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# Wages and Utilities in a Closed Economy- A StrategicAnalysis

The broad objective of this paper is to initiate through a mathematical model, the study of causes of wage inequality and relate it to choices of consumption and the technologies of production in an economy. The paper constructs a simple Heterodox Model of a closed economy, in which the consumption and the production parts are clearly separated and yet coupled through a tatonnement process. The equilibria of this process correspond directly with those of a related 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 Heterodox model also allows the formulation and explicit construction of the consumer choice game, where individual utilities serve as the strategies with total or relative wages as the pay-offs. We illustrate, through two 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 influences not only prices but also production and wages. Thus, consumer choice may serve as an effective tool for wage redistribution.

• 2 publications
• 5 publications
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## 1. Introduction

As the present millenial edition of the global economy unfolds, many authors and agencies have pointed out several undesirable features which 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 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 build a simple mathematical model, called the Heterodox Model, which illustrates some of the key features of the dependence of wages on the production and consumption parts of the economy. Our model has two parts, the production as determined by a technology matrix , which utilizes labour classes and produces goods and determines quantities of goods produced, labour utilization and wages. This part assumes prices as a given, i.e., which cannot be changed. The second part is the consumption part, which is modelled as a Fisher market and a utility matrix . This part of the economy assumes the production part as a given, i.e., wages and quantities of goods produced, and allocates goods based on wages (i.e., endowments or disposable incomes) held by each labour class, and determines prices.

This paper shows the connection between , the utility matrix and the wages obtained by various labour classes, as implemented by . In other words, it traces the connection between personal consumption choice, with prices of goods, their production and finally wages received. Next, it shows that having a ”private and real” utility , and posting or posturing a different into the economy does indeed alter wages and has the potential to improve both the social welfare as well as the relative welfare for certain classes. This sets up the consumer choice game, where the manipulation of is the strategy, and the relative or total welfare, as measured by the allocation of goods and their utilities according to , are the pay-offs.

The manipulation of has been studied before, e.g., in connection with the impact of advertising on the competition between firms and their profitability. It has been used by pressure groups to label certain products, e.g., coffee, as ”compliant” with a desirable idiom, e.g., fair wages to the coffee-bean picker. Our analysis is largely that of a closed economy and the study of its modes as functions of the parameters of the system. In classical terms, it is to develop and study, as a strategic game, the dependence of market equilibria on various parameters which define the economy. It is useful to point out earlier work on the Fisher Market Game (c1, ),(c3, )(c8, ), where it was shown that going to the market with postured utilities (instead of the real ones, ) may indeed bring rewards in terms of more favourable allocations. However, that model relates only to the consumption side of the economy.

Our aim is complement this by a simple production model to define a closed economy completely, so as to be able to find a consistent set of prices of goods, wages for each labour class, allocations of goods amongst classes so that an equilibrium exists in the economy. Then, building on the Fisher market concepts, we model the effects of strategizing on consumer preferences on wages, production and allocations. This work extends the Fisher Market Game conclusion that to show that, indeed consumer choice may be used to change wages as well.

The model borrows from many existing models, specially so from the Arrow-Debreu model and its earlier cousin, the Fisher model, and philosophically, from Sraffa’s accounting methods for calculating prices and wages (c4, ) and labour inventory using the theory of value, the marginal production principle for calculating wages, and finally the use of utility functions to compute allocation of goods.

The rest of the paper is organized as follows. In Section 2 we describe the Heterodox Model as composed of two interconnected systems, the consumption model and the production model . The consumption model consists of the key parameter , the utility matrix, and the inputs as the endowments of agents, and , the quantity of goods produced. The ”output” variables are , the prices, and , the allocation of goods to labour classes (or agents). The production model has the key parameter , the technology matrix, and

, the size of individual labour classes. The input variable is the price vector

. The outputs are the wages , and the production vector .

