# Wages and Utilities in a Closed Economy

Sanyukta Deshpande and Milind Sohoni

###### Abstract.

The broad objective of this paper is to propose a mathematical model for the
study of causes of wage inequality and relate it to choices of consumption,
the technologies of production, and the composition of labor in an economy.
The paper constructs a Simple Closed Model, or an SCM, for short, for closed
economies, in which the consumption and the production parts are clearly
separated and yet coupled. The model is established as a specialization of the
Arrow-Debreu model and its equilibria correspond directly with those of the
general Arrow-Debreu model. The formulation allows us to identify the
combinatorial data which link parameters of the economic system with its
equilibria, in particular, the impact of consumer preferences on wages. The
SCM model also allows the formulation and explicit construction of the
consumer choice game, where expressed utilities of various labor classes serve
as strategies with total or relative wages as the pay-offs. We illustrate,
through examples, the mathematical details of the consumer choice game. We
show that consumer preferences, expressed through modified utility functions,
do indeed percolate through the economy, and influence not only prices but
also production and wages. Thus, consumer choice may serve as an effective
tool for wage redistribution.

economic models, wages, inequality, strategic behavior, pricing, labor
markets, consumer choice, utility maximization, Fisher markets, duality,
technology, Arrow-Debreu market, equilibria

††conference: ; ;

##  1\. Introduction

As the present millenial edition of the global economy unfolds, many authors
and agencies have pointed out several undesirable features that have emerged.
These are the paucity of ”good” jobs, rising inequality, excessive
consolidation, and the possibility of linkages between modern production and
consumption processes with climate change.

This paper deals with the first two problems, viz., the allocation of jobs and
wages and rising inequality in wage incomes in the economy, and its structural
determinants. For this purpose, we construct a simple mathematical model,
called the _SCM Model_ , which illustrates some of the key features of the
dependence of wages on the production and consumption choices within the
economy. Our model is divided into two parts: the production sector,
determined by a technology matrix $T$ that utilizes $m$ labor classes to
produce $n$ goods, where prices are known and revenue maximization is used to
determine the quantities of goods produced, labor utilization, and wages. This
part of the model assumes prices as a given input that cannot be changed. The
second part is the consumption sector, modeled as a Fisher market and a
utility matrix $U$. This sector of the economy assumes the production sector
as a given, i.e., wages, which are now disposable incomes, and quantities of
goods produced, and allocates goods based on disposable incomes held by each
labor class, and their expressed utility matrix $U$, and determines prices.

This paper demonstrates the connection between $U$, the utility matrix, and
the wages obtained by various labor classes, as implemented by $T$. In other
words, it traces the connection between personal consumption choices, prices
of goods, their production, and finally, the wages received. Furthermore, it
shows that having a ”private and real” utility $U^{t}$ and posting or
posturing a different $U$ into the economy does indeed alter wages and has the
potential to improve both social welfare and the relative welfare for certain
classes. The setup of the SCM model then allows us to define the _consumer
choice game_ (CCG), where the manipulation of $U$ is the strategy, and the
relative or total welfare, as measured by the allocation of goods and their
utilities according to $U^{t}$, are the payoffs.

There exists a rich body of literature on the general theory of equilibrium
(Harsanyi et al., 1988; Samuelson, 1953). The theory seeks to model closed
economies and explains the prices of goods and services in a market and how
different markets are interconnected. Closed economy models have been studied
extensively, starting with Walras’s model and the most prominent being the
Arrow-Debreu (A-D) market (Starr, 2011). Although the existence of equilibrium
is proved in models like the A-D market, most of these have seen state
policies as the strategy space, not consumer choice (Arrow and Debreu, 1954).
Additionally, many of these models suffer from computational intractability or
not providing sufficient ground to model behaviors and strategy spaces as
analytical games, which is the primary focus of this study. In the celebrated
text, (Samuelson, 1948; Backhouse, 2015), Samuelson presents the theory of
consumer behavior using ordinal utilities, backed by the principles of
”maximizing behavior of consumers and firms” and ”stability of equilibrium”,
which also form the base of our work. Although utility functions as strategies
was pointed out (Chipman, 1982), it was for the determination of prices and
allocations, not wages.

The manipulation of $U$ has also been studied in connection with the impact of
advertising on the competition between firms and their profitability. The
connection between branding, i.e., managing utilities, and revenues and market
shares is well understood (Wernerfelt, 1991; Delgado-Ballester and Munuera-
Alemán, 2001), but without an understanding of how consumption links to
production. Pressure groups utilized this concept to label certain products,
such as coffee, as ”compliant” with desirable ideals, such as fair wages for
coffee-bean pickers (Levi and Linton, 2003). Our paper offers a model of how
such advice is likely to play out in the economy. Our analysis focuses largely
on a closed economy and the study of its modes as functions of the system’s
parameters.

On the production side, the effects of technology on wages have also been
studied in Ricardian and Keynesian economics, mainly in the context of
globalization and unemployment (Ray, 1998; Keynes and Keynes, 2010). Building
on the theory of comparative advantage, other models such as the Heckscher-
Ohlin-Samuelson model (Bergstrand, 1990), the specific factors model (Eaton,
1987), and the most recent Kremer-Maskin model (Kremer and Maskin, 1996), have
analyzed the link between production and technological advancements made
available through trade. As long as the division of inputs and outputs of the
key blocks of the SCM model are maintained, such variations in the production
function may be incorporated into the SCM model. They may help link the
effects of globalization and trade with variable consumer choice.

In the computational science community, there has been growing interest in the
fields of market equilibria and how strategic agents may manipulate market
equilibria under various settings of the economy (Brânzei et al., 2017;
Amanatidis et al., 2022; Barman et al., 2019; Brânzei et al., 2014; Codenotti
and Varadarajan, 2004; Brânzei et al., 2016). Specifically, in connection with
our work, the Fisher Market Game (Adsul et al., 2010; Garg, 2017) has shown
that going to the market with postured utilities $U$ (instead of the real
ones, $U^{t}$) can bring rewards in terms of more favorable allocations. The
strategic effects of feigning have also been studied under various settings
(Babaioff et al., 2014; Chen et al., 2012, 2011). However, these models only
relate to the consumption side of the economy. In another related body of
work, the Shapley-Shubik Market Game (Shapley and Shubik, 1977) explaining the
formation of prices is studied extensively, with algorithmic developments on
the convergence of proportional dynamics in exchange economies (Codenotti et
al., 2005; Wu and Zhang, 2007; Zhang et al., 2020; Brânzei et al., 2021;
Birnbaum et al., 2011; Cheung et al., 2018; Bei et al., 2019).

This paper aims to complement these studies by introducing a simple
macroeconomic production model helping define a closed economy in its
entirety. This allows us to find a consistent set of prices of goods, wages
for each labor class, and allocations of goods among classes, resulting in an
equilibrium in the economy. Utilizing microeconomic concepts like the Fisher
market, we model the effects of consumer choice strategy on wages, production,
and allocations. By coupling the co-dependent production and consumption sides
of the economy, this work extends the conclusion of the Fisher Market Game to
show that consumer choice may indeed be used to derive wages as well. In
classical terms, our work aims to develop and study the dependence of market
equilibria on various strategic factors that define the economy, set up via
the SCM model and executed via the CCG.

The work draws heavily from existing models discussed earlier, particularly
the Arrow-Debreu model and its predecessor, the Fisher model (Arrow and
Debreu, 1954). It also takes inspiration from Sraffa’s accounting methods for
calculating prices and wages (Abraham-Frois et al., 1979), the use of the
theory of value to calculate labor inventory, the use of dual variables for
calculating wages, and the use of utility functions to compute the allocation
of goods (Starr, 2011).

The rest of the paper is organized as follows. In Section 2 we describe the
SCM Model $\mathcal{H(C,P)}$ as composed of two interconnected systems, the
production model ${\mathcal{P}}$, and the consumption model ${\mathcal{C}}$.
The production model ${\mathcal{P}}$ has the key parameter $T$, the technology
matrix, and $Y$, the size of individual labor classes. The input variable is
the price vector $p$, while the ”outputs” are wages $w$ of agents, and the
production vector $q$. The consumption model ${\mathcal{C}}$ consists of the
key parameter $U$, the utility matrix, and the inputs as the wages $w$, and
production $q$. The ”output” variables are $p$, the prices, and $X$, the
allocation of goods to labor classes (or agents).

We define the market equilibrium in the SCM model, i.e., SM equilibrium as
$\eta=\\{p,q,w,X\\}$ with its parameters satisfying both production and
consumption system constraints. We also define two global optimization
functions $f_{\mathcal{C}}$ and $f_{\mathcal{P}}$, which couple
${\mathcal{C}}$ and ${\mathcal{P}}$. They allow for a possible tatonnement as
an iterative interaction between ${\mathcal{C}}$ and ${\mathcal{P}}$, which we
only briefly discuss.

In Section 3, we cast the SCM model ${\mathcal{H}}$ as an Arrow-Debreu (A-D)
market. We show the equivalence of SM equilibria, with the equilibria of the
corresponding A-D market, proving the existence of equilibrium in the SCM
model. Thus, this connects the two concepts and also gives an explicit
description of the dependence of the A-D equilibria on the parameters of the
economy.

In Section 4, we define the Consumer Choice Game (CCG) as the game between
various labor classes with the expressed utility matrix $U$ as the strategy
space and the private utilities and wages as the payoffs. For this, we
associate key combinatorial data, viz., the Fisher solution forest, with a
given market solution. This data determines the regime of interactions between
various agents and helps compute the dependence of the payoffs on the matrix
$U$ locally. If $U$ changes dramatically, then the market may move to a
neighboring Fisher forest, and a new regime, and a new local payoff function.

In Section 5, we further analyze the combinatorial data arising from an
equilibrium point. First, we show through a 2-player example, the
decomposition of the strategy space, i.e., the $U$-space into various regions
indexed by Fisher forests. This also leads to a correspondence ${\mathcal{N}}$
between the strategy space and the space of possible pay-offs. In other words,
${\mathcal{N}}\subseteq\mathbb{R}^{2}\times\mathbb{R}^{2}$. We show that
${\mathcal{N}}$ is largely a 2-dimensional manifold.

In Section 6, we take up the case of a small market of soaps. We utilize the
SCM model theory to demonstrate the use of expressed utilities $U$, market
structures as described through Fisher solution forests, and lastly,
technological updates and labor migration carried out through modifications to
$T$ and $Y$, as devices to alter payoffs and allocations in the market.

Finally, in Section 7, we conclude by pointing out what was achieved, its
economic significance, and possible future directions.

##  2\. The SCM model

In this section, we formally set up the Simple Closed Model (SCM). Section 2.1
develops the notation and highlights the internal variables of SCM. Section
2.2 defines the production and consumption segments of the economy and poses
the two as correspondences. Using these, Section 2.3 defines the market
equilibrium formally.

###  2.1. The basic notation and assumptions

  * •

A good is a fixed unit of service or an output of a manufacturing plant, made
available in a fixed time interval, called epoch. The set of goods, indexed by
$j$, is denoted by ${\mathcal{G}}=\\{g_{1},\dots,g_{n}\\}$. The column vectors
$q$ and $p$ denote the amount of good $g_{j}$ manufactured in the epoch and
its price, respectively.

  * •

The entire population in the economy consists of agents divided into distinct
classes, according to their expertise and training. Set
${\mathcal{L}}=\\{L_{1},\ldots,L_{m}\\}$, indexed by $i$, denotes these
classes of labor, with each class $L_{i}$ capable of devoting $Y_{i}$ units
(e.g., person-years) of labor in every epoch. Thus, $Y$ forms a $m\times 1$
column vector, indicating labor availability.

  * •

Each good $g_{j}$ has exactly one production process or technology $T_{j}$ and
each $T_{j}$ is represented as a column vector
$T_{j}:{\mathcal{L}}\rightarrow\mathbb{R}$. The class of all such technologies
is denoted by ${\mathcal{T}}$, which is represented as a $m\times n$-matrix
$T$ with column $T_{j}$. Thus, in matrix $T$, the entry $T_{ij}$ is the amount
of labor $L_{i}$ needed to produce a unit of $g_{j}$.

  * •

For each labor class $i$, there is a linear utility function
$U_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$, that maps $x\in\mathbb{R}^{n}$,
the bundle of goods allocated. We assume that utilities are the same for all
persons in one labor class and are measured in the units - happiness per
person per unit weight. For example, $u_{ij}$ is the happiness derived by a
person from class $L_{i}$ by consuming a unit of good $g_{j}$.
As $U_{i}$ is linear, we write it as $U_{i}(x)=\sum_{j\in[n]}u_{ij}x_{j}$,
where $u_{ij}\geq 0$ are constants. We assume that for every good $i$ and
buyer $j$, there exist buyer $j^{\prime}$ and good $i^{\prime}$ respectively,
such that $u_{ij^{\prime}},u_{i^{\prime}j}>0$. We denote the matrix formed by
$u_{ij}$ entries as U.

Further, without loss of generality, we assume that for each labor class,
there are technologies which utilize them, although the demand for a specific
good is generated by the economy. We also assume that the entries of $T,Y$,
and $U$ are all in general position and satisfy no algebraic relation (with
rational coefficients) amongst themselves.

Under this notation, an economy with given specifications $(T,Y,U)$ (and
implicit description of $\mathcal{G,L}$) is parameterized by a non-negative
tuple $(p,q,w,X)$ in each epoch, where

  1. (1)

$q$ is a $n\times 1$ column vector with $q_{j}$ is the amount of good $g_{j}$
manufactured,

  2. (2)

$p$ is an $n\times 1$ column vector and $p_{j}$ is the price of each good
$g_{j}$,

  3. (3)

$w$ is an $1\times m$ row vector where $w_{i}$ are the wages received by each
person in labor class $L_{i}$

  4. (4)

$X$ is a $m\times n$-matrix and $x_{ij}$, the total amount of good $G_{j}$
consumed by labor $L_{i}$

The tuple $(p,q,w,X)$ must satisfy additional production and consumption
conditions specified below.

###  2.2. Production and Consumption

####  2.2.1. Production

The production space ${\mathcal{P}}(T,Y;p)$ is the collection of all wages $w$
and quantities $q$ such that Eq. (1)- (3) are satisfied.

(1) $\displaystyle Tq\leq Y$ (2) $\displaystyle q_{j}\cdot(p_{j}-(wT)_{j})\geq
0$ (3) $\displaystyle q,w\geq 0$

Note that Eq. (1) says that the quantities of goods produced are limited by
labor constraints, while Eq. (2) says that unprofitable goods are not
produced. Consider next, the global maximization function $f_{\mathcal{P}}$ as
$\sum_{j}p_{j}q_{j}$, i.e, revenue maximization and the linear program
$P_{\mathcal{P}}$ and its dual LP $DP_{\mathcal{P}}$ as given in Eq. (4).

