Exploring Harrod Domar and Solow Models of Economic
Growth
Abstract
This paper models economic growth as described by the Harrod Domar and Solow models using the
System Dynamics framework. The modelling exercise highlights the boundaries of the Harrod Domar
model and presents alternative solutions to overcome these boundaries. As an extension to the Harrod
Domar model, this paper demonstrates the endogenous effect of savings and population changes on
economic growth. Solow’s enhancement of the Harrod Domar model has been simulated to highlight the
capital output constraint in determining income growth. The System dynamic modelling toolset in this
paper provides markedly different modelling perspectives that are unavailable in static models. In
addition, the models developed for this paper allow easier testing of sensitivities to a range of
macroeconomic variables.
Keywords: Harrod Domar, Solow, Economic Growth, Demographic transition
Contents
Exploring Harrod Domar and Solow Models of Economic Growth
Section 1 Introduction
1,4, Development Economics - Traditional modeling techniques & their limitations
Section 2 Harrod Domar Model
2.1 Background
2.2 Model description
2.3 Inbuilt equations in the Harrod Domar model
2.4 System Dynamics — Modelling Harrod Domar equations
2.5 Reconstructing Harrod Domar through the descriptive text
2.6 Reconciling Harrod Domar model description with the equations
20 Literature - Harrod Domar in System Dynamics representation
2.8 Discussion.
Section 3 Harrod Domar — Population extension:
3.1 Background
3.2 Introducing per capita variables
3.3 Sensitivity of input variables on economic growth
3.4 Endogeneity in Harrod Domar variable:
3.4.1 Savings Rate
3.4.2 Population Growth Rate
3.4.3 Modelling endogeneity of savings rate and population rates in the Harrod Domar
equation 17
3:5 Discussion.
Section 4 Solow’s extension to the Harrod Domar model
4.1 Background
4.2 Model Description.
4.3 Changes required for implementing Solow’s extension
4.4 Endogeneity of the Capital Output Ratio
4.5 Discussion.
Section 5 Discussion
References
July 16
July 16
Section 1
Introduction
1.1 Development Economics - Traditional modeling
techniques & their limitations
Advances in economics till the 1940’s centered on the mechanisms and sources of economics and
prosperity (rather than those deprived by it) in general. As a consequence of this, development
economics has not been a natural focus area for most economists till the high period of 1940s
+(Krugman, 1995). Given the wider range and reasons for economic outcomes, the understanding of
multi-faceted subject of development is relatively recent (Hosseini, 2003). The multi-dimensional
nature of Developmental economics makes it an ideal field for applying System Dynamics methods.
Most standard development economics text books, explain economics theory through models that tend
to be static. For instance, the Harrod Domar model or the Solow model? (Ray, 1998), including Solow’s *
(Solow, 1956) original paper, have been explained through both text, block diagrams and mathematical
equations. Some of these equations have multiple variables, all moving across time. These models help
give insight and also help in offering a perspective toward development. But the static models are
severely limited in bringing forth the dynamics between variables and also the endogeneity among
variables. Furthermore, most of the static models covered in economics theory do not offer test cases
that can help enhance the practical working of a model.
The inherent dynamic ability that the System Dynamics framework contributes is especially useful in the
context of economics. Economics system spaces are complex. Most economic models deal with
imperfect information about the state of the real world and have interconnected and/or ambiguous
variables. These attributes of dynamic complexity make economics models suitable for System Dynamics
based frameworks 5(Sterman, 2000).
July 16 4
Section 2
Harrod Domar Model
2.1 Background
Harrod and Domar developed their models independently but their assumptions and results were
similar. The background to their work (1939 Harrod, 1947 Domar) was that industrialized economies had
faced cataclysmic upheavals — the Great Depression followed by the Second World War. The Harrod-
Domar model therefore perhaps, has attempted to explain as to how economies would need to grow (or
would be left to stagnate) over time.
2.2 Model description
The Harrod Domar model, has two main actors. The model assumes a closed economy in which
households consume & save. Firms, the other “actor” in the Harrod Domar world, produce both capital
goods and consumer goods based on investments. The investment in firms, in turn, is a result of the
shortfall between the consumption expenditure that households have and the income the households
earn. The income that households earn, is the income generated by firms as a result of investments.
The schematic for this description has been shown in Figure 1 §(Ray, 1998). It may be noted that the
schematic has “stocks, “inflows” and “outflows”.
Inflow Investments
Firms
Outflow Inflow
Wages, Profits, Rents Consumption Expense
Inflow Outflow
Households
Outflow Savings a
Harrod Domar model Sch ic (Debraj Ray, Di Ec ics, Page 52, Figure 3.1)
Figure 1
July 16 5
2.3 Inbuilt equations in the Harrod Domar model
The key equations that govern the Harrod Domar model include (Ray, 1998):
1) YO=HCH+S0)
Equation 1) states that the income denoted by (Y(t)) (at time t ) is used by households to consume
(C(t)) and save (S(t)). This has been depicted on the left side of the schematic in Figure 1.
2) Y(t) =C(t) +10)
Equation 2) above states that income (Y(t)) is also a derivation from consumption (C(t)) and
investment (/(t)).
3) S(t) =1(t)
It follows from Equations 1 & 2 that the identity could be derived. In literature, this identity has also
been referred to as the macro-economic balance (Ray, 1998)’. From this equation, it implies that all
savings in a Harrod Domar world, translate to investments.
