Aggregate risk in a competitive economy with a financial market Open Access

The analysis of entrepreneurial innovation and evolutionary changes in the production sphere is one of the central subjects of evolutionary economics, especially in the line of thought of Schumpeter’s theory of economic development (Schumpeter, 1934). Innovation is recognised as a major force to achieve organisational success in an intensively competitive environment. However, in a large part of mainstream formalisations of the neo-Schumpeterian theory of economic development (Nelson and Winter, 1982; Nelson and Winter, 2002), the relationship between the real and financial sectors of the economy has not been examined in a satisfactory way because of the lack of specification of the conditions under which there is possibility of risk minimisation within innovative economic evolution.

The multi-period models’ risk is seen through the prism of the utility theory. The basic tool for analysing the actions of individuals under risk conditions is the function of von Neumann-Morgenstern’s utility and its modifications (Neumann and von Morgenstern, 1944; Savage, 1972; Wakker and Deneffe, 1996). Their respective characteristics determine the individual attitudes toward risk. Due to the assumptions usually considered in the multi-periods models, von Neumann–Morgenstern utility function can be naturally applied within the examination of properties of the behaviour of the risk-averse market participants in those type of models (Werner, 1998, 2005).

At the same time, one of the traditional criteria for assessing the effects of collective economic undertakings is the principle of the social optimum of Pareto. These two sequences of terms, which allow us to precisely define one’s attitude toward risk and, on the other hand, Pareto-optimal economic properties, are not disjoint. Their relationship in the equilibrium theory of Arrow and Debreu is described by Arrow (1964) and Werner (1998).

Multi-period models are natural extensions of the Arrow–Debreu model, (1954) which is regarded as a basis for the analysis of the competitive markets (Mas-Colell et al., 1995). In these kinds of models the role of financial markets within the processes on the market of real commodities can also be studied (Werner, 1998). In this context, the models in which agents’ activities under uncertainty (Arrow, 1964; Radner, 1972) or under risk (Arrow, 1965; Werner, 1998) are examined, have been developing for many years (Kaplan and Garrick, 1981; Duffie, 1988; Chew and Mao, 1995; Bell, 1995; Magill and Quinzii, 2002; Netzer, 2009; Robatto and Szentes, 2017; Miyagishima, 2022).

The multi-period equilibrium economies considered by Magill and Quinzii (2002) are examples of models in which the activities of market participants in real markets are coordinated by the financial market while the division of total resources are created for the financial markets. The model presented in the paper is a modification of a two-period financial economy with corporations (Magill and Quinzii, 2002, pp. 329–361). The production sector of the model presented is determined by the existence of a finite number of firms, while the ownership structure of firms is defined as a private ownership, where consumers are the owners.

The different models of risk (Bedford and Cooke, 2001; Cox, 2009; Werner, 2009; Heller and Nehama, 2021) as well as the concepts of aggregation of risk or diversification of risk were born with regard to various applications of analysis of risk (Bjørnsen and Aven, 2019; Hassel et al., 2022). However, the problem of risk classification also has its roots in the sources of risk. Some multi-period models in which financial markets are taken into account enable to analyse the implications of agents’ risk-aversion (Werner, 1998, 2009) and analysis of market participants’ attitudes towards risk leads to its aggregation with respect to some characteristics of the model and, consequently, to its minimising in an optimal allocation (see also Ćwięczek, 2013).

The problem of risk minimisation is strictly related to the necessity to define of the measure of risk (Artzner et al., 1999; Denuit et al., 2005; Jia and Dyer, 1996; MacKenzie, 2014) which usually depends on the model considered and on the initial conditions. Modern methods of risk measurement are based on complex methods that combine statistical-econometric tools and network theory (Denkowska and Wanat, 2021).

