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This paper discusses the theoretical validity of Thomas Piketty’s fundamental laws about income distribution in the context of a standard neoclassical growth model. We take Uzawa’s two-sector growth model as the platform of our analysis, as it allows us to make a distinction between the technological elasticity of factor substitution of the production function and the aggregate distributive elasticity of substitution. We examine the properties of the non-steady growth path through both analytical and numerical investigations. We conclude that some of the numerical simulations corroborate Piketty’s theory without assuming that the economy is on a steady growth path. However, if the elasticities of factor substitution in the individual sectors are less than one as many empirical studies show, then the economy approaches the state where all products are completely distributed to workers. This contradicts Piketty’s diagnosis about the current distributional inequality. In addition, the aggregate income distribution is stable for a relatively long time, and differences in the initial conditions are preserved during this period. This means that the comparative statics of the steady states might not present an adequate description of the economy’s behavior in a period of time that is practical. Our final evaluation of Piketty’s proposition is that it is better understood as a theory inferred from historical data and not one necessarily deduced from standard neoclassical growth theory.

This paper uses Uzawa’s two-sector model^{1} with non-neutral technological progress to examine Thomas Piketty’s fundamental laws of capitalism proposed in his book Capital in the Twenty-First Century^{2}. His argument derived from the laws is that economic inequality will be accelerated when the growth rate of income decreases and the capital/output ratio increases. Although Piketty’s theory encompasses not only functional income distribution but also distribution of wealth and both of them are inseparably interwoven to explain the current distributional inequality in his book, we will restrict our scope of the argument to the long-run trend of functional distribution of income. Aggregate distribution of income between wages and profits is closely related to the macroeconomic investment―savings balance, and we will exclusively focus on this distributional mechanism. Given the present unequal distribution of wealth, however, “anything that increases between-inequality ... is very likely to increase overall inequality’’^{3}.

We use a neoclassical model because Piketty himself employs some terms of neoclassical theory, such as the steady state, the elasticity of factor substitution and the production function, to demonstrate that economic inequalities do arise even in the framework of a standard neoclassical growth model. In fact, his second fundamental law of capitalism can be directly deduced from the Solow-type growth model^{4}. We employ the two-sector model, because Piketty believes “the right model to think about rising capital-income ratios and capital shares in recent decades is a multi-sector model of capital accumulation, with substantial movements in relative prices, and with important variations in bargaining power over time’’^{5}.

Although Piketty’s “sector’’ in this context encompasses a much wider variety of sectors, such as labor unions, than that of the neoclassical two-sector model, Uzawa’s formulation is, at least from a theoretical point of view, a natural extension toward a more general approach, because Piketty thinks that the difference in capital intensities among industries is important to understand the behavior of the aggregate capital/output ratio^{6}. The two-sector model has only one kind of capital goods; however, the behaviors of the capital/output ratio and of the aggregate profit share, both of which constitute Piketty’s fundamental laws, are determined by technological conditions in the two sectors.

The two-sector model also enables us to make a distinction between the technological elasticities of factor substitution of the production function in the individual sectors and the aggregate distributive elasticities of substitution. These two kinds of elasticities are often confused with one another. The latter is simply another expression of the behavior of the aggregate distributive shares and is not determined solely by the technological elasticities of individual industries. Therefore, we believe that this is a necessary step toward a more general discussion of the functional distribution of aggregate income.

As well known, the one-sector version of the neoclassical growth model has the steady state, or the balanced growth path, if Inada’s derivative condition is satisfied, and the equilibrium is stable under the ordinary set of assumptions. However, if we introduce technological changes into the model, then the only type of technological progress that generally assures the existence of the steady state is Harrod neutral, unless the production function is a Cobb―Douglas type. Although we have abundant literature on induced technological progress, beginning with [^{7}. In addition, non-neutral technological progress has literally non-neutral effects on income distribution. Given these reasons, we need to analyze the economy with non-neutral technological progress and on a non-steady growth path. In this paper, we analyze the dynamic behavior of the economy under the assumption of non-neutral technological progress to verify Piketty’s theory for more general states of the economy that have no inner equilibrium^{8}. Lacking any comparable equilibrium state within the finite space, we have to examine the dynamic process itself directly.

