The question of how, or whether, nominal variables such as inflation or "aggregate demand" affect real economic variables such as unemployment or real GDP has been one of the central concerns of macroeconomics since the field of study emerged as an independent discipline. A cursory read of any business newspaper will give you numerous examples of assumed links between the two kinds of variables. 

Here are some examples:

The often assumed link between "more spending" and "more production" is an example. Spending is fundamentally a nominal variable as it's measured in dollars (or some other currency), while production is a real variable (such as more buildings built, more roads paved, more widgets manufactured, *et cetera*). When people assert that more spending leads to more production, they are working with an implicit model which links nominal variables to real variables.

Monetary policy, unlike fiscal policy, can only directly affect nominal variables. Any mechanism which attributes real effects to monetary policy therefore has to somehow bridge the gap between nominal and real variables. There are many different ways to do this, but the most popular by far has been the so-called *Phillips curve* and its variations.

When countries are accused of "currency manipulation", this is an implicit attribution of real effects to a policy that affects nominal variables. What matters for the terms of trade between two countries is the *real exchange rate* between them. If one unit of country A's currency, say A, is worth e units of country B's currency, say B; then the units of e are A/B and to make it into a dimensionless quantity we have to consider e p\_B/p\_A where p\_A, p\_B denote the price levels in countries A and B respectively. This quantity is what is often referred to as the real exchange rate. If you think about it, you'll realize this quantity is really what matters in whether the exports of country A to country B are relatively cheap or expensive. Unfortunately, there's no apparent reason that a policy that affects the nominal variable e also affects the real variable e p\_B/p\_A since it can simply be offset by a movement in p\_A.

—If the above examples don't make much sense to you, they will after I explain the distinction between the two types of variables.

## Nominal variables

"Nominal" in macroeconomics is used to describe a quantity that is measured in units of some currency rather than in "real units" such as mass, length, size, *et cetera*. If a company has produced a million mobile phones with given specifications, that's a real variable. If those mobile phones sold for a billion dollars, that's a nominal variable. In particular, *relative prices between goods* are always real variables. For instance, I can say that one iPhone 12 is currently worth around 15 barrels of Brent crude oil, and this claim is independent of any currency unit. One of the most commonly used nominal variables in macroeconomics is the "price level", which is a rather nebulous quantity that's defined to be an appropriately weighted average of all the prices in an economy, with weights changing depending on context.

The reason for this distinction is that there is a "firewall" between nominal and real variables. If we doubled all dollar prices of everything in the world (including salaries, debt contracts, *et cetera*) simultaneously and also replace each dollar bill outstanding with two identical dollar bills, there is a sense in which nothing would change, since all goods and services would still have the same relative price as they did before. In some sense it would be equivalent to what happens when a public company does a [stock split](https://en.wikipedia.org/wiki/Stock_split). This argument suggests that we need some kind of mechanism to get changes in nominal variables, such as the money supply or nominal spending, to have an effect on real variables such as relative prices between goods.

There is an easy way to do this which is still popular, and this is what I will focus on in this essay. The easy way is to assume that for whatever reason, nominal prices are *sticky*. For instance, debt and employment contracts might be in units of dollars and not indexed to inflation, and there might be some "menu costs" which discourage businesses from changing the price of the goods and services they sell too frequently. If it is costly to change some prices quoted in nominal units and not others, or even if it's costly to change all of them but the cost profiles vary across goods and services, there's an obvious mechanism for a change in nominal variables such as total spending to influence real variables.

The most naive model of this kind is the [AD-AS model](https://en.wikipedia.org/wiki/AD%E2%80%93AS_model), often the first model people are taught when learning macroeconomics. The idea of the model is that for whatever reason the real output produced in an economy increases as the price level increases. It could be because wages and salaries are sticky and therefore the higher price level allows companies to hire more workers, for example. This relationship gives us the "aggregate supply curve", which we couple with an "aggregate demand curve" that represents the total amount of spending in the economy. For example, at constant nominal GDP (nominal GDP is the total value *in units of currency* of the production of an economy in a given time period), it might look like

\textrm{price level} \times \textrm{real GDP} = \textrm{nominal GDP} 

These two curves intersect somewhere, and we can judge the effects of more or less spending on the economy by changing the nominal GDP defining the aggregate demand curve and consequently moving the curve around on the graph.

[Here is a sample graph of the AD-AS model](https://i.pinimg.com/originals/f4/1e/4d/f41e4d056a09c8dc4fb2875c699cb752.jpg). In this case, an increase in spending from AD to AD2 leads to both an increase in the price level PL and an increase in real GDP.

This model is what people outside academic macroeconomics often have in mind, even if implicitly, when they think about the effects of nominal shocks on real variables. However, this model has serious shortcomings. For starters, without explicit parameters describing the shape of these two curves it is actually incapable of making quantitative predictions. It is a static model, so it can't handle questions such as "what happens today if the government promises to increase spending a year from now?". Most importantly, however, the ingredients going into the model are suspicious. For instance, instead of specifying an explicit way in which prices are sticky and then deriving a theoretical relationship between prices and real variables, we simply stipulate the aggregate supply relationship out of thin air.

