In some mammalian genes, the paternally and maternally derived alleles are expressed differently: this phenomenon is called genomic imprinting. Here we study the evolution of imprinting using multivariate quantitative genetic models to examine the feasibility of the genetic conflict hypothesis. This hypothesis explains the observed imprinting patterns as an evolutionary outcome of the conflict between the paternal and maternal alleles. We consider the expression of a zygotic gene, which codes for an embryonic growth factor affecting the amount of maternal resources obtained through the placenta. We assume that the gene produces the growth factor in two different amounts depending on its parental origin. We show that genomic imprinting evolves easily if females have some probability of multiple partners. This is in conflict with the observation that not all genes controlling placental development are imprinted and that imprinting in some genes is not conserved between mice and humans. We show however that deleterious mutations in the coding region of the gene create selection against imprinting.