The Theoretical Incoherence Of Amount Utilization Arguments

One quite oftentimes runs into arguments that rely on assuming total employment, as well as and hence relating that policy decisions. In my view, such arguments are fundamentally weak; nosotros demand to refer to actual model results to hash out policy. In this article, I explicate why an endeavor to apply a NAIRU declaration to a Job Guarantee is misguided. The analysis is unusual: instead of discussing a unmarried model, the conduct of an entire course of didactics of reasonable economical models is analysed. This reflects the mental attitude towards model doubtfulness that animates robust command theory.

NOTE: I get got added comments almost the generalisation of these results. [2017-11-24] These comments volition last repeated inwards a separate post.

Since my thesis is that total job arguments are mathematically incoherent, I had piffling alternative but to lapse into a stilted mathematical writing style. My apologies.

Also, every bit I complaint below, the declaration is perchance to a greater extent than full general than exactly applying to an economic scheme amongst a Job Guarantee. The economical construction mattered to a greater extent than inwards my initial reach at the proof, but the logic changed every bit I cleaned upwardly the proof.

The Premise

I assume that the reader is familiar amongst a description of a Job Guarantee. I announce the percent of the workforce that is employed past times the Job Guarantee every bit $J$. For simplicity, nosotros assume that the ease of the workforce is employed. In other words, $J$ acts inwards a similar fashion every bit the unemployment charge per unit of measurement inwards the electrical flow institutional structure.

One could imagine the next argument, which is an endeavor to apply existing beliefs almost total job to the analysis of a Job Guarantee. We suppose that at that topographic point is a variable $J^*(t)$ that has the next properties.

1. The variable $J^*(t)$ cannot last affected past times policy choices, inwards detail financial policy choices.
2. Wage inflation acceleration is given past times some role $f(J(t) - J^*(t))$, amongst $f(0) = 0$ as well as $f$ is strictly decreasing. That is, inflation accelerates upwards (downwards) if the gap $J(t) - J^*(t)$ is negative (positive). (We reckon that $J^*$ is an analogue of NAIRU.)

My conjecture is that such a variable $J^*$ cannot be for internally consistent, reasonable economical models. I cannot seek out this conjecture for all possible internally consistent models, but it is possible to demonstrate it for a non-empty laid of economies. The declaration beingness that it is possible to extend the laid of economies past times relaxing simplifying assumptions. I justify this belief below, ane time the proof technique has been demonstrated.

Theoretical Economies versus Model

To begin, I would complaint that my analysis hither appears to last unusual. We demand to distinguish 2 theoretical concepts: a (theoretical) economy, as well as economical models. The reader is presumably familiar amongst economical models; stock-flow consistent models, or dynamic stochastic full general equilibrium (DSGE) models would last 2 classes of models. Influenza A virus subtype H5N1 theoretical economy is a laid of mathematical constraints for which nosotros tin expose a laid of economical models that abide by those constraints.

I practise non get got a formalisation of the concept of economic scheme constraints, but the constraints that I get got inwards hear would last things similar production functions as well as national accounting constraints. Influenza A virus subtype H5N1 detail economical model is a mathematical model which meets those constraints, spell at the same fourth dimension filling inwards behavioural rules. In other words, an economy is a mathematical description that sets the rules of the game for which behavioural rules alive in. This distinction does appear inwards 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 guild to agree the existing literature inside my framework, nosotros demand to pull that distinction.

One key signal is that a theoretical economic scheme has a number of variables that are assumed to be inwards all models of that economy. These are diverse non-behavioural variables: the grade of employment, average wages, prices, etc. These variables are presumably augmented past times variables that are used to determine behaviour, which are model-specific. For example, ane economical model mightiness job the moving average of toll changes every bit inflation expectations (adaptive expectations). Another model may get got an inflation expectation variable that is determined past times some complex optimisation procedure. The key signal is that both models get got to get got an actual toll level, which maps to the toll grade of the theoretical economy.

