Balanced-budget rules: Chaos and deterministic sunspots

Chaos and capacity utilization under increasing returns to scale

“…I say that an admissible P * is optimal if for every admissible P Given thatē t may be only piecewise continuous, the household's optimal control problem is non-standard (ē t is a discontinuous state variable). However, using a standard argument as in Stockman (2010), one can show that the usual sufficient conditions for optimality hold:…”

“…function x(t) that is continuous and continuously differentiable almost everywhere withẋ ∈ H(x). 1 I will say that a solution x to the differential inclusion H is generated by H. Solutions to the differential inclusion may involve switching between the branches f and g. One interpretation to these switches is to say that the economy follows a deterministic sunspot process as in Gardini et al (2009) and Stockman (2010). Such equilibria can be constructed that satisfy the standard definition of a perfect foresight equilibrium, i.e., the set of perfect foresight equilibria contains these type of switching equilibria.…”

“…Note that since dynamical systems generated by a differential inclusion are non-standard, one has to first define chaos for such systems. Stockman (2010) extends the notion of Devaney chaos to a dynamical system generated by a differential inclusion. 3 Raines and Stockman (forthcoming) extend the notions of (1) Li-Yorke chaos and (2) distributional chaos to a dynamical system generated by a differential inclusion.…”

“…For an introduction to differential inclusions seeSmirnov (2002). 2 See Raines and Stockman (forthcoming),Stockman (2009) andStockman (2010), respectively, for an analysis of Euler equation branching in each of these models.3 SeeDevaney (2003). 4 SeeLi and Yorke (1975) and for distributional chaos seeSchweizer and Smítal (1994) andSchweizer et al (2000).…”

Chaotic sets and Euler equation branching

“…A solution to a differential inclusion is a function x(t) that is continuous and continuously differentiable almost everywhere withẋ ∈ H(x). 1 We will say that a solution x to the differential inclusion H is generated by H. Solutions to the differential inclusion may involve switching between the branches f and g. One interpretation to these switches is to say that the economy follows a deterministic sunspot process as in Gardini et al (2009) and Stockman (2010). Such equilibria can be constructed that satisfy the standard definition of a perfect foresight equilibrium, i.e., the set of perfect foresight equilibria contains these type of switching equilibria.…”