Using mainly comparative statics in his book, Piketty has to assume the steady growth rate of income^{9}.

Since we mainly focus on the state that has no inner equilibrium in the following argument, we have to investigate the properties of the dynamic trajectories directly. In the standard approach to analyze the dynamical property of the neoclassical growth models, we reduce all equations of the model to a single dynamic equation of the capital/labor ratio,

Instead, we reorganize the model by using the wage shares of each sector, because they stay in the closed interval of

We make Uzawa’s two-sector growth model with non-neutral technological progress the starting point of our analysis. The asymptotic trajectories of the neoclassical one- sector growth model with non-neutral technological progress are discussed in [^{10}. As for the two-sector version of the asymptotic approach, only [^{11}.

Since Uzawa’s model assumes the classical savings function, it is easy to manipulate the equations. However, there is more to this assumption; it allows us to treat the savings rate as an endogenous variable^{12}. If the savings rate from profits is larger than that from wages, the economy must distribute more income to capital owners to generate more savings when capital accumulation accelerates. This is the fundamental mechanism of macroeconomic income distribution^{13}. We believe that any dynamic model of aggregate income distribution must incorporate this property of the economy.

Our model consists of 11 variables and five parameters as given below.

(Subscript 1 denotes the investment goods sector, and subscript 2 denotes the consumption goods sector. Parameters are marked with an asterisk in the above list. We assume all the parameters take a non-negative value.)

We assume that all markets in the economy are perfectly competitive and both the rate of profit and the real wage rate are completely arbitrated between the sectors. There is no idle capacity and no unemployed labor force in the economy.

^{14}.

According to the marginal productivity theory,

Full utilization of capital equipment and full employment of the labor force are expressed as follows.

Since we assume the classical savings function, all profits are saved and all wages are consumed.

In the above two equations in (6), if either the left or the right one holds, the other is automatically satisfied, because the model follows Walras’ law, which suggests that the existence of excess supply in one market must be matched by excess demand in another so that it balances out. All products produced in the investment goods sector are devoted to capital formation of the economy, and the labor force grows at the rate of

and

Since the 11 equations given above contain 11 variables in total, our model is mathematically complete.

In this section, we derive conditions that satisfy Kaldor’s stylized facts before analyzing the case of non-steady growth. The stylized facts basically consist of the following four conditions.

a. Constant rate of profit.

b. Constant capital/output ratio.

c. Capital grows faster than the labor force.

d. Constant growth rate of labor productivity.

To determine the conditions under which the Kaldorian steady state exists, we integrate all the equations into the following three differential equations^{15}.

and

where

^{16}

where

factor substitution is generally an endogenous variable. However, we have to specify the sign of the third derivative of the production function to define its dynamic behavior, but that seems to have no practical meaning in terms of economic theory. Therefore, we assume here that the elasticity of factor substitution in each sector is constant over time. This means that we implicitly assume a CES type production functions. If we consider

For our model to have a growth path corresponding to Kaldor’s stylized fact, the value of (11) should be zero. Then, since

Since the right-hand side of this equation is a function of

Therefore, for the steady growth path to exist, the following three conditions should be satisfied:

and

under the following constraints for the solution to be an inner equilibrium.

It is evident from (17) that either

(and)

Substituting

Therefore, for

Following the same process as in the above case, we have

If either of these two conditions is satisfied, then, from (19), the growth rate becomes

As for the case of

Now, we have the following four cases for the existence of a steady-growth path. The first two conditions were derived by [

Case a:

Case b:

Case c:

Case d:

Next, we look into the dynamic behavior of

For Case a:

For Case b:

For Case c:

For Case d:

These four differential equations contain either

Since the denominator of this equation is obviously positive, we look into the sign of the numerator.

Therefore, if either of the following two conditions,

are satisfied,

The behavior of the capital/output ratio is determined by the following equation, where

From the above equation, it is evident that

Inputting the above four conditions of cases a, b, c, and d into (32), respectively, its value becomes zero for all cases, and that means

growth path.