We also define two global optimization functions and , which couple and . They also set up the tatonnement as an iterative interaction between and

In Section 3, we further analyse the tatonnement process and exhibit certain non-convergent trajectories. Next, we associate combinatorial structures associated with equilibria and understand how they vary with the parameters of the economy, i.e., and . We use these results to define the , the consumer choice game, where are fixed, and is the strategy space. We do this over a collection of open sets and show that explicit description of the game is obtained over these open sets.

In Section 4, we use the above results to illustrate a particular market of three labour classes and three goods, and examine the vicinity of a particular fixed point . We use the combinatorial data associated with the fixed point, viz., the Fisher solution forest, to explicitly construct the consumer choice game, i.e., the use of as strategies and the total utility as the pay-offs. We show that even within this open set, the strategic choice of makes eminent sense.

In Section 5, we cast the Hetrodox market as an Arrow-Debreu market. We show the equivalence of solutions of the Heterodox model, i.e., its fixed points, with the equilibria of the corresponding A-D market. 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 6, we go back to the combinatorial data arising from a fixed point. First, we show through a 2-player example, the decomposition of the strategy space, i.e., the -space into various regions indexed by Fisher forests. This also leads to a correspondence between the strategy space and the space of possible pay-offs. In other words, . We show that is largely a 2-dimensional manifold.

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

## 2. The Heterodox model

We first list the basic parameters and internal variables of the Heterodox model.

### 2.1. The basic notations and assumptions

• A good can be both, a fixed unit of service or an output of a manufacturing plant, made available in a fixed time interval, called epoch, for example, a year. The set of goods is denoted by . Also, is a () column vector with is the amount of good manufactured and is an row vector with being the price of each good ,

• The whole population in the economy consists of agents divided into distinct classes, say according to their training. Thus, let be these classes of labour, with each class willing to devote units (e.g., person-years) of labour in every epoch. Thus, Y forms a vector.

• Each (manufactured) good has exactly one production process or technology and each is a linear map . These tell us the amount of labour required from each labour-class to produce one unit of good . The class of all such technologies is denoted by , which is represented as a -matrix with column . Thus, in matrix , is the the amount of labour needed to produce .

• For each , there is a function , such that if is the bundle of goods allocated to , then its utility to is given by . In fact, the utility is linear and . We assume that for every good j, there is a buyer such that and for every buyer , there is a good such that 0.

• We assume that utilities are same for all persons in one labour class and are measured in happiness per person per kilo units. For example, is the happiness derived by a person from class by consuming a unit of good . We denote the matrix formed by entries as U.

Further, without loss of generality, we assume that for each labour class, there are technologies which utilize them. We also assume that the entries of and are all in general position and satisfy no algebraic relation amongst themselves, with rational coefficients.

We assume that for each epoch in an economy, there is a non-negative tuple , where

1. is a column vector with is the amount of good manufactured,

2. is an column vector and is the price of each good ,

3. is an row vector where are the wages received by each person in labour class and finally,

4. is a -matrix and , the total amount of good consumed by labour

### 2.2. The consumption space C

Consumption in our economy is modelled as a Fisher market. Recall that, in a Fisher market, there are buyers (, as in our case), and goods . Each agent is endowed with money , and each good has quantity for sale.

Solution of Fisher market is equilibrium prices and allocations such that they satisfy the following two constraints

Market Clearing: Allocations are such that all goods are completely sold and the money of all the buyers is exhausted, i.e.
,  and  ,

The consumption space is defined as: .
Next, on the set we define a solution to the Fisher market as one which satisfies:

Optimal Goods: Each buyer buys only those goods which give her the maximum utility per unit of money i.e if , then
While the above condition appears to be a multi-objective optimization problem, it is known that solutions to Fisher market are optimal points of the Eisenberg-Gale maximization function, a money weighted combination of the utilities of the buyers. Adsul etc. (simplex, ) too have given a convex program which captures the Fisher market solution as global optima. We denote such a global function as .

We now illustrate the Fisher market through an example.

Example - 1. Consider a 2 buyers, 2 goods market with , and . The equilibrium prices of this market are and the unique equilibrium allocation is .
We also note that when utilities and are generic, i.e. satisfy no algebraic relation amongst themselves with rational coefficients, then, the optimal solution to the Fisher market defines a unique weighted forest (c1, ): Let V(H) = and . For instance, the forest corresponding to Example 1 given above has four nodes and three edges () with weights respectively i.e. the bipartite graph is a unique tree with three edges.

