The derivation of a Hardy field induces on its value group a certain function Ψ If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over ℝ. Such “ordered vector spaces with Ψ-function” are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function Ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over ℝ. We analyze in detail finitely generated extensions of a given H-couple.