Essential Elements of Continuous Time Dynamic Optimization

where x ∈ N is a vector of decision or choice variables, α ∈ A is a vector of parameters, f (·) is the twice continuously differentiable objective function, that is, f (·) ∈ C (2), and φ(·) is the indirect or maximized objective function. This is terminology you should be more or less familiar with from prior courses. Because we will deal repeatedly with vectors and matrices as well as the derivatives of scalarand vector-valued functions in this book, we pause momentarily to establish three notational conventions that we shall adhere to throughout. First, all vectors are treated as column vectors. To denote a row vector, we therefore employ the transpose operator, denoted by the symbol ′. Thus x ∈ N is taken to be an N-element column vector, whereas x′ is an N-element row vector. Note also that vectors appear in boldface type. Second, if g(·) : N → M is a C (1) vector-valued function, thereby implying that g(·) def = (g1(·), g2(·), . . . , gM (·))′, then at any x ∈ N , we define the M × N Jacobian matrix of g(·) by