### 2.3. The production space P

The production space is the collection of all wages and quantities such that:

 (1) Tq≤Y (2) qj⋅(pj−(wT)j)≥0 (3) q≥0

Note that the first condition states that the quantities of goods produced are limited by labour constraints, while the second says that unprofitable goods are not produced.

The global maximization function is defined as , i.e, revenue maximization. However, we must specify how wages get decided. For this, we consider the following relaxation LP program:

 (4) maxqpTqs.t.Tq≤Y qj≥0∀j∈G\par

To find the wages, we consider its dual program -

 (5) minλ1λT1Ys.t.λT1T≥pT\par

Or

 [λ1λ2]⋅[T−I]=[\parp]

where and correspond to the dual variables associated with the first and second inequalities respectively. Here, and are and vectors respectively.
We take as the wages determined by . Karush-Kuhn-Tucker (KKT) conditions for the primal and dual program have the following implications-

1. If , then .
Therefore, it follows from (5) that i.e. .

2. If i.e. , then as .

3. If , then , i.e. corresponding is zero.

4. Similarly, if , then .

In particular, we see that the inequality in the dual program, viz. , is opposite of the required constraint. However, in accordance with 3, complementary slackness implies . The conditions reduce T to a square matrix where all constraints are satisfied and tight for the active goods and classes. Let the corresponding prices, production and wages be . We can find the dual variables i.e. wages as a product of and . We see that the tight equation can also be derived through the marginal law of production. We also see that total money is conserved in the economy, i.e. holds true. This sets up a remarkable result which connects wages and production amounts as the dual variables of each other.

### 2.4. A tatonnement process

We now set up the tatonnement. The basic objective of the tatonnement process is to arrive at an equilibrium such that (i) are the outputs of the consumption side Fisher market if input the money vector , and quantities , and (ii) are the optimal solutions on the production side on input . The process begins with a candidate and checks first if is indeed an equilibrium. If not, it updates alternately, the consumption side and the production side.

The detailed description of the iterator function is given below.

1. Input . Put .

2. We first check if the state is an equilibrium. This is done by first checking if is an optimal solution to in the process with input . Next, we check if satisfy the optimality conditions for the function with inputs (where , the total wages. If it does indeed satisfy both conditions, we declare the point as a Heterodox equilibrium.

3. If is not in equilibrium, we follow the iterative steps below.

4. Using , we first compute by optimizing . This is the -th production-side update.

5. We next find through the process using the input .

6. Note that does not set prices for goods not produced. These are set, assuming that a small is indeed produced and predicting its price. Thus if are the maximum bang per buck values for the players, then

 pj=maxiuijbi

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 as before and its modification is called , i.e., the -th consumption-side update.

7. This completes the definition of . We go back to Step 2.

We now illustrate two examples of equilibria obtained through the above iterative process.
Example 2 : Let us consider a 3 classes - 3 goods market with following specifications for technology, utility and labour availability.

 T=⎡⎢⎣1023400.52.52⎤⎥⎦;U=⎡⎢⎣1.50.4100.581.10.20.51.40.6⎤⎥⎦;Y=⎡⎢⎣111⎤⎥⎦

Starting with the price vector , the tatonnement process converges to an equilibrium point in 3 iterations, with the prices, production and wages in each iteration given by-

 ⎡⎢ ⎢ ⎢ ⎢⎣p1p2p3p4⎤⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣2.04083.87060.70371.59613.0271.29731.50992.86361.22731.50992.86361.2273⎤⎥ ⎥ ⎥⎦;
 ⎡⎢ ⎢ ⎢ ⎢⎣q1q2q3q4⎤⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣0.26320.05260.368400.250.18750.26310.05260.36840.26310.05260.3684⎤⎥ ⎥ ⎥⎦;
 ⎡⎢ ⎢ ⎢⎣w1w2w3w4⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣0.316360.520070.1635700.680.320.0866930.38650.526810.0866930.38650.52681⎤⎥ ⎥ ⎥⎦

