CC BY
9
Экономико-математические методы и модели
УДК 330.35
Dmitry V. Kovalevsky
OPTIMAL GROWTH PATHS UNDER DIFFERENT UTILITY SPECIFICATIONS,
WITH APPLICATION TO THE STRUCTURAL DYNAMIC ECONOMIC MODEL SDEM-2
1. Introduction
We discuss two different model setups (system-dynamic vs. optimization setup) of the model SDEM-2 (Structural Dynamic Economic Model, version 2), originally proposed in [2, 6]. The «performance» of various actor (in case of SDEM-2 — entrepreneur) control strategies governing the evolution of the economy in the system-dynamic mode is evaluated by computing the utilities on corresponding trajectories in phase space. It is shown that the decision-strategies of entrepreneurs are fundamentally dependent on the structure of the chosen utility functions.
2. SDEM-2: model description
2.1. Model equations (general case). SDEM-2 is a model of a closed economy with exogenous population dynamics. For simplicity, we assume the exponential growth of population with a constant rate XL . The population is divided into two social classes: entrepreneurs and wage-earners. Entrepreneurs comprise a constant fraction 8 of population (0 < 8 < 1), while wage-earners comprise the remainder (1 — 8) of population.
We assume the situation of full employment. The structure of consumption in the economy is twofold. Wage-earners consume everything they earn. We denote the average wages of a wage-earner as w. Entrepreneurs do not earn wages — instead, each entrepreneur consumes the «dividend» d. Then the per capita consumption c in the economy can be presented in the form
c = (1 - 8)w +8d, (1)
The per capita output y in the economy depends on two primary production factors — per capita physical capital k and per capita human capital h. However, unlike in standard economic growth models (cf. [1]), these two forms of capital are assumed to be non-substitutable, and the corresponding production function takes the Leontief form
y = min(vk, |h) (2)
where v, | are constant factors. An in-depth discussion of the background behind Eq. (2) can be found in [3, 4, 6].
In the case vk > |h (abundant physical capital, scarce human capital) there is idle physical capital in the economy. In the contrary case vk < |h (scarce physical capital, abundant human capital), there is unemployment. However, in the major part of the paper we restrict ourselves to consideration of the development path at which there is neither idle physical capital nor unemployment (we refer to it as to as to the balanced growth path):
ybal(t) = vk(t) = |h(t). (3)
We now turn to calculation of dividend and investment. Entrepreneurs own the output y, from which they first have to make a mandatory payment of wages (1 - 8)w to wage-earners. Entrepreneurs are then free to choose the way in which they distribute the remainder between the dividend d, and investment in physical (4) and human (ih) capital. The balance equation is:
y = (1 - 8)w +8d + ik +ih. (4)
Therefore entrepreneurs have two degrees of freedom in their decision-making, and any two of three variables {d, ik , ih} may be regarded as entrepreneur control variables governing the dynamics of the economy.
We note in this respect that in our previous paper devoted to treatment of SDEM-2 in system-dynamic mode [5] we chose another set of variables as control variables, namely, the pair {pd, CTk} quantifying the fractions of net income n = y - (1 - 8)w allocated for dividend d and investment in physical capital ik respectively. However in the current paper we found it more convenient to regard d and ik themselves as control variables.
The dynamics of the economy is governed by a system of three first-order ordinary differential equations:
k = ik - (Xk + XL )k, (5)
h = к — + ^ l )h
w = ^ (qwtarg - w)
(6) (7)
sophisticated parameterizations where q depends on the current level of employment have been proposed in [2, 6]. For further discussion of Eq. (7) we refer the reader to [2—4, 6]. For the particular case of balanced growth wtarg(t) will be calculated below and will appear in Eq. (10).
2.2. Balanced growth. We now consider a particular case of the dynamic system (5)—(7) when the balanced growth, Eq. (3), is maintained at every instant. Striving to maintain the balanced growth, entrepreneurs lose one of two degrees of freedom: if, say, the dividend d(t) is chosen, the other control variable ik(t) (investment in physical capital) is no longer their free choice. Since physical and human capital obey a very simple linear relationship (3), the dynamic equation for human capital can be used for the calculation of the proper value of ik(t) (for a given dividend d(t)) and then eliminated from the dynamic system. The resulting system has two state variables (k and w) and one control variable (dividend d).
We introduce an auxiliary constant
тар
= M-v — ^k - vXh - (Ц + v)xl ,
(8)
Eqs. (5)—(6) have the conventional form of capital dynamics equations, Xk and Xh being the depreciation rates of physical and human capital, respectively.