The welfare effects of financial innovations in incomplete markets were analysed by Cass and Cittana (1998). Magill and Quinzii (2002) studied the consequences of trading in multi-periods economies with incomplete markets for equilibria of an economy. Carvajal et al. (2012) examined, among others, the economies with innovations, in which in all equilibria, financial markets were incomplete.

In the paper, previous analyses of market participants’ attitudes toward risk and the problem of minimising risk are extended to the risk of introducing innovation. We analyse real and financial innovations in the newly specified two-period competitive economy with the complete financial market of bonds and equity contracts (Section 3). In the model under study, we show that there is a Pareto optimal allocation in which every consumer is risk-free (Section 4). The above result is obtained under the assumptions that total endowments are not the sources of risk, every consumer is risk-averse, the production and consumption sets are convex and equal in all the future states of nature and it is known what new commodities or what new assets will appear on the market in a second period and what their prices in every state will.

The paper consists of six parts. In Section 2, a model of the multi-period economy with the financial market of bonds and equity contracts is presented. Section 3 is devoted to the modelling of innovations in the model considered. Section 4 contains the analysis of aggregate risk in the economy presented. Section 5 of the paper is devoted to discussion, while Section 6 presents conclusions.

We consider a modification of the model presented by Magill and Quinzii (2002, pp. 378–426) in which three kinds of market are considered: the commodity market, the bonds market and the stock market. The model presented is a two-period economy in which time and uncertainty are given by an event tree, consisting of the initial period $t0$ and the finite number of states of nature $s=1,…,S (S∈{1, 2,…})$ in a period $t1$⁠, where $t0<t1$⁠. It is assumed that in period $t0$ there is only one state of nature denoted as $s=0$⁠. Hence the modelled economic activity extends over two consecutive periods, $t0$ and $t1$ due to which $S+1$ states of nature are considered. Similarly as in Magill and Quinzii (2002) we assume that, for $s=1,…,S$⁠, probability $Ps>0$ of the occurrence of the state $s$ is exogenously given and $∑s=1SPs=1$⁠.

We start the presentation of the economic system which will be later of use with the definitions of the mentioned three kinds of markets considered in the model.

We put the following assumptions: in every state $s∈0,1,…,S$⁠, there are $ℓ∈{1,2,…}$ available goods on the commodity market, space $Rℓ$ is the commodity space in the sense of the definition formulated by Arrow and Debreu (1954) as well as price vector $ps=ps,1, …, ps,ℓ∈Rℓ$ is exogenously given (number $ps,l$ is the price of the commodity $l∈{1,…,ℓ}$⁠). Thus, space $Rℓ(S+1)$ is the commodity space, whereas vector:

is the price system in the model presented further. Under the above assumptions pair $R=RℓS+1, p$ is called the commodity market.

Let $J∈{1,2,…}$ be the number of bonds, $q¯=q-1,…,q-J∈0,∞J$be the vector of bond prices at date $t0$ (number $q¯j$⁠, for $j∈{1,…,J}$⁠, means the price of the bond $j$⁠), $V=vsjS×J$ be the matrix of payment of bonds at date $t1$⁠. Bond market is defined as a triple $F- = RJ,q¯, V$⁠.

For every state $s∈1,…,S$⁠, row $Vs$ of matrix $V$ consists of the payment of all bonds in state $s$ at date $t1$⁠, while column $Vj$⁠, for every $j$⁠, describes the payoff of bond $j$ in every state at date $t1$⁠. Let vector $zi=z1i,…,zJi ∈RJ$ be the portfolio of agent $i$⁠. That means that coordinate $zji∈R$⁠, for every $j$⁠, denotes the number of bonds $j$ in portfolio $zi$⁠. If at date $t0$ agent $i$ buys portfolio $zi$⁠, then $zji>0$⁠; if agent $i$ sells bond $j$⁠, then $zji<0$⁠. Thus portfolio $zi$ at date $t0$ is an input-output vector, whereas its values under vector $q-$ at this date is equal to $q-∘zi:=∑j=1Jq¯jzji$⁠. More properties of the matrix $V$⁠, the reader can find in Lipieta and Ćwięczek (2022).

