucas Critique, Extended

Testing the Analytic Framework

Forecasting Turning Points

Long-Run Growth Forecasts

Declining growth – productivity or something else?

CAS Simulation of Growth and Inequality

Introduction

A Radically Different Approach

Simulation Specification

Outputs from CAS Simulation

Deconstruction of Simulation

1945 to 1962 – Highly Progressive Taxation

1962 to 1975 – Capital Constrained Economy

1970s, 80s and 90s – Releasing the Virtuous Cycle

1995 to 2010s – Income Constrained Economy

Looking Forward

Simulation Findings

Conclusions and Recommendations

Bibliography

Key Features of Simulation

Technical Specification of Simulation

Adopted simplifications and potential future refinements

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