(4) $P_{\mathcal{P}}:\ =\left[\begin{matrix}\displaystyle\max_{q}&p^{T}q\\\
\textrm{s.t.}&Tq&\leq&Y\\\ &\displaystyle\ q_{j}&\geq&0&&\forall
j\in[n]\end{matrix}\right];\>\ \ DP_{\mathcal{P}}:\
=\left[\begin{matrix}\displaystyle\min_{w}&wY\\\ \textrm{s.t.}\ &\displaystyle
wT&\geq&p^{T}\\\ &\displaystyle\ w_{i}&\geq&0&&\forall
i\in[m]\end{matrix}\right]$

Utilizing these, we further define the production correspondence as:

(5) $CP(T,Y;p)=\\{q,w|q,w\text{ are simultaneous solutions to
$P_{\mathcal{P}}$ and $DP_{\mathcal{P}}$}\\}$

By complementary slackness, any $(q^{\prime},w^{\prime})\in CP(T,Y;p)$
satisfies $q^{\prime}_{j}((w^{\prime}T)_{j}-pj)=0$ and
$(Y_{i}-(Tq^{\prime})_{i})w^{\prime}_{i}=0$. Thus, we see that if
$q^{\prime}_{j}>0$ then $(w^{\prime}T)_{j}-p_{j}=0$, and hence
$(q^{\prime},w^{\prime})\in{\mathcal{P}}(p)$ as well. Thus,
$CP(p)\subseteq{\mathcal{P}}$.

Moreover, $w^{\prime}_{i}>0$ implies that $(Tq^{\prime})_{i}=Y_{i}$ and more
importantly, if $Y_{i}>(Tq^{\prime})_{i}$ then $w_{i}=0$, i.e., if a
particular labor class constraint is not tight then the wages of that class
are zero. This implies that the production process conserves money, i.e.,
$pq^{\prime}=w^{\prime}Tq=w^{\prime}Y$, and the revenue of the producers goes
to the labor classes as wages. Also note that the inequality in the dual
program, viz., $w^{T}T\geq p^{T}$, is opposite to the profitability constraint
Eq. (2).

The choice of the above optimization model as a model for production is
similar to many earlier economic models, the most popular being the production
frontier model where the feasible region of $P_{\mathcal{P}}$ from Eq. (4) is
used to define the feasible space of the production of goods (Ray, 1998). The
optimal point is determined either by a family of indifference curves or an
explicit (concave) utility function. The use of optimal production is also the
basis of the comparative advantage argument used by Ricardo to illustrate why
trade is beneficial to both parties. When the optimal $\bar{q}$ is a vertex of
the production frontier, the wages $\bar{w}$ determined by the theory of
marginal productivity matches with our assignment of wages as dual variables.
We illustrate the consumption model, and specifically, a trade case in
Appendix A.1 and A.2, respectively.

####  2.2.2. Consumption

We model consumption as a Fisher market (Adsul et al., 2010). Recall that, in
such a market, there are $m$ buyers (i.e., ${\mathcal{L}}$ labor classes, in
our case), and $n$ goods ${\mathcal{G}}$. Each class $L_{i}$ is endowed with
money $W_{i}=w_{i}Y_{i}\geq 0$, and each good $G_{j}$ has quantity $q_{j}\geq
0$ for sale. The utility function is given by the matrix $U=(u_{ij})$.

Solution of the Fisher market are equilibrium prices $p=(p_{j})$ with
$j\in[n]$ with $p_{j}q_{j}>0$ for some $j$, and allocations $X=(x_{ij})$ with
$i\in[m],j\in[n]$ and $x_{ij}\geq 0$ such that they satisfy the following two
constraints:

  * •

Market Clearing: All useful goods are completely sold and the money of all the
buyers is exhausted, i.e.
$\forall j\in[n]$, $\sum_{i\in[m]}x_{ij}=q_{j}$ and $\forall i\in[m]$,
$\sum_{j\in[n]}p_{j}x_{ij}=w_{i}Y_{i}$

  * •

Optimal Goods: Each buyer buys only those goods which give her the maximum
utility per unit of money i.e if $p_{j}x_{ij}>0$, then
$\frac{u_{ij}}{p_{j}}=\max_{j^{\prime}\in[n],u_{ij^{\prime}}\neq
0}\frac{u_{ij^{\prime}}}{p_{j^{\prime}}}$.
This value is denoted as the bang per buck $bb_{i}$ and it is defined only for
buyers $i$ with $W_{i}>0$ and then $0<bb_{i}<\infty$. A generalization to splc
utilities is given in Appendix A.2.

The consumption correspondence is defined as:

(6) $CC(U;q,w)=\\{x,p|x,p\text{ satisfy the Fisher market conditions}\\}$

Given $(x,p)$ as above, the set $X=\\{(i,j)\>|\>x_{i,j}>0\\}$ is called the
Fisher graph of the solution $x$ and records the combinatorial structure of
the allocation $x$. It is well known that solutions to the Fisher market are
optimal points of the Eisenberg-Gale maximization function, a money-weighted
combination of the utilities of the buyers (Jain and Vazirani, 2010). This
allows us to define a global optimization function $f_{\mathcal{C}}$ for the
consumption process.

As with the production process, money is also conserved in the consumption
process- all money with the buyers, i.e., the labor classes, ends up with the
producers as revenue. The optimal goods condition is also applicable to goods
$j^{\prime}$ with $q_{j^{\prime}}=0$. The Fisher market sets the prices of
such goods high enough so that no buyer finds it profitable to shift
consumption even if an infinitesimal amount of such goods was produced. Hence,
this consumption part sees price as the only constraint and no artificial
shortage of desirable goods. Also, note that scaling the utilities of a buyer
by a fixed constant does not change the equilibrium or the prices, since it is
the relative importance of goods for a given agent that determines the
relation between the prices of the goods she consumes. We illustrate the
consumption process through an example in Appendix A.3.

###  2.3. Equilibrium in SCM

For the economy specifications $(T,Y,U)$, we next utilize correspondences
$CP(T,Y)$ and $CC(U)$ to an equilibrium in the SCM model. The production
happens via $CP$, which takes price $p$ as the input and generates productions
and wages $(q,w)$ by maximizing revenues, while $CC$ uses $(q,w)$, to generate
the prices via the Fisher market.

###### Definition 2.1.

Given the market $SCM(T,Y,U)$ for the data $T,Y,U$, the tuple $\eta=(q,w,p,X)$
is a simple closed model equilibrium, or simply, SM equilibrium, if $(q,w)\in
CP(T,Y;p)$ and $(p,X)\in CC(U;q,w)$.

The structure of SCM and the definition of SM equilibrium also allow us to
naturally define a weak _tatonnement_ process (Uzawa, 1960). The basic
objective of the tatonnement is to arrive at an equilibrium $\eta=(p,q,w,X)$
such that (i) $p,X$ are the outputs of the Fisher market if the inputs are
$w,q$, on the consumption side, and (ii) $q,w$ are the optimal solutions to
$P_{\mathcal{P}},DP_{\mathcal{P}}$ on input $p$, on the production side.
Briefly, the process begins with a candidate solution $\eta$ and checks if
$\eta$ is indeed an equilibrium. If not, it alternatively updates the
consumption side and the production side by solving the global optimization
functions $f_{\mathcal{C}}$, and $f_{\mathcal{P}}$. However, the _tatonnement_
does not always converge. It may alternate between multiple states of the
economy if production and consumption do not agree on a common set of active
goods and classes but cyclically choose two or more states. We provide more
details on its description, and analysis, along with examples in Appendix A.3.

##  3\. Existence of SM Equilibrium

This section proves the existence of equilibrium $SM(T,Y,U)$. We construct an
Arrow-Debreu (A-D) market $AD(T,Y,U)$ from the SCM model $SCM(T,Y,U)$, and
show the equivalence of the equilibrium points in the SCM model and market
equilibria in the corresponding A-D market. We follow Starr, R. M. (2011),
(Starr, 2011) for the description of the Arrow-Debreu market, and provide
specific page numbers for reference111We assume splc utilities in this
section.

###  3.1.  Arrow-Debreu Market Notation and Conditions

The A-D market is given by the following set of specifications.

  1. (1)

The Arrow Debreu (A-D) market consists of $n$ goods, $m$ households viz.,
$H_{1},\ldots,H_{m}$, and $l$ firms, viz., $F_{1},\ldots,F_{l}$.

  2. (2)

Each firm $F_{j}$ has a non-empty possible production technology set
$Y_{j}\subseteq\mathbb{R}^{n}$, where $n$ is the number of goods. A vector $y$
in $Y_{j}$ represents a technically possible combination of inputs (raw goods)
and outputs (produced goods). A negative entry in $y$ represents an input, and
a positive entry, an output. We assume that for every $j$, the set $Y_{j}$ is
convex, bounded, closed, and includes $\bar{0}$, the zero vector. We also
assume that there can be no outputs without inputs and there does not exist
any way to transform outputs into inputs. (page 115, 165)

  3. (3)

For a given price vector $p\in\mathbb{R}^{n}$, we define $S_{j}(p)$ as the
optimal supply function of firm $j$

(7) $S_{j}(p)=\\{\ y^{*}|y^{*}\in Y_{j},\ p\cdot y^{*}\geq p\cdot y\ \ \forall
y\in Y_{j}\ \\}$

Then $S_{j}(p)$ is nonempty and convex for every $p\in\mathbb{R}^{n}_{+}$ and
upper hemicontinuous as a function on $\mathbb{R}^{n}_{+}-\bar{0}$. In other
words if $(p_{i})\rightarrow p$ and $(y_{i})\rightarrow y$ are convergent
sequences with $y_{i}\in S_{j}(p_{i})$ for all $i$, then $y\in S_{j}(p)$.
(page 170, 298)

  4. (4)

For each household $H_{i}$, there is a set $X_{i}\subseteq\mathbb{R}^{n}_{+}$
denoting the set of possible consumption plans of agent $i$. Each $X_{i}$ is
closed, convex, and unbounded above. (page 125)

  5. (5)

The economy has an initial endowment of resources which is denoted as
$r\in\mathbb{R}^{n}$. Each household has an initial endowment of goods given
by $r^{i}\in\mathbb{R}^{n}$ so that $\sum r^{i}=r$. Let $Y=\sum_{j}Y_{j}$ and
$y\in Y$. The vector $y$ is said to be attainable if $y+r\geq 0$. This denotes
the possible production of the economy. We require that there is a $c>0$ such
that $|y|<c$ for all attainable vectors $y$, the existence of which is
guaranteed. (page 127, 167)

  6. (6)

Firm $j$’s profit function $\pi_{j}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is
given by:

(8) $\pi_{j}(p)=\max_{y\in Y_{j}}(p\cdot y)=p\cdot S_{j}(p)$

Each household $i$ has shares in firm $j$’s profit given by $\alpha_{ij}\geq
0$ such that $\sum_{i\in H}\alpha_{ij}=1$. (page 142)

  7. (7)

Each agent is endowed with a convex preference quasi-ordering $\geq_{i}$ on
$X_{i}$. This may be represented by a continuous real-valued function $u_{i}$
which is the utility function for each $i$. We assume that there is always
universal scarcity i.e. for every $x_{i}\in X_{i}$, there exists an
$x^{\prime}_{i}\in X_{i}$ so that agent $i$ values it more than $x_{i}$, i.e.,
$x_{i}^{\prime}>_{i}x_{i}$, or simply, $u_{i}(x_{i}^{\prime})>u_{i}(x_{i})$.
We also assume that the sets $A_{i}(x_{0})=\\{x|x\in X_{i},\
x\geq_{i}x_{0}\\}$ and $G_{i}(x_{0})=\\{x|x\in X_{i},\ x_{0}\geq_{i}x\\}$ are
closed. (page 125, 126, 130)

  8. (8)

Income of agent $i$ is defined as:

(9) $M_{i}(p)=p\cdot r_{i}+\sum_{j\in F}\alpha_{ij}\pi_{j}(p)$

The budget set of agent $i$ is defined as:

(10) $B_{i}(p)=\\{x|x_{i}\in\mathbb{R}^{n}\mbox{ and }p\cdot x\leq
M_{i}(p)|x|\leq c\\}$

This set should be convex for every $p\in(\mathbb{R}^{n}_{+}-\bar{0})$. (page
175, 181)

  9. (9)

Each agent has a demand set given by:

(11) $D_{i}(p)=\\{x_{i}|x_{i}\in B_{i}(p)\cap X_{i},\ \ x_{i}\geq_{i}x\ \
\forall\ x\in B_{i}(p)\cap X_{i}\\}$

Moreover, this set is convex. (page 132)

  10. (10)

$B_{i}(p)\cap X_{i}$ is continuous (lower and upper hemicontinuous), compact
valued, and nonempty for all $p$ in the unit simplex $\Delta$, i.e., $p\geq 0$
and $\sum_{i}p_{i}=1$. Also, $D_{i}(p)$ is upper hemicontinuous, convex,
nonempty, and compact for all $p\in\Delta$. (page 175, 301,303)

  11. (11)

For all $H_{i}$,

(12) $M_{i}(p)>\inf_{x\in X_{i}\cap\\{x||x|\leq c\\}}p\cdot x\ \forall\
p\in\Delta$

(page 133)

  12. (12)

Finally, a notation: The excess demand correspondence at prices $p$ is defined
as $Z(p)=\sum_{i=1}^{m}D(p)-\sum_{j=1}^{l}S_{j}(p)-r$. Thus,
$Z(p)\subseteq\mathbb{R}^{n}$ records excess of demand over the supply
(including initial resources). (page 147)

Arrow and Debreu prove (see Ross, Theorem 24.7(Starr, 2011), page 308) that
under the assumptions and specifications mentioned above, there exists a
competitive market equilibrium, i.e., a sequence of production vectors
$(y^{*}_{1},\ldots,y^{*}_{l})$, with each $y^{*}_{i}\in Y_{i}$, a price vector
$p^{*}=(p^{*}_{1},\ldots,p^{*}_{n})$, and a set of allocations
$x^{*}_{1},\ldots,x^{*}_{m}$ with each $x^{*}_{i}\in X_{i}$, all of which
satisfy the following conditions:

  1. AD1

Each firm maximizes profits, i.e., $y^{*}_{j}$ belongs to $S_{j}(p^{*})$ for
each $j$.

  2. AD2

Each household maximizes utility, i.e., $x_{i}$ maximizes $u_{i}(x_{i})$ over
the set $B_{i}(p^{*})$ i.e. $x_{i}\ \in D_{i}(p^{*})$.

  3. AD3

Prices are non-negative, bounded and not all are zero. Without loss of
generality, we can assume that $\sum_{j}p_{j}=1$.

  4. AD4

(i) $Z(p^{*})\leq 0$. Moreover, (ii) if $Z_{k}(p^{*})<0$ then $p^{*}_{k}=0$.

###  3.2. SCM model as an A-D instance

Given the data $T,Y,U$ for a small market, we shall now build a suitable A-D
market. Recall that there are $m$ labor classes and $n$ processed goods, and
$Y$ is the $m\times 1$ vector recording the available labor in each class.
Also, recall that $T$ and $U$ are $m\times n$ matrices recording the
technologies available and the utilities of the classes where $T_{ij}$ refers
to the number of labor-units of type $i$ required to produce one unit of good
$j$, and $u_{ij}$ denotes the utility of good $j$ to class $i$. We now
construct the market $AD(T,Y,U)$.

  * •

The total number of goods in $AD$ are $n+m$, viz.,
$\\{g_{1},\ldots,g_{n},r_{1},\ldots,r_{m}\\}$, where $r_{i}$ corresponds to
the labor of class $i$. We call labor inputs ‘raw’ goods.

  * •

The set of firms in $AD$ is $F=\\{f_{1},\ldots,f_{n}\\}$, where $n$ is the
number of columns of $T$. The firm $f_{j}$ produces good $g_{j}$.

  * •

The number of households is $m$ corresponding to the labor classes, and each
agent $H_{i}$ begins with an endowment $Y_{i}$ of the good $r_{i}$ above.