It is assumed that, there are $K∈{1,2,…}$equity contracts at date $t0$ which can be bought or sold in this period. Every contract $k∈1,…, K$ generates in state $s∈{1,…,S}$ at date $t1$ income $dsk$⁠. Stock market is defined as a triple $F̿= ℝK, q̿, Dt1$⁠, where $K$ is the number of equity contracts at date $t0$⁠, $q̿= q̿1,…, q̿K∈0,∞K$ is the vector of prices of equity contracts at date $t0$ and $W=wskS×K$ is the matrix of payments of equity contracts at date $t1$⁠.

Financial market is structure $F=(RJ+K,q,V,W)$⁠, where $J$ means the number of bonds, $K$ is the number of equity contracts in state $s=0$⁠,$q=q¯,q̿∈ℝJ+K$ is the vector of prices of securities in state $s=0$⁠, where $q-∈0,∞J$is the vector of prices of $J$ bonds and $q̿∈0,∞K$ is vector of prices of $K$ equity contracts in state $s=0$⁠, $V=vsjS×J$is the matrix of payments of bonds in period $t1$ and $W=wskS×K$ is the matrix of payments of contracts in period $t1$⁠.

Let $hi=h1i,…,hJ+Ki ∈RJ+K$ be a portfolio of securities of agent $i$⁠. It is assumed that portfolio $hi$ is of form $hi=zi,δi$⁠, where $zi$ is the portfolio of bonds and $δi$ is the portfolio of equity contracts of agent $i$⁠. We say that financial market $F$ is complete, if and only, if $S=J+K$ and $detVWS×(J+K)≠0$⁠. Consider matrix $Φ=−q¯-q̿VW$⁠. Under the above notation, the following subspaces of $RS+1$ can be considered: $Φ:=τ∈RS+1:τ=Φ·hT: h∈RJ+K$⁠, $Φ⊥≔μ∈RS+1:μ·Φ = 0$⁠. Now the purchase of portfolio $hi=zi,δi$ at date $t0$ generates income stream $τ=τ0,τt1 ∈RS+1$⁠, where $τ0=-q∘hi$⁠, $τt1=τ1,…,τS$ and $τs=φs∘hi$⁠, $φs$ is the $s-th$ row of matrix $VW$ consisting of the payments of $J$ bonds and $K$ equity contracts in state $s∈{1,…,S}$ at date $t1$⁠. More precisely, $τs=φs∘hi$ is the income received from the sale of the portfolio $hi$ in the state $s$ at date $t1$⁠.

Space $Φ⊥$ is called the space of state prices or the space of the present-value vectors. Thus, every vector:

is a present-value vector and every coordinate$μs$⁠, for $s=1,…,S$⁠, is interpreted as the present-value of one unit of income in state $s$ at date $t1$⁠. If market $F$ is complete, then $dimΦ⊥=1$ and consequently vector $μ$ of the form (2) is uniformly determined.

Let us notice that, by definition (Magill and Quinzii, 2002, p.70), under given $(q,V,W)$ there are no arbitrage opportunities on the financial market $F$⁠, if there does not exist a portfolio of bonds $z∈RJ$ and a portfolio of equity contracts $δ∈RK$ such that $Φ·(z,δ)>0$⁠. So, the absence of arbitrage in the approach presented can be written as follows:

The property (3) also means that there is a vector of positive state prices $ν∈R++S+1$ such that $ν·Φ=0$ (compare to Magill and Quinzii, 2002, pp. 71–72).

The activities of economic agents on commodity markets are coordinated by the financial market which enables borrowers to take out loans by market participants, thus financing consumers’ and producers’ activities on commodity markets.