Since

Therefore, it grows at the rate of

For cases a and b,

If one of these four conditions is satisfied, we have a growth path corresponding to Kaldor’s stylized facts. However, in actual economies, there seems to be no need for every type of technological progress and the values of elasticity of factor substitution to satisfy always these strict conditions. Next, we investigate what behavior an economy exhibits in the case of non-steady growth.

As evident from the forms of Equations ((9) and (10)), there are two additional cases of corner equilibria, even if the system has no inner equilibrium. If

The dynamic system for the two singular points is given by the following equations respectively.

Therefore, if a positive equilibrium exists, it is stable in the vicinity of

Case 1.

Let

Therefore, we assume the parameters in (34) take values that satisfy

The Jacobian of the system given by (9), (10), and (11) evaluated at

It is evident from the array of elements in this matrix that the trajectories of

The characteristic roots are as follows.

Therefore, when

Proposition 1.

If

The two broken lines in

Let us consider the behavior of other variables in the vicinity. Let

Therefore,

This means that

because the wage share is fixed at one at the equilibrium. Therefore,

and we see that the capital/output ratio decreases. Since ^{17}.

Case 2.

The stable equilibrium value of

The characteristic equation of (42) is

Therefore, the characteristic roots are

From the above conditions, we see that

In the final state of the economy that the trajectories approach toward,

Proposition 2.

If

Case 3. The other unstable cases

In the above cases, we examined only stable equilibria, but there also exist unstable trajectories that have no convergence point. For these cases, since any dynamic path of ^{18}, when

From the above three cases, we have the following proposition.

Proposition 3.

If the system has no inner equilibrium, and

We convert the differential system given by (9), (10), and (11) to the corresponding difference system by the fourth-order Runge-Kutta method with

dynamic behavior of the system when it is not on the steady growth path^{19}. As for the initial values of^{20}. To determine the initial equilibrium conditions, we assume that the CES production functions with constant returns to scale at ^{21}.

where

The two variables that are given the time derivatives explicitly in the original system, ^{22}. The causality condition implies equilibrium in the factor markets and the profit maximization of firms. The other variables are then determined at their equilibrium values by the following three equations. The last one assures equilibrium in the goods markets.

Therefore, there are five parameters,^{23}. We set ^{24}. We also suppose that

We set the growth rate of the labor force at 1.0%, which is presumably close to the average annual rate of population growth in the majority of developed countries. For the parameters of technological progress, we assume, according to the finding of Pol Antràs, that the rates of labor-augmenting technological progress of both sectors exceed the rates of capital-augmenting technological progress^{25}. We also assume

Case 1.

The results for the main variables are summarized in Panel 1(a), where “time’’ represents ^{26}. It is also interesting that the wage share increases while the capital/output ratio (the inverse of^{27}. This is caused by the relatively high initial value of

Piketty’s argument includes the comparative statics of the steady state in addition to analyses of the dynamic process. Since our system has no inner steady state, we have to rely on the comparative dynamics, which compares one particular trajectory with some other trajectories. Panel 1(b) shows the result of our comparative dynamics. We assume two economies, both of which share the same set of parameters except for the growth rate of the labor force

In this case, ^{28}. A smaller value of

Panel 1(c) is the outcome from a comparative dynamics with two different initial values of^{29}. Because of capital deepening, the marginal productivity of capital is low,

(a)

(b)

(c)

Panel 1. (a) Simulated Trajectories toward

whereas that of labor is high, and this causes a high wage share. The other economy exhibits typical properties of developing countries. Capital is accumulating much faster than that of the other economy, and therefore, the profit rate is high, whereas the wage share is low to generate enough savings to finance such a faster capital accumulation. Therefore, this kind of comparative dynamics must be interpreted as a comparison between two different economies, and it is better not to take the result as what could be observed in a single economy. In any case, the result of this case also contradicts Piketty’s theory.

^{30}. The trajectories cover a wide range of the domain, and the system exhibits its global stability. All of the trajectories converge to

Although the final state is unrealistic, it is possible to suppose that the economy would approach such a state over hundreds of years. Therefore, it is arguable that such

an economy shows a steady upward trend of the wage share in the long run, and if this is the case, it is difficult to explain the recent incorrigible declines of the labor share in many countries as long-run phenomena. Rather, these declines should be considered as an ephemeral transit phase in terms of the economic process. Therefore, some institutional explanation, such as retrenchment in a welfare state and/or globalization of the economies^{31}, might offer a better explanation.

