The Theoretical Incoherence Of Full Employment Arguments

One quite often runs into arguments that rely on assuming full employment, and then relating that policy decisions. In my view, such arguments are fundamentally weak; we need to refer to actual model results to discuss policy. In this article, I explain why an attempt to apply a NAIRU argument to a Job Guarantee is misguided. The analysis is unusual: instead of discussing a single model, the behaviour of an entire class of reasonable economic models is analysed. This reflects the attitude towards model uncertainty that animates robust control theory.

NOTE: I have added comments about the generalisation of these results. [2017-11-24] These comments will be repeated in a separate post.

Since my thesis is that full employment arguments are mathematically incoherent, I had little choice but to lapse into a stilted mathematical writing style. My apologies.

Also, as I note below, the argument is perhaps more general than just applying to an economy with a Job Guarantee. The economic structure mattered more in my initial pass at the proof, but the logic changed as I cleaned up the proof.

The Premise

I assume that the reader is familiar with a description of a Job Guarantee. I denote the percentage of the workforce that is employed by the Job Guarantee as $J$. For simplicity, we assume that the rest of the workforce is employed. In other words, $J$ acts in a similar fashion as the unemployment rate in the current institutional structure.

One could imagine the following argument, which is an attempt to apply existing beliefs about full employment to the analysis of a Job Guarantee. We suppose that there is a variable $J^*(t)$ that has the following properties.

1. The variable $J^*(t)$ cannot be affected by policy choices, in particular fiscal policy choices.
2. Wage inflation acceleration is given by some function $f(J(t) - J^*(t))$, with $f(0) = 0$ and $f$ is strictly decreasing. That is, inflation accelerates upwards (downwards) if the gap $J(t) - J^*(t)$ is negative (positive). (We see that $J^*$ is an analogue of NAIRU.)

My conjecture is that such a variable $J^*$ cannot exist for internally consistent, reasonable economic models. I cannot prove this conjecture for all possible internally consistent models, but it is possible to demonstrate it for a non-empty set of economies. The argument being that it is possible to extend the set of economies by relaxing simplifying assumptions. I justify this belief below, once the proof technique has been demonstrated.

Theoretical Economies versus Model

To begin, I would note that my analysis here appears to be unusual. We need to distinguish two theoretical concepts: a (theoretical) economy, and economic models. The reader is presumably familiar with economic models; stock-flow consistent models, or dynamic stochastic general equilibrium (DSGE) models would be two classes of models. A theoretical economy is a set of mathematical constraints for which we can find a set of economic models that respect those constraints.

I do not have a formalisation of the concept of economy constraints, but the constraints that I have in mind would be things like production functions and national accounting constraints. A particular economic model is a mathematical model which meets those constraints, while at the same time filling in behavioural rules. In other words, an economy is a mathematical description that sets the rules of the game for which behavioural rules live in. This distinction does appear in some treatments of stock-flow consistent models, but is arguably absent from many mathematical descriptions of DSGE models: behavioural assumptions are freely substituted for accounting constraints. In order to fit the existing literature within my framework, we need to draw that distinction.

One key point is that a theoretical economy has a number of variables that are assumed to exist in all models of that economy. These are various non-behavioural variables: the level of employment, average wages, prices, etc. These variables are presumably augmented by variables that are used to determine behaviour, which are model-specific. For example, one economic model might use the moving average of price changes as inflation expectations (adaptive expectations). Another model may have an inflation expectation variable that is determined by some complex optimisation procedure. The key point is that both models have to have an actual price level, which maps to the price level of the theoretical economy.

Alternatively, we can argue that economic models are defined by four classes of mathematical constraints.

1. Things that have to be true: accounting identities.
2. Constraints that define the laws of motion of an economy: production function, etc.
3. Plausibility constraints: the money stock cannot be arbitrarily large relative to nominal GDP.
4. Behavioural constraints that determine the precise economic trajectory in a simulation.

A theoretical economy is defined by the first three types of constraints, particular models fill in the final set.

