# KNOW–A Real-World Ontology for Knowledge Capture with Large Language Models

Arto Bendiken
Haltia, Inc.
<EMAIL_ADDRESS>

###### Abstract

We present KNOW–the Knowledge Navigator Ontology for the World–the first
ontology designed to capture everyday knowledge to augment large language
models (LLMs) in real-world generative AI use cases such as personal AI
assistants. Our domain is human life, both its everyday concerns and its major
milestones. We have limited the initial scope of the modeled concepts to only
established human universals: spacetime (places, events) plus social (people,
groups, organizations). The inclusion criteria for modeled concepts are
pragmatic, beginning with universality and utility. We compare and contrast
previous work such as Schema.org and Cyc–as well as attempts at a synthesis of
knowledge graphs and language models–noting how LLMs already encode internally
much of the commonsense tacit knowledge that took decades to capture in the
Cyc project. We also make available code-generated software libraries for the
12 most popular programming languages, enabling the direct use of ontology
concepts in software engineering. We emphasize simplicity and developer
experience in promoting AI interoperability.

_K_ eywords Ontology $\cdot$ Real-world knowledge $\cdot$ Commonsense
knowledge $\cdot$ Knowledge representation $\cdot$ Knowledge graphs $\cdot$
Code synthesis $\cdot$ Large language models $\cdot$ Neuro-symbolic AI

##  1 Introduction

While not yet widely known in industry, the neuro-symbolic approach to
artificial intelligence (AI)–in particular, the union of large language models
(LLMs) and knowledge graphs (KGs)–has in academia already established itself
as the most promising state-of-the-art pathway towards building practical,
trustworthy, explainable, and interoperable AI systems. We will briefly
outline the promise of the neuro-symbolic synthesis and explain why common
ontologies are needed for reliability and interoperability.

###  1.1 Why LLMs Need KGs

Despite limited successes with prototypes which escaped the lab, over the
course of the next several years industry attempts to deploy standalone LLMs
in production will be increasingly frustrated by the limitations and
deficiencies of said models: the limited context window and its poor scaling
characteristics, the lack of introspectability and justifiability, the
propensity towards hallucinations, as well as ultimately the implicit and
static nature of the very knowledge encoded during pre-training and the
difficulty of amending it afterwards.

These various problems can be mitigated or solved by a hybrid approach where
the LLM accesses and manipulates a symbolic knowledge base where facts are
captured and represented in explicit form.

###  1.2 Why KGs Need LLMs

Conversely, large language models do already encode internally much of the
commonsense tacit knowledge that in previous decades proved the most
formidable challenge to symbolic approaches of knowledge representation and
AI. Until recently, any toddler ultimately knew more about the nature and
causal structure of our physical and social worlds than any computer. The
ambition of the decades-long Cyc project [Lenat and Guha, 1989] that sought to
capture and encode all commonsense knowledge was amply matched by the
interminability of the endeavor.

An LLM’s “knowledge soup”–a large reservoir of loosely organized encyclopedic
knowledge [Sowa, 1990]–is quantitatively derived from statistical
distributions in an LLM’s pre-training dataset and after training is
implicitly encoded in the billions of weights that constitute the model
[Templeton et al., 2024]. As mentioned, it is not particularly introspectable
nor is it even consistent, but it is nonetheless useful: it largely sidesteps
the need to explicitly represent and encode truly basic tenets of the world,
such as that the noon sky is blue, that the arrow of gravity points downwards,
and that humans tend to live in families and build housing to seek shelter
from the elements.

###  1.3 The Neuro-Symbolic Synthesis

In this way, neural networks in the form of LLMs provide the missing
ingredient for a hybrid that realizes long-standing visions for symbolic AI
beyond narrow expert system use cases. And conversely, the explicit knowledge
representation in KGs complements and enables LLMs to transcend their
limitations and ultimately be safely deployed in real-world use cases beyond
technology demos.

To realize the promise of this neuro-symbolic synthesis, however, the language
model must understand the restricted vocabulary (that is, the set of symbols)
used for knowledge representation. This is where the need for an ontology
comes into play. The ontology restricts and guides knowledge capture [Çöplü et
al., 2024], defines and describes the semantics of what has been captured
[Allemang and Sequeda, 2024], and enables the LLM to give better, more precise
responses.