Alternatively, nosotros tin debate that economical models are defined past times iv classes of mathematical constraints.

1. Things that get got to last true: accounting identities.
2. Constraints that define the laws of motion of an economy: production function, etc.
3. Plausibility constraints: the coin stock cannot last arbitrarily large relative to nominal GDP.
4. Behavioural constraints that determine the precise economical trajectory inwards a simulation.

Influenza A virus subtype H5N1 theoretical economic scheme is defined past times the showtime iii types of constraints, detail models fill upwardly inwards the concluding set.

My give-and-take hither is largely informal. I rely on mutual usage for most definitions used. 

Economy Assumptions

I am forthwith pinning downwardly the course of didactics of theoretical economies that are to last analysed. Since it is somewhat unclear how to define constraints on economies, I volition phrase the constraints every bit beingness on economical models themselves.

We demand to distinguish some theoretical economic scheme variables every bit beingness "inputs." These are exogenous variables, or external shocks. Other economic scheme variables are determined past times these inputs. If nosotros get got a stochastic (random) model, the detail realisation of those random variables has to last fixed. There is a sure awkwardness that arises amongst shocks to behavioural parameters, which are apparently model-specific. For our purposes here, all behavioural parameters get got to last clamped down. 

• The economical models are fourth dimension invariant. That is, if nosotros apply a fourth dimension shift to "input" variables, the model variables inwards the model output are similarly shifted. (A formal definition tin last flora inwards the command theory literature.)
• Accounting identities agree inwards a criterion manner.
• Models are of a unopen economy, amongst a fixed workforce as well as reasonable individual sector production function. There is no investment or capital. The production role tin last allowed to vary, but it should get got the holding that to a greater extent than people working inwards a menstruation implies greater output.
• There is an implicit policy objective of long-term toll grade stability, as well as policy is laid consistent amongst that objective. In particular, the Job Guarantee wage is a fixed value (denoted $W_j$). Income from this wage is non taxed. 
• Workers are either working for the individual sector, or inwards the Job Guarantee pool. There are no other welfare programmes inwards place. Implicitly, this assumes that people practise non halt working completely.
• Government (real) consumption is assumed to last a constant inside whatever model scenario.
• Interest rates are pegged at 0%. This is done for simplicity; otherwise nosotros demand to worry almost involvement income effects. The internet termination is that solid sector pre-tax income exactly equals individual sector wage income addition Job Guarantee payments, addition dividends.
• We announce the average individual sector wage every bit $W$. Taxes are paid via a non-regressive revenue enhancement charge per unit of measurement denoted $\tau_W$. (Non-regressive implies that $\tau_{W_1} > \tau_{W_0}$ if $W_1 > W_0$.) Average revenue enhancement rates e'er prevarication inwards the interval $[0,1)$. The after-tax wage is $(1-\tau_W)W$, as well as denoted $\hat{W}$.
• Job Guarantee output is non sold inwards the market. Workers are paid, but all output is given away. Therefore, total solid after-tax income is either saved (as non-interest bearing money) or used to purchase goods produced past times the individual sector.
• There is a trade-off betwixt paid piece of work as well as the Job Guarantee that is based on the gap betwixt the fixed Job Guarantee wage as well as $\hat{W}$. In particular, for whatever giveneconomic model, at that topographic point would last a value $\hat{W}_l$ at which the entire piece of work strength would select to piece of work inwards the Job Guarantee programme. (This value may last model-dependent.)
• Market clearing. We assume that the job concern sector volition arrive at at to the lowest degree plenty goods to run into the demand from the authorities as well as household.
• We assume that solid holdings of authorities coin cannot last negative, as well as that they cannot last arbitrarily large versus nominal income. The showtime status is reasonable: governments to a greater extent than oftentimes than non practise non finance indefinite solid deficit spending. The minute imposes a behavioural assumption: that nosotros volition never reckon a province of affairs where someone has coin holdings that is 1000000% of GDP; the presumption is that they would endeavor to purchase upwardly all available production, causing a toll shift. Alternatively, nosotros debate that authorities liabilities cannot grow to last infinitely large relative to GDP, or that the coin multiplier is bounded away from zero.
• We assume that job concern sector authorities coin holdings cannot last negative, or arbitrarily large versus solid coin holdings. We cannot get got the job concern sector acting every bit a dark hole inwards the national accounts (an awkward number for DSGE model accounting).
• There are upper as well as lower limits (both positive existent numbers) for the ratio of the wage $W$ to the toll grade of individual sector output.