  * •

The production function of $f_{j}$ is $PY_{j}$ (to differentiate it from the
total labor available, viz., $Y_{i}$) and it arises from the column $j$ of
$T$. We define $v_{j}$ as the $(n+m)$ vector $g_{j}-\sum_{k}T_{kj}r_{k}$ to
represent that $T_{kj}$ units of labor type $k$ are used to make one unit of
good $g_{j}$ and the production technology set $PY_{j}$ as
$PY_{j}=\\{\lambda\cdot v_{j}|\lambda\in[0,L]\\}$ where $L$ is a large number.
Thus, the firm $f_{j}$ may produce $\lambda$ units of good $g_{j}$ after
consuming the necessary resources $\lambda\cdot(\sum_{k}T_{kj}r_{k})$.

  * •

Recall that household $H_{i}$ owns a fraction $\alpha_{ij}$ of the firm
$f_{j}$. We specify these numbers arbitrarily with the prescribed conditions
that $\alpha_{ij}\geq 0$ for all $i,j$ and that $\sum_{i}\alpha_{ij}=1$ for
all $j$. The exact numbers will be irrelevant since we will see that in
equilibrium, the firms make zero profits.

  * •

The space $X_{i}$ of all possible consumption plans is $\mathbb{R}^{n+m}_{+}$,
the set of non-negative vectors denoting the consumption of the finished goods
and the primary goods.

  * •

The utility matrix serves to define the continuous real-valued utility
function $u_{i}$ for each agent $i$. If $x_{ij}\geq 0$ is the amount of good
$j$ allocated to agent $i$, then $u_{i}=\sum_{j}u_{ij}(x_{ij})$. Utilities are
zero for labor unit hours, i.e., $u_{ij}=0$ for $j>n$, as it is only the firms
that have any use for labor. Since the utilities are piece-wise linear, it is
clear that the principle of non-satiation holds if, for every $i$, there is a
$j$ such that the derivative $u^{\prime}_{ij}$ is positive, a property which
we assume. Also, since $u_{ij}=0$ for $j>n$, i.e., the agents have no use for
labor, we may restrict $X_{i}$ to those elements $x$ with $x_{j}=0$ for $j>n$,
i.e., effectively $X_{i}=\mathbb{R}^{n}_{+}$.

This completes the specification of the A-D market $AD$. We verify that the
specification satisfies conditions 1-13 of 3.1. Condition 1 is obvious. Given
that $PY_{j}=\\{\lambda v_{j}|\lambda\in[0,L]\\}$ is a line segment, we may
check that conditions 2-3 are satisfied. Condition 4 requires us to compute
$S_{j}(p)$, the optimal supply for a firm $j$, given a price vector $p$. We
see that $S_{j}(p)$ is as given below:

(13) $S_{j}(p)=\left\\{\begin{array}[]{rl}0&\mbox{ if $p\cdot v_{j}<0$}\\\
PY_{j}&\mbox{ if $p\cdot v_{j}=0$}\\\ L\cdot v_{j}&\mbox{ if $n\cdot
v_{j}>0$}\\\ \end{array}\right.$

It is easily seen that $S_{j}$ is upper hemicontinuous.

Condition 5 is straightforward. As regards Condition 6, it is easy to find a
bound $c\in R$ for the overall production in our economy: one such bound is
$m\cdot n$ times the product of the largest labor class and the largest entry
in the matrix $T$. Conditions 7-9 are easily satisfied by our piece-wise
linear concave utility functions. Condition 10 requires us to compute the
optimal bundle $D_{i}(p)$ for an agent $i$ and argue for the convexity of this
set. For an splc utility function, it is easily seen that this is a convex
polyhedral set. It is easily seen that this is upper hemicontinuous.
Conditions 12-13 are trivial. Thus, our market model $AD(T,Y,U)$ satisfies the
A-D conditions of Section 3.1.

###  3.3. The equivalence of equilibria

We next show an equivalence between the equilibria of the market $SCM(T,Y,U)$
and the A-D equilibria of $AD(T,Y,U)$. To begin with, we explain the nature of
equilibria for $AD(T,Y,U)$ as assured by the A-D result, and set it in the
context of our notation for the SCM model. The A-D equilibria is given by the
$(n+m)$-vector $P^{*}$ of prices , the $(n+m)$-vector of productions
$(Y^{*}_{j})$ and the $m\times(n+m)$-matrix of allocations $(X^{*}_{ij})$.

The price vector $P^{*}$ specifies prices for all the $(n+m)$ goods, i.e., for
labor as well as finished goods. Whence, we may denote $P^{*}=(p^{*},w^{*})$,
where $p^{*}$ is the $n$-vector recording the prices of goods, and $w^{*}$ is
the $m$-vector recording the wages of the $m$ labor classes. Similarly, the
vector $Y^{*}_{j}$ must be the multiple $q^{*}_{j}v_{j}$. Finally, the
allocations $X^{*}_{ij}$ for $j>n$ may be assumed to be zero since they do not
alter the utility of any household and only (possibly) reduce its budget.
Thus, we may assume $X^{*}$ to be specified instead by $x^{*}$, an $m\times n$
matrix. Thus, an A-D equilibrium is effectively specified by the tuple
$(q^{*},w^{*},p^{*},x^{*})$, i.e., in the same format as the SM equilibrium.

###### Proposition 3.1.

An SM equilibrium $(q^{*},w^{*},p^{*},x^{*})$ is also an AD-equilibrium.

###### Proof.

Let us prove AD1, i.e., each firm maximizes profits. For this, consider a firm
$j$ and the variable $q_{j}$. By the fact that $w^{*}$ satisfies
$DP_{\mathcal{P}}$, we know that $w^{*}T\geq p^{*}$, i.e., for a column $j$,
either the firm is unprofitable or its profit = $p_{i}q_{i}-w_{i}T_{i}q_{i}$
is zero. Thus there is nothing to prove when $q^{*}_{j}=0$. On the other hand,
when $q^{*}_{j}>0$, we know that $(w^{*}T)_{j}=p^{*}_{j}$ and that the profit
is zero, irrespective of $q^{*}_{j}$. Thus the firm $j$ maximizes profits.

Next, let us come to AD2, i.e., each household $H_{i}$ maximizes its utility
given its budget as the only constraint. The budget of the household $j$ is
precisely $w^{*}_{j}*Y_{j}$, and this is what the household is endowed with
when it enters the Fisher market. The market clearing condition for the buyer
ensures that all money is spent. The maximum bang-per-buck (optimality)
condition ensures that, given the prices, only the best possible goods are
bought. Thus, while the allocation $x^{*}_{ij}$ respects conservation
constraints, there is no allocation that improves the utility for household
$j$, given its budget constraint.

AD3 is obviously ensured.

For AD4, first, consider AD4(i), i.e., that the supply exceeds demand and the
price of a good is non-zero only when supply exactly meets demand. For this,
consider first the labor goods $r_{i}$’s. While the supply is $Y_{i}$, the
condition $Tq^{*}\leq Y$ ensures that supply exceeds demand. Now, if there is
any excess supply of labor class $r_{i}$, then the dual $DP_{\mathcal{P}}$
ensures that the wage $w^{*}_{i}=0$. For the produced goods $g_{j}$, the
market clearance for the seller ensures that the whole amount $q^{*}_{j}$ is
allocated, i.e., $\sum_{i}x^{*}_{ij}=q^{*}_{j}$. Thus the market clears for
every good. Note that $p^{*}_{j}>0$ always holds for an SM equilibrium. This
concludes the proof. ∎

###### Proposition 3.2.

An AD equilibrium $(q^{*},w^{*},p^{*},x^{*})$ is also an SM equilibrium.

###### Proof.

Let us first examine the condition AD4(i) that $Z(p^{*})\leq 0$. For a labor
good $r_{i}$, this precisely states that $Tq^{*}\leq Y$. Note that we have
chosen a large $L$ to construct $Y_{j}$, hence we must have $0\leq
q^{*}_{j}<L$. On the production side, AD4(ii) ensures that
$w^{*}_{i}(Y_{i}-(Tq^{*})_{i})=0$, i.e., $w^{*}$ exhibits complementary
slackness with respect to the conditions $Tq^{*}\leq Y$.

Now let us examine AD1 for a firm $f_{j}$. Since $Y_{j}=\\{\lambda
v_{j}|\lambda\in[0,L)\\}$, the profit of the firm is given by
$q^{*}_{j}(p_{j}-(w^{*}T)_{j})$. Thus if $p_{j}>(w^{*}T)_{j}$ then the optimal
value would have been $q^{*}_{j}=L$. Since we know that $q^{*}_{j}<L$, we have
that $(w^{*}T)\geq p_{j}$, i.e., the variables $w^{*}$ satisfy the dual
program $w^{*}T\geq p$ and thus the tuple $(q^{*},w^{*})$ indeed satisfy the
optimal production condition for SM equilibrium.

Next, let us evaluate $Z(p^{*})$ for a good $g_{j}$. If indeed
$Z(p^{*})_{j}<0$, then we are assured $p^{*}_{j}=0$. But $Z(p^{*})_{j}<0$
indicates an excess of supply of good $g_{j}$. Assuming that there is an $i$
such that $u_{ij}>0$ and $w_{i}>0$ tells us that the allocation $(x_{ij})_{j}$
is not an optimal allocation since, some extra good $g_{j}$ is available for
free and which is of value to buyer $i$. This tells us that $Z(p^{*})_{j}=0$
for all useful goods and that there is market clearance from the seller side.
By the optimality of the consumption plan, i.e., AD2, given the budget, we
know that all money is spent, i.e., there is market clearance on the buyer
side as well.

Finally, given the prices $p^{*}$, for an agent $i$ with $w^{*}_{i}>0$, define
$b_{i}=\max_{j}(u_{ij}/p_{j})$, i.e., the bang-per-buck for labor class $i$.
Note that if $b_{i}=\infty$, then the optimal allocation for this class would
have been infinite too. Since the AD equilibrium exists and is finite, we must
have $p_{j}>0$ for all useful goods $g_{j}j$ and $b_{i}<\infty$. Now, from the
optimality constraint AD2, it is clear that whenever $p_{j}x_{ij}>0$, we must
have $b_{i}=u_{ij}/p_{j}$ and that if $x_{ij}=0$, we must have
$b_{i}>u_{ij}/p_{j}$. This proves the optimality condition of $(p^{*},x^{*})$
and that $(p^{*},x^{*})$ satisfy the optimal consumption requirement of SM.

This concludes the proof. ∎

###### Theorem 3.3.

There is an equivalence between the equilibria of the SCM market parametrized
by $(T,Y,U)$ and the corresponding A-D market $AD(T,Y,U)$. As a result, for
any SCM market given by the above data, there exists an SM equilibrium.

###### Proof.

The equivalence follows from Proposition 3.1, and Proposition 3.2. The
existence of an AD equilibrium is guaranteed by Theorem 24.7 of (Starr, 2011).
∎

##  4\. The Structure of SM Equilibria and the Consumer Choice Game

In this section, we analyze the combinatorial structure of SM equilibria and
its connection with market parameters, where we show that these structures are
local invariants. This understanding is then used to define the _Consumer
Choice Game_ , i.e., CCG.

###  4.1. Generic equilibrium and combinatorial data

For an equilibrium point $\eta$, we define $I(\eta)=\\{i\in[m]|w_{i}>0\\}$.
i.e., the labor classes whose wages are positive. Similarly, we define
$J(\eta)=\\{j\in[n]|q_{j}>0\\}$, i.e., the goods which have non-zero
production. Similarly, we define $F(\eta)$ as $\\{(i,j)|X_{ij}>0\\}$, i.e.,
the indices $(i,j)$ such that labor class $i$ has a positive consumption of
good $j$.

We next associate $I(\eta),J(\eta)$ and $F(\eta)$ as the combinatorial data
with equilibrium point $\eta=(p,q,w,X)$ for the parameters $(T,Y,U)$ of the
economy. The combinatorial data identify key features of the equilibrium,
e.g., the labor classes with non-zero wages, the goods produced, and the
Fisher forest, i.e., the price-determining consumption of goods. We define the
notion of ‘generic-ness’, which allows us to construct the equilibrium from
its combinatorial data, and to extend such equilibria at a point to its
vicinity.

###### Definition 4.1.

We say that $\eta$ is a generic equilibrium if (i) for $j\not\in J(\eta)$, we
have $(wT)_{j}>p_{j}$, and (ii) for $(i,j)\not\in F(\eta)$, we have
$u_{ij}/p_{j}<\max_{k}u_{ik}/p_{k}$.

Let us now fix $T,Y$ and vary $U$ over $\mathcal{U}=\mathbb{R}^{m\times n}$.
Given a $U\in\mathcal{U}$, and an equilibrium point $\eta$ with the economy
specifications $T,Y,U$, we say that $\eta$ sits over $U$, since it is for this
element of $\mathcal{U}$, that $\eta$ was observed. Theorem 4.2 relates to the
existence of generic equilibria.

###### Theorem 4.2.

Let $T,Y,U$ be matrices in general position, i.e., there be no algebraic
relationship between the entries, with rational coefficients. Given an
equilibrium $\eta$ over $U$, there are arbitrarily close $U^{\prime}$ and
equilibria $\zeta$ sitting over $U^{\prime}$ which are generic. Moreover, if
$\eta$ has $m$ wage-earning labour classes, i.e., $|I(\eta)|=m$ and $n$ goods
produced, i.e., $|J(\eta)|=n$, then the number of connected components ($k$)
of the solution Fisher forest $F(\eta)$ is at least $n-m+1$.

###### Proof.

See Appendix B.1. ∎

It is an important question if the data $(I,J,F)$ do indeed determine $\eta$,
the equilibrium. This is summarized in the Theorem 4.3.

###### Theorem 4.3.

Again, let $T,Y,U$ be in general position and $\eta$ be a generic equilibrium
over $U$ with the combinatorial data $(I,J,F)$, then the parameters of $\eta$,
viz., $p,w,q$ are solutions of a fixed set of algebraic equations in the
coefficients of $U$. For an open set of the parameter space of $\mathcal{U}$,
the equilibria, as guaranteed by Theorem 3.3, are generic and have the same
combinatorial data as $\eta$.

###### Proof.

See Appendix B.2. ∎

###  4.2. The Consumer Choice Game

We now define the consumer choice game $CCG(T,Y)$, parametrized by the
technology matrix $T$ and the labor inventory $Y$, which are henceforth
assumed to be fixed. The players are the labor classes, i.e.,
${\mathcal{L}}=\\{L_{1},\ldots,L_{k}\\}$. The strategy space $S_{i}$ for
player $i$ is the utility ”row” vector $(u_{i*})\in\mathbb{R}^{n}$. These rows
together constitute the matrix $U$. This strategy space is denoted by
${\mathcal{U}}$. We also assume that there is a ”real” utility matrix $U^{t}$
which is used to measure outcomes. The motivation is to advance the
implications of the _Fisher market game_ , (Adsul et al., 2010), by defining
strategies in the SCM model.

Given a play $U$, the outcome is given by an $\eta(U)=(q,w,y,p,X)$, an
equilibrium over $U$ obtained in the SCM model. The payoffs,
$b_{i}(X)=\sum_{j}(U^{t})_{ij}x_{ij}$, i.e., the equilibrium allocations
evaluated by each player on their true utilities, define the preference
relations for each player.