We present herein the first iteration of our ongoing effort to design a
commonsense, everyday ontology suitable for real-world use cases for neuro-
symbolic AI. We have appropriately titled our ontology KNOW, a backronym
standing for Knowledge Navigator Ontology for the World. Our ontology is open
source–indeed, it is placed into the public domain with no strings attached.

##  2 Domain and Scope

Our domain is human life, both its everyday concerns and its major milestones.
We have limited the initial scope of the ontology to only established human
universals: spacetime (places, events) plus social (people, groups,
organizations). Top-level classes are illustrated in Figure 1. We anticipate
extending this scope in future work.

Figure 1: Top-level classes in the ontology (not comprehensive)

Human universals “comprise those features of culture, society, language,
behavior, and psyche for which there are no known exception” [Brown, 1991].
That is, human universals are common to all human societies ever studied. They
include, for example, language, gossip, jokes, toys, tools, promises,
symbolism, semantic categories, and logical notions.

We as humans are embedded in spacetime. To understand and interact with the
physical world in which we live, time and space are the fundamental concepts
that provide the basic framework for navigating and orienting ourselves in the
world as well as the dimensions in which all physical events occur. From
spacetime, human universals, and language essentials, we derive the base
concepts Place and Event.

Examples of places common to all human societies include landmarks and places
of worship. For contemporary developed societies, we also include airports,
cafes, hospitals, hotels, and restaurants. Some examples of events for the
same include birthdays, appointments, holidays, meetings, and parties.

As highly social creatures, the stature of the social reality–people and
relationships–that we inhabit is nearly that of spacetime. Indeed, people have
been known to prioritize consensus social reality over objective physical
reality in their understanding of the world and actions in it. In this social
dimension, then, we include the base concepts of Person and Group, the latter
denoting a group of people. Unlike a biologist’s classification of a person as
a human, hominid, anthropoid, primate, mammal, and vertebrate animal, we are
here concerned only with commonsense understanding111Indeed, the distinction
here is perhaps most clearly illustrated by considering the question of
animals: scientific taxa and evolutionary lineages aside, from an early age
children tend to consider themselves as something categorically distinct from
animals. and omit these taxonomic superclasses–at least in this first
formulation of the ontology.

##  3 Use Cases

Our primary use cases revolve around everyday human concerns. For example, and
in particular:

Social relationships and roles. Family, friends, acquaintances, coworkers,
authority figures, and the various roles people play in society.

Basic human needs. Shelter, heating, cooling, water, food, healthcare, safety,
etc. [Gupta, 2008]

Emotions and mental states. Happiness, sadness, anger, fear, love, desire,
beliefs, intentions, etc.

Daily activities and routines. Work, school, household chores, leisure
activities, transportation, etc.

Time and events. Temporal concepts (past, present, future, duration, etc.),
and common events and life milestones that occur in human life (birth,
education, employment, marriage, illness, death, etc.)

For secondary use cases, we also wish to extend support to fictional uses,
especially modeling fictional social graphs–including in speculative fiction.
This means, for example, that we don’t constrain the value range of the age
property on Person. And on the other hand, we call the class Person instead of
Human; on the gripping hand, however, we haven’t at present included an Agent
class, as that would veer away from practicability. Similarly, Place isn’t
inherently something constrained only to Earth’s surface, though the
supermajority of uses will be exactly that. When practical uses and fictional
uses are in tension, we have aimed to make the fictional use possible yet the
practical use convenient.

##  4 Software Libraries

Large language models have proven capable code generation tools, and are
deployed both for advanced code completion in code editors as well as in more
ambitious prompt-driven uses. Given how code synthesis is seeing an
unprecedented surge in interest, we have explicitly designed the ontology to
be usable towards this important end. This affects, for example, property
naming, which is designed to accord with naming conventions for coding.

We have made available code-generated open-source software development kits
(SDKs) for the 12 most popular programming languages, enabling the direct use
of ontology concepts in software design and implementation. The initial set of
programming languages comprises C, C++, C#222And other programming languages
on the Microsoft .NET platform., Dart, Go, Java333And other programming
languages on the Java Virtual Machine (JVM) platform., JavaScript, Python,
Ruby, Rust, Swift, and TypeScript.

Our primary objective for the SDKs is to be able to represent ontology
concepts using the native composite data abstraction for each language. For
example, we represent ontology classes as structures in C, Go, and Rust; and
as classes in Dart, Java, Python, and Ruby.

