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Extra info for General equilibrium, overlapping generations models, and optimal growth theory

Example text

When you do so, you arrive at a point on an edge of the box. 18 illustrates the case aA1/aA2 < aB1/aB2. The set of Pareto optima consists of the heavy line segments 0A E and E0B. 5 Pareto Optimality and Optimality: Minkowski’s Separation Theorem Under special assumptions, an allocation is Pareto optimal if and only if it is optimal with respect to a social-objective function that is a weighted sum of consumers’ utilities, Ii=1 ai ui (xi ), in which the weights, ai , are nonnegative numbers, not all of which are zero.

4 The Calculation of Pareto Optima; Homogeneous Functions 33 N → R is a monotone transformation of f if g (x) = Recall that g: R+ N→R h(f (x)), for all x, where h: R → R is increasing. A function f : R+ is homothetic if it is homogeneous of positive degree, for suppose that f is homogeneous of degree b, where b > 0. Then g(x) = [f (x)]1/b is homogeneous of degree 1, since g(tx) = [f (tx)]1/b = [t b f (x)]1/b = t[f (x)]1/b = tg(x). Since the function h: R → R defined by h(s) = s 1/b is an increasing function and g(x) = h(f (x)), it follows that f is homothetic.

2f (x) ∂ 2f (x) ⎠ ... ∂xN ∂x1 ∂xN ∂xN is the N × N matrix of second partial derivatives of f with respect to its variables. If f has only one variable, so that N = 1, then Df (x) = dfdx(x) is d 2f (x) dx 2 d 2f (x) < dx 2 the first derivative of f at x, D 2f (x) = f at x, D 2f (x) semidefinite if is negative definite if d 2f (x) dx 2 is the second derivative of 0, and D 2f (x) is negative ≤ 0. 30 If X is a convex set of N-vectors and f : X → R is twice differentiable, then f is concave if the matrix D 2f (x) is negative semidefinite, for all x.

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