Multiple Equilibria and Thresholds Due to Relative Investment Costs

Abstract

A dynamic model of the firm is studied in which investment costs depend on the magnitude of the investment relative to the stock of capital goods. It is shown that in general nonunique steady states can exist which can be stable or unstable. It is possible that unstable steady states occur in the concave domain of the Hamiltonian. For a particular specification, a scenario occurs with two stable steady states and one unstable steady state. The two stable steady states are long run equilibria; which one of them is reached in the long run depends on the initial state. In case the Hamiltonian is locally concave around the unstable steady state, this steady state is the threshold that separates the domain of initial conditions that each of the stable steady states attracts. The unstable steady state is a node and investment is a continuous function of the capital stock. If the unstable steady state lies in the nonconcave domain of the Hamiltonian, this steady state can either be a node or a focus. Furthermore, continuity can (but need not) be retained similarly to the concave case, a fact which has been entirely overlooked in the literature.