Price Theory and the Price Level
More than you ever wanted to know about the price level, price indices, price the
From Covid lockdowns to recent discussions of tariffs, there have been a lot of discussions about the price level and inflation. At times, these discussions focus on theoretical issues and different theories of price level determination. At other times, these discussions tend to focus on the measurement of the price level. Unfortunately, this commentary occasionally runs into a problem that I rarely see addressed. That problem is that theoretical arguments about price level determination necessarily assume a theoretically correct measurement of the price level. However, in reality, our measures of the price level are flawed.
This isn’t some conspiracy theory about the government purposely misleading the public or adjusting the measurement of some particular price index to get a desired result. Rather, my claim is rooted in price theory. Empirical analysis requires the use of a price index. A price index is an estimate or approximation of what economists mean by the term “price level.” The proper measure of the price level can be determined by price theory. However, a number of practical considerations make it difficult for our estimates of these price indices to perfectly replicate the theoretical measure. At best, we can get approximations of the theoretical ideal.
This means that in some sense, all of our price indices are flawed. Some of these flaws are more important than others. Nonetheless, all is not lost. Understanding the precise ways in which measurement might differ from theory can be useful in interpreting the data.
Measurement Requires Price Theory
Let’s start with this. People like to draw clear distinctions between theory and empirics. The current trend in the economics profession is not only to draw a clear distinction, but also to elevate empirical work to a higher status than theory. There is a “let the data speak” mentality prevalent in the profession. However, data do not speak. For data to be useful, the data must be used to test a hypothesis. In order to have a hypothesis to test, you need some theory that generates that hypothesis. Furthermore, you might want to be aware of alternative theories that might generate the same hypothesis, lest you might incorrectly conclude that your chosen hypothesis is the only explanation.
However, the point is deeper than that. If you are going to do good empirical work, you need to make sure that you are measuring things properly. To measure things properly, you also need theory. This is where price theory is crucial for empirical work. Price theory guides measurement. When you use data to test a hypothesis, improper measurement might cause you to incorrectly reject a hypothesis.
One example of this that deserves more attention is William Barnett’s work on monetary aggregation. There is a significant literature in macroeconomics that calls into question not only the value of using monetary aggregates in the conduct of monetary policy and the evaluation thereof, but also rejects conventional explanations of price level and nominal income determination. What Barnett showed is that economic theory could be used to derive monetary aggregates that reflected the differences in the flow of monetary services provided by different types of money. The conventional way of measuring monetary aggregates to that point (and which continues today) is to simply add up all the things that we think constitute “money.” Barnett’s approach demonstrated that these existing aggregates only make sense when every component of the aggregate is a perfect substitute for all other components. Basic common sense tells us that is not true.
Barnett discussed the empirical significance of his approach is an easily accessible speech. Subsequent empirical work by people like me (shameless plug) and my former colleague, Mike Belongia, shows that puzzling empirical results aren’t so puzzling when you measure things properly.
I think we can provide some similar lessons when we think about the price level.
A Definition
Let’s start with what economists really mean when they talk about the price level. The term price level is meant to measure exactly what it says, the general level of prices. In other words, it is meant to measure what we might think of as a weighted average of prices. However, even this definition is imprecise. For example, suppose that all prices (including wages) rise by 2% from one year to the next. This would change the price level, but it wouldn’t make anyone better off. Real income would be the same. Relative prices would not change. People would consume the same basket of goods as before. That seems pretty consistent with what we are thinking about when we think about the price level. The real world is messy though. Things don’t typically happen like that.
In reality, even when prices are changing on average, some prices are going up by more than others. Some prices might even be falling. In this case, a simple weighted average of prices might not convey what we think it does. The reason is that even if one’s real income doesn’t change, relative price changes are going to change the choices people make and the pattern of consumption. Different patterns of consumption correspond with different levels of utility, or satisfaction, with the choices made. Now we are starting to depart from what we typically think of when we think about the price level.