In the paper, we consider three kinds of firms: individually owned firms, partnerships and corporations, distinguishable depending on the structure of ownership. Finite set $B=b:b=1,…,B$⁠, where $B∈1, 2,…$⁠, denotes set of firms. Set $Ysb⊂Rℓ$ consists of production plans of firm $b$ in state $s∈0,…,S (ysb=(ys,1b,ys,2b,…, ys,ℓb)∈Ysb)$⁠, feasible to realisation with respect to technologies; it is assumed that $Ysb≠∅$⁠, for every $b$and $s∈0,…,S$ (Magill and Quinzii, 2002, p. 333). Set $Yb =Y0b,…,YSb⊂RℓS+1$ is the production set of firm $b$ (⁠$yb=y0b,…,ySb∈Yb is a production plan of firm b)$⁠, $Y=(Y1,…,YB)$⁠. Number $dsb=ps∘ysb$ is the income firm $b$ in every state $s∈0,…,S$⁠, vector $db=(d0b,dt1b)∈RS+1$⁠, where $dt1b=(d1b,…,dSb)∈RS$⁠, is the income stream of firm $b$⁠. The matrix of income is of the form:

and vector $ds=ds1,…,dsB∈RB$ denotes the income of all firms in state $s∈0,…,S$⁠.

Assume that financial market $F$ is complete. Each firm $b$ aims at realizing a production plan $yb∈Yb$ which maximises, at prices $p$⁠, the profit function of firm $b$⁠:

which to every production plan $yb∈Yb$ assigns the present-value of the income stream induced by vector $yb$⁠, where $dsb=ps∘ysb$⁠, $s∈0,…,S$⁠.

Production system is a relational system $P=B,R,Y$⁠. Let us recall that the expected value of vector $μ1d1b, …, μSdSb$ under probabilities $P1, …,PS$ is equal to $∑s=1SPsμsdsb$⁠, i.e. $Eμ1d1b, …, μSdSb=∑s=1SPsμsdsb$⁠. Number $μsdsb=μs∑l=1ℓps,lys,lb$ is the present-value of the income stream in state $s=1,…,S$⁠, induced by the realisation of production plan $yb$⁠.

Finite set $A=a:a=1,…,A,A∈1, 2,…$⁠, is a set of consumers. Vector $za=z1a,…,zJa∈RJ$ denotes a portfolio of bonds of consumer $a$⁠, vector $δa=δ1a,…,δKa∈RK$ is a portfolio of equity contracts of consumer $a$⁠. Set $Xsa⊂Rℓ$ consists of consumption plans of consumer $a$ in state $s∈0,…,S xsa=xs,1a,xs,2a,…, xs,ℓa∈Xsa$⁠; it is assumed that $Xsa≠∅$⁠, for every $a$and $s∈0,…,S$ (Magill and Quinzii, 2002, p. 38). Set $Xa=X0a,…,XSa⊂RℓS+1$ is the consumption set of consumer $a$ (⁠$xa=x0a,…,xSa∈Xa$is a plan of action of consumer$a$⁠; it is also called consumer’s $a$ consumption plan); $Xa=(X0a,…,XSa)⊂Rℓ(S+1)$ is the consumption set of consumer $a$⁠, $X=(X1,…,XA)$⁠. Function $usa:Xsa→R$ is a utility function in the state $s∈0,…,S$⁠, $u=u1,…,uA$⁠. Vector $ωsa=(ωs,1a,ωs,2a,…, ωs,ℓa)∈Rℓ$ means an initial endowment of consumer $a∈1,…,A$ in state $s∈0,…,S$⁠, vector $ωa=ω0a,…,ωSa∈Rℓ(S+1)$ is the initial endowment of consumer $a$⁠, $ω=ω1,…,ωA∈RS+1$ is the vector of total endowments. Consumption system is a relational system $C=A,R,F,X,u,ω$⁠.

