Merton’s financial multi-agent consumption

1. Introduction

Consumption and utility pricing models underlie a broad number of financial models based on the seminal Merton paper [1]. For example, Constantinidis and Duffie [2], Cochrane [3] as well as an extensive list of research papers and books have profusely pointed out to its applications. In particular, the Consumption Capital Assets Pricing Model (CCAPM) has used a consumption rationality [3,4] to define a kernel pricing model based on a single and representative agent summarized by 1 = E ( M ∼ t + 1 ( 1 + ℜ ∼ t + 1 ) | ℑ t ) with M ∼ t + 1 , a pricing kernel, 1 + ℜ ∼ t + 1 the return on an initial investment of one dollar and ℑ t , a filtration, summarizing all the relevant and common past information at time t. This approach, unlike the Arrow-Debreu pricing model [5–10] is utility based and therefore its probability measure is a function of the risk preferences embedded in the definition of its utility [1,10] formally stated in terms of a representative investor, k, endowed with initial wealth $W_0^k>0$ and optimizing both her utility of consumption c t k and her investment policy by choosing an investment portfolio, a t k . When the decision to consume is financial, the consumer decision is defined by a financial commitment while the price of consumption is random as it depends on other agents’ financial commitments to consume (as well as competitive, risk and supply factors). In this case, both current consumption and future wealth are random, unlike the conventional Merton model defined by the following problem which we state in its infinite horizon version: where r is a time preference for consumption, d ( S t 0 , S t ) is the wealth increment of the investor’s trading strategy a t k , and U k ( c t k ) is the investor’s utility of consumption. This formulation assumes that the consumer is a price taker whose decision to consume has no effect on the prices of consumption; that there are no agents that can significantly affect the market price of consumption and finally there exists “a representative agent” whose decision defines the market demand for consumption. A number of extensions to the Merton’s fundamental model have considered various types of investors (e.g., time inconsistent investors [11]), constraints (e.g., borrowing and consumption constraints [12,13]), as well as a multi-agent framework (e.g., [14–18]). In particular, Karatzas et al. [14] reduce a multi-agent consumption/investment decision problem to the problem of finding an appropriate representative agent. Anufrieva and Dindo [15] focus on asset price dynamics in a market with heterogeneous agents. The agents do not consume and only decide whether to invest in a risk-free bond or in a risky financial asset. Using wealth dynamics as a selection device they characterize the long run asset returns and wealth distributions for a general class of investment behaviors. Gorton et al. [16] incorporate a standard agency problem into a classical asset pricing model to simultaneously endogenize the pricing kernel and the cash flows. Tapiero [17] considers a wealth heterogeneous multi-agent financial pricing CCAPM model, which is based on a distinction between what agents are willing to pay for consumption and what they actually pay, pointing to a growth of inequality among consumers. Mbodji et al. [18] study the Merton problem for the case of two investors sharing a final wealth. Each agent has a different utility function and discount factor.

Unlike previous papers, we consider the Merton problem as a stochastic differential Cournot game resulting in a unique consumption price, a function of aggregate demand for consumption. In a multi-agent and dynamic financial framework, both savings and consumption are then strategic if some consumers can affect the aggregate demand by their investments and consumption policies. In a stochastic differential Cournot game framework, consumers’ wealth matters and provides additional insights on the competitive effects of wealth and savings when the price of consumption is defined by aggregate demand and aggregate supply. Consequently, as with the conventional Cournot model, an agent in our model is strategic in the sense that her decisions alter the market price. However, since an agent does not know what the other agent’s decision is, an additional strategic risk compounds the consumer’s investment risks (and thereby her future wealth). Technically, the solution of the Cournot stochastic differential game is usually addressed with only a few agents, which leads to duopoly and oligopoly market pricing models. In this paper, due to the number of agents, we have defined the “Cournot game” as that of an individual agent (consumer) “gaming the market”. Such an approach is concurrent with the financial approach that views investors as gaming the financial market to “beat its price”. Of course, consumers that are poorly endowed financially have a negligible ability to game the market and are therefore price takers while only appreciably endowed agents are able to sway the market.