Let us now construct the pay-off functions in the vicinity of a generic
equilibrium point $\eta(U)$ with the combinatorial data $(I,J,F)$. We first
see that there is an open set $O_{I,J,F}\subseteq{\mathcal{U}}$ containing $U$
which has the same combinatorial data $(I,J,F)$. The exact inequalities
defining $O_{I,J,F}$ arise from the requirement that the Fisher forest $F$
have non-negative flows in all edges of $F$, that the edge $(i,j)\not\in F$
has an inferior bang-per-buck, and that $(wT)_{j}-p_{j}>0$ for $j\not\in J$.
As an example, consider an edge $(i,j)\in F$, and the requirement that the
flow in this edge be positive. Now, the flow in this edge is a suitable linear
combination of the wages $w$’s, prices $p$’s, and quantities $q$’s. As we have
argued before, these, in turn, are smooth functions of the entries of $U$.
Thus the condition that flow in the edge $(i,j)$ be positive is the
requirement that $f(U)>0$ for a suitable smooth function $f$ on $U$.

Thus, there is indeed such an open set $O_{I,J,F}$, and the pay-off functions
are solutions of algebraic equations in the entries of $U$, the coefficients
of which depend on the combinatorial data $(I,J,F)$. This gives us Theorem 4.4
below.

###### Theorem 4.4.

For a generic equilibrium point $\eta(U)$ with the combinatorial data
$(I,J,F)$, there is an open set $O_{I,J,F}\subseteq{\mathcal{U}}$ containing
$U$ and a smooth family $\eta^{\prime}(U^{\prime})$ of equilibria for each
$U^{\prime}\in O_{I,J,F}$ such that (i) $\eta^{\prime}(U)=\eta(U)$ and (ii)
the combinatorial data for $\eta^{\prime}(U^{\prime})$ is $(I,J,F)$.

The general pay-off function is to be pieced together by such a collection of
open sets, indexed by combinatorics. On non-generic $U^{\prime}$, the
equilibrium $\eta(U^{\prime})$ will have multiple feasible allocations and
this determines a correspondence between the strategy space ${\mathcal{U}}$
and $\mathbb{R}^{k}$, the pay-off space. Even for a generic $U$, there may be
multiple equilibrium points, viz., $\eta_{1},\ldots,\eta_{k}$, and each of
these will define an analytic sheet of the correspondence over the generic
open set. Before we show these characteristics in detail in Section 5, we
demonstrate the theory described so far using an example.

###  4.3. Illustration

We describe an economy $\mathcal{H}_{3}$ with three labor classes
${\mathcal{L}}=\\{1,2,3\\}$ and three goods ${\mathcal{G}}=\\{1,2,3\\}$ and
construct a market equilibrium $\eta$ and the CCG around it, i.e., the payoff
functions at $\eta$. For this equilibrium $\eta$,
$I(\eta)=J(\eta)=\\{1,2,3\\}$.

(26) $\displaystyle T=\left[\begin{array}[]{ccc}0.5&0&0\\\ 0.5&2&0\\\
0.5&4&8\end{array}\right]\>Y=\left[\begin{array}[]{c}5\\\ 20\\\
100\end{array}\right]\>U^{t}=\left[\begin{array}[]{ccc}1.5&0&0\\\ 1.5&2&0\\\
0&2&1\end{array}\right]\>U(\alpha,\beta)=\left[\begin{array}[]{ccc}1.5&0&0\\\
\alpha&2&0\\\ 0&\beta&1\end{array}\right]$

The economy has specifications $T$ (Technology matrix), $Y$ (Labor
availability), $U^{t}$ (True Utility matrix) and a parametrized matrix
$U(\alpha,\beta)$ as given in Eq. (26). Matrix $T$ has columns as technologies
$T_{1},T_{2}$ and $T_{3}$ and rows as the labor classes $L_{1},L_{2}$ and
$L_{3}$. We see that $T_{2}$ and $T_{3}$ do not need $L_{1}$ at all, while
$T_{1}$ utilizes $L_{3}$ about 16 times more effectively. The matrix $U$ is
assumed to have two parameters; $\alpha$, to be used by $L_{2}$, and $\beta$,
to be used by $L_{3}$. Note that $U^{t}\in U(\alpha,\beta)$. We compute (i)
the dependence of the pay-offs on $\alpha$ and $\beta$, and (ii) the sub-
domain of $U(\alpha,\beta)$ over which the chosen forest
$F=\\{(1,1),(2,1),(2,2),(3,2),(3,3)\\}$ in Fig. 1 is the equilibrium forest.

Figure 1. Fisher Market Forest

The production vector $q$ and wages $w$ follow from
$P_{\mathcal{P}},DP_{\mathcal{P}}$, independent of $U$ for fixed combinatorial
data $I=J=\\{1,2,3\\}$, as given in Eq. (36).

(36) $\displaystyle q=T^{-1}Y=\left[\begin{array}[]{c}10\\\ 7.5\\\
8.125\end{array}\right];\
w^{T}=[T^{-1}]^{T}\cdot\left[\begin{array}[]{c}p_{1}\\\ p_{2}\\\
p_{3}\end{array}\right]=\left[\begin{array}[]{c}2p_{1}-p_{2}/2+p_{3}/8\\\
p_{2}/2-p_{3}/4\\\ p_{3}/8\end{array}\right]\>$

This describes wages in terms of prices. Note that the support of $U$ gives us
utilities for all edges of the forest, and therefore allows us to compute the
Fisher market forest. Using the bang-per-buck relations between prices gives
$p_{2}=\beta p_{3}$ and $p_{1}=(\alpha\beta/2)p_{3}$. Next, the net class
wages $W$ and the net revenue collected by the producers $P$ are calculated
as:

$\displaystyle W_{1}$
$\displaystyle=y_{1}w_{1}=5p_{3}*(\alpha\beta-\beta/2+1/8);$ $\displaystyle
P_{1}$ $\displaystyle=q_{1}p_{1}=5p_{3}\alpha\beta$ (37) $\displaystyle W_{2}$
$\displaystyle=y_{2}w_{2}=20p_{3}*(\beta/2-1/4);$ $\displaystyle P_{2}$
$\displaystyle=q_{2}p_{2}=15p_{3}\beta/2$ $\displaystyle W_{3}$
$\displaystyle=y_{3}w_{3}=100p_{3}/8;$ $\displaystyle P_{3}$
$\displaystyle=q_{3}p_{3}=65p_{3}/8$

Setting $p_{3}=1$, since the forest $X$ is actually a tree, the conservation
equations give us the amount spent $A_{ij}=p_{j}*x_{ij}$, and hence
allocations $X$, as given in Eq. (38).

$\displaystyle A_{11}$ $\displaystyle=W_{1}=5p_{3}*(\alpha\beta-\beta/2+1/8)$
$\displaystyle x_{11}$ $\displaystyle=10-5/\alpha+5/4\alpha\beta$
$\displaystyle A_{21}$ $\displaystyle=P_{1}-W_{1}=5p_{3}*(\beta/2-1/8)$
$\displaystyle x_{21}$ $\displaystyle=5/\alpha-5/4\alpha\beta$ (38)
$\displaystyle A_{22}$ $\displaystyle=W2-A_{21}=15p_{3}(\beta/2)-5p_{3}(7/8)$
$\displaystyle x_{22}$ $\displaystyle=7.5-35/8\beta$ $\displaystyle A_{32}$
$\displaystyle=P_{2}-A_{22}=5p_{3}(7/8)$ $\displaystyle x_{32}$
$\displaystyle=35/8\beta$ $\displaystyle A_{33}$
$\displaystyle=W_{3}-A_{32}=65p_{3}/8$ $\displaystyle x_{33}$
$\displaystyle=8.125$

Finally, the total payoffs $b_{i}(\alpha,\beta)$ of each labor class $i$ are
the net “true” utilities received from their total consumption, as given in
Eq. (39).

$\displaystyle b_{1}(\alpha,\beta)$ $\displaystyle=1.5x_{11}(\alpha,\beta)$
(39) $\displaystyle b_{2}(\alpha,\beta)$
$\displaystyle=1.5x_{21}(\alpha,\beta)+2x_{22}(\alpha,\beta)$ $\displaystyle
b_{3}(\alpha,\beta)$
$\displaystyle=2x_{32}(\alpha,\beta)+x_{33}(\alpha,\beta)$

The conditions that wages, prices, and allocations, i.e., the money flows
$x_{ij}$, need to be positive, give us the following constraints: (i)
$8\alpha\beta-4\beta+1>0$, (ii) $4\beta-1>0$ and (iii) $60\beta-35>0$. Under
these conditions that define an open set, forest $F$ arises as the equilibrium
forest, giving the fixed combinatorial data $(I,J,F)$. Note that the payoffs
in Eq. (39) are also valid whenever $\eta$ remains the market equilibrium,
i.e., where $\alpha$ and $\beta$ are in the open set. Thus, we have an
$SM$-equilibrium
$\eta(\alpha,\beta)=(w(\alpha,\beta),q(\alpha,\beta),p(\alpha,\beta),X(\alpha,\beta))$,
parametrized by $\alpha$ and $\beta$.

This illustrates that the local combinatorial data is sufficiently explicit to
enable the computation of the pay-off functions. Moreover, significant
benefits may accrue to players if they utilize the freedom of posturing their
utility functions.

##  5\. Correspondence associated with CCG

In this section, we illustrate the connection between the strategy space
$\mathcal{U}$ and the combinatorial data of Fisher forests. We utilize a small
example to demonstrate the theory and refer to Appendix C.1 for a general
market scenario.

We consider a two-class economy with specifications $T$ (Technology matrix),
$Y$ (Labor availability), $U^{t}$ (True Utility matrix), and
${\mathcal{U}}=\mathbb{R}^{2\times 2}$ (Strategy matrix) as given in Eq. (48).
Since Fisher solutions do not change with independent row scaling of the
utility matrix, the description of $U$ suffices in representing
$\\{\mathcal{U}\\}$, with $\alpha=\frac{u_{11}}{u_{12}}$ and
$\beta=\frac{u_{21}}{u_{22}}$.

(48) $\displaystyle T=\left[\begin{array}[]{ccc}0.25&0\\\
0.25&1\end{array}\right]\>Y=\left[\begin{array}[]{c}2\\\
4\end{array}\right]\>U^{t}=\left[\begin{array}[]{cc}1&1\\\
1&1\end{array}\right]\>U=\left[\begin{array}[]{cc}\alpha&1\\\
\beta&1\end{array}\right]$

Assuming $0<\alpha,\beta<\infty$, we now cover the entire strategy space with
the finite collection of Fisher Forests. This enable us to define payoff
functions for the whole strategy space. In a two-class, two-goods economy,
there are nine possible ways to allocate the goods produced among the classes.
These include six forests and one cycle, where both classes participate; and
two forests where only one class is active. Out of these nine possibilities,
the inputs $(T,Y)$ allow for six distinct combinatorial structures while
satisfying the production and consumption side constraints, as given in Fig.
2.

Figure 2. Feasible combinatorial structures for equilibrium $SM(T,Y,U)$, with
for $J(\eta)=\\{1,2\\}$.

We see that the actual combinatorial data $(I,J,F)$ and the SM equilibrium
values $(p,q,w,X)=SM(T,Y,U)$ depend on the choice of $\alpha$ and $\beta$. As
guaranteed by Theorem 4.3, there is an open set for each forest, a ”zone”, of
the $\alpha$-$\beta$ space over which the forest is the Fisher forest of the
equilibrium. For fixed combinatorial data $(I,J,F)$, the corresponding zone-
defining conditions can be derived from the positivity of wages and
allocations, and the maximum bang-per-buck conditions, as seen in effect in
Section 4.3. Conditions specific to the feasible combinatorial structures are
given in Table 1, with a complete description of corresponding $(p,q,w,X)$ in
Appendix C.2.

| Forest-1 | Forest-2 | Forest-3 | Forest-4 | Forest-5 | Cycle
---|---|---|---|---|---|---
Zones | $\alpha\geq\beta>1/4$ | $\beta\geq\alpha>1/2$ |  $1/4<\beta<1/2$ and $\beta\geq\alpha$ | $\alpha\leq 1/2\leq\beta$ | $\beta\leq 1/4$ | $\alpha=\beta$
Table 1. $SM(T,Y,U)$ zone descriptions for the structures in Fig. 2

We also classify the generic equilibrium points here. Recall that $\eta$ is a
generic equilibrium if (i) for $j\not\in J(\eta)$, we have $(wT)_{j}>p_{j}$,
and (ii) for $(i,j)\not\in F(\eta)$, we have
$u_{ij}/p_{j}<\max_{k}u_{ik}/p_{k}$. Therefore, we see that an equilibrium
point belonging to, say zone-1, is generic if and only if $\alpha>\beta$. We
can define an open set for zone-1, viz. , $\alpha>\beta>1/4$, comprising only
of generic equilibrium points. Similarly, for zone-4, the open set would be
$\beta>1/2>\alpha$.

Figure 3. The equilibrium $SM(T,Y,U(\alpha,\beta))$ categorized into multiple
zones, as a function of $(\alpha,\beta)$.

Finally, we may summarize this discussion through Fig. 3, where the five zones
are marked corresponding to the forests. The arrows indicate the direction of
optimization for the dominant class i.e. the class that has control over the
ratio $\alpha$ or $\beta$. The interior of each zone is an open set consisting
of generic points and the boundaries correspond to the non-generic equilibrium
points.

Fig. 3 shows that by changing $\alpha$ or $\beta$, it is possible to transit
from one forest to another, by crossing the non-generic (boundary) points
where both forests are feasible. It is shown earlier that the set of
allocations in the Fisher market is hemicontinuous with respect to initial
endowments and utility functions (Vazirani and Yannakakis, 2011) . Here, we
show that although multiple allocations are possible at such points, utilities
on the boundary points are bounded by the utility limits of the forests on
both sides. In other words, allocations and utilities at the transitions are
convex combinations of the boundaries of those obtained in the adjoining
zones. To make this precise, we make the following definition.

###### Definition 5.1.

Let $x\in\mathcal{U}$ be a point on the boundary of two zones, say Zone A and
Zone B and let $\eta=(p,q,w,X)$ be a typical point above $x$, i.e., $\eta$ is
an equilibrium for the parameter $x$. Let $\mathcal{X}(x)$ be the collection
of all allocations of equilibria above $x$ in the U-space. We say that $x$ is
a manifold point if the set $U_{i}(\mathcal{X}(x))$ is a bounded interval and
its bounds are obtained as the limits $\lim_{q\rightarrow x}U_{i}(X(q))$ and
$q\in$ Zone A and $q\in$ Zone B.

For a point $x$ located on the line $\alpha=\beta$, with
$\alpha(x)=\beta(x)=\mu>0.5$, i.e. on the transition between Zone 1 and Zone
2, the corresponding equilibrium $\eta(x)=(p,q,w,X)$ has: (i)
$I(x)=J(x)=\\{1,2\\}$, and (ii) $F(p)=\\{(1,1),(1,2),(2,1),(2,2)\\}$, i.e. a
cycle. Furthermore, there exists a set of possible allocations $\mathcal{X}$
at $\eta(x)$, and the bounds on $U_{i}(\mathcal{X})$ are precisely those
achieved as $\lim_{q\rightarrow x}U_{i}(X(q))$ for $q$ in zone 1 and 2, making
it a manifold point. Similarly, we find that $x$ located on
$1/4<\alpha=\beta<1/2$, i.e., between Zone 1 and 3, is a manifold point. For
the points on the boundaries with Forest 4, however, we see that the limits
$\lim_{q\rightarrow x}U_{i}(X(q))$ are equal from both zones, thus giving a
unique allocation $X$. On the other hand, we see that the points on the
boundary $\beta=1/4$ are not manifold points. This is because, for each point
on this line, there is a unique allocation $X$ leading to a unique $U_{i}(X)$
which does not equal $\lim_{q\rightarrow x}U_{i}(X(q))$ for $q$ in zone 1 or
3.