In consumption system $C$⁠, the budget set of every consumer $a∈A$ is defined as: $βa:=xa,za,δa∈Xa×ℝJ×ℝK: p0∘x0a-ω0a,…,ps∘xSa-ωSaT≤Φ·za, δaT$⁠,

where $Φ=-q--q̿VW$⁠. The budget set is assumed to be not empty, whereas $S+1$ inequalities determine compliance of expenditures and incomes of consumer $a$ on the commodity market and on the bond market. If $xa,za,δa∈β(a)$⁠, then it is said that portfolio $ha=za,δa$ of consumer $a$⁠, finances the consumption plan $xa$⁠. Function $Ua:Xa→R$⁠, of the form:

where $xa=x0a,…,xSa∈Xa$⁠, is called the expected utility of consumer $a$⁠, where $Eu1ax1a, …,usa(xsa)=∑s=1SPsusa(xsa)$⁠. Hence the form of the expected utility is additively separable over time is (cf. LeRoy and Werner, 2001), as by (5):

The aim of consumers is to maximise the expected utility functions of the budget set.

The economy presented in the paper is a combination of production and consumption systems in which agents trade a number of securities, i.e. bonds and equity contracts on the financial market. It is assumed that the agents at date $t0$ correctly determine their future possibilities (i.e. their production set or their consumption sets, utilities and endowments, respectively), their future actions on the market of securities as well as the future prices of goods in every state at date $t1$⁠. However, at date $t1$ there is only one state of nature from set $1,…,S$ and the information about which state occurs is available to all market participants just at date $t1$⁠.

We consider an economy which is a combination of consumption system $C$ and the production system $P$ in which some firms reached the corporate stage. Thus, in that economic system, it includes consumers and firms active on the commodity markets of $ℓ$ commodities, $J$ bonds, $K$ equity contracts. It is assumed that among the firms at date $t0,$ there is at least one corporation as well as every corporation issues exactly one equity contract. Under the latter assumption, $K$ is also the number of corporations on the market, $1≤K≤B$⁠. Let $B=b: b=1,…,B$ be the set of all firms, $b:b=1,…,K$ be the set of corporations, where $K∈1,…, B.$If $B-K>0$⁠, then set $b:b=K+1,…,B$ means the set of the firms which do not have the corporate stage at date $t0$ and they do not issue equity contracts in state $s=0$⁠. We assume that each firm $b∈1,…,B$ is owned by a group of shareholders from set $A$ and number $θba$ is an initial share of shareholder $a∈1,…,A$in firm $b∈1,…,B$ as well as:

moreover, if, for some $b∈1,…,B$ and $a∈1,…,A$⁠, $θba=1$⁠, the firm $b$ is individually owned by shareholder $a$⁠; if, for some $b∈1,…,B$ and $a∈1,…,A$⁠, $θba<1$⁠, the firm $b$ is a partnership or a corporation. For every $a∈1,…,A$⁠, vector $θa=θ1a,…,θBa∈[0, 1]B$ is an exogenously given portfolio of initial shares of agent $a$ in the set of firms $1,…,B$ in state $s=0$⁠. For every $a∈1,…,A$⁠, vector $δa=δ1a,…,δBa∈RB$ is a portfolio of the shares of agent $a$ in the set of firms $1,…,K$ in state $s=1,…,S$⁠, where additionally:

Now, it is needed to modify, in comparison to Section 2.4, the definition of the financial market to adjust it to the considered situation. We admit that $q=q-,q̿∈RJ+B$ is the vector of prices of securities in state $s=0$⁠; especially, $q-∈0,∞J$ is the vector of prices of $J$ bonds and $q̿∈RB$⁠, where $q̿1,…,q̿K>0$and $q̿K+1=…=q̿B=0$ if $K<B$⁠, is vector of prices of $K$ equity contracts in state $s=0$⁠:

which means that payment $wsb$ of equity contract$k∈1,…,K$⁠, in every state $s∈1,…,S$ at date $t1$⁠, is equal to income $dsk$ of corporation $k$ in state $s$ (see equation (4)).

