I used to give a big discount for longer periods of time until I realized how much income I was losing. Yes, it does bring peace of mind to be solidly booked, and it's nice to avoid the problem of turnovers, but is it worth it?
Figure out your savings if you rent for two months -- would you reduce the cleaning crew visits, laundry expenses, welcome basket costs, soaps & other amenities? If so, reduce the price by that amout. You can also place some value on the labor you save with only one guest instead of 8 separate weekly rentals - - email & phone time, dealing with payments and refunds, background checks, and your own time at turnover, and reduce the monthly rate accordingly. Figure out your time and a reasonable hourly rate and subtract that from the rate.
I also get antsy if I have a long rental and I can't get into the house to putter around and do some on-going maintenance; with weekly turnovers, I can get in every week to do or fix something. Yes, I could ask a long-term tenant to let me in to work, but that is intrusive and awkward.
Still, this is your first year and you are just getting your feet wet -- it might be easier for you. If so, I would make sure your contract protects you in case they change their minds at the last minute - all your eggs are in one basket. What provisions are there if they leave after one week or one month? What if they are the "guests from hell" we read about in this community -- can you evict them if they violate the lease?
Best of luck! This is a nice problem to have.
I believe the discount should depend on how difficult it would be to rent the weeks individually. And, as a previous poster pointed out, how much you save by renting a block of weeks.
I offer a 5% discount for multiple week rentals. I save the weekly cleaning and laundry services if my guest does not want these services during an extended stay. Other than these services I save myself the effort of answering inquries and processing rental materials. I will offer this reasoning to explain my paltry discounts to guests. They are generally understanding.
In your first years of renting I would recommend renting to as many guests as possible. You shouldd be building your rental base. It may be difficult to turn a two month rental away, but it may be in your best interest to do so.
And here's a way to determine if you should even offer a discount to a renter: google your potential renter. I have found some very well-to-do people asking for large discounts, people who own 2.5 million dollar homes, people with highly placed positions, people with their own companies . . .people who clearly do not need discounts to vacation at my home!.
Good luck with whatever you decide!