We compare the mechanical properties of two elastic models for a two-dimensional intrinsically curved semiflexible biopolymer. Model 1 allows a signed curvature and the model 2 allows a positive definite curvature only. We show exactly that these two models have different ground states. We discretize both models and use the Monte Carlo method to simulate the models, and find that they have also different mechanical properties at finite temperature. Under the same force and with the same intrinsic curvature and bending rigidity, the extension of model 1 is smaller than that of model 2 at low force, but becomes larger at large force. Moreover, the extension of model 1 undergoes a discontinuous transition when the intrinsic curvature is sufficient large, but the extension of model 2 is always a smooth concave function of force, so that there is no phase transition. The difference between the two models is due to the fact that under external force model 2 disfavors looped configurations and favors the configurations with moderate extensions.