4) K(t+1)=(1— 6)*K(6) +1(6) (6 is the depreciation; K(t) and K(t + 1) denote capital at
times t andt + 1)
5) Ss _ S (s is the savings rate)
Y(t)
6) 6= oO (@ is the capital output ratio)
7) z = g + 6 (g denotes the growth rate; Equation 7 is derived through substitutions in equations
via equations 8,9 and 10 below)
8) K(t+1)=(1- 6)*K()+S(t)
9) 6*Y(t+1) =(1—6) *O* V(t) +s*Y(t)
¥(t+)-¥O©) _
y(t)
denoted as g)
The subsequent sections will seek to reconcile the description of the Harrod Domar model in
this section with the schematic representation in Figure 1 as well the equations 1 through 10.
(3) — 6 (the left hand side of this equation is the percent change in income
2.4 System Dynamics — Modelling Harrod Domar
equations
Equations 1 through 10 that govern the Harrod Domar Model are complete and consistent. But
these equations do not provide insights into the casualty between the variables. For example the
first equation describes income as being made up of two components — savings and consumption.
Equation 1 however does not provide enough detail to provide a direction for the flow inbuilt
inside the equation.
A system dynamics representation for equations 1 through 10 is provided in Figure 2 that seeks to
bring casualty to these equations.
July 16 6
x
Li Firm Capital
Savings Savings cycle Deprecsaion
Consumption iN
I sales, oo
Savings % estment Depreciation %
4+Growth %
System Dynamics: Modeling Harrod Domar equations
Figure 2
Using the System Dynamics approach, each equation (i.e. Equations 1-10) in the Harrod Domar model
has been used to build a stock and flow representation for the model. The SD model in Figure 2 is a
complete and exact representation of equations 1 through 10.
With reference to Figure 2 and all other System Dynamics representations of Harrod Domar, Table 1,
provides a list of the entities and the units that constitute the System Dynamics representation of the
Harrod Domar model. It may be noted that the “units” for each of the entities are just an exact “unit”
logical representation for the description of that entity.
Entity Description Unit
Capital Output Ratio | Ratio of Firm Capital to Income Year
Consumption Household consumption per period Money/Year
Depreciation Markdown to account for asset’s useful life Money/Year
Depreciation % Markdown of asset value in percentage terms Ratio
Firm Capital Accumulated firm capital Money
Growth Percent change in Wage-Profits-Rents per period Ratio
Household Capital Accumulated household capital Money
Investment Investments in firms per period Money/Year
Savings Savings per period Money/Year
Savings % Ratio of Savings to Wages Ratio
Wages-Profits-Rents | Wages, profits, rents that households earn per period Money/Year
Per Capita Values Per person values for flow or stock variable Units/Person
System Dynamics: Description of each entity with unit used
Table 1
July 16
The System Dynamics model in Figure 2 adds value to the equations 1 through 10 as it gives a better
insight into the casual relationships that exist. For instance, it is clear now that Savings and Consumption
exist because of Wages. Wages in turn are a result of Investment being transformed through the capital
output ratio.
But, with reference to Figure 1 (Harrod Domar schematic) and Figure 2 (Harrod Domar SD
representation), it is to be noted that the “Households” “stock” (from Figure 1) is completely missing
from Figure 2. That there is a gap in the description of the model (as shown in Figure 1) in terms of stock
and flow connectivity and the equations that represent this model is a discrepancy that System
Dynamics allows the modeler to uncover.
The output graphs for two key variables —- Wages-Profits-Rents (Income) and Firm Capital — using the
System Dynamics model in Figure 2, have been shown in Figure 3.
Income Firm Capital
140000
120000
100000
80000
60000
40000
Income - Money/
20000
°
Income and Firm Capital output graphs using System Dynamics
Figure 3
For reference, control variables for the model were provided the following input:
Savings Rate (s = 0.35 ); Capital Output ration (9 = 3 years); Depreciation (5 = 0.10); Starting Firm
Capital (1 “Money”)
Growth rate given by the formula (= — 6 = g)is 1.67%, matches with the output produced by the
System Dynamics model in Figure 3.
2.5 Reconstructing Harrod Domar through the descriptive
text
This section reconciles the difference between the descriptive text of the Harrod Domar model and
equations 1 to 10 that represent the model mathematically.
“Households” have a prominent place within the Harrod Domar model description as well as within the
schematic shown in Figure 1 (text book description of the Harrod Domar mode). Wages, Savings and
Consumption, shown as auxiliary variables in Figure 2, in reality, carry the units of a flow (i.e.
July 16 8
“Money/Year”). In order to appreciate this in a System Dynamics framework, a new “Household capital”
stock was introduced in the System Dynamics model of Figure 2. The three auxiliary variables (Wages,
Savings & Consumption) were converted into flows (or rates). Without disturbing the sanctity of
equations 1 through 10, and through incorporating the “Household capital” stock and three additional
flows (Wages, Savings & Consumption) a new System Dynamics representation was attempted.
Building an exact replica for the schematic (in Figure 1) along with keeping the sanctity of equations 1
through 10 faced two inherent contradictions:
1. Consumption “flow” has been shown as an inflow in the schematic in Figure 1 but this
representation is completely missing out from equations 1 through 10.
2. Similarly, Wages-Profits-Rents “flow” has been shown as an outflow in the schematic in Figure 1.
Again, this information is not represented at all in equations 1 through 10.