For now, however, let's switch from the theoretical to the empirical side of the issue. What relationship has *actually* been observed between, for example, inflation and unemployment?

## The Phillips curve

In 1958, William Phillips published a paper titled *The Relation between Unemployment and the Rate of Change of Money Wage Rate*, which lay the groundwork for what would later become the [Phillips curve](https://en.wikipedia.org/wiki/Phillips_curve). At first, the Phillips curve was an empirical finding which suggested an inverse relationship between the inflation rate—in other words the rate of change of the price level in an economy—and the unemployment rate. ([Here is the original Phillips curve identified by William Phillips](https://d1yhils6iwh5l5.cloudfront.net/charts/resized/62621/large/zcha.png), wherein he looked at the rate of change of nominal wages instead of nominal prices as a whole. This doesn't, however, alter the basic findings.)

The Phillips curve is somewhat different from the aggregate supply curve of an AD-AS model, since it links unemployment not to the price level directly but to its *rate of change*, i.e. to inflation. There's a simple way to modify the model so that it fits the Phillips curve rather than the previous aggregate supply relationship we postulated, however, and broadly the conclusions we draw from it remain similar.

Throughout the 1960s, macroeconomists treated the Phillips curve as an established and fixed relationship between inflation and unemployment which could be exploited by government policymakers. Depending on the economic conditions, government could adjust aggregate demand so as to move along the Phillips curve, trading off inflation against unemployment as they saw fit. This approach appeared to work well for a decade or so, but critics warned that we couldn't trust such an empirical regularity to be robust under government attempts to exploit it. In 1968, Milton Friedman delivered a now famous AEA presidential address titled [*The Role of Monetary Policy*](https://www.aeaweb.org/aer/top20/58.1.1-17.pdf) in which he warned, for theoretical reasons, that the Phillips curve in its existing form would soon break down. His argument was simple: he thought that the reason the Phillips curve worked was because of the "money illusion", the fact that under normal conditions people can't tell if an increase in their income is nominal or real, so they can be fooled that a nominal increase in their income is in fact a real increase and increase their work effort as a consequence. However, once people caught on that the nominal increase is indeed nominal and part of a general increase in the price level, this effect would disappear.

In formal terms, if u denotes the rate of unemployment and \pi denotes the rate of inflation, Friedman believed that the relationship between the two variables looked less like u = a - b \pi for constants a, b with b > 0, and more like u = a - b (\pi - \pi^e) where \pi^e denoted people's expectations of how much inflation there would be. Before people's expectations of inflation \pi^e adjust to the new higher inflation rate, the relationship looks like an inverse relationship between u and \pi since \pi^e appears to be constant, but in the long run \pi^e adjusts to equal \pi on average and the naive relationship breaks down.

Friedman's forecast proved prescient as the United States headed into what is today called "stagflation", a situation in which there is both high inflation and high unemployment. This was inconsistent with the naive Phillips curve, but consistent with Friedman's interpretation of it, which is today called the expectations-augmented Phillips curve. The general point that observed correlations in macroeconomic time series may not be stable under attempts by policymakers to manipulate them is now called the [Lucas critique](https://en.wikipedia.org/wiki/Lucas_critique), and due to the unreliability of this kind of analysis, the field of macroeconomics has moved towards looking for "microfoundational models". These are game-theoretic models containing agents who are optimizing some utility function and interacting with one another. The objective of considering these kinds of models is to build in features into these models which are believed to be robust rather than doing vector autoregressions or other kinds of naive time-series analyses which treat historical correlation structures as robust when they are likely not so. 

## The modern Phillips curve

In the 1990s, a research program in macroeconomics called New Keynesian macroeconomics emerged. The goal of this research program was to build microfoundational models which would not be vulnerable to the Lucas critique while still retaining the fundamental linkage provided by the Phillips curve between nominal and real variables. Instead of postulating an arbitrary relationship between inflation and unemployment or fitting curves to historical scatter plots, they started with fundamental market imperfections such as sticky prices and derived the Phillips curve that would result from those imperfections from first principles. One exposition on this approach based on Calvo pricing [can be found here](https://www.karlwhelan.com/MAMacro/nkpc-details.pdf). It led to the "New Keynesian Phillips curve", which is of the form 

c u\_t = \beta \mathbb E[\pi\_{t+1} | \mathcal F\_t] - \pi\_t 

for some parameters 0 < \beta < 1 and c > 0. Often we have \beta \approx 1, so roughly speaking this version of the Phillips curve links unemployment to the rate of change of inflation, so to the *acceleration* of the price level rather than its rate of change. This variant of the Phillips curve is the most popular one in academic macroeconomics today, and it features prominently in a wide range of New Keynesian models called DSGE (dynamic stochastic general equilibrium) models. 