To understand what I mean, think about this as follows. When the general public thinks of the price level or looks at a price index, they see this as a measure of the cost of living. In fact, this is the correct theoretical interpretation. Policymakers recognize this and often adjust spending in particular programs in conjunction with the changes in the price index in an attempt to correct for changes in the cost of living. But suppose that the price level rises by a modest amount and relative prices change such that the consumer is actually a little better off than before. Would the rise in prices really measure the change cost of living? No, it would overestimate the rise in the cost of living. More likely, imagine that the price level rises modestly and relative prices change such that the individual is a little worse off. Would the rise in prices really measure the change in the cost of living? No, it would underestimate the change in the cost of living.
In other words, if the price level is to measure the cost of living, then what we need is not just any weighted average of prices, but rather a weighted average of prices associated with a particular level of utility. In other words, consumers are often indifferent (based purely on their preferences) between different bundles of goods. When relative prices change, consumers choose different bundles. Some make them better off. Some make them worse off. Some leave them indifferent between their current and previous bundle. The objective of any attempt to measure the price level should be to adjust the weights of the index in response to relative price changes such that utility remains constant. In the event that there are no relative price changes, the weights should remain the same.
Okay, that sounds great. It also sounds impossible. After all, utility is an abstract concept. We can’t really measure it. How can we adjust the weights in response to something that we cannot measure?
The answer, as you might expect, is to rely on the guidance of price theory.
Using Theory to Inform Measurement
In price theory, we often frame the basic consumer problem as follows. A consumer has a particular amount of income and is able to observe the prices of the goods that he or she would like to purchase. Given income and prices, there are various combinations of those goods that are affordable. This is the consumer’s budget constraint. But which of these various combinations do they choose? Well, that is determined by their preferences. They choose the bundle of goods that maximizes their utility. Economists can use this sort of framework to derive the demand functions for each of the various goods that consumers might want to consume. These demand functions determine the amount of the good that the consumer demands, given the prices of those goods and the consumer’s income. We call these Marshallian demand curves.
An alternative way of thinking about the consumer’s problem, however, is as follows. Suppose that the consumer wants to achieve a particular level of utility. There are various bundles of consumption goods that can provide the same level of utility. The consumer’s objective in this case would be to choose the bundle that minimizes the cost of achieving that level of utility. Economists can use this sort of framework to derive demand functions for each of the various goods consumer might want to consumer, but this time as a function of prices and the level of utility. We call these Hicksian demand curves.
Let’s stop and think about these Hicksian demand curves. They measure the quantity that consumers wants to buy of a particular good, holding utility constant. Isn’t this exactly what we are looking for? As a matter of fact it is.
Consider a simple two-good example. Suppose that there are only two goods in the world, apples and bananas. We can use the cost-minimization approach to the consumer’s problem to derive demand curves for apples and bananas.
The total expenditure on apples and bananas can be written as
Where E is the total expenditure, P_a is the price of apples, Q_a is the quantity of apples, P_b is the price of bananas, and Q_b is the quantity of bananas. This equation is true by definition. If there are only two goods and you add up the dollar amount spent on both goods, you get the total dollar expenditure.
However, it is important to realize that the demand for each of these goods depends on the price of the goods. Given the Hicksian demand functions for each of these goods, we can write the consumer’s total expenditure as
Where we have replaced the quantities with the demand functions. What this shows us is that, for a given level of utility, if we know the prices of the goods, we know how much of each good the consumer will buy. Thus, given the prices of the goods, we can determine the total expenditure of the consumer, holding that consumer’s utility constant. In fact, we can plug in any price of apples or bananas that we wish and it will give us the money cost of the corresponding basket of goods without changing the utility that the consumer receives. That is precisely what we are trying to measure when we measure the cost of living, or the price level!
Think about how this expenditure function compares to our practical discussion of what we are trying to measure. We said that the price level should be thought of as the weighted average of prices in which the weights adjust such that the consumer is neither better off or worse off. As I have shown, the expenditure function is a weighted average of prices, where the weights are the quantities that would be consumed, given those prices, without influencing the consumer’s well-being.