And the allocations are given by the forest-

Example 3 : Let us now consider a market with the following market specifications -

 T=[0.2500.251];U=[10.811.2341];Y=[24]

The tatonnement process converges to the following output of prices, production and wages, when it starts with =

 ⎡⎢⎣q1q2q3⎤⎥⎦=⎡⎢⎣048282⎤⎥⎦
 ⎡⎢⎣p1p2p3⎤⎥⎦=⎡⎢⎣0.30850.250.103940.0842320.103940.084232⎤⎥⎦
 ⎡⎢⎣w1w2w3⎤⎥⎦=⎡⎢⎣010.663070.336930.663070.33693⎤⎥⎦

The solution forest and the allocations are -

## 3. Equilibria and the Consumer Choice Game

In this section, we illustrate the working of the tatonnement algorithm where it fails to converge. Next we analyse the combinatorial structure of equilibria, where we show that these structures are local invariants. This is then used to define the consumer choice game.

### 3.1. Analysis of the Tatonnement process

The tatonnement process does not always converge. We illustrate with an example, where it alternates between two or more states. In each state, a different production set and/or set of active classes is chosen, though there may be overlaps of multiple active goods/classes. This happens when production and consumption do not agree on a common set of active goods and classes, but cyclically choose two or more states.
Example 4 : Let us now consider a 3 classes - 3 goods market with following specifications.

 T=⎡⎢⎣0.0510.90.50.80.150.40.50.4⎤⎥⎦;U=⎡⎢⎣0.20.30.80.90.20.40.250.850.33⎤⎥⎦;Y=⎡⎢⎣101010⎤⎥⎦

The production, prices and wages, as computed by the function described before, are-

 Prod=[4.359.78014.7010.3];Prices=[0.110.050.140.040.120.05];Wages=[0.520.4800.1200.88]

The above two states toggle cyclically. In this case it is not possible for a state to exist where all goods and classes are simultaneously active. The solution involves two states, where class-1 and good-1 are active in both of them and other goods and classes alternate between the two. In general, we see the following necessary condition for an equilibrium to exist with a given set of active goods and classes. - For all goods () that are not active in the economy,

 pj=maxiuijbi (U- Level)<∑iTijwi% (T- Level)

As we see earlier, if a good is not produced, it is allotted a Fisher like price, which is the -level defined above. All produced goods have their prices greater than or equal to their levels. It is clear that if -Level for an unproduced good is more that its -Level i.e. , then it being profitable, that good becomes active in the next iteration, by perhaps pushing a less efficient good out of production.

As seen from the examples, this method does work similar to the tatonnement process given in Walrus’ theory of general equilibrium (c11, ). It too starts with a price vector, computes production and wages and gives a 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 the above example, it can be observed that the process may not always converge, and there may be toggling states. Moreover, it can be shown that equilibria whose Fisher forests are disconnected are unlikely to arise from the above process, even though they are fixed points. We thus make a distinction between a heterodox equilibrium or a fixed point and a limit point of the tatonnement process.

However, as we show in Section 5, the Heterdox market has an equivalent Arrow-Debreau market. Whence, via the general theory of existence of equilibria, i.e., via Prop. 5.1 and 5.2 of the equivalence of the two, for any parameter set , satisfying certain broad conditions, a Heterodox equilibrium, i.e., a fixed point, always exists, but this need not be unique and it need not arise as a limit point.

### 3.2. Generic equilibrium and combinatorial data

We now associate a suitable combinatorial data with an equilibrium point for the parameters of the economy. Define , and . The combinatorial data identify key features of the equilibrium, e.g., the labour classes with non-zero wages, the goods produced, and the Fisher forest, i.e., the price-determining consumptions. We now 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 3.1 ().

We say that is a generic equilibrium if (i) for , we have , and (ii) for , we have .

Let us now fix and vary over . Given a , and an equilibrium point with the parameters , of the economy, we say that sits over , since it is for this element of , that was observed. Theorem 3.2 relates to the existence of generic equilibria.

