Do Higher-Order Functions provide more power to Functional Programming? My original question was: Is Kappa calculus less powerful than Lambda calculus? 

Does the lack of Higher-Order functions on a programming language excludes some programs that could only be written in functional programming (and therefore in a turing machine)?

By "powerful" I mean a Turing Complete programming language. For example, a language that can not contain infinite recursive expressions, ie infinite recursion or an infinite while loop, is not as powerful as Lambda-Calculus and Turiung Machines, since it would otherwise be a contradiction to the Halting Problem.

I can't find the reference right now, byt I remember something about an algorithm that on a non-lazy functional programming language has an $\Omega( n \log n )$ complexity, while the same algorithm in imperative programming is $\Omega( n )$
Even the full simply typed lambda calculus with products is not Turing-complete (cf. http://en.wikipedia.org/wiki/Simply_typed_lambda_calculus#General_observations), and since the kappa calculus is a fragment of it (if I understand the Wikipedia page correctly), it isn't either.