The fact that we cannot literally measure utility isn’t of first-order importance (it is of some importance as I will explain below). We have a general guide to measure the price level that is derived explicitly from price theory.
From Theory to Measurement
We now need to move from theory to measurement. How can we construct a price index that approximates what we mean by the price level?
Let’s look at our expenditure function. What we are trying to track is the change in the monetary cost of a constant utility basket of goods. We could start by simply tracking money expenditures over time. In a world in which prices are increasing by the same percentage every period, this would be correct. The reason is that equiproportional changes in prices leave relative prices, and thus quantity demand unchanged. Since prices are increasing by the same percentage, money expenditures will increase by the same percentage as well. Since the basket of goods hasn’t changed, neither has utility. Thus, the constant-utility basket of goods requires more money to purchase. The price level went up.
In reality, however, it is very unlikely that all prices are increasing by equal percentages over time — and even if they were, measuring changes in the price level would be trivial. From one period to the next, prices might be rising or falling on average, while at the same time relative prices are changing. If we are tracking changes in expenditures over time, and relative prices are changing, then the quantity demanded will also be changing. To the extent to which the changes in consumption bundles correspond to different levels of utility, this poses a problem for measurement.
Nonetheless, from our expenditure function, we can say the following. Suppose that there is a change in the price of apples. In that case, the first-order effect is that total expenditures will rise proportional to the quantity of apples. However, there is also an offsetting second-order effect. The rise in the relative price of apples will result in a decline in the quantity demanded for apples.
What this means is that if we want to develop a price index to measure the price level, we could do the following. We could track expenditures over time, but hold the quantities of each good consumed constant. This price index is a first-order approximation of our expenditure function. As long as relative price changes are small, the second-order effects will be sufficiently small that this is good enough.
Nonetheless, this does create an obvious flaw. The expenditure function has a positive first-order effect and a negative second-order effect. This means that the expenditure function is concave in the price of any one particular good. (Picture a line that is increasing at a decreasing rate.) Our price index is a linear approximation of this curve. That means that our measured price index is tangent to the true expenditure function. As a result, the bigger the change in one particular relative price, the bigger the bias in the price index compared to the theoretically correct price level.
This flaw is known as substitution bias. It gets this name because when the price of a particular good goes up, the quantity demanded of the good goes down. (People substitute away from the higher priced good.) This attenuates the increase in the price level, properly measured, but not in our price index. This is where the ability to measure utility matters. Ideally, we would be able to measure the Hicksian demand curves to adjust the weights of the price index each period.
Nonetheless, our first-order approximation isn’t bad.
Theory When There is Dynamic Decision-Making
Substitution bias is a well-known issue with commonly used price indices. However, there is a larger problem that is often overlooked: the role of dynamic decision-making.
In my previous example, I assumed that there are only two goods: apples and bananas. An additional, yet explicit, assumption is that there was a single period for decision-making. Thus, the expenditure function is a function of the current prices of the two goods.
However, imagine how things change if we introduce a second period to our model. Suppose that decision-makers live for two periods and consume apples and bananas in each period. By expanding the model to include a second period, our expenditure function would now include the current price for apples in period one, the current price for apples in period two, the current price for bananas in period one, and the current price of bananas in period two.
It follows that the theoretically correct measure of the price level includes the current prices of all current and future consumption. As a practical matter, this is difficult. We do not have the current prices of future consumption for every type of consumption good. (We do have futures markets, but not for all goods.) However, we do have proxies. For example, the relative price of apples today in terms of apples tomorrow is measured by the real interest rate. Furthermore, other asset prices might help to supplement the current prices that we do have in order to get a better estimate of the price level.
Asset prices, however, are missing from the most commonly used price indices. These price indices tend to focus on current consumption service flows or current output prices.