Where possible (when supported by the language), we also define abstract
interfaces (C#, Dart, Go, Java, TypeScript), traits (Rust), or protocols
(Swift) for the same. These are in fact the highest-value uses of our SDKs,
since they make possible semantic interoperability between third-party
libraries that each can depend on our SDK so as to achieve type compatibility
between themselves.

A secondary objective for the SDKs is to enable data interchange via
serializations in JSON and RDF form, where appropriate for the language and
when a predominant RDF framework is available: for example, we support Jackson
and Jena for Java, and RDF.rb for Ruby.

It is our intention to continue, through open-source community contributions,
to extend language support over time so as to cover at least the top 20 most
popular programming languages and establish the initiative as a cornerstone
for polyglot programming.

##  5 Compare and Contrast

We compare and contrast previous work including Schema.org and Cyc.

###  5.1 Schema.org

The previous ontology nearest to our endeavor is undoubtedly Schema.org [Guha
et al., 2016], and we make sure to map our concepts (classes and properties)
to corresponding Schema.org concepts where possible. Though there is
inevitably some overlap, the two projects have different focuses and
ultimately different audiences.

Whereas Schema.org grew out of microdata (structured data markup on web pages)
and thus focuses on describing things such as creative works, KNOW is
specifically designed for neuro-symbolic uses and focuses on describing life,
beginning with social relationships.

For example, during the first decade of the Schema.org initiative, it seems
nobody has as yet needed to describe family relationships in greater detail
than parent/child and sibling. KNOW, on the other hand, directly includes
properties for father/mother, brother/sister, uncle/aunt, nephew/niece, and so
on–materializing these most important personal relationships explicitly,
without the need for graph traversal nor entailment at the point of lookup.

KNOW also differs in its insistence on modeling the world in a commonsense,
everyday manner. We are not concerned with “correct” taxonomy, but merely
pragmatic taxonomy, leading to a relatively flat class hierarchy. A further
difference is due to our support for code generation, described in a previous
section.

###  5.2 Cyc (OpenCyc, ResearchCyc)

In seeking to model commonsense knowledge, our most esteemed predecessor is
the already mentioned Cyc project. The differences, though, are also clear.
For one, we don’t aspire to be an upper ontology; for another, the
expressiveness of our ontology is merely the description logics of OWL, not
the undecidable higher-order logic (HOL) used in Cyc.

Our initiative also uses open source instead of proprietary licensing, and a
public and collaborative development model instead of a closed and internal
one. Our development cycle is iterative–acknowledging that worse is better
[Gabriel, 1989]–and driven by real-world use cases.

##  6 Availability and Usage

The ontology is developed using an open-source ethos and a public,
collaborative, and iterative development process at GitHub
(github.com/KnowOntology), with documentation and downloads made available on
a dedicated website (know.dev).

###  6.1 Base IRI and Prefix

The base IRI for the ontology is https://know.dev/, and our recommended prefix
for CURIE construction is know. The base IRI has been designed to be memorable
in support of our emphasis on developer experience. In Turtle format:

    @base <https://know.dev/> .
    @prefix know: <https://know.dev/> .

##  7 Conclusion

We have outlined a first draft of KNOW, an ontology for capturing and
representing everyday knowledge to augment LLMs in real-world generative AI
use cases such as personal AI assistants, and explained how and where it fits
into the emerging neuro-symbolic–or neuro-semantic, as it may be [Hermelen,
2024]–synthesis.

It is worth keeping in mind the starting point that most practitioners working
with LLMs today are as yet unaware of knowledge graphs, semantic technology,
or even the need for an ontology for knowledge capture. Practitioners’ work
generally proceeds, as it has for decades, based on ad-hoc properties conjured
up on the spot when they were first needed, with little regard for stronger
semantics or wider interoperability. This, though, will become increasingly
untenable as an approach over the next several years.

While being based on a best-practice Semantic Web technology stack (that is,
utilizing the RDF and OWL standards), by emphasizing wider availability and
applicability, and prioritizing simplicity and the developer experience, we
expect the KNOW initiative to serve as something of a Trojan horse for the
dissemination of semantic technology in generative AI use cases more broadly
as well. Much remains to be done, but the promise here is ultimately that of
interoperable AI, where knowledge captured by one system can flow to and be
utilized in other vendors’ systems as well.

## Acknowledgements

We would like to thank Tolga Çöplü and Stephen Cobb for their review and
constructive feedback, which significantly enhanced the clarity and quality of
this paper.

