In this paper and its sequel, we attempt to understand the rheological differences between lyotropic and thermotropic liquid‐crystalline polymers by contrasting the rheology of three different liquid‐crystalline polymers at concentrations ranging from moderate (12%–30%) to highly concentrated (30%–50%) to densely packed (50%–100%). This first paper presents the steady‐state first and second normal stress differencesN1andN2as functions of the shear rate γ̇ for solutions of poly(γ‐benzyl‐glutamate) with molecular weight 238 000, in the solvent metacresol, at concentrationsCranging from 12.5% to 40% by weight. Predictions ofN1andN2for this range of concentration are obtained from the Doi molecular theory for rod‐like nematics, using both an approximate, and a nearly exact, method for solving the Doi equation. The predictions ofN1, and to a lesser extentN2, agree qualitatively with measured values for all concentrations. In particular, the range of shear rates over whichN1is negative shifts upward with increasing concentration: if we define γ̇maxas the shear rate for whichN1reaches a positive maximum valueN1max, then we find that theory agrees with experiment in that both γ̇maxandN1maxincrease monotonically with concentrationC. These increases occur because the strength of the nematic interaction increases withC, which implies that for high concentrations,N1becomes negative only at high‐shear rates.