The higher upwardly laid of assumptions is a mixture of things that get got to last truthful (accounting identities hold) likewise every bit simplifying assumptions. However, none of them should last controversial, they apply to many classes of existing economical models.

We as well as hence halt upwardly amongst some to a greater extent than poorly-defined assumptions. If I formalised this entire argument, I promise that these assumptions could given proper definitions. My feeling is that nosotros would demand to pivot the properties of the production role (which is currently fairly open-ended) to brand some decisions to a greater extent than precise.

Fiscal Coherence. By analogy to the governmental budget constraint of the DSGE literature, I assume that revenue enhancement rates are sufficiently high to preclude arbitrarily large wages. There exists a wage bound $W_h$ for which the revenue enhancement accept is hence large that the individual sector is unwilling to piece of work to run into demand. The fact that this a nominal upper bound arrives from the supposition that revenue enhancement charge per unit of measurement schedule follows a fixed schedule inwards nominal terms, as well as the Job Guarantee wage is fixed, implying that at that topographic point is a maximum nominal expenditure on that programme past times the government. The exact grade of $W_h$ depends upon the behavioural assumptions of a given model. I believe that if nosotros were to formalise this constraint, it is that the marginal revenue enhancement charge per unit of measurement is eventually greater than the supremum of all possible steady soil authorities consumption levels every bit a percent of GDP, which implies that imposed taxes must last much greater than authorities spending for all sufficiently high wage levels.

As an example, imagine that the top marginal revenue enhancement charge per unit of measurement is 40%, as well as at that topographic point is something inwards the authorities consumption role dominion that limits authorities spending to 10% of GDP. Arbitrarily large nominal incomes (GDP) would imply a financial surplus of 30%, which should presumably tamp downwardly inflationary pressures at some point. 

Remark. We tin formalise this concept inwards a fauna strength fashion past times assuming that the average revenue enhancement charge per unit of measurement converges to 1 inwards a uniform fashion, as well as that nosotros assume that all models comprise an supposition that workers volition last unwilling to piece of work inwards the individual sector if they are unable to purchase the output of their labour.  This supposition is stronger than nosotros need, but other formulations would require a to a greater extent than formal definition of the production function.

Steady State Definition. There is a concluding (heroic) supposition almost the economic scheme entering a steady state. We assume that the economic scheme e'er enters a steady soil inwards model simulations. In particular, nosotros halt upwardly with:

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

We as well as hence brand a theoretical leap to assume that nosotros refer to those constants every bit beingness a steady-state solution. We could re-start the simulation amongst those values, as well as they would rest unchanged inwards the novel simulation. If the reader is uncomfortable amongst this, nosotros would demand to job $\delta - \epsilon$ arguments approximately the limiting values of series, which would last inelegant and tiresome.

Generalisations of These Assumptions

(Added 2017-11-24).

Remark. We practise non demand to assume that economical models prevarication inwards the course of didactics of models described here. It may only last plenty to specify that they tin last approximated past times such a model inwards steady state. This is apparently a much wider course of didactics of model behaviour. We tin formalise the notion of approximating a organization past times applying the definitions used inwards robust command theory. (My doctoral thesis is the only reference I tin offering off-hand that covers the issues I see.)

Remark. The steady soil supposition is needed to allow us to apply algebra inwards the proof. If nosotros were willing to delve into to a greater extent than advanced mathematics, it seems possible that nosotros could specify atmospheric condition inwards frequency domain terms, as well as exactly appear at the null frequency ("DC") factor of signals. Since most fourth dimension serial are expected to last bounded away from null for infinite periods of time, at that topographic point would last obvious convergence issues for the Fourier transform. We would demand to practise our analysis on finite intervals, as well as and hence accept limits. This would last tedious, but the generalisation may termination inwards a to a greater extent than elegant proof.