## References

  * [Allemang and Sequeda, 2024] Allemang, D. and Sequeda, J. (2024).  Increasing the LLM Accuracy for Question Answering: Ontologies to the Rescue!  arXiv:2405.11706 [cs].
  * [Brown, 1991] Brown, D. E. (1991).  Human universals.  Human universals. Temple University Press, Philadelphia, PA, US.  Pages: x, 220.
  * [Gabriel, 1989] Gabriel, R. P. (1989).  Worse is better.  https://www.dreamsongs.com/WorseIsBetter.html.
  * [Guha et al., 2016] Guha, R. V., Brickley, D., and Macbeth, S. (2016).  Schema.org: evolution of structured data on the web.  Communications of the ACM, 59(2):44–51.
  * [Gupta, 2008] Gupta, V. (2008).  Six ways to die.  https://www.appropedia.org/Six_ways_to_die.
  * [Hermelen, 2024] Hermelen, F. v. (2024).  Neuro-Symbolic is not Enough, We Need Neuro-*Semantic*.  https://www.slideshare.net/slideshow/neuro-symbolic-is-not-enough-we-need-neuro-semantic/269377374.
  * [Lenat and Guha, 1989] Lenat, D. B. and Guha, R. V. (1989).  Building Large Knowledge-Based Systems; Representation and Inference in the Cyc Project.  Addison-Wesley Longman Publishing Co., Inc., USA, 1st edition.
  * [Sowa, 1990] Sowa, J. F. (1990).  Crystallizing Theories out of Knowledge Soup.  Intelligent Systems: State of the Art and Future Directions, pages 456–487.  Publisher: Ellis Horwood USA.
  * [Templeton et al., 2024] Templeton, A., Conerly, T., Marcus, J., Lindsey, J., Bricken, T., Chen, B., Pearce, A., Citro, C., Ameisen, E., Jones, A., Cunningham, H., Turner, N. L., McDougall, C., MacDiarmid, M., Freeman, C. D., Sumers, T. R., Rees, E., Batson, J., Jermyn, A., Carter, S., Olah, C., and Henighan, T. (2024).  Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet.  Transformer Circuits Thread.
  * [Çöplü et al., 2024] Çöplü, T., Bendiken, A., Skomorokhov, A., Bateiko, E., and Cobb, S. (2024).  Prompt-Time Ontology-Driven Symbolic Knowledge Capture with Large Language Models.  arXiv:2405.14012 [cs].


# Assessing the practicability of the condition used for dynamic equilibrium
in Pasinetti’s theory of distribution.

A. Jayakrishnan  Anil Lal S  Ielektron Technologies, IIT Madras Research Park,
Taramani, Chennai, India - 600 113
Email<EMAIL_ADDRESS>College of Engineering,Trivandrum,
Kerala, India - 695016.
Email<EMAIL_ADDRESS>
###### Abstract

In this note an assessment of the condition $K_{w}/K=S_{w}/S$ is made to
interpret its meaning to the Passineti’s theory of distribution[1]. This
condition leads the theory to enforce the result $s_{w}\rightarrow 0$ as
$P_{w}\rightarrow 0$, which is the Pasinetti’s description about behavior of
the workers. We find that the Pasinetti’s claim, of long run worker’s
propensity to save as not influencing the distribution of income between
profits and the wage can not be generalized. This claim is found to be valid
only when $W>>P_{w}$ or $P_{w}=0$ with $W\neq 0$. In practice, the Pasinetti’s
condition imposes a restriction on the actual savings by one of the agents to
a lower level compared to its full saving capacity. An implied relationship
between the propensities to save by workers and capitalists shows that the
Passineti’s condition can be practiced only through a contract for a constant
value of $R=s_{w}/s_{c}$, to be agreed upon between the workers and the
capitalists. It is showed that the Passineti’s condition can not be described
as a dynamic equilibrium of economic growth. Implementation of this condition
(a) may lead to accumulation of unsaved income, (b) reduces growth of capital,
(c)is not practicable and (d) is not warranted. We have also presented simple
mathematical steps for the derivation of the Pasinetti’s final equation
compared to those presented in [1].

