Agricultural production studies are usually conducted using classical econometrics that make it difficult, if not impossible, to impose conditions derived from economic theory on flexible functional forms. Therefore, such conditions need not hold in estimation. We apply Bayesian econometrics to estimate a flexible production function using U.S. agricultural data under alternative restrictions dictated by theory, including a fully theoretically consistent model that satisfies all restrictions at each point of the data. The probability density functions of parameters, output elasticities, scale elasticity, and productivity gains vary substantially across models. Output elasticities from the fully theoretically consistent model suggest that agricultural production in the United States has typically been more responsive to land than to other factors of production. However, output became more responsive over time to changes in materials, and less responsive to changes in land and labor. By the early 2000s, the elasticity of output to materials was very similar to the elasticity of output to land. The elasticity of output to capital remained relatively stable over the entire sample. The estimates from the fully theoretically consistent model also show a slight increase in the average rate of productivity gains between the 1960s and the 1970s, a relative slowdown in the 1980s and 1990s, and a relative stabilization afterwards. All inputs are substitutes in production; all the elasticities of substitution have increased through the decades, though at different rates.