Further corresponding to the interior, i.e., generic points of the zones, we
have payoffs $b_{1},b_{2}$ defined uniquely, which are continuous functions of
$\alpha,\beta$, as guaranteed by Theorem 4.4. Moreover, these are invertible
functions on their restricted domain of $\alpha,\beta$, which makes the sets
of possible payoffs $b^{p}_{1}$, $b^{p}_{2}$ open. It follows that the
correspondences $\mathcal{N}_{i}=(\alpha,\beta,b^{p}_{1},b^{p}_{2})$ are open
for each forest $i$. Moving further, we see that for points on $\alpha=\beta$,
associated with a cycle, the corresponding $b^{p}_{1}$ and $b^{p}_{2}$ belong
to open sets. For example, on the boundary of forests 1 and 3, the open set
$U^{p}_{1}$ is given by $1/4<\alpha<1/2$ and $(8\alpha-2,8-2/\alpha)$. The
region looks like half of a parabola, bounded from all sides. Thus, the
interior of every possible solution or zone is open and the neighborhood of
each point is homeomorphic to open subsets of $\mathbb{R}^{2}$. For forests,
the homeomorphism is given by the inverse of utility functions and for the
cycle, the sets are open in $\mathbb{R}^{2}$.

Finally, we claim that the boundaries of the forests and the cycle form
1-dimensional entities which serve as boundaries to the described
2-dimensional manifolds. Each boundary can be given by a unique equation in
$\mathbb{R}^{2}$. On line $\alpha=\beta$, there are two boundaries, one coming
from $\alpha<\beta$ and another from $\alpha>\beta$. In between these two, a
2-dimensional plane is situated on each part of the segment $\alpha=\beta$,
i.e. on the boundary of forest-1 and 3 and forests 1 and 2. When considering
the closures of the open sets, we see that correspondences intersect along
these boundaries.

We therefore establish that in the region $\beta>1/4$, payoff function $b_{i}$
is a 2-dimensional manifold with boundary, consisting of all ‘manifold’
points. Moreover, a correspondence $\mathcal{N}$ =
$(\alpha,\beta,b^{p}_{1},b^{p}_{2})$ can be defined between the strategy space
and the payoffs space. A general version of markets is dealt with in Appendix
C.1, where we argue that the same results follow.
Strategic Analysis: As shown in Fig. 3, class-2 is dominant in the sense that
it has a strategy to become the only active class in the economy, just by
reducing $\beta$. However, it is not in its interest to completely drive out
class 1. If the state is in any zone with both classes active, $\beta$ can be
decreased to reach zone 5, where class-2 gets utility 4. But, due to a
discontinuity in the utility function, for any $\alpha$, class-2 achieves the
highest payoff (getting arbitrarily close to 10), whenever its $\beta$
approaches 1/4, but is, strictly more than 1/4. Thus, the best strategy for
class-2 is to keep class-1 active and pose $\beta$ as close to, but greater
than 1/4. Technically, we see that the discontinuity of the utility functions
results in the non-existence of Nash equilibria.

Although this economy rules out the possibility of a Nash equilibrium by
making one class clearly dominant, the general scenario has a possibility of
the existence of Nash equilibria and is discussed in Appendix C.1.

##  6\. The role of strategy

In this section, we utilize the SCM model to discuss the case of a specific
example - the small market of soaps. We demonstrate the use of $U_{i}$, the
utility vector of an agent or labor class $i$, as devices to change
allocations. We also illustrate how $T$, the set of technologies, and $Y$, the
availability of labor determine $X$, the market structure, and the payoffs in
the small market.

###  6.1. The market of soaps

We consider a society that produces a single product, such as household soaps.
The labor classes involved in the production of this product may be classified
as follows: $L_{3}$, a low-technology worker in the factory, or an individual
producer at the household scale; $L_{2}$, an intermediate-technology worker
who can adapt processes to address consumer choice, coordinate and improve
production and distribution processes or is able to lead small and medium
enterprises; $L_{1}$, a high-end technology worker who can bring international
investments and machinery, can produce at a very large scale, and can run
elaborate distribution and marketing, networks. The matrices $T,Y,U^{t}$ are
as given in Section 4.3 and are reproduced again in Eq. (61). This is a
simplified representation of the data on page 16, (Date, 2004), a report on
soap manufacturing in India.

(61) $\displaystyle T=\left[\begin{array}[]{ccc}0.5&0&0\\\ 0.5&2&0\\\
0.5&4&8\end{array}\right]\>Y=\left[\begin{array}[]{c}5\\\ 20\\\
100\end{array}\right]\>U^{t}=\left[\begin{array}[]{ccc}1.5&0&0\\\ 1.5&2&0\\\
0&2&1\end{array}\right]\>U(\alpha,\beta)=\left[\begin{array}[]{ccc}1.5&0&0\\\
\alpha&2&0\\\ 0&\beta&1\end{array}\right]$

The technology matrix $T$ lists the ways of manufacturing soaps in this
economy. The last column $T_{3}$ uses only the $L_{3}$ labor class, while
$T_{2}$ is a boutique manufacturing process that utilizes both $L_{2}$ and
$L_{3}$ labor classes but with a reduced requirement of $L_{3}$ and a less
than proportionate the use of $L_{2}$. $T_{1}$ is the most efficient method in
terms of manpower. It uses a half-unit of $L_{1}$ to substantially reduce the
number of both $L_{2}$ and $L_{3}$. In real life, $T_{1}$ encompasses various
activities such as packaging, transport, advertising, rents on expensive
machinery, and other activities related to mass production and distribution.

The payoff functions $b_{1},b_{2}$ and $b_{3}$ of individual labour classes as
functions of the strategy vector $U(\alpha,\beta)$, are also as in Section 4.3
and are reproduced below:

(62) $\displaystyle b_{1}(\alpha,\beta)$
$\displaystyle=1.5x_{11}(\alpha,\beta)$ (63) $\displaystyle
b_{2}(\alpha,\beta)$
$\displaystyle=1.5x_{21}(\alpha,\beta)+2x_{22}(\alpha,\beta)$ (64)
$\displaystyle b_{3}(\alpha,\beta)$
$\displaystyle=2x_{32}(\alpha,\beta)+x_{33}(\alpha,\beta)$

Note that, the allowed flexibility in $U$ is restricted to $\alpha$ and
$\beta$, merely for the purpose of illustration and easy computability of the
payoff functions.

####  6.1.1. Utility amendments:

The payoff functions may be used to construct a finite strategy bimatrix game
for the labour classes $L_{2}$ and $L_{3}$. We choose $3$ values for $\alpha$,
viz., $\alpha_{1}=1,\alpha_{2}=1.5,\alpha_{3}=3$ and $3$ values for $\beta$,
viz., $\beta_{1}=1,\beta_{2}=2,\beta_{3}=3$, which lie within the open set of
valid values, derived in Section 4.3. For nine combinations of $\alpha,\beta$,
we record the net payoffs $b_{i}(\alpha,\beta)$ for all three classes in Table
2. The center of each table, i.e., the values $b_{i}(\alpha_{2},\beta_{2})$
correspond to $U=U^{t}$, i.e., true expression of utilities.

$\begin{array}[]{|c||c|c|c|}\hline\cr L_{1}&\beta_{1}&\beta_{2}&\beta_{3}\\\
\hline\cr\hline\cr\alpha_{1}&1.875&1.688&1.625\\\
\hline\cr\alpha_{2}&2.250&{\bf 2.125}&2.083\\\
\hline\cr\alpha_{3}&2.625&2.563&2.542\\\
\hline\cr\end{array}\>\>\>\begin{array}[]{|c||c|c|c|}\hline\cr
L_{2}&\beta_{1}&\beta_{2}&\beta_{3}\\\
\hline\cr\hline\cr\alpha_{1}&0.594&0.859&0.948\\\
\hline\cr\alpha_{2}&0.500&{\bf 0.750}&0.833\\\
\hline\cr\alpha_{3}&0.406&0.641&0.719\\\
\hline\cr\end{array}\>\>\>\begin{array}[]{|c||c|c|c|}\hline\cr
L_{3}&\beta_{1}&\beta_{2}&\beta_{3}\\\
\hline\cr\hline\cr\alpha_{1}&0.169&0.125&0.110\\\
\hline\cr\alpha_{2}&0.169&{\bf 0.125}&0.110\\\
\hline\cr\alpha_{3}&0.169&0.125&0.110\\\ \hline\cr\end{array}$

Table 2. Net payoffs $b_{i}(\alpha,\beta)$ for classes $L_{i}$,
$i\in\\{1,2,3\\}$, with ”true” values in bold

In the true state, there is a large disparity in the payoffs received by the
three labor classes. $L_{2}$ earns 6 times that of $L_{3}$ and $L_{1}$ earns
2.833 times that of $L_{2}$ and 17 times that of $L_{3}$. It is also evident
that there is a significant benefit to be gained by reporting $U\neq U^{t}$,
i.e., modified utility vectors. In fact, $L_{3}$ can increase its payoffs by
$35\%$ by reducing $\beta_{2}$ to $\beta_{1}$, $L_{2}$ can then minimize its
losses by $21\%$ by reducing $\alpha_{2}$ to $\alpha_{1}$, and this causes a
loss of $12\%$ to $L_{1}$. This implies that $L_{3}$ should devalue good
$g_{2}$ in comparison to $g_{1}$, while $L_{2}$ devalues $g_{1}$ in comparison
to $g_{2}$. In other words, the Nash equilibrium for this game is
$(\alpha_{1},\beta_{1})$.

####  6.1.2. The boutique market structure:

We now consider an alternate market structure, with
$F_{b}=\\{(1,1),(1,2),(2,1),(3,1),(3,3)\\}$, as shown in Figure 4.

Figure 4. Fisher Market Forest: boutique market structure

Here the production of the boutique soap, $g_{2}$, is completely consumed by
$L_{1}$. All classes consume the mass-produced soap, $g_{1}$, while only
$L_{3}$ consumes its own household production, $g_{3}$. We define
$U_{b}(\alpha,\beta)$ as the utility function for this consumption structure,
where the “true” utility matrix $U^{t}$ is $U_{b}(\frac{4}{3},\frac{2}{3})$,
as in Eq. (65).

(65) $U_{b}(\alpha,\beta)=\left[\begin{array}[]{ccc}1&\alpha&0\\\ 1&0&0\\\
1&0&\beta\end{array}\right]\>\>\>\>\>\>U^{t}=\left[\begin{array}[]{ccc}1&4/3&0\\\
1&0&0\\\ 1&0&2/3\end{array}\right]$

The price vector corresponding to $U_{b}(\alpha,\beta)$ would be
$p(\alpha,\beta)=[1,\alpha,\beta]^{T}$. Since the technology matrix $T$ and
the labor composition of the population, $Y$, remain the same, the quantity of
each good produced remains unchanged at $q=[10,7.5,8.125]^{T}$.

It turns out that this market structure is not feasible for
$U(\frac{4}{3},\frac{2}{3})$, i.e., at “true” prices, this consumption pattern
will not be observed. However, it is seen for $U(1,\frac{2}{3})$, where the
boutique soap $g_{2}$ is valued (and priced) the same as the mass-produced
$g_{1}$. It is also seen at $U(1,1)$, where $L_{3}$ chooses to value its
household production $g_{3}$ as equal to $g_{1}$. The corresponding payoffs
are given in Table 3.

$\begin{array}[]{|c||c|c|c|}\hline\cr(\alpha,\beta)&b_{1}&b_{2}&b_{3}\\\
\hline\cr\hline\cr(1,2/3)&3.124&0.503&0.125\\\
\hline\cr(1,1)&3.187&0.375&0.147\\\ \hline\cr\end{array}$

Table 3. Total payoffs for all the classes, under different settings of
$(\alpha,\beta)$

The reasons for the unsustainability of “true” prices in $X_{b}$ are
straightforward: Given that all boutique production, i.e., of $g_{2}$, must be
consumed by $L_{1}$, either the amount produced, i.e., $q_{2}$, must be small
or the price $p_{2}$ must be low as compared to $p_{1}$. This is easily
verified by setting the labor composition to $Y_{b}$ equal to
$[5,10,100]^{t}$, leading to a lower production of $g_{2}$. With this,
allocation $X_{b}$ becomes feasible and the payoff equal
$b=[2.129,0.750,0.128]$.

####  6.1.3. Technological updates:

Here, we examine the impact of changes in technology, $T$, while preserving
its hierarchical structure of labor classes. We introduce a new technology
matrix, $T^{\prime}$, which differs from the original matrix, $T$, as given in
Eq. (66).

(66) $T=\left[\begin{array}[]{ccc}0.5&0&0\\\ 0.5&2&0\\\
0.5&4&8\end{array}\right]\>\>\>\>\>\>T^{\prime}=\left[\begin{array}[]{ccc}0.5&0&0\\\
0.5&2&0\\\ 1&4&8\end{array}\right]$

This change may occur by internalizing some of the informal labor required in
the life-cycle of good $g_{1}$, which is mass-produced soap. This could
include incorporating additional garbage collection and disposal cycle for the
production good $g_{1}$, which is not done for good $g_{2}$ and $g_{3}$. By
internalizing this process, $T^{\prime}$ represents a possible new technology
matrix. Table 4 shows the comparison of the new market equilibrium, i.e.,
$SM(T^{\prime},Y,U)$ with that of the original one, i.e., $SM(T,Y,U)$.

$\begin{array}[]{|c||c|c|c||c|c|c|}\hline\cr&\vrule\lx@intercol\hfil
T\hfil\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil
T^{\prime}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\text{production}&10&7.5&8.125&10&7.5&7.5\\\
\hline\cr\text{payoffs}&2.125&0.75&0.125&2&0.75&0.125\\\ \hline\cr\end{array}$

Table 4. Comparison of the goods production and class payoffs, with
technologies $T$ and $T^{\prime}$

Table 4 shows that while the production of good $g_{3}$ has decreased, it is
the payoffs of labor class $L_{1}$ that have decreased while the production
and payoffs of other labor classes and goods remain unchanged.

####  6.1.4. Labor migration:

Finally, we examine the case with changes in $Y$, representing the relative
strengths of the labor classes. This may occur, for example, through
education. For instance, we can consider changes in $Y_{21}$ and $Y_{32}$
obtained by moving 5 units from $L_{2}$ to $L_{1}$ and $L_{3}$ to $L_{2}$
respectively, as given in Eq. (67).

(67) $Y=\left[\begin{array}[]{c}5\\\ 20\\\
100\end{array}\right]\>\>\>Y_{21}=\left[\begin{array}[]{c}10\\\ 15\\\
100\end{array}\right]\>\>\>Y_{32}=\left[\begin{array}[]{c}5\\\ 25\\\
95\end{array}\right]$

If $U^{t}$ is being asserted by every labor class, then while the quantities
of production may change with $Y$, the payoffs of the various classes remain
unchanged. However, if there is a deviation from the expected values of
$\alpha\neq 1.5$ or $\beta\neq 2$, then these distortions are further
exacerbated by migrations. The payoffs of $L_{1}$ are not affected by
migrations. For the other two classes, out-migrations benefit a class if it is
already making a strategically beneficial choice of its utility function. Out-
migrations or in-migrations have a small impact on the net payoffs of non-
participating classes. The observations can be summarized in Table 5.

