This blogpost is a decidedly simple model answering something that has been bothering me for a while: if you think that some subset of investors have a sluggish recognition of sectoral growth trends, are there additional excess returns you can profit off of? My model here says yes. This is relevant as there are many who are more plugged in to AI engineering, semiconductor manufacturing, electrical & earths refining infrastructure, and other such trends than the average person; those who have seen the light about the coming fundamental transformation to the human experience.

Some people like my friend Connor are trying to insure themselves through the market against the oncoming AI revolution, whilst others like @bubbleboi are trying to profit through investing early into the space through public markets. Vulgar financial theory says this is unlikely to work as markets are efficient, but if there are stubbornly sluggish savers (inattentive investors) then this might be possible in the sector. I personally think this is possible as the technology is cognitively challenging to understand, evolves very fast with unclear growth trends, and because retail & discretionary investment plays such a big role whilst most Americans have not even heard of Anthropic. Disclaimer: this is not investment advice, and my apologies if there are other, similar pieces of work out there that I did not cite.

I am working on some longer and more detailed models for blogposts at the moment, but I keep having a problem in that I keep making those models overly detailed and turning them into paper drafts. Still, I hope to have a post out next Monday modelling my reasoning about why the market response to Trump’s attack on Federal Reserve independence was so small.

Subscribe now

Model

I am keeping this model super simplistic for maximum clarity and because I do not think you need a complicated model to illustrate this point.

We assume CARA utility & Gaussian payoff shocks because such collapses market demand to one line: $q\propto\left(\mathbb E_t[v_{t+1}]-p_t\right)$ scaled by a constant risk term, so market clearing yields a closed-form price as a weighted average of beliefs. The mean-zero shock is just normalization: any common non-zero mean can be absorbed into the conditional-mean state $m_t=\mathbb E_t[v_{t+1}]$, so it shifts levels but cancels out of the predictable component, whilst types disagreeing on the mean just adds another wedge of drift. Dropping normality (or allowing richer tails) mainly makes demand depend on higher moments; it changes premia levels and state-dependence, but not the mechanism: as long as some investors update slowly, a belief wedge $g_t=m_t-\hat m_t$ persists, prices reflect a weighted average, and returns drift until the wedge decays.

Primitives

One risky sector asset is in unit net supply. Its payoff next period is

The “news” state follows the below law of motion:

There are two investor types: attentive investors $A$ (mass $\lambda\in(0,1)$) and inattentive investors $I$ (mass $1-\lambda$). Attentive investors observe $m_t$ instantaneously, whilst inattentive investors only update sluggishly, as follows

Preferences & demands

Each type chooses holdings $q_t^i$ via one-period mean–variance optimization with risk aversion $\gamma>0$:

Market clearing is

Beliefs about the conditional mean are

Equilibrium price & predictability

Placing the demands & beliefs inside the market clearing equation gives the equilibrium price:

Define the belief gap (the wedge created by slow recognition of signals) as

Then the expected excess payoff under the true state is

Share

What This Shows

First, we have predictable drift. When news jumps up, $m_t$ rises immediately but $\hat m_t$ follows with delay, so $g_t>0$ and the expected excess payoff says expected returns are high until the inattentive catch up. Second, we have persistence. After a one-time jump in $m_t$ (with $m_{t+s}=m_t$ for $s\ge1$), the gap decays geometrically: $g_{t+s}=(1-\kappa)^s g_t$. Slow updating ($\kappa$ small) creates slow mean reversion in expected returns.

Some basic comparative statics:

• Smaller $\kappa$ (more friction) implies a larger and longer drift.

• Smaller $\lambda$ (fewer attentive dollars) implies a larger drift amplitude.

• Larger $\gamma\sigma^2$ shifts the level of premia, but the predictable component is still $(1-\lambda)g_t$.

Most importantly, it shows there can be an increasing wedge, not a decreasing one. An endogenized model would likely have this simply be an inaction region with a bang-bang solution such that this doesn’t last forever, but that suggests a leap-up; otherwise, it might be better to get in now than before from a mean-variance standpoint due to the wedge.

Anyways, I hope you enjoyed this short post. Stay tuned for more!

Leave a comment

However, given the observed characteristics of productivity shocks in the economy (namely their autoregressive nature), I want to clarify that the common criticism of RBC models as “attributing recessions to the economy forgetting how to make cars in response to an oil shock” to be an uncharitable strawmanning of the models. Thus, I will be steelmanning this mechanism after explaining the core functionings of a base model as well as an extension I created to show how easy of a basis for DSGE it is (and thus why it is popular).

This post will not be a full explainer of RBC models which some of you have been begging for, but I will promise one soon enough. In the meantime, read McCandless’ very accessible The ABCs of RBCs.

This is going to be a shorter post because I am reaching an incredibly busy period in my life and I do not have the temporal resources to post as frequently as I would like for the time being (I guess this is what you get when you star blogging just as PhD application season rolls around). That being said, I am working on a couple of long-form responses to the current discourse on mathematics in economics, heterodox economics, economic assumptions, rational expectations and preferences, complexity economics, and related topics to my first semi-viral piece on the subject, the first of which should be dropping later this week. I find that I can offer the most efficient combination of expertise and valence on that set of subjects.

Share

The Simplest RBC

This model is taken from the canonical King, Plosser, and Rebelo (1988).1 I present it in discrete time, much to my chagrin, for simplicities sake.

We start with the log-linearized system for real variables, which is quite simple to derive with Cobb-Douglas production and constant relative risk aversion (CRRA) utility. This version has both labour and capital, and contains an intensive margin for labour, so a labour-leisure choice.

Notice here the auto-regressive shock of order 1, also known as an AR(1) shock.

We can re-write this into a series of state equations for the components of real output driven by (7):

which allows us to solve for a mapping to real output:

This allows us to derive the classical reduced-form law for output of

with the final term being a completely transitory shock.

We have

with the first coefficient giving the output share driven by capital persistence and the second being the output share given by technology propagation.

We see clearly that technology is the shock, capital carries its persistence, and therefore therefore output behaves like an AR(2) in practice, since both capital and technology evolve with inertia. This technology, capital, and output persistence roughly matches the data a large amount of the time, especially in the time period of the creation of the model. This is the core mechanism of the model; all else in DSGE has some ancestry in this. RBC output follows a second-order autoregressive process driven by technology, where capital accumulation provides the endogenous propagation, and any other model with capital and productivity shocks will just essentially follow this by necessity (if it is brought to reduced form via the same set of representative agent and rational-expectations assumptions).

We can observe my main point here with a little interpretation: if technology is productive efficiency, such as human capital, knowledge, organization, and supply chains, then this propagation mechanism makes a lot of sense. There is no “the economy forgets how to make cars” in this model, just persistence of the inefficiencies. When there is an oil shock, there are impacted supply chains, fired chemical engineers, and depreciated industrial facilities. This is what is meant by productivity: not blueprints for goods, but the real knowledge and technical capacity to make them, which is often fragile.

This then motivates studying supply chains, capital investment, networks, and forgetting by-not-doing, among other such “real” factors of the economy. As much as I focus on the nominal side of things, it is important to remember that ultimately, real phenomena occur, even though generally supply shocks are not seen as major shocks in the data. It is then understanding why this model fails that is valuable, and that requires models of both nominal and real frictions (which often the same, such as coordination and contracting failures).

Leave a comment

A Beckerian Twist

I want to show how easy this model is to modify now by including so-called Beckerian home production, or Beckerian commodities. The origins of this incredible modelling tool are from Becker (1965). One could imagine that households may produce substitutes at home, or bundle leisure time with purchased consumption goods to produce final utility commodities.

Writing out the equations of state:

We then log-linearize:

The Beckerian term c_t^* captures the idea that market goods (c_t) and home goods (h_t) are imperfect substitutes in generating utility. Agents allocate time between market labor (n_t^M) and home labor (n_t^H) optimally. Shocks to A_t^H shift the opportunity cost of home time, generating realistic labor supply comovements with market output. If η goes to infinity, c_t^* goes to the linear aggregate ω c_t + (1-ω) h_t, the case of perfect substitutes. If η goes to 1, home and market production become complements, producing more propagation and smoother consumption.

We can then write the state dynamics:

Finally, we obtain a reduced-form law for real output:

In the Beckerian RBC model, output evolves not only through capital accumulation and productivity, but also through shifts in how households allocate their time between market and home production. A rise in market productivity raises wages, drawing time out of home activities and into market work, which boosts measured output but may temporarily reduce home-produced goods. As this reallocation adjusts gradually, output inherits persistence from both capital and time-use dynamics. The result is a richer, more realistic propagation mechanism—business cycles reflect not just technology shocks, but the slow, interdependent adjustment of both physical capital and household time across sectors.

Adding Beckerian home production to an RBC model fundamentally broadens the scope of what “economic activity” means within the business cycle. In the standard RBC framework, households supply labor to firms, earn wages, and use their income to purchase market goods, deriving utility from those goods while disliking work. Time spent outside the market is simply “leisure.” In Becker’s formulation, however, households use both goods and time to produce commodities—home-produced services such as meals, childcare, or maintenance—that enter directly into utility. This makes time an input in both market and non-market production, creating a genuine allocation problem of time rather than a binary work-versus-leisure choice.

The result is that productivity or wage shocks now affect not only how much people work, but also how they reallocate time between market labor, home production, and leisure. When market productivity rises, households may shift time toward market work and away from home production, reducing the output of home commodities and changing effective consumption. This adjustment margin introduces new propagation mechanisms: home production smooths consumption, adds persistence to output, and makes labor supply more elastic through substitution across time uses rather than across pure labor hours.

Consequently, the Beckerian extension enriches the interpretation of business cycles—it suggests that cyclical fluctuations reflect not only changes in capital and technology, but also the continual rebalancing of how households deploy their finite time between market and home activities.

There are a few key uses of this extension. The most key, to me, is that it explains the labour and consumption choices of the old and the unskilled: they have relatively less marginal productivity in work than in the home, so in response to shocks, they will often choose to quit or retire, raising reservation wages and thus adding wage pressure (as in one of my favourite recent papers) or creating consumption effects that do not show up in the data (because we do not spy on your grandma baking pies).

I hope you enjoyed this edition of Modelling Mondays! I try and write something like this every week. I am also going to be issuing the aforementioned post on the “methodology of economics” discourse later this week, as well as a newsletter summarizing some goings-on in economics and beyond that I find interesting. If you enjoyed this post, please consider a free or paid subscription and please share it wherever you can.

If I omit any from this list, it is only because there is such a glut of worthy winners in economics that I cannot possibly write about all of them.

Likely Winners This Year

The first tranche of likely winners is from the world of Industrial Organization. Although Timothy Bresnahan is a worthy winner in this category, one trio of model-makers already features a ‘B’ name and is much more likely to win.

Whilst Paul Krugman formed the first foray into spatial economics (after Robert Lucas, of course), his prior victory makes it unlikely to me that Masahita Fujita and Anthony Venables win the prize for their shared work. Treb Allen, Costas Arkolakis, David Atkin, and Esteban Rossi-Hansberg, alongside Dave Donaldson and Gilles Duranton, all have a great chance of winning together in some combination, they created the core of modern empirical spatial economics and much of the advanced theory underlying it all, but they are all too early in their careers to win over other potential winners, and I cannot really judge who amongst them would win, and in what combinations. My pick goes to three members of the spatial field who are the most senior and have the most influence.

International (macro)economics is of incredible relevance due to recent events. There are many directions the Prize Committee could take to reward members of the field that discusses trade, immigration, currency, and financial flows, but I have identified two likely groups.

My last pick is for the three architects of the New Keynesian model and much of modern monetary economics. I say, despite the model’s occasional failings, that they are more likely than the ‘Fiscalists’ of John Cochrane and Eric Leeper, especially as Christopher Sims already has a prize.

Although Heterogenous-Agent New Keynesian models are wholly better and unify both perspectives, they are either too confusing or too new to win yet. Truman Bewley and Mark Huggett will likely never get recognition for developing incomplete markets models (and of course S. Rao Aiyagari will sadly never even have the potential, as he was taken from us well before his time), whilst Guido Lorenzoni, Veronica Guerrieri, Emi Nakamura, Jon Steinsson, Ricardo Reis, Gregory Kaplan, Ben Moll, and Gianluca Violante all have many more years of contributions ahead of them and all are sure to win Nobel Prizes for their countless current contributions to macroeconomics. Further, Ricardo Cabellero and Fernando Alvarez cannot win for their work on menu costs before a certain fantasy woodland creature is recognized first.

No, we first must have a prize for RANK before we can have one for HANK and a prize for exogenous frictions before we have one for endogenous ones, and there are three elder economists who all deserve one for that model above all else.

Steven Berry, James Levinsohn and Ariel Pakes

This team developed the Berry-Levinsohn-Pakes model of demand that forms the basis of much of modern structural, empirical industrial organization economics. That subfield has exploded in popularity recently, especially as cracks in the Credibility Revolution have formed and people begin to see that cookbooked reduced form models are not the right way to do economics. As the job market for economists worsens, expect to see a lot more BLP-based job market papers…

Samuel Kortum, Edward Glaeser, and Stephen Redding

If Jonathan Eaton were alive, one would surely predict that he and Samuel Kortum would win the price together. Their Eaton-Kortum model of trade is the ancestor of all modern spatial models, even if the lineage seems tenuous sometimes. His untimely death might be best symbolized by a half-share for Kortum, as was done with David Card because his coauthor Alan Kreuger had passed on.

Edward Glaeser is probably the most famous urban economist currently alive today, and although he is more junior, his influence makes him a likely choice for the committee if they want a spatial-themed prize.

I chose Redding as the representative of the modern approach to spatial economics most likely to win because he is the most senior of that group, and thus has less time to win it in another combination.