The rest of the components of the financial market is the same as in Section 2.4. The financial market satisfying the above assumptions will be denoted as $F=(RJ+B,q,V,Dt1)$⁠. Financial market $F$ is complete, if and only, if $S=J+K$ and $detVWS×(J+K)≠0$⁠.

Let market $F$ be complete. Two periods competitive economy with financial market $F$ is a relational system $EF=P,C, θ$⁠.

Economy $EF$ operates as follows. There are $A$ consumers and $B$ firms on the market of $ℓ$ commodities. Among the firms, there is at least one corporation. All firms are owned by the shareholders from set $1,…,A$ and the shares are described by exogenously given vector $θa=θ1a,…,θBa∈R+B$ satisfying assumption (6). Every investment project $yb=y0b,…,ySb∈Yb$⁠, $b=1,…,B$⁠, is financed by the shareholders. Under exogenously given price vector $p$ of the form (1), the producer’s $b$ choices in every state induce income stream $db=d0b,dt1b∈RS+1$ where $dt1b=d1b,…,dSb$ and $dsb=ps∘ysb$for $s=0,…,S$⁠.

Thus, the matrix of incomes is of form (4), where number $dsb$ is the income of producer $b$⁠, $b=1,…,B$⁠, in state $s$⁠, $s=1,…,S$⁠. Every column $b$ of matrix $Dt1$ consists of the incomes of firm $b$ in every state in period $t1$⁠, whereas every row $s$⁠, consists of the incomes of all firms in the state $s$⁠. The shareholders of corporations at period $t0$⁠, after the choice of the production plans by firms to be realized, can sell their portfolios of shares. Thus, if the shareholders, who at date $t0$ influenced on the choice of the production plans, sell their initial portfolios of shares between dates $t0$ and $t1$⁠, then they will do not have receive the income in period $t1$⁠. Consumer $a$ can buy a portfolio $za$ of bonds at date $t0$⁠, $za=z1a,…,zJa∈RJ$⁠. Within the first stage of period $t0$ agent $a$ can sell his shares in corporations described by vector $θ-a∈ℝB$⁠, under given exogenously prices $q̿$⁠, where:

and $θa=θ1a,…,θBa∈RB$ is the portfolio of initial shares of agent $a$ in the set of firms $B$ in state $s=0$⁠. Within the second stage of period $t0$⁠, consumer $a$ can buy, also under prices $q̿$⁠, new portfolio $δa=δ1a,…,δBa∈RB$⁠, for which assumption (7) is valid. Thus, the shareholders on the stock market at date $t0$ sell and buy the portfolios of equity contracts only from set ${1,…,K}$⁠, represented in the model by their shares in those corporations. It is assumed that for $b∈{1,…,K}$ number $δba>0 δba<0$⁠, if agent $a$ buys (sells) equity contracts of corporation $b$⁠. As an initial shareholder of firm $b$⁠, agent $a$ shares $d0bθba$ as his part of the initial costs of the firm $b$⁠. The sale of initial share $θ-ba$ induces income $q̿bθ-ba$⁠, while the purchase of new share induces cost $q̿bδba$⁠. Share $δba$ makes agent $a$ get income stream in period $t1$ equal to $δbadt1b$⁠, where $dt1b=d1b,…,dSb$ and $dsb=ps∘ysb$ for $s=1,…,S$⁠. Hence, the transactions of agent $a$ on the financial market $F$ generate in period $t0$⁠, for $d0=d01,…,d0B$⁠, income:

At date $t1$⁠, the income stream of consumer $a$ is equal to:

Combining the above, the inequalities defining the budget set of consumers $a$ are of the form:

Inequalities (10) and (11) lead to the definition of a budget set of consumer $a$⁠:

where: $ϖ0a=ω0a+y0∘θa$⁠, for $y0=y01,…,y0B$⁠, $ϖsa=ωsa+ys∘θa$⁠, for $ys=ys1,…,ysB$ and $s=1,…,S$⁠, $ha∈RJ+B$⁠: $ha=(h1a,…,hJa,…,hJ+Ba)$ where:

and $Φ=-q--q̿VW$⁠, where $W$ satisfies (8).