Overlooking these two contradictions, a System Dynamics representation, that attempts to match the
schematic (Figure 1) and equations 1 through 10 has been modelled. The block diagram for this model
has been shown in Figure 4 and the output graph for Firm Capital and Wages-Profits-Rents have been
shown in Figure 5. The input values for the control variables have been left unchanged (i.e. Savings Rate
(s = 0.35 ); Capital Output ration (@ = 3 years); Depreciation (6 = 0.10))
Reconciling Harrod Domar description with Harrod Domar equations
Figure 4
The graphs in Figure 5 show that both the Income (Wages-Profits-Rents) and the Firm Capital has
collapsed within 50 years. Since the model in Figure 4 contravenes two rules highlighted earlier,
unsurprisingly, the result in Figure 5, comes as a complete contradiction compared to the results shown
in Figure 3.
It should also be noted that “Household” capital “stock” in the model described by Figure 4, finds itself
isolated and unused. In fact, the software used to build the model complains and highlights this fact as a
warning. A stagnant “Household” capital “stock” at a time when other entities in the model are changing
July 16 9
contradicts Solow’s observation that all entities in a Harrod Domar world grow (or change) at the same
speed
Firm Capital Income
Time - Yea Time -Years
Reconciling HD description with equations
Figure 5
2.6 Reconciling Harrod Domar model description with
the equations
For improvising the models described in Figure 2 and Figure 4 (Sections 2.4 and 2.5 respectively), an
additional System Dynamics model was constructed to reconcile the contradictions presented in Figure
4 (section 2.5, i.e. extraneous “flow” links for wages & consumption). The improvised model has been
shown in Figure 6. Figure 6 tells us that:
a. Consumption is now flowing out to a cloud (and not flowing into “Firm Capital”)
a. Wages-Profits-Rents are a flow but this flow originates from a cloud with inputs from the “Firm
Capital”
b. Wages-Profits-Rents, are now an inflow into a newly created “Household stock” as described in
the original Harrod Domar
c. Savings and Consumption have now been shown as outflows from the “Household stock”
However, given the lack of mathematical equations connecting the “Household stock” to the
flows, the model in Figure 6 informs that the “Household stock” would cease to play a role in
determining flow of “Wages”, “Consumption” and “Savings”
In reality, Consumption would need to be connected with the “Firm Capital” and Wages too would need
to be derived from the production/ value addition that firms bring. But this would mean enhancing the
basic Harrod Domar framework.
The improvised model shown in Figure 6 has produced exactly the same output as was shown in Section
2.4.
July 16 10
Income Growth Depa
~ Sig ea Output
= Ratio oO
= + a
Wages-Prof
its-Rent
ON
Savings Ss Household
Capital
Consumption
Savings
Investment cycle
[Savings a ee (linvestment
Reconciling Harrod Domar description with true stock and flow representations
Figure 6
2.7 Literature - Harrod Domar in System Dynamics
representation
Betz et al (Betz, 2015)® have presented used System Dynamics framework and various models in the
economics literature to explain the evidence left by the 2008 financial crisis.
The schematic (Figure 7) shows the Harrod Domar modeling used by Betz et al. In the schematic below,
Betz et al, give the following descriptions:
1) Clouds represent sources of flow
2) Arrow represent direction
3) Rectangles denote Stocks
4) Triangle over oval denote rates
Although not present in the original Harrod Domar model, Betz et al, in addition to (t), the ratio for
savings to investment, have introduced two other variables: (a) a ratio for change in output to change in
capital; («) as ratio of increased output (AY) to increased capital (AK). Betz et al have not supported the
model representation with data sets/ output graphs. Their focus largely was to look at the Harrod
Domar model (and other economic models) in the context of the 2008 financial crisis.
July 16 1
Wy is supply ohoupat io is stock of apical
Harrod Domar used by Betz et al
Figure 7
2.8 Discussion
Harrod Domar model and equations give an elegant and aggregative view for a country’s economy. The
model links the growth rate in income (also referred to as the economic growth rate) to two
fundamental variables — the ability of the economy to save and the influence of the capital output ratio
to convert this savings into income (referred to as Wages-Rents-Profits). System Dynamics
representation provides insight on the boundaries of the original model. In particular, the original
Harrod Domar model equations do not account for any linkages such as savings and consumption,
between household capital and firm capital.
By pushing up the savings rate it is possible to accelerate economic growth. The capital output ratio too
influences the rate of growth. By requiring lesser capital to produce income (more efficiency, lesser 0),
economic growth can be accelerated as well.
System Dynamics representation of the Harrod Domar models provides the following important
contributions:
a. Identifies boundaries in the Harrod Domar model - one key finding is that the Household stock
described in the model is never really used in the equations used in the model
b. Helps users, test and simulate the effect of changes in input variables on parts of the model
including the main output variable i.e. economic growth
The following Sections build on the extension to the basic Harrod Domar model and highlight the
sensitivities of input variables as well.
July 16 12
Section 3
Harrod Domar — Population extensions
3.1 Background
This section introduces two important extensions to the Harrod Domar model. The first extension brings
population as a variable in the model. Bringing population into the Harrod Domar equations make the
model more complete in describing real economies. The second extension in this Section discusses
important effects of applying the Harrod Domar population extension to the real world.
3.2 Introducing per capita variables
Population is an integral part for describing a real world economy. Especially for developing countries,
population plays an influential role in determining a country’s prospect in lifting its population out of
poverty.
If population (P) grows at a rate of (n), so that P(t + 1) = P(t)(1 + n) for all time (t), equation 7 can
be rewritten as:
11) 5 = (1 +g) *(1 +n) — (1 — 6) (g), now denotes the per capita growth rate; (n) is the
population growth rate. Equation 11 is derived through substitutions via equations 7,8,9 and 10)
12) ; = g+n-+6 is an useful approximation of equation 11, given the relatively small values of (g)
and (n).