Gene Grossman and Elhanan Helpman

This group is probably the most timely, if the Nobel in Economics is to mimic the Nobel in Physics of last year. Grossman and Helpman’s model of the political economy of trade is incredibly relevant to the world of Trump’s tariffs, so I am making sure to put this team down as an option. I don’t know much of their other work, however, so I cannot speak to how likely this option is without the relevance factor.

Perhaps Philippe Aghion will receive the prize as well this year, as he is the lead economist in Europe arguing for actual solutions to the looming pension crisis that will fully destroy the peninsula unless drastic and immediate action is taken.

Maurice Obstfeld and Kenneth Rogoff

Obstfeld and Rogoff are the intellectual architects of modern open-economy macroeconomics. Their synthesis of monetary macroeconomics and international financial economics created a unified theory that has largely displaced the Mundell-Fleming framework. The Obstfeld-Rogoff model provides the microfounded backbone for nearly all current analyses of exchange rates, global imbalances, and monetary policy spillovers, which are all incredibly timely in the Big 2025.

A prize for them would ironically also be a second prize of sorts for Ben Bernanke, another pioneer of the analysis of global trade flows as a core paradigm for modern international economics.

Guillermo Calvo, Michael Woodford, and John Taylor

Calvo is arguably the architect of the modern approach to introducing nominal frictions and monetary non-neutrality to economics models. His Calvo Fairy has managed to stick around for over forty years as a core element of modelling in macroeconomics, and challenges to it in the form of menu costs. His reconciliation of nominal frictions with the Lucas critique and rational expectations was a key step towards the New Neoclassical Synthesis. He worked in international as well as monetary settings, so he might fit in the previous prize, too.

Michael Woodford is arguably the second-most influential living macroeconomist. He is the ultimate theoretical canonizer of the New Keynesian model which central bank policy dynamic stochastic general equilibrium models evolved out of, as well as one the leading lights for the reform of the model and new monetary policies in the wake of the Great Recession. He has written papers in the international context, just like Calvo, but most of his work concerns his own idiosyncratic-and-yet-rich approach to the New Keynesian model. His approach to ‘natural’ variables, expectations, and optimal policy underlies how nearly all applied macroeconomists have thought over the last twenty-five years.

The only man more influential is probably my third pick, John Taylor. Taylor is the father of inflation targeting and arguably also the grandfather of modern nominal frictions (if Calvo is the father). The Taylor rule still dominates the way economists think about how a central bank acts, and for years dominated how central banks thought of themselves! Taylor’s staggered contracts, although intractable, form the spiritual microfoundation of modern nominal frictions. His work on policy interdependence is, like the other two winners, also relevant to the international stage as well. A great review of the case for Taylor can be found here.

I will also comment that Julio Rotemberg would have been a great third candidate for a prize for the microfoundations of nominal frictions, especially as such is relevant for the modern day continuous-time approach, but sadly he has passed on before his time, and cannot win alongside Calvo like is just.

No Modelling Mondays post is planned for this week, but I plan to check in on everyone after the announcement of the winner. My schedule has gotten very hectic, but I hope to get back on track with more frequent posts soon. If you want to receive some follow-up posts on future winners or economists I think deserve to win, please become a paid subscriber or upgrade your current subscription to a paid one.

Subscribe now

I discovered in the process of writing this post,1 however, that I was really building a hybrid model of Old and New Keynesian viewpoints. I therefore will use this model not only to illustrate why we use nominal rigidities in modern macroeconomics but also why we care so little about what Keynes directly had to say about the functioning of a macroeconomy and the origins and persistence of business cycles. I will demonstrate to you with this simple model that the sort of rigidities he envisioned, and that modern Vulgar- or Post-Keynesians admonish the New Keynesians for abandoning, reflect reality far less than the precious nominal rigidities of the New Keynesians.

I will avoid discussions of sales vs. production in this post other than to say that the intermediation of the two, inventory, also reflects my point here in the data and does not change my point much. I will be demonstrating this in a later post, however.

I will be working here in continuous time, but with Calvo adjustments instead of Rotemberg, as the latter makes little sense for real goods and, furthermore, as I discovered when writing this post this makes much more sense for the sort of disequilibrium system I will end up getting. I’m not a fan of log-linearization as it is capable of obscuring important factors around things like divine coincidence but here it is fine because I just need to show off the key laws of motion.

I take important elements of my model from a diverse array of sources. My continuous-time New Keynesian basics come from Woodford (1996, 2003) and Gali (2005), although the latter is discrete time, it was the textbook for the quarter of my first-year (of a doctoral program) macroeconomics course that focused on New Keynesian models here at the University of Chicago.2 My language disequilibrium and “notional vs. effective [realized]” demands comes from Clower (1965) and Barro & Grossman (1971). Christiano, Eichenbaum & Evans (2005) have a much more complicated and rigorous output-adjustment mechanism than mine, but they are worth mentioning. Blinder & Maccini (1991) and Khan & Thomas (2007) probably get the closest to my approach.

There is a much more beautiful and rich menu-cost literature for menu cost adjustments of quantity and price; the former can be found in Caplin and Leahy (1997) and Kiyotaki & Moore (1997). The price literature is too extensive to list here; key contributors include Alvarez,3 Caballero, and Lippi. Golosov & Lucas (2007) and Alvarez & Lippi (2014, 2021) form the basis of the modern approach to the subject which I focus on in my academic work, whilst Adao, Nakamura & Steinsson (2020) form another important contribution to the understanding of Calvo models with microfoundations. I will stop this discussion here before it becomes seventeen paragraphs long.

However, as usual I worked from first principles on my microfoundations and especially my couching of this model in continuous time with infinitesimal complimentarity conditions; I am sure others have tried similar in discrete time but I could not find anything in my Chicago setting.

I took all of your feedback to heart and I will do my best here to name all the variables in the text and provide more explanations, which I have avoided doing so far because I am autistic about them not rendering because shamefully on their part there is no in-line LaTeX in Substack, so they will not match. Just be aware that when I use the $$ delimiters, inside is the exact LaTeX that I would otherwise write.

Subscribe now

Model

The entirety of the household side that pins down the Investment/Savings-Loanable Funds Market (I-S/L-M) curve, although it is technically just the I-S equation, is identical in both models, so I will only have to repeat it once. The firms and technology are also identical aside from price setting. I am also going to be very curt with my derivations because they really are something you only need to see once in these kinds of models and I already have; once you know how you get there you it is best to just focus on the state equations. However, what is complicated in this model is that individual firm supply will be differentiated in the sticky-output case, so to make matters clearer, I will write out how we aggregate firm marginal cost to demonstrate the heterogeneity is not from that factor.

We also choose to abstract from investment and government as these will complicate the derivations whilst not impacting the model (as they just modify the relation of consumption to demand).

We also must now make certain objects clear. We demarcate notional demand, realized demand, notional output, and realized output as

respectively. The first quantity expresses the desired consumption level of the representative agent and is used for intertemporal optimization, the second describes the realized consumption of the agent (as mentioned above), the third being the aggregated desired output of the firms, and the fourth is the realized output of the economy. This distinction will be of utter importance later, but for I will simply explain that the second and fourth quantities must always equilibriate, but the first and third need not, instead there being infinitesimal complementarity systems that define flow constraints that ensure the notional quantities attract back towards the manifold of equal realized quantities, in differential equation terms. I will note whenever I use a specific quantity and why, as well as one of the aforementioned structures, and under what systems notional and actual quantities may never deviate.

This structure is already inherent to the New Keynesian model, but you might not know it yet. I will demonstrate to you that this is the case when I show that model.

Note also that we aggregate output, consumption, and prices with the relevant Dixit-Stiglitz constant-elasticity-of-substitution (CES) aggregators with elasticity ε > 0.

Households

Write the representative household with lifetime utility

with discount rate ρ > 0, risk aversion σ > 0, and labor elasticity 1/φ. We abstract from the budget constraint here to avoid commenting on the notional versus realized distinction too soon. The budget constraint and the no-Ponzi-game, no-bubbles condition will grant the consumption Euler equation (or Ramsey rule or Keynes-Ramsey rule or whatever you want to call it), for notional demand,

where r_t is the real interest rate. We use notional demand here because households plan given their first-order condition before the learn whether or not the system has equilibriated. Let π_t be inflation in the sense that it is the time-derivative of log-linearized P_t and i_t​ the nominal rate, so we have

Let us define the demand gap between the natural logarithms4 of notional demand and “natural” flexible-price, efficient steady-state demand (which equal the output of that system always), respectively, as

The household side then yields the I-S curve

where we include a demand-pull shock for later convenience.

We also derive the wage level from labour market clearing as follows. Let us write

the labour-market clearing condition. All firms hire from the same labour market so wage is common to all firms and determined by the aggregated labour demand and the realized household first-order conditions such that

as a household’s disutility of labour depends on their actual consumption flow. Then we write the log-linearized wage as

as we wrote out all other factors of demand.

Firms and Technology

We have to be more careful in this section because whilst normally we can just view all firms as one on the production side, in the sticky-output model we cannot do that. We work with realized output section since we are discussing the technology and costs, and not the output plan. We endow all firms with identical production technology, a Cobb-Douglas production function in capital and labour, and with common capital stock such that firm output is given by

such that holding capital fixed in the short run at k_t = k we have that real marginal cost is proportional

Then we use log-linearization, aggregation, and market clearing to attain

We then set

such that, collecting all cost-push shocks into u_t,5 we have the real marginal cost gap

via subtracting the natural steady state value of the real marginal cost.

Flexible output, sticky price

This is the classical New-Keynesian case, which normally I would just write out for you out of convenience, but I want to show you the dynamical constraint structure which makes the sticky-output equilibrium possible in a more familiar setting first. First, realize here that notional quantities are always equal to realized quantities since output may always instantaneously adjust to meet demand and thus maximize profits given prices. We have the re-pricing problem for a firm at time t who is choosing a price to maximize the present value of profits until the next reset opportunity, which arrive at rate λ:

Let

be the optimal reset price gap. Up to the first order, this should be equal to the present value of expected real marginal-cost gaps. Now, consider defining notional inflation, or the inflation rate if all firms could increase their price flexibly, aggregated as standard. This rate should be equal exactly to the optimal price reset gap, as firms wish to counter their real marginal cost increases one-to-one.

Now, just to rush through some well-known derivations, we have

the effective discount rate of profits around the flexible-price optimum. This allows us to write the familiar New Keynesian Phillips Curve via the standard aggregation of the price index under Poisson resets

Now, whilst the above can be derived as usual, let us discuss the aforementioned differential Karush-Kuhn-Tucker complementarity system to illustrate how that concept works. Since we need the realized quantities to clear instantaneously, as firms meet demand at posted prices, we obtain a differential equality for all aggregated goods quantities; this gives a degenerated differential complementarity problem using two non-negative variation processes

whilst relying on notional output always matching realized output due to flexibility. The only admissible solution to this system with finite quantities is equality for all time.

Subscribe now

Sticky output, flexible price

Notional quantities are not always equal to realized quantities in this case, but inflation will always adjust to equalize the value in each market. The CES aggregator grants demand such that for any firm, we derive due to flexible prices that each instant we have

such that via aggregating we obtain

We have the re-setting problem for a firm at time t who is choosing a quantity of output to maximize the present value of profits until the next reset opportunity, which arrive at rate λ:

Write the Calvo aggregation law

and linearize the reset problem to obtain a formula for the aggregated optimal reset gap from natural output:

Together these yield the aggregate supply (AS) curve

with

Another way to think of the flow constraint for inflation is the differential complementarity problem

which generates our flow identity for inflation. I write it this way to emphasize something key: under the New Keynesian model, supply and demand always instantly equilibriate, with prices slow to adjust, whilst here, prices jump to equilibriate supply and demand, the former of which is slow to adjust.

We can combine out inflation flow with the prior I-S equation to write

This directly shows how demand can be a stable system in this model.

The New Keynesian aspects of these models may be familiar to all; if not, there are far better resources than I on their functioning and interpretation. Our I-S curve gives us the response of household (notional) demand to higher interest rates and lower inflation; in the New Keynesian model, higher marginal costs push through this to lower demand via the price channels using the New Keynesian Phillips curve, whilst in this stick-output model the story is more complicated due to the nature of the “disequilibrium”. First, higher demand is (may be) attracted back down towards the realized output via the inflation channel and the (although attenuated) aggregate supply channel. Second, higher marginal costs lower demand through the inflation channel, as in the New Keynesian model, although the relative size of the effects between the two models is a function of the relative sizes of the elasticity of substitution in aggregation and production. However, in both cases, notional demand does indeed have a tendency (or should have a tendency given the Taylor rule) stabilize unto itself.

Second, in the New Keynesian Phillips curve, higher marginal costs raise inflation which does (may, given policy, see below, but generally in the cases policy is as theory requires) attract back to itself as it rises, whilst inflation in this new model may jump as necessary. In this model, supply pulls itself back down quite strongly, with amplification to the order of greater than one times the Poisson rate of Calvo Fairy arrivals; this causes an extremely sluggish output response to shocks. In the original model, supply co-moves (adjusts) with demand.

Analyzing this model and contrasting it further its father, whether it be through a solution or through stability analysis would be interesting, but I will leave that to another post based on the response to this post as it can get complicated fast; please let me know if such compels you!

Central Bank Policy

We may write some form of Taylor rule to close both systems (it forms the L-M side of traditional Keynesian theory). The specifics of such a rule are not important here, as we are neither discussing stability nor optimal monetary policy.

What we see here as a particularly striking difference is the way such a rule is incorporated. In the traditional model, it stabilizes not only the aggregate demand curve but also the Phillips curve because it becomes the basis of a stable dynamical system via the size of its attenuation of current inflation on its time-derivative, but in this model, it stabilizes just notional demand; its effects on realized supply are second-order (through the I-S curve). This says that monetary policy plays less of an important role in demand stabilization than in New Keynesian theory (and the real world). Instead, real and rigid factors of adjustment (as well as moving to meet demand) mostly govern that half of the market process.