We study two cases of the problem characterized by (i) a finite market with some agents setting the price while the others are price takers; (ii) an infinitely large market which is compared with only one monopolistic agent setting the consumption price. In terms of the micro-economic consequences of the multi-agent framework, we show that, given the same level of wealth and trading strategy along with constant risk aversion, the agent (no matter how rich she is), as determined by the conventional Merton model, spends on consumption more in a transient state and less in a steady state than a strategic price-setting agent who is able to affect prices to ensure greater consumption in the long run. In terms of macro-economic consequences, we show that this strategic behavior promotes social inequality – the rich becomes richer and the poor becomes poorer. On the other hand, when the market is extremely large, we find that if the mean rate of return of investments is high – consumer strategies can lead to social equality.

The paper proceeds as follows. A financial, multi-agent, differential Merton game model is defined in Section 2 while Section 3 determines the Nash feedback equilibrium conditions and studies underlying steady-state equilibrium consumption strategies. These strategies are detailed in Section 4 and characterized by consumers with an Increasing Absolute Risk Aversion (IARA). Expected dynamics of consumers’ wealth are discussed in Sections 5. The effects of a trading (investment) strategy, inflation and market volatility on the agent wealth are considered in Section 6. Our results are summarized in Section 7.

2. The continuous time financial merton model

We consider an agent k, k = 1, …, K whose wealth, W ^k (t), is invested proportionately, 0 ≤ s ^k (t) ≤ 1, for future consumption while its residual, (1 − s ^k (t))W ^k (t), is consumed currently. An agent investment strategy consists then in a portfolio allocation which is distributed between a financial index and bonds in proportions 1 − a ^k (t), 0 ≤ a ^k (t) ≤ 1, respectively. This results in the following wealth process: (1) In Eq. (1) R ^k (t) is the rate of return on the investment portfolio: (2) where R _f is the fixed rate of return, 𝜆 is the equity investment mean rate of return, and 𝜎 is its volatility while B (t) is a standard Weiner process. That is: (3) With e ^k = W ^k (1 − s ^k ) being an agent’s financial consumption at time t and aggregate expenditure (4) equation (3) reduces to (5) while the price of consumption, 𝜋(t), is by definition: (6) Assume that aggregate supply equals aggregate demand, its price 𝜋^∗, is then a function of aggregate expenditure, i.e., 𝜋 = 𝜋^∗(E (t)) and c k ( t ) = e k ( t ) π ∗ ( E ( t ) ) . For example, let the Cournot price be π = γ + β C ( t ) , where 𝛾 is the minimal price when the market is an infinitely large and 𝛽 is the price sensitivity to changing consumption. Then from (6) we have 𝛾C (t) + 𝛽 = E (t) or C ( t ) = E ( t ) − β γ . Thus, π ∗ ( E ( t ) ) = γ E ( t ) E ( t ) − β and c k ( t ) = e k ( t ) π ∗ ( E ( t ) ) = e k ( t ) E ( t ) − β γ E ( t ) . The utility of consumption of an agent k, is therefore a function: (7) Consequently, at any time t, the consumer chooses a control e ^k (t) from a set (8) Normal consumption persists unless W ^k (t) ≤ w ⁰, where w ⁰ is the minimal wealth preventing the consumers bankruptcy. Given a confidence level 𝜂, the propensity to avoid bankruptcy is set by a downside chance constraint: (9) Equivalently, letting the cumulative probability distribution of W ^k (t) be F _{e ^k , W ^k} (. ), (10) Given an agent k initial wealth, W ^k (0) = w ^k , k = 1, …, K, her objective is to define an admissible control process e k = { e k ( t ) : t ∈ [ 0 , ∞ ) } , i.e., 𝛺 ^k -valued process satisfying (8) and (10) from a set of admissible controls 𝛤 ^k that maximizes the agent expected discounted future utility. Agent’s k control, unlike the Merton model, depends on the cumulative effect of all agents, E. Thus, the following problems result for all agents: (11) Below we estimate the multi-agent optimal consumption strategy.

