Business Cycle Models with Embodied Technological Change and Poisson Shocks

The first part analyzes an Endogenous Business Cycle model with embodied technological change. Households take an optimal decision about their spending for consumption and financing of R&D. The probability of a technology invention occurring is an increasing function of aggregate R&D expenditure in the whole economy. New technologies bring higher productivity, but rather than applying to the whole capital stock, they require a new vintage of capital, which first has to be accumulated before the productivity gain can be realized. The model offers some valuable features: Firstly, the response of output following a technology shock is very gradual; there are no jumps. Secondly, R&D is an ongoing activity; there are no distinct phases of research and production. Thirdly, R&D expenditure is pro-cyclical and the real interest rate is counter-cyclical. Finally, long-run growth is without scale effects. The second part analyzes a RBC model in continuous time featuring deterministic incremental development of technology and stochastic fundamental inventions arriving according to a Poisson process. In a special case an analytical solution is presented. In the general case a delay differential equation (DDE) has to be solved. Standard numerical solution methods fail, because the steady state is path dependent. A new solution method is presented which is based on a modified method of steps for DDEs. It provides not only approximations but also upper and lower bounds for optimal consumption path and steady state. Furthermore, analytical expressions for the long-term equilibrium distributions of the stationary variables of the model are presented. The distributions can be described as extended Beta distributions. This is deduced from a methodical result about a delay extension of the Pearson system