Solutions

I solved the models in my head earlier today, but I feel like I should code up graphical displays to go alongside the solutions; if you wish to see these in a later post, please let me know down in the comments section below!

Leave a comment

Final Adjustments

Now that we have outlined our two models, I would like to christen the second one the “Middle Keynesian” model, as it captures essential facets of both Old and New Keynesian thought.

First, it restores the disequilibrium logic of Old Keynesian economics: real rigidities prevent the instantaneous clearing of goods markets, so money—not quantities—equilibrates value. The nominal side clears, but real production lags behind demand, producing a world of excess demand and constrained output.

Second, supply adjusts sluggishly because production itself is frictional to adjust. Firms cannot costlessly or continuously alter real output, so prices step in to balance aggregate demand with available goods. This interplay between slow physical adjustment and fast nominal adjustment generates the model’s dynamic aggregate supply curve.

Third, the framework is tempered by New Keynesian sensibilities: agents optimize intertemporally, frictions are explicitly modeled rather than assumed behavioral or irrational, and monetary policy operates through a Taylor rule rather than through the ad hoc loan or money-market mechanisms of the IS–LM era.

In this way, the Middle Keynesian model embeds the Old Keynesian disequilibrium within a continuous-time New Keynesian architecture, preserving the microfoundations and dynamic consistency of modern macroeconomics while recovering the classical Keynesian intuition that demand governs output whenever production, not prices, is the inflexible margin.

Some choice Keynes quotes to support all this will be necessary as he as treated almost as religiously as Marx by some economists (and many non-economists):

“Now if for a given value of N the expected proceeds are greater than the aggregate supply price, i.e. if D is greater than Z, there will be an incentive to entrepreneurs to increase employment beyond N and, if necessary, to raise costs by competing with one another for the factors of production, up to the value of N for which Z has become equal to D. Thus the volume of employment is given by the point of intersection between the aggregate demand function and the aggregate supply function; for it is at this point that the entrepreneurs’ expectation of profits will be maximised. The value of D at the point of the aggregate demand function, where it is intersected by the aggregate supply function, will be called the effective demand.” — Keynes, The General Theory of Employment, Interest and Money (1936, Ch. 3).

“Our independent variables are, in the first instance, the propensity to consume, the schedule of the marginal efficiency of capital and the rate of interest, though, as we have already seen, these are capable of further analysis.”
— GT, Ch. 18.

“Moreover, the contention that the unemployment which characterises a depression is due to a refusal by labour to accept a reduction of money-wages is not clearly supported by the facts.”
— GT, Ch. 2.

“If, on the contrary, money-wages were to fall without limit whenever there was a tendency for less than full employment, the asymmetry would, indeed, disappear. But in that case there would be no resting-place below full employment until either the rate of interest was incapable of falling further or wages were zero. In fact we must have some factor, the value of which in terms of money is, if not fixed, at least sticky, to give us any stability of values in a monetary system.”
— GT, Ch. 21.

“A reduction of money-wages will somewhat reduce prices. It will, therefore, involve some redistribution of real income (a) from wage-earners to other factors entering into marginal prime cost whose remuneration has not been reduced, and (b) from entrepreneurs to rentiers to whom a certain income fixed in terms of money has been guaranteed… The transfer from wage-earners to other factors is likely to diminish the propensity to consume… if rentiers represent on the whole the richer section of the community and those whose standard of life is least flexible, then the effect of this also will be unfavourable… Probably it is more likely to be adverse than favourable.” — GT, Ch. 19.

All this is to say that although I am not a Keynes scholar I think I have captured many important elements reasonably well in my own language. I will make a post later about how I think economists have whitewashed Keynes into sticky-wage monetarist instead of an aggregate-demand-obsessed behaviouralist. You can find the best evidence for these views in chapters 19-21 of The General Theory of Employment, Interest and Money (1936).

Canning Keynes

This model might seem more natural, whether you be Keynesian or just a businessman. After all, prices are easy to set, but production is hard to adjust. Yet there are strong reasons to reject this sort of rigidity and this model, even in the face of the oft-loathed New Keynesian model. I will be a little loose with my citations here, as I am tired and do not remember everything about the details of these papers perfectly.

The truth is simple: reality chose sticky prices. The New Keynesian view emerged precisely because prices adjust infrequently, not quantities. We know from the micro price-setting data literature that median price durations last 4-10 months, whilst roughly 70% of output adjustment in the short run occurs via price rigidity, not production inertia. Further, prices change less often than output or employment, even in volatile sectors. [Bils & Klenow (2004); Nakamura & Steinsson (2008); Klenow & Kryvtsov (2008); Klenow & Malin (2010); Midrigan (2011); Eichenbaum, Jaimovich & Rebelo (2011); Cavallo (2018)]

Production and employment data tell us output and labour respond rapidly to shocks; whilst production is flexible at high frequency. Inventory and utilization adjustment occur within weeks and firm-level output elasticities are near one in the short run. Additionally, employment flows are enormous even in recessions: around 40% of workers separate or join each quarter. All this is to say that real adjustment is fast, not sticky. [Blinder & Maccini (1991); Davis & Haltiwanger (1999); Ramey & West (1999); Foster, Haltiwanger, and Syverson (2008)]

The literature on inflation-output dynamics provides further evidence: empirical Phillips curves show inertia in inflation, not output gaps. Inflation responds sluggishly to activity, consistent with sticky prices, not sluggish quantities. The data simply reject the Middle Keynesian timing of adjustment: price level jumps are rare; output jumps are common. [Christiano–Eichenbaum–Evans (2005); Gali (2015)]

There are plenty of other theoretical reasons (rejection of central bank importance, disequilibrium, sluggishness of productive response to shortfalls or negative productivity, etcetera) that come from the model that I have declined to elaborate upon further here; see if you can elucidate upon them yourself in the comments!

This economy is dual to the New Keynesian economy and gives a clear demonstration of why we rejected rigidities in output adjustments (at least of this sort) and why we rightfully reject Keynes’ way of thinking today. In the Middle Keynesian model, nominal markets clear instantly and real markets slowly. Yet, this implies arbitrage across goods in nominal value, which would require instant information and perfect credit liquidity—precisely what breaks down in recessions. In reality, credit and liquidity constraints make nominal adjustment slow, not instantaneous.

Firms choose quantities continuously, even under adjustment costs, but face contractual and informational lags when changing prices. Output rigidities are typically endogenous to nominal contracts or costs, not primitive frictions—thus better modeled via sticky prices with intermediate real rigidities.

Keynes thought that production itself would lag because of inefficiencies or behavioural responses to changing demand, but in reality contracting frictions explain output responses better, especially in microfoundations. Those rigidities are better when modelled on price, which helps us get the high persistence and low pass-through of inflation found in the data.

In the Middle Keynesian world, prices clear value markets instantly, so monetary policy has little power over nominal aggregates beyond the instant. Output responds mechanically to demand shocks only until supply catches up—then prices jump, inflation overshoots, and stabilization policy becomes hyper-reactive. The model predicts volatile inflation, short-lived output gaps, and instant nominal adjustment—none of which we observe. On the other hand, in the New Keynesian world, prices move slowly, so nominal demand shocks propagate through real channels for multiple quarters. Inflation is smooth, expectations matter, and output gaps close gradually, which matches VAR and micro evidence. Monetary policy can manage the cycle precisely because nominal rigidities persist, whithering the fiscal policy of the vulgar Keynesians.

I also hope this reassures those that are obsessed with sticky output as (bus)laymen that economists aren’t missing something. Rational inattention is a better explanation for your views than asymmetric information over us about the pattern of observable macroeconomic adjustments.

Not New Keynesianism Neither

I will be generous here; a lack of sticky wages means we are missing a key part of (some interpretations of) the vision of Keynes. It may be that a wage-Phillips curve helps to speed output adjustment and temper price adjustments smoothly. I am already a supporter of sticky wages as a microfoundation; they form a core basis for most of my current research6 and are part of the reason I do not consider myself a New Keynesian. We can see that Calvo frictions are not really satisfactory to explain what sort of rigidities there really are and how firms respond to them; perhaps an impulse-control model of fixed-cost adjustment of prices, wages, and output would best explain the macrodata and also the origins of observed frictions in the microdata—I certainly embrace this view.

However, it remains to be said that the disequilibrium and “excess demand-rigid supply” side of Old Keynesianism cannot be rescued by even a modern neoclassical dynamic-microfounded context; it is simply not a correct element of a good model of the world.

I expect many of my Post-Keynesian friends will tell me I have an incorrect interpretation of Keynes; in that case, I ask them to draw up a better model in the modern continuous-time macroeconomic context that is able to recreate their vision of Keynes and the data more accurately.

There is much wrong with New Keynesian models. That fact does not make them any less of a massive improvement over Old Keynesian paradigms.

I hope you enjoyed this Modelling Monday! This post was more intensive and of greater relevance than I expected when I first set out to write it, but I suppose I should expect such from the kind of posts I have been writing as of late. After all, real, hard-nosed macroeconomic modelling is indeed the point of this blog.

If you are new here, you might want to check out the first post in this series; my aim is put out one of these every Monday if I am able. I also aim for a bi-weekly newsletter on academic macroeconomics and some form of discourse, model, or paper explainer, as well as topical posts on current economics debates, economic (or socio- or political-economic) news, “pop”-culture tie-ins, and my thoughts on related subjects such as mathematics and history of economics. I also aim to do paid posts with hotter takes and even deeper and more useful explanations and analysis, as well as other things I would not say without a paywall. My next post is likely to be on the Lucas Critique, Mean Field Games, or the Kyle model of insider trading. If you are interested in any of this content, please consider a paid or free subscription; paid subscriptions are what motivates me to spend even more time working on this project.

If you enjoyed this post, please share it with your friends and on social media; referrals earn rewards around here.

I hope this post will inspire much discourse; if you have comments, questions, or suggestions, please leave them in the comments below, on Notes, or on the platform formerly known as Twitter.

Leave a comment

If you enjoy Uncommon Noise, it would benefit me greatly if you invited friends to subscribe and read with us. If you refer friends, you will receive benefits that give you special access to Uncommon Noise.

How to participate:

1. Share Uncommon Noise. When you use the referral link below, or the “Share” button on any post, you'll get credit for any new subscribers. Simply send the link in a text, email, or share it on social media with friends.

Refer a friend

2. Earn benefits. When more friends use your referral link to subscribe (free or paid), you’ll receive special benefits.

• Get 1 month of complimentary subscriber posts for 5 referrals.

• Get a 3 months of complimentary subscriber posts for 10 referrals.

• Get a 6 months of complimentary subscriber posts for 25 referrals.

Visit the leaderboard

If you want to learn more, check out Substack’s FAQ.

Thank you for helping others learn about Uncommon Noise!

Yet their relevance has almost completely waned at this point in academic economics. The problem is not that demographics or intergenerational transfers are unimportant—if anything, they matter more now than ever—but that overlapping generations models are too blunt an instrument. They impose rigid structures on preferences and lifespans, while ignoring the far richer forms of heterogeneity and uncertainty that modern macroeconomists now routinely model. Where once overlapping generations seemed the only way to “get” heterogeneity, Krusell-Smith- and Bewley-style models now provide more flexible, data-consistent frameworks.

Subscribe now

Read more

I do admit I came up with a bit of a terrible pun first: “Clairo-asean Bargaining”, and that is what ultimately forced me to write this post. I thought back to when I had attended concerts in Forest Hills, including seeing Clairo open for boygenius, and wondered to myself: there are easily around twenty times the amount of attendees at this event compared to impacted local residents. Why can’t they work out a deal to keep the concert open later? It would benefit the good of all, would it not? Surely financially it can be worked out?

The economics framework one is taught to look at this problem in, especially in the grand tradition of Chicago Price Theory, which ostensibly is still my instinct when it comes to microeconomics,2 is Coasian bargaining and his eponymous theorem. Many a story I was told of Coase bringing his famous Theorem to the land of Knight and Viner and Stigler and converting the old guard one-by-one until a new world was born, starting with Friedman Himself. Coase’s Theorem, articulated by Ronald Coase in his seminal 1960 paper The Problem of Social Cost, posits that in the presence of well-defined property rights and negligible transaction costs, parties affected by externalities will negotiate voluntary agreements—known as Coasian bargaining—to achieve an economically efficient allocation of resources, irrespective of the initial assignment of those rights. This theorem challenges traditional Pigouvian approaches to externalities, which advocate for government interventions like taxes or subsidies, by emphasizing that private negotiations can internalize external costs or benefits without state involvement. For instance, if a factory’s pollution harms a neighboring fishery, the theorem predicts that the parties will bargain to a Pareto-optimal outcome: the polluter might compensate the fishery to continue operations at an efficient level, or the fishery could pay the polluter to reduce emissions, depending on entitlement. Coasian bargaining thus relies on rational, self-interested actors engaging in costless exchanges, assuming complete information and no barriers to agreement, thereby highlighting the potential for market-based resolutions to social inefficiencies.

In observed reality, however, Coase’s Theorem serves more as an ideal benchmark than a direct descriptor of empirical outcomes, given pervasive transaction costs, imperfect information, and ambiguities in property rights that often preclude efficient bargaining. High negotiation expenses, such as legal fees, coordination challenges among multiple parties (as in collective action problems), or asymmetric information leading to holdouts and strategic behavior, frequently result in suboptimal equilibria, as evidenced in real-world environmental disputes like acid rain mitigation or fisheries management, where government regulations or tradable permits have proven necessary to approximate efficiency. Nonetheless, the theorem’s insights underpin practical applications, such as emissions trading schemes under the Kyoto Protocol or cap-and-trade systems in the United States, which simulate low-transaction-cost environments by clarifying rights and facilitating exchanges, thereby bridging theoretical elegance with pragmatic policy design to address market failures in a complex, friction-filled world. My personal view has always been that you use it to demonstrate that our world is full of transaction costs, imperfect information, and complementarity problems, and then you address them through a well-informed mechanistic approach.

