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Investment and Growth

On Using the Harrod–Domar Model

Pronab Sen (pronab.sen@gmail.com) is Country Director, International Growth Centre, India.

 

The simplicity of the Harrod–Domar model of growth, which is at the heart of most planning and growth models that exist today, has enabled a significant widening of the range of participants in debates surrounding the needs and prospects of growth in developing countries. Three of the more obvious oversimplifications of the Harrod–Domar model are identified and discussed, and reasonably simple correctives are provided which can be applied even by laypersons to alter their initial assessments and arrive at more realistic and technically justifiable conclusions.

An earlier version of this article appeared as a Planning Commission Working Paper with the title: “A Note on Growth Projections for Capital-Constrained Economies.”
 

There are many good reasons for which most of the planning models that exist today are based on the Harrod–Domar model.1 For most developing economies, which are usually capital-constrained, the supply-side issue of creating adequate productive capacity is of dominating concern. The Harrod–Domar model encapsulates this concern admirably and provides a powerful analytic tool to assess future prospects and the requirements for growth. Over the years, there have been many extensions and refinements of this model, but
none have captured the imagination in quite the same way as the original. The Harrod–Domar model today has gone beyond the toolkit of the professional economist and has entered the domain of popular discourse.

The essence of the Harrod–Domar model lies in a simple and yet powerful formulation, which links growth to the rate of capital accumulation or investment:2

g = i/k … (1)

where g = growth rate of GDP;

i = investment rate = investment/GDP;

k = incremental capital–output ratio.

The central concept in this formulation is “k”—the incremental capital–output ratio or ICOR—which is a summary expression for the technical conditions and structural configuration of the economy that captures the relationship between investment and additional output. It is also frequently, and not entirely justifiably, treated as a measure of the efficiency of capital. Nonetheless, the value of equation (1) is that, once there is an estimate of the ICOR (k), future projections for the economy become very simple. Whether these are done by estimating the maximal growth rate that can be achieved given the likely trajectory of the investment rate or by computing the investment requirements for any target rate of growth makes little difference since the two are uniquely related by equation (1). Thus, the investment requirement (i*) for any target rate of growth (g*) is given by:

i* = k.g* = i.g*/g … (2)

where i = estimated historical investment rate; g = estimated historical growth rate

Although the ICOR is an extremely abstruse concept covering, as it does, an extraordinarily complex set of relationships, its apparent simplicity has made it an integral part of the popular discourse on economic matters. At one level, this is undoubtedly a good thing in that debates on growth prospects or its requirements have become more democratised and not just confined to the arcane world
of planners and growth theorists. At
another level, however, there has also been a trivialisation of the concept, its measurement and application, which has the potential of leading to severe
errors in assessing the needs and prospects of future growth.

The purpose of this article, therefore, is to draw attention to a few critical issues in the measurement of the ICOR and the method of application of the general growth equation (1) to mitigate the impact of the more common errors that are made in this respect. The objective is not to go into all the details or the complexities, which can be left to the experts in the field, but to provide relatively simple methods by which even laypersons can correct their initial assessments and arrive at more realistic and technically justifiable conclusions.

Continuous vs Discrete Time

The first point that needs to be recognised is that the growth equation (1) has been derived under three basic assumptions:3

(i) The economy is on a steady-state growth path, (ii) investments instantaneously translate to additional capacity, and (iii) time is continuous—that is, it is infinitely divisible.

Each of these assumptions has a significant bearing on how the growth equation should be applied and the modifications that need to be made.

