The response (R) of V1 neurons to elongated stimuli at the preferred orientation varies sigmoidally as a function of contrast (C). This contrast response function (CRF) accelerates at low C and saturates at high C. It is characterized by the maximum firing rate (Rmax), the steepness and the contrast at which R equals half Rmax (C50). Experiments reveal a large diversity in these parameters. What are the respective contributions of the feedforward input, the recurrent intra-cortical input and the intrinsic properties of the neurons in shaping the CRF? To answer this question we have developed a rate model of a hypercolumn in V1, in which we assumed power-law input-output transfer-function (TF) for the neurons. Assuming that the exponent of the TF is larger than 1, the width of the orientation tuning curve (OT) of the output is narrower than the OT width of the input, and it is approximately contrast-invariance (CI). The response varies sigmoidally with C, the acceleration at low C reflects the TF non-linearity, whereas the saturation is strongly affected by the intra-cortical inhibition and by the saturating properties of the input. We have found similar behavior while testing the model with conductance-based neurons, in which power-law TFs emerge naturally as a result of the presence of noise in the inputs. A great deal of the diversity observed in the shape of the CRF can be accounted for in our model by assuming a large diversity in the input and cellular properties. Based on these findings we predict a correlation between the steepness and C50 of the CRF of neurons in V1.