Studying this particular failure of the theorem therefore had special interest to me, then. Could I find and identify the failure modes of Coasian bargaining in this case, even when transaction costs would be low due to government intervention? Could I find a combination of economizing behaviours, government action, and markets that could resolve this issue so the show truly could go on? I’m an economist, I solve problems with economics, even when they seem trivial—so could I solve this, or at least explain it?

I was talking this over with my roommate (who blogs as well) and he speculated that housing price impacts were part of the story here—people who live in the area worry about the housing price impacts perhaps even more so than the noise. That really got me thinking about the model you will now see.

I set out to build a model capable of showing a “failure” of Coase’s Theorem in a realistic scenario where the failure occurs because the budget of the rights-buyer could not handle a second-order lump transfer to make up for impact on the mark-to-market valuation of their property for just one period, even though such taking place would benefit the wider society, and because they had imperfect mechanisms to rebate heterogenous preferences. I came away, however, with a story that says the main problem in achieving the transfer of rights is still the willingness-to-pay of the current occupants, in fact, the impact to property values was not so huge as to make lump-sum compensation infeasible, and further, a variety of different ways to choose the ‘market’-determined rebate worked to a high degree in actually rewarding those with a high willingness to pay. The story ended up being more interesting statically than dynamically. Embedded in this story as well is rationing, sorting both over residents and over purchasers, city housing market structure, and most importantly, YIMBYism versus NIMBYism.

I also wanted to use this as an opportunity to talk about Eric Adams and the New York City mayoral election, but he dropped out before I could finish this piece. I’ll leave more direct thoughts on that ‘till the end or a follow-up post.

If you’re just here for the pop-culture commentary, searing-hot urbanism takes, or pictures of the oh-so-stunning Claire Cottrill, I highly encourage you try and stick around as I work through the model in the following section. I am a macroeconomist by trade, so whenever I build a microeconomic model like this, it’s from first (Chicago) principles and contains a lot of my own idiosyncrasies. I tend to go through all my decisions and considerations one-by-one, so hey, maybe you will learn a thing or two. The model is exceedingly simple in my view, and you can probably follow along with it.

Modelling Music Festivals

The goal in this model is that we want to relate Coasean bargaining to willingness-to-pay in a dynamic setting such that even if New York City implemented a market3 for noise rights such that transaction costs were zero for the festival organizers offering rebates to locals, there might still be the outcome of noise rights belonging to Forest Hills (Gardens) residents in perpetuity despite the allocation of maximum welfare (under chosen welfare weights) being the noise rights belonging to the festival organizers in perpetuity. We want to illustrate that regardless of the absence of transaction costs, the inability for the festival organizers to perfectly rebate the locals in every period due to heterogenous preferences ensures that the drop in housing prices means the incumbents will demand a one-time transfer to make up for the present value of the drop in the mark-to-market value of their property which may be prohibitively large as to prevent the noise rights from being transferred, even if flow costs were manageable.4 Please note that I inconsistently use noise and lateness throughout this section, as well as rights vs. the activity itself. The former two are equivalent and we assume that all rights transferred are fully exercised.

I am almost tempted to include a more sophisticated housing market with mortgages in an attempt to make this more of a macroeconomics problem, but here we will treat (potential) residents as having the opportunity to sell (or buy) as a Poisson process. There is a fixed mass of neighbourhood residents who each have identical preferences over wealth but have differentiated preferences over living in the neighbourhood of the venue and noise such that they receive different constant flow payoffs from living in the neighbourhood versus outside of the neighbourhood, a disutility to noise based on concert length if they live in the neighbourhood adjacent to the venue, and a linear separation payoff from selling their house. The resident’s problem will thus be an optimal stopping problem. Atypically for me, I will write this problem in discrete time and not continuous time, as music festivals happen but once a year.

Our primitives are two neighbourhoods, one adjacent to the venue and one further away that represents the rest of the city. The adjacent neighbourhood has less housing stock than the rest of the city. We work in discrete time with a homogenous discount factor and assume that only a fraction of households receives the opportunity to move each period, with the shocks being independent and identically-distributed Bernoulli random variables (I study impulse control problems and heterogenous-agent models, don’t yell at me for putting the Calvo Fairy into everything!). There is a starting value for the lateness of the festival and the festival organizer may only increase its lateness if she compensates the residents of the local neighbourhood enough for the extra noise and raises enough in sales tax to keep city welfare at least as high as when the festival ran only as late as the baseline. We treat the price of housing in the other neighborhood as a zeroed numéraire.

It is here we make our first major simplification in modelling assumption. We say the adjacent neighbourhood is a “small open neighbourhood”, to modify a term from international macroeconomics; the pool of potential entrants is far deeper than the pool of current incumbents and the neighbourhood is measure zero in the whole city. This ensures that the distribution of potential buyers is stationary. This is equivalent to the scenario where we have constant inflow from the stationary distribution into a closed city with exact replacement at a rate equal to or greater than the arrival rate of purchasing opportunities, which again forces the distribution of potential buyers to be stationary. This ensure that as the lateness policy changes, price instantly updates to its stationary value and we only have to worry about the evolution of the distribution of neighbourhood residents and not the housing price. I specify that all residents are part of the city so my welfare analysis makes sense.

Please note that sometimes substack has trouble rendering long-ass posts with lots of equations in the app or in an email. I suggest a desktop or web browser to read this post best.

Residents and the Housing Market

Residents of the neighbourhood receive a differentiated constant payoff, a disutility representing noise aversion linear in the festival lateness, and a flow cost of property tax, whilst residents of the rest of the city only receive the differentiated constant payoff (different than if they were in the local neighbourhood). We write the flow payoffs for the residents as well as the relative amenity gap as

We can write out the value functions for residents based on neighbourhood starting at the initial period as explicit sums given histories of option draws or given a geometric distribution over the first optimal transaction opportunity, but I have been doing macroeconomics solely recursively since day one of my macroeconomics classes and thus will go straight to writing the Bellman equations for neighbourhood residents

and city residents

We can use these two equations plus the stationary distribution for preferences and the stock of the neighbourhood to price the difference of the local property value. In brief, we assume that the incumbents don’t always sell when given the opportunity and that as the marginal entrant is drawn from outside the neighbourhood, the incumbents are price-takers and price should be a function of the stock and the marginal willingness-to-pay. This ensures a shift in distribution over time but still allows the price to be pinned down by the “deeper” outside market. If you are satisfied with this description, feel free to skip the next paragraph.

Let me now be more detailed about the market structure. In this setting, the housing market works like a deep market: the neighborhood has a fixed stock of homes and is small relative to the wider city, so prices are determined by the willingness-to-pay of marginal entrants from outside, not by the incumbents already living there. Incumbents face turnover shocks that give them the chance to sell, but only those with relatively low willingness-to-pay for staying—such as the most noise-averse—actually accept; more tolerant incumbents stay put. Concurrently, high-willingness-to-pay outsiders are eager to buy, and the equilibrium price adjusts so that exactly the neighborhood’s stock of houses is filled. This means that on the margin the sellers who leave and the new entrants who arrive are effectively identical types: the cutoff resident indifferent between staying and leaving matches the cutoff outsider just indifferent to moving in. Thusly, only some houses actually change hands each period, but the equilibrium price is pinned by the outside distribution of demand, while the internal distribution of incumbents matters for flows, compensation, and political feasibility rather than for pricing itself.

Write now

so using the obstacle at the option draw, we have a

Let the distribution of potential entrants be parametric. Then we can use the inverse quantile of the distribution of potential entrants to pin down the price using the share of neighbourhood stock relative to the whole city stock,5 which should lie in the open interval:

Note that the equilibrium price is a function of the lateness of the festival, id est, at equilibrium

If the lateness increases, we have that the distribution of the relative amenity gaps should shift downwards, and thus neighbourhood property values should fall. Now, let’s identify the probability mass of potential entrants who cross the cutoff when the lateness shifts. We can write such as

The average one-time compensation that may make incumbents whole from the price change effect of increased lateness is the mark-to-market change (using the distribution of incumbents), as it discounts the price change by the churn in the market as well as the change in the probability of acceptance of the offer:

We can further specify the distribution to make the story clearer. We could use a logarithmic transform for the lateness disutility coefficients combined with a multivariate normal-log-normal distribution to write out an explicit form, up to some one-dimensional Gaussian expectations, for the price function and its derivative whilst still preserving non-negativity, but just using a multivariate Gaussian distribution will grant us even cleaner forms, albeit with some potential for negativity amongst the lateness disutility coefficients.

We write

so then

and we have

This allows us to write

and thus

Here we see a very natural interpretation of the flow value of the neighbourhood expressing some form of preference for its quietness. After all, Forest Hills is quite an old neighbourhood by New York City standards, and people move there because most of the time it is quaint, quiet, charming, and out of the way. I noted this myself when I attended a Phoebe Bridgers concert there in high school and then later saw boygenius for the second of three times that summer (you may be noting a theme her). If the correlation between preferring the neighbourhood and preferring quiet is positive, as we might expect, whilst preferences for neighbourhood are more varied than that for quietness, then this comparative static will have property values falling in the lateness of the festival.

The Market for Concert Tickets

We write linear demand functions for the quantity of concert tickets demanded by residents and outsiders, respectively:

Market clearing then gives, for the fixed supply of festival tickets,6

I feel like this is a fair assumption to make as the constraint to tickets is usually venue capacity, and festivals almost always sell out.

We determine the share of outsiders (and its derivative in lateness) to be:

I assume that residents crowd in faster in lateness, as a later night out in their own city is more convenient for them than for outsiders. Furthermore, locals find it easier to camp out for the concert and get to the barricade, so they have an amenity reason for their higher marginal demand. Additionally, locals are more likely to be younger and thus music festival attendees. Then we have

We need to consider the financial structure of the festival organizer as well. Festivals are usually pretty inconsistent in organizational fidelity from year to year, but we made the simplifying assumption earlier that this one was infinitely lived, could credibly commit to hosting the event forever, and faced no changes in cost to production nor quantity sold of tickets from year to year.

Furthermore, the organizer can borrow money once (at the first period) to capitalize the first running of the festival under the new rights regime,7 provided that the present value of their future cashflows will cover the cost of servicing their debt year to year to fulfill the no-arbitrage and no-Ponzi-games transversality conditions. If they do so, for all time starting after the first period they will have to pay down a debt service cost. I know I promised I wouldn’t find a way to turn this into a macroeconomics problem, but I lied. Sorry!

The organizer has to pay operating costs each concert but can sell merchandise as well as food and beverages to generate additional profit, because the fixed ticket amount means they cannot increase production nor change price, so they must find other ways to draw revenue. This merchandise revenue is rising in the outsider share as we assume home production covers the food and beverage needs of the locals as well as outsiders being more likely to be fans who purchase merchandise (and thus further raising the costs of hiring the artists) as well as be older, as young people cannot afford to travel as much whilst city-dwelling young people make maximum demand for the events in their area.8 We further should note that lateness does not increase artist merchandise sales (those are probably lumpy by-day) but does increase food and beverage sales as well as increase the proportion of younger attendees, who tend to purchase less on the margin (I need to point out direct counter-evidence to this is provided by the Twitter feed of my Stan Twitter burner every festival season—including from many who were at this very event!).

The organizer themselves follows a simple profit function with a floor at every period past the first:

The profit of the first period is much analogous except for the borrowed cash and no debt payment (we can also say no minimum profit for simplicity, if needed).

Operating costs might on the surface rise with lateness, but if we take Ms. Cottrill’s reaction to her abrupt and forcible stoppage last Sunday night as any indication, it might actually be that locally those costs are falling in lateness as artists prefer to have more time to finish performing their sets.

I want to briefly discuss the intertemporal problem before the market for rights exists. We can also potentially assure ourselves that as more noise-friendly folk move to the neighbourhood, they will purchase more at the festival, so every year we get closer to the equalization of distributions, the more of this extra cash-flow gets brought in, but for now let us set that scaling function to be universally zero. I will discuss more on the transitional dynamics of the distributions later.

Given the rights are fully sealed to the residents, we have a simple outside-option problem for the organizers, as they can alternatively hold the festival elsewhere, earning a floor of profits by doing that, rising in the amount they may borrow. Specifically, I will call that amount the net present value of investing the money in a risky new festival with expected yearly returns off of that borrowing of greater than the risk-free rate of interest. Fuck, you caught me doing macroeconomics again. Old habits die hard, I guess. We also should declare a borrowing limit as festival organizers are not able to access the same sort of lending markets as other firms. Then the present value of the whole profit stream gives the problem

subject to the constraint on minimum profits each period, where the organizer will not hold the festival if the outside option is taken, and further subject to transversality conditions and the positivity and budget constraints.9

Transfers and Allocations

Consider if the organizers wanted to pay some form of transfer to induce an allocation of noise rights to themselves. They would do this by a covenant that promises transfers in perpetuity to the residents of the property. Assembling this themselves would be costly and thus a transaction cost would have to be paid. We can assume this cost would be utterly prohibitive, as they would have to hire a team of predocs to analyze the data on the neighbourhood despite the synthesis of data in the American Community Survey. Then, let us assume the Eric Adams government has decided to implement a noise market without anything but information frictions as to the full distribution of the preferences in exchange for a tax on merchandise sales with incomplete pass-through on to the purchasers.

The problem with this noise market, as exists in the real world, is that it only allows for the same perpetual payouts for each property (after all, we are in Zohran Mamdani’s New York City now) and they remain the same each year for all eternity. Perhaps he would cancel Adams’ noise market and implement a “community bargaining process” that settles on some named transfer size. I’ll credit him by saying this prevents the most ornery and most abrasively noise-loving of residents from demanding their preferences be met in the transfers so we can have a reasonable positive transfer size, although I bet it would also lead to a transfer size that is too large for true welfare maximization from the whole-city standpoint. We will discuss welfare later, but for now assume that whatever market or process we have informational (or some other kind thereof) frictions that will not allow true Coasean bargaining where each incumbent’s exact disutility is met—not only would that be too costly, but it makes this model too trivial. Coase’s Theorem is interesting not because it says what can be accomplished, but that our lack of witness to such private rights markets in the real world is the result of transaction costs and other such frictions (as well as incumbency willingness-to-pay generally being high) being so large and so prevalent.