Consider first the third assumption regarding continuous time. Although in a philosophic sense this assumption is indeed true, the data on which all estimation is done are available only for discrete periods, such as a quarter or a year. Thus there is a disjunction between the theory and the data to which it is applied. Since not much can be done about the data, the theory needs reconsideration. In such a situation, the growth equation needs to be suitably modified to take
account of the discreteness of data availability. The starting point for such a reformulation is the definition of the ICOR:

k = It/ΔYt ... (3)

where It = investment; Yt = GDP

In continuous time, dividing both the numerator and the denominator by Yt and rearranging terms yields the growth equation (1). Unfortunately, the definition of the ICOR given in equation (3) is frequently forgotten and the continuous time formulation encapsulated in equation (1) is used. This is in fact the basis for the most common way to estimate the value of “k,” in which the average investment rate during a particular period is divided by the growth rate of GDP over the same period. In what follows, the nomenclature “k” shall be used to denote such an estimate of the ICOR, which we shall refer to as the measured ICOR.

In discrete time, however, such a procedure is just incorrect. The correct measure would be to define:

 

k* = It /Yt-1 =i(1 + g)

... (4)

ΔYt /Yt-1 g

 

Thus, the relevant equation for estimating the investment requirement for any growth target, or the analogue of equation (2), is given by:

i* = i.g*.(1 + g)/g ... (5)

where again i and g respectively refer to the estimated historical investment rate and growth rate. This change can make a significant difference while making projections. Consider a situation where the historical growth rate of an economy (g) has been 6% per annum and the average investment rate (i) 24%, yielding a value of k of 4. The value of k* in such a situation would be 4.24 from equation (5). Thus, the investment requirement for maintaining a 6% growth rate in the
future using the commonly-used continuous time relationship would remain at 24% of GDP, but in the discrete time formulation, it would be significantly higher at 25.44%. As a result, if an investment rate of 24% is targeted, the growth rate actually experienced would probably not be 6%, but around 5.7%. There are many ways of correcting for this problem,4 but the simplest may be to recognise that, from equations (1) and (4):

k* = k(1 + g) ... (6)

Hence, only a small modification is required to the standard method. A comparative picture of the differences to the investment requirements arising from the normally used continuous time formulation given by equation (2) and the discrete time formulation given by equation (5) is presented in Table 1.

Finally, it should be pointed out that this particular modification to the Harrod–Domar model has no economic content at all. It is merely a mathematical correction, which is necessitated by the fact that data are available only for discrete periods of time. The shorter the period, the lower will be the extent of correction required. Nevertheless, as can be seen from Table 1, the empirical significance of this modification is considerable.

Gestation Lags

The second assumption, that of the instantaneous transformation of investment into productive capacity, obviously does not apply in real life. All investments are subject to gestation lags of some duration or the other. Once these gestation lags are taken into account, the “true” or technical ICOR turns out to be significantly different from the implicit or measured ICOR as defined earlier, and this affects the way in which future projections should be made.

Suppose that the average gestation lag for investments in the economy is “n” years, then the “true” relationship between investment and output is given by the following relationship:5

ΔYt = It-n/k’ ... (7)

where k’ = “true” or technical ICOR which essentially states that the increase in GDP in the current year is determined not by investments made in the current year, but by those made n years earlier. Rearranging terms and dividing through by Yt-1yields:

k’ = i/g(1 + g)n-1 ... (8)

From equations (1) and (8) it can, therefore, be stated that

k’ = k/(1 + g)n-1 ... (9)

In other words, the “true” ICOR (k’) is almost always lower than the measured ICOR. To put this in perspective, consider again the example taken earlier where k = 4 and g = 6%. If the value of the average gestation lag, n = 2.8 years,6 equation (9) implies that the value of k’ will be 3.6. The longer the gestation lag, the lower will be the value of the “true” ICOR for any given value of the measured ICOR. Furthermore, it should also be noted that the difference between k and k’ will be wider; the higher is the growth rate that has been experienced in the past.7 For instance, had the historical growth rate been 7% instead of 6%, the value of k’ would have been 3.54.