This isn’t merely a theoretical curiosity. There are practical implications here that are important when it comes to testing theories of price level determination and conducting policy when price indices are missing the asset prices that price theory tells us should be included.
The Determination of the Price Level, the Non-Neutrality of Money, and the Problem of Taking the Theory to the Data
Given this big, broad overview, we are now prepared to discuss issues with using our existing price indices to test economic theory.
Especially in popular writing, economists tend to equate the price level with the price index (I’m guilty of this sometimes as well) when the price index is simply the empirical approximation fo the price level. However, the distinction is important.
As economists, we should have a theory of how the price level is determined. In other words, we should be able to make predictions about what causes the general level of prices to go up or go down.
Price theory is about relative prices. Nonetheless, price theory can provide us with a theory of the price level: it is just the relative price of money.
Consider a gold standard. Under the gold standard, the unit of account is defined as a particular quantity of gold. Suppose the unit of account is the dollar and the dollar is defined as 1/35 of an ounce of gold. This implies that the price of one ounce of gold is $35. And yet, there is a market for gold. It follows all dollar prices must adjust fluctuations in the supply and demand for gold such that the market clears. In other words, the supply and demand for gold determines the price level.
Consider our current system. The central bank controls the supply of the money base. All other types of money are claims to the dollars that the central bank creates. The nominal price of a dollar is fixed at (you guessed it) one dollar. There is a supply and demand for the monetary base. All dollar prices have to adjust to clear the market. In other words, the supply and demand for central bank money determines the price level in our current system.
Some might quarrel with this idea, but that is not the point. What I am arguing is that these are theories of price level determination. Before accepting or dismissing these theories, we might want to put them to the empirical test.
In order to test the theory, we need to clearly understand the predictions of the theory. We also need some empirical estimate of the price level. This is our price index. Finally, we need to understand the limitations of such tests, given what we know about the limitations of the price index.
An Example
Let’s think about a particular example. Suppose that the monetary base has been constant for some time. Now, imagine that the central bank increases the monetary base by 10 percent. What would we expect to happen theoretically? What would we expect to observe empirically?
Assuming that everything else is constant, the degree to which the price level would rise depends on the shape of the money demand curve. If money demand is relatively inelastic, then the price level would be expected to rise by more than 10 percent. If money demand is relatively elastic, the price level would be expected to rise by less than 10 percent. If money demand is unit elastic, the price level would be expected to rise by exactly 10 percent.
Empirically, of course, everything else is not held constant. A lot of other things are simultaneously happening. Maybe that is an unexpectedly good harvest of apples in Washington. The relative price of apples would decline. Given the linear approximation of the expenditure function, this relative price change would result in a measured decline in the price index, all else equal. Now imagine that there are many relative prices changes going on. Assuming that our theory of price level determination is correct, all of these relative price changes are going to introduce measured changes in the price index separate from the broader effect of the increase in the money supply.
To make matters even worse, money tends to be non-neutral in the short-run. What this means is that monetary policy tends to affect relative prices in the short-run and thereby have real effects of economic activity and not just the purchasing power of money. By inducing relative price effects, this will tend to produce changes in the measured price index as well.
It is here that we have perhaps the biggest difficulty with limitations of our existing price indices. There is a commonly held view that the way in which monetary is transmitted through the economy is via the change in one crucial relative price: the real interest rate. Because the real interest rate — and asset prices more generally — are excluded from our commonly used price indices, these price indices will tend to systematically overestimate or underestimate changes in the price level, depending on the direction of policy. For example, when the money supply increasing, the typical macroeconomic theory would imply that the real interest rate falls and asset prices rise. The prices of current consumption flows might be slower to adjust. If so, the measured price index will tend to underestimate the true increase in the price level because the index excludes asset prices.
At the very least, this introduces noise into the empirical analysis that might make it more difficult to evaluate a particular hypothesis.
More seriously, the link between the limitations of our price indices and monetary policy itself might suggest that policymakers are more prone to policy errors. Policymakers might observe a sluggish response of the price index and assume that they need more expansionary policy when, in fact, the price index is underestimating the price level.
