Why is government corrupt? You'll notice that I did not ask, "Is government corrupt." We've had enough experience to go right to the main question. I might have softened it with the phrase tends to be to acknowledge that not everyone in government is corrupt, at least not in the conventional sense. Lord Acton's statement was, "Power tends to corrupt, and absolute power corrupts absolutely" -- although I wish he had added, "Power also attracts the corrupt."
At any rate, we have a question on the table. Let's go beyond the easy answer. Obviously, any government is intrinsically corrupt because it uniquely has the widely approved ("legitimate") power -- for "the common good" -- to use physical force against others who never used force against anyone. The state's personnel should know that they benefit at others' expense on the basis of this dishonest claim.
For most of the public, of course, that's not corruption. Thank the government's own schools in large part for that. As philosopher Michael Huemer points out, in challenging political authority (as others have), most people have no problem with government officials doing things by force that would be condemned if anyone else did them. No one thinks it would be proper for non-officials to extort money from their neighbors to feed hungry people or to pursue another good cause. For the government, the rules are different. Why? Huemer demolishes the popular answers. (See his The Problem of Political Authority.)
Let's leave that aside and look at what everyone would consider corruption (at least if the opposing team did it). That includes politicians or bureaucrats who benefit financially or otherwise from performing certain actions. When such transactions become known, we call them scandals. These involve privileges in the proper sense of the word, not the pseudo-privileges implied by the weird terms overprivileged and underprivileged.
So why is such corruption endemic around the world? It doesn't take much imagination to see why; it relates to the more fundamental kind of corruption already discussed. Certain politicians and bureaucrats are close to the power the government has to hand out other people's money, both directly and indirectly. A subsidy is an example of direct distribution. A tariff or competition-stifling regulation is an example of indirect distribution. In both sorts of cases money is politically and coercively transferred from some people -- taxpayers, consumers, disfavored businesses -- to a favored person, business, or industry. Inflation, brought to you by the Federal Reserve's money creation and government borrowing, is another way to transfer wealth through the shifting of purchasing power.
Since some officials have access to such transfer power, other people outside the government will seek to benefit by currying favor with them. Reluctantly, some business people may do so just to keep up with competitors who are doing it.
History is full of stories of benefits provided to politicians and bureaucrats in return for political favors: outright bribes, campaign contributions, expensive vacations, post-retirement jobs, sinecures for family members (even relatives of presidents, if you can believe that!), and more. The buyer of favors might even be a foreign official who wants better treatment for himself or his government. Let's also acknowledge that the favor sought might be something that libertarians would approve of, such as the repeal of an unjust government policy, such as a tax, regulation, or trade embargo. Ironically, corruption could serve a good cause.
Of course, some people in government will not be tempted by corrupt offers, but others will be, and still others will be disappointed that no offers are tendered. It will be hard for voters to tell who's who. As for those who accept such offers, it would not necessarily mean they are malicious. They may really believe they are promoters of the "public good" and deserve to stay in power and enjoy its perks. I'm not saying that's a good excuse. We know how the road to hell is paved.
We shouldn't be shocked by scandals, but we should be perturbed. Attempted reforms such as mandatory disclosure and the like have not shown notable success. Proximity to power holds temptations that many will find too good to resist. As the Public Choice school of political economy teaches us, people don't become saints the moment they take government jobs.
The only way to change this is to radically reduce, check, and decentralize (if we can't eliminate) government power.