My discussion here is largely informal. I rely on common usage for most definitions used. 

Economy Assumptions

I am now pinning down the class of theoretical economies that are to be analysed. Since it is somewhat unclear how to define constraints on economies, I will phrase the constraints as being on economic models themselves.

We need to distinguish some theoretical economy variables as being "inputs." These are exogenous variables, or external shocks. Other economy variables are determined by these inputs. If we have a stochastic (random) model, the particular realisation of those random variables has to be fixed. There is a certain awkwardness that arises with shocks to behavioural parameters, which are obviously model-specific. For our purposes here, all behavioural parameters have to be clamped down. 

• The economic models are time invariant. That is, if we apply a time shift to "input" variables, the model variables in the model output are similarly shifted. (A formal definition can be found in the control theory literature.)
• Accounting identities hold in a standard manner.
• Models are of a closed economy, with a fixed workforce and reasonable private sector production function. There is no investment or capital. The production function can be allowed to vary, but it should have the property that more people working in a period implies greater output.
• There is an implicit policy objective of long-term price level stability, and policy is set consistent with that objective. In particular, the Job Guarantee wage is a fixed value (denoted $W_j$). Income from this wage is not taxed. 
• Workers are either working for the private sector, or in the Job Guarantee pool. There are no other welfare programmes in place. Implicitly, this assumes that people do not stop working completely.
• Government (real) consumption is assumed to be a constant within any model scenario.
• Interest rates are pegged at 0%. This is done for simplicity; otherwise we need to worry about interest income effects. The net result is that household sector pre-tax income just equals private sector wage income plus Job Guarantee payments, plus dividends.
• We denote the average private sector wage as $W$. Taxes are paid via a non-regressive tax rate denoted $\tau_W$. (Non-regressive implies that $\tau_{W_1} > \tau_{W_0}$ if $W_1 > W_0$.) Average tax rates always lie in the interval $[0,1)$. The after-tax wage is $(1-\tau_W)W$, and denoted $\hat{W}$.
• Job Guarantee output is not sold in the market. Workers are paid, but all output is given away. Therefore, total household after-tax income is either saved (as non-interest bearing money) or used to purchase goods produced by the private sector.
• There is a trade-off between paid work and the Job Guarantee that is based on the gap between the fixed Job Guarantee wage and $\hat{W}$. In particular, for any giveneconomic model, there would be a value $\hat{W}_l$ at which the entire work force would choose to work in the Job Guarantee programme. (This value may be model-dependent.)
• Market clearing. We assume that the business sector will produce at least enough goods to meet the demand from the government and household.
• We assume that household holdings of government money cannot be negative, and that they cannot be arbitrarily large versus nominal income. The first condition is reasonable: governments generally do not finance indefinite household deficit spending. The second imposes a behavioural assumption: that we will never see a situation where someone has money holdings that is 1000000% of GDP; the presumption is that they would attempt to buy up all available production, causing a price shift. Alternatively, we argue that government liabilities cannot grow to be infinitely large relative to GDP, or that the money multiplier is bounded away from zero.
• We assume that business sector government money holdings cannot be negative, or arbitrarily large versus household money holdings. We cannot have the business sector acting as a black hole in the national accounts (an awkward issue for DSGE model accounting).
• There are upper and lower limits (both positive real numbers) for the ratio of the wage $W$ to the price level of private sector output.

The above set of assumptions is a mixture of things that have to be true (accounting identities hold) as well as simplifying assumptions. However, none of them should be controversial, they apply to many classes of existing economic models.

We then end up with some more poorly-defined assumptions. If I formalised this entire argument, I hope that these assumptions could given proper definitions. My feeling is that we would need to pin the properties of the production function (which is currently fairly open-ended) to make some decisions more precise.