Equation 12 is an important per capita extension of the Harrod Domar model. The equation informs that
the per capita growth rate is now influenced by three variables. In addition to the savings rate (s),
depreciation and the capital output ratio (@), per capita growth is also influenced by the population
growth rate (n). A high rate of population growth adversely affects the economic growth rate.
The per capita enhancements to the Harrod Domar model resulting from equations 11 and 12 above
have been made to Figure 6. The enhanced model has been shown in Figure 8 (a).
July 16 13
ae #
- . VY
Growth Rate Wages-Prof.
its-Rent
Savings % ¥
‘i : mr
Net People
added
yy
(oe)
(a)
Per Capita Growth
——_ i Se :
~~
Capital Output = Depreciation %
Ratio
z)
iii [Depreciation +
ai
Firm Capita | '5"F
core
(b)
Per Capita implementation (a) savings/,
endogenous
Figure 8
rates as (b) savings/, rates as
3.3 Sensitivity of input variables on economic growth
Giving values to the input variables in the Harrod Domar equation (equation 12), provides a realistic
perspective of using the Harrod Domar equation to model a real economy.
The Harrod Domar world view described in equation 12, requires inputs on four variables:
Savings rate, Capital Output ratio, Population Growth rate, Depreciation.
July 16 14
Table 2 shows the absolute effect on per capita growth if each of these variables is changed
independently in steps of 5%.
The tables shows that:
a
On average, changes in the savings rate and capital output ratios have a bigger influence on
economic growth than population growth
A decrease in the capital output ratio influences economic growth the most — this can only be
expected as capital output is a variable that appears in the denominator
b)
c) Population growth has the least influence compared to the other variables
Base
Variable Name -5% -10% Line 5% 10%
Savings -0.42 -1.00 0.17 0.75 1.33
Capital Output 0.78 1.46 0.17 -0.39 -0.89
Population
Growth 0.24 0.32 0.17 0.09 0.02
Sensitivity of growth rate to univariate changes in each of the three input variables in the Harrod Domar per
capita growth equation. Values show percentage changes.
Table 2
3.4 Endogeneity in Harrod Domar variables
The simplicity from the growth equation (equations 7 and 12) in the Harrod Domar model hides the
interrelationships amongst the input variables that determine growth and also the influence of the
growth process on these input variables.
3.4.1 Savings Rate
A sensitivity analysis of the four input variables in Table 2 informs that the savings rate is among the top
drivers of growth. But can the average savings rate be influenced easily by policy? It is intuitive to
understand that a wealthy person (i.e. more per capita income) is in a better position to save more than
a poor person (i.e. less per capita income). In fact lower levels of income may disallow a person from any
savings at all. As an economy grows it increases the per capita income levels which in turn sets the pace
for increased economic growth. Enhancing per capita income levels beyond the minimum threshold
required for subsistence and thus allowing the savings rate to increase, becomes an important policy
objective® (Ray,1998).
The dependence of the savings rate on per capita income levels informs that the savings rate is not an
exogenous variable but it is rather an endogenous variable.
In the simulation exercise described in Figure 8 (b), the savings rate along with the population rate has
been transformed to be a dependent variable. The savings rate changes to corresponding changes in per
capita income have been shown in Figure 9.
July 16 15
Savings % Vs Per Capita Income
35%
30%
Savings %
25%
20%
100,000 113,783 177,447 504,302 1,491,951
Per Capita Income, Money/Year/Person
—=Savings %
Savings as a dependent variable used in simulation — inverted U curve
Figure 9
3.4.2 Population Growth Rate
In addition to the savings rate, population growth rates are also influenced by the per capita income
levels of an economy” (Tsen et al, 2005). The feedback relationship between the per capita income level
and the population growth rate is not as straightforward (and intuitive) as the relationship between
savings rate and per capita income. Social scientists however, appreciate the variation in a country’s
population growth rates with level of economic development. The demographic transition — a change
from higher to lower rates of mortality and fertility — with the level of development has been well
documented “(World Bank, 2000).
Death rates in poor countries (low per capita income) are higher given malnutrition and poorer health
care access. To compensate, the birth rates in such countries are higher too. The combination of higher
mortality and fertility rates keeps overall population growth rate in check. As income levels start to rise,
the death rates fall but birth rates adjust at a slower pace. This leads to an increase in the net population
growth rate — in some countries this has created “baby boom” generations *°(Bloom et al, 2011). In the
longer run, fertility rates revert down, bringing down the net population growth. The inverse “U” shaped
behavior of population growth has been noted in different countries 73(Valli et al, 2011).
In the simulation exercise described in Figure 8 (b), the population growth rate (in addition to the
savings rate) has been transformed to be a dependent variable. The population rate changes to
corresponding changes in per capita income have been shown in Figure 10.
July 16 16
Population Growth % Vs Per Capita
Income
1.5%
es
= 1.0% ———
z 0.5%
2 0.0%
2 100,000 113,783 177,447 504,302 1,491,951
3
a
§
&
Per Capita Income, Money/Year/Person
—— Population Growth
Population growth rate as a dependent variable changes as the per capita income varies
Figure 10
3.4.3 Modelling endogeneity of savings rate and population rates in the Harrod Domar
equation
The System Dynamics framework allows setting up of an enhanced version of the Harrod Domar model
discussed in Figure 6 to allow for changes in per capita income growth levels to influence the savings
rate and the population growth rate. Simultaneous testing of these two variables was conducted to
understand and test the discussion highlighted in 3.4.1 and 3.4.2.
Of the four variables that influence economic growth from Harrod Domar’s per capita extension
(Equation 12: < = g+n+6), the depreciation rate has been set to a standard 10% per year and the
capital output ration has been set to 1.6, at the lower end of the capital output scale in emerging
economies *4(Taguchi et al, 2014).