$\begin{array}[]{|c|c|c|c|}\hline\cr class&regime&Y_{21}&Y_{32}\\\
\hline\cr\hline\cr L_{2}&\alpha<1.5&\uparrow&\downarrow\\\ \hline\cr
L_{2}&\alpha>1.5&\downarrow&\uparrow\\\ \hline\cr L_{3}&\beta<2&-&\uparrow\\\
\hline\cr L_{3}&\beta>2&-&\downarrow\\\ \hline\cr\end{array}$

Table 5. Increase/decrease in the payoffs of strategic classes, with the
migration effect

##  7\. Conclusion

This paper shows that consumer choice is indeed an important determinant of
the wage distribution in an economy. This connection provides an important
tool for wage-earners to understand how they can adapt their consumption so as
to support a more equitable distribution of wages. It does this by providing a
modeling and analytic framework which allows us to explore concretely the
thread between consumer choice, prices, production, and wages.

The paper also helps us understand the pricing of many everyday items, e.g.,
smartphones, where two similar devices may have very different prices, and
also that these prices may dramatically change based on a fluid consumer
choice. It also suggests that preferring goods and services provided by small-
branded and local/regional players, rather than buying the ”best” may be a
better strategy to ensure better wages.

Next, the key data required of the economy, viz., $T$ and $Y$, is a labor
inventory of the production processes of the economy and is part of some of
the standard data sets of countries. Such an inventory could be used to
develop a tool allowing each household to compute its labor footprint, i.e.,
an understanding of how household consumption brings employment across the
economy. One conjecture is that the consumption preferences of many wage-
earners possibly do not support their own employment. Such an understanding
may be useful to these very classes in modifying their personal consumption.
Also note that the labor footprint, while very similar to the GDP calculation,
does not need monetization. This is important in its own way.

Technically, the computation of an equilibrium, given a $T,Y,U$ is an
interesting problem. The A-D connection implies that already-known efficient
algorithms will shed some light. The tatonnement process of this paper needs
to be strengthened and its ”computing” power needs to be enhanced. A study of
the pay-off correspondence and the existence of Nash equilibria in general
$(T,Y,U)$ markets need to be undertaken.

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##  Appendix A Section 2 details

###  A.1. Production and Consumption sides: Illustrations

###### Example A.1.

(Production) Consider the following economy with two labor classes and two
goods as follows:

(68) $T=\left[\begin{array}[]{cc}1&0\\\
5&10\end{array}\right]\>\>\>Y=\left[\begin{array}[]{c}\delta\\\
10\end{array}\right]\>\>\>$

The number of people in labor class is a parameter $\delta$. The price vector
$p=[1,1]$. The labor constraints give us the equations:

(69) $\begin{array}[]{rcl}q_{1}&\leq&\delta\\\
5q_{1}+10q_{2}&\leq&10\end{array}$

We see that for $\delta\leq 2$, $q_{1}=\delta$ and $q_{2}=1-0.5\delta$, gives
us the maximum revenue, which is $p_{1}q_{1}+p_{2}q_{2}=1+0.5\delta$. The
wages must satisfy $wT=p$ whence they become $w_{1}=0.5$ and $w_{2}=0.1$. The
net wage bill is $0.5\cdot\delta+10\cdot 0.1$ which equals the revenue.

However for $\delta>2$, we see that $q=[2,0]^{T}$ and the revenue is fixed at
$2$. Also note that $(Tq)_{1}=2<\delta$. This forces $w_{1}=0$. The wages $w$
must satisfy $wT\geq p$ and since $q_{1}>0$, we must have $(wT)_{1}=p_{1}$.
This gives us $w_{2}=0.2$. Finally $(wT)_{2}=2>p_{2}=1$ which, of course, is
complementary to $q_{2}=0$.

###### Example A.2.

(Trade and Production) Consider societies $A$ and $B$ with populations $10$
and $60$ respectively, and with two commodities, viz., $g_{1}$, i.e., rice and
$g_{2}$, i.e., computers. Suppose that in $A$ it takes $4$ units of labor
years to produce a bag or rice and $2$ units of labor years to manufacture a
computer. For $B$, these numbers are $6$ and $15$ respectively. Suppose that
the utility function for both societies are the same and are given by family
$u(q_{1},q_{2})=\sqrt{q_{1}q_{2}}$, where $q_{i}$ is the production of good
$g_{i}$.

Thus we have:

(70)
$T_{A}=\left[\begin{array}[]{cc}4&2\end{array}\right]\>\>\>Y_{A}=\left[\begin{array}[]{c}10\end{array}\right]\>\>\>T_{B}=\left[\begin{array}[]{cc}6&15\end{array}\right]\>\>\>Y_{B}=\left[\begin{array}[]{c}60\end{array}\right]$

Thus, the production frontier $P^{A}$ of society $A$ is given by

(71) $P^{A}=\\{(q^{A}_{1},q^{A}_{2})\>|\>q^{A}_{1},q^{A}_{2}\geq 0\mbox{ and
}4q^{A}_{1}+2q^{B}_{2}\leq 10\\}$

while that of $B$ is given by

(72) $P^{B}=\\{(q^{B}_{1},q^{B}_{2})\>|\>q^{B}_{1},q^{B}_{2}\geq 0\mbox{ and
}6q^{A}_{1}+15q^{B}_{2}\leq 10\\}$

We can see that the optimal production is given by the vectors
$q^{A}=(1.25,2.5)$ and $q^{B}=(5,2)$. The joint production frontier for the
two societies together is given by:

(73) $T=\left[\begin{array}[]{cccc}4&2&0&0\\\
0&0&6&15\end{array}\right]\>\>\>Y=\left[\begin{array}[]{c}10\\\
60\end{array}\right]$

The production frontier in terms of rice and computers, both societies taken
together, is given by the convex hull of $\\{(0,0),(0,9),(10,5),(12,0)\\}$.
The optimal production is given by the point $\bar{q}=(10,5)$ which is better
than the sum $q^{A}+q^{B}$, and is the basis of trade.

###### Example A.3.

(Consumption) Here, we illustrate the Fisher market through a small example.
Consider two buyers $L_{1}$ and $L_{2}$, and two commodities $g_{1}$ and
$g_{2}$, with $1$ unit of each available for sale. Let $x_{ij}$ be the
allocation of good $j$ to buyer $i$. Suppose that $L_{1}$ prefers $g_{1}$
twice as much as $g_{2}$, while $L_{2}$, prefers $g_{2}$ thrice as much as
$g_{1}$. In other words, we may write $U$ as:

(74) $U=\left[\begin{array}[]{cc}2&1\\\ 1&3\end{array}\right]$

Suppose $W_{2}=10$ is fixed. Let us consider first the case when
$W_{1}=10+\delta$. For $\delta\leq 10$, we claim that the price vector is
$\bar{p}(\delta)=(10+\delta,10)$ and that $x_{11}=x_{22}=1$ with
$bb_{1}=2/(10+\delta)$ and $bb_{2}=3/10$. In other words, $L_{1}$ consumes
$g_{1}$ and $L_{2}$ consumes $g_{2}$. For the price of $g_{1}$ as $10+\delta$,
$g_{2}$ remains the most attractive good for $L_{2}$. Similarly, while
$10+\delta=p_{1}\leq u_{12}\cdot p_{1}=2\cdot 10$, $g_{1}$ remains the most
attractive good for $L_{1}$. Moreover, as $\delta$ increases, the price of
$g_{1}$ rises and absorbs all the money of $L_{1}$. Thus the prices $p$ are
$p=(10+\delta,10)$ for $\delta\leq 10$ and the Fisher graph remains fixed at
$\\{(1,1),(2,2)\\}$. The net revenue is $20+\delta$, i.e., the net money with
the buyers.

Things change when $\delta>10$, and $L_{1}$ starts consuming $g_{2}$ as well.
Thus $x_{11}=1$ and $x_{12}>0$ while $0<x_{22}<1$, and the Fisher graph now
becomes $\\{(1,1),(1,2),(2,2)\\}$. By the optimal goods condition for buyer
$1$, we have $p_{2}/u_{12}=p_{1}/u_{11}$. This fixes the price of $g_{1}$ as
twice that of the price of $g_{2}$, since $L_{1}$ is consuming both. Thus, if
the price of $g_{2}$ if $p_{2}$, then the price of $g_{1}$ is $2p_{2}$. The
market clearing condition gives us two equations:

(75) $\begin{array}[]{rcl}10+\delta&=&2p_{2}+x_{12}\cdot p_{2}\\\
10&=&(1-x_{12})\cdot p_{2}\end{array}$

Solving for $p_{2}$ and $x_{12}$ gives us:

(76) $\begin{array}[]{rcl}p_{2}&=&\frac{\delta+20}{3}\\\
x_{12}&=&\frac{\delta-10}{\delta+20}\end{array}$

The prices $p$ are $p=(\frac{2\delta+40}{3},\frac{\delta+20}{3})$. The net
revenue is as before, i.e., $20+\delta$.

###  A.2. SPLC utilites

An important extension of the linear utility Fisher market is the separable
piecewise-linear concave (splc) utility Fisher market. Here, the utility
function of each agent is given by an splc continuous function
$u_{i}:(\mathbb{R}^{+})^{n}\rightarrow\mathbb{R}^{+}$, where $u_{i}(\bar{x})$
is the utility of the consumption $\bar{x}=(x_{1},\ldots,x_{n})$ to agent $i$,
and which in turn is given by

(77) $u_{i}(\bar{x})=\sum_{j}u_{ij}(x_{j})$

where each $u_{ij}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is splc. For this
case, the optimality condition is revised as follows:

Optimal Goods: For every $x_{ij}>0$, let $I_{ij}$ be the interval of slopes of
the function $u_{ij}$ at the point $x_{ij}$, i.e.
$I_{ij}=[u_{ij}^{\prime}(x_{ij}^{+}),u_{ij}^{\prime}(x_{ij}^{-})]$. Let
$BB_{ij}$ = $I_{ij}/p_{j}$ (whenever $I_{ij}\neq[0,0]$), the interval of
possible bangs-per-buck. Note that the interval $I_{ij}$ is a point if
$x_{ij}$ lies in the interior of a particular linear segment of the function
$u_{ij}$. It is a proper interval only when $x_{ij}$ is on the intersection of
two linear segments. The optimality condition then is the requirement that
$\cap_{j:x_{ij}>0}BB_{ij}\neq\phi$. This corresponds to the existence of a
common bang-per-buck value for all consumption by agent $i$.

###  A.3. The Analysis of _Tatonnement_

A detailed description of the _tatonnement_ iterator function, as described in
Section 2, is given below.

  1. (1)

Input $\eta_{0}=(p_{0},q_{0},w_{0},X_{0})$. Put $n=0$.

  2. (2)

We first check if the state $(p_{n},q_{n},w_{n},X_{n})$ is an equilibrium, by
checking if $(q_{n},w_{n})\in CP(T,Y;p)$ and $(p_{n},X)\in CC(U;q,w)$. If so,
then $\eta_{n}$ is an SM equilibrium. If not, we follow the iterative steps
below.

  3. (3)

Using $p_{n}$, we first compute $(q_{n+1},w_{n+1})$ by optimizing
$f_{\mathcal{P}}$. This is the $n$-th production-side update.

  4. (4)

We next find $(p_{n+1},X_{n+1})$ through the process $f_{\mathcal{C}}$ using
the input $(q_{n+1},W_{n+1})$.

  5. (5)

Note that $f_{\mathcal{C}}$ does not set prices for goods not produced. These
are set, assuming that a small $\epsilon$ is indeed produced and predicting
its price. Thus if $b_{1},\ldots,b_{k}$ are the maximum bang per buck values
for the players, then

(78) $p_{j}=\max_{i}\frac{u_{ij}}{b_{i}}$

This tells us that when these Fisher-like prices are offered, for (at least)
one player, the maximum bang-per-buck ratio equals the ratio these prices
give, making the player buy the good. The computation of $p$ as before and its
modification is called $\mathcal{C}(n)$, i.e., the $n$-th consumption-side
update.

  6. (6)

This completes the description of $\eta_{n+1}$. We go back to Step 2.

We illustrate an example of SM equilibrium reached through the _tatonnement_.

###### Example A.4.

Let us consider a market with 3 classes and 3 goods with the following
specifications for technology, utility, and labor availability.

(79) $T=\left[\begin{array}[]{ccc}1&0&2\\\ 3&4&0\\\
0.5&2.5&2\end{array}\right];\>U=\left[\begin{array}[]{ccc}1.5&0.41&0\\\
0.58&1.1&0.2\\\ 0.5&1.4&0.6\end{array}\right];\>Y=\left[\begin{array}[]{c}1\\\
1\\\ 1\end{array}\right]\>$

Starting with the price vector $[0.7379,\ \ 0.9379,\ \ 0.3617]$, the
tatonnement process converges to an equilibrium point in 3 iterations. These
are represented as rows in the following matrices to show the corresponding
prices, production, and wages in each iteration:

(80) $\left[\begin{array}[]{c}p_{1}\\\ p_{2}\\\ p_{3}\\\
p_{4}\end{array}\right]=\left[\begin{array}[]{ccc}2.0408&3.8706&0.7037\\\
1.5961&3.027&1.2973\\\ 1.5099&2.8636&1.2273\\\
1.5099&2.8636&1.2273\end{array}\right],\ \left[\begin{array}[]{c}q_{1}\\\
q_{2}\\\ q_{3}\\\
q_{4}\end{array}\right]=\left[\begin{array}[]{ccc}0.2632&0.0526&0.3684\\\
0&0.25&0.1875\\\ 0.2631&0.0526&0.3684\\\
0.2631&0.0526&0.3684\end{array}\right];$ (81)
$\left[\begin{array}[]{c}w_{1}\\\ w_{2}\\\ w_{3}\\\
w_{4}\end{array}\right]=\left[\begin{array}[]{ccc}0.31636&0.52007&0.16357\\\
0&0.68&0.32\\\ 0.086693&0.3865&0.52681\\\ 0.086693&0.3865&0.52681\\\
\end{array}\right]$

And the allocations $X$ are given by the forest-

Next, we provide an example where the process fails to converge.

###### Example A.5.

Let us now consider a market with 3 classes and 3 goods with the following
specifications.

(82) $T=\left[\begin{array}[]{ccc}0.05&1&0.9\\\ 0.5&0.8&0.15\\\
0.4&0.5&0.4\end{array}\right];\>U=\left[\begin{array}[]{ccc}0.2&0.3&0.8\\\
0.9&0.2&0.4\\\
0.25&0.85&0.33\end{array}\right];\>Y=\left[\begin{array}[]{c}10\\\ 10\\\
10\end{array}\right]\>$

Tatonnement does not converge here, instead, the following two states are
reached that stay in a loop -

(83) $q=\left[\begin{array}[]{ccc}4.35&9.78&0\\\
14.7&0&10.3\end{array}\right];\>p=\left[\begin{array}[]{ccc}0.11&0.05&0.14\\\
0.04&0.12&0.05\end{array}\right];\>w=\left[\begin{array}[]{ccc}0.52&0.48&0\\\
0.12&0&0.88\end{array}\right]$

In Example A.5, two states toggle cyclically. In this case, it is not possible
for a state to exist with all goods and classes simultaneously active. The
solution involves two states, with class-1 and good-1 active in both the
states and other goods and classes alternate between the two. In general, we
see the following necessary condition for an equilibrium: For all goods ($j$)
that are not active in the market,

(84) $p_{j}=\max_{i}\frac{u_{ij}}{b_{i}}\text{ (U-
Level})<\sum_{i}T_{ij}w_{i}\text{ (T- Level})$

As we see earlier, if a good is not produced, it is allotted a Fisher-like
price in the _tatonnement_ , which is the $U$-level defined above. All
produced goods have prices greater than or equal to their $U$ levels, as
guaranteed by the Fisher market. The $T$-level, on the other hand, is defined
as the amount of money required to produce a good. It is then clear that if
$U$-Level for an unproduced good $j$ is more than its $T$-Level i.e.
$p_{j}>(w\cdot T)_{j}$, then being profitable, good $j$ becomes active in the
next iteration, by perhaps pushing a less efficient good $j^{\prime}$ out of
production. If subsequent iterations assign good $j^{\prime}$ a $U$-level
price larger than its $T$-level, then it may become active again.