These two characteristics of the relationship between k and k’ given by equation (9) point to the dangers of making simple-minded comparisons between countries from their implicit ICORs. To illustrate the point, consider the case of India and China, both of which have recorded measured ICORs of around 4 during the 1990s and early 2000s. On this basis, it may be tempting to infer that the two countries are roughly comparable in terms of the efficiency of capital utilisation. Such an inference would not be correct. Over the period concerned, India has recorded an average growth rate of 6.6%, whereas China has achieved about 9%, which are also reflected in their respective investment rates of 26.5% and 37%. Moreover, the estimated gestation lag for India is 2.8 years and for China it is almost four years. The difference in the gestation lags arises primarily out of sectoral differences in the structure of growth in the two countries. In India, the fastest-growing sector during the period concerned has been services, which has very short gestation lags, whereas in China it has been manufacturing. Using equation (9) to correct for these differences in parameters yields a “true” ICOR for India of 3.57 and for China of 3.09—which is indicative of a fairly significant difference.8

In projecting future growth requirements, however, it is important to realise that the “true” ICOR (k’) cannot directly be plugged into the growth equations (1) or (4). It needs to be reiterated that, with gestation lags, future growth is determined by the investment made “n” periods earlier. Thus, the relationship between investment and a target GDP level in the future is determined by:

ΔYt+n = It/k’ ... (10)

Rearranging terms and dividing by Yt:

It/Yt = k’.ΔYt+n/Yt ... (11)

Therefore, the investment rate required (i*) to attain a future target growth rate (g*) is given by the expression:

i* = k’.g*.(1 + g*)n-1 ... (12)

Substituting for k’ from equation (9) yields:

 

i* = k.g*.(1+g*)n-1 = i.g*.(1+g*)n-1

... (13)

(1+g)n-1 g.(1+g)n-1

where i = historical investment rate

    g = historical growth rate

As may be seen, if no acceleration in the growth rate is being targeted, that is, g* = g, then the investment requirement remains at the historical rate (i). However, as the target growth rate is increased, the difference between the two widens. In order to appreciate the magnitudes involved, the extent of differences that arise from recognition of gestation lags at different growth targets is given in Table 2.

In the table, it has been assumed that the gestation lag remains invariant at 2.8 years regardless of the targeted rate of growth. This is not realistic since it implicitly assumes that the growth rates of all constituent sectors of the economy increase proportionately with the increase in the target growth rate of the GDP. The manner in which these proportions will change, and thereby affect the average gestation lag, cannot be generalised and will depend upon a number of factors, not the least of which is the sufficiency of infrastructural facilities in the country. Since infrastructural investments typically have the longest gestation lags, developing economies, which need to build up infrastructure rapidly, tend to exhibit longer average lags than more developed economies and also tend to experience an increase in the lag length as the pace of growth is sought to be accelerated.

On the other hand, gestation lags also reflect policy, procedural and institutional rigidities which affect the pace of implementation of investment activities in the country. It is, therefore, entirely possible that the average gestation lag of an economy can be reduced through a streamlining of policies and procedures and with better governance. Given these possibilities, it becomes desirable to examine the numerical implications of changes in the lag length. This involves respecifying equation (13) as:

 

i* = i.g*.(1 + g*)n*-1

... (14)

g.(1 + g)n-1

 

where n* = expected gestation lag in the future

The results of such an exercise are presented in Table 3.

As may be seen, differences in gestation lags can have a significant impact on the investment requirements of an economy, and the impact increases as higher growth rates are targeted. This characteristic arises from one basic difference between the two growth equations (2) and (13), which needs to be noted. Equation (2) relates g* and i* in a linear manner, whereas equation (13) is inherently non-linear. Nevertheless, both the curves pass through the origin, which implies that in both cases a zero growth rate is associated with zero investment. The shapes of these two relationships are presented graphically in Figure 1:

Depreciation of Capital Stock

Possibly the most critical weakness of the commonly used Harrod–Domar formulation, is that it does not explicitly take into account the fact that the productive capital stock of an economy does not have an infinite life. All capital assets have a finite life, which is determined partly by wear and tear and partly by obsolescence. In the data normally available from the National Accounts, this is sought to be captured by the concept of “consumption of fixed capital,” whereby a particular fraction of the existing capital stock is assumed to drop out of the production processes in each time period. However, it should be noted that depreciation was explicitly taken into account in the original Harrod–Domar model, but under a steady-state assumption, and with a constant depreciation rate, it can be shown that the depreciation term cancels out and the growth equation (1) gets re-established. Thus, under the specific assumptions of the Harrod–Domar model, ignoring the possibility of depreciation is perfectly valid. Nevertheless, when making projections for the future or estimating the requirements of accelerated growth, not taking into account the real-life fact of depreciation can lead to serious errors.