Fiscal Coherence. By analogy to the governmental budget constraint of the DSGE literature, I assume that tax rates are sufficiently high to prevent arbitrarily large wages. There exists a wage limit $W_h$ for which the tax take is so large that the private sector is unwilling to work to meet demand. The fact that this a nominal upper limit arrives from the assumption that tax rate schedule follows a fixed schedule in nominal terms, and the Job Guarantee wage is fixed, implying that there is a maximum nominal expenditure on that programme by the government. The exact level of $W_h$ depends upon the behavioural assumptions of a given model. I believe that if we were to formalise this constraint, it is that the marginal tax rate is eventually greater than the supremum of all possible steady state government consumption levels as a percentage of GDP, which implies that imposed taxes must be much greater than government spending for all sufficiently high wage levels.

As an example, imagine that the top marginal tax rate is 40%, and there is something in the government consumption function rule that limits government spending to 10% of GDP. Arbitrarily large nominal incomes (GDP) would imply a fiscal surplus of 30%, which should presumably tamp down inflationary pressures at some point. 

Remark. We can formalise this concept in a brute force fashion by assuming that the average tax rate converges to 1 in a uniform fashion, and that we assume that all models incorporate an assumption that workers will be unwilling to work in the private sector if they are unable to buy the output of their labour.  This assumption is stronger than we need, but other formulations would require a more formal definition of the production function.

Steady State Definition. There is a final (heroic) assumption about the economy entering a steady state. We assume that the economy always enters a steady state in model simulations. In particular, we end up with:

• wage inflation converging to a constant;
• $J$ converging to a constant;
• real inventories converge to a constant.

We then make a theoretical leap to assume that we refer to those constants as being a steady-state solution. We could re-start the simulation with those values, and they would remain unchanged in the new simulation. If the reader is uncomfortable with this, we would need to use $\delta - \epsilon$ arguments around the limiting values of series, which would be inelegant and tiresome.

Generalisations of These Assumptions

(Added 2017-11-24).

Remark. We do not need to assume that economic models lie in the class of models described here. It may only be enough to specify that they can be approximated by such a model in steady state. This is obviously a much wider class of model behaviour. We can formalise the notion of approximating a system by applying the definitions used in robust control theory. (My doctoral thesis is the only reference I can offer off-hand that covers the issues I see.)

Remark. The steady state assumption is needed to allow us to apply algebra in the proof. If we were willing to delve into more advanced mathematics, it seems possible that we could specify conditions in frequency domain terms, and just look at the zero frequency ("DC") component of signals. Since most time series are expected to be bounded away from zero for infinite periods of time, there would be obvious convergence issues for the Fourier transform. We would need to do our analysis on finite intervals, and then take limits. This would be tedious, but the generalisation may result in a more elegant proof.

Proof

Lemma The growth rate of wages in a steady state is zero.

Proof. By definition, the growth rate of wages in a steady state is some constant $c$. 

If $c < 0$, then $W(t)$ is constantly declining. There would be some time point $t^*$ for which $W(t^*) < \hat{W}_l$. Since after-tax incomes cannot be greater than pre-tax incomes, this implies the entire work force would prefer to take a Job Guarantee job at $t^*$. That in turn implies zero private sector output. However, there is non-zero demand for goods (government consumption, and the spending of Job Guarantee wages). The market clearing assumption rules out such an outcome; wages cannot fall to such a low level.

If $c>0$, then $W(t)$ is constantly rising. This implies that $W(t)$ will eventually surpass $W_h$, which is precluded by assumption. The level of $W_h$ depends upon the functions determining the incentive to work, and the tax schedule. (This argument is obviously a short-cut; we could apply a formal fiscal coherence definition to prevent unbounded wage growth.) $\square$

Remark. The previous lemma was a statement of the obvious. The fact that an economy with non-zero government spending (and a JG) cannot spiral into a deflation where nobody is working for the private sector should not raise eyebrows. The argument that we cannot have a continued positive inflation rate solely as the result of a progressive tax system is probably controversial in some quarters. However, it is very difficult to see how continued inflation could be sustained if the tax take is twice government spending (for example), and there is essentially no welfare state spending (since Job Guarantee wages are fixed, and inconsequential when compared to private sector wages).

