# In GDP growth studies, are factors of production measured as stock or flows?

Consider the following cases:

• One machine (capital) is used by multiple producers within one year. For instance, farmers renting tractors from the same company. In this case, in terms of GDP, there is one unit of capital which was used to produce value added by many producers.

• One worker (labour) which have two jobs. For instance, in a factory during the day, and as a Uber driver during the night. In this case, in terms of GDP, there is one unit of labour, used to produce value added in different goods/services.

The common feature is that the same factor of production (stock) is employed in more than one activity over a given period (flow).

Is the measurement of capital and labour used in the standard "determinants of growth" studies based on the stock or flow approach?

In other words, are capital and labour measured in terms of the flow of services that these factors produce, or in terms of the stock of factors available in an economy over time? It is clear to me that the latter approach is plagued with problems, and the former is clearly more accurate.

My experience is that the use of stock is more common.

• To me capital is always a stock while labor is always a flow. The corresponding stock to labor would be labor force, but it is quite clear that the size of the labor force is not a "factor of production". The flows of capital would be investment, depreciation, obsolescence and destruction. In studies of the determinants of growth based on the Solow model, the data for the stock of capital are quite often measured by adding up the series of investment and depreciation of the previous years. – Ululo Feb 18 '17 at 16:06

In Nicholas Stern (1991)'s The Determinants of Growth first equation (I a), without preconceived ideas, one reads

$\frac{\dot{K}}{K} = \frac{d K}{d t} \frac{1}{K}$

Which means explicitly (in discret values)

$= \frac{K_{t+dt}-K_{t}}{dt} \frac{1}{K_{t}}$

If $t$ is in years, and that we are interested in the variation of $K$ over $1$ year (i.e. $dt=1$), one obtains

$= -1 +\frac{K_{t+1}}{K_{t}}$

Even if the above equality could represent either the variation of a stock or the variation of a flow, the story behind this is about stocks. To summarize

• $K_{t}$ is the stock of capital in the economy at year $t$

• $dK = K_{t+dt}-K_{t}$ is a flow of capital occuring at year $t$, differently put an investment in capital.

• $\dot{K}=\frac{d K}{d t}$ is the absolute dynamic of investments in capital in economy $\forall t$

• $\frac{\dot{K}}{K}$ is the relative dynamic of investments in capital in economy $\forall t$

Depending on the object you are looking at, it can be both measured in flow or in stock. To know with what you are dealing with, you must understand the story behind equations.

To the practice side of the issue, you cannot make realistic analogy between Solow's capital and, say, tractors. The least abusive perception of $K$ would be cumulated flows of investment in terms of money since, for example in micro-based CGE modelling, it is conceived as representative of the shareholding structure of the economy, and is thus linked to income formation via dividends (see the zero profit condition). In this non-realistic framework, although non-abusively linked to the everyday-life object, $L$ is very similar to $K$, is a self owned stock, and gives right to wages, that are flow objects, as dividends are. Still on the practice side of the issue, the difference between $L$ and labor force is the fact that the former is effectively employed, while the latter is not. Put differently, the link between the two is the labor market, $L$ is a demand, while labor force is a supply, and both form jointly a labor market whose equilibrium is reached via deflation/inflation of wages.

Finally, to my knowledge, the fact that the same factor of production (stock) is employed in more than one activity over a given period (flow). is not really a neoclassic native notion, I would say, once again when one refers to CGE modelling, because of the capital market clearing condition. It is more a monetary-macro-financial object, and is, e.g. all what debt is about when dealing with capital. However I do not see any analogy with $L$ in this case.

• Thanks, but I don't think this answer the question (where is the discussion on labour?). The accumulation and depreciation of capital suggests $K_t$ is better understood as a stock. No doubt about that. However, in terms of production, what seems to matter more is the flow definition of factors of production. That is the point of the two examples I give. The question relates not to theory but to the practice side of the issue. – luchonacho Feb 18 '17 at 17:46
• @luchonado. See my refined answer. – keepAlive Feb 18 '17 at 23:56

To add to Tnerual's answer, even when natural resources are added typically stocks are used, at least in theoretical work. The flow of the used up resource is converted to a stock variable by assuming a fixed percentage of depletion each year (see for example Nordhaus Lethal model of 1992 here)

In your example, we have a multi-sector economy with (presumably)

1. a farming sector
2. a tractor producing sector (manufacturing)
3. transit/transportation sector (services)

Also, there (seem to be/) are two different types of labor available to the economy:

1. unskilled workers (farmers, factory workers)
2. skilled drivers (services sector)

Inputs to each sector are not spelled out in detail but it wouldn't be far-fetched to assume that the farming sector used land, unskilled labor and tractors, while the manufacturing sector used unskilled labor and tractors (let's assume-for the sake of the argument-that we are dealing with special multipurpose tractors that can plough the land and mine for ore...). As far as the services sector is concerned, we'll assume that it requires skilled labor (drivers) and tractors (... some really multipurpose piece of machinery)

In all honesty, it is a daunting task to analyze this economy. I wouldn't know where to begin... So let's begin by writing down the production side of this economy:

The production function for the farming sector should look something like this:

$Q^f = Q^f(T, H_u^f, M^f)$

where the superscript $f$ denotes the sector ('f'-arming) and $T$ denotes effective arable land area (in square km-we use no superscript to denote the sector it provides services to, since in our economy, the land is used productively only in farming), $H_u^f$ is unskilled labor (in thousands of man-hours per year) used in farming and $M^f$ denotes the number of tractors used in the farming sector. Note that the number of man-hours used up in the farming sector can be also decomposed as $L_u^f \times e_u^f$ ie the number of unskilled workers used ($L_u^f$) times the average effective number of man-hours one worker puts in during a whole year ($e_u^f$). The output of the farming sector is measured in billions of 'metric tons/tonnes' of grain per year.

