Which is harder will really depend on who you are and what your strengths are.
The Duke economics program is perhaps more nationally acclaimed than the engineering program (the biomedical engineering department is also well known). This said, the engineering department as a whole is exceptionally good too.
I know @duncan.parsons studies mechanical engineering there, so he might be able to shed some more light on this.

It’s a good question and one a lot of people at Duke ask. Nobody will doubt that engineering is a lot harder - I’d suggest it’s about twice much work as economics because each course is more rigorous and you are required to complete more courses - five math papers, two physics, one chemistry and 15 mechanical engineering classes. Compare this to economics, where one math class, five core and five electives are requires will earn you a degree. I haven’t taken the economics classes but from seeing peers and hearing from friends that did double degrees in engineering and economics (unusual and very impressive), most engineering classes are harder than any economics class. I picked the hardest thing I could study and did it. It’s a worthy investment for opening doors down the line.

Most students that seriously consider engineering (have the mental capacity and work ethic) will either study engineering or get a double degree. A good qualitative alternative to engineering would be a double degree in computer science and economics, or a hard science (biology, chemistry or physics) and economics.

You’d choose mechanical engineering if you want to build things, explore the mechanical world and learn the skills that allow you to enter the engineering industry. Engineering is the best undergraduate degree to prepare you for a career in technical fields. It opens a lot of doors both in business, consulting and technology. For example, it’s a pathway into graduate study about neuroscience using engineering and technology to better understand how our brain operation and efficiency.

It is the most well-respected, rigorous degree and a lot of science and applied math will prepare you better for the finance and consulting industries than an economics degree. This works both ways too, because you become a better thinker, and employers also understand the rigor of your degree.

@jeffrey_li Studies economics there. He might be able to expand on the other side to the question.