Aggretation is a necessary evil in that what you lose in detailed information gets made up for in managability. So far it's done remarkably well at explaining quantitative movements in variables ex post. Now what we need is to move away from calibration and make some ex ante predictions.
Being more tractable is not a good justification. As I said, there is no reason to believe that the economy acts like an individual at the aggregate level, whether this makes the math more tractable or not. We are making a very strong assumption, which may even be contradicted by our model (if the utility function does not have certain properties).
As for explanation, it's incredibly difficult to tell how well these simulations replicate the data. Yes, sometimes the model seems to follow the downturns and upturns in the data. Other times, it fails to do so. However, there are so many degrees of freedom and arbitrary calibrations in these models that fitting the data is not a very good way to judge the theory. Many RBC theorists seem to aknowledge this (in a paper on Argentina, Kydland even rejects the data, rather than his model).
Where exactly does praxeology diverge from mainstream micro? In the case of price determination, we get demand schedules from maximising utility subject to a budget constraint, and supply schedules from maximising profit subject to a cost structure; then solve from the equilibrium condition. How does praxeology do it differently?
I'm so glad you asked! Buyers and sellers form valuations of the goods they would like to buy and sell w.r.t money, and given their expectations of future prices, these form the maximium buying prices and minimum selling prices they would be willing to buy or sell goods for. These form discrete demand and supply series, or as Rothbard called them, "schedules." Giiven that it is in the interests of all more competitive participants in a market to not needlessly outbid their competitors but to seek the best deal they can get, this explains why there is a tendency for them to prefer trading at the price between the bids of the marginal seller and marginal buyer (which is further narrowly determined if bids of the most competitve submarginal buyer and seller also lie within this range). It is not a theory of simultaneous determination; the price forms as a consequence of a competitive process in which submarginal buyers and sellers are gradually outbid/undersold.
For those buyers and sellers rendered submarginal, who however may have entered a market under the impression based on their expectations making insufficeint bids; this must lead to drastic revisions in their own plans of provision. But something very similarly, this must also have been the case for all supramarginal traders(all buyers with max. buying prices exceeding the clearing price), since if they knew in advance they could have bought at the clearing price, they could have allocated the savings made elsewhere, either to hold the money or to spend on other goods/services. Perfect anticipation would actually imply acute elasticity therefore for both buyers and sellers in the vicinity of the clearing price, before sharp inelasticity as soon as the price was reached. This also explains why price determination tends to be more rapid in markets populated by traders more accustomed to dealing with each other, and hence what their respective impressions of the going price may be.
Hence the Austrian theory of price does not require perfect anticipation of future prices or the demand/supply situation on a market to explain how these prices are formed as a result of a competitive process. This is most vividly the case for firms. They make factor bids based on expectations of future prices, not by solving an optiisation problem using "given" factor and product prices. Their plan revisions must come about not only as a consequences of failures of anticipations reflected in their profit-loss statements having conducted an entire production process, but also intermittently through the failure of buy/sell bids on each market.
Furthermore, a key principle underlying the Austrian price theory is diminishing marginal utillity (in purely ordinal form), explaining why max bids for further purchases of goods tend to drop for buyers (the opposite holds for sellers), and this informs the downward sloping nature of the demand series. This is not the case with neoclasical price theory. What is fundamental there is that indifference surfaces are convex and continuous (hence importantly implying a diminishing marginal rate of substitution), allowing for a unique tangency with a flat budget surface constraint. Positive, monotonic utillity and diminshing marginal utility are "nice" interpretive assumptions that play no fundamental role in this theory price determination (since the MRS will stay invariant given all functional transformations, not just monotonic, as a trivial consequence of he chain rule of calculus, inspite of what micro lecturers seem to inanely repeat). Consequently it loses much of its connection to economic interpretation. At best it's just a way to represent tradeoffs in an impossible situation of perfect anticipation, often assuming exchangers as infinitesimally minute price takers for convenience.
If you seriously want to learn how the Austrian theory of price works in greater detail than what I've very scantily said above, I recommend the following:
Man, Economy and State, Murray Rothbard, pp.103-161 (for an excellent, basic introduction)
Positive Theory of Capital, Eugen von Bohm Bawerk pp.193-234 (indispensable read), also pp.129-189 may help you begin to appreciate where we're coming from in terms of value theory (note of course BB was corrected later with regard to cardinal utility by Mises and Franz Cuhel).
I'm currently myself working on producing an essay that will compactly clarify and present in a slightly more "formal" manner this price theory, as well as discuss in greater detail its practical, analytical and philosophical advantages. Let me know if you'd like me to end it.
"When the King is far the people are happy." Chinese proverb
For Alexander Zinoviev and the free market there is a shared delight:
"Where there are problems there is life."