The pressure on health care resources is rising. Therefore, creating a sustainable health care system is necessary. Cost minimization and productivity growth is demanded. An undesirable side effect of cost minimizing behaviour could be a reduction in quality. The current evidence on the relationship between cost and quality is ambiguous. The relevance of this research is to find evidence on the correlation between the quality of health care provided by hospitals and their costs in order to figure out what part of costs and cost inefficiency can be explained by quality. Operating costs, quantitative and case-mix adjusted qualitative (performance indicators) output variables based on data of 22 Norwegian hospitals between 2008-2014 were used to estimate the hospitals’ cost functions and cost efficiencies using Stochastic Frontier Analysis. A true fixed effects method was applied to respect the longitudinal aspect of the data and quadratic terms of the qualitative output variables were added in order to allow for a flexible relationship between cost and quality. Sign, level and significance of estimated coefficients are inconsistent over the cross-section and panel data models and the models with or without quadratic terms. The most consistent result is a negative correlation between cost and quality for the patient safety indicator pulmonary embolism/deep venous thrombosis. Furthermore, negative correlations between cost and quality based on the patient safety indicators sepsis and obstetric trauma are estimated in two models. A positive, but inconsistent correlation between cost and quality based on 90-day mortality and quality based on bed sores was found. A U-shaped relationship between cost and quality was found for the quality indicators of 90-day mortality and an inverted U-shaped relationship was found for the patient safety indicator pulmonary embolism/deep venous thrombosis and the patient safety indicator obstetric trauma. Based on the most basic cross-section model, the average cost efficiency is 89.7%. There is no significant inefficiency term found in the more complex models. The results are inconsistent leading to a remaining uncertainty in the evidence on the cost/quality trade-off. Unobserved heterogeneity and model specification might be important factors of influence in this field of research. The finding of a negative cost/quality relationship indicates that health care organizations could possibly aim for cost minimization by improving quality based on patient safety indicators. There is weak evidence for the existence of a cost/quality trade-off for the quality indicator 90-day mortality and bed sores. Practical implications for health care authorities are that unobserved heterogeneity should be considered when ranking and rewarding hospitals based on their activity and quality.