Case 2.

Panel 2(a) is a typical outcome of the simulation with the same parameter values and initial conditions as case 1 barring the elasticity of factor substitution of the consumption goods sector: ^{32}.

In our simulation, the economy must constantly decrease the rate of capital accumulation. Therefore, a temporary excess supply of investment goods always appears and the price of investment goods falls so that the market equilibrium is maintained. This

(a)

(b)

Panel 2. (a) Simulated Trajectories toward

decline of the price of investment goods means a rise in the real wage rate in terms of investment goods. Consequently, the firms of this sector reduce the number of workers and replace some of them with capital. Since the elasticity of factor substitution of this sector is less than one, the production of investment goods decreases, and the wage share in this sector increases. However, since the elasticity of factor substitution in the consumption goods sector is greater than one, the wage share of this sector decreases as firms employ more workers. The total effect of this process is a decline in the aggregate wage share.

The wage share decreases toward a fixed value―0.5 in this case―which implies that the aggregate elasticity of factor substitution tends to be nearly one in the long-run. In other words, the Cobb―Douglas function is an appropriate form of the aggregate production function in the long run, whereas each sector has a CES-type function^{33}. This holds good for any cases with a stable equilibrium.

In the two charts of Panel 2(a),

Let us look into the results of the comparative dynamics for this case, which is summarized in Panel 2(b). We set

If we increase the value of ^{34}.

tories start from different initial values. The trajectories cover a wide range of the domain, and the system exhibits its globally stable property in this case as well. All of the trajectories converge on the lower-right corner equilibrium. The elasticity of the investment goods sector is less than one, and the wage share of this sector increases constantly over time, whereas the elasticity of substitution of the consumption goods sector is greater than one and the wage share of this sector constantly decreases.

It should be noted here that the aggregate income distribution is remarkably stable for a considerable length of time as we have seen in the first case above. The wage share needs about 200 years in case 2 to decrease from 0.737 to 0.697 as shown in the upper-right chart of Panel 2(a)^{35}. It is less than the ^{36}. Therefore, it can be argued that the widely recognized stability of the functional distribution is theoretically observable even when the economy is not on the steady growth path.

Case 3. The saddle point

Panel 3(a) summarizes a simulated result for another set of parameters, where all parameter values are the same as in case 1, except for the rate of labor-augmenting technological progress in both sectors to analyze a saddle case:

As indicated in the panel, the aggregate elasticity of factor substitution, ^{37}. However, it should converge to one eventually, because the wage share approaches the fixed value. This is because the aggregate elasticity of factor substitution is just another expression of the behavior of the aggregate income shares, and it is meaningless to say that either one of them causes the other^{38}.

As shown in the lower-right chart of Panel 3(b), the capital/output ratio rises steadily. However, the wage share rises until around the 2510th period. This is caused by rapid decreases in the rate of profit, because the initial rate of capital growth of 0.1 is much higher than the steady state level of 0.02, and the economy must distribute more income to workers to lower the rate of capital growth. The volume effect does not outweigh the price effect in this situation.

This result contradicts Piketty’s inference^{39}. There can be a case where his proposition derived from the assumption of the steady state economy contradicts the dynamic behavior of the economy that is not on the steady growth path, especially when its growth rate slows down. If such a situation continues for a considerable length of time in the real world, like more than 400 years in this case, his theory loses explanatory power. After this early stage, the trajectories in Panel 3(a) are consistent with Piketty’s theory, although that can happen only in the very distant future^{40}.

As for the comparative dynamics of this case, the trajectory of the wage share with the lower growth rate of labor force is consistently higher than that of the other case with the faster growth rate as shown in the upper-left chart of the Panel 3(b). This is because the elasticities of both sectors are lesser than one and a faster rate of increase in the ratio of factor prices are favorable to workers. The result of the comparative dynamics for the case 2 also contradicts Piketty’s theory.