Per capita income is used to determine the savings rate as well as net population growth rate. Figure 9
and Figure 10 highlight the values taken by savings rate and population growth rates. The shape of these
curves resembles as closely as possible the inverted “U” curve discussion in 3.4.1 and 3.4.2.
Figure 11a shows the effect of the simultaneous changes in the savings rate and population growth rate
on per capita income. Keeping the Savings Rate constant, Figure 11b provides additional insight in the
role that population growth plays in deciding per capita income.
Simulating the effect of savings rate and population growth rate, provides a better understanding of the
discussion in 3.4.1 and 3.4.2:
a. Per capita growth increases with savings, but it slides down at both lower as well as higher
savings levels. It is to be noted that lower savings levels are a result of an inability to save
whereas at higher level of savings there is a lack of inclination to save.
b. From Equation 12, it is clear that population growth reduces a nation’s total growth. For per
capita income to grow, total income growth has to be higher than population growth. From
Figure 11b, below the “Trap” level, population growth is less, so per capita income growth is
increasing. But if one starts just above the “Trap” level, population growth has the ability to
outstrip total income growth and the economy can actually become poorer in per capita terms.
In addition, if the economy has the growth thrust to move beyond (right of the) the “Threshold”
July 16 17
level, the economy would go into sustained growth. Policy to boost savings and or population
control measures can guide an economy to sustained growth’ (Ray, 1998).
Population Growth % Vs Per Capita Income
Population Growth Income Per Capita Growth
== eTotal Income Growth ——Savings
Simultaneous savings rate & population growth changes & effect on per capita growth
Figure 11a
Population & Total Income Growth
6% 856
F 5% ax &
€ =
2 4% 7% 8
s G
© 3% 7% 8
2 s
8 2% Se — ess
8 =
5 as ox &
a 1% Trap Threshold * 8
0% 5%
100,000 325,248 420,958 702,199 1,132,323
Per Capita Income (Money/Year/Person)
——Populstion Growth Rate = ——=Total Income Growth
Population growth changes & effect on per capita growth
Figure 11b
3.5 Discussion
x
<
= 7.0% 34%
6
G 6.0% 32%
5 5.0% 30%
& 4.0% 28%
2 3.0% 26%
£ 2.0% 24%
= 1.0% 22%
S 0.0% 20%
2 100,000 113,783 177,447 504,302 1,491,951
®
§
2 Per Capita Income, Money/Year/Person
Savings %
Extending the basic Harrod Domar model and using a System Dynamics framework to model per capita
levels gives a practical tool for analysts and policy makers to plug and simulate real inputs and observe
the effects on some of the most important macroeconomic variables. Testing the interconnectedness
and endogeneity of variables on per capita income levels provides insights into the workings of an
economy, beyond the understanding provided by standard static models.
July 16
18
Section 4
Solow’s extension to the Harrod Domar model
4.1 Background
Solow provided arguments to think beyond the “long range” values of variables in the Harrod Domar
model — i.e. the assumption of a stable capital output ratio “(Solow, 1956). Solow’s argument is based on
the law of diminishing returns 7° (Ricardo, 1815) to individual factors of production. Capital and human
effort (labor) work in tandem to produce. If there is plenty of labor relative to capital, a small amount of
capital will suffice. On the other hand, if labor is in short supply capital intensive measures such as
automation are required to raise output levels ‘”(Ray, 1998). Solow, proposed that the capital output
ratio (@) is an endogenous variable — not exogenous as was assumed in the basic Harrod Domar model.
4.2 Model Description
Solow’s proposed endogeneity of the capital output ratio being influenced by the level of income and
the level of capital is an additional layer of dynamism for the basic Harrod Domar model. The modular
nature of System Dynamics tools provides modelers the ability to seamlessly add this feature and
enhance the existing model.
Solow’s extension focuses on the per capita output to the per capita capital stock. Per capita capital
stock in turn is derived through a continuation of the basic Harrod Domar equations — in particular,
equation 4.
Equation 4 is K(t + 1) =(1— 6) * K(t) + I(t) 6 is the depreciation; K (t) and K(t + 1) denote capital
at times t and (t + 1).
Dividing through by population (P) growing at constant rate (n), so that, P(t + 1) = (1 + n) P(t), and
using the relationship between savings rate, total income and investment (/(t) = s * Y(t)), equation 4
transforms to:
13) 1 +n) *k(t+1) = (1-6) * k(t) +s * y(t) where k(t), y(t), s denote per capita values of
capital stock, income and savings rate
Solow’s extension connects the per capita values of the capital stock and income using a Production
function 48(Ray, 1998). The production function represents the technical knowledge of the economy,
bringing capital and labour together to produce output. One of the popular forms *° (Koch, 2013) of the
production function — the Cobb Douglas production function — has been used in the simulation exercise
in this paper.
The Cobb Douglas production function has been represented in the simulation through the following
equation:
14) Y(t) = a*A* K% « P!~% where Y(t) is the total income, a is the ratio of capital income to
total income ”° (Piketty,2015), K is the firm capital, A is the technology multiplier and P is the
total labor.
July 16 19
The Cobb Douglas function is notable for its constant returns to scale function (output doubles if the
input variables — capital and labor double) of labor and capital.
Another property of the Production function is the diminishing returns to per capita capital increases.
Figure 12, shows the Production function run for a simulation of the Solow extension to the Harrod
Domar model. This function plots the changes in wages per capita in response to changing per capita
capital.