Eq. (7), the wage dynamic equation, is a dynamic model of wage negotiation process between trade unions of wage-earners and entrepreneurs. As a result of these negotiations, wages w tend to adjust to a constant fraction q (0 < q < 1) of a changing target wage rate wtarg(t) dependent on the current state of the economy.
The target wage rate wtarg(t) is a hypothetical wage rate in the imaginary («target») state of the economy which, as wage-earners perceive, is an «ideal» state of the economy corresponding to the current level of output. This imaginary, «ideal» state is characterized by the following properties («ideal» variables are marked by tildes): a) physical and human capital is maintained at
the constant level (k = 0, h = 0); b) nothing is spent on dividend (d = 0); c) the state of the economy is a stationary balanced state (balanced in a sense of Eq. (3)); d) the imaginary balance is achieved at the current real level of output.
For the sake of simplicity the parameter q in Eq. (7) is assumed to be constant. More
and, after some straightforward algebra, come to a set of two dynamic equations describing the case of balanced growth:
k =
1
ц + v
[та 0k — ц{(1 — 0)w +6J}], (9)
W = Xw
q тар
ц(1 — 0)'
k — w
(10)
2.3. Non-dimensional variables and model constraints. Variables k and {w, d} have different units: if we adopt the method of quantifying the macroeconomic parameters in goods units (as opposed to monetary units) then [k] = good, [w] = [d] = good/year. We introduce new variables with tildes
- (1 — 0) ц
w =-w,
та0
d = d, та0
non-dimensional time
x = Xwt,
(11)
(12)
(13)
and a non-dimensional parameter
та0
У =
и0
(ц + v)
(14)
Note that k, w and d can be regarded as non-dimensional if we agree to assign a non-dimensional value to the unit «good».
For the optimization setup of SDEM-2 (see Sec. 2.4 below) it is also convenient to introduce utilities expressed in non-dimensional variables. In the optimization model setup we assume that the utility function depends on the entrepreneurs' dividend d only, and that entrepreneurs choose the time-dependent path of the dividend (and hence investment) in such a way that the discounted utility is maximized. We will consider the utility functions of two different forms: (i) linear in d, and (ii) logarithmic in d. Therefore the corresponding utilities take the form
w
Ulin =J d (t) exp(-8t )dt, (15)
0
w
Ulog =J ln [d(t)]exp(-8t)dt, (16)
0
where 8 is the discount rate.
To come from Eqs. (15), (16) to non-dimensional utilities, we introduce a non-dimensional discount rate
Д = —
K,
(17)
and make use of Eqs. (12), (13). Straightforward algebra then leads us to the following equations:
Ulin = c0inU,lin,
where
(18)
" * 9- (19)
Ц9
да
Uiin = J ¿(x) exp(^x)dx; (20)
„lin _ та0
and
where
u log = c0og +—ulog,
c0og = -ln 0 8 9ц
(21)
(22)
w
U*log = J ln [d(T)] exp(-Ax)dx. (23)
0
Note that the new utilities with asterisks (U»lin, U,log) are non-dimensional. It is clear from Eqs. (18) and (21) that the problems of maximization of Ulin and U*lin (and, similarly, of Ulog
and Ullog) are equivalent.
From now on, we remove the tildes from variables w and d:
w ^ w, d ^ d.
(24)
After introducing the non-dimensional variables, the dynamic system takes the form
k = y(k - w - d), w = qk - w.
(25)
(26)
We now apply an additional constraint: k > 0, i.e. entrepreneurs always choose the dividend in such a way that the physical capital (and hence the human capital, and output) does not decrease. According to Eq. (25), this is equivalent to the constraint d < k - w. On the other hand, the dividend cannot be negative: d > 0. The capital should also be positive at every instant: k(t) > 0. In view of the constraint k > 0, it is sufficient to impose the latter restriction on the initial condition only: k0 > 0. Similarly, the wage rate w(t) should be positive, w(t) > 0, and, as is easily seen from Eq. (26), it is also sufficient to impose this restriction on the initial condition only: w0 > 0.
Putting the above constraints together, we come to a problem defined by the set of Eqs. (27)—(30):
k = y(k - w - d), w = qk - w, 0 < d < k - w,
0 < w0 < k0.
(27)
(28)
(29)
(30)
But how do the entrepreneurs choose the time path for the control variable d(x)? Below we consider two alternative strategies: a system-dynamic setup and an optimization setup.