We will assume the variance of the current residents as to the peace & quiet preference is less than that of the larger population and the mean of the noise preference in incumbents is greater. We would increase the mean of the stationary flow payoff by the amount of those transfers to obtain

and thus, property values would be shifted up a level. Specifically, we would write

We then have

Assume

such that the price would normally be guaranteed to decline. This is reasonable for reasons discussed earlier, as it essentially amounts to noise preferences being more varied than preferences for the neighbourhood, all whilst we assume they are still correlated. The current incumbents wish to be compensated for their flow loss in utility as well as the mark-to-market decline in their property value.

We now need an appropriate rule for selecting the size of the transfers. This rule will be ad-hoc and account for what I feel neighbourhood residents would actually act upon; that is, I will construct this rule according to the contradictions with compensating mean flow, the mark-to-market price change, the quantiles of the distribution, and incumbent preferences. We can assume reasonably that the cost of compensating the incumbent’s flow disutility in perpetuity is too costly if the difference is large enough such that it raises property values (more on this later), which is more likely if the housing stock is small. I will further describe each rule as corresponding to a different kind of Coasean ‘market’ outcome (or government-imposed non-Coasean alternative mechanism) that arises because of the way we actually conduct the “commodification” of rights in the real world and tie them to political and social circumstances in modern cities. The reason why I do this (and assume the many market frictions) is because I want to discuss potential outcomes more than I want to watch the Coase Theorem operate perfectly and instantly resolve the transfers issue through the market or set up an actual market structure to determine the allocation, as these outcomes are what should interest you the most (I personally just like the math, man).

We have several exact options for the rule: we can perfectly cancel the mean (or some other quantile or function of the moments of the incumbent distribution) disutility of the incumbents but expect them to pay a lump sum up front to return their mark-to-market gains to the festival,10 we can perfectly cancel some statistic of the incumbent flow again whilst compensating them for the mark-to-market decline, if that opposite scenario so happens, inversely, we can cover the current mean flow whilst crediting the incumbents for their property’s price increasing, we can set the cashflow to cancel the mark-to-market property value decline, and we can cancel the stationary disutility (or some other quantile or function of the moments of the stationary distribution) and perhaps combine with compensation further for the mark-to-market property value decline.

I consider the first scenario to be supremely unlikely, as I cannot find a covenant or easement in the real world so valuable as to increase a property’s value (aside from perhaps the cashflows of Wrigley Field-adjacent Wrigleyville parking spots), so let us now examine the second scenario, which arises when

This then implies that

still, so the mark-to-market flow drop will kill the rights transfer. There are generally two ways to compensate this: via a large lump-sum transfer once and for all to the current incumbents, or an increase to the flow transfer in perpetuity. Let’s look at this sort of lump sum later when we look at compensation for the stationary distribution. We otherwise just need to compensate the factor introduced by the differencing with the “tail-quantile term”. I back this out later, but right now I will just say this can end up running a cost of $5-$15 from reasonable estimates of the effects of the tail, implying a yearly flow cost of between $5,000 and $15,000, not even taking into account the main compensation I am about to discuss. Comparing, the mark-to-market compensation would be at most $10,000 as a total lump paid once.

The older, richer residents of Forest Hills Gardens probably place high premia on their peace, quiet, and undisturbed sleep. This leads me to estimate a mean between $50 and $200, which will produce flow costs of $50,000 to $300,000 per annum. That is almost certainly unsustainable for the purchase of just an extra hour of the festival. Given what I know about festival pricing, this would not be recouped, and thus no rights would be transferred.

How would this price come about? I call this “pricing via a community meeting”: the flows and transfers are designed such that the neighbourhood assembles into a coalition who together negotiate a transfer of rights by voting, with the relevant statistic picked to assemble them determining their price off of their stated preferences. There are obviously a lot of things wrong or over-simplified with this story, like why they commit or why they cannot game this process, but I think it most accurately reflects the NIMBY-dominated city we live in; where community meetings determine everything, run by the most strongly-asocial and strongly-anti-urban life of the retired, grumpy old homeowners you can possibly imagine who oppose all new construction or urban activities. This is the New York City of the city council, who are truly the main villains of the city. It reflects and respects specifically the local preferences of each neighbourhood over the city residents as a whole who I envision gaining the most welfare through a lengthened concert.

The third scenario runs into similar issues as the first, just in perpetuity instead of one time, so we proceed to the fourth. I will focus on one-time compensation as that is simplest here. That simply involves paying a lump-sum transfer

which is the cumulative probability of incumbents accepting the sale times the current stock times the sellable proportion times the discounted mark-to-market gains. We have bivariate normal tail probability but I use the approximation

as we have a small shift. I will talk about the organizer’s borrowing ability here first. Say the organizer current collects minimum profit. If interest rates are low enough, this sum is insignificant, the alternative venture risky enough, and their preferred rights purchase small enough, the festival organizers will purchase the rights. Say, if housing prices change by $1,000-$1,500 (this is only a few basis points of the median house price in Forest Hills Gardens), you have a 20% chance of churning every year, as the New York City housing market is extremely tight, there are 1,000-2,000 affected units in Forest Hills Gardens and the surrounding area, and we discount at 95% yearly, then even if the home gets 2%-3% harder to sell we have a lump-sum payment of around $30,000-$60,000. This is not insignificant, but if we consider that the extra revenue brought in by merchandising, although marginally decreasing in the lateness due to the share effect, might run around $15,000 (think of one-twentieth of the 20,000 attendees purchasing a $15 beer, accounting for the fact that marginally more attendees will be younger), then it seems like a clear economic decision even if lending rates available to festival organizers are high (damn, I wish the bargain hadn’t made sense because of interest rates, so I could tie this back to a macro story). Even if the extra hour actually costs money because every damn teenager in the Queens shows up and sneaks in a handle of Sköl, perhaps the headliner artist really does prefer the extra time and lowers their fee (this is me choosing to believe the best about the musicians I like; that they are irrationally inspired by the muses and full of love). I am not sure if Clairo would have done so, but perhaps one can calibrate a scenario in which she forms the margin that way.

I call this story the “irrational” story or the “Eric Adams story”. Firstly, in a world in which NIMBYs are so focused on their housing prices, they irrationally ignore their own disutility just to focus on the compensation to their precious property values. Secondly, with a mayor who is unfathomably based, one might expect him to simply ignore the complaints people have about noise; after all, this is the City of Yes. He deigns it fair to compensate them for the drop in their house prices, but they do not earn money from technically owning the rights and disliking concerts near where they live, because No is not Yes. Notably, it is the only story in which we make the show go on!

The final market just neutralizes future flow of some statistic of the stationary distribution to zero. Due to the higher variance in preferences for the neighbourhood, you will still have plenty of residents sorting in who dislike noise, so you will still have a price drop. This is a good illustration of the importance of the self-sorting angle, but also its limits in neutralization of the price effect—neighbourhoods are full of many amenities which are not dominated by one noisy night, but this still affects willingness to pay on the margin. Under this formulation, of course, you would have no sale. Completely neutralizing the wedge imposed by the quantile tail term in addition to flows would impose a cost of between $50-$150 per unit every year for a flow cost of $50,000 to $225,000 per year in transfers. That’s huge and almost certainly not economical. What is a better compromise in this “stationary-focused” story is to reimburse the mark-to-market loss as well as pay the mean back to each parcel in perpetuity. We can recover a mean of $30-$120 from reasonable estimates of the effects of the tail, implying a yearly flow cost of between $30,000 and $120,000. Even if that implies a mark-to-market recovery transfer of at most $15,000, it is still an unsustainable level of yearly transfers. I actually never expected to see this result; I figured the property value channel was much stronger, but I suppose this is why I practice economics! Challenging your intuitions with precise models is what economics is all about.

I call this last story the Zohran Mamdani story. Compensation based on the whole distribution of the city at once seems fairer, but when one realizes most of the city shares these horrific anti-urbanist tendencies, one will not see a huge improvement over the local city councilor. Even if denying the NIMBYs will mean tax revenues for most of the city, he still will not do it.

Subscribe now

Government and Welfare

I want to conclude the modelling section with a brief discursion onto welfare. I will not formally model the welfare, but my view would be that the combination of increased tax revenues from sales and the utility value to the concertgoers would be enough to ensure that the welfare-maximizing allocation transfers more rights to the organizers. There are a couple very complicated papers in microeconomic theory I would use to formally describe this notion that seems inherently contradicting, but perhaps I will just say that I assign higher welfare weights to the concert attendees than I do to people who live in a planned community in New York City of such beauty. Formally, I suppose one could decompose the utility functions and remove the premia against the concert running late to promote a more fun society or argue that the limited size of the concert venue ensures that the full willingness-to-pay of the whole population isn’t properly reflected. Maybe, in fact, we should weight the welfare of the grumpy, the curmudgeonly, and the elderly less (just kidding on the last one, or am I)?

Perhaps a related point is that the government should just transfer the rights or pay for the concert itself, if we believe a number of stories as to low wealth in the highest-utility individuals who attend concerts. These sorts of events mean the world to a lot of young people, including the teenaged Zane—now, which mayoral candidate will actually support what the youth really want?

Raising the housing stock in the desired neighbourhood also effectively helps lower the needed rebate—I have not estimated the margin of this, but it is useful to keep in mind.

How I got to some epiphany

The main reflection I must come to as an economist is this: Coase’s Theorem is really hard to break. I hit it with second-order effects of the externality, imperfect information, lack of true markets, time value, and still the only failure mode that existed was a low willingness-to-pay. I had really expected the opposite to occur: the second-order effects would dominate, and it would essentially be the short-term transaction costs of the mark-to-market change occurring because no one can estimate anything beyond some simple statistic of the distribution that induce this. Instead, imperfect markets and compensation still work perfectly. My intuitions as well about the elasticity of housing prices failed, too, but again, this is why one of my many mottos is Model Everything, and why at the University of Chicago our unofficial saying is, “That’s all well and good in practice, but how is it in theory?”

Perhaps one of the reasons for the lack of dominance is that I modelled this under absolutely perfect information. If agents simply had pessimistic beliefs as to the property value impacts but themselves did not mind all that much, as might very well be the case, the mark-to-market lump sum would have dominated the rebates and we could be having a discussion of the government funding the arts here (my arts).

We just see that the problem is almost inherently socio-political: we cannot maximize our own view of welfare in light of NIMBYism pervading our society. It’s a lot more than a frictional, or market power, or regulatory capture story. It’s also a story about the attitudes and norms that we have let invade the greatest city on the face of the earth, attitudes that reject all that makes a city vibrant and living, attitudes that will kill it within two decades if we do not root them out and destroy them—soon. Yes, this is me promising to defeat the NIMBYs on behalf of bisexual teenaged lo-fi indie rock fans everywhere. Thank me later. I’m no hero, that’s understood.

Sometimes what makes market urbanism and such bargaining over externalities fail is the preference or welfare-weighting composition, and you cannot mechanism your way out of that situation. Making property values less of an issue is one thing, but making co-existing in a shared urban space less of an issue is something I can’t (yet) work out mathematically, so we are all going to have to work this one out together. We need to let each other and especially the youth make noise. “…I’d rather be alone than a stranger” is a good motto to follow, don’t you think?

My point about the New York City mayoral election, although a little depreciated at this point, is simple: if you are young like me and want(ed) the candidate who actually supported a vibrant, open urbanism that let you build things and experience things together, you should have chosen Adams, not Mamdani. I know Zohran Mamdani is young, has an attractive wife he met on Hinge, is a good communicator, and promises lots of fun socialist ideas that seem great if you don’t think about them at all, but he has a terrible track record of actually saying yes to things. His plans are hell-bent on upholding the worst members of our council (see the Master Plumbers discourse), entrenching NIMBYism, enforcing the rent controls that completely lock you out of the housing market in favour of untaxed old people, destroying the main assets in the schooling system, and wasting precious tax dollars that could go to transit on government-run Erewhon. He will destroy this city, if given the chance. We cannot let that happen. I left New York City, the city of my birth, to come to Chicago to learn how to fix the world, only to watch as Brandon Johnson, who is identical save switching rent control for total public-sector union dominance, utterly destroy this city in a matter of years, a worse job even than the COVID-19 did on Chicago. I am so terrified for the capital of the world, and I feel so helpless to stop it, even at this point

On the other hand, please appreciate Eric Adams whilst you still have him. City of Yes was empirically a massive success. Adams actually relates very well with the diversity of the city you don’t see online. He is independent and exceedingly funny. He reminds me of Bloomberg, who is horrifically underrated as mayor. Eric Adams is the kind of guy to just extend a curfew. Zohran Mamdani and his Warrenite staffers are not. If you’re like me and like music late into the sad hours of a Sunday, if you’re a true romantic, a real lover, the choice is obvious… was, I guess

I do want some credit for never making a “walking out the door with your bags” or “sell all my things, become nomadic” joke about the housing sales in my model. It was really hard to resist, guys.

I really hope you enjoyed reading this post; I worked very hard on it for the last few days. If you don’t know who I am or what I do here, this is a brief guide: I am a fourth-year joint undergraduate/masters at the University of Chicago studying mathematics and economics. I have been taking PhD classes in macroeconomics for a couple of years now. My technical focus is on impulse control problems, computationally-difficult heterogenous agent models, and mean field games. My research interests for the layman are how inflation and the labour market affect each other, how precautionary saving and wealth, income, and skill inequality affect recessions, and how nominal versus real shocks affect the economy differently. I used to have a fairly prominent Twitter but I abandoned that for a few years and only recently rejoined the site and began blogging again. My hope with this blog is to provide a window into the most advanced work going on in macroeconomic theory right now, as well as provide formally-modelled economic thinking on a variety of topics from the useful to the mundane and communicate how the actual work of economics is done.