###### Theorem 3.2 ().

Let be matrices in general position, i.e., there be no algebraic relationship between the entries, with rational coefficients. Given an equilibrium over , there are arbitrarily close and equilibria sitting over which are generic. Moreover, if has wage-earning labour classes, i.e., and goods produced, i.e., , then the number of connected components () of the solution Fisher forest is at least .

###### Proof.

Appendix-B. ∎

It is an important question if the data does indeed determine , the equilibrium. This is summarized in the next theorem.

###### Theorem 3.3 ().

Again, let be in general position and be a generic equilibrium over with the combinatorial data , then the parameters of , viz., are solutions of a fixed set of algebraic equations in the coefficients of . For an open set of the parameter space of , the equilibria, as guaranteed by Prop. 5.1, 5.2, are generic and have the same combinatorial data as .

###### Proof.

The combinatorial data does give us the relationships and . From this it follows that for otherwise there would be an algebraic relationship between and . However, if , and , then is determined by and by . Since the forest is connected, is determined upto a scalar multiple and thus the whole system is solved. In summary, if and , there is a unique sitting above this combinatorial data. However, in the general case, we must first append to the variables , a suitable subset , as in Appendix B.2. The ’s and the remaining ’s are expressible as homogeneous linear combinations of these prices. Next, to the linear set of equations we add the independent money conservation equations to solve these simultaneously. Unfortunately, the conservation equations involve terms ’s and are quadratic in the chosen variables with coefficients in the entries of . 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 , but with finitely many solutions. By standard algebraic geometry results, other than a over a closed algebraic set, these solutions depend smoothly on the entries of . ∎

### 3.3. The Consumer Choice Game

We now define the consumer choice game , which is parametrized by the technology matrix and the labour inventory , which are henceforth assumed to be fixed. The players are the labour classes, i.e., . The strategy space for player is the utility ”row” vector . These rows together constitute the matrix . This strategy space is denoted by . We also assume that there is a ”real” utility matrix which is used to measure outcomes.

Given a play , the outcome is given by an , an equilibrium over obtained in the Heterodox market. The payoffs, , 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 with the combinatorial data . We first see that there is an open set containing which has the same combinatorial data . The exact inequalities defining arise from the requirement that the Fisher forest have non-negative flows in all edges of , that the edge has an inferior bang-per-buck, and that for . As an example, consider an edge , 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 ’s, prices ’s and quantities ’s. As we have argued before, these in turn, are smooth functions of the entries of . Thus the condition that flow in the edge be positive is the requirement that for a suitable smooth function on .

Thus, there is indeed such an open set , and the pay-off functions are solutions of algebraic equations in the entries of , the coefficients of which depend on the combinatorial data . This gives us Theorem 3.4 below.

###### Theorem 3.4 ().

For a generic equilibrium point with the combinatorial data , there is an open set containing and a smooth family of equilibria for each such that (i) and (ii) the combinatorial data for is .

The pay-off function in general is to be pieced together by such a collection of open sets, indexed by the combinatorics. On non-generic , the equilibrium will have multiple feasible allocations and this determines a correspondence between the strategy space and , the pay-off space. Even for a generic , there may be multiple equilibrium points, viz., , and each of these will define an analytic sheet of the correspondence over the generic open set.

We now demonstrate the theory described so far through a market.

## 4. An Example

In this section, we describe an economy with three labour classes and three goods, viz., and construct the consumer choice game where two of the labour classes engage in strategic behaviour.

Let . Let be as given below:

 T=⎡⎢⎣100120124⎤⎥⎦Y=⎡⎢⎣110100⎤⎥⎦Ur=⎡⎢⎣100110011⎤⎥⎦S=⎡⎢⎣100α100β1⎤⎥⎦

There are 3 labour types, with numbers 1, 10 and 100 respectively. only prefers good , only and prefers and as shown in the true utilities . The example can be understood as an instance of a market with three socio-economic classes and a good such as footwear which is produced in three different ways or qualities. In such cases, the given utility matrix catches the general preference towards the goods produced by different classes.

