The Importance of Units in Economics

One basic logical principle that gets emphasized in the physical sciences, but – oddly – not in economics, or even in elementary math, is the importance of carrying through the units in an analysis. Sometimes this does pop up in limited form – for example the occasional discussion about whether certain variables should be interpreted as a stock (a plain unit) or a flow (a unit per period of time) – but largely the question is ignored. Reading through models in economics, units are rarely specified, and even more rarely do they matter in any meaningful way. Variables are simply bare scalars, able to be added or subtracted without bothering about the meaning of the sum.

Rather than mounting a comprehensive argument, which I’ll do at some point in the future, here I’ll just give a few prominent examples of how to do this well, and how to do it poorly, which should indicate most of the important points. Most of the good examples, unfortunately, are usually not taught this way. But the following section indicates a few important concepts in economics that can and should be taught this way, followed by a very widespread example of the incoherence that results from disregarding units.

Good Examples of Unit Analysis

Revenue

Starting very simply, revenue is defined as R=PQ: Revenue equals price times quantity. But for this to be a coherent definition, we need units on each of these. What are these numbers counting, exactly?

Quantity, obviously, is a physical unit. Apples, or computers, or pounds of grain, or some other homogenous physical unit. Price might be thought to be a dollar-unit, since prices are expressed in dollars. But that would give us revenue in dollar-item terms, which makes little sense. Price in this context has a dollar-per-item unit. Multiplying this by quantity, with an item unit, produces revenue, which has a dollar unit. Writing this out we have:

$$ R$ = P \frac{$}{i} \times Qi $$

Writing it this way might seem otiose if a simple unitless R=PQ does just fine. But it does encourage a certain type of discipline, namely, making explicit what you can’t add together. One important principle of unit analysis is that you can’t add inconsistent units – or, at least, you can’t sum the scalars together. This is particularly important when defining aggregates, and an important reason why economics has to be conducted in value terms, not physical terms.

For the revenue equation, this is important because each separate commodity is its own unit. There is no economically meaningful generic “item” unit; there are millions and billions of separate units in an economic system. Suppose we have five apples and three jars of jam. What do we have? Simply five apples and three jars of jam. If these are regarded as separate commodities, the quantities cannot be added together to give us eight generic items. Hence, aggregation has to be done in value terms, which is to say, with consistent units.

In addition to this, writing the revenue equation with units also shows us the steps we have to take in order to meaningfully aggregate in value terms. Not only are the quantity units inconsistent across various items, so are the price units. From the perspective of a consumer, spending money on single items, price units are simply dollars, able to be summed into a dollar-unit budget. But from the perspective of the economist, looking at sticker prices rather than single exchanges, prices take the unit of dollars-per-item, which can no more be summed together across various items than item-quantities themselves. Any summation into an aggregate must first eliminate the item units, as the revenue equation does. Only then can revenues be summed into a GDP (or some other) aggregate.

The Equation of Exchange

The equation of exchange, MV=Py, states that income (the quantity of money times the velocity of money) necessarily equals output (the price level times real output). The units on some of these quantities are somewhat opaque, and usually ignored, so it’ll be worth going through them explicitly.

First let’s simplify by combining rewriting MV=Y, based on the definition y=Y/P: Real income is defined as nominal output divided by the price level. M is obvious enough: the quantity of money is measured in dollars. We’ll skip velocity and leave it for last, since that’s also how it’s calculated numerically. Nominal output Y is often written in value terms, with a dollar unit, but more properly GDP is a flow: GDP is a yearly measure, so the units on Y will be $/t.

From this it follows that the unit on velocity V is 1/t, so that both sides together are in $/t units. This is not an intuitive unit, hence sometimes the difficulty explaining it, and perhaps the reluctance of some economists to use it. The best way, perhaps, is to think of the product MV as having $/t units, and thus expressing the flow of the quantity of money in a given amount of time. Velocity, therefore – dividing that flow by M – expresses the rate at which one dollar flows through the economy; the number of times it changes hands in the period measured by GDP.