In order to accommodate the process of consumption of fixed capital stock or depreciation, the basic model used so far has to be modified quite significantly. As a starting point, it is necessary to define the concept of net investment, which is the difference between the gross investment made in a particular period and the depreciation of the existing capital stock during the same period. Thus:

NIt = It - Dt ... (15)

where NIt = net investment in time period t

It = gross investment in time period t

Dt = “depreciation” of existing capital stock in time t

Equation (15) essentially states that the gross investment (I) made during any particular period can be conceptually split into two components—one component (D) is basically for replacement of the capacities which have gone out from the production structure, and the other (NI) for the creation of additional capacities. Clearly, the additional output produced during the given time period (ΔYt) should depend only upon the additional capacity created (NIt) and not upon the replacement investment, which merely restores the productive capacity to its pre-existing level. Thus, it becomes necessary to redefine the relationship between investment and output growth by introducing the concept of the net ICOR:

v = NIt/ΔYt = (It – Dt)/ΔYt ... (16)

where v = net ICOR

It may be noticed that equation (16) is an analogue of equation (3) and can be rewritten in the form:

v = k – Dt/ΔYt ... (17)

In order to operationalise this concept, it becomes necessary to add some more structure to the second term in the right-hand side of equation (17). For simplicity it is assumed that a constant fraction “d” of the capital stock existing at the end of the previous time period goes out of the production process during the current period:

Dt = d.Kt-1 ... (18)

where Kt-1 = capital stock at the end of period t-1

Furthermore, the relationship between the capital stock existing at a particular point in time and the level of output is given by the concept of the average ACOR, which is defined as:

a = Kt/Yt ... (19)

In a steady-state path, the ACOR (a) is a constant, and can be estimated directly from the data on GDP and capital stock routinely found in the National Accounts. Using equations (18) and (19), the definition of net ICOR given by equation (17) can be modified as follows:

v = k – a.d.Yt-1/ΔYt = k – a.d/g ... (20)

Substituting for k from equation (1) yields:

v = (i – a.d)/g ... (21)

While making future projections, it needs to be recalled that the gross investment made at any point in time not only has to provide the additional capacities necessary for the target growth path, but also replacement for capacities which have been lost due to “depreciation.” Therefore, the analogue of equation (2) in this case becomes:

i* = v.g* + a.d ... (22)

where the first term on the right hand side of equation (22) represents the investment going into creating new capacity and the second term represents replacement investment. Substituting from equation (21) and collecting terms then yields the final expression for estimating the investment requirement for any future target rate of growth:

i* = i.(g*/g) – a.d.(g* - g)/g ... (23)

where as earlier, “i” and “g” represent the historical investment and growth rates respectively, and “a” and “d” the measured values of the ACOR and the average annual depreciation rate.

The first point to be noted from equation (23) is that if the economy is in a steady state or no significant change is expected in the growth rate in the future, the second term on the right hand side becomes zero or negligible, and the standard growth equation (2), or the gross investment approach, gets re-established. If, however, the target growth rate (g*) is aimed to be significantly different from the historical experience (g), the investment requirements arising from the net investment approach given by equation (23) will be very different from that arising from the gross investment approach, with the difference increasing as the divergence between the two growth rates widens. Moreover, it should also be noted that, for whatever reason, if the future growth rate is expected to be lower than the historical, the net investment approach will require a higher investment rate than the gross investment approach. A numerical example, using parameters which approximate recent Indian experience, may help to clarify these points, and is presented in Table 4.