Lemma The growth rate of consumer prices in steady state is zero.

Proof. Apply assumption on the wage/price ratio being bounded. $\square$

Remark. The previous lemma is just a way of getting rid of consumer prices in analysis; the only inflation that matters is wage inflation.

Lemma In steady state, the fiscal balance has to be zero (as is the household financial balance).

Proof In steady state, we know that government spending and taxes have to be fixed (given the previous results on price and wage stability, as well as the assumption that $J$ and government consumption are constant). Therefore, the fiscal balance is constant. If this constant is non-zero, it implies that the absolute value of household sector money holdings are eventually unbounded relative to household income, which is assumed to be impossible. (The arguments for the household sector fiscal balance to be zero are similar.) $\square$

Lemma. The variable $J^*(t)$ has to be a constant $J^* \in R_+$.

Proof.  We take any steady state solution. By the definition of steady state used here, $J(t)$ is a constant, and wage acceleration is zero for all $t$. By applying the definition of $J^*$, we see that it has to be constant. $\square$

Theorem The variable $J^*$ cannot exist (with the above assumptions in place).

Proof. We have shown that $J^*$ has to be constant. For simplicity, we take a scenario that assumes that government consumption is zero (discussed below). We determine the steady state wage, denote it $W_0$. Since the household sector (and government sector) exhibit fiscal balance, we have the following relationship:
$$
(1-J^*)W_0 \tau^0_{W_0} = J^* W_j,
$$
with $\tau^0_{W_0}$ being the average tax rate associated with average wage $W_0$ for this scenario (index 0).

This implies that
$$
W_0 \tau^0_{W_0} = \frac{J^*}{1-J^*} W_j.
$$

If we then run a second scenario, we see that:
$$
W_1 \tau^1_{W_1} = \frac{J^*}{1-J^*} W_j = W_0 \tau^0_{W_0}.
$$

We see that if the tax rate $\tau^1_{W_1} > \tau^0_{W_2}$, then $W_1 < W_0$. This means that a rise in tax rates between the two scenarios creates a double whammy for after-tax incomes: the tax rate is higher, and the pretax income is lower.

We then can decide to have the tax rate in the second scenario to be determined by a completely flat average tax rate of $\tau^1_W = \frac{(1+\tau_{W_0}^0)}{2}$, which lies in $[\frac{1}{2},1)$. We can then verify that the after-tax income $\hat{W}_1 \leq \frac{\hat{W_0}}{2}$.  Since we can divide the after-tax wage in half in one iteration, it is clear that we can repeat the process, and drive the after-tax wage below $\hat{W}_l$ in a finite number of iterations. This contradicts the assumption that the after-tax wage remains above that threshold. $\square$

Remark. We have just demonstrated that there cannot be a natural rate of $J$ that is independent of fiscal policy settings. In order to achieve such a result, we need to have a model where behaviour is unaffected by things like prices or utility maximisation.

Remarks on the  Steady State Assumption

The steady state assumption is presumably controversial.

The fact that it features price level stability should not be surprising. So long as the marginal tax rate is sufficient high and spending is not completely indexed to the price level, the fiscal surplus will be highly restrictive. This is only surprising for a modelling tradition that has no variables that act as anchor points for nominal variables.

The second line of attack is that wages remain bounded, but the system oscillates or is allegedly "chaotic." For example, a Minsky-ite might argue that economic models should feature endogenous business cycles. However, it is extremely difficult to compare such models: since all scenarios feature business cycles, how can we compare them other than going over the entire time history? If we are looking at "full employment" argumentation, there is always an implicit assumption that we have a static scenario, so we can compare steady state values.

Remarks on Solution Generality

One could look at the host of assumptions used and argue that I have found a special case. However, that underestimates the logic of the argument.

If I were to use an "all else equal" argument, it would look like this.

1. We have shown that in a steady state, the fiscal balance is zero.
2. Fix an initial steady state solution.
3. Raise the tax rate, and "hold all else equal."
4. Since "everybody knows" that raising tax rates increases the fiscal balance in steady state, we end up with a steady state with a fiscal surplus, which was shown to be impossible.