Let us consider labour class 2 and 3 as the players who exercise their strategies by choosing the variables and . This defines the strategy space as shown above. Note that . We compute (i) the dependence of the pay-offs on and , and (ii) the sub-domain of over which the chosen forest below is the equilibrium forest.

We first solve for the production part.
We see that gives:

 q=T−1Y=⎡⎢⎣100−0.50.500−0.250.25⎤⎥⎦Y=⎡⎢⎣14.522.5⎤⎥⎦

thus, the production is determined. Next, we use ,

This describes wages in terms of prices. All this does not need the equilibrium forest .

For the consumption and allocation processes, let us assume that the solution forest is given by:

Note that this forest is motivated by the utility matrix given earlier. Using Fisher market constraints of optimum utility, we can write these equations :

 p1/α=p2 and p2/β=p3

Conserving the total money while allotting the goods, the equations result in the following money flows, with the conditions that , :

The flows mentioned on edges are the amounts spent by classes on the corresponding goods. This conditions reduce to the requirements that and . Under these conditions, will arise as the equilibrium forest.

Assuming total money in the economy as 1, we find class wages and allocations as functions of and

 w1=β(2α−1)2αβ+9β+45,   x11=1−0.5α
 w2=5(2β−1)2αβ+9β+45,   x21=0.5α,  x22=4.5−52β
 w3=502αβ+9β+45,   x32=59β,  x33=22.5

where are the class wages and dividing those by the number of people yields the values per person in each class.

We see that have significant impact on wages and allocations and thus, can be used as strategies. For example, if class-2 decides to keep the value of at 0.75 as opposed to 1, the wage share of class-1 decreases and thereby that of class-2 improves. Moreover, the allocations also increase.

For this forest, i.e., in the region and , the pay-offs based on the true utility are given below as functions of .

 u1=1−0.5α
 u2=4.5+0.5α−52β
 u3=59β+22.5

It is clear that decreasing and are the strategies for class-2 and 3. In fact, the impact of on is more significant than on , and it would be in the interest of and to squeeze through the use of .

Moreover, We see that multiple equilibrium forests are possible here, depending on and , including the one given above. For each of them, the number of active classes may be different and thus the utility functions will vary. In fact, for a sufficiently small value of , class-2 and thereby class-1 receive no wages. In this case, only good-3 is produced and its utility for class-3 is 25. This exceeds the utility of 23.5 which comes from the forest described above.

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. Connecting the Heterodox model with the A-D Market

The objective of this section is to construct an Arrow-Debreu (A-D) market from the heterodox model , and show the equivalence of the equilibrium points in the Heterodox model and the market equilibria in the A-D sense. See Appendix C for a detailed description of the standard A-D market.

### 5.1. Heterodox Model as an A-D instance

We shall now build a suitable A-D market, given the data for . We assume that and are , i.e., there are labour classes and processed goods, and is the vector of labour class size. Recall that, refers to the number of labour-units of type required to produce one unit of good . The labour availability is given as a vector . We now construct as follows.

• The set of firms in is , where is the number of columns of . The firm produces good .

• The total number of goods are , viz., , where corresponds to the labour of class . We call labour inputs as ’raw’ goods.

• The number of agents is , and each agent begins with an endowment of good above.

• The production function of is which arises from the column of . Define as the vector to represent that units of labour type are used to make one unit of good . Then where is a large number. Thus firm produces some multiple of .

• Agent owns a fraction of the firm . The exact numbers will be irrelevant since we will see that in equilibrium, the firms make zero profits. Hence, income of each agent , is defined as (price i.e. wage) (initial endowment of labour units)(wages) (labour input of one agent).

• The utility matrix serves to define the continuous real valued utility function for each agent . If is the amount of good allocated to agent , then . Utilities are zero for labour units hours, i.e., for , as it is only the firms which have any use for labour. Since the utilities are linear, it is clear that the principle of non-satiation holds.