Henceforth we omit in our notation, where it is obvious, the time dependence t. Recalling that consumer k is uninformed of others’ initial wealth, the consumer’s value function is: (12) and by the principle of dynamic programming: (13) where an instantaneous expected capital gain/loss from investment and consumption in (13) is (14) With respect to (14), Eq. (13) defines equilibrium feedback expenditures in the following form: (15) The first order condition for optimal consumption expenditure, e ^k = e ^k (w ^k ), yields, (16) Equation (16) implies that the optimal individual consumption is determined as in Merton’s model by equating the marginal utility of consuming the current period output, ∂ U ∂ c k ∂ c k ∂ e k , to the marginal utility of allocating it to capital and enjoying an augmented consumption over the following periods, ∂ E t ∂ e k d V k = [ ( 1 + λ + a k ( R f − λ ) ) ∂ V k ∂ w k − σ 2 ( w k − e k ) ( 1 − a k ) 2 ∂ 2 V k ( ∂ w k ) 2 ] d t . The consumer’s utility, however, depends on aggregate consumption, that is, the first term in (16), ∂ U ∂ c k ∂ c k ∂ e k (the marginal utility), is a component that makes consumer behavior strategic, where (17) contains two terms, 1 π ∗ ( E ) and e k [ π ∗ ( E ) ] 2 ∂ π ∗ ( E ) ∂ E . The first term is a given constant in the conventional Merton’s model while the other, is not present in the Merton’s model and is due to the effects of the aggregate consumption on the resulting price and thereby on the individual consumption of agent k. Note, the weaker the dependence of the price on the aggregate consumption ∂ π ∗ ( E ) ∂ E , the smaller the second term becomes in (17) and the closer the multi-agent model is to the conventional Merton one.

The second order condition is: (18) Consider a steady-state Nash solution satisfying (16) and (18). From Eq. (5) we observe that E[dW ^k ] = dE[W ^k ] = 0, if (19) In other words, a consumption equilibrium may have an expected steady state wealth E [ W k ] = W ^ k which is the solution of Eqs. (19) in E[W ^k ]. If the steady state exists, ${\hat{W}}^k>0$ , it also satisfies (16), i.e., if a ^k < 1: (20) Comparing (19) and (20) we find and with a ^k < 1 we have (21) Accordingly, the greater the market volatility the lower the expected wealth W ^ k . On the other hand, if a ^k = 1, then from (5) we readily find that the Nash equilibrium steady state wealth is given by w k R f 1 + R f = e k , where the optimality conditions (16) reduce to: (22) which with respect to (17) results in a system of equations, (23) Consequently, if $\frac{\mathrm{\partial }U}{\mathrm{\partial }{e}^k}>0$ , (22) has a feasible interior solution only if $\frac{\mathrm{\partial }{V}^k}{\mathrm{\partial }{w}^k}>0$ and vice versa. Therefore differentiating (22) with respect to w ^k , and dividing the result by ∂ U ∂ e k , we obtain ∂ 2 U ( ∂ e k ) 2 ∂ e k ∂ w k − ( 1 + R f ) ∂ 2 V k ( ∂ w k ) 2 ∂ U ∂ e k = 0. Using the Arrow-Pratt index of absolute risk aversion A ( . ) = − U ′ ′ ( . ) U ′ ( . ) , we then have: (24) Next substituting ∂ U ∂ e k from (22), (24) is reduced to the following condition for the equilibrium consumption e ^k , providing equivalence between a utility-based risk aversion and a value function index of risk aversion: (25) For example, let A (e ^k ) = A be a constant risk aversion and A _v (w ^k ) be the consumer’s value function absolute risk aversion. Then their relationship is reduced to: (26) i.e., a change in agent’s k financial consumption (expenditure) is proportional to the agent’s index of risk aversion of the value function (as defined by the agent’s Hamilton-Jacobi equation). With respect to ∂ e k ∂ w k = R f 1 + R f and A ^k (e ^k ) = A, the value function at a steady state satisfies the ordinary differential equation, (27) whose solution is (28) where C ₁ > 0 and C ₂ > 0 are the integration constants determined from the HJB (15).

