In randomized controlled trials with treatment non-compliance, instrumental variable approaches are used to estimate complier average causal effects. We extend these approaches to cost-effectiveness analyses, where methods need to recognize the correlation between cost and health outcomes. We propose a Bayesian full likelihood approach, which jointly models the effects of random assignment on treatment received and the outcomes, and a three-stage least squares method, which acknowledges the correlation between the end points and the endogeneity of the treatment received. This investigation is motivated by the REFLUX study, which exemplifies the setting where compliance differs between the randomized controlled trial and routine practice. A simulation is used to compare the methods' performance. We find that failure to model the correlation between the outcomes and treatment received correctly can result in poor confidence interval coverage and biased estimates. By contrast, Bayesian full likelihood and three-stage least squares methods provide unbiased estimates with good coverage.