Proof

Lemma The growth charge per unit of measurement of payoff inwards a steady soil is zero.

Proof. By definition, the growth charge per unit of measurement of payoff inwards a steady soil is some constant $c$. 

If $c < 0$, as well as hence $W(t)$ is constantly declining. There would last some fourth dimension signal $t^*$ for which $W(t^*) < \hat{W}_l$. Since after-tax incomes cannot last greater than pre-tax incomes, this implies the entire piece of work strength would prefer to accept a Job Guarantee labor at $t^*$. That inwards plow implies null individual sector output. However, at that topographic point is non-zero demand for goods (government consumption, as well as the spending of Job Guarantee wages). The marketplace clearing supposition rules out such an outcome; payoff cannot autumn to such a depression level.

If $c>0$, as well as hence $W(t)$ is constantly rising. This implies that $W(t)$ volition eventually surpass $W_h$, which is precluded past times assumption. The grade of $W_h$ depends upon the functions determining the incentive to work, as well as the revenue enhancement schedule. (This declaration is apparently a short-cut; nosotros could apply a formal financial coherence definition to preclude unbounded wage growth.) $\square$

Remark. The previous lemma was a declaration of the obvious. The fact that an economic scheme amongst non-zero authorities spending (and a JG) cannot spiral into a deflation where nobody is working for the individual sector should non enhance eyebrows. The declaration that nosotros cannot get got a continued positive inflation charge per unit of measurement exclusively every bit the termination of a progressive revenue enhancement organization is in all likelihood controversial inwards some quarters. However, it is really hard to reckon how continued inflation could last sustained if the revenue enhancement accept is twice authorities spending (for example), as well as at that topographic point is essentially no welfare soil spending (since Job Guarantee payoff are fixed, as well as inconsequential when compared to individual sector wages).

Lemma The growth charge per unit of measurement of consumer prices inwards steady soil is zero.

Proof. Apply supposition on the wage/price ratio beingness bounded. $\square$

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

Lemma In steady state, the financial residual has to last null (as is the solid financial balance).

Proof In steady state, nosotros know that authorities spending as well as taxes get got to last fixed (given the previous results on toll as well as wage stability, likewise every bit the supposition that $J$ as well as authorities consumption are constant). Therefore, the financial residual is constant. If this constant is non-zero, it implies that the absolute value of solid sector coin holdings are eventually unbounded relative to solid income, which is assumed to last impossible. (The arguments for the solid sector financial residual to last null are similar.) $\square$

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

Proof.  We accept whatever steady soil solution. By the definition of steady soil used here, $J(t)$ is a constant, as well as wage acceleration is null for all $t$. By applying the definition of $J^*$, nosotros reckon that it has to last constant. $\square$

Theorem The variable $J^*$ cannot be (with the higher upwardly assumptions inwards place).

Proof. We get got shown that $J^*$ has to last constant. For simplicity, nosotros accept a scenario that assumes that authorities consumption is null (discussed below). We determine the steady soil wage, announce it $W_0$. Since the solid sector (and authorities sector) exhibit financial balance, nosotros get got the next relationship:
$$
(1-J^*)W_0 \tau^0_{W_0} = J^* W_j,
$$
amongst $\tau^0_{W_0}$ beingness the average revenue enhancement charge per unit of measurement associated amongst 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 nosotros as well as hence run a minute scenario, nosotros reckon that:
$$
W_1 \tau^1_{W_1} = \frac{J^*}{1-J^*} W_j = W_0 \tau^0_{W_0}.
$$

We reckon that if the revenue enhancement charge per unit of measurement $\tau^1_{W_1} > \tau^0_{W_2}$, as well as hence $W_1 < W_0$. This agency that a ascent inwards revenue enhancement rates betwixt the 2 scenarios creates a double whammy for after-tax incomes: the revenue enhancement charge per unit of measurement is higher, as well as the pretax income is lower.