As seen from the examples, this method does work similarly to the tatonnement
process given in Walrus’ theory of general equilibrium (Uzawa, 1960). It too
starts with a price vector, computes production and wages, and produces the
next set of prices based on these market variables. It is clear that the
process terminates if and only if it attains an equilibrium. From Example A.5,
it can be observed that the process may not always converge, and there may be
toggling states. Moreover, it can be observed that equilibria whose Fisher
forests are disconnected are unlikely to arise from the above process, even
though they are fixed points.

##  Appendix B Section 4 details

###  B.1. Proof of Theorem 4.2

See 4.2

###### Proof.

Let us consider an economy with $m$ labor classes and $n$ goods. Without loss
of generality, we examine the existence of equilibrium so that all labor
classes and goods are active. Let us $n-m$ as the deficit $`def$’. We first
consider the case $k-1<def$ and prove that no equilibrium can exist in this
case. We then consider the case $k-1\geq def$ and state the possibilities of
equilibrium by giving a system of solvable equations. With this, we prove that
for any generic or non-generic equilibrium, it is possible to find an
arbitrarily close generic equilibrium by changing the control variables i.e
$U^{\prime}$. We conclude the proof by giving an example of a $2\times 3$
case.

$k-1<def$: Let us assume that there exist $p,q,w$ such that $p=wT$, $Tq=Y$ and
$p$ is the Fisher market solution for the market set up by $U,q$ and $w$ i.e.
there exists an SM equilibrium ($p,q,w,X$). Using the prices, it is possible
to generate the Fisher forest. We assume that there are $k$ trees in the
forest so that $k-1<def$.
Setting up the profitability conditions, we have,

(85) $\left[\begin{array}[]{cccccc}T_{11}&T_{21}&...&T_{m1}&|&p_{1}\\\
T_{12}&T_{22}&...&T_{m2}&|&p_{2}\\\ .&.&.&.&|&.\\\ .&.&.&.&|&.\\\
T_{1n}&T_{2n}&...&T_{mn}&|&p_{n}\end{array}\right]\left[\begin{array}[]{c}w_{1}\\\
w_{2}\\\ .\\\ .\\\ w_{m}\\\ -1\end{array}\right]=\left[\begin{array}[]{c}0\\\
0\\\ .\\\ .\\\ 0\end{array}\right]$

We have $k\leq def$. Let us assume that the goods $g_{1},\ldots,g_{k}$ are in
distinct components and for the rows $R_{1},\ldots,R_{k}\in\mathbb{R}^{n}$,
where $R_{i}(i)=1$, $R_{i}(j)=0$ if $g_{j}$ is not in the component of
$g_{i}$, and finally $R_{i}(j)=\alpha_{ij}$ a monomial in the entries of $U$
which relates the price of $g_{i}$ with $g_{j}$. If we are to assume that
$p_{1},\ldots,p_{k}$ as temporarily known, then the equation $wT=p$ leads us
to the equation:

(86)
$\left[\begin{array}[]{cccc}p_{1}&\ldots&p_{k}&-w\end{array}\right]\left[\begin{array}[]{c}R_{1}\\\
\vdots\\\ R_{k}\\\ T\end{array}\right]=0$

Thus, in the case of $k=def$, we can rearrange the matrix so that the
determinant of the following matrix is zero.

(94)
$\displaystyle\left[\begin{array}[]{cccccccc}T_{11}&T_{21}&...&T_{m1}&|&p_{1}&.&.\\\
T_{12}&T_{22}&...&T_{m2}&|&\alpha p_{1}&.&.\\\ .&.&.&.&|&.&p_{2}&.\\\
.&.&.&.&|&.&\beta p_{2}&.\\\ .&.&.&.&|&.&.&.\\\ .&.&.&.&|&.&.&.\\\
T_{1n}&T_{2n}&...&T_{mn}&|&.&.&\gamma p_{k}\end{array}\right]$

Here, $\alpha,\beta..,\gamma$ represent the algebraic relationships between
prices of goods belonging to the same components of the forest. Each column on
the right corresponds to a tree. We have rearranged $p$ so that there are $k$
columns. It is clear that the $p$’s can be taken out from the determinant so
that the condition translates to an algebraic relationship between
$\alpha,\beta..,\gamma$ i.e. ratio of entries of $U$ and coefficients of $T$.
This contradicts the assumption that $U$ and $T$ are independent. In case
$k<def$, we can delete $def-k$ number of rows from the above matrix so that it
becomes a $(m+k)\times(m+k)$ matrix. The determinant of this matrix is zero,
which again contradicts the assumption. Thus, there does not exist any
equilibrium when $k-1<def$.

$k-1\geq def$: We again consider (86), which is a system of $n$ homogeneous
equations in $m+k$ unknowns. Thus, if the first $m+k-n=k^{\prime}$ prices were
fixed, then all prices and wages are defined using these $k^{\prime}$ prices.
Here, $k^{\prime}\geq 1$.

Next, let us consider the conservation of money in each of the $k$ components
$G_{i}$ and use these to solve for $q$. We of course have $wT=p$, whence
$wY=wTq=pq$. Thus the overall conservation of money is already available and
there are only $k-1$ independent equations. Next, We have the $m$ equations
$Tq=Y$. Now, for each component, the money available is given as
$\sum_{L_{j}\in G_{i}}Y_{j}w_{j}$. On the goods side, we have $\sum_{g_{j}\in
G_{i}}p_{j}q_{j}$. By the earlier argument, these translate into $k$ algebraic
equations in the variables $p_{1},\ldots,p_{k^{\prime}}$ and
$q_{1},\ldots,q_{n}$, which are homogeneous in $p$’s. By dividing throughout
by $p_{1}$, we get $k-1$ equations in $k^{\prime}-1$ variables. This gives us
a total of $m+k-1$ equations in $n+k^{\prime}-1$ variables. Substituting
$k^{\prime}=m+k-n$ gives us $m+k$ equations in $m+k-1$ variables. These may be
solved to obtain all quantities. Section-6 shows the existence of a
disconnected $(2\times 2$ market i.e. $k-1=1>def=0$, which confirms that
$k-1\geq def$.

Let us now consider the case where this equilibrium is generic. We recall the
definition of generic equilibrium points- we say, $\eta$ is a generic
equilibrium point if (i) for $j\not\in J(\eta)$, we have $(wT)_{j}>p_{j}$, and
(ii) for $(i,j)\not\in F(\eta)$, we have $u_{ij}/p_{j}<\max_{k}u_{ik}/p_{k}$.
In this case, the set of tight equations is the same as the set given by
$(I,J,F)$ i.e. $Tq=Y$, $wT=p$, and the maximum bang per buck and money
conservation constraints given by the Fisher market. Therefore, it is possible
to change a $u_{ij}$ by $\delta$ and retain the equations- $(i,j)\not\in
F(\eta)$, $u_{ij}/p_{j}<\max_{k}u_{ik}/p_{k}$, which result in a generic
equilibrium point. It is clear that the equations $(wT)_{j}>p_{j}$ hold true.
On the other hand, when we have a non-generic point, i.e. $(wT)_{j}=p_{j}$ for
$j\not\in J(\eta)$ or $u_{ij}/p_{j}=\max_{k}u_{ik}/p_{k}$ for $(i,j)\not\in
F(\eta)$, the number of tight equations is more than those given by its
$I,J,F$. We can then relax an equation to reach a generic point.

In all, we have proved that no equilibrium exists if $k<def+1$. Moreover, we
can see that the condition $k-1\geq def$ is necessary but not sufficient, as
there are more conditions like the nonnegativity of variables and the
optimality of prices. We also note that the condition of a generic equilibrium
point is crucial. To see this, consider a non-generic equilibrium point with
$(i,j)$ satisfying the bang per buck condition for buyer $i$ such that
$x_{ij}=0$, i.e., $(i,j)\not\in F(\eta)$. This increases the number of
actually connected components, thereby adding an extra tight constraint, which
violates the generic economy argument. Thus, if there are $z$ such zero-weight
edges, the number of connected components is at least $def+1+z$. We now
complete the proof showing a $2\times 3$ case in Example B.1.

###### Example B.1.

Let us consider an economy with the following specifications.

(95) $T=\begin{bmatrix}1&0.5&0.8\\\ 0.5&1.5&0.9\end{bmatrix}\ \ \ \ \
U=\begin{bmatrix}1&0.75&0.8\\\ 0.4&0.9&0.7\end{bmatrix}\ \ \ \ \
Y=\begin{bmatrix}1\\\ 1\end{bmatrix}$

Here, two labor classes, with one labor unit each, can produce three goods
using the above technology constraints. Let us consider $p,q,w$ as follows.

(96) $p,q,w^{T}=\left[\begin{array}[]{c}0.5235\\\ 0.8324\\\
0.6470\end{array}\right],\left[\begin{array}[]{c}0.5639\\\ 0.2426\\\
0.3934\end{array}\right],\left[\begin{array}[]{c}0.2952\\\
0.4565\end{array}\right]$

We see that these variables follow the equations-

(97)
$p^{T}=\left[\begin{array}[]{ccc}0.5235&0.8324&0.6470\end{array}\right]\>=w^{T}\cdot
T=\left[\begin{array}[]{cc}0.2952&0.4565\end{array}\right]\cdot\left[\begin{array}[]{ccc}1&0.5&0.8\\\
0.5&1.5&0.9\end{array}\right]$

And

(98) $Y=\left[\begin{array}[]{c}1\\\ 1\end{array}\right]\ =\>T\cdot q=\
\left[\begin{array}[]{ccc}1&0.5&0.8\\\
0.5&1.5&0.9\end{array}\right]\cdot\left[\begin{array}[]{c}0.5639\\\ 0.2426\\\
0.3934\end{array}\right]$

According to the definition of equilibrium in the SCM model, we see that $q$
and $w$ are dual variables of each other and follow the market constraints. On
the consumption part, when $w$ and $q$ are given as inputs to the Fisher
market, the solution is the price vector $p$ with the following allocations-

(99) $X=\left[\begin{array}[]{ccc}0.5639&0&0\\\
0&0.2426&0.3934\end{array}\right]$

That is, class 1 only consumes good 1 while class 2 buys goods 2 and 3. This
shows an equilibrium in a general $m\times n$ economy with an unequal number
of active goods and classes, with the number of connected components
$=n-m+1=2$.

∎

###  B.2. Proof of Theorem 4.3

See 4.3

###### Proof.

The combinatorial data does give us the relationships $w_{I}T_{I,J}=p_{J}$ and
$T_{I,J}q_{J}=Y_{I}$. From this, it follows that $|I|\leq|J|$ for otherwise
there would be an algebraic relationship between $T$ and $Y$. However, if
$|I|=|J|$, and $k=1$, then $w$ is determined by $p$ and $q$ by $Y$. Since the
forest $F$ is connected, $p$ is determined up to a scalar multiple and thus
the whole system is solved. In summary, if $|I|=|J|$ and $k=1$, there is a
unique $\eta$ sitting above this combinatorial data. However, in the general
case, we must first append to the variables $q_{J}$, a suitable subset
$\\{p_{1},\ldots,p_{k^{\prime}}\\}$. The $w$’s and the remaining $p$’s are
expressible as homogeneous linear combinations of these $k^{\prime}$ prices.
Next, to the linear set of equations $T_{I,J}q_{J}=Y_{I}$ we add the $k-1$
independent money conservation equations to solve these simultaneously.
Unfortunately, the conservation equations involve terms $p_{i}q_{i}$’s and are
quadratic in the chosen variables with coefficients in the entries of $U$.
Once these are solved, all other variables are known and the equilibrium point
is reconstructed. Thus, over a given combinatorial data, we get an algebraic
system with coefficients in $U$, but with finitely many solutions. By standard
algebraic geometry results, other than over a closed algebraic set, these
solutions depend smoothly on the entries of $U$. ∎

##  Appendix C Section 5 details

###  C.1.  Correspondence and the Nash Equilibria in General $2\times 2$
markets

We now consider a general scenario involving 2 classes and 2 goods where $T,Y$
are given as constants. Here, given $T$ and $Y$, we rigorously solve for
markets, and compute the conditions for a forest to be in equilibrium. We
argue that wages are continuous functions of utilities and payoffs and
allocations are continuous for each forest. Moreover, we generalize the
results given in section 6 by proving the existence of a correspondence
between the strategy space and utilities and also that utilities on the
boundaries are linear combinations of those of the forests on both sides. We
also look at the necessary conditions for Nash equilibria to exist.
Market and specifications:
Let us consider a $2\times 2$ market with inputs $T$, $Y$. Let $U^{t}$ be the
true utility matrix. Let $U$ stand for the strategy matrix with
$\frac{u_{11}}{u_{12}}=\alpha$, $\frac{u_{21}}{u_{22}}=\beta$.

Let $p_{1}/p_{2}$ = $\alpha$ where $\alpha$ is the appropriate ratio of
utilities when the forest is connected or the ratio of some technology inputs
when the forest is disconnected. For example, if class-1 buys good 1 and
class-2 buys good-2, we have $p_{2}T_{21}^{-1}=p_{1}T_{12}^{-1}$. Assuming
total money in the economy constant and equal to 1, we can solve for $q$ in
terms of $T,Y$.

(100) $\displaystyle q_{1}=T^{-1}_{11}Y_{1}+T^{-1}_{12}Y_{2}$ (101)
$\displaystyle q_{2}=T^{-1}_{21}Y_{1}+T^{-1}_{22}Y_{2}$

We assume that $T$ and $Y$ are such that $q_{1},q_{2}$ are positive. Solving
for wages, we get

(102) $\displaystyle w_{1}=\frac{Y_{1}(\alpha
T_{11}^{-1}+T_{21}^{-1})}{Y_{1}(\alpha T_{11}^{-1}+T_{21}^{-1})+Y_{2}(\alpha
T_{12}^{-1}+T_{22}^{-1})}$ (103) $\displaystyle w_{2}=\frac{Y_{2}(\alpha
T_{12}^{-1}+T_{22}^{-1})}{Y_{1}(\alpha T_{11}^{-1}+T_{21}^{-1})+Y_{2}(\alpha
T_{12}^{-1}+T_{22}^{-1})}$

Like the case described before, we know that for a $2\times 2$ market, there
are six possible forests and one cycle. Also, there are two possible $(1\times
1)$ markets. The possible $(2\times 2)$ forests are-

where each is a solution under specific conditions, which place it in a
specific zone. For both classes to be active, we first require
$\frac{p_{1}}{p_{2}}>\min\left(\frac{-T_{21}^{-1}}{T_{11}^{-1}},\frac{-T_{22}^{-1}}{T_{12}^{-1}}\right)=\min\left(\frac{T_{21}}{T_{22}},\
\frac{T_{11}}{T_{12}}\right)$. For connected forests such that class $i1$ buys
only good $j1$ and class $i2$ buys $j1,j2$, we have $w_{i1}<p_{j1}q_{j1}$ and
$w_{i2}>p_{j2}q_{j2}$. This ensures that all allocations are positive. Lastly,
we require the bang per buck conditions to hold.
For example, consider the first graph given above, where
$p_{1}/p_{2}=u_{21}/u_{22}=\beta$. This graph constitutes a generic
equilibrium if and only if,

  1. (1)

$\frac{p_{1}}{p_{2}}=\beta>\frac{T_{21}}{T_{22}},\ \frac{T_{11}}{T_{12}}$ i.e.
wages are positive.