###### keywords:

Propensity to save , Full saving capacity , Actual saving , Worker’s share on
profit , Theory of distribution

††journal: Journal

##  1 Introduction

Pasinetti’s theory of distribution[1] has introduced a correction to the
logical slip in Kaldor’s theory[2] of not allocating profit on worker’s
saving. Passineti argued that whenever a worker saves a part of his income, he
must be allowed to own a portion of the total capital. Otherwise the workers
do not save at all. The ownership of assets generates a profit share to the
workers in the same manner as that of capitalists. The distribution is
expressed as

$P=P_{c}+P_{w}$ (1)

Where $P_{c}$ and $P_{w}$ are profit shares to capitalists and workers
respectively. This correction leads to workers getting a feel of ownership
along with a consequent increase in their total income. Now, the total income
of workers becomes $P_{w}+W$, where $W$ is worker’s wage. The propensities to
save by the workers ($s_{w}$) and the capitalists ($s_{c}$) result in the
partial savings of capitalists $S_{c}=s_{c}P_{c}$ and workers
$S_{w}=s_{w}(P_{w}+W)$ and the total saving $S$ as

$S=S_{c}+S_{w}=s_{c}P_{c}+s_{w}(P_{w}+W)$ (2)

The purpose of this note is to interpret the meaning of the following
condition Pasinetti has used for the dynamic equilibrium of economic growth.

$\frac{S_{w}}{S}=\frac{K_{w}}{K}$ (3)

where $K_{w}$, $K_{c}$ and $K=K_{w}+K_{c}$ are the quantity of capital owned
by workers, capital owned by the capitalists and total capital respectively.
The condition in equation(3) is referred to in this note as Pasinetti’s
condition for the dynamic equilibrium. Another assumption used in the
derivation of the Pasinetti’s final equation for the theory of distribution is
a constant profit ratio given by

$\frac{P_{w}}{K_{w}}=\frac{P_{c}}{K_{c}}=\frac{P}{K}$ (4)

Dividing equation(1) through out by $K$ we can write

$\displaystyle\frac{P}{K}-\frac{P_{w}}{K}=\frac{P_{c}}{K}$
$\displaystyle\frac{P}{K}-\frac{P_{w}}{K_{w}}\frac{K_{w}}{K}=\frac{P_{c}}{K}$
$\displaystyle\frac{P}{K}-\frac{P}{K}\frac{K_{w}}{K}=\frac{P_{c}}{K}~{}~{}~{}(\because\mbox{constant
profit ratio in equation-\ref{asump2}})$
$\displaystyle\frac{P}{K}\left[1-\frac{K_{w}}{K}\right]=\frac{P_{c}}{K}$ (5)

Now, the Pasinetti’s condition for dynamic equilibrium in equation(3) is
applied to simplify the above expression to obtain the following equation.

$\displaystyle\frac{P}{K}\left[1-\frac{S_{w}}{S}\right]=\frac{P_{c}}{K}~{}~{}~{}~{}~{}\mbox{or}~{}~{}~{}~{}~{}~{}\frac{P}{K}\left[\frac{S_{c}}{S}\right]=\frac{P_{c}}{K}$
(6)

By taking $S=I$ and $S_{c}=s_{c}P_{c}$, we get

$\displaystyle\frac{P}{K}\left[\frac{s_{c}P_{c}}{I}\right]=\frac{P_{c}}{K}\implies
P_{c}\left[\frac{P}{K}-\frac{1}{s_{c}}\frac{I}{K}\right]=0$

In a capitalist economy $P_{c}\neq 0$ and hence we write the Pasinetti’s final
equation for distribution as

$\frac{P}{K}=\frac{1}{s_{c}}\frac{I}{K}$ (7)

It may be noted the mathematical steps used for obtaining the above final
equation is simple which students of economics can more easily understand and
comprehend in comparison to the one given in [1]. Now we will look at the
difference between Pasinetti’s model and Kaldor’s model. In Kaldor’s theory
$P_{w}=0$, hence by equations(1)&(2), we have

$\frac{P}{K}=\frac{P_{c}}{K}=\frac{S-s_{w}W}{Ks_{c}}=\frac{1}{s_{c}}\frac{\left(I-s_{w}W\right)}{K}$
(8)

The Kaldor’s theory of distribution with $s_{w}$ taken as equal to zero is

$\frac{P}{K}=\frac{1}{s_{c}}\frac{I}{K}$ (9)

