Wind farm design and operation are traditionally driven by economic objectives, often overlooking environmental impacts. Although greenhouse gas emissions associated with wind energy are substantially lower than those of fossil-fuel-based plants, they are not negligible when assessed over the entire life cycle. This study investigates the impact of introducing a carbon-related objective alongside classical objectives, such as annual energy production and levelized cost of energy, into wind farm optimization. The analysis considers wind farm layout optimization, wind farm flow control, and the combined optimization of layout and control strategies, namely co-design. A range of case studies is examined, from illustrative examples to a large-scale offshore wind farm under varying site conditions. The results indicate that selecting a carbon-related objective yields economic performance very similar to the one obtained with conventional economic objectives, and vice versa, a significant (unexpected) finding with potentially far-reaching implications. This suggests that economic and environmental goals are largely aligned, and that prioritizing one does not necessarily compromise the other. Despite similar overall performance metrics, the resulting layouts differ in terms of turbine positions depending on the chosen objective and on whether layout and control are optimized jointly or sequentially.