In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as reduced form of the boundary value problem for the Lam\'e system, we consider a Neumann problem for harmonic functions in the half-space with a cavity \(C\). Zero normal derivative is assumed at the boundary of the half-space; differently, at \(\partial C\), the normal derivative of the function is required to be given by an external datum \(g\), corresponding to a pressure term exerted on the medium at \(\partial C\). Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum \(g\) which describes a constant pressure at \(\partial C\), we recover a simplified representation based on a polarization tensor.