Given V ^k satisfying (28), the Nash equilibrium steady-state controls e ^ k , k = 1 , … , K (as well as, states W ^ k ) are found from the system of K equations (22) by accounting for w ^ k R f 1 + R f = e ^ k : (29) or equivalently: (30) Note, if a steady state exists, $\frac{\mathrm{\partial }U}{\mathrm{\partial }{e}^k}>0$ , then ∂ 2 U ( ∂ e k ) 2 < 0 (or equivalently dividing by ∂ U ∂ e k , if − A < 0, which always holds unless the agents are risk-loving). That is, the derivative of the left-hand side of (30) with respect to e ^k is negative, the same holds for the derivative of the right-hand side.

Let K = 2, i.e., equations (30) result in and consider two types of agents: price-setting and price-taking. Unlike price-setting (strategic) consumers and similar to the representative agent defined by the Merton’s model, price takers assume they do not affect the market, ∂ π ∗ ( E ) ∂ e k = 0 . Note, that ∂ π ∗ ( E ) ∂ e k = ∂ π ∗ ( E ) ∂ E ∂ E ∂ e k = ∂ π ∗ ( E ) ∂ E . Therefore ∂ π ∗ ( E ) ∂ e k = 0 implies, given identical conditions, the left hand side of (29) is smaller for a price-taking agent when we consider Cournot pricing, . We thus observe a number of possibilities. If an agent becomes price-taking, then to balance the decreased left-hand side of (30) she needs to reduce e ^k when risk-aversion A is either relatively low or, inversely, relatively high, i.e., . Otherwise, moderate risk aversion will imply increased e ^k by the price taker.

Proposition 3.1.

Let all agents be characterized by a constant risk aversion A. If A is either relatively low or high, given the same wealth and trading strategy, price setting agents spend more and are expected to achieve a greater steady-state wealth than agents that are price taking ( ∂ π ∗ ( E ) ∂ e k = 0 ) . On the other hand, it is vice versa under a moderate risk aversion.

Proposition  3.1 implies that given the same initial wealth and trading strategy, under relatively low/high risk aversion and Cournot pricing property, price-setting consumers would consume less than price-taking consumers at transient states which will enable them to become richer in the long run and enjoy a greater consumption at a steady state. Of course, price-setting agents are normally endowed with much higher wealth. Therefore, they would not need to hold back their expenditures even at the transient states to grow richer. On the other hand, when risk aversion is moderate the price-setting agents will spend more in the transient state and if they are initially rich may still arrive at a greater wealth in the long run than the price-taking agents.

Let again the Cournot price be π = γ + β C . Then C = E − β γ and c k = e k π ∗ ( E ) = e k E − β γ E . Consider a general utility function under the Cournot pricing and a ^k < 1. Then ∂ c k ∂ e k = 1 γ ( 1 − β E + e k β E 2 ) and ∂ 2 c k ( ∂ e k ) 2 = 1 γ ( 2 β E 2 ( 1 − e k E ) ) . Since constraint (10) prevents the consumers bankruptcy, when the number of consumers K and thereby consumption E grow, we readily obtain the following result.

Proposition 3.2.

Let V solve (15) in V ^k . If K tends to infinity, constraint (10) is not binding, (31) and (32)The wealth consumption equilibrium by consumer k, is then: (33) Otherwise, if the chance constraint (10) is binding, an equilibrium consumption e ^k is determined by solving F e k , w k − 1 ( 1 − η ) = w 0 .

Proposition  3.2 implies an equilibrium solution when the market is large. In this case, its Jacobian matrix is simplified and the stability of the steady state defined by (19) is straightforwardly determined by differentiating the expected value of (19) with respect to E[W ^k ] and requiring the result to be negative at its steady state: (34) which holds for W ^k > 0 when . This implies that if (34) holds and , then (E[W ^k ] − E[e ^k ])(1 + 𝜆 + a ^k (R _f −𝜆)) − E[W ^k ] > 0, and therefore, $\frac{dE[W^k]}{dt}>0$ and the expected wealth of this consumer tends to W ^ k . Otherwise if $W^k(0)>{\hat{W}}^k$ then and the expected wealth of this consumer tends to W ^ k , as well. Based on Proposition  3.2, these results are summarized as follows.