This is one reason why clean identification matters for macroeconomic questions as much as it does for price theoretic questions. For example, if I am correct that the relative price of the medium of account is what determines the price level, it might help to find some sort of natural experiment to test this hypothesis. FDR’s devaluation of the dollar is one such experiment. Given the supply and demand for gold, there is some real price of gold that clears the market. For the given real price that clears the market, a dramatic increase in the nominal price of gold (i.e., the devaluation of the dollar) would result in a similarly dramatic increase in the price level (the real price hasn’t changed). Given the magnitude of the devaluation (from approximately $20 to $35), the measured change in the price index should actually approximate the theoretically correct change. The reason is that the large magnitude should mask any noise from changes in relative prices and measured changes in the price index that are independent of the change in policy. This is precisely what was observed.
Supply Shocks and the Price Level
Things get even more complicated when we start to think about supply shocks more generally. For example, consider the following two scenarios. In one scenario, the supply of apples and bananas are distributed evenly across the two periods. In the second scenario, there are very few apples available in the first period and an abundance of apples available in the second period. We can think of the first scenario as a normal state of affairs. We can think about the second scenario as similar to a supply shock.
Consider how the price level would differ between these scenarios. Recall that the properly measured price level would be a weighted average of the current prices of both current and future consumption of apples and bananas. All else being equal, the major change in scenario two is that the current price of apples would rise and the current price of future apples would decline relative to the first scenario. Notice, however, that the theoretically correct measure of the price level would therefore have offsetting effects. In other words, using a price index that approximated the theoretically correct measure of the price level would show only a modest observed effect from supply shocks.
However, as a practical matter, the price indices used to approximate the price level are not able to include the current prices of future consumption and neglect other prices that could proxy for such effects, like asset prices. Instead, many of the commonly used price indices only use the current prices of consumption service flows. This necessarily implies that supply shocks will have larger effects on these price indices than would show up in a theoretically correct measure.
Accounting Versus Economics
Finally, to make matters worse, extensive discussions on how price indices are calculated tend to lead a lot of observers astray. A great deal of commentary on the monthly fluctuations in the consumer price index often compound the problems inherent in measurement by substituting accounting for economics. A lot of popular commentary will state things like “food prices drive higher inflation” when what they really mean is that if you break down the index into its components, food prices went up by more than others. Note, however, that is a relative price change!
This is one reason why economists sometimes get aggravated by discussions surrounding inflation. We have very clear hypotheses about what explains the general trend in the price level. Accounting exercises confuse measurement with theory itself.
Conclusion
Now let’s return to my original point. What makes this issue complicated is that economists often use the terms “price level” and “price index” interchangeably (at least in arguments for the popular press and in interviews). Economists can be forgiven for this. Price indices are meant to be empirical representations of the theoretical concept of the price level and price theory is used to inform the construction of price indices. Price theory also provides predictions about how and why the price level changes over time. Since price indices are imperfect empirical representations of the price level, it can be hard to test these theories and predictions when so much is going on simultaneously, excess (or deficient) money growth is relatively small, and when money has non-neutral effects.
What makes matters worse is that the limitations of price indices are not generally well-understood. The general public often criticizes measured prices indices as failing to capture the “true” cost of living. One needn’t rely on conspiracy theories about corrupt government agencies to make such claims. Even assuming the best of intentions, one can get measures of the price level that can potentially deviate significantly from the theoretical construct. Although substitution bias is a second-order effect, there is some reason to believe that the exclusion of asset prices might be an issue of the first-order. If so, there might be some validity to the complaints about our commonly used price indices. Whether or not this matters largely depends on how all of this washes out in the long run.
What is most important is to remember that proper measurement requires price theory. Understanding the limitations on our measurement and the potential problems that can arise are also important. But again, those are lessons that can only be drawn from understanding the deviations between the theoretical ideal from price theory and practical measurement.