If I were to bury the above argument with enough bloviation, it might appear just as rigorous as most non-mathematical economic analysis. However, the logic is incorrect.

If we know the fiscal balance is constrained to be zero, it is possible that other variables squirt to new values that allow an unchanged fiscal balance yet with higher tax rates. Many of the questionable assumptions taken were simplifications to force the adjustment onto a single variable: average wages. (This is why I set government consumption to zero: to eliminate the effect of the price of private sector output on the fiscal balance.) The only way to preserve fiscal balance with a higher tax rate with the given assumptions is to force wages lower, by a comparable amount. We just cudgel the economy with higher taxes until the after-tax wage rate drops below a plausibility threshold value.

All we need to do to extend this proof is to see what other variables can change, and then create plausibility limits for their movements. However, it seems likely that we have to start imposing restrictions on the production function, and the proof is far more cumbersome. In order to create an elegant proof, we need to find a way of expressing such plausibility assumptions in a clean fashion.

However, it appears clear that finding a counter-example model would be very awkward. One needs to find a model in which fiscal policy settings (consumption, tax rates) must always have no effect on the steady state solution. It seems obvious that would result in a model that most observers of fiscal policy debates would find implausible. (This may provide an avenue for a more general framing of the plausibility constraints.)

Furthermore, the proof technique sidesteps heterodox complaints about equilibrium analysis. There is absolutely no requirement that any two steady states are near to each other in the state space; all that is required is that the system converges to some steady state regardless of initial conditions. There is no temporal relationship between successive steady states; they are the limiting results of completely independent model simulations. Arguments that a steady state cannot exist are somewhat plausible, but in such an environment, it is exceedingly difficult to compare policy choices.

Finally, there exists misunderstandings regarding the post-Keynesian usage of the expression stock-flow consistency. As illustrated here, it is not just that model accounting adds up. Instead, it means that stock variables are accounted for, and they are not allowed to reach implausible values. The assumption that the money stock cannot grow in an unbounded fashion is a key reason why $J^*$ cannot exist.

Remarks on Robust Control

The above argumentation is certainly not using existing robust control theory. However, it reflects the spirit of sensible robust control theory: our analysis should not be tied to particular model. We instead should try to cover a wide class of models within a single analysis.

Applying robust control principles to economics appears to require a re-thinking of the mathematical formalism. We need to distinguish between constraints that must apply, and behavioural constraints. We then need to have a high level description on what constitutes reasonable economic behaviour. As an example, we see that it would be unreasonable for workers to remain in the private sector if the Job Guarantee wage was ten times the average private sector wage; any model that suggested that outcome should be ruled out of contention.

Other Remarks

This section is a group of observations regarding the mathematical exposition. I am following the mathematical writing style that these remarks are stand-alone observations, and there is no narrative arc connecting them. That is, they can be safely read in any order, and issues with one remark do no impinge on the others.

Remark. As I have emphasised, this proof is informal, relying on common usage of terms. Once the argument is formalised, it is unclear whether or not there is a Job Guarantee matters; all that is needed is a lower bound for nominal after-tax wages. If this is indeed correct, this argumentation could be extended to take on the concept of NAIRU in the present institutional structure.

Remark. The assumption that government consumption is zero in the theorem is jarring. It was done solely for algebraic simplicity, I believe it can be relaxed. The problem is the assumption that government purchases are done in a purely price-taking fashion, as is the case in other economic models. The assumption that government spending is completely indifferent to price levels seems obviously incoherent with a belief that price level stability is the primary economic objective. A simple alternative is to impose the condition that total government spending in nominal terms on consumption is constant.

Remark. As to be expected, this model is Chartalist. If we allow the Job Guarantee wage and tax brackets to be indexed to the price level in some fashion, it would not be surprising that steady states could feature non-zero inflation rates. Very simply, the argument is that if the objective is price level stability, fiscal policy has to be set in a fashion that is consistent with that objective. Attempting to use monetary policy to counteract an incoherent fiscal policy is a questionable strategy.