Pasinetti’s theory as shown in equation(7), obtained without making any
assumption on the propensities to save by the workers and Kaldor’s theory
obtained with $s_{w}=0$ as shown in equation(9), both seem to be exactly the
same. Pasinetti has given an interpretation to this match between the
equations as a long run insignificance of worker’s propensities to
save($s_{w}\rightarrow 0$). In this manner he claims that, in the long run
worker’s propensity to save does not influence the distribution of income
between profits and wage. We argue that the comparison with the assumption
$s_{w}=0$ of Kaldor’s model by Pasinetti to arrive at a conclusion about the
long run worker’s propensities to save is not correct. Truly, Pasinetti’s
model starts with a single difference from Kaldor’s model, which is in the
share of profit to workers($P_{w}$). In Kaldor’s model $P_{w}=0$, whereas it
is greater than zero in Pasinetti’s model. However, difference between the two
models widens by a larger extent, due to inclusion of the Pasinetti’s
condition for dynamic equilibrium (equation-3). We now demonstrate the
differences between Pasinetti’s and Kaldor’s models. For this, an alternate
form of the Pasinetti’s final equation is first derived by subtracting the LHS
and RHS of equation(6) from $P/K$ as in the following.

$\displaystyle\frac{P}{K}-\frac{P}{K}\left[\frac{S_{c}}{S}\right]=\frac{P}{K}-\frac{P_{c}}{K}\implies\frac{P}{K}\frac{S_{w}}{S}=\frac{P_{w}}{K}$

Putting $S_{w}=s_{w}(P_{w}+W)$ and taking $S=I$, we get for $P_{w}\neq 0$

$\displaystyle\frac{P}{K}=\frac{1}{s_{w}}\left(\frac{P_{w}}{W+P_{w}}\right)\frac{I}{K}$
(10)

By equating the right hand sides of equations(7) and (10), we get

$\displaystyle s_{c}=s_{w}\left(1+\frac{W}{P_{w}}\right)$ (11)

It shows that the application of the Pasinetti’s condition results in the
propensities getting constrained to satisfy equation(11) and therefore the
claim on insignificance of $s_{w}$ does not remain generalized. The above
equation can not be achieved naturally, as the propensities depend on the
behavior the capitalists and the workers. Practically, the satisfaction of the
condition in equation(11) can be realized through a force of contract or
agreement between workers and capitalists.

##  2 Observations on the relationship between propensities to save

Some observations and the meanings of the equation(11) between the
propensities to save by workers and capitalists are enumerated in the
following.

  1. 1.

In the limit as $P_{w}\rightarrow 0$, the value of $s_{w}$ should tend to
zero, so as to get a finite value for $s_{c}$ as per equation(11). This
enforces with confirmity, the idea that workers do not save at all if no
portion of the profit is shared to them. This idea is incorporated in to the
Pasinetti model by merely including the term $P_{w}$ in equation(2), which
does not enforce $s_{w}\rightarrow 0$ as $P_{w}\rightarrow 0$. Here the
logical correction by enforcing $s_{w}\rightarrow 0$ as $P_{w}\rightarrow 0$
is resulted from the uses of the Passineti’s condition and the constant profit
ratio.

  2. 2.

Pasinetti’s model is silent about the spending of $W-C_{w}$ when $P_{w}=0$,
where $C_{w}$ is the worker’s aggregate consumption.

  3. 3.

For fixed values of $s_{c}$ and $P_{w}$, an increase of worker’s wage leads to
a reduction of $s_{w}$. This is an inconsistency. Here, an increase of
worker’s income due to rise of worker’s wages results in a lesser savings.

  4. 4.

The equations(7) and (9) are same only in terms the variables $s_{c}$ and $I$.
But the other external variables involved in the two models occur with
different ranges/proportions. Hence the interpretation made by Pasinetti on
the external variable $s_{w}$ as a consequence of the same form in
equations(7) and (9) can not be generalized.

  5. 5.

The value of $s_{w}$ as not influencing the distribution of income between
profits and the wage is operational only in conditions such as $P_{w}=0$ when
$W\neq 0$ or $W>>P_{w}$.

The above observations related to interpreting the Pasinetti’s final equation
and relationship between propensities motivated to look into the insight of
this condition for the dynamic equilibrium used by Pasinetti.