We as well as hence tin create upwardly one's hear to get got the revenue enhancement charge per unit of measurement inwards the minute scenario to last determined past times a completely apartment average revenue enhancement charge per unit of measurement of $\tau^1_W = \frac{(1+\tau_{W_0}^0)}{2}$, which lies inwards $[\frac{1}{2},1)$. We tin as well as hence verify that the after-tax income $\hat{W}_1 \leq \frac{\hat{W_0}}{2}$.  Since nosotros tin split the after-tax wage inwards one-half inwards ane iteration, it is clear that nosotros tin repeat the process, as well as motion the after-tax wage below $\hat{W}_l$ inwards a finite number of iterations. This contradicts the supposition that the after-tax wage remains higher upwardly that threshold. $\square$

Remark. We get got exactly demonstrated that at that topographic point cannot last a natural charge per unit of measurement of $J$ that is independent of financial policy settings. In guild to reach such a result, nosotros demand to get got a model where conduct is unaffected past times things similar prices or utility maximisation.

Remarks on the  Steady State Assumption

The steady soil supposition is presumably controversial.

The fact that it features toll grade stability should non last surprising. So long every bit the marginal revenue enhancement charge per unit of measurement is sufficient high as well as spending is non completely indexed to the toll level, the financial surplus volition last highly restrictive. This is only surprising for a modelling tradition that has no variables that human activity every bit anchor points for nominal variables.

The minute describe of assail is that payoff rest bounded, but the organization oscillates or is allegedly "chaotic." For example, a Minsky-ite mightiness debate that economical models should characteristic endogenous job concern cycles. However, it is extremely hard to compare such models: since all scenarios characteristic job concern cycles, how tin nosotros compare them other than going over the entire fourth dimension history? If nosotros are looking at "full employment" argumentation, at that topographic point is e'er an implicit supposition that nosotros get got a static scenario, hence nosotros tin compare steady soil values.

Remarks on Solution Generality

One could appear at the host of assumptions used as well as debate that I get got flora a special case. However, that underestimates the logic of the argument.

If I were to job an "all else equal" argument, it would appear similar this.

1. We get got shown that inwards a steady state, the financial residual is zero.
2. Fix an initial steady soil solution.
3. Raise the revenue enhancement rate, as well as "hold all else equal."
4. Since "everybody knows" that raising revenue enhancement rates increases the financial residual inwards steady state, nosotros halt upwardly amongst a steady soil amongst a financial surplus, which was shown to last impossible.

If I were to bury the higher upwardly declaration amongst plenty bloviation, it mightiness appear exactly every bit rigorous every bit most non-mathematical economical analysis. However, the logic is incorrect.

If nosotros know the financial residual is constrained to last zero, it is possible that other variables squirt to novel values that allow an unchanged financial residual yet amongst higher revenue enhancement rates. Many of the questionable assumptions taken were simplifications to strength the adjustment onto a unmarried variable: average wages. (This is why I laid authorities consumption to zero: to eliminate the number of the toll of individual sector output on the financial balance.) The only way to preserve financial residual amongst a higher revenue enhancement charge per unit of measurement amongst the given assumptions is to strength payoff lower, past times a comparable amount. We exactly cudgel the economic scheme amongst higher taxes until the after-tax wage charge per unit of measurement drops below a plausibility threshold value.

All nosotros demand to practise to extend this proof is to reckon what other variables tin change, as well as and hence create plausibility limits for their movements. However, it seems probable that nosotros get got to start imposing restrictions on the production function, as well as the proof is far to a greater extent than cumbersome. In guild to create an elegant proof, nosotros demand to expose a way of expressing such plausibility assumptions inwards a create clean fashion.

