Remarks on fair wealth accumulation in Russia

Abstract

The paper addresses the topic of wealth accumulation in Russia. This phenomenon plays an important role for the understanding and forecasting the future economic and social development of the country. The “westernized” paradigm calls for hard honest work during the life and approves getting a reward in a form of wealth in the end. When brought to Russia, this paradigm faces the orthodox traditions and rules together with the post-soviet mental patterns. In this paper, we consider how the pattern “first accumulate wealth, then consume it” competes with its opposition, the pattern “first consume wealth, then accumulate it” in Russia. We base our discussion on the consumers’ simple optimization problem, which exhibits a bifurcation between those two patterns depending on the relation between the consumption “impatience” and the wealth growth rate. We also suggest a framework to model the phenomenon of unfair wealth through impulse type of wealth development.

Appendix: simple consumer’s optimization model and its solution

Here, we consider a model of an individual consumer whose fully rational behavior is defined by classical paradigm of optimizing the integrated discounted consumption.

Let W(t) and c(t) ϵ [c _min, c _max] be his wealth and consumption at time t, respectively, and let r be the constant wealth growth rate. Then, the wealth stock accumulates according to, e.g., (Acemoglu 2009),

$$ \frac{{{\rm d}W(t)}}{{{\rm d}t}} = rW\left(t \right) - c(t). $$

(4)

Assume that the individual acts so that to maximize the integrated discounted utility of the form

$$ J = \frac{1}{1 + \alpha}\mathop \int \limits_{0}^{T} e^{- \beta t} c\left(t \right){\rm d}t + \frac{\alpha}{1 + \alpha}e^{- \beta T} W(T), $$

(5)

where the discounting factor β plays a role of the “impatience” parameter, $\alpha \in [0,\infty)$ is a weight parameter, so that consumer prescribes weight of $\frac{\alpha}{1 + \alpha} \in [0,1]$ to the accumulated wealth in the relation weight of $\frac{1}{1 + \alpha} \in [0,1]$ which he prescribes to the integrated discounted utility.

Due to linearity of the problem with respect to c and W, we can measure the wealth in proportion to its initial value setting the latter to 1. Then, c is measured as a share of W. Thus, we consider an optimization problem of the form

$$ {\rm Maximize} \, J\left[{c\left(\cdot \right)} \right] = \frac{1}{1 + \alpha}\mathop \int \limits_{0}^{T} e^{- \beta t} c\left(t \right){\rm d}t + \frac{\alpha}{1 + \alpha}e^{- \beta t} W\left(T \right), s.t. \frac{{{\rm d}W\left(t \right)}}{{{\rm d}t}} = rW\left(t \right) - c\left(t \right), W\left(0 \right) = 1, c\left(t \right) \in [c_{\rm min},c_{\rm max}], t \in \left[{0,T} \right] $$

(6)

Pontryagin maximum principle (Pontryagin et al. 1962) provides a solution to problem (6).

Namely, let ψ be the adjoint variable. The Hamilton–Pontryagin function takes the form

$$ \mathcal{H}\left({W,c,\psi} \right) = \frac{1}{1 + a}e^{- \beta t} c\left(t \right) + \psi \left(t \right)\left({rW\left(t \right) - c\left(t \right)} \right), $$

(7)

where the adjoint equation together with the transversality condition becomes

$$ {\dot{\psi}}(t) = - r\psi (t), \psi (T) = {\frac{\alpha}{1 + a}}e^{- \beta T}. $$

(8)

The maximum condition calls for $c\left(t \right) = c_{\rm max},$ if $\psi \left(t \right) < \frac{1}{1 + a}e^{- \beta t}, c\left(t \right) = c_{\rm min},$, if $\psi \left(t \right) > \frac{1}{1 + a}e^{- \beta t},$ and in case $\psi \left(t \right) \equiv \frac{1}{1 + a}e^{- \beta t}$, consumption c(t) is singular.

Comparison of ψ(t) and $\frac{1}{1 + a}e^{- \beta t}$ reveals the following solution patterns.

• 1. If r > β and α < 1, for $t \in \left[{0, \tau} \right]$ consumption becomes $c^{*} \left(t \right) = c_{\rm min}$ and wealth becomes $W^{*} \left(t \right) = \left({1 - \frac{{c_{\rm min}}}{r}} \right)e^{rt} + \frac{{c_{\rm min}}}{r},$ and for $t \in \left[{\tau,T} \right]$ consumption becomes $c^{*} \left(t \right) = c_{\rm max}$ and wealth becomes $W^{*} \left(t \right) = \frac{{c_{\rm max}}}{r} - \frac{{e^{rt} \left({\frac{{- c_{\rm min} - re^{r\tau} + c_{\rm min} e^{r\tau}}}{r} + \frac{{c_{\rm max}}}{r}} \right)}}{{e^{r\tau}}}$; where $\tau = T + \frac{ln\alpha}{r - \beta}$ is switching time.

• 2. If r > β and α > 1, consumption becomes $c^{*} \left(t \right) \equiv c_{\rm min}$ for all $t \in [0,T]$, and wealth becomes $W^{*} \left(t \right) = \left({1 - \frac{{c_{\rm min}}}{r}} \right)e^{rt} + \frac{{c_{\rm min}}}{r}$ for $t \in [0,T]$.

• 3. If r < β and α > 1, for $t \in \left[{0, \tau} \right]$ consumption becomes $c^{*} \left(t \right) = c_{\rm max}$ and wealth becomes $W^{*} \left(t \right) = \left({1 - \frac{{c_{\rm max}}}{r}} \right)e^{rt} + \frac{{c_{\rm max}}}{r}$, and for $t \in \left[{\tau,T} \right]$ consumption becomes $c^{*} \left(t \right) = c_{\rm min}$, and wealth becomes $W^{*} \left(t \right) = \frac{{c_{\rm min}}}{r} - \frac{{e^{rt} \left({\frac{{- c_{\rm max} - re^{r\tau} + c_{\rm max} e^{r\tau}}}{r} + \frac{{c_{\rm min}}}{r}} \right)}}{{e^{r\tau}}}; \tau = T + \frac{ln\alpha}{r - \beta}$ is switching time.

• 4. If r < β and α > 1 consumption becomes $c^{ *} \left(t \right) \equiv c_{\rm max}$ for all $t \in [0,T]$, and wealth becomes $W^{ *} \left(t \right) = \left({1 - \frac{{c_{\rm max}}}{r}} \right)e^{rt} + \frac{{c_{\rm max}}}{r}$ for $t \in [0,T]$.

Note, that parameter a indicates the importance of the terminal wealth value and determines whether switching occurs during the planning period [0, T] (see Table 3).

Table 3 Dependence of switching time in accordance with a and relationship between r and β