##  3 An insight into Pasinetti’s condition

In the section we obtain a proof to show that the Pasinetti’s condition can
not represent a dynamic equilibrium. In the dynamic equilibrium of growth,
savings translated into investments in each period is equal to the rate of
capital growth in that period. Mathematically,

$\displaystyle
S_{w}=\frac{dK_{w}}{dt};~{}~{}~{}S_{c}=\frac{dK_{c}}{dt}~{}~{}~{}\mbox{and}~{}~{}~{}S=\frac{dK}{dt}$
(12)

Therefore

$\displaystyle
K_{w}(t)=\int_{0}^{t}S_{w}dt;~{}~{}~{}~{}~{}K_{c}(t)=\int_{0}^{t}S_{c}dt~{}~{}~{}\mbox{and}~{}~{}~{}K(t)=\int_{0}^{t}Sdt$
(13)

And

$\displaystyle\frac{K_{w}}{K}=\frac{\int_{0}^{t}S_{w}dt}{\int_{0}^{t}Sdt}$
(14)

As per the Pasinetti condition in equation(3), this ratio is equal to
$S_{w}/S$. This is possible only if the instantaneous savings $S_{w}(t)$ and
$S(t)$ satisfy the conditions

$\displaystyle S_{w}(t)$ $\displaystyle=$ $\displaystyle Cf(t)$ (15)
$\displaystyle S_{c}(t)$ $\displaystyle=$ $\displaystyle Df(t)$ (16)
$\displaystyle S(t)$ $\displaystyle=$ $\displaystyle S_{w}+S_{c}=Ef(t)$

where $E=C+D$. Clearly, at any point in time the ratio of propensities is
$S_{w}/S_{c}=C/D=R$. This ratio rule has to be strictly followed at all the
time periods without deviating. For keeping a constant value for $R$, the
actual amount saved by an agent(worker or capitalist) should be different from
the amounts that an agent is willing to save based on his Full Capacity to
Save(FCS). Let $FCS_{w}(t)$ and $FCS_{c}(t)$ be the Full Capacity to Save by
the workers and capitalists respectively. In order to write a mathematical
form for the actual savings by workers and capitalists for obtaining a
constant contract ratio $R=S_{w}/S_{c}$ corresponding to a certain capacity
ratio $R_{1}=FCS_{w}(t)/FCS_{c}(t)$, we define two conditional functions

$\displaystyle M_{1}=\max(R_{1}-R,0)$ $\displaystyle M_{2}=\max(R-R_{1},0)$

Now the actual savings by workers and capitalists restricted using $R$ can be
written as

$\displaystyle S_{w}(t)$ $\displaystyle=$ $\displaystyle
S_{w}(FCS_{w}(t),FCS_{c}(t),R)=FCS_{w}(t)-FCS_{c}(t)M_{1}$ (17) $\displaystyle
S_{c}(t)$ $\displaystyle=$ $\displaystyle
S_{c}(FCS_{w}(t),FCS_{c}(t),R)=\frac{FCS_{w}(t)}{M_{2}+R_{1}}$ (18)

The selection of savings in the above manner leads to lesser saving by one of
the agents, who has a higher capacity/willingness to save. The agent who has a
higher capacity to save is forced to save less during the process of
negotiating a constant value for $R$. The unsaved capacity to save by workers
and capitalists, denoted as $US_{w}(t)$ and $US_{c}(t)$ are given by

$\displaystyle US_{w}(t)$ $\displaystyle=$ $\displaystyle FCS_{c}(t)M_{1}$
(19) $\displaystyle US_{c}(t)$ $\displaystyle=$
$\displaystyle\frac{FCS_{c}(t)M_{2}}{M_{2}+R_{1}}$ (20)

The value of $US(t)=US_{w}(t)+US_{c}(t)$ is either greater than or equal to
zero and it becomes equal to zero only when both $M_{1}$ and $M_{2}$ are equal
to zero. Thus, $US(t)>0$ is common, $US(t)=0$ is only rarely possible and
$US(t)=0~{}\forall t$ is almost impossible. The consequences of $US(t)>0$ are
(a) reduction in the growth of capital due to lesser total investments in all
the time periods, (b) reduction in total profit $P$ and (c) reduction in
profit share for one of the agents. Another important drawback arises is from
the fact that, the unsaved capacity of an agent in a period will become a
component in the Full Capacity to Save of that agent in the next period. This
higher capacity to save due to the accumulated unsaved amounts from the
previous periods may be further restricted from saving in the subsequent
periods as per equations(17) and (18). This may be a matter of a continuous
divergence of interests of the agents, rather than attainment of a dynamic
equilibrium. In this way the Pasinetti’s condition for dynamic equilibrium as
given by equation(3) is not correct. We consider that the contract between the
workers and capitalists to practically implement the Pasinetti’s condition
encompasses losses from different factors. Again, using the condition of
contract, we can derive the following

$\displaystyle\frac{S_{w}}{S_{c}}=\frac{C}{D}=\frac{P_{w}}{P_{c}}$
$\displaystyle\frac{s_{w}(W+P_{w})}{s_{c}P_{c}}=\frac{P_{w}}{P_{c}}$

For $P_{c}\neq 0$, the above equation implies

$\displaystyle s_{c}=s_{w}\left(\frac{W+P_{w}}{P_{w}}\right)$ (21)

From the equations(11) and (21), it is clear that the relationship between the
propensities is resulted from a contract for a constant value of $R$ to be
agreed upon between the workers and the capitalists.