So far so good, so let’s expand the right-hand side back into Py. The price level P, and index numbers more broadly, are sometimes taught as unitless, which would make make the expansion trivial. This isn’t quite right though. Properly speaking, money units are dated, because the value of the money changes. The units on M are not dollars simpliciter, but 2019 dollars (or dollars in whatever year the analysis is happening in). Nominal GDP is also expressed in 2019 dollars per year. Similar to the problem of summing inconsistent item units, this also cautions us against summing inconsistently dated values. A 5¢ Coke price from 1940 can’t be meaningfully added to a $2 Coke today. To set up the problem as \( 0.05 \frac{$}{Coke} + 2\frac{$}{Coke} = 2.05\frac{$}{Coke} \) is invalid because the dollar units, rather than the item units, are inconsistent. Expressed properly, \( 0.05 \frac{$_{1940}}{Coke} + 2\frac{$_{2019}}{Coke} \) can’t be simplified further without another equation.

That other equation is the point of a price level, which can only be considered unitless if we ignore the essential distinction between dated value units. The unit on a price level is current dollars per base-year dollar. The units on P, in other words, is $₂₀₁₉/$_B.

Defining the price level in these units shows more clearly the meaning of real output. Real output does not take physical units. Dividing by the price level does not recover item units. Instead, it expresses a dollar sum in terms of a different year, the base year, in order to render differently-dated aggregates comparable. The units on real output, therefore, are base-year dollars per year, $_B/t.

Putting these together, therefore, the equation of exchange is properly written:

$$ M $_{2019} \times V \frac{1}{t} = P \frac{$_{2019}}{$_B} \times y \frac{$_B}{t} $$

Notice that there are no physical item units.

Exchange Rates

The most tedious part of international economics is the tangle of symmetrical directional relationships. A rise in the exchange rate means that the price of foreign goods in terms of domestic currency rises (same direction), and also that the price of domestic goods in terms of foreign currency falls (opposite direction). It implies both appreciation (of the foreign currency) and depreciation (of the domestic currency). Sometimes you multiply a price by the exchange rate (to convert from foreign prices to domestic prices) and sometimes you divide (to convert the other way). Students very easily get confused as to which direction things are going if they just try to memorize with a “same or opposite” strategy. Being scrupulous and disciplined about units can help keep relationships like these going in the right direction.

First we have to stipulate a consistent meaning of the exchange rate, since sometimes it’s reported in dollars-per-foreign currency (1.15 $/€) and sometimes it’s reported the inverse (109.57 ¥/$) to keep the scalar value above 1. So we stipulate conventionally that the unit on an exchange rate is always expressed with the home country in the numerator – so, 0.0091 $/¥ rather than the inverse.

With this knowledge, being consistent with units makes it simple to remember whether to multiply or divide a price when converting. Given an exchange rate – 1.15 $/€, say – and a price – 10€ – we know our starting units (euros), as well as the units we want to end up with (dollars). Given this, the correct operation with the exchange rate is obvious: we have to multiply, so that the € in the price cancels out with the € in the denominator of the exchange rate, leaving us with $11.50. The process works the other way too: Given a dollar price (say $10), we have to divide the exchange rate in order to cancel out the $ and bring the € into the numerator, which gives us €8.70. Setting up a sample problem like this makes all the tangle of rises and falls settle straightforwardly into place.

Failure in Unit Analysis: Cobb-Douglas and Solow

Most modeling in economics disregards units almost entirely. Sometimes consistent units can be read back into the analysis, as in the three examples above. Most of the time, however, it cannot.

The Cobb-Douglas production function, the center of neoclassical growth theory, is perhaps the most significant example of the neglect of units:

$$ y=AK^\alpha L^\beta $$

Output, this equation states, is a function of capital (K), labor (L), and technology (A).

Mathematically speaking, a functional form like this has a number of advantages: it’s linearly homogeneous; it can be easily log-linearized for empirical application; returns to scale can easily be summarized by α+β; product exhaustion holds under constant returns to scale (i.e. α+β=1), and so on. But as an economic statement, it’s practically nonsensical.