Wages Per Capita Vs Capital Per Capita
3.1 Production function
Mn/Year/Preiod)
A
<— -
o. =
--
Wages Per Capita (Money
i ‘i
=" Capital output = 2.6 — Capital output = 2.8
-
0.2 1.0 2.3 3.8 53 6.5 75 8.3 8.8 9.3 9.6
Capital per capita (Mn)
Wages Per Capita (Mn)
Solow’s extension to the Harrod Domar model
Population Growth=1.35%, Alpha=0.7, Savings=35%, Depreciation=10%
Figure 12
4.3 Changes required for implementing Solow’s extension
Figure 13 shows the modified System Dynamics representation of the Solow’s extension to the Harrod
Domar model. Figure 13 is the enhanced version of the per capita Harrod Domar model discussed in
Figure 8. This enhanced version, in line with equations, 13 & 14, now has the capital output ratio as a
variable dependent on labor participation, the ratio of capital income to total income, level of
technology and firm capital.
July 16
20
Per Capita Growth =
oO
Net Peop}t adde ({Depreciation
1? Savings % RI
‘apital
(Investment
ioe)
Capital per capita
Solow’s extension to the Harrod Domar model
Figure 13
4.4 Endogeneity of the Capital Output Ratio
The range of values that the capital output ratio takes to changes in per capita income has been shown
in Figure 14. The capital output ratio becomes relatively insensitive to the higher levels of per capita
income (as well as per capita capital as shown in Figure 12). Although the output per person continues
to rise, due to a relative shortage of labor, the ratio of output to capital used in production falls.
Capital Output Ratio
PRN Nw
cououso
0.5
224,953 1,255,885 2,221,493 2,841,203 3,189,876 3,375,290
Per Capita Income (Money/Person/Year)
Capital Output Ratio - Years
—Capital Output Ratio
Change in Capital Output ratio to changes in per capita income
Figure 14
July 16 21
Growth in Solow’s extension loses momentum if capital is unable to grow fast relative to labor. This is
shown in Figure 15
Per Capita Income Growth
30%
25%
PoP oN
eas
RRR
a
BS
0%
Per Capita income Year over Year %
——Per Capita Growth
Per Capita Growth in Solow’s extension
Figure 15
4.5 Discussion
Solow’s extension allowed for modeling the important concept of diminishing return to capital. System
Dynamics tools added insights to the modeling process by incorporating the endogeneity of the capital
output ratio to list of other endogenous variables. Solow’s extension introduced additional new
variables that include: labor participation, level of capital income to total income, level of technology.
July 16 2
Section 5
Discussion
Successive additions to the simple Harrod Domar, four variable world view show how a simple model
extends itself beyond its original boundaries to incorporate additional perspectives. The Harrod Domar
model demonstrates both the usefulness and limitations of modelling. Simple models are useful as they
provide a clear and quick understanding based on limited number of important variables. But the
limitations too become clearer as a simple model is validated against a range of inputs. Among others,
equations supporting consumption & savings links between households and firms are missing in the
original Harrod Domar model. Additional variables and changing nature of the variables (from being
exogenous to endogenous), limits the usefulness of the original model.
Starting from the first Harrod Domar model (Figure 2), successive extensions incorporated increasing
detail. The nature of variables changed from absolute to per capita. Interconnectedness among the
variables provided a completely different dimension. System Dynamics tools provided insights by
seamlessly incorporating the endogeneity of variables. As part of future work, it will be useful to test &
tune the models in this paper to real inputs based on macroeconomic variables of different countries.
References
1 The fall and rise of development economics, Paul Krugman, MIT, 1995
? Why Development is more complex than growth, Hamid Hosseini, King’s College, March, 2003
3 Development Economics, Figure 3.1, Page 52, Debraj Ray, 1998
* A contribution to the theory of Economic Growth, The Quarterly Journal of Economics,, pp. 65-94, Robert M.
Solow, February, 1956
5 Business Dynamics, pp 21, John D. Sterman, 2000
® Development Economics, Figure 3.1, Page 52, Debraj Ray, 1998
7 Development Economics, Debraj Ray, 1998
® Disequilibrium Systems Representation of Growth Models—Harrod-Domar, Solow, Leontief, Minsky, and Why the
U.S. Fed Opened the Discount Window to Money-Market Funds, Frederick Betz, Modern Economy, 2015
°* Development Economics, Debraj Ray, Page 59, 1998
10 The Relationship between Population and Economic growth in Asian Economies, Wong Hock Tsen, Fumitaka
Furuoka, ASEAN Economic Bulletin, Vol 22, No 3 (2005), pp 314-30, 2005
4 Beyond Economic Growth, World Bank, September, 2000
22 Implications of Population aging for Economic Growth, David E. Bloom, David Canning, Gunther Fink, PGDA
working paper 64, Harvard, January 2011
33 Economic and population growth: an inverted-U shaped curve?, Vittorio Valli, Donatella Saccone,
University of Turin, June 2011
14 A revisit to the incremental capital-output ratio: the case of Asian economies and Thailand, Hiroyuki Taguchi,
Suphannada Lowhachai, Journal of Economic policy in emerging economies, 2014 Vol.7, No.1, pp.35 - 54
+5 Development Economics, Debraj Ray, pp.60-62, 1998
16 Essay on the influence of a Low Price of Corn on the Profits of Stock, David Ricardo, 1815
17 Development Economics, Debraj Ray, pp.64, 1998
18 Development Economics, Debraj Ray,pp 65, 1998
*° The Cobb-Douglas Function: Simple Derivations and How Students Might Accept Strange Dimensional-
Properties, Matthias Koch, Vienna University of Economics and Business, June, 2013
20 Economics of Inequality, Thomas Piketty, 2015
July 16 23
Question (ranges
from 1-never to 5-
frequently)
Examples
example— “That science log is high level learning!”
