The basic equations for a layer of an electrically conducting fluid are considered in the magnetohydrodynamic approximation. An electric field is applied parallel to the layer. Because of various instabilities, turbulent fluid motions will be generated in general. The goal of the analysis presented in this paper is the derivation of upper bounds for the average turbulent resistivity of the layer. In order to demonstrate the concept of the upper bound theory, only the simplest case will be treated and the boundary conditions will be assumed in such a way that the analogy with the problem of bounds on the momentum transport in a turbulent Couette layer can be utilized.