“New“ Growth Theory and Knowledge Externalities in Capital Accumulation

Abstract

This chapter probes into the pioneering approach of the so-called “new” growth theory, i.e. Romer’s (Journal of Political Economy, 94, 1002–1037, 1986) knowledge externalities in private capital accumulation. After listing the empirical and theoretical shortcomings of the “old” growth theory, the main approaches of the new growth theory are briefly outlined. In Sect. 5.3, knowledge externalities associated with private capital accumulation are introduced into our basic OLG model and the fundamental equation of motion is then derived from the FOCs for utility and profit maximization, and under market clearing. In the subsequent section the deficiencies of the old growth theory are reconsidered from the perspective of the knowledge externalities of new growth theory. In Sect. 5.5, public debt is introduced in our new growth model and the effects of variation in the politically fixed net deficit ratio on capital and public debt are investigated. Finally, it is shown that stochastic shocks to total factor productivity in the CD production function, together with investment adjustment costs, can in fact generate GDP time-series which resemble empirical evidence.

Appendix

1.1 Stochastic Growth Theory and Stochastic Processes

Chapters 1, 2, 3, and 4 tried to explain the empirical macroeconomic dynamics of deterministic general equilibrium models. In deterministic models, all relevant variables are fully explained by the model equations and external influences (i.e. weather, political revolutions etc.) are assumed to be negligible.

The production function in Eq. 5.30 adopts a different approach. The function contains a stochastic (= random) variableS that stands for all those influences, which are not included in a deterministic production function, but could nevertheless influence production in one way or another. For example, in agricultural production, the stochastic variable could be used to capture the expected impact of weather on the harvest rate, or in industrial production, the stochastic variable may represent the impact of an unexpected change in technology on production output.

The stochastic variable can be fully described by its first two moments; the expected value (= 1) and its variance. Both are time-independent and constant (= stationary).

Production function (5.30) is a stochastic version of the (deterministic) production function (5.1): The stochastic variable alters the production output, after the producers have decided on labor and capital inputs. Since the expected value of the stochastic variable is constant over time and equal to unity, no systematic factors which might influence the production output (and the production function), are neglected. It is an unbiased estimator, i.e. the probability of a realization being larger than one is equal to the probability of it being smaller than one.

In empirical studies (especially in the econometric literature) a number of some specifications for stochastic processes have been established. These are for example random walks, autoregressive (AR) processes and autoregressive moving average (ARMA) processes.

A random walk, which reproduces the development of stock prices quite well, has the following form:

$$ {{\tilde{Y}}_t}={{\tilde{Y}}_{t-1 }}+{{\tilde{S}}_t}. $$

(5.32)

This process is not stationary; the variable $\tilde{Y}$ has a time trend. However, the first differences (or the first derivative) of this stochastic process have the properties of the stochastic variables S – they are time independent and identically distributed with mean 1 and exhibit a constant variance.

If a variable $\tilde{Y}$ follows an autoregressive process, it can be described by its past values. An autoregressive process of order p, AR(p), can be written as follows:

$$ {{\tilde{Y}}_t}={\alpha_1}{{\tilde{Y}}_{t-1 }}+{\alpha_2}{{\tilde{Y}}_{t-2 }}+\ldots +{\alpha_p}{{\tilde{Y}}_{t-p }}+{{\tilde{S}}_t}. $$

(5.33)

Every finite autoregressive process can be transformed into a moving average process (for any number of components). A moving average process of order q, MA(q), represents a relationship between a variable $\tilde{Y}$ and the weighted mean of lagged stochastic shock variables:

$$ {{\tilde{Y}}_t}={\beta_0}{{\tilde{S}}_t}+{\beta_1}{{\tilde{S}}_{t-1 }}+{\beta_2}{{\tilde{S}}_{t-2 }}+\ldots +{\beta_q}{{\tilde{S}}_{t-q }}. $$

(5.34)

When both processes are combined, the resulting specification is called an ARMA(p,q) process. The simplest version is ARMA(1,1).

$$ {{\tilde{Y}}_t}={\alpha_1}{{\tilde{Y}}_{t-1 }}+{\beta_0}{{\tilde{S}}_t}+{\beta_1}{{\tilde{S}}_{t-1 }} $$

(5.35)

From the definition in Eq. 5.35 and the specifications in Eq. 5.36, it becomes immediately obvious that the GDP dynamics in Eq. 5.30 follow an ARMA(1,1) process. Thus:

$$ \tilde{Y}\equiv \ln Y,\tilde{S}\equiv \ln S,{\alpha_1}\equiv [1+(\delta +\chi )(\alpha +\gamma -1)],{\beta_0}\equiv 1,{\beta_1}\equiv (\delta +\chi )-1. $$

(5.36)