“| had one student try to explain that he was going to be sitting next to certain
students at lunch in a reinforcing loop because he was doing Kid A, Kid B, Kid
A, Kid B. Clearly not the correct use...but he recognized a similar pattern in his
own life!”
"A precious current example: a student was saying how the more he & his
parents paid attention to their baby, the cuter the baby seemed. Then the cuter
the baby seems, the more they want to pay attention to him. A classmate
helped to construct a causal loop.”
“We are making a light & sound show in science. Students are constructing
musical instruments that demonstrate variability in pitch and volume, and many
of them cited using recycled materials as having less impact on the world than
using previously unused material would.”
“Beginning with rudimentary use of BOTGs in reading response, students have
now gone deeper, working to identify, understand, and communicate the
interdependent nature of character's thinking: actions."
“will independently create a BOTG or stock flow to represent something they are
curious about or interested in”
e Students would use cause and effect language (and thumbs to demonstrate
increase/decrease) to describe playground issues and their thinking about
literature.
2nd-3rd Grade
e “using behavior[-over-]time graphs to track eating/exercise habits... yes! 3rd
graders!”
e “I had a student [who] kept a personal journal of BOTGs that she created at
home. She noticed trends and changes over time in her house, and she
graphed them (example: TV usage).”
e “Students often refer to the tools or habits to explain situations they find
themselves in.”
e “Students are beginning to think about the habits and how that [affects] their
lives and choices they make”
Multiple Grades
e “Student initiated BOTGS at home in regards to likes and dislikes of trying new
foods, excitement about home celebrations”
e "Many of my students have experienced friendship issues and changes in
friends that happen. They have been able to use BOTGs to see
patterns/changes and discovered what might have been causing those
changes in their friendships.”
e “My students also have used BOTGs to help understand/explain what helps
them learn and what distracts them from learning. This opens up
conversations about what behaviors can help them stay focused and learn.”
Developing Understanding of Dynamic Systems Within Early Childhood Settings 24
Survey Resp Types of A Used to Gather Evidence of Learning
Type (choose as many as are used) Number of Respondents
Parent Observation 2/18
Portfolios 7/18
Pre-test and Post-test 4/18
Rubric Assessment 4/18
Self-assessment with Rubric 4/18
Teacher Observation 17/18
Evidence of Student Learning Given by Teachers
Preschool
e “l was teaching a herpetology class. Once a week | took students out to count the lizards and
snakes that we saw and mark them on a BOTG. During that class | had a herpetologist from
the U of A come to work with the kids. He happened to show them a behavior over time
graph of his research. | was so proud when my 4 & 5 year olds were able to analyze his
graph.”
e “One day, a child asked to take the BOTG we created at story time to the writing table. The
teacher handed the BOTG to her and observed what happened next. The child went to the
table, and proceeded to create a replication of the BOTG on her own piece of paper. When
the teacher inquired as to what she was doing, she stated, "| want to make the graph so | can
take it home and tell my daddy about the story." Her mother was the one to drop her off and
pick her up from school each day, so she had not been able to share prior BOTGs with her
dad like she had with her mom at pick up time.”
K-1st Grade
e “We start the year learning about realism and fantasy stories. This is such an easy way to
learn this skill with a visual BOTG. | put some of the pictures from the story on the graph and
then we decide what is real about the story and what is fantasy. It can be a simple thing to
use and so effective for those low learners!”
e “The story structure archetype and the reinforcing loop have been most effective when we dig
into stories. They are able to sequence better with these systems [tools] and it increases
their understanding.”
e “Just one detour back up to the question of how often I've noticed students mentioning ST
habits/tools: sometimes, things become such a matter of routine that they are unremarkable.”
e “At the beginning of this school year my students launched a project involving problems with
a drinking fountain. Over time, they decided that a water bottle filling fountain would be a
great solution. Having gained the support of our district's facilities director, the principal next
came to us to ask about its installation. At that moment, [there] was instant conversation
regarding best placement, and the final decision was to locate it where it would be [most]
accessible by all students, and additionally where it would not cause interference with our
normal routines. Independent application is the best demonstration of learning!"
e “Using a BOTG to investigate the plot of various literature and then applying what they have
learned to write their own stories with BOTGs.”
Developing Understanding of Dynamic Systems Within Early Childhood Settings 25
e "Using systems-thinking strategies has made me a more engaged teacher and has helped
motivate me through my own insecurities and my worries about below grade-level students
making progress.
| remember watching two students (both of whom struggled with decoding and writing) build a
connection circle as they discussed the book Mr. Tiger Goes Wild. Their depth of thinking,
ability to argue in a positive way about text details, and confidence in their ability to do well
was amazing.
| feel like these strategies allowed me to teach deep, critical thinking skills in a way that
allowed below grade-level readers/writers and above grade-level readers/writers to engage
with content on more equal footing. As | would watch students with low decoding and writing
skills become excited/motivated, my own excitement and motivation to teach would
increase."
2nd-3rd Grade
e “Having a student explain thoughts behind a BOTG. | thought the student was ‘wrong’ or
‘missing the point’ but as he explained why he made the graph the way he did, | realized he
had a deeper understanding of the concept. He actually added a small nuance that | had not
considered.”
e “When my students entered the third grade, a teacher noticed that a former student of mine
created a stock and flow to help answer a question on a standardized test. This shows me
that the information transferred from grade to grade, and that the tools can be utilized in
many situations.”
e “Students did the in/out game to reinforce the concept of stock/flows. Students were highly
engaged during the lesson. Students participated in high-level thinking, discussion, and even
healthy debate as they participated in the simulation.”
