"Because I like money" is probably not the best response. "I am worth that much" is entirely reasonable if your research indicates that that is the going rate for your skills in that area, though you should probably be able to cite sources if challenged. I am going to assume that your skills and the position are in close alignment, such that you are not offering high-value skills for a low-value job, and expecting to be paid for the skills.
One of two things are happening (or maybe a combination): the recruiter thinks they can convince you that you are worth less than market rate; or the recruiter needs to feel better about purchasing your services at market rate, when they know that you previously worked for less. I think that both can approached with the same strategy, which would be to provide reasons why you previously worked for less. Possible examples, if they are true in your case:
- I gained valuable skills and experience in my last job, and my research indicates that that this is the market value of my current skill-set.
- My personal situation when I was last searching for work was such that I needed a contract sooner, rather than holding out for my actual market value.
- Due to the short duration of the contract, the additional pay would at best have covered the cost and risk of extended negotiations.
- I was looking for a shorter contract, and most of what was available was longer than I was looking for. To get a contract of the length I was looking for, I adjusted my ask.
- The last contract offered a number of non-pay features which are not currently included in this contract. They were _; would you be interested in negotiating them?
You should be looking to address both possibilities, showing that you know your actual market value and showing some reason why you last accepted a much different rate.
The recruiter's question is interesting ("How can I justify that?"), because it would seem to indicate that they need to justify the pay delta to their management. Assuming the recruiter asked the pay question and is negotiating pay (this seems to be your case), the easiest solution would just be for them to report the results of the negotiation, and not go into the details of your previous rate(s). Of course, it could be that the recruiter is using I to make you feel empathy for them as a tactic, but I don't think confronting them on that possibility is a path to successful negotiations.
Justifying the 50% increase is may be harder simply because it means finding another (and preferably different) reason why you accepted a below market rate two contracts ago, if it was indeed below market rate. Maybe it was a longer contract? In which case maybe you had much less experience, or you could have learned a lot, or its rate was calibrated based on a different time in the market, etc.. Sticking to just your previous rate, and not two contracts previous, would probably be best if at all possible.