Remark. One could argue that higher marginal tax rates did not prevent inflation historically. However, that was in an environment where all variables ended up being indexed to realised inflation rates. This behaviour obviously destabilises the system, and is inappropriate to deal with analysis that presupposes that the objective is price level stability.

Remark. The fact that Job Guarantee income is not taxed, but private sector wages are, is used to simplify the proof. This unfairness is regrettable.

Remark.  It is possible to imagine someone arguing that mathematical arguments do not matter, we know that $J^*$ has to exist for some reason. The only response is that this person does not have an internally consistent mathematical model that respects the given behavioural assumptions, particularly the accounting constraints. It is easy to construct models where the inflation rate has to be determined by $J^*$, however, they would feature behaviour that either impervious to tax rates (and so economic incentives do not matter for employment decisions), or feature the household or business sector generating money balances with an arbitrarily large magnitude. We cannot know which assumption fails until the model is constructed and analysed.

Remark. The transition to a constant-growth framework seems straightforward. We need to have the Job Guarantee wage and tax brackets growing at a constant rate, and we would need to use these variables as scaling constants in analysis. This would increase the amount of algebra required, and obscure the working of the proof.

Remark. The time invariance assumption does not seem to appear in the analysis. However, I believe that is required for the manipulations involving the steady state to work (which are currently informal). We need to be able to cleanly transition from solutions converging to a steady state, and a solution to the economic model with constant values. Time invariance makes this process much simpler to work with.

Remark. This result should be worrisome to those who believe that they can take a purely empirical approach to macroeconomics. For any model in the class studied, it is clear that we can calculate inflation acceleration and $J$ for all time for a scenario $s$. Furthermore, unless the model has some very unusual dynamics, $J(t)$ and inflation acceleration will be both auto-correlated. We can then apply the same statistical techniques used to estimate NAIRU in the real world to this model data. It seems likely that they will produce an estimate for a "natural rate" of $J$ which converges to some value; call this $\hat{J}_s^*$. However, as can be seen from this proof, the estimate $\hat{J}_s^*$ has no predictive value for any other scenarios generated by the same model, for every single model in this class.

Remark. The proof is based on a construction that shows we can always raise taxes to drive up $J$ if it is less than 100% (if necessary, by driving the whole labour pool into the Job Guarantee programme). The implication appears to be that we can lower $J$ by cutting taxes. However, that process cannot be repeated indefinitely: sooner or later, we have to hit the theoretical minimum for tax rates (which is presumably model-dependent). One could attempt to rescue "full employment" arguments by assuming that we are always and everywhere at the absolute theoretical minimum for tax rates. However, such a belief is hard to square with observed data. I may expand this discussion in a later article.

Remark. One could argue that there is an irreducible minimum value to $J$. Such an argument does not appear to make much sense for a sensible Job Guarantee scheme, but it is more plausible if we translated back to the unemployment rate. There are strong arguments that there will always be a certain part of the workforce who are transitioning between jobs at any given time, and show up as unemployed in the survey. I believe that this minimum level was estimated to be 2% or so. However, the existence of such a minimum tells us nothing about $J^*$: if it is a threshold that cannot be surpassed, we can never observe $J$ below that level. Trying to fit this concept into a NAIRU-like definition is difficult. It is equivalent to saying that accounting identities always hold, and if they do not, it would be inflationary. In any event, the theoretical lower bound for tax rates would presumably interact with this barrier to create a theoretical minimum value for $J$; however, if we are in steady state, we still will have price stability.

Remark. Neil Wilson made a comment with observations about the non-uniformity of workers. This is a real world issue, However, non-uniformity implies that wage incomes are not all equal, and it would be possible to to have the same average wage rate corresponding to different average income tax levels. That violates the assumptions about the tax rate. However, if we can approximate the more complex model in steady state with a model that meets these assumptions, the conjecture is that the result would still apply.

(c) Brian Romanchuk 2017