The main problem all of the different variants of the Phillips curves suffer from is that they have very poor fit with data: For instance, [here is a scatter plot of US inflation and unemployment from 1948 to 2008](https://i.gyazo.com/c03ce703a20fa5d2528f4770558544f6.png).

In case you thought that poor fit is just the naive Phillips curve performing badly, [here is a scatter plot of the New Keynesian Phillips curve](https://i.ibb.co/gm2Xb5k/nk-phillips.png) instead (again for the US), from 1948 to 2019. Where, exactly, is the Phillips curve?

Remarkably, this poor empirical performance hasn't caused much of a dent in the reputation of the Phillips curve. The frequently given reason is that the poor performance may simply be an identification problem arising from endogeneity. 

To explain this problem, recall that the New Keynesian Phillips curve states that c u\_t = \beta \pi\_{t+1} - \pi\_t + \varepsilon\_{t+1} where \mathbb E[\varepsilon\_{t+1} | \mathcal F\_t ] = 0. This *does not* mean that we can identify this curve by running a regression of the left hand side on the right hand side, because we actually don't know that the error term \varepsilon is exogenous, i.e. that it does not depend on the trajectory of inflation. For example, suppose that the objective of the central bank of a country is to minimize a policy objective of the form

\mathbb E \left[ \sum\_{k=t}^{T} (u\_k^2 + \lambda \pi\_k^2) \mid \mathcal F\_t \right] 

This would roughly fit the "dual mandate" of the Federal Reserve: they want to keep both unemployment and inflation low, with some implicit tradeoff constant \lambda > 0 between their two objectives. Furthermore, suppose that the central bank has trouble precommitting to monetary policy in advance, and they are constrained in their choices by the New Keynesian Phillips curve. In this case, since the central bank is unable to precommit to anything, we can differentiate the policy objective at time t with respect to pi\_t to see that the central bank will choose inflation such that

u\_t \frac{\partial u\_t}{\partial \pi\_t} + \lambda \pi\_t = 0 

or, substituting the Phillips curve relation for u\_t,

u\_t = c \lambda \pi\_t 

In other words, even though this model in fact satisfies the New Keynesian Phillips curve relationship exactly, if we simply look at a scatter plot of unemployment versus inflation in equilibrium, we will just end up identifying the relationship u\_t = c \lambda \pi\_t which has nothing to do with the Phillips curve—it actually slopes in the wrong direction! If \lambda \approx 0 and the central bank cares much more about unemployment than it cares about inflation, we will end up identifying a vertical "Phillips curve". It's not hard to play with this toy example to produce pretty much any sample correlation between inflation and unemployment that you want without ever violating the New Keynesian Phillips curve relationship.

The debate over whether the identification problem explanation is simply a cop-out trying to save a doomed model or if it's a legitimate explanation which shows that naive curve-fitting is insufficient to prove that the Phillips curve does not exist, has raged back and forth, and right now there seems to be no consensus on the issue. Regardless of the absence of consensus, however, the New Keynesian Phillips curve continues to be ubiquitous in contemporary DSGE models.

In some sense, we're now in the opposite situation compared to where we were in 1958. In 1958, we had an empirical regularity without any actual explanation of it; today, we have plenty of explanations and yet the original motivating empirical regularity has disappeared.

## Forecasts

The endogeneity critique above is in some sense too powerful, and if we take it seriously then it means our attempts to test the Phillips curve are doomed unless we can get the Federal Reserve to randomize monetary policy just so we can see what happens. In the absence of such a natural experiment, however, we have to be content with what we have available to us.

One way to test the Phillips curve is to forecast the future correlation between inflation and unemployment, or the rate of change of inflation and unemployment, corresponding to the naive variant and the New Keynesian variant of the Phillips curve respectively. Here are the corresponding two questions:

Another way is to ask if current unemployment gives us information about current inflation as follows: Suppose we run two linear regressions \pi\_t = \alpha + \beta \pi\_{t-1} + \varepsilon\_t and \pi\_t = \alpha + \beta\_1 \pi\_{t-1} + \beta\_2 u\_t + \varepsilon\_t. While the second regression is guaranteed to have a higher R^2 than the first, it will only have a higher [adjusted](https://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2) R^2 if the added variable u\_t actually "adds information" that is more significant than chance. Therefore, we can ask if over some time horizon whether this condition will be met or not as a way to proxy for whether the Phillips curve is actually useful for forecasting inflation dynamics. The corresponding question can be found below.

## Conclusion

The Phillips curve remains the most popular way to link nominal variables to real variables in macroeconomic models, and the fact that it seems to have disappeared from the data has not bothered people as much as perhaps it should have. While the endogeneity defense can be used to argue that the Phillips curve is not doomed yet, the fact that we have moved from the clean correlation that Phillips observed in 1958 to an increasingly desperate search for identification strategies suggests that even if the Phillips curve is still here, the way we ought to think about it has definitely changed in the past half century.