Competitive Market Analysis when have to extrapolate price per acre and price per square foot of building?

Question

I am looking to buy a house in rural San Francisco Bay Area and I am trying to carry out Competitive Market Analysis on my own.
Unfortunately, in the area where I have found one house that interests me, I am unable to find many other similar comparable properties that were recently sold. So I am trying to extrapolate what would be "fair price" to pay for the property by extrapolating data from other recently sold properties that have different land lot size and different building size.
Basically, I have created equation system where for each recently sold property I have added an equation to the system:
4,311*x+1.47*y=$2,050,000 ... house 1
4,000*x+45.21*y=$3,850,000 ... house 2
4,300*x+1.22*y=$1,950,000 ... house 3

And then I am attempting to get the approximate value for x which is implied value per one square foot of building. And y which is implied value per one acre of land. These properties look very similar to each other and I have thrown away those who have been recently remodeled or would need major rework.
However, I see two problems with my approach as

it has two unknown variables. I believe I should start to solve the equation system by first substituting variable x which is value per square foot of building. So my first question - is there a database where I could get information on how much one square foot of building that was built in 1990 and last time remodeled in 2000 would cost? Or maybe I can use new construction cost and subtract prorated remodeling costs on when I expect to remodel house? Hard to get this used building price per square foot data data from authoritative source.
the implied value for one acre of land obviously changes depending on how many acres I would be willing to buy. What function should I use to adjust appropriate price per acre of land depending on how many acres I would be willing to buy? It seems that each marginal acre becomes significantly cheaper based on recently sold properties.

FYI, the property I am looking at six months ago was listed for $3.6M. Then listing price went down to $2.8M. Seller offered me almost two digit discount on the latest $2.8M listing price, but even with this discount applied it still seems a little high. Hence I am trying to do CMA on my own.

Justin Cave · Accepted Answer

If we assume that your equations are a match for reality (the primary drivers of the price are the size of the home and the size of the lot and that the price is a linear function of both home size and lot size), then you've got 3 equations and 2 variables so there are plenty of techniques to come up with an approximate solution.  The simplest and most common would be to do a linear least squares approximation.  This minimizes the sum of the square of the difference between the predicted value and the observed value.  You can do this by hand or in Excel or you can use a variety of online calculators.
When I put your equations into Wolfram Alpha, it comes back with the approximate solution
x = $460.38 
y = $44,430

Of course, your equations are only rough guides to reality.  One square foot of space isn't quite equal to another if one is original 1950's construction and one was just remodeled.  One acre of land isn't quite equal to another for any number of reasons-- the marginal value of the 20th acre isn't as high as the marginal value of the 1st acre, land with a lot of mature trees is more valuable than the same land with a few seedlings, some land has commercial value, some land has scenic value, etc.  Those sorts of things tend to be much harder to quantify, though, so it is very hard to say from a distance that your utility function ought to have utility grow with, say, y^0.75 to account for the diminishing value of additional acres or whether it should be some other non-linear function.  You can obviously play around with different factors based on your personal preferences and see how that affects the equations you've come up with.

mhoran_psprep · Answer

Your model is too simple.
You aren't considering:

Location. Differences in location of a few miles can have a significant difference in home prices. It can make a difference in schools, community amenities, commuting. Being in the location with desirable name can cost more. View can also factor into the value of the land.
Quality of the materials. Price per square foot is meaningless unless you know what stuff is making up those costs.
Floor plans. People care about the size, shape, location of certain things in the house. Some want a big kitchen, some want a gigantic kitchen, others only order takeout.

Other items you are putting too much emphasis on:

Remodel date/year of construction. I am not sure how the year of the remodel makes a difference. I have seen bad remodels, I have seen great remodels. You might have to gut last years remodel, and only tweak one made 10 years ago.
Size of house. People decide if the items in the house make sense for them. In some cases an extra 500 square feet but one less bedroom makes the house not work for them.

There are people that can help. There is an entire group of people that can help you evaluate the price of a home you want to buy. They are real estate agents. Some work only for buyers. They have access to data. They have the ability to get you inside homes that are currently on the market. So you can see the floor plans and the materials, and the layout.

Competitive Market Analysis when have to extrapolate price per acre and price per square foot of building?

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