Micro-beams are building blocks for many micro- and nano-structures as well as micro-electro-mechanical systems (MEMS) and cannot accurately be modeled by classical continuum theories due to significant size effects at the length scales associated with these structures. Size effects can be taken into account by the so-called higher order continuum theories. In this study, Euler-Bernoulli micro-beams are analyzed with the Modified Strain Gradient Theory (MSGT), which extends the classical local continuum theories of grade one with the introduction of three additional length scale parameters. In this contribution, finite element implementation is briefly demonstrated by using Galerkin discretization techniques for Euler-Bernoulli beams. The size effect for gold-micro beams is demonstrated and the length scale parameters of gold micro-beams for MSGT are identified form the existing experimental data from literature for the first time. As a novel aspect, significant size effect is demonstrated for the length-scales associated with the state of the art gold micro-beam structures developed for NEMS and MEMS applications, which reveals the necessity of the use of higher order theories at these length scales.