The impact of changes in relative weights on the optimal solution of a maximization problem
Starting point for the analysis is the standard maximization problem under constraints: \[ \max_{x_ 1,...,x_ n}\sum^{n}_{1}w_ iu_ i(x_ i)\quad s.t.\quad f(x_ 1,...,x_ n)\leq 0, \] where \(u_ i(\cdot)\) denotes a concave, monotonically increasing and twice differentiable utility function defined for a positive variable, \(w_ i\) denotes a positive weight attached to \(u_ i(\cdot)\) and f(\(\cdot)\) denotes a twice differentiable function which is increasing and convex. The main subject of the paper is the analysis of the effects of a variation of the weights on the optimal solution of the optimization problem cited above. As most interesting results the author is able to prove the following propositions. Result A: Let \(x^ o\) be the optimal solution for \(w^ o\) and x' be the optimal solution for w'. Then we have \(\sum^{n}_{1}(w^ o_ i-w_ i')(u_ i(x^ o_ i)-u_ i(x_ i'))\geq 0\). This result obviously implies that w and \(x^*\) are positively correlated. Result B: Suppose for example that f(\(\cdot)\) is both convex and Schur- convex and \(u_ i(.)=u(.)\). Then we have \(w_ n\geq w_{n-1}\geq...\geq w_ 1\Leftrightarrow x^*_ n\geq x^*_{n-1}\geq...\geq x^*_ 1\). This result is especially interesting for economic applications. Let for instance the \(w_ i\) be interpreted as discount factors in an intertemporal consumer choice problem. Then result B implies that the optimal consumption stream is decreasing over time.
scientific article; zbMATH DE number 3978835