The focus on the purchase price surprises me. I think one of the major factors is what car company you are dealing with. For me there are many factors other than the purchase price. Here are some.
My experience leasing started with BMW (once) and GMC (once) and Lexus 8 times.
The reason I didn't go back to BMW was how they treated me at the end of the lease. They tried to get me to pay for normal wear and tear, like small scratches on the bumper. It got nasty. After I would not pay they offered a settlement. No more BMW.
The reason I did not re-lease with GMC was the second lease price. The first lease I got a great price, they probably had extra inventory, the second lease they wanted about 30 percent more for they same model car. No thanks.
Started with Lexus. Got a great deal the first time and every time since, and they make the very best cars, too.
Now the other factors to watch for:
1) what happens at the end of the lease. Is there an end of lease fee? How much? will they waive it if you lease another car? Lexus does.
2) How many miles per year are you getting? How much do extra miles cost if you go over your lease? Usually they will offer 10K, 12K or 15K per year. The lease price is more for more miles, but this is one point you can negotiate. Same with the cost of extra miles. Sometimes you will pay less for extra miles if you buy them during your lease rather than at the end. Also, most companies will waive small overages, and sometimes, larger ones if you lease another vehicle.
3) The length of the lease is also a major factor. There is usually one lease length that is the sweet spot. This is based on the residual value. You may think the lease amount will decrease on a straight line from a one year lease to a 2 year, 3 year, etc. This is not always true. Get the lease prices for 2 year and 3 year leases. 4 year if you want. You may find out as I have that 3 year leases are usually the best deal, at least with Lexus. You don't really care about the length of the lease, at the end, you are going to give it back and get another one.
4) There are other things that the dealer can vary besides the purchase price, specifically the residual value and the "drive off" amount. The same purchase price will produce different payment amounts based on differences in the residual value used to calculate the lease amount. Same is true with the drive off amount. I never plan to buy out my leases at the end so I try to get the highest residual value I can. The higher the residual value the lower the payment. Also, one of the ways the dealer reduces the purchase price is with the drive off amount, which can include a down payment and fees like license, sales tax, etc. Why would I want to make a down payment on a car I am never going to own? And why pay the fees and taxes up front when I don't have to? I always take a zero drive off amount. This makes the lease payment higher, but I have no cash out of pocket in the car.
5) It also helps to understand what happens when you lease a car from a dealer. First of all, the dealer is just acting as an agent for the leasing company. The leasing company may by owned by the manufacturer (best), like Lexus Finance, or it may be an independent financer like a bank. In the latter case the dealer is probably getting a kick back from the finance company, but you are paying it in your lease deal. Second, the dealer has no interest in the car after the lease is signed. What really happens is that the dealer is selling the car to the leasing company, just like they would sell it to you, and you are leasing it from the leasing company. You are never leasing from the dealer. Same when you return the car. The dealer may act as an agent for the leasing company when you return the car, but all decisions in regard to car condition, buyout, return fees, etc. are made by the leasing company, not the dealer.
Finally, my approach to leasing a car is different from what is posted above. I focus only on the monthly payment. The other numbers are completely irrelevant because I am going to drive the car for 36 months and then give it back. I want no cash out of pocket up front and I want to get the best monthly payment I can. The previous poster is correct that you should not name your desired payment, most of the time, but it can work to your advantage in some cases, especially if you are a little low. It gives them a target to shoot for, and they can manipulate some items, as I mentioned above. BTW: some will argue that the residual value is not variable. It is. No one knows what the car will actually be worth in 3 years. With my Lexuses the retail value at the end of the lease has always been higher than my residual value. The actual value of my last car was so much higher than the residual value, I actually bought out the car and sold it to a friend. The friend got it for a few thousand less than the retail value, and I made a few thousand cash.
An example, from my current lease. It turned out that Lexus had, because of the dock strike, only a small number of 2016 cars in this country in the fall of 2015. Therefore, the demand was high and prices were too, and deals were scant. I had the advantage of having leased 7 previous cars from the dealer. BTW: I never worry about color of the car. While I have color preferences, they are no where near as important as the features. The features I wanted were just about everything they had to offer, although there were some of only minor importance. It turned out that they had just the car I wanted. When I first inquired, the salesman quoted me $896 per month. My previous lease had been $800 for a car worth about $3000 less list price. I told him absolutely not and that I wanted to deal with the sales manager. The sales manager always makes all the decisions anyway so why not get to them as fast as possible, whether buying or leasing. He came back from the sales manager and asked if I would pay $796 for the car (monthly payments). I said yes and got the newer more expensive car for $4 per month less than my old lease. When I saw the lease, I understood why. The residual value of the new car was $8000 higher than the last one, even though the list price only went up by $3000. They manipulated the residual value to get me the price I was willing to pay. Later, I found out they were also anxious to sell that car because they thought the color combination would be unpopular. I could care less.
One more thing. You probably know that buying a new car from last years model year can save you a lot of money. Same is true of leasing. When I lease cars for my wife and daughter I frequently take the previous model year (i.e. 2015 in December 2015 after the 2016s have come out). My reasoning is that it is still a new car, and they are only going to drive it for 3 years and give it back. It is not always a good deal to buy these cars because the car's value drops significantly the minute you drive it out of the dealer's lot, but when leasing, that becomes the problem of the leasing company, not you.
Whew, didn't plan on saying this much, but hope it gives you some suggestions.