This completes the specification of the A-D market . An A-D equilibrium of the are prices , production and allocations such that (i) each firm maximizes profits under the given global labour constraints, and (ii) each agent maximizes its utility under the expenditure constraint of the wages received from its endowments priced at wages and (iii) demand meets supply for each good when the corresponding price is nonzero i.e. all produced goods are exhausted. If supply is more than demand, the price is zero.

### 5.2. Heterodox equilibrium as an equilibrium point in AD

For the market , we denote an Heterodox equilibrium as . Let us assume that in the market, in the above equilibrium, it is the first labour classes and the first goods which are active. We let denote the prices, production and wages of active goods and labour i.e. , and . As defined in the Heterodox model, refers to the modified prices of the unproduced goods.

By the feasibility of the equilibrium and the activity conditions, we have,

 T¯q≤Y
 ¯wT≥¯p

and

 T′q=Y′
 wT′=p

where is the reduced technology matrix and is the reduced Y vector in accordance with the active goods and classes. Let be the amount of labour used. The variables are such that the production is optimal given and prices are solutions to Fisher market. Also, by the price-setting mechanism of unproduced goods , and the choice of as the dual variables, we have

 pj=maxiuijbi (U- Level)<∑iTijwi% (T- Level)

where refers to the bang per buck ratio of ’s agent.

###### Proposition 5.1 ().

Let be the market with a Heterodox equilibrium point as described above. Let and . Then, is an equilibrium in the A-D market.

###### Proof.

Along with the optimization constraints, we need to prove that total supply of all goods, producible and raw, is greater than or equal to the goods demanded or consumed. Moreover, if supply is more than the demand, the corresponding price/wage is zero. This translates to saying that all ‘produced’ and ‘raw’ goods satisfying should be exhausted and all ‘raw’ goods satisfying should receive zero wages. As described above, .

• Let us first consider the active goods and classes. We prove that each firm maximizes the profit, given the global constraints. For firm , let maximize the profit, given that it lies in the set of production possible technologies. Here, is a vector of the amount of labour units consumed in making amount of good . For all , should satisfy

 (6) maxy  piqi−w⋅li=piqi−(w⋅T′∗,i)qis.t.   qi,li≥0,

Using the Technology matrix, here we have given by .
We note that since , the expression equals . This means that whenever , firm gives an optimal production value, irrespective of .

• We now consider . Since there is no production () and consumption of labour, as given by the duality of and in the Heterodox model. Hence, the maximum occurs at , . Therefore, the optimization function is multivalued and given is an optimal point satisfying the global constraints. We also see that implies that the raw ‘used’ goods are exhausted completely. On the other hand, there is a supply of raw ‘unused’ goods but no demand resulting in zero prices i.e. wages. This establishes that all firms maximize their profits and for raw goods: supply meets demand for ‘used’ goods and prices are zero for ’unused’ goods.

• Next, we prove that each agent finds an optimal consumption set by maximizing her utility under the expenditure constraint . The optimization program given below exactly conveys this requirement. Given , we set up the equation for each agent -

 maxx∑jxijuij s.t.  ¯p⋅xi≤Mi(¯p)=wi, 0

KKT conditions for this program imply that the optimal point satisfies , where and are the Lagrange multipliers associated with constraint 1 and 2 respectively. This means that whenever is positive, , and . In other words, whenever agent buys goods , we have , which is a Fisher condition.
Since the utility function is convex, we observe that the Heterodox output for allocations, i.e., maximizes the above function, as given by the sufficiency of KKT. Moreover, the Heterodox output for consumption is such that all produced goods are completely exhausted. Since utility for raw goods is zero, we see that each agent maximizes her payoff by buying the right set of produced goods. Thus, we prove that all agents maximimze their payoffs and demand equals supply of the reduced set of producible goods.

• This establishes that so defined using the Heterodox model is an equilibrium point in the A-D market.

### 5.3. A-D equilibrium as an equilibrium in the Heterodox market

Now, let be an equilibrium point in the A-D market. We assume that classes and goods are active. Let be the corresponding prices and wages. Let the optimal production vector for each firm be so that the total output of firms is . Let give the consumption.