However, it appears clear that finding a counter-example model would last really awkward. One needs to expose a model inwards which financial policy settings (consumption, revenue enhancement rates) must always get got no number on the steady soil solution. It seems obvious that would termination inwards a model that most observers of financial policy debates would expose implausible. (This may render an avenue for a to a greater extent than full general framing of the plausibility constraints.)

Furthermore, the proof technique sidesteps heterodox complaints almost equilibrium analysis. There is absolutely no requirement that whatever 2 steady states are close to each other inwards the soil space; all that is required is that the organization converges to some steady soil regardless of initial conditions. There is no temporal human relationship betwixt successive steady states; they are the limiting results of completely independent model simulations. Arguments that a steady soil cannot be are somewhat plausible, but inwards such an environment, it is exceedingly hard to compare policy choices.

Finally, at that topographic point exists misunderstandings regarding the post-Keynesian usage of the appear stock-flow consistency. As illustrated here, it is non exactly that model accounting adds up. Instead, it agency that stock variables are accounted for, as well as they are non allowed to reach implausible values. The supposition that the coin stock cannot grow inwards an unbounded fashion is a key argue why $J^*$ cannot exist.

Remarks on Robust Control

The higher upwardly argumentation is sure non using existing robust command theory. However, it reflects the spirit of sensible robust command theory: our analysis should non last tied to detail model. We instead should seek to encompass a broad course of didactics of models inside a unmarried analysis.

Applying robust command principles to economic science appears to require a re-thinking of the mathematical formalism. We demand to distinguish betwixt constraints that must apply, as well as behavioural constraints. We as well as hence demand to get got a high grade description on what constitutes reasonable economical behaviour. As an example, nosotros reckon that it would last unreasonable for workers to rest inwards the individual sector if the Job Guarantee wage was x times the average individual sector wage; whatever model that suggested that outcome should last ruled out of contention.

Other Remarks

This department is a grouping of observations regarding the mathematical exposition. I am next the mathematical writing trend that these remarks are stand-alone observations, as well as at that topographic point is no narrative arc connecting them. That is, they tin last safely read inwards whatever order, as well as issues amongst ane remark practise no impinge on the others.

Remark. As I get got emphasised, this proof is informal, relying on mutual usage of terms. Once the declaration is formalised, it is unclear whether or non at that topographic point 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 last extended to accept on the concept of NAIRU inwards the introduce institutional structure.

Remark. The supposition that authorities consumption is null inwards the theorem is jarring. It was done exclusively for algebraic simplicity, I believe it tin last relaxed. The job is the supposition that authorities purchases are done inwards a purely price-taking fashion, every bit is the instance inwards other economical models. The supposition that authorities spending is completely indifferent to toll levels seems apparently incoherent amongst a belief that toll grade stability is the primary economical objective. Influenza A virus subtype H5N1 uncomplicated alternative is to impose the status that total authorities spending inwards nominal terms on consumption is constant.

Remark. As to last expected, this model is Chartalist. If nosotros allow the Job Guarantee wage as well as revenue enhancement brackets to last indexed to the toll grade inwards some fashion, it would non last surprising that steady states could characteristic non-zero inflation rates. Very simply, the declaration is that if the objective is toll grade stability, financial policy has to last laid inwards a fashion that is consistent amongst that objective. Attempting to job monetary policy to counteract an incoherent financial policy is a questionable strategy.

Remark. One could debate that higher marginal revenue enhancement rates did non preclude inflation historically. However, that was inwards an environs where all variables ended upwardly beingness indexed to realised inflation rates. This conduct apparently destabilises the system, as well as is inappropriate to bargain amongst analysis that presupposes that the objective is toll grade stability.

Remark. The fact that Job Guarantee income is non taxed, but individual sector payoff are, is used to simplify the proof. This unfairness is regrettable.