(compare to Magill and Quinzii, 2002, pp. 379–380). The number of securities (bonds and equity contracts) on financial market $F$ is the same in every state. That is why activities on the stock market of those shareholders who hold corporations in their portfolios influence on their budget set at every state.

Sequence $(x,y,h)$⁠, where $x=x1,…,xA∈X1×…×XA$⁠, $y=y1,…,yB∈Y1×…×YB$⁠, $h=h1,…,hA∈RJ+BA$ satisfies (13), is called an allocation in economy $EF$⁠. Allocation $(x,y,h)$ in economy $EF$ is called feasible, if:

and:

A set of all feasible allocations in economy is denoted by $M$⁠. Equations (14) and (15) are called equilibrium conditions in every state on the real market or the financial market, respectively. On the basis of the above we put the following:

Definition 1. Sequence $x*,y*,h*,p*,q*$ satisfying:

is called an equilibrium state in economy $EF$⁠.

The thesis of the below lemma is the immediate consequences of its assumptions.

Lemma 1. Let sequence $x*,y*,h*,p*,q*$ satisfy conditions (16) and (17). Suppose that, for some $s0∈0, 1,…S$⁠, $ζs0≔ ∑a=1Axs0a*-∑b=1Bys0b*-∑a=1Aωs0a≤0∈Rℓ$⁠, $ps0∘ζs0=0$ as well as $Ys0b0-[0,∞)ℓ⊂Ys0b0$⁠, for some $b0∈1,…B$⁠. Then:

(1) $y˜s0b0*:=ys0b0*+ζs0∈ Ys0b0,$

(2) $y˜s0b0*∈arg⁡max⁡πb0(yb0):ys0b0∈Ys0b0,$

(3) $∑a=1Axs0a*-∑b=1, b≠b0Bys0b*-y˜s0b0*-∑a=1Aωs0a=0.$

Put, for $a=1,…A:$

Then $Θa∈RJ+B$⁠. Let $H≔∑a∈Aha*∈RJ+B$⁠.

Lemma 2. Let sequence $x*,y*,h*,p*,q*$ satisfy conditions (14), (16) and (17). If vector $H$ satisfies the following:

then there exists a mapping $f: RJ+B→RJ+B$ such that sequence $x*,y*,h∼*,p*,q*$⁠, where $h∼*=h∼1*, …, h∼A*$ and $h∼a*≔fha*$ is an equilibrium state in economy $EF$⁠.

Proof. See Supplementary Material.

Let us notice that if the market clearing condition on the market of bonds is not satisfied, then, for $a=1,…A,$vectors$H and Θa$are linearly independent. Moreover, under the assumption of Lemma 2, if $rank H Θ1… ΘA(J+B)×(A+1)<J+B$⁠, then there are infinitely many mappings $f$ satisfying the thesis of the lemma and, consequently, infinitely many equilibrium states in economy $EF$⁠, determined by sequence $x*,y*,h*,p*,q*$⁠.

Lemmas 1 and 2 lead to the following:

Proposition 1. Let sequence $x*,y*,h*,p*,q*$ satisfy conditions (16) and (17). If

• for every $s∈0, 1,…S$⁠, $ζs=∑a=1Axsa*-∑b=1Bysb*-∑a=1Aωsa≤0$⁠, $ps∘ζs=0$⁠;

• for every $s∈0, 1,…S, Ysb0-[0,∞)ℓ⊂Ysb0$⁠, for some $b0∈1,…B$⁠; and

• $H=∑a∈Aha*=0$ or$H$ satisfies condition (19),