In general there are two kinds of jobs:
Hourly rate jobs: Your client wants you to track your time so they can pay you a specific amount per hour worked.
Fixed price jobs: Your client pays a set amount per project, no matter how long you have to work to finish it.
I personally charge less for an hourly rate than for a fixed-price rate. When you are working for a fixed price and encounter unexpected aspects of the job, you could be working a long time for a ridiculously small amount. In an hourly job the unexpected is covered. Also your client is trusting you to accurately record the time you spend on the job. This trust should be rewarded with a better price.
When you are agreeing on an hourly rate, give an approximate budget but be clear that it depends on the time needed to complete the job.
Some clients will ask for your hourly rate, and then ask you to complete a job in a fixed amount of time. Okay, this is obviously a fake hourly job; it is a fixed price. If you are charging different for an hourly than for a fixed price rate, then you must let your client know it. If a client tries to negotiate a lower amount than your fixed price, it is better to refuse the job. It is a very bad signal if a client is trying to take advantage of this “kind of negotiation”. Negotiation is okay and healthy, but this is dishonest. I don’t appreciate dishonesty in people that will have to pay me.
If you ask me, it is okay if you want to offer a deal to your client and do a job for a cheaper price, but it is a deal because you have offered it. This situation is not acceptable if it is a result of your client asking for an hourly job and then limiting the time that will be compensated.
I can accept the time limit if it seems reasonable plus a little.
What does plus a little mean?
As I said, the good thing about an hourly rate is that the unexpected is always covered. So if you are going to agree on an hourly rate (cheaper than a fixed price) and your client wants a time limit, calculate the time you will need and double it, explaining that you will likely use less time than you quoted and that if he wants a fixed price, you can do it but the conditions will be different. You can explain about unexpected factors to let your client know why you are charging different rates.