If you would like to read more or to track my career journey, please consider subscribing for free or obtaining a paid subscription. I currently aim to put out one post containing a formal model on Mondays as part of my Modelling Mondays series (you can read the first here and the next post will be a basic model of sticky output in continuous time to show why we don’t use that concept in macroeconomics all that much), as well as a newsletter containing news from the academic macroeconomic world every Friday or so. I also constantly respond to economics discourse in the news as online, as well as irreverent pop culture or high art events, and occasionally I write down my thoughts on methodological or normative economic issues less formally. I also aim to put out paper or literature explainers to bridge the gap between common understanding and what my peers and colleagues do. My plans for paid posts are more of the same, as well as personal updates and more immediately-utilitarian takes on current macroeconomic events.

My current subscribers: please let me know if you liked this post, and if you want to see more stuff like it, whether it be pop-culture commented upon this way or just more urban economics or microeconomic theory or politics. I have some old photos of these concerts I might share in a paid personal post as well. Also, please subscribe to Maia Mindel’s blog if you enjoy the combination of economics and Clairo—she was a friend from my early Twitter years. If you have any questions or comments, no matter how insignificant, feel free to leave them below.

Leave a comment

The response to my first posts has been amazing and my blog has grown so much faster than I ever expected. If you like this post, please like and share it wherever you can, as well as consider a paid subscription. I may be offering free trials, deals, or referrals soon.

My goal with this post is to show a basic mechanism for money illusion in a stochastic search-and-matching model in continuous time. This is an ideal setting to discuss its effects on unemployment and one can see the mechanism that famously lead to the Phillips Curve breaking down during the Great Inflation of the 1970s. This model will lack many important features, especially features related to the way information works in the model, that would make it Lucas Critique-robust, but those features are not necessary for the mechanism and I will add them at a later point when expanding or closing the model. It will instead treat information and attention as mutually the same; a competent economist could proceed from there.

Those of you who know me more personally know that both my current working papers are based on ‘sticky’ versions of Diamond-Mortensen-Pissarides, as my current research interest is evaluating the ways that downwards nominal wage rigidity affects both employment volatility and unemployment-price spirals. I have very (Market) Monetarist roots, so I always want to include some form of money, or money illusion, in my models, but as it stands those papers are very much still in the tradition of Bob Lucas or Christopher Sims where agents view everything in real terms and bonds are preferred to money. However, I was recently thinking of how I would insert money illusion into such a model. My hope would be to do so with only one extra state variable, as one of my models already contains a true, non-block-recursive, Mean Field Game across three states with a Clarke generalized Jacobian,2 so computation is already very tedious, and the curse of dimensionality is already starting to bite hard in my finite-differences scheme. I think I have found a way that fits in well to both my HANK-SAM3 model with stickiness and my GHF-DMP4 model with stickiness. I will not solve the model here, nor fully analyze nor calibrate it, but I will treat the model and discuss what considerations went in to its building process.

I also hope to walk my readers through how I construct a model when starting from scratch; I work from first principles often here. There are not nearly as many blogs on the econ blogosphere that discuss how economists work versus ones that discuss the issues they work on, and even fewer blogs that discuss macrotheoretic modelling. My hope is to fill that gap in the e-literature as well as spread my particular idiosyncrasies like a plague. Hopefully this walk-through grants a clarity to how modern macrotheorists work—although I am not yet a professional, being but a fourth-ear undergraduate, I am a good bridge between the autodidacts and the men of the ivory tower. Please do forgive any mathematical or economic mistakes, shortcuts, or roughness that I did not intentionally place. Note that many of the equations will not be viewable in an email, so please read this post on a desktop or in the app.

Model

I will construct my ‘toy’ model (although it is actually almost a research model and could probably be turned into a paper publishable in a field journal if I didn’t have so much on my plate already with my other papers and applications) in partial equilibrium because, as you will see, figuring out the closure of the labour market in this model if we were to, say, wrap it in a New Keynesian model would be a bit tedious for a Sunday morning blog post. This will already be a very Chicagoan, or should I say Friedmanite, paper and I do not think Uncle Milty would mind this at all, as he was famously an advocate for partial equilibrium modelling even as the Freshwater School turned away from that paradigm. That’s not to say that the Phillips Curve is an insignificant part of this paper, but in a ‘simplified’ closure of the model using Rotemberg pricing, we would essentially be solving for the price of unit labour productivity using the Phillips Curve, which doesn’t do anything but scale the law of motion for real wage, so partial equilibrium analysis should suffice here. A paper by one of my mentors, Erik Hurst, and others, which I cite as one of my favorite papers currently which did much to shift my thinking on the post-pandemic inflation and forms the core of one of my current working papers, is itself unclosed like this and it is considered a cutting-edge macro-labour model that concerns nominal effects, so I do not think I am too out of left field on this one.

The utility of developing such a model is that it would partially reconcile the lack of immediate pass-through of inflation to worker behaviour observed versus the pass-through to firms, at least under low-inflation regimes. Regime-dependence is one of the main stories of the last fifteen years in macroeconomics, so a future inclusion of such dependence would be even more powerful.

Environment

We operate in continuous time. Our price level moves according to the exogenous (for now, as discussed above) law of motion

whilst aggregate productivity moves according to

Normally, I like endowing all workers with an idiosyncratic productivity because this allows using elasticities to recreate certain observed shifts in the labour market with regards to quits and on-the-job search (see the aforementioned Afrouzi, Blanco, Drenik, and Hurst, 2025) and a simple resolution of the Shimer Puzzle, but here we have the natural logarithm of idiosyncratic match quality as a diffusion process with law

Nominal wages are constant in a match until they are reset according to a memoryless Poisson process that may be controlled in rate. There is a stock of information regarding the true rate of inflation that lies on the closed unit interval with law of motion

We will leave our separation hazard an exogenous function for now, and wage reset opportunities will arrive according to both state-dependent Calvo opportunities and a Poisson intensity control-based reset opportunity, and resetting will take place given a menu cost. I will leave discussing real-wage mappings until later. The information stock will imperfectly persist across resets and also reset to zero at according to a (possibly state-dependent) Poisson process. Matching is standard Cobb-Douglass, with a universal unemployment flow utility, vacancy cost, and free entry. The important thing to note here is that I decided to model money illusion as workers and firms perceiving different drifts in the match real wage: workers do not comprehend always the extent of the drift, whilst firms are always perfectly informed. I explain why later, and the consequences of this decision.

I am making the decision to not include endogenous separation in this model for now due to the additional complexity it introduces by adding in free boundaries and thus transforming the backwards optimization problems into Hamilton-Jacobi-Bellman Variational Inequalities instead of Hamilton-Jacobi-Bellman Equations. This is a strange move for me, as I am typically focused on the unemployment flows in these models and both my papers have specifically focused on endogenous separations in these sorts of models. They would actually not be hard to write in to the model, but would make the computations harder, and for right now are not relevant to my analysis. I can easily modify the model to insert them later and will likely do so at a later time.

The model will indeed be at least triangular because I am choosing to make resetting of wages rigid but controlled through intensity control and menu costs, which therefore makes the forwards distributional dynamics dependent on value functions. If we had continuous re-bargaining à la standard Diamond-Mortensen-Pissarides, we would simply only have exogenous diffusions in two variables and thus, if our wage-resetting rule was closed-form, the system would be fully block-diagonal as the Kolmogorov Forwards Equation could be written to not depend on the value functions. If wage resetting was an exogenous process this would also hold, although if we had a fixed-point problem for the reset wage (say because of Nash bargaining) the forwards dynamics would get more complicated as the value functions would begin to matter. Introducing endogenous separations also would force us to have a triangular system.

The system will be block-recursive, however, as we allow firms to have full information over their matched worker’s information stock and the firms have no controls (outside of potentially the wage mapping), and for the present purposes of analysis we need not study the forwards problem nor even write down the Kolmogorov Forwards Equation, especially as we are not clearing a labour market (distinct from a matching market). We will not have a true Mean Field Game until we introduce general equilibrium or some feedback mechanism or on-the-job search or some other function of the distribution into the backwards optimality problems.

The most difficult part of this problem, however, had to do with ensuring we could write down a problem with stationarity when percent inflation is nonzero, as a problem in nominal terms is a problem in terms of the price level, or numéraire, which is non-stationary. Say we had written the worker’s problem in full nominal terms. Just using a co-numéraire by dividing out the price level from the worker’s problem would inherently strip much of the money illusion mechanic out of the problem without verifying that all other factors were real, so it would have been unacceptable to just do that without setting up the hazards to rely on real wages only. Ensuring our reset rule for wage is real would be key, but it would be not enough even if we make our state-dependent probabilities not dependent on nominal wage.5 We thankfully could have a non-stationary worker’s problem as long as we somehow ensured that the controls entering the firm’s problem and the distribution were stationary, of which there should be several ways to accomplish this, including through showing something that I nicknamed “numéraire invariance of the first-order-condition” for the intensity and menu cost, which essentially was an expansion on the co-numéraire method. However, in this case, there was a simpler solution.

I instead solved this problem by framing the problem as one of “actual versus perceived” real wage drift instead of having the worker’s problem be nominal and the firm’s problem be real. I rely on some unique properties of this particular partial-equilibrium model in order to do this. First off, there is no inter-temporal optimization beyond the choice of resets, and later, separation. This lack of saving or borrowing or consumption means that the behaviour of the worker is the same as in the nominal model, as I will explain. This will give us the ability to say that this model is a good representation of money illusion, which might not hold in a model where there are differing choices based on nominal quantities. Additionally, the payoffs are real and linear, and the costs allow for consistent deflating. Furthermore, no other nominal variable enters besides percent inflation—which might not hold for a general equilibrium model. Lastly, we will target a real wage when we reset. Given that and denominating the exogenous hazards as depending only on real wage, we can show that the endogenous hazard similarly depends only on real wage, so the backwards generator operator is the same as the nominal problem, and so are the flows, so they are the same problem. Adding in nominal costs or indexing or inter-temporal optimization in nominal terms would break this and lead to us needing to rewrite the model (or perhaps it will just ensure time-dependence). Whether a model with convex utility payoffs, inter-temporal optimization, or a full general equilibrium necessarily prevents us from framing the problem this way is a discussion for later. It should technically be possible to write it again this way, but we must then be satisfied that the optimization carried out is the same as what we care about for money illusion, id est optimization when an agent cannot percieve that his purchasing power is eroded. That a drift can solely capture that and such we do not need consumption choices to be optimized solely in nominal terms is somewhat unclear to me at the moment. I lean towards yes, but I am not fully sure.

It is also unclear to me if the triangularity is needed, as it might be that feedback or stochasticity in the price level, or market tightness, or the real interest rate, or inflation, either via the distribution or just via backwards-constructed equilibrium objects, would ensure that the equivalence fails, whether just because it is general equilibrium or because it is a Mean Field Game. I am also unsure of the validity of the assumption of real indexation for the costs or hazards, although there are ways around that problem (inflation-indexed resetting, anyone?).

I further prevented time-inhomogeneity by taking advantage of the fact that our aggregate productivity growth is Harrod-neutral to normalize the problem to a balanced growth path, scaling both the units of the states as well as the flow payoffs to ensure the equations are autonomous. The validity of this may again be a concern later in a general equilibrium with proper intertemporal consumption and savings, but it stands for now as the main objection on might raise to it, efficiency units being in the wage variable, is only a concern, by definition, outside of the steady state; if it is a major concern to you that the “stationary” (re-bargained) wage distribution depends on aggregate productivity, then realize that distribution is describing wages on the balanced growth path, growing in aggregate productivity, and if such disturbed them, their normalized counterparts would not be stationary. You might also be curious if the exogenous separations or state-dependent Calvo depend on aggregate productivity in transition, but we can write those in via giving time indexation to those functions if we want to model that (although it might be more annoying to figure out the proper specifications than in the non-normalized problem).

Worker’s Problem

We may strip the time indices off of the state variables for simplicity in writing. We work over the natural logarithm of real wages less the natural logarithm of aggregate productivity here to mirror what we have for match productivity. Treat all value functions as being scaled by aggregate productivity as well, which has a normalized starting value of one. Together, these will ensure the problem is autonomous. We can write the employed worker’s value function as follows:

I could also choose to include some form of indexation exogenously depending on the information stock to complicate the wage drift, or further some controlled indexation by the firm (this could lead to a game of asymmetric information where the firm has to screen the information stock of the worker by offering them a menu of incentive-compatible indexation contracts), but here I will decline to focus on core mechanisms.

You may be tempted to ask if we should include illusion over the real growth rate as well. That I am not sure of, especially in this normalization. Perhaps a clever commenter can give me an idea.

We are going to not worry about the Shimer Puzzle for now and take the easy closure for the unemployed worker’s problem with respect to starting wage, and further assume all workers start symmetrically with no6 information on inflation:

Firm’s Problem

We write down the firm’s problem as follows. It sets no controls outside of the wage mapping. Its Hamilton-Jacobi-Bellman problem is

The indicator function is technically an abuse of notation; it should really be a Dirac-delta function. The free-entry condition grants

which pins down market tightness, which is rising in the expectation as the firm’s match probability falls in market tightness.

If we have some reduction in complexity (either via constant resets or removing the worker controls), we could write down a very simple law of motion for unemployment (and likewise a steady-state value), as is standard in Diamond-Mortensen-Pissarides. These could either be ordinary differential equations or partial differential equations, given our choice of state-dependence.

Distribution

I will write down the Fokker–Planck equation for employed workers here despite it not yet being relevant to my analysis. It is not difficult to derive. First, some housekeeping. A reminder that:

where the hat denotes the un-scaled natural logarithm of real wage. We need this to ensure the time-homogeneity of our Kolmogorov Forwards Equation and our boundaries.

with the same abuse of notation as in the previous section. Although this looks terrible, with Barles-Souganidis-compliant upwinded finite-differences schemes for both the backwards and forwards equations are quite easily solvable with Howard’s algorithm or a modified policy iteration.

