To be honest, the two universities’ admission numbers for Economics are broadly similar - Cambridge has an admission success rate of about 14%, an average of 7 applicants per place. Bear in mind that the Cambridge economics course has a reputation for being particularly math-heavy (see this advisory published by King’s College, for example: http://www.kings.cam.ac.uk/study/undergraduate/subject-resources/6409). If you’re strong in math and enjoy it, by all means go for it, but consider carefully whether the emphasis is for you!

You’re correct - Oxford doesn’t have a single-honours degree for Economics. Instead, there are three degrees where you can study Economics alongside another subject. The closest you can get to a ‘pure’ Economics degree is probably Economics and Management (E&M), but it is statistically one of Oxford’s most competitive courses, with an admission rate of 8%.

Alternatively, you can opt for two other degree combinations at Oxford. The most famous of these is Philosophy, Politics and Economics (PPE), which is one of the largest and most popular subject groups in Oxford with a relatively higher admission rate (14%). The other is History and Economics, with a similar admission rate (13%). If you want to focus solely on Economics, it is possible to take any of these three degrees, take only the minimum required number of modules in the other subject, and focus the rest of your options on Economics. In PPE, for instance, you’re allowed to drop one of the three subjects from your second year onwards, allowing you to focus exclusively on the remaining two.