Two Types of Innovation and Their Economic Impacts: A General Equilibrium Simulation

Abstract

In the traditional two-sector growth model, we show that the real wage rate and the rate of profit converge to positive values when the “constant returns to scale” is assumed. When the “decreasing returns to scale” is assumed, however, the real wage rate converges to zero. Thus, we examine how the trajectories are modified by the creation of a third sector, under the “decreasing returns to scale”. First, we examine the downstream innovation: i.e. the third sector produces a new luxury. This innovation is temporarily effective since it raises the average rate of profit, while the rate converges to the same positive value as in the basic model. Next, we introduce the third sector which produces a new energy: the upstream innovation. This innovation is temporarily effective in raising the real wage rate and the rate of profit so long as it takes place in the early stage. These rates, however, converge to zero. Although the effect on the rate of profit in the downstream innovation is greater than the upstream innovation, it is because the total investment in the latter is greater than the former. Thus, we conclude that the upstream innovation has stronger economic impact.

Appendix

The variation of the real wage and the rate of profit was also one of the main concerns of the classical economists, starting from Smith (1776), Malthus (1798), Ricardo (1817) to Marx (1867). They examined this variation in terms of labor theory of value. Pasinetti (1960) presented a mathematical formulation of Ricardo model, which was extended to Ricardo-Marx model by Fukiharu (1987) in the framework of the continuous type capital accumulation process. In what follows, the relation between Fukiharu (1987) and the Uzawa-type basic model of the present paper is examined in the framework of the discrete type capital accumulation process.

The Ricardo-Marx capital accumulation process in Fukiharu (1987) in the continuous version may be reformulated as in what follows utilizing the same variables and functions as in the main text.

$$ \frac{f_1\left[{N}_1\left(t+1\right),{C}_1(t)\right]}{\overline{x}}={N}_1\left(t+1\right)+{N}_2\left(t+1\right) $$

(22)

$$ {f}_2\left[{N}_2\left(t+1\right),{C}_2(t)\right]={M}_1\left(t+1\right)+{M}_2\left(t+1\right) $$

(23)

$$ \overline{x}\ {N}_2\left(t+1\right)=p\left(t+1\right){M}_1\left(t+1\right) $$

(24)

$$ p\left(t+1\right)=\frac{\ \frac{\partial }{\partial {N}_1}{f}_1\left[{N}_1\left(t+1\right),{C}_1(t)\right]}{\frac{\partial }{\partial {N}_2}\ {f}_2\left[{N}_2\left(t+1\right),{C}_2\ (t)\right]} $$

(25)

$$ r\left(t+1\right)=\frac{f_1\left[{N}_1\left(t+1\right),{C}_1\ (t)\right]-\overset{\_}{x}{N}_1\left(t+1\right)}{C_1(t)p\left(t+1\right)+\overset{\_}{x}{N}_1\left(t+1\right)}=\frac{p\left(t+1\right){f}_2\left[{N}_2\left(t+1\right),{C}_2(t)\right]-\overset{\_}{x}{N}_2\left(t+1\right)}{\ {C}_2(t)p\left(t+1\right)+\overset{\_}{x}{N}_2\left(t+1\right)} $$

(26)

A few remarks are in order. One parameter, $\overline{x}$: fixed real wage rate in terms of consumption good, and one variable, p(t + 1): relative price of capital good in terms of consumption good, p ₂/p ₁, are newly introduced. The real wage rate is fixed at the subsistence level. The classical economists did not assume the full employment. The Eq. (24) implies the exchange between the first sector and the second sector. Noting that dN _i/dy _i = 1/$\frac{\partial }{\partial {N}_i}{f}_i\left[{N}_i\left(t+1\right),{C}_i(t)\right]$ (i = 1, 2), the Eq. (25) implies the “marginal” labor theory of value, which was also assumed in Pasinetti (1960, p. 83). Ricardo appears to have accepted this type of labor theory of value (Negishi 1981). The Eq. (26) implies that the rate of profit, r, is equalized between the two sectors.

Under Eqs. (22)–(26), the capital accumulation process, C _i(t + 1) = C _i(t)(1 − g ₁) + M _i(t + 1) (i = 1, 2) is examined (t = 1, 2 …). Formulating these relations in terms of differential equations, Fukiharu (1987) asserted that the rate of profit on this capital accumulation process is constant when the “constant returns to scale” is assumed.

We have the following simulation result (Fukiharu 2017d):

• 1. Under Eq. (3) when $\overline{x}$= 1, C ₁(1) = 100, C ₂(1) = 200, and g ₁ = 0.05 are assumed, we have r(t) = 0.2 (t = 1, 2, …).

• 2. Under Eq. (3) when $\overline{x}$ = 1, C ₁(1) = 10,000, C ₂(1) = 10,000, and g ₁ = 0.05 are assumed, we have r(t) = 0.2 (t = 1, 2, …).

• 3. Under Eq. (9) when $\overline{x}$ = 1, C ₁(1) = 100, C ₂(1) = 200, and g ₁ = 0.05 are assumed, we have r(t) (t = 1, 2 …) as the dashed curve in Fig. 12.

• 4. Under Eq. (9) when $\overline{x}$= 1, C ₁(1) = 10,000, C ₂(1) = 10,000, and g ₁ = 0.05 are assumed, we have r(t) (t = 1, 2 …) as the solid curve in Fig. 12.

Fig. 12

The trajectories of the rate of profit under decreasing returns to scale with capital depreciation. Source: Fukiharu (2017d)

Considering Marx (1867), it may be strange to have the rising rate of profit on the capital accumulation process. It must be noted that these results stem from the large depreciation of capital: g ₁ = 0.05. In this capital accumulation process, in fact, “capital deccumulation” emerges due to the capital depreciation: i.e. C _i(t) (t = 1, 2 …) decreases monotonically. Indeed, we have the following:

• 5. Under Eq. (9) when $\overline{x}$= 1, C ₁(1) = 100, C ₂(1) = 200, and g ₁ = 0 are assumed, we have r(t) (t = 1, 2, …) as the dashed curve in Fig. 13.

• 6. Under Eq. (9) when $\overline{x}$= 1, C ₁(1) = 10,000, C ₂(1) = 10,000, and g ₁ = 0 are assumed, we have r(t) (t = 1, 2 …) as the solid curve in Fig. 13.

Fig. 13

The trajectories of the rate of profit under decreasing returns to scale without capital depreciation. Source: Fukiharu (2017d)

Indeed, in these cases, capital accumulation takes place: i.e. C _i(t) (t = 1, 2 …) increases mototonically.

Thus, we may conclude that in the framework of classical economics, under “decreasing returns to scale”, the rate of profit declines on the capital accumulation while it remains constant under “constant returns to scale”.