Let's turn our attention to the production side of the manufacturing sector:

$Q^m = Q^m(H_u^m, M^m)$

Obviously, the superscript denotes the sector ('m'-anufacturing) and $H_u^m$ and $M^m$ denote hours of unskilled labor and number of tractors used up in the production of the manufacturing sector, respectively. The output of the manufacturing sector is thousands of 'tractors' per year.

Finally, the services sector has a production side that looks like this:

$Q^s = Q^s(H_s, M^s)$

In the case of the transportation/transit industry, the labor input is dedicated skilled labor ($H_s$ ie skilled labor that is solely used in the services sector-remember the model structure layout from the beginning).'Transportations' also use 'tractors' ($M^s$-remember the highly versatile machinery we assumed this economy is endowed with, in the beginning of this note). The output of the *services * industry is measured in millions of 'passenger km' per year.

Now that we have presented the production side of each sector in the economy we will display the productive potential of the economy in a more compact form:

$Y_f = Q^f(T, H_u^f, M^f) + P^m\times Q^m(H_u-H_u^f, M^m) + P^s\times Q^s(H_s, Q^m(H_u-H_u^f, M^m)-M^f-M^m)$

Note how $Y_f$ is the value of the economy's output measured in terms of the output of the 'farming' sector ie $P^m$ and $P^s$ are relative prices ('metric tons/tonnes of grain per tractor' and 'metric tons/tonnes of grain per passenger km'). Also note how the input of unskilled labor in the manufacturing sector has been replaced by $H_u^m=H_u-H_u^f$, that is to say, the manufacturing sector uses all the unskilled labor that is left over from the farming sector (we assume that the stock of unskilled labor ($H_u$) is fixed for the period of study). Additionally note that something similar happens to the input of machinery in the transportation sector ie $M^s=Q^m-M^f-M^m$. Finally, note how the output of the services sectors is dependent upon the output of the manufacturing sector ($Q^m$ as an input in $Q^S$).

Before proceeding we'll make the following simplifying assumptions for our model: the effective land area that is available is constant over time ($T=\bar{T}$) ie there are no wars of conquest going on neither is there any technological improvement on the soil (no fertilizers, hybrids etc); the available unskilled and skilled labor force does not grow ie workers simply replenish their numbers and there is no productivity improvement ($H_u=\bar{H_u}$ and $H_s=\bar{H_s}$). Finally, there is only one variety of tractors produced and it is this model that is used in all relevant sectors of the economy.

Is the description so far an accurate portrayal of the economy you have in mind?

• Thanks for the answer, but I am not sure this addresses the question. You are using "number of tractors" used in the farming sector. What if those tractors are shared by multiple companies? Renting of machine is a very common phenomenon. Hence, number of tractors might not be accurate enough. – luchonacho Sep 20 '17 at 16:05
• Yes, that is relevant. The thing is that when using a measure of capital in studies of growth determinant, we should use the flow of services from capital, rather than the capital stock, just as we use number of hours of work, rather than number of workers. I am not sure that is mainstream though. – luchonacho Sep 21 '17 at 8:02

I realize this question might be too old to reply to, but it was just bumped to the homepage. But anyway:

I think other answers might be missing the forest for the trees a bit. Most basic economic models could be defined with either "stocks" or "flows" as you've defined them. Ultimately, it's about measuring productivity in a generalized production function. You can define the inputs however you'd like- labor could be defined in "full time employee equivalents," or "hours worked," or a vareity of other things. Capital, as many other answers have noted, is incredibly hard to pin down in some cases. That said, it can often be defined and measured through depreciation, investment, "hours of machine X used," "number of tractors used," etc..

More generally, both are pinned down by their relative factor income shares: an early estimation of which led to the creation of the Cobb-Douglas Production Function, which is an instructive example. For that, you define labor as the number of person hours worked (so if you worked for two companies, four hours each, it would count as 8 total "units" of labor), and capital as the total value of all machinery, buildings, equipment, etc.. You then look at how productive resources are allocated between the two, and back out income shares. For this, you never needed to really touch the "stock vs flow" issue, you just had to make sure your definitions were consistent.

There are a few apparent misconceptions in this question that need addressing.

1. In the actual exercise of national accounting, nothing is "measured" except for currency flows. The degree of sophistication of accounting comes down to the degree of nuance with which said flows are categorized. For your purposes, this is a good thing, because it means the confusing issue about "two companies, one unit capital" is irrelevant.
2. Most common macro growth models represent both capital and labor as stocks, from which value-added (read: goods and services, or "GDP") flows. Within the context of a given model, "investment" is essentially the mechanism through which this value-added is transformed into more capital, while the stock of labor is typically assumed to grow via natural processes or netted-out by normalizing the model on a per-worker basis. In either case, this growth is not a flow, any more than punching multiple taps into a keg is a flow. At the abstract level of most models, labor and capital are fundamentally stocks.

With that out of the way, you ask:

are capital and labour measured in terms of the flow of services that these factors produce, or in terms of the stock of factors available in an economy over time

It may be more accurate to say that capital and labor are priced according to the flow of services they produce. They are typically "measured" according to their share of GDP, which isn't quite the same as a count of tractors or a number of employed persons because the point of growth models isn't to capture nominal values at all. It's to understand the aggregate, dynamic behavior of the system. Scale invariance (homogeneity of degree one) is usually assumed as a matter of convenience, and so "counting" as such is often irrelevant.

That's probably not a very satisfactory answer, but as some others have touched upon, this high degree of abstraction allows avoiding almost philosophical arguments about what a "unit of labor" is, or how "sectors" are defined, that are actually quite complicated and difficult to resolve in a clear and universal way.