##  4 Conclusions

An assessment of the practicability of the Pasinetti’s condition
$(K_{w}/K=S_{w}/S)$ used for the dynamic equilibrium of economic growth in his
theory of distribution is carried out and the following conclusions have been
made.

  1. 1.

Pasinetti’s condition leads to enforcement of the worker’s behavior given by
$s_{w}\rightarrow 0$ when $P_{w}\rightarrow 0$.

  2. 2.

The Pasinetti model deviates to a larger extent from Kaldor’s model due to the
inclusion of the Pasinetti’s condition. Hence the claim by Pasinetti on the
long run worker’s propensity to save as not influencing the distribution of
income between profits can not be generalized. His claim is found to be valid
only when $W>>P_{w}$ or $P_{w}=0$ with $W\neq 0$.

  3. 3.

Pasinetti’s condition implies a relationship between the propensities to save
by workers and the capitalists. The relationship between the propensities can
not be achieved naturally and requires the force of a contract for a constant
value of $R$ to be agreed upon between the workers and the capitalists.

  4. 4.

The implementation of Pasinetti’s condition is found to restrict the actual
savings by one of the agents to lower levels compared to its full capacity for
saving.

  5. 5.

This condition leads to a reduction of total savings and hence reduces the
growth of capital. The amount of unsaved income tend to increase with time. So
the Pasinetti’s condition can not be described as a dynamic equilibrium of
economic growth.

  6. 6.

It is concluded that the Pasinetti’s condition is not practicable and hence
not warranted for describing the dynamics of growth of economies.

## References

  * [1] L. L. Pasinetti, Rate of profit and income distribution in relation to the rate of economic growth, The Review of Economic Studies 29 (4) (1962) 267–279.
  * [2] N. Kaldor, Alternative theories of distribution, The review of economic studies 23 (2) (1955) 83–100.

## Appendix A: Illustrative examples on the calculation of actual savings
using equations(17) and (18)

Example-1: Given $R=\frac{1}{5}$, $FCS_{w}=5$ and $FCS_{c}=20$.
Now, $R_{1}=\frac{5}{20}=\frac{1}{4}$,
$M_{1}=\max(R_{1}-R,0)=\max\left(\frac{1}{20},0\right)=\frac{1}{20}$
and $M_{2}=\max(R-R_{1},0)=\max\left(-\frac{1}{20},0\right)=0$
and we have by equations(17) and (18)

$\displaystyle S_{w}(t)$ $\displaystyle=$ $\displaystyle
FCS_{w}(t)-FCS_{c}(t)M_{1}=5-20\times\frac{1}{20}=4$ $\displaystyle S_{c}(t)$
$\displaystyle=$
$\displaystyle\frac{FCS_{w}(t)}{M_{2}+R_{1}}=\frac{5}{0+\frac{1}{4}}=20$

In this case the workers will save only $4$ units which is less than their
full capacity by keeping $R=\frac{4}{20}=\frac{1}{5}$ as the required value.

Example-2: Given $R=\frac{1}{4}$, $FCS_{w}=4$ and $FCS_{c}=20$.
Now, $R_{1}=\frac{4}{20}=\frac{1}{5}$,
$M_{1}=\max(R_{1}-R,0)=\max\left(-\frac{1}{20},0\right)=0$
and $M_{2}=\max(R-R_{1},0)=\max\left(\frac{1}{20},0\right)=\frac{1}{20}$
and we have by equations(17) and (18)

$\displaystyle S_{w}(t)$ $\displaystyle=$ $\displaystyle
FCS_{w}(t)-FCS_{c}(t)M_{1}=4-20\times 0=4$ $\displaystyle S_{c}(t)$
$\displaystyle=$
$\displaystyle\frac{FCS_{w}(t)}{M_{2}+R_{1}}=\frac{4}{\frac{1}{20}+\frac{1}{5}}=16$

In this case the capitalists will save only $16$ units which is less than
their full capacity by keeping $R=\frac{4}{16}=\frac{1}{4}$ as the required
value.