Wage Reset Mapping

There are a large variety of different wage bargaining or setting schema available in the literature, but I do not see any problem with just using good, clean, old-fashioned Nash bargaining:

which grants the usual first-order condition

An important question for analysis later is whether we can determine if money illusion affects the Nash split and thus the bargained-for wages, or whether it only affects the distribution through hazards (to resets or possibly endogenous separations).

Boundary Conditions

These are often the most technical parts of these models, and without free boundaries they are a lot less relevant here, and are only really of interest to my most serious of readers. I’ll just write this: it seems natural to have reflecting boundaries in idiosyncratic match productivity and advection in effective real wage. This ensures zero outflow (because of upwinding and lacking external inflow) in wage as well as makes endogenous separations less immediately necessary. This is all subject to change if we introduce those; as then we will have to deal with value matching, smooth pasting, and flux out of the boundaries, beyond the scope of this introductory post. Keeping all outflow via hazards for now simplifies that aspect of the distributional equation immensely.

It seems prudent to me to assign reflecting to the lower bound of the information stock but also to the upper bound of the information stock, even given that it can Calvo-reset to zero, to avoid a stationary equilibrium defaulting to full information, although that would be an interesting change to model. Such a scenario essentially describes only this sort of information over the drift as being relevant during transitions, with steady-states either having full (really, high, because of the partial loss on resetting) or zero information (if we absorb the opposite boundary). That will not be interesting to analyze in steady state as it defaults to the model I am already studying in one of my working papers—you can all wait for that to come out when it does. It might, however, be more realistic, as it essentially represents that learning only happens during shocks—perhaps I will try it if I compute this model.

Stationary Analysis

I want to include some simple analysis of the steady-state to show off how key aspects of the model interact. I will be omitting some details of stationary distributions or cross-sections here, but they should not be relevant for the following points.

Comparative Statics

Let us first evaluate how our aggregate variables affect employment. Differentiate free-entry to obtain

Furthermore, as firm payoffs shrink in wage and wage drift is negated, we have

The first term is the direct “erosion channel” whilst the second is the effect of marginally-higher separations. Note also that I am being rough with my notation here for the hazard. Then, via the Leibniz rule, we may write

I will not directly prove this here, although I could easily. Instead, to honour my mentor Fernando Alvarez, who just received a conference in his honour, I will claim that I have a special papal invocation to interchange limits and integrals as the pope is from Chicago and leave it at that.

That result is a simple intuition for why we say inflation “greases the wheels of the labour market” and decreases unemployment, at least in the first order.

The next point is easiest to see if the reset hazard is a constant and the information stock is fully deterministic, but instead I will just be mathematically imprecise to explore this claim by partially fixing the reset. Setting up a full generalized hazard framework is annoying, if not deeply satisfying. I claim that the magnitude of the marginal effect of inflation on market tightness falls in the learning coefficient and falls in the reset hazard. The higher the learning coefficient, the greater the perceived drift in magnitude and thus the higher the reset targets. The higher the reset hazard, the faster the propagation of the reset wage into the actual wage. Together, these lead to a lower firm value. This effect is compounded if resetting is endogenous, as resetting will also take place more often, and functions through reducing the firm value using hazards.

Here is a quick justification, abstracting away from the wage-bargaining channel, although it would preserve much the same:

Then, using a rough approximation for the expected flow, we have

Putting it all together, we have

Inattention attenuates the worker channel: a smaller time average of information stock implies a smaller reset rate and thus the negative indirect term shrinks, so the positive direct profit effect of inflation dominates more strongly, and therefore the marginal effect of inflation on market tightness is higher. If we have a full information stock, the worker’s resetting channel offsets more of the direct effect.

This provides the core mechanic of the model: a lack of information on inflation slows and shrinks in magnitude the wage resetting of the worker, tightening the labour market and further lowering the opposing effect on tightness of more inflation through the resets channel by attenuating the effect of inflation on that channel. A finer justification for why a higher perceived erosion of wage lifts firm value on the margin directly whilst lowering it through raising reset intensity and size can be found in the next section; it is through something I’ll call the “annuity channel”.

If we look at the time average of 1 minus the information stock in-between resets, we have the following “sufficient statistic” for the short-run expected cumulative perceived erosion weight of inflation on the wage:

This then grants

This essentially measures the cumulative length of the “combined spells” of resetting and learning.

Under strict illusion, workers never learn and never reset, which maximizes the benefits of inflation to the labour market. This, however, is not even guaranteed to be directionally true under endogenous separations. If there is just learning, we gradually approach neutrality over time. If there are just resets, there are random “meaningless” re-anchors to a productivity-dependent wage. As both the learning and reset rates go to infinity, we reach a fully neutral model with immediate inflation passthrough (continuous re-bargaining).

Distributional Considerations

I’m not going to write it all out here because it is quite standard, but inflation and growth reduce the stationary mean of effective wage, whilst raising resets increases that mean. This means the stationary gap between wage and reset wage shifts left in inflation or growth and right in the hazard. Furthermore, this is scaled with endogenous hazards, as the reset hazard rises in inflation or growth.

A Dimensional Reduction

We can reduce this model’s dimension by one using a simple trick that I developed for one of my papers. I will not be tedious and repeat the proof here, but essentially, if we allow that

we can write

which not only makes the problem easier to solve, but allows us to possibly say something direct about the Nash-bargained wage (or sometimes the “wage gap”, if separable). We can determine that the endogenous reset hazard is also written in the difference of match productivity and effective wage and information stock, which enables us to study the present value of the future gap given the fixed real wage of the match; the menu cost just affects the intensity.

We can show that the wage of reset generally depends on the information stock as well as the gap; furthermore, this is what induces the Nash-bargained wage to depend on inflation. I am going to write a technical note on this later for paid subscribers because I think the derivation is interesting but requires more time and space than is suitable in this post, but the short of it is that we may write the “annuity factors”

and

These scale the present value of a unit of flow such that the value of the split in the Nash first-order condition can be written in a more simplified manner. You will see that the hazard and the perceived decay rate are the two ways in which money illusion enters the problem, and furthermore, that these enter through raising the future value and duration of a match in the eyes of a worker, such that less information induces a lower reset wage.

We further have a simpler closed-form first-order condition if the exogenous rates are constant, but as this is standard and the interesting result is already elaborated upon above, I will not elaborate more other than to say that eliminating the productivity-wage gap from the reset wage should not eliminate the effects of the learning stock, which should simply scale it via the annuity factors. We can use this to see easier that there are two channels: the value of the wage is perceived as decaying slower, so a worker demands a lower real wage on the margin as it raises the value of their part of the split, and as lower information on inflation makes the worker less intent on resetting, the match is likely to last longer, which raises the annuity and affects the split such that the reset wage is lowered.

We can instead scale in match productivity, convert it to an idiosyncratic productivity for the worker, and scale the unemployment value the same way to obtain that the reset wage should be separable in the gap and can be written as

This result then allows further analysis of the channel of money illusion onto wage bargaining and thus the wage distribution, which I may elaborate upon in a follow-up post. The main issue here is that then we need to close the model for the expectation of a firm in the vacancy equation with our distribution of unemployed workers, which ends block-recursivity as then the distribution feeds into the value function, unless we convert market tightness to a price (and thus eliminate our free entry condition).

Subscribe now

Further Lessons

The most major problem with this model is that my specification for information is exogenous and thus the model would fail the Lucas Critique. A way to fix this would be to add a distinction between attention and information. Let our current specification be that of attention, let the intensity of the attention growth (or forgetting) be controlled but costly, and then add another state variable for information: a current guess of the inflation rate. I view this not as the consciously-held information of the worker as to the precise rate of erosion of their wages, but instead as the information the agent has on the inflationary regime: is it higher or lower than their current perception, larger or smaller in magnitude? Then, to incentivize paying more attention, let there be a stochastic arrival of a ‘comeuppance’: we switch the current guess to the actual rate of inflation and raise attention by a scaled convex function of the difference between true inflation and the guess, so if true inflation is higher and thus values are lower, this will be a negative flow, so paying more now attention minimizes this gap. Let also Calvo attention resetting depend on the state of the guess and introduce resetting to full attention with stochastic arrivals rising in true inflation. Let this stochastic arrival include a choice to pay a menu cost to pay full attention or no attention (we probably have to include a non-zero minimum for attention and perhaps a greater-than-one upper bound). This induces an (s, S) band such that we have defined “regimes” of inflation: low, medium, high, which each induce their own separate attentive behaviour. Finally, have the inflation guess arrive according to some signaling process that grows more intensive as inflation rises. Together, this creates a model where inflation grows in relevance to workers and affects their decisions and thus labour market dynamics more as inflation grows.

We could simplify this by just adding the ‘comeuppance’ (via switching to full attention) and endogenous attention growth without adding the guess, but that will eliminate the “over-attention” channel as well as regime-dependent forgetting.

Firms in my current model have perfect information over the attention of the workers, but we could potentially complicate this by aggregating over the distribution, although this kills block-recursivity.

Further direct analysis of this model is difficult because of the fixed point for the reset wage, but we can utilize kernel or perturbation methods to deal with this. I hesitate to say more as the contents of one of my worker papers already deigns to cover this.

Solving this model should be easily done with Howard’s algorithm or modified policy iteration, even though there are fixed points, as we can treat the reset wages as policies, compute, then update them using the first-order condition of the Nash equilibrium. If anyone wants to see this model calibrated, please let me know. There are ways to do it that are easy but that do not reflect microdata, and ways that would make this model go from blogpost to paper.

Some further major modifications to the model would be to add closure to the model. There are canonical general equilibrium closures of Diamond-Mortensen-Pissarides, but I would prefer to embed the model into a (heterogeneous-agent) New Keynesian framework with assets. Adding this intertemporal optimization furthers us from true money illusion, but my hope is that the core labour-market mechanisms remain. We could also discuss money illusion in firms, but in general the main mechanism would be how the real interest rate and true inflation scale the perceived interest rate.

We should also add aggregate shocks to our model. This makes the distribution far more complicated and enlarges the state space but makes the model far more empirically relevant, especially as regards the signal-shock correlation.

Retaining inflation attention or knowledge for the unemployed also would be beneficial, especially if we make productivity a state for all workers regardless of unemployment.

Perhaps the most crucial change to the model would be to add endogenous separations. This would enable the core of labour market dynamics to be depicted the model. The free boundary problem is more interesting to analyze, but when computing the model using finite differences, it is far more frustrating, especially as the numerical instability will grow as dimension does. A simple reduction, going back to aforementioned Afrouzi, Blanco, Drenik, and Hurst (2025), is to implement a stochastic separation cost and let separation opportunities arrive at a Poisson rate, thus smoothing the boundary and making computation easier.

I think this example, however, already provides important lessons for all sorts of models. It was a fun exercise to construct it, especially as I forced myself to rework how to create a balanced growth path from first principles. Time inhomogeneity is usually not something that appears in my models, after all. I do think some of this may end up in one of my papers someday.

As for practical economic lessons from this exercise? Perhaps that the ability to grasp the extent of inflation matters most if inflation is high. Low-inflation regimes already erode wages little, so a lack of attention meaningfully scales less.

I’m hoping to make this a continual series, a core of my blog called Modelling Mondays, where I will build an original model or go through a pre-existing one to provide an in-depth look at how we build and think of models in economics. Please let me know if this appeals to you.

Also let me know if this model itself appeals to you, or if the topic of money illusion does, so I know if I should follow up on these threads quicker. I have some explainers on mean field games, rational expectations and the Lucas Critique, and a brief guide to the evolution of modern macro in the pipeline, as well as a critique of complexity economics and agent-based models in depth, as well as a post analyzing some models from informational finance I find particularly problematic. Lastly, I will be writing some newsletter, behind-the-scenes, and personal content for my paid section, as well as a post on a certain indie rock singer who had been a topic of conversation this last summer as well.

Leave a comment

The response to my first post was amazing and my blog has grown so much faster than I ever expected. If you like this post, please like and share it wherever you can, as well as consider a paid subscription. I may be offering free trials, deals, or referrals soon.

Newtonian Mathematics is a Nonsense Term

The article itself contains a bunch of nonsense I find prudent to dismiss before any further discussion. I wish I could embed the tweets of others on this subject, but as that is impossible I will write responses myself.

The original article was deeply embarrassing to its author because it contains a clear admission that Harford could not understand the fairly-basic mathematics in his masters-level classes.

Calculus is easy and can be expected of any college student. Acting as if it is an unreasonable tool is rank absurdity. Harford, upon being confronted with something he did not understand, proceeds to declare it incorrect instead of questioning his own limited abilities, the height of hubris. His criticism uncovers the first common element of anti-mainstream critiques: a lack of understanding motivating the critique. If Harford had a basic mathematics education, he would know calculus, and thus not be frightened by undergraduate microeconomics, and further he would know that the proof that underlies Arrow’s impossibility theorem is the same set-theoretic logic that underlies the existence of calculus. This lack of understanding often makes neoclassical economics seem as if it is physically describing a system instead of formalizing forces we all agree exist: optimization, scarcity, information.1

Harford seemingly thinks of neoclassical microeconomics as something derived from classical physics because it uses constrained optimization, instead of the basic and correct understanding that constrained optimization is how we represent the decision-making of utility-maximizing individuals who face some form of budget constraint. I am not being uncharitable here. His complaint is literally just so on “Newtonian Mathematics”:

That this critique is invalid should be obvious: individuals try and make good decisions for themselves (without this, you either have to reconstruct decision rules as an a priori assumption, which seems to me to impose even more contingent and unrealistic assumptions, or you have to assume people behave randomly, which doesn’t impact models as much as you might imagine), firms want to profit the most they can, households have limited resources with which to conduct their business. There are numerous experiments confirming these to hold, but even if you were to take the behavioural or reduced-form approach, you would still be afoul of the “criticism” of Harford.