Proposition 3.3.

If either $\frac{{{\mathrm{\partial }}^{2}V}^k}{(\mathrm{\partial }{w}^k{)}^2}>0$ and $\frac{\mathrm{\partial }U}{\mathrm{\partial }{c}^k}\frac{\mathrm{\partial }{c}^k}{\mathrm{\partial }{e}^k}-(1+\lambda +a^k(R_f-\lambda ))\frac{\mathrm{\partial }{V}^k}{\mathrm{\partial }{w}^k}>0$ , or and , then an expected steady state wealth, ${\hat{W}}^k>0$ , exists but it is not necessarily stable.

Proposition  3.3 implies that there are four possibilities related to the signs of ∂ U ∂ c k ∂ c k ∂ e k − ( 1 + λ + a k ( R f − λ ) ) ∂ V k ∂ w k and ∂ U ∂ c k ∂ c k ∂ e k . These possibilities determine the evolution of an agent’s wealth and depend on the choice of the Cournot price, 𝜋 = 𝜋(C) and the utility, U (c ^k ). We shall show that all four possibilities occur under the same utility function when the market is large.

Recalling (31), we note that the sufficient optimality condition can be expressed in terms of a consumers risk aversion. Specifically when dividing (31) by − U ′ ( c k ) = − ∂ U ∂ c k and employing the Arrow-Pratt index of absolute risk aversion, we have (35) Therefore, the more an agent is risk averse, the more likely the sufficient condition (31) defining a unique solution of (32) holds. The example below is based on an Increasing Absolute Risk Aversion (IARA) and therefore emphasizes the existence of a solution to agent k consumption expenditure.

Consider a quadratic consumption utility function U ( x ) = x − x 2 2 n , where n is a parameter, $n>max\{1,\frac{1}{n\gamma }\}$ . Then U ( c k ) = 1 γ e k ( 1 − β E ) ( 1 − 1 2 n γ e k ( 1 − β E ) ) , and the interior equilibrium (32) is: (36)Consequently, the dynamic programming (HJB) Eq. (15) is reduced to: (37) When e ^k = 0, it simplifies into (38) whose solution is: (39) where A 1 k , A 2 k are integration constants and Assume next a quadratic form for the value function, V = b 0 + b 1 w + 1 2 b 2 w 2 . Consequently, an interior solution, e ^k , is determined by (36) and the HJB equation transforms into: (40) which is solved by comparing the corresponding terms on both sides of (40). Introducing the notations, (41) a kth consumer with current wealth w ^k = w, a ^k = a, 𝜒 ^k = 𝜒, A 1 k = A 1 , A 2 k = A 2 , m 0 k = m 0 and m 1 k = m 1 , has then a linear wealth consumption expenditure e ^k = e: (42) Note, when m ₁ < 1 and m ₀ < 0, we have e < w, if $w>\frac{-m_0}{1-m_1}>0$ . On the other hand, when − m 0 1 − m 1 < 0 , we can set w ⁰ = 0. Next, e = 0 when $w=\frac{-m_0}{m_1}=w^{\ast }>0$ . Further, the integration constants A ₁, A ₂ in (39) are found by requiring that the value function is continuous at w ^∗ and its derivative is continuous at this point as well: (43) with (44) This ensures the value function is twice differentiable except for a single point of w = w∗ (where its first derivative is continuous). Since Ito’s Lemma is valid for functions with a second derivative that have a finite number of discontinuities, we conclude the following theorem.

Theorem 4.1.

Let constraint (10) be not binding, w ∗ = − m 0 k m 1 k and w ^k = w.

If $m_1^k>0$ and $m_0^k>0$ , then the consumer’s k value function and the equilibrium expenditure are (45)