The fact that blame-minimalization seemed somewhat plausible as a first idea already tells us that the question might be unanswerable:
All these variables to be maximized or minimized are relevant for the respective theories in which they are central because they determine some sort of quality or at least some sort of measurement that enables the respective entity to either be better or worse than actualy alternatives (competition between companies or politicians) or at least it may serve to enable others to hold the entity responsible for being better or worse than actual or imaginable alternatives (and therefore maybe be replaced by another one etc. etc.).
This automatically leads to a selection-mechanism that shapes the respective entities - though the similarity to natural selection is at least in some cases pretty limited.
Since a theory of such entities is suposed describe them and therefore the shapes they have/take, it must at least in some way deal with (facts that depend on) the way those entities are being shaped.
But if there it is really at least to some degree true that bureaucracies tend to limit blame(worthyness), then this entails that they tend to limit their own responsibility (in some regards). What follows is that - at least to some degree - they lack the respective shaping-mechanism, the selective pressure that follows from sometimes being blamed and therefore sometimes being replaced by an alternative.
What follows (and seems very plausible at least to me) is that the dominant shaping-mechanism of a bureaucracy is just the sum of decisions that lead to its installment (again, this is only the case to the degree that this lack of blame-worthyness is given).
Such decisions (correct me in so far as I am wrong, of course) are based on specific laws and regulations that ought to be implemented.
Therefore, when we try to formulate a theory of bureaucracies, we deal with a whole lot of individual differences when it comes to the defining characteristics of the entities that ought to be described - due to the differences in terms of those laws and regulations.
I therefore suspect it will at least stay very hard to formulate an adequate theory of bureaucracy - at least if we stay in a realm of such simplicity where the maximization or minimzation of a single variable is our starting point.
Of course we might say "the degree to which rule x is adequately implemented is being maximized" might be a fitting description, but the blame-avoidance makes it impossible to say that this rule actually describes the entities we want ot describe and not just the intentions that went into their respective installation.
(sorry if my writing is kind of inefficient - I fear this might be the case - as I am not a native speaker)
Like a lot of people, I've become interested in the performance of public health bureaucracies over the course of the COVID-19 pandemic. But while I've seen a lot of discussion of the specific flaws and failures of the FDA, the CDC, the WHO, etc., I haven't seen much discussion of the logic of bureaucracy in general.
What I'm hoping to find, or at least to better understand why I can't find it, is a general theory of bureaucracy, in the same way that economics already has for people, firms and democratic politicians. Any time we want to explain what a person, a firm, or a democratic politician does, we say they are a utility-maximizer, a profit-maximizer, and a median-voter-grabber, respectively. These models aren't necessarily perfect, but they are a clear starting point for almost any analysis of the respective subjects. They offer a broad, general understanding consistent across many contexts and time periods.
The power of these broad, general models is that they give a very clear starting point for analyzing problems and solutions to those problems. When individuals are choosing badly, change the incentives. When businesses are choosing badly, tax or subsidize something—change what's profitable. When politicians are choosing badly, change what voters want. These solutions aren't necessarily easy to design or implement. But it's nice as a base and a starting point.
Having access to these kinds of very general explanatory models minimizes the kind of confusion that many of us may have felt watching the decisions of public health authorities—a sense of "Bwuh?!" When a person talks about giving back to the community but never donates to charity, I made not be happy with them, but I can understand it in terms of utility maximization. When a firm pollutes the air or makes a product with cheap, unreliable parts, I may not be happy with them, but I can understand it in terms of profit maximization. When a politician adopts an extreme position in the primaries and then moves to a moderate position during the general election, I may not be happy with them, but I can understand it in terms of the median voter model. But the choices of bureaucracy are confusing. Even after being consistently surprised by them, it's relatively hard to not be surprised by the next thing they do.
Without access to these models, I think that a sense of "Bwuh?!" can turn into a sense of indignant fury and hatred for other people. If you don't have a scientific explanation for why people don't always do good things and sometimes do bad things, then what else can you do but get angry at them? But if I have access to these general explanatory models, I can talk about incentives, Pigovian taxes, voting theory, etc. I can be calm and constructive when the world seems to be falling apart.
At a glance, bureaucracies seem much more like a business to me than they do an individual or a political party. So I'd like to have something analogous to profit-maximization with which to view the behavior of bureaucracy. With regard to public health bureaucracies specifically, the theory of blame-minimization has been popular. The FDA, for example, arguably has an incentive to delay approval for new medicines because they get all of the blame for any unexpected side effects and none of the credit for the lives saved by faster approval.
But blame-minimization doesn't have the same status as being the general basic explanation for what bureaucracies do as profit-maximization has for what firms do. And I think that's just because blame-minimization doesn't work as a general basic explanation for what bureaucracies do. For example, the CDC claimed a lot of authority over the rental housing market, but a blame-minimizer arguably would want to avoid claiming authority for things. If you're in charge of something, you can be blamed for it. A blame-minimizer would allow people to get evicted and say, "Hey, there was nothing I could do about it, that's out of my jurisdiction."
And blame-minimization falls apart as a potential equivalent to profit-maximization when we try to apply it to other bureaucracies. Is the army a blame-minimizer? What about the Federal Reserve? How about unions and school boards?
Obviously, all of these institutions would always rather minimize blame, all else held even. But so would all individuals, businesses, and political parties, but we don't model them as blame-minimizers. To make money, for example, you have to accept the risk of blame when things go wrong, and we expect that firms will favor profit-maximization over blame-minimization when the former is inconsistent with the latter. By comparison, we also expect people to be work-minimizing, all else held even, yet people get jobs and start businesses. Just because something seems to be a pure bad and should always be minimized doesn't mean that its minimization is the overriding goal of the system.
When economists tried to construct general basic explanatory models of bureaucracy in the past, they came up with the budget-maximizing model, which feels like an attempt to create something directly analogous to profit-maximization. I think that's a reasonable direction to try, but it doesn't seem to have found common acceptance among economists. It's hard to make sense of a lot of bureaucracies and bureaucratic decisions as being things that cause a budget to be maximized. The empirical literature, unsurprisingly, finds that the model is kind of true and kind of false, and it doesn't seem like there's much interest in it today.
In 2013, Brad DeLong pointed out that we know very little about what causes bureaucracy to work well or poorly. Economists have fleshed-out theories of market failure and government failure but nothing comparable for bureaucratic failure. There doesn't seem to have been much progress made on the question since DeLong's post. And so even today, we aren't just unhappy about suboptimal behavior on the part of our bureaucracies, we are continually surprised by how surprising their behavior is.
DeLong said, and many others suspect as well, that we are entering into an age of increasing bureaucratic involvement in many aspects of life. It seems important to figure out the basic general rules that influence bureaucratic behavior so that we can make bureaucracies work well. But I don't have a theory that does the job.
So I'm asking you. What might be the quantity whose maximization or minimization is the basic general explanation of what bureaucracies do? And if no such quantity is apparent...why not? What is it about bureaucracy that seems to escape the methodology of economics, even when bureaucracies resemble firms in a lot of ways? And is this something that people are interested in exploring here?