As may be seen, the investment requirements arising from the net investment approach are significantly different from those given by the gross investment, or Harrod–Domar approach, except in a steady state (represented by the historical growth rate of 6%). The above results can be interpreted in the following manner:

Of the total investment made in the country in a particular year, an investment amounting to 10.88% of GDP has to be applied towards replenishing the capital stock and thereby leaving the over-all productive capacity unchanged. Thus, this magnitude of investment has to be made simply to keep the economy at its current level of production with no growth at all. Any investment in excess of this level leads to an increase in the production capacity, and hence to GDP. It requires about 2.19 percentage points of GDP of such additional investment to produce each percentage point of additional GDP growth—that is, a net ICOR of 2.19. In contrast, the gross investment approach implies that every percentage point of GDP growth requires an investment rate of 4 percentage points. This difference can best be understood graphically by plotting equations (2) and (23), as is shown in Figure 2.

In brief, therefore, the net investment approach suggests a higher investment requirement than the gross investment (or Harrod–Domar) approach for growth rates lower than the historical experience and vice versa for higher growth rates. Hence, savings (or investible resources, in general) may not be as insuperable a constraint to accelerating growth as is commonly made out to be. However, if external or exogenous conditions dictate a slowdown, then care has to be taken to ensure that investments do not fall excessively.9

An Integrated Model

In the preceding sections, three separate modifications to the standard Harrod–Domar formulation have been proposed. Of these, two—discrete time and gestation lags—operate towards increasing the investment requirement for any acceleration in the growth rate as compared to that given by the Harrod–Domar model. The third—consumption of capital or depreciation—works in exactly the opposite direction. Since all three influences are likely to obtain in real life, it appears desirable to integrate them into one single model, so that the combined or net effect of the three can be estimated. The starting point for such an integration is the definition of the “true” or technical net ICOR in the presence of gestation lags, which involves combining equations (7) and (16):

v’ = NIt-n/ΔYt = (It-n – Dt-n)/ΔYt ... (24)

where v’ = net ICOR with gestation lag

From equations (18) and (19), this may be rewritten as:

v’ = (It-n – a.d.Yt-n-1)/ΔYt ... (25)

Dividing through by Yt-1 yields:

v’ = i.(1 + g) – a.d ... (26)

    g.(1 + g)n

As earlier, when projecting forward, the investment rate required (i*) to attain a target growth rate (g*) with a gestation lag of n* is given by:

i* = v’.g*.(1 + g*)n*-1 + a.d ... (27)

Substituting for v’ from equation (26) and arranging terms yields the final expression for the integrated model:

 

i*= i.g*.(1+g*)n*-1 – a.d.|g*.(1+g*)n*-1 - 1| ... (28)

g.(1+g)n-1 | g.(1+g)n |

where all the symbols bear the definitions mentioned earlier. Although equation (28) appears extremely complex, it can be easily evaluated in numerical terms. The results of this exercise are presented in Table 5.

As can be seen, the net investment effect arising out of a recognition of depreciation dominates the other two effects at the assumed parameter values. As a consequence, the investment requirements for accelerating the growth rate of the economy for a country like India is likely to be significantly lower than that indicated by a simple-minded application of the Harrod–Domar model. The situation could well be different at other combinations of parameter values, and hence no generalisations should be made.

Conclusions

The purpose of this article is to contribute to the wider debate on growth prospects and requirements in developing economies, particularly in India, by providing fairly simple correctives for the more obvious oversimplifications that are embodied in the manner in which the Harrod–Domar model is usually applied. In attempting to do so, it has been necessary to indulge in a fair amount of algebra. Although much of this may appear intimidating, it was felt essential that the rigorous derivations be provided in order for discerning readers to verify the validity of the results. Nevertheless, the reformulations of the growth–investment relationship proposed in this article—given by equations (5), (14), (23) and (28)—can be numerically evaluated using only a pocket calculator, provided that the parametric values are available.