Remark.  It is possible to imagine someone arguing that mathematical arguments practise non matter, nosotros know that $J^*$ has to be for some reason. The only answer is that this somebody does non get got an internally consistent mathematical model that respects the given behavioural assumptions, specially the accounting constraints. It is slowly to build models where the inflation charge per unit of measurement has to last determined past times $J^*$, however, they would characteristic conduct that either impervious to revenue enhancement rates (and hence economical incentives practise non affair for job decisions), or characteristic the solid or job concern sector generating coin balances amongst an arbitrarily large magnitude. We cannot know which supposition fails until the model is constructed as well as analysed.

Remark. The transition to a constant-growth framework seems straightforward. We demand to get got the Job Guarantee wage as well as revenue enhancement brackets growing at a constant rate, as well as nosotros would demand to job these variables every bit scaling constants inwards analysis. This would increment the amount of algebra required, as well as obscure the working of the proof.

Remark. The fourth dimension invariance supposition does non seem to appear inwards the analysis. However, I believe that is required for the manipulations involving the steady soil to piece of work (which are currently informal). We demand to last able to cleanly transition from solutions converging to a steady state, as well as a solution to the economical model amongst constant values. Time invariance makes this physical care for much simpler to piece of work with.

Remark. This termination should last worrisome to those who believe that they tin accept a purely empirical approach to macroeconomics. For whatever model inwards the course of didactics studied, it is clear that nosotros tin calculate inflation acceleration as well as $J$ for all fourth dimension for a scenario $s$. Furthermore, unless the model has some really odd dynamics, $J(t)$ as well as inflation acceleration volition last both auto-correlated. We tin as well as hence apply the same statistical techniques used to approximate NAIRU inwards the existent basis to this model data. It seems probable that they volition arrive at an approximate for a "natural rate" of $J$ which converges to some value; telephone band this $\hat{J}_s^*$. However, every bit tin last seen from this proof, the approximate $\hat{J}_s^*$ has no predictive value for whatever other scenarios generated past times the same model, for every unmarried model inwards this class.

Remark. The proof is based on a construction that shows nosotros tin e'er enhance taxes to motion upwardly $J$ if it is less than 100% (if necessary, past times driving the whole labour puddle into the Job Guarantee programme). The implication appears to last that nosotros tin lower $J$ past times cutting taxes. However, that physical care for cannot last repeated indefinitely: sooner or later, nosotros get got to hitting the theoretical minimum for revenue enhancement rates (which is presumably model-dependent). One could endeavor to rescue "full employment" arguments past times assuming that nosotros are e'er as well as everywhere at the absolute theoretical minimum for revenue enhancement rates. However, such a belief is hard to foursquare amongst observed data. I may expand this give-and-take inwards a afterwards article.

Remark. One could debate that at that topographic point is an irreducible minimum value to $J$. Such an declaration does non appear to brand much sense for a sensible Job Guarantee scheme, but it is to a greater extent than plausible if nosotros translated dorsum to the unemployment rate. There are potent arguments that at that topographic point volition e'er last a sure component of the workforce who are transitioning betwixt jobs at whatever given time, as well as demo upwardly every bit unemployed inwards the survey. I believe that this minimum grade was estimated to last 2% or so. However, the existence of such a minimum tells us zippo almost $J^*$: if it is a threshold that cannot last surpassed, nosotros tin never notice $J$ below that level. Trying to agree this concept into a NAIRU-like definition is difficult. It is equivalent to maxim that accounting identities e'er hold, as well as if they practise not, it would last inflationary. In whatever event, the theoretical lower bound for revenue enhancement rates would presumably interact amongst this barrier to create a theoretical minimum value for $J$; however, if nosotros are inwards steady state, nosotros silent volition get got toll stability.

Remark. Neil Wilson made a comment amongst observations almost the non-uniformity of workers. This is a existent basis issue, However, non-uniformity implies that wage incomes are non all equal, as well as it would last possible to to get got the same average wage charge per unit of measurement corresponding to dissimilar average income revenue enhancement levels. That violates the assumptions almost the revenue enhancement rate. However, if nosotros tin approximate the to a greater extent than complex model inwards steady soil amongst a model that meets these assumptions, the conjecture is that the termination would silent apply.

(c) Brian Romanchuk 2017