Growth and Income Distribution Under Induced Innovation

Abstract

To explain the long-run fluctuations of growth and income distribution consistent with the data shown in Piketty (Capital in the twenty-first century. Belknap Press of Harvard University Press, Cambridge, 2014), two types of technological progress, labor-saving and capital-saving, must alternately emerge. With a modification of induced innovation theory, this chapter first analyzes the firm’s choice of type of technological progress based on the present value maximization criterion. Incorporating endogenous type choice into the standard neoclassical growth model, we next examine the long-run dynamics of the macroeconomy. Assuming that the innovation possibility frontier, the key concept in induced innovation theory, shifts up or down depending on the rate of capital accumulation, we modify the model so that it is capable of explaining the aforementioned alternating emergence. Finally, we derive the socially optimal type of innovation to compare it with the market outcome. As a result, it is shown that the innovation attained by the market is not socially optimal. This implies that policies in the capital markets such as taxation and possibly regulation are necessary.

Mathematical Appendix

In this mathematical appendix, we prove the following proposition, which is relevant to the firm’s maximization in a decision about the direction of technological progress.

Proposition A

Given the time paths of $I_{t}$, $K_{t}$ and $\tilde{k}_{t}$, the firm’s maximand (2.27) with respect to the direction of technological progress is reduced to $\dot{M} \equiv dM/dt$ by approximation.

Proof

Since the time path of $I_{\tau }$, and $K_{\tau }$ for $[t,t + h]$ is assumed to be given, maximizing (2.22) is equivalent to maximizing

$$ \tilde{V} = \int\limits_{t}^{t + h} {M_{\tau } K_{\tau } e^{ - r(\tau - t)} d\tau } . $$

(2A.1)

According to the trapezoidal rule, this maximand is approximated by

$$ \tilde{V} = \frac{{(M_{t} K_{t} + M_{t + h} K_{t + h} e^{ - rh} )h}}{2}. $$

(2A.2)

When the firm makes a decision about the direction of technological progress for the time interval $[t,t + h]$ at time t, the initial value $M_{t} K_{t}$ and the terminal stock $K_{t + h} e^{ - rh}$ are given by assumption. Therefore, maximizing (2A.2) is equivalent to maximizing $M_{t + h}$. But, when h is sufficiently small, the first order approximation of $M_{t + h}$ is given by

$$ M_{t + h} = M_{t} + h\dot{M}_{t} ,\quad where\quad \dot{M}_{t} = \frac{dM}{dt}. $$

(2A.3)

Since $M_{t}$ and h are given, maximizing $M_{t + h}$ is reduced to maximizing $\dot{M}_{t}$. This completes the proof.