In his view, if we even attempt to formalize behavioural economics mathematically, we are somehow doing it wrong, because developing a representation of a decision rule for a human being is impossible because… reasons, I guess? As for statistics, it must be pointed out that constrained optimization also underlies that entire field, but alas, as derivatives are a false construct, we must be stuck reading poetic descriptions of economics,2 for some reason.

There are two motivations for the nonsense views that Harford holds. One is apolitical, and is described perfectly by Paul Krugman in this classic Slate article. The sum of this motivation is simple: people who lack technical skills but think of themselves as deep commentators would like to be able to comment on the economy, a thing they witness everyday, like it’s culture. This is rife on both the right and the left. Back in the day, gentleman Austrians, Marxists and even normal Keynesians may have been able to comment on the economy in natural language, but those times are gone, yet those of arrogance still do this (in addition to heterodox economists, sociologists, journalists, and culture writers are constantly guilty of this as well). This often leads to a critique that psychology, institutions, or culture aren’t correctly incorporated into mainstream economics, as those sort of thinkers both over-emphasize those and also don’t understand what their incorporation into a model means. This is the kind of fallacy I hope Harford, whose pop-economics books I used to recommend in place of Freakonomics or Basic Economics, is committing.

The other motivation is more insidious: political radicals, understanding that mainstream economics will never support their nonsense beliefs, need to critique the science that rules them invalid. Isabella Weber, notorious leftist crank economist, wrote the supportive tweet on the article that initially drew my ire. Unable to discover actual evidence to support her views, she must attack the concept of evidence itself. This is common regardless of right or left. Leftists, adhering to the religious traditions of Marxists, tend to obsess over the minor critiques levied at economics by Great Men like Schumpeter and Romer, or try to rehash debates from before the birth of my advisor, instead of reading research that came out post-1970. Some libertarian rightists reject using math and data as a violation of human nature, whilst authoritarian rightists tend to reject economics as a conspiracy (where have I heard that one before). They all tend to adhere to a deliberate misunderstanding of economic assumptions as failing to capture behavioural idiosyncracies or real-world institutions instead of acknowledging how models deliberately incorporate those things. This leads me to my next point on the validity of those critiques in general.

Cope! The Friedman F-twist is Valid

Milton Friedman’s 1953 essay on economic methodology is the classic foundation of how the field works. It, alongside the introduction to Gary Becker’s Economic Behavior, formed the first elements of the syllabus to my first ever college economics class, Honors Economic Analysis at the University of Chicago (wow, what a journey it has been these past two years). There have been valid rebuttals to this style of economics written throughout the years from as diverse sources as Mark Blaug and Daron Acemoglu, but in general nearly all critiques you hear of this approach are horribly misinformed. I will ignore philosophy of science debates here and argue solely on the consequences to model outputs that common, “unrealistic” economic modelling assumptions produce.

Let me briefly summarize the essay first. Friedman’s claim is deceptively simple: the value of a scientific theory lies not in the realism of its assumptions, but in the accuracy and scope of its predictions. This is the positivist creed distilled for economics. A theory should be judged by whether it yields valid and useful predictions about phenomena not yet observed. The assumptions may be descriptively false, idealized, even wildly so, provided they capture the essential forces. Three easy derivatives:

1. Unrealistic assumptions are not a weakness but often a necessity.

2. The benchmark for theory is predictive success, not psychological plausibility.

3. Normative debates should be grounded in robust positive science

Friedman invokes physics like billiard-ball models of firms maximizing profits, or the air resistance simplifications in mechanics, which do not mirror the literal decision-making of agents or molecules, but they predict aggregate motion with accuracy, and, sharply distinguishes “positive economics” (what is) from “normative economics” (what ought), the former of which requires methodological clarity: theories must be falsifiable and assessed by empirical adequacy.

Seventy years later, economists still live inside Friedman’s methodological frame, which remains the methodological backbone of economics, because it explains what we do—construct simplified, abstract, mathematically precise models, and judge them only by their power to illuminate and predict phenomena. Without that shield, modern economics would collapse under the weight of the complexity of the issues it must analyze. With it, we continue to generate insight, guide policy, and refine theory. We are still Friedman’s children. Every grid discretization, every log-linearization, every tractable preference specification is an “unrealistic” assumption. We defend them not by realism but by empirical adequacy and policy relevance. I will address the most common critiques of the most common assumptions using this framework.

The modelling assumptions that people tend to complain about can be divided into three categories: household-side, firm-side, and market-side. Household-side criticisms tend to revolve around the assumptions that households optimize. First, a general response to this: without this assumption, there is no way to study economics at all. No field that uses ad-hoc explanations for each phenomenon, which is needed if we cannot derive behaviour using utility maximization, is a valid scientific field. Agents may not literally optimize, but they behave as-if they do, in the words of Friedman, and as we can write down a set of constraints, preferences, and factors that explain economic behaviour through optimization every time, we can analyze problems across through that lens much easier than than if we had the entire space of laws-of-motion, so to speak. It is the same as doing chemistry through the lense of models of the atom or physics through geometry, calculus, and group theory: we can describe all problems that way and thus describe the factors that create outcomes or make them deviate.

The other criticism of the household side is that there are behavioural or “complexity” factors that standard models don’t incorporate. Behavioural economics shows deviations from strict rational optimization. Yet these deviations are meaningful only against the benchmark of optimization. Behavioural models retain scarcity and often replace maximization with satisficing or reference-dependent utility, or a utility curve that isn’t strictly von Neumann-Morgenstern rational. Friedman’s standard still applies: only replace optimization when doing so improves predictive power. Further still, behavioural models often try to write down a utility function that grants the observed behaviour when optimized according to, which shows that optimization is highly complimentary towards behavioural considerations. Becker’s models of the family do precisely this: incorporate “psychological” preferences into a neoclassical model. Complexity arguments, like those concerning bounded rationality or computational limits, claim agents cannot optimize. Yet these too support, not undermine, the assumption of optimization is an as-if device. Firms behave as if they maximize profit because competition selects survivors; households behave as if they optimize because persistent sub-optimality is unsustainable in equilibrium. Furthermore, again there are ways to model these forces in an optimization framework via information constraints or costs to actions, which further shows the robustness and accuracy of the framework.

Complaints about firm behaviour strike me as more valid, as most economic models contain firms that are highly simplified, either in output or profit structure, input type, in “decision-making”, in supply linkage, or in lack of heterogeneity. Yet each of these factors may be modelled if relevant: you can have differentiated or idiosyncratic productivity types, models of internal management, you can add transaction costs or frictions to price setting or wage setting, search across labour markets or for other types of inputs, different goods aggregators, heterogenous capital types, intermediate or labour-retailer firms, network linkages, heterogenous sizes… the list goes on. Whenever one of these is discluded, it is to make the model more tracteable and focused on the key elements of the question (this is often why capital is not included in models of the business cycle). Supply chains or other network considerations are probably the most incomplete in our ability to portray them, but otherwise, common concerns like the timing of purchases of inputs or firm optimization decisions are things that we deliberately choose not to include because modelling them is irrelevant to the question. If we can make wage adjustment or price adjustment sticky, then why do we need to make labour purchasing or production sticky as well? Overpaying for labour is the same as buying too much labour at current prices. Not being able to adjust your price down to sell off stock is the same as having supply you can’t sell at current prices.

Market-clearing concerns are the most complicated to address. First, economists believe, rightly so, that equilibrium is the right concept to analyze with. When we choose to model over- or under-supply, it is through stickiness, rationing, or entry or exit of the supply to the market, like in search-and-matching models, rather than through markets not actually dis-equilibriating or behaviors not actually adjusting. Having actions change, markets clear, and deriving a price is inherently more useful to the analysis of actual behaviour and phenomena than having a price and pre-set actions and waiting for markets to clear. Furthermore, tâtonnement, the process by how this price is arrived at, is usually not relevant in dynamic models. It has not really been investigated since the 1960s since we have moved past looking at ad-hoc laws of motion moving towards static models and instead look at how dynamic models transition outside of steady state, where prices move step-by-step as behaviour adjusts in order to keep markets cleared. Even if you think a price might be not exactly so day-to-day, prices should not be so unstable that this is an issue. Indeed, I work on studying this exact phenomenon, and alongside two very notable professors at Chicago Booth and one at MIT I am investigating the issue of stability in tâtonnement for the first time in decades. What we see is that the process of adjustment essentially is a device to ensure closure and that there is robustness of the actual equilibrium across different processes of arriving to it, which says a lot about how important they are. If markets don’t equilibriate smoothly, better to derive a law of motion than impose one out-of-equilibrium. If we fear that there is something missing, we can add a friction (whether it be structural, behavioural, or informational) to the framework rather than tossing it itself.

Equilibrium is also not tautological, as it is simply a way to rule-out the explanations that are not mutually feasible, rather than an imposed result. It can allow for all sorts of inefficiencies, including that actions cannot be adjusted rapidly (like in the aforementioned supply chain example). Equilibrium then provides the engine for analyzing positive and normative implications as it maps exogenous shocks and parameters to endogenous outcomes and outputs. Economics without equilibrium and market clearing is not economics, but yapping without constraint. Like optimization and scarcity, these structural assumptions are not claims about literal reality, but devices to guarantee coherence, falsifiability, and predictive power.

You Guys Seriously Have Never Heard of Robert Lucas?

I hope readers of my blog have heard of the Lucas Critique. If not, just know that it essentially took macroeconomics away from using a priori, ad-hoc dynamics to deriving aggregate macroeconomic dynamics through microfounded behaviour. It is the single most important development in the history of the methodology of the field. However, it seems, for some reason, that post-Keynesians, Marxists, etcetera do not grasp that we do this. I have seen one particularly-ignorant account continually claim that the problem in mainstream economics is that we use thousands of a priori state equations. We actually do quite the opposite—it is the (post-)Keynesians and Marxists who do this. Massive state-equation models with zero optimization were made verboten by Lucas, and for good reason: they do not accurately predict the data, especially when the environment of the model meaningfully changes. I realize now I need to write an in-depth explainer on the evolution of the field, especially as concerns the “model complexity vs. behavioural complexity” evolution of methodology from Solow-Swann to Mean Field Games, in a separate blog post, but for now please keep this in mind: economics has long been devoted to making models where the derived behaviour captures complexity, not the many ad-hoc supply and demand equations of yore.

Subscribe (if you are rational)

Complexity Economics Is A False Idol

I am going to be bad here and promise another blog post instead of just explaining it here, but essentially, one who understands the research horizon in the field will realize that so-called complexity economics essentially moves in the wrong direction in terms of research. Instead of stochasticity and optimization generating PDEs or SDEs, we just have intricate laws-of-motion generating ODEs. Instead of backwards-facing optimization alongside forwards laws of motion, we just have “complex” economies with many forwards state equations. Instead of more complicated frictional structures affecting optimization, we have extra a priori forces imposed at-will.

No where is this more apparent than in the conflict of agent-based models versus the modern tool of choice for dealing with situations where a myriad of agents matter, the mean field game. Both are a choice for dealing with the curse of dimensionality in regular dynamic N-player games. ABMs essentially only model the forwards evolution of the economy as decision rules are imposed, as opposed to classical representative-agent models that only have the recursive optimality of behaviour (and thus blow up when dealing with the complexities of best responses in high finite dimensions). Even when ABMs have optimization, it is limited and cannot be properly forward-looking. Instead, these “complexity models” ignore equilibrium to focus on the sort of dynamics, like tâtonnement, that I dismissed earlier, and thus are incapable of meaningfully answering questions. Instead, modern heterogenous-agent economics, which has gained steam as a way to reconcile the failures of DSGE macroeconomics to predict important post-2008 phenomena, uses the mean field game to do both at once in the equilibrium context: we have recursive optimization paired with the forward evolution of the distribution of all agents, dispelling the curse of dimensionality whilst preserving both optimizational and distributional factors in economic outcomes, enabling comparative statics and impulse response calculations in a disciplined, closed system without overfitting.

The fruits of the modern, heterogenous-agent approach are clear, whilst complexity economics has not really produced anything. Time for us to move on and solve the problems of rational expectations in mean field games rather than be stuck the deserts of New Mexico.

Final Notes

Some threads to check out: Anthony Lee Zhang on complexity economics and “non-Newtonian” mathematics, and Ben Golub steelmanning Harford. There is also the excellent point made by Thomas Bourany about complexity economics versus modern heterogenous-agent methods. I am too tired to write more tonight (plus there is still two-and-a-half hours of Fuji left to race), but a final thought: modern intermediate micro and macro rely on Lagrange, Euler, Hamilton, and Bellman. We are already past Newton,3 so we really should call a spade a spade here: our opponents want an economics that is post-logical.

Anyways, please consider subscribing to join me as I begin my blogging project.

Subscribe now

My intent is to write both free and paid explainer posts, as well as a free periodical newsletter on current research and personal updates (or musings) as well as a paid periodical newsletter covering one paper in depth and/or updates on my own research.

A little background on me: I am a fourth-year undergraduate at the University of Chicago, majoring in mathematics and economics. I am also completing jointly a masters in economics at the same institution. I have been taking PhD classes in economics since the beginning of my third year, and attending the seminars and workshops since my first, so I have been behind the fluttering veil for a while now. My work is on the more technical and mathematical side of business cycle theory, especially as concerns heterogenous-agent models, generalized hazard function “reset” models, search-and-matching, mean field games, and macro-labour interactions. I prefer to keep things as analytical as possible before taking on the computational or statistical, and I never take shortcuts when it comes to the latter.

I used to big on Twitter on my old account and blog at In The Aggregate, as well as have a podcast; but no longer. My current X account is set up to be a burner that I can abandon easily, as it is often a personal account, but rest assured that I am more committed to this blog.