  2. (2)

$\beta>\frac{T_{21}^{-1}Y_{1}}{T_{12}^{-1}Y_{2}}$ i.e. allocations are
positive.
$w_{1}=Y_{1}(p_{1}T^{-1}_{11}+p_{2}T^{-1}_{21})<p_{1}q_{1}=p_{1}(T_{11}^{-1}Y_{1}+T_{12}^{-1}Y_{2})$
or
$w_{2}=Y_{2}(p_{1}T^{-1}_{12}+p_{2}T^{-1}_{22})>p_{2}q_{2}=p_{2}(T_{21}^{-1}Y_{1}+T_{22}^{-1}Y_{2})$

  3. (3)

And, $\beta<\alpha=\frac{u_{11}}{u_{12}}$ i.e. class-1 gets more bang per buck
from good-1 than good-2.
$\left(\frac{u_{11}}{p_{1}}>\frac{u_{12}}{p_{2}}\right)$.

It can be seen that these conditions define an open set in the strategy space
$\mathcal{U}$ which has a unique combinatorial data $(I,J,F)$. In the above
example, it is given by then $I=\\{1,2\\},J=\\{1,2\\}$ and $F$ as forest-1. We
note that the third condition can be relaxed i.e. $\beta\leq\alpha$, which
results in a non generic equilibrium at $\alpha=\beta$. This is because
multiple allocations, including the forest-1 allocations are possible for this
condition. We discuss the cycle i.e. the set $\alpha=\beta$ in the next
section.

In addition to the forests described before, we note that it is possible for
wages of a particular class to become zero. When
$p_{1}/p_{2}\leq\min\left(\frac{T_{21}}{T_{22}},\
\frac{T_{11}}{T_{12}}\right)$, only one class is active and it produces
exactly one good. We refer to its corresponding domain in the $U$-space as
Zone 7. In this case, if class $i$ produces good $j$, it is possible to
compute its production and the utility $U_{i}$ that it gives. We note that
$U_{i}$ is smaller than the total utility of both classes combined, as there
is more net production when both classes are active.
Continuity analysis:
For the forests described above, the combinatorial data $(I,J,K)$ is defined
by the following conditions

Forest 1 | Forest 2 | Forest 3 | Forest 4 | Forest 5 | Forest 6
---|---|---|---|---|---
$\alpha\geq\beta=\frac{p_{1}}{p_{2}}$ | $\frac{p_{1}}{p_{2}}=\alpha\geq\beta$ | $\beta\geq\frac{p_{1}}{p_{2}}=\frac{T_{22}^{-1}Y_{2}}{T_{11}^{-1}Y_{1}}\geq\alpha$ | $\beta\leq\frac{p_{1}}{p_{2}}=\frac{T_{21}^{-1}Y_{1}}{T_{12}^{-1}Y_{2}}\leq\alpha$ | $\beta\geq\alpha=\frac{p_{1}}{p_{2}}$ | $\frac{p_{1}}{p_{2}}=\beta\geq\alpha$
$\beta>\frac{T_{21}}{T_{22}},\frac{T_{11}}{T_{12}}$ | $\alpha>\frac{T_{21}}{T_{22}},\frac{T_{11}}{T_{12}}$ | $\frac{T_{22}^{-1}}{T_{11}^{-1}}>\frac{T_{21}}{T_{22}},\frac{T_{11}}{T_{12}}$ | $\frac{T_{21}^{-1}}{T_{12}^{-1}}>\frac{T_{21}}{T_{22}},\frac{T_{11}}{T_{12}}$ | $\alpha>\frac{T_{21}}{T_{22}},\frac{T_{11}}{T_{12}}$ | $\beta>\frac{T_{21}}{T_{22}},\frac{T_{11}}{T_{12}}$
$\beta>\frac{T_{21}^{-1}Y_{1}}{T_{12}^{-1}Y_{2}}$ | $\alpha<\frac{T_{21}^{-1}Y_{1}}{T_{12}^{-1}Y_{2}}$ |  |  | $\alpha>\frac{T_{22}^{-1}Y_{2}}{T_{11}^{-1}Y_{1}}$ | $\beta<\frac{T_{22}^{-1}Y_{2}}{T_{11}^{-1}Y_{1}}$

We also graphically represent these six forests by marking their zones. We
number these forests from 1 to 6, ordered as given above. 7th zone refers to
the case where only one class is active and in equilibrium. Here, the arrows
indicate the direction of optimization for the classes, considering $U^{t}$ as
the strategy matrix comprising of all ones.

Here, $c1=\frac{T_{21}^{-1}Y_{1}}{T_{12}^{-1}Y_{2}}$ and
$c2=\frac{T_{22}^{-1}Y_{1}}{T_{11}^{-1}Y_{1}}$ and we have assumed $c2>c1$.
The first graph refers to the condition $\det(T)>0$ i.e.
$d1=\frac{T_{11}}{T_{12}}>d2=\frac{T_{21}}{T_{22}}$ and second refers to
$\det(T)<0$. There are four possible schemes corresponding to $c1,c2$ and
$\det(T)$, of which we have considered two with $c2>c1$. In the first scheme,
class-1 always gets wages and vice versa. Note that the region for the 7th
zone can extend depending on $d1$. If $d1>c1$, as we will see, the continuity
of wages implies zone 4 becomes a part of zone 7.

We see that there are clear ranges of $\alpha$ and $\beta$ which define the
solution forest. First, we observe that as wages are rational functions
involving only $p1/p2$ i.e. the relevant $\alpha,\beta$ in case the forest is
connected, wages are always continuous within a forest. For disconnected
forests, wages are constant. It is important to note that wages are bounded,
$w_{1}+w_{2}=1$. At the boundary of zone 4, $p1/p2$ i.e. $\alpha$ or $\beta$
equals $c1$, which determines the wages. In other words, as $\alpha$ or
$\beta$ continuously approach $c1$, limit of $w_{i}$ equals $w_{i}$ for zone
4. Similarly, zone 3 allows for smooth transitions. It is clear that wages are
continuous at the boundary $\alpha=\beta$ also, as one relevant variable
smoothly transitions into another. Similarly, in zone 7, wages continuously
converge to zero.

Next, we observe that allocations and utilities are continuous within each
forest, where they are functions. For disconnected forests, they are
constants. For connected forests, where there are three edges, we see that 2
allocations are linearly dependent on $p_{1}/p_{2}$ and one is constant. For
example, forest 1 has the following allocations-

(104) $\displaystyle
x_{11}=w_{1}/p_{1}=Y_{1}T_{11}^{-1}+Y_{1}T_{21}^{-1}/\beta$ (105)
$\displaystyle x_{21}=q_{1}-x_{11},\ \ x_{22}=q_{2}$

Since utilities (true and strategic) are linear functions of allocations,
those are continuous too. Moreover, they are either linear or rational
functions of $\alpha,\beta$. Though we have considered a special strategy
matrix with all ones here, the result is general.
Transitions and the cycle:
Firstly, we note that except for the $\alpha=\beta$ line, other boundaries
allow for a continuous function where for each pair of ($\alpha,\beta$), a
unique allocation/utility exists. Let us see this by changing $\alpha$ or
$\beta$ continuously to approach a forest/cycle from another. We see that when
a tree is disconnected by making an edge weight zero, allocations and
utilities continuously transform. For example, in zone -1, when $\beta$
approaches $c1$ from above, $p_{1}/p_{2}$ equals $c1$ where zone-4 starts, and
$x_{11}$ becomes $q_{1}$. Therefore, forest 3,4 include their boundaries while
defining utility functions i.e. boundaries have same utilities as the forests.
In case of $\alpha=\beta$, first an edge is added to get a cycle, where
multiple allocations are possible and then another edge is removed.

The governing equations for a connected forest, say forest 1 are :
$w_{1}=x<p_{1}q_{1}$, $y<p_{1}q_{1}$ and $z=p_{2}q_{2}$ along with
$x+y=p_{1}q_{1}$. When we increase $\beta$ to approach the cycle, the
equations become $x,y<p_{1}q_{1}$, and $z,v<p_{2}q_{2}$ along with
$x+y=p_{1}q_{1}$ and $z+v=p_{2}q_{2}$. Thus, the maximum value of $z$ is
$p_{2}q_{2}$ where zone 1 is overlapping and minimum value is 0 where
$v=p_{2}q_{2}$ when zone 6 overlaps.

The net utilities for forest 1 and 6 are-

(106) $\displaystyle(b_{1},b_{2})_{1}=\left(\frac{w_{1}}{p_{1}},\ \
q_{1}+q_{2}-\frac{w_{1}}{p_{1}}\right)$ (107)
$\displaystyle(b_{1},b_{2})_{6}=\left(q_{2}+q_{1}-\frac{w_{2}}{p_{1}},\ \
\frac{w_{2}}{p_{1}}\right)$

And for the cycle, these can be computed in terms of $v,z$-

$\displaystyle(b_{1},b_{2})$
$\displaystyle=\left(\frac{w_{1}-v}{p_{1}}+\frac{v}{p_{2}},\ \
\frac{w_{2}-x}{p_{1}}+\frac{z}{p_{2}}\right)$ (108)
$\displaystyle=\left(\frac{w_{1}}{p_{1}}+v\left(\frac{1}{p_{2}}-\frac{1}{p_{1}}\right),\
\ \frac{w_{2}}{p_{1}}+z\left(\frac{1}{p_{2}}-\frac{1}{p_{1}}\right)\right)$

Firstly, we note that the ratio $1/p_{2}-1/p_{1}$ is a constant for any fixed
$\alpha,\beta$. Let this be a positive constant. Now, it is clear that when
$v=0$, $b_{1}$ for the cycle is minimum which coincides with that of forest-1.
Similarly, when $v=p_{2}q_{2}$, the maximum coincides with that of forest-6.
This establishes that utilities at the cycle are linear combinations of those
on the boundaries, so that for each $\alpha$ on the $\alpha=\beta$ line,
utilities make a smooth transition. This makes us view the cycle as a linear
combination of two forests depending on values of $\alpha$ and $\beta$. Though
we have considered $U^{t}$ made up of all ones, this result is valid for any
matrix $U^{t}$.

We see that the results generalize the example given in section 6. Using the
same proof, we can now argue that there exists a correspondence $\mathcal{N}$
between the strategy space and utilities. Also, continuous payoffs for the
forests and the range for the line $\alpha=\beta$ allow for a 2 dimensional
manifold whose boundaries are given by those of the forests. Next, we look at
the conditions for a Nash equilibrium to exist.
Existence of a Nash equilibrium:
We observe that allocations are functions of these variables, either linear or
inversely related ( $U_{i}=a+b\alpha$ or $c+d/\alpha$), when the forests are
connected. In those cases, maximization can occur only at the boundaries i.e.
at the points where the transition of graphs occurs. When the forests are
disconnected, allocations and utilities are constants for a range of
$\alpha,\beta$. Therefore, if any Nash equilibrium exists, it should be at the
transition points or forest 3 or 4.

Let us now analyze the first scheme. Looking at the arrows, we know that
forest-3 cannot be an equilibrium, which leaves $\alpha=\beta$ and forest-4 as
possibilities. Class-2 prefers forest-4 to forest-2, and similarly, class-1
prefers forest-4. For a point in forest-4 to be an equilibrium, $\beta>d1$ and
$\alpha>c1$. In this region, given any $\alpha$, class-2 prefers forest-4 to
forest-1,3 and 5. We now have to find a region for $\beta$ so that $\alpha$
maximises $b_{1}$ in that region. Note that when $c1>\beta>d1$, $b_{1}$ is
constant as a function of $\alpha$ in zone 6. When it enters zone-2, $b_{1}$
increases until it is in zone 4. Thus, we prove Nash equilibria exist in the
region $d1<\beta<c1$ and $c1<\alpha$. Similarly, it can be proved that Nash
equilibria exists in scheme-2 in zone-3 for $d2<\alpha<c1$ and $c2<\beta$.
This equilibrium is, however, subject to the condition that the forest exists
i.e. the region is feasible. For example, in scheme-1, forest 4 exists when
$d1<c1$ i.e. $\frac{T_{11}}{T_{12}}<\frac{T_{21}^{-1}Y_{1}}{T_{12}^{-1}Y_{2}}$
and $\beta<c1<\alpha$.

This also settles the question of the existence and uniqueness of equilibrium
in a $2\times 2$ case.

###  C.2. Description of zones in Table 1

| Forest-1 | Forest-2 | Forest-3 | Forest-4 | Forest-5
---|---|---|---|---|---
Prices | $\beta=\frac{p_{1}}{p_{2}}>1/4$ | $\alpha=\frac{p_{1}}{p_{2}}>1/2$ | $1/4<\beta=\frac{p_{1}}{p_{2}}<1/2$ | $\frac{p_{1}}{p_{2}}=\frac{1}{2}$ |  $p_{1}$ = 0
Alloc - 1 | $x_{11}=8-\frac{2}{\beta}$ |  $x_{11}=8-\frac{4}{\alpha}$, $x_{12}=2$ | $x_{12}=8\beta-2$ | $x_{12}=2$ | $x_{11},x_{12}=0$
Alloc - 2 | $x_{21}=\frac{2}{\beta},\ \ x_{22}=2$ | $x_{21}=\frac{4}{\alpha}$ | $x_{21}=8,\ \ x_{22}=4-8\beta$ | $x_{21}=8$ | $x_{22}=4$
$b_{1}$ | $b_{1}=8-2/\beta$ | $b_{1}=10-4/\alpha$ | $b_{1}=8\beta-2$ | $b_{1}=2$ | $b_{1}=0$
$b_{2}$ | $b_{2}=2+2/\beta$ | $b_{2}=4/\alpha$ | $b_{2}=12-8\beta$ | $b_{2}=8$ | $b_{2}=4$
Wages-1 | $w_{1}=1-\frac{2}{1+4\beta}$ | $w_{1}=1-\frac{2}{1+4\alpha}$ | $w_{1}=1-\frac{2}{1+4\beta}$ | $w_{1}=1/3$ | $w_{1}=0$
Wages-2 | $w_{2}=\frac{2}{1+4\beta}$ | $w_{2}=\frac{2}{1+4\alpha}$ | $w_{2}=\frac{2}{1+4\beta}$ | $w_{2}=2/3$ | $w_{2}=1$
Zones | $\alpha\geq\beta>1/4$ | $\beta\geq\alpha>1/2$ |  $1/4<\beta<1/2$ and $\beta\geq\alpha$ | $\alpha\leq 1/2\leq\beta$ | $\beta\leq 1/4$
Table 6. $SM(T,Y,U)$ characteristics for the forests in Fig. 2