The optimization of spatial resolution is a critical consideration in the design of small-diameter positron emission tomography (PET) scanners for animal imaging, and is often addressed with Monte Carlo simulations. As a faster and simpler solution, we have developed a new analytical model of the PET detector response function, and implemented the model for a small single-slice, multilayer PET scanner. The accuracy of the model has been assessed by comparison with both Monte Carlo simulations and experimental measurements published in the literature. Results from the analytical model agreed well with the Monte Carlo method, being noise free and two to three orders of magnitude faster. The only major discrepancy was a slight underestimation of the width of the point spread function by the analytical method as inter-crystal scatter is neglected. We observed good agreement between the predictions of the model and experimental measurements. For two large-diameter scanners additional discrepancies were seen due to photon acollinearity, which is not considered in the model. We have shown that the simple and fast analytical detector response function model can provide accurate estimates of spatial resolution for small-diameter PET scanners, and could be a useful tool for several applications, complementing or cross-validating other simulation methods.