Of the five basic parameters that have been used—namely the investment rate (i), the growth rate (g), the ACOR (a), the “depreciation” rate (d) and the average gestation lag (n)—the first three are also amenable to fairly straightforward calculation by laypersons themselves from data readily available in the National Accounts. The other two, however, are a little more complex. As far as the “depreciation” rate is concerned, the National Accounts do provide an estimate of “consumption of fixed capital,” which is an aggregation of asset-wise depreciation rates based on a priori specifications of the operational life of each form of asset. This, however, is an inadequate measure of the rate of attrition of capital stock, which has to include not only the physical deterioration of assets but also the scrapping of assets on grounds of economic non-viability or technological obsolescence. Measurement of the more complete concept is extremely difficult to do, and it may be more practical to simply settle for the limited version for which data readily exists.10

Estimating the average gestation lag in the economy, however, presents a much more intractable problem. There are no readily available estimates that are routinely published, nor a universally accepted methodology that can be applied to available data. There are a number of econometric methods available, but most of these are fairly complex and not easily usable by non-experts.11 However, as shown in this article, given the substantial difference that this factor makes to future assessments, it would not be proper to ignore it altogether either. In the interest of generating wide-spread debate on growth prospects and requirements of the economy, it may, therefore, not be a bad idea to standardise a methodology for obtaining gestation lags and insist that the statistical system publish such estimates on a regular basis.

Notes

1 The so-called Harrod–Domar model was independently developed by Roy F Harrod (1939) and Evsey Domar (1946).

2 This particular formulation is not in either of the original papers, but is the most popular form in usage today. The original versions were strictly classical in terms of investments being determined by savings. This assumption led to the “razor’s edge” characteristic of the Harrod–Domar model, which spawned an
entire literature. We abstract from the razor’s edge problem in this article to focus on issues which we believe are more germane to the current discourse.

3 There are of course many other assumptions, such as continuous full capacity utilisation and full employment, unchanging sectoral structure of the economy, etc, but these need not concern us at present.

4 A procedure that is commonly used is to compute the average investment rate with a one-period lag. Another procedure, which is true to the definition of the ICOR given in equation (3), involves calculating

It = Kt – K0 ,

where K is the capital stock at a given point in time

ΔYt = Yt – Y0

and dividing one by the other.

5 Gestation lags usually imply that the investment necessary to create new capacity are spread over a number of time periods, and hence should be specified in the form of distributed lags rather than as a one-time investment leading to capacity after “n” periods as has been done in equation (7). While this is certainly true, the assumption of a steady state permits this kind of a simplification in the specification.

6 The average gestation lag of 2.8 years has been taken from the Indian experience on the basis of estimates made by the author.

7 The first partial derivatives of k’ with respect to both n and g are negative.

8 It should be further noted that structural differences in growth patterns affect not only the average gestation lag, but also the aggregate ICOR, which is a weighted average of sectoral ICORs. Thus, if the ICOR for manufacturing is higher than that for services, the “true” sectoral ICORs for China would be even lower than those for India.

9 Economic slowdown arising from external or exogenous factors, such as a slowdown in the world economy, oil price shocks or agricultural failure, can easily translate into recession if the domestic investment activity falls sharply. This is a real possibility since private investment reacts adversely to decreasing capacity utilisation, thereby setting off a downward spiral. In such situations, public investment has to be stepped up to ensure that the aggregate investment rate does not fall below the level warranted by the estimates given above.

10 In fact, in this article also, the value of the “depreciation” rate has been taken from the National Accounts.

11 The estimate of the average gestation lag for India (and also for China) used in this article is derived from some very sophisticated exercises that were carried out in the Planning Commission during the formulation of the Ninth Five Year Plan.

References

Domar, E (1946): “Capital Expansion, Rate of Growth, and Employment,” Econometrica, 14(2), pp 137–47.

Harrod, R F (1939): “An Essay in Dynamic Theory,” The Economic Journal, 49(193), pp 14–33.

Updated On : 7th Jun, 2019

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