A type of complete financial market with finite and countable heterogeneous investors, that is, investors equipped with heterogeneous elasticities of intertemporal substitution, heterogeneous time discount rates, and also heterogeneous beliefs, is constructed and two main results are established. First, long-run behaviors, specifically golden rules or modified golden rules, about consumption path and wealth accumulation are investigated under uncertainty and in the sense of uniform topology. Second, inefficacy of temporary taxation policies, which are chosen to be consumption tax and wealth tax, is confirmed in the current financial market.

Our goal in this paper is to explore the golden rule or modified golden rule properties of consumption and wealth-accumulation dynamics, as well as the effects of temporary taxation policies, which are chosen to be consumption tax and wealth tax, in a type of complete financial market with finite and countable heterogeneous investors (see, [

In the past several decades, portfolio turnpikes (see [

Indeed, the current paper confirms the following strong conclusion: both optimal consumption path and optimal wealth dynamics are long-run golden rules in the sense of uniform topology and in the corresponding nonstationary environment, regardless of the fact that there are many heterogeneous investors in the economy. In other words, the uniform topology golden rules demonstrated in the present paper are robust to the types of investors in the market as long as they all exhibit the same type of CRRA preferences. Nonetheless, these golden rules are not turnpikes because they are sensitive to initial conditions of the corresponding dynamics [

The second goal of this paper is to study the effect of taxation policies, which are specifically chosen to be consumption tax and wealth tax, to optimal consumption strategy. As in the literatures of Yano [

In addition, although both this paper and Jin [

Finally, we would like to indicate the differences and also similarity between our investigation and the well-known papers of Sandroni [

The rest of the paper is organized as follows. Section

Suppose that there are

We define the canonical Lebesgue measure

Now, based upon the probability space

The initial conditions

The following linear growth and local Lipschitz continuity conditions are satisfied, respectively,

(i) Provided Assumption

(ii) Assumption

(iii) Here, and throughout the current paper,

The real symmetric matrix

Now, as a preparation for solving individual optimization problem defined in the following section, we, as usual, provide the following formal definition,

We call the control variable

Here, and throughout the current paper, we just consider

We, as usual and without loss of any generality, suppose that the agents exhibit constant relative risk aversion (CRRA) preferences, and the individual optimization problem reads as follows:

Employing the classical technique of dynamic programming, the individual optimization problem defined in (

Provided Assumption

See Appendix

By Proposition

To prove the golden rules, we need the following assumptions.

The initial conditions

There exist constants

The inequality in (

It follows from Assumption

Given the above assumptions, the following lemma is derived.

Given the optimal wealth dynamics defined in (

See Appendix

Moreover, we give the following assumption.

There exist constants

Thus, similar to the proof of Lemma

Given the original wealth dynamics and the optimal wealth dynamics defined in (

Noting that

For any given constants

And for the sake of simplicity, we need the following assumption

There exist constants

Now, based on Lemmas

Provided Assumptions

See Appendix

This theorem is about the asymptotic properties of two wealth processes,

Moreover, based upon Lemma

Provided Assumptions

Notice, by Proposition

In particular, if we are given the following case.

There is a coefficient

Indeed, as corollaries of Theorem

Based upon the assumptions and conclusions of Theorem

See Appendix

If we define

Based upon the assumptions and conclusions of Theorem

See Appendix

We now denote by

Based upon the assumptions and conclusions of Theorem

See Appendix

In (

We do not prespecify any relationship between consumption process

We first introduce the following assumptions.

There is a constant

For

Consumption path

Consequently, we get the following theorem.

Based upon Assumptions

See Appendix

One can easily tell the differences between Corollary

As is well known, theory about golden rule or modified golden rule and different types of turnpike theorems (i.e., Neighborhood Turnpike Theorem, see Yano [

Finally, the current paper can be naturally extended according to the following three lines: first, one can explore uniform topology turnpikes about consumption path and wealth accumulation in financial market, that is, one can search for the conditions under which the uniform topology golden rules demonstrated in the present paper are also uniform topology turnpikes; second, one can study the existence and uniqueness of uniform topology golden rule or turnpike of any given

We do so by first defining the following process:

Applying Itô’s rule to (

The idea of the proof comes from Higham et al. [

By (

Noting that

We define

By using Itô’s rule and (