###### Proposition 5.2 ().

Let be the market described above with a A-D equilibrium point . Then, is an equilibrium point in the Heterodox model.

###### Proof.

We let , be the ‘active’ vectors consisting of all positive entries from . Similarly, let denote the prices of active goods.

• We first look at the conditions satisfy being a part of A-D equilibrium. For each firm which is active, is its optimal solution where . In other words, has to maximize subject to the non-negativity constraints. We can now consider these three cases -
,
and
.

Since are given, we note that since firm produces finite amount , we can only have . If this is not true, then any finite cannot maximize , which contradicts the definition. In other words, the function is strictly increasing as increases, thus giving an unbounded solution. When we artificially put a bound on , the optimal solution is at an unattainable production plan. Moreover, the fact that firm is active i.e. it is not making any losses, translates to the condition . Therefore, we get that for all active firms/goods , or where represents the reduced T matrix corresponding to active goods. Continuing with the same concepts, for the inactive firms we must have , for any feasible .

• The analysis for agents’ optimal consumption is exactly similar to that given in the earlier section, where the optimization program catches Fisher market conditions. Next, by the definition of an equilibrium point in AD, we know that total supply equals usage/consumption for all such that . We have, zero utilities for raw goods i.e. labour hours. This forces that initial endowment of raw goods should equal the amount of raw goods consumed while producing other goods. This confirms that . For the inactive classes, as , firms don’t utilize those. Therefore, there is supply but no demand for these goods. As the equilibrium production plan is attainable, we have . In all, we have that for all labour classes and for all classes that are active. Similarly, we have that for all goods, and for active goods. Along with these two conditions, we have that the allocations and prices follow Fisher market conditions i.e. all goods and endowments are exhausted and every buyer maximises her utility and buys only those goods which give her maximum bang per buck value.

• Building from the observations, we see that are dual variables of each other and optimal for the following programs, as they satisfy the complementary slackness conditions.

 (7) max¯q¯p⋅(T¯q)s.t.T¯q ≤Y¯qj ≥0min¯w¯w⋅Ys.t.¯wT ≥¯p¯wi≥0

Moreover, we see that for (unproduced goods) must satisfy

 maxiuijbi (U- Level)≤pj<∑iTijwi (T- Level)

If T-Level, then it violates the constraint of optimality of production for firm . Similarly, if U-Level, we have for some player . This means that is less than the bang per buck that good offers, which contradicts the optimality condition of . Therefore, we see that the A-D model allows for a band for each that corresponds to an unproduced good. In the Heterodox model, we fix prices of such goods equal to their U-Levels, which belong to this band for every . In short, Heterodox model modifies the equilibrium prices of inactive goods in A-D, while keeping all other variables and optimality conditions the same.

Thus we see that the A-D equilibrium point satisfies all conditions for a fixed point in the Heterodox settings. In other words, when this point is given as an input to the iterator defined earlier, the production, prices, wages and allocations remain unchanged. ∎

This proves that given any set of , an equilibrium exists in the Heterodox model. As described before, Heterodox model can be considered an instance of the A-D model, where the existence of equilibrium is proved. Using the proof given above, the equilibrium (with little modifications) is a fixed point in the Heterodox model too.

In the next section, we analyse a 2-player scenario and look at the decomposition of strategy space and also examine players’ strategic behaviour.

## 6. A 2×2 market

Let us consider a two class economy with the following specifications - (Technology matrix), (Labour availability), (True Utility matrix) and (Strategy matrix)

 T=[0.2500.251]Y=[24]Ur=[1111]U2=[u11u12u21u22]

Since Fisher solutions do not change if the rows of are scaled independently, we see that effectively, is given by:

 U2=[α1β1]

We assume that . Let us solve this case completely, i.e., decompose into various zones by their combinatorial signatures. We also analyse the case when we transit from one zone to another, and finally, when one of the labour classes is shut out of the market.
Whenever both classes are active, (production vector) is given by :

 q=T−1Y=[40−11]Y=[82]

and

 w=[w1w2]=[