Multiple Grades
e “My students can verbally and through writing better explain their thinking and reasoning. The
visual tools [aid] in explaining for those students lacking vocabulary and language.”
Developing Understanding of Dynamic Systems Within Early Childhood Settings 26
Appendix D
Water Challenges
For each challenge, start with all the extra water in the
inflow “cloud.” Before beginning, graph the goal (what
the water should do over time). Then, graph what
actually happens in a different color. After completing a
challenge, continue with the next one. Feel free to
repeat any challenge to improve results.
_—
Dice ong i
ty
(ME 4 =
Challenge #1
Start with 200 ml of water in the
cylinder, then raise the water from
200 ml to 800 ml while water is
always going in and always going
out.
800
ml of water
in cvlinder 500
Water Challenge #1 Results
200
Time (in seconds)
Challenge #2 Water Challenge #2 Results
Start with 800 ml of water in the 800
cylinder, then lower the water
from 800 to 200 ml, while water is ml of water
always going in and always going
out.
in cvlinder 500
200
Time (in seconds)
Challenge #3 Water Challenge #3 Results
Start with 500 ml of water in the 800
cylinder, then keep the water at
the 500 ml level for at least 30 ml of water
seconds, while water is always
going in and always going out.
in cvlinder 500
200
Time (in seconds)
Developing Understanding of Dynamic Systems Within Early Childhood Settings
27
Challenge #4
Create the water level shown on
the graph while water is always
going in and always going out.
Water Challenge #4 Results
800
ml of water
in cvlinder 500
200
Time (in seconds)
Reflection: Match the graph to the description.
A
1. Inflow < Outflow
Graph describes this situation because
2. Inflow > Outflow
Graph describes this situation because
3. Inflow = Outflow
Graph describes this situation because
Developing Understanding of Dynamic Systems Within Early Childhood Settings
28
Challenge #5
Start with 500 ml in the cylinder. Predict on the graph: What will happen if you adjust the
inflow and outflow as shown? Notice the outflow stays the same while the inflow starts at
slow and gets faster over time. Use the graph on the right to predict and then record what
actually happens over time.
Water Challenge #5 Water Challenge #5 Results
Fast
nor o » 800
3 og
2 wigan Outflow 2 £500
jedium ss
3 ob:
iy = 200
s Slow
Time
Time
Challenge # 6: Create your own challenge
Create a challenge similar to #5 that graphs what you will do with the flows and predict what
will happen to the stock. Make sure to label the inflow and outflow lines.
Fast Water. Challenge:#6 Water Challenge #6 Results
3 B . 800
= Medium $ 3 500
3S coe
& —
<
S Slow £ £ 200
Time Time
Reflection: Consider that the water in the cylinder is a stock and the water going in and out
are flows. What else in your experience is similar to the stock of water? How do the flows
work in that system?
Developing Understanding of Dynamic Systems Within Early Childhood Settings
29
References
Base, Graeme (2001). The Water Hole. Harry N. Abrams; First Edition edition.
Bowman, B.T., Donovan, M.S., & Burns, M. S. ed. (2001). Eager to Learn. Washington DC:
National Academy Press.
Brown, Chris and Newell, Barry, ANU. Original water apparatus design. Modified with permission
for water challenges. 2014.
Elya, Susan Middleton and Sweet, Melissa (2010). Rubia and the Three Osas. Disney-Hyperion.
Copple, C. and Bredekamp, S. eds. (2009). Developmentally appropriate practice in early
childhood programs serving children from birth through age 8 (3 ed.). Washington: NAEYC.
Dias, Ron and Disney, RH (2005). Cinderella. Golden Books, English Language edition.
Galinsky, E (2010) Mind in the Making: The seven essential life skills every child needs, Harper
Collins: NY.
Hays, Michael and Seeger, Pete (1994). Abiyoyo, Aladdin Paperbacks.
Katz, L.G., and Chard S.C. (2000). Engaging children’s minds: A project approach. Stamford,
Connecticut: Ablex Publishing Corporation
Keats, Jack Ezra (1998). Goggles. Puffin Books, Reprint edition.
Mackinnon et al. (2007). The Gingerbread Man. Usborne Pub Ltd.
Miller, J.P., Illustrator (2001). The Little Red Hen. Golden Books, English Language edition.
National Governors Association Center for Best Practices & Council of Chief State School
Officers. (2010). Common core state standards. Washington, DC: Authors.
National Research Council (2001). Eager to learn: Educating our preschoolers. Committee on
Early Childhood Pedagogy. Barbara T. Bowman, M. Suzanne Donovan, and M. Susan Burns,
editors. Commission on Behavioral and Social Sciences and Education. Washington,
DC: National Academy Press.
Quaden, Rob and Ticotsky, Alan (2004). The Shape of Change. Creative Learning Exchange.
Sykes, M. (2014). Doing the Right Thing for Children: Eight Leadership Qualities of Leadership.
RSt. Paul, MN: Redleaf Press.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes.
Cambridge, MA: Harvard University Press.
Waters Foundation (2010, 2014). Habits of a Systems Thinker. Waters Foundation.
Wood, D., Bruner, J., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child
Psychology and Child Psychiatry, 17, 89-100.
Wood, D., & Middleton, D. (1975). A study of assisted problem-solving. British Journal of
Psychology, 66(2), 181-191.
Developing Understanding of Dynamic Systems Within Early Childhood Settings 30