Consider first what appears to be the more straightforward quantity, labor. The labor force, after all, can be straightforwardly represented with a ‘people’ unit, since for the purpose of tallying, people can be regarded as homogenous (and therefore summable). And this does seem to be what the Solow Model, for example, has in mind. y/L, for example, indicates per-capita income, with ‘people’ in the units’ denominator, and similarly for K/L. Firm-level analyses, however, often prefer to use man-hours, or people×t, since – intuitively – output should depend more directly on how much people are working than how many people are working. This decision is not a matter of convenience, however: if we introduce a t unit on the right-hand side, this will also change the interpretation of the left-hand side, unless we cancel it out with a t in the denominator of some other unit on the right-hand side.

Many have even suggested a productivity-adjusted man-hour measure, cognizant of the fact that it might not be analytically useful to regard one man’s labor as homogenous with another man’s. Can man-hours of plumbing be meaningfully summed with man-hours of welding? Or programming? Or bus-driving? It is apparent that the only way to sum up labor in economically meaningful terms is with a value unit, namely, value added. The unit on labor cannot be meaningfully interpreted as ‘people’ or ‘man-hours’ or any physical unit.

Much the same will be true of capital, K. What is the sum of twenty semi trailers plus two smelting plants plus a thousand hammers? What is the sum of a worn-out hammer and a new hammer? Can even these be regarded as homogenous for the purposes of economic analysis if they contribute differentially to production, or fetch different prices on the market?

In a multi-product world, the only meaningful unit on either capital or labor is a dollar unit, not a physical unit – a proposition decisively established during the Cambridge Capital Controversy half a century ago, but blithely ignored in the aftermath. There can be no equivocation between value-units and physical units except in the single-good world of the Solow model.

Dollar units on K and L would at least hold out some hope of interpreting output y consistently with the usual $/t unit, despite neoclassical growth theory’s self-image as a moneyless model. But even this does not restore the model to coherency, for at least three reasons. First, and more trivially, a dollar unit for K and L only returns a dollar unit on the left-hand-side under constant returns to scale – i.e. α+β=1. Otherwise the unit will be $^α+β – dollars raised to the power of α+β – and what is the interpretation of a quantity of that?¹

Second, what is the unit on A? In growth accounting, A is the residual, the component of economic growth not accounted for by growth of K or L. Sometimes this is interpreted as “technology”. So what are the units on technology? What are the units on a thing that we don’t even know the nature of? If the model is to be coherently interpreted as in the previous paragraph under constant returns to scale, A must be unitless. Otherwise whatever units it has will also come through to the left-hand side, giving y the unit of dollars-times-whatever unit we put on A. If we assign dollar-units to A, treating it as an unknown factor of production, we end up with $² units on the left-hand side.² But – significantly – A is not unitless; A has unknown units. What possibility do we have, therefore, of constructing a meaningful model of economic output if we don’t even know what kind of thing we’re looking at?

Third, and most importantly, even if we assume constant returns to scale and admit a unitless A, value units on K and L render the model circular. Specifically, the interest rate is defined in the Cobb-Douglas framework as the marginal product of capital, ∂y/∂K. Now the marginal product of capital evidently depends on the value of the capital stock. But the value of the capital stock is the present discounted value of the stream of income produced by that capital equipment. And the rate at which that income is discounted is necessarily… the interest rate. There are more equations than unknowns: if we interpret K as the value of the capital stock, rather than the physical stock of capital, the system is overdetermined. The moneyless interpretation in terms of homogenous “product” units is not an oversight that can be easily remedied by replacing them with value units; it is central to the model’s internal coherence. And yet, a physical interpretation of capital and labor removes any connection of the model to a world with heterogeneous product.

In empirical practice, K is usually recorded in value units and L is recorded in people-units, sometimes adjusted for human capital depending on the model. This compromise ends up with the worst of both worlds: the circularity of using a monetary definition of capital, and the inconsistency of units.

Conclusion

Many such unit-failures exist in economics; more, perhaps, than successes, especially given the habits of thought encouraged by econometrics, where the units are quite irrelevant for a regression. If economics is to be truly scientific, however – an illumination of structural relationships rather than a mere catalogue of quantitative relationships – consistency in units is an essential discipline, both for teaching and for model-making.

Footnotes

1. And, for that matter, we have to assume that α and β are unitless themselves – not an unproblematic assumption – lest we get even more ridiculous units on the left-hand side.
2. It is important, however, that A is not a factor of production; otherwise, returns to scale would be α+β+1, which includes the exponent on A.