^{41} that depicts the trajec-

tories approach

Case 4. The inner equilibrium

Panel 4 shows the case where the system has an inner equilibrium so it can be compared with the other three cases. In this case, we assume

As observed in the upper chart of the panel, it takes a long time for the system to reach the vicinity of the inner equilibrium. For example, starting from 0.516, the wage share needs about 850 years to increase by 0.1 points and reach 0.616. During such a long period, other institutional and/or political factors are far more important to explain the trend of income distribution than the purely economic process.

same production functions. We set

We set four different values on

These results of our simulation are summarized in the table below, and Piketty’s argument appears to be verified in some cases. However, since the results of numerical simulations are generally affected by the relative magnitudes of the parameters, we should not derive any decisive conclusion from our limited numerical experiments. Most importantly, the location of the initial state relative to the equilibrium point is crucial, especially in the case of the saddle point.

The remarkable stability of aggregate income distribution over time was distinctly observed in our simulations. This is true, even when the corner equilibrium is a saddle point or when the system has no inner equilibrium. This stability is mainly brought about by the mechanism of factor substitution. If the capital/output ratio stays at the same level when the growth rate and the profit rate fall, then the profit share decreases. However, a decrease in the profit rate causes the ratio of factor prices to be increased. This causes more of the labor force to be replaced with capital and the capital/output ratio to increase. Therefore, the effect of a decrease in the profit rate on income distribution is partly offset by the increase in the capital/output ratio. In the case where the aggregate elasticity of factor substitution is one, this mechanism works perfectly, and the aggregate income distribution naturally stays at the same level. The sectoral composition of outputs also contributes to the stability of income distribution through an adjustment of relative prices in the case of multi-sector models.

Therefore, we may argue that it is difficult to explain a major change in the trend of income distribution as a purely economic process. Rather, the standard neoclassical theory verifies the robust stability of income distribution regardless of the existence or non-existence of the inner steady state. This result also suggests that the stylized stability of income distribution can be explained without assuming a Cobb―Douglas type production function and/or Harrod-neutral technological progress even in the long run. In contrast, external shocks that reset the initial conditions, such a change in the tax regime, might have far more important effects on the trend of income distribution^{42}.

The most crucial point of our analysis in this paper is to determine the case that offers the most appropriate description of the real economy. As Robert Rowthorn notes in his

Process Analyses | Comparative Dynamics | ||
---|---|---|---|

Case 1 | Early Stage | negative | negative |

Latter Stage | negative | ||

Case 2 | Early Stage | affirmative | affirmative |

Latter Stage | affirmative | ||

Case 3 | Early Stage | negative | negative |

Latter Stage | affirmative |

critical paper on Piketty’s work ( [

Therefore, if the recent incorrigible declines of wage shares in many countries should not be considered as short-run phenomena, we should pay more attention to institutional and/or political aspects of the problem than to the technological factors. In this regard, Piketty’s theory is better understood as a theory based on historical data and not one deducible from standard neoclassical growth theory, and his second fundamental law, which plays a crucially important role in his theoretical explanation, can be taken as a “bridge’’ that we, economists, cross for historical and socioeconomic studies on the subject. In this sense, the conclusion of the present paper endorses his statement that “The history of the distribution of wealth has always been deeply political, and it cannot be reduced to purely economic mechanism’’ ( [

However, our results are obtained solely by using Uzawa’s two-sector model with the classical savings function as well as by using limited numerical simulations. Further investigations with a more general framework and numerical simulations with various settings of the parameters are indispensable to confirm our conclusions.

The author expresses his grateful appreciation to Takehiro Nagaoka for helpful suggestions on the numerical simulation in the present paper.

Morita, M. (2016) Non-Neutral Technological Progress and In- come Distribution―Piketty’s Fundamental Laws in a Neoclassical Two-Sector Model. Theoretical Economics Letters, 6, 1267-1298. http://dx.doi.org/10.4236/tel.2016.66119

Since we assume the production functions are homogenous of degree one,

where,

From (1)-(5), and (A.1),

where

Differentiating this equation by time,

where

Next, differentiating (A.4) by

Since

Substituting (A.6) into (A.8), we have

Differentiating the wage shares by

Substituting (A.6) and (A.9) into this equation, we have the next equation as follows.

Substituting

Next, differentiating

Substituting (15) and

into (A.12), we have,

Substituting (A.6) and (A.9) into (A.13), we have

Substituting

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