0
w
2.4. Closing the model: two model setups. The
system-dynamic model setup implies that entrepreneurs choose d(x) at every instant according to some certain formalized control strategy, the value of a control variable at every instant being determined by the current state of the dynamic system (and possibly by its past). In general, the control strategy may be presented in the form
d(т) = к(т) - w(t).
(33)
For the HN strategy, the capital does not
grow at all and remains constant (k = 0). The
dynamics of k(x), w(x)and d(x)for HN strategy is shown on Fig. 1. One can see that in the long
term both dividend and wages converge to stationary values, and the economy comes to stagnation.
13 6
"a sT лТ
d(t) = D (k(t), w(t)) (31) I
(as adopted in [6]); or, alternatively, in the form
d(T) = D2 (k(t), w(t), d(t)) (32)
(as considered in detail in [4]).
When specified explicitly, Eq. (31) (or, alternatively, Eq. (32)) closes the system (27)—(30). The problem therefore becomes fully defined.
Alternatively, we may proceed in line with the conventional wisdom of neoclassical economic growth theory and assume that entrepreneurs choose the time path of the dividend in such a way that their intertemporal utility is maximized. In this optimization model setup the system (27)—(30) needs to be supplemented with the utility maximization condition, maximizing either Ulin (Eq. (20)) or U*log (Eq. (23)), thereby also yielding a fully defined dynamic optimization problem.
3. Two examples of entrepreneur control strategies
To obtain numeric solutions of the set of model equations, we first ascribe the following values to the model parameters: | = 0,3 year1, v = 0,2 year—1, Xk = Xh = 0,05 year—1, XL = 0,02 year—1, Xw = 0,2 year—1, q = 0,75, 8 = 0,95. We then choose k0 = 1,0 and w0 = 0,57 as non-dimensional initial conditions.
We consider two different control strategies:
1. The «here-and-now» (HN) control strategy: entrepreneurs choose at every instant the maximum possible value of dividend still obeying the constraint k > 0 :
2,0
1,5
1,0
0,5
0,0
20
40 60
Years
80
100
Fig. 1. Physical capital k, wages w and dividend d under the «here-and-now» (HN) entrepreneur control strategy
2. The «moderate dividend growth» (MDG) strategy: entrepreneurs choose the initial value of the dividend equal to d0 = 0,193, then allow the dividend to grow at a constant rate equal to 0,4 % per annum. Explicitly,
d(t) = d0 exp(ax), d0 = 0,193, a = 0,02 (34)
(note that t in Eq. (34) is non-dimensional time). For the MDG strategy, k(T), w(t) and d(T) grow in the long term (Fig. 2).
2,0 „ 1,5
^ 1,0
sT
■ii
0,5 0,0
100
Fig. 2. Physical capital k, wages w and dividend d under the «moderate dividend growth» (MDG) entrepreneur control strategy
0
The «performance» of different control strategies in system-dynamic mode can be compared by computing the associated utilities. It turns out, however, that the result depends — even qualitatively! — on the functional form of the utility. For example, if we choose 8 = 0,02 year1 and compare linear utility functions of the form (20), we find for the two strategies specified above: U¿N* = 2,66, UMDg* = 2,42, U¿N* > UMnDG*. On the other hand, if we choose
8 = 0,01 year1 and compare logarithmic utility functions of the form (23), we find
UHN* =_25,9, uMDG* =_24,9, uHN* < uMDG*.
Thus, depending on the structure of the chosen utility function, one or the other strategy would be «better». The reader should not be confused by negative signs of non-dimensional logarithmic utilities computed above, since dimensional utilities would exert the constant positive shift as in Eq. (21). This analysis also leads us to an interesting counterintuitive conclusion: in the case of the linear utility function, the HN strategy ultimately leading to stagnation would be
valued higher than the MDG strategy leading to persistent growth!
4. Conclusions and outlook
To further explore the dynamic behavior of the economy described by the model SDEM-2, one should examine a number of alternative entrepreneur control strategies and their impact on the asymptotic properties of the solutions. The linear structure of dynamic equations enables a derivation of the optimal solutions in closed form for several different utility functions. An in-depth analysis of the SDEM-2 model in system-dynamic and optimization modes as outlined above will be performed in a forthcoming accompanying paper.
Acknowledgments. The author would like to thank Prof. Klaus Hasselmann for helpful comments. This study was supported by the Nansen Scientific Society (Bergen, Norway, project no. 5/2011), by private donation by Prof. Klaus Hasselmann (project no. 6/2011), and by the Russian Foundation for Basic Research (projects no. 10-06-00369 and 12-06-00381).
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