This paper examines an innovative and practical way to model the supply of agricultural crops. This will be done by extending the technique developed by Howitt (1995), Positive Mathematical Programming (PMP), using Bayesian estimation. A key problem in the use of the PMP model is the relative difficulty of finding calibrating parameters such that the first and second order conditions are satisfied; with the added difficulty that many of the conditions needed to be satisfied are not exactly known. Thus the use of Bayesian analysis is a useful tool to try and determine these parameters. By employing a Markov chain Monte Carlo (MCMC) algorithm, specifically a Metropolis-Hasting Algorithm, a posterior distribution for the calibrating parameters can be found such that the resulting supply model will not only reproduce an optimum close to observed acreages, but also produce reasonable elasticities due to the prior information. The value of this style of estimation for a crop supply model lies in the limited amount of data needed to estimate the model.