If you buy a house at a foreclosure for $100,000... how is the government allowed to value that house at $180,000 for tax reasons.
How is what the house cost not exactly the value of it.
When a house is sold at a foreclosure auction, the selling bank usually does not provide the guarantees that a normal house seller provides. Furthermore, the previous owner may have neglected the property, and/or spitefully damaged the property. Bank-owned properties are often neglected and/or vandalized. Banks are usually too short-sighted to properly market the real estate they own, and do a poor job of making it easy to buy the property.
Thus, foreclosure sales usually happen at a price that is significantly below the "fair market value" of sales between competent households. It is common for a house that is worth $ 125,000 (even in a depressed market) to sell for only $ 100,000 in a short sale or foreclosure.
It is possible that this property sold for an even larger discount. It is also possible that the tax assessor is (inadvertently) comparing a run-down property with well-maintained properties that have extra expensive features, without fully adjusting for the properties' conditions and features. In the latter scenario, the property owner can ask the tax assessor to re-consider the assessment. Usually this request is called an "appeal".
The real answer why the government is "allowed" to do something is because they are the government and they make the rules. There are lots of laws that I think make no sense.
I ran into a similar situation to yours. I bought a house during a time when the market in my area was way down. The previous owner had paid $140,000 but I got it for only $80,000. The government appraised it for, I forget the exact number, but over $100,000. I appealed, and the argument I made to the appeal board was that the law says it is supposed to be appraised for "fair market value". The definition of "fair market value" is the amount that a willing buyer would pay to a willing seller, absent special conditions like a sweetheart sale to a relative. The house had been advertised for a higher price and the seller had to drop the price several times before getting an offer, and finally accepting mine. This is pretty much the definition of "fair market value". The appeals board replied that it was not FMV because the market was bad at this particular time and so I got a good deal. I said that that's the definition of market value: it goes up and down as market conditions change. If the market happened to be up when someone bought a house and they had to pay a high price, would the government assess the house at a lower value because that was an unusually high price? I doubt it. They ended up reducing the assessed value, but not to what I actually paid.
All that said, arguably a foreclosure sale might be considered special conditions. Prices at a foreclosure sale tend to be lower than "ordinary" sales. In a foreclosure, the bank is usually trying to get rid of the property quickly because they don't want to be in the property-management business, they want to be in the money lending and management business.
Of course you could say that sort of thing about conditions surrounding many sales. Maybe the price is low because the seller needed cash now to start a business. Maybe the price is high because the buyer was too lazy to shop around. Maybe the price is low because the buyer is a very skilled negotiator. Etc etc.
My watch just broke and while I was shopping for a new one I found two listings for the exact same watch, I mean exact same manufacturer and model number, identical picture, on the same web site, one giving the price as $24 and the other as $99. What is the market value of that watch? I presume anyone who saw both listings would pick the $24 one, but I presume some number of people pay the higher price because they never see the lower price. In real life there isn't really one, exact, fair market value. That's an abstraction.
The property tax valuation and the fair market price are NOT one and the same. They track each other, correlate to each other, but are almost NEVER the same number.
In some parts of the USA, a municipality has to re-assess property tax values every ten years. In these places, the tax value of a property is on something like a 10-year moving average, NOT on the volatile daily market price.
EDIT: It is easy to fall into the "trap" of thinking that property tax valuation is intended to represent fair market value. It's INTENT is to provide an accurate (or, as accurate as possible) RELATIVE VALUATION of your property compared to the other properties in the municipality. The sum of all the property values is the tax base of the municipality. When the town budget (which is paid in part via property taxes) is set, the town simply divides the tax base into the budget total to arrive at the ratio of tax-to-collect, to the tax base, also called the "tax rate per thousand dollars of valuation." i.e. if the town tax base is US$10,000,000, and the town budget is US$500,000, then the ratio is 0.05, or $50 per thousand dollars of valuation. If your property is assessed at US$100,000, then you would pay 100 x $50, or $5000 in property taxes that year.
Since this is the goal of the property tax valuation, NOT deciding what your house is worth on the open market, then we are left with the question of "why use the market value of a house for property assessment?" and the answer is that of all the various schemes and algorithms you can try, "fair market value" is the easiest and most accurate...IF TIME FLUCUTATIONS ARE TAKEN OUT. For example, if I buy a house in a development for $250,000 today, and next summer the housing market crashes, and you buy the identical house next door to me for $150,000, it does NOT stand to reason that you should pay less taxes than me, because your house is "worth" $100,000 less. In fact, BOTH our houses are worth $100,000 less.
What matters most in property tax valuation isn't the actual number, but rather, is YOUR valuation the same as other essentially similar properties in your tax base? Getting the RELATIVE ratio of value between you and your neighbors correct is the goal of property tax valuation.
Keep in mind, there are times that house is in such bad shape that it's going to need 6 months of renovations, in which case you might ask the town if they are willing to reappraise a lower value until the work is completed. Keep in mind, you'll get a new appraisal when permitted (I mean pulling a permit from the town) work is done. I finished my basement and the town was eager to send the appraiser over even before work was fully complete.
From my perspective I suspect that if the government use the paid price, people will start to buy at very low nominal prices in order to pay less taxes, and will repay the seller by other means.