Does Adam Smith ... support a Central Bank/Federal Reserve?
Adam Smith was firmly in favor of central banking (although not necessarily in the same way as understood today). In Chapter 3 Book 4 of The Wealth of Nations he writes:
“In order to remedy the inconvenience to which this disadvantageous exchange must have subjected their merchants, such small states, when they began to attend to the interest of trade, have frequently enacted, that foreign bills of exchange of a certain value should be paid not in common [coin] currency, but by an order upon, or by a transfer in the books of a certain bank, established upon the credit, and under the protection of the state”.
Regarding the Federal Reserve, it is impossible to say. Fed is not just central bank but also research institution and it does banking regulation and supervision. Adam Smith, contrary to widely held beliefs was not principally anti-regulation, in fact in WoN he supports all sorts of pragmatic regulations such as the act of navigation which he rigorously defended.
As argued by Reisman (1998) [emphasis mine]:
Adam Smith was not a single-minded advocate of a laissez-faire market in which
the minimal State had no more than a protective function. Rather, he was a
pragmatic social thinker who in each case selected the tool that was the best
suited to his meta-objective of rapid economic growth. .... In
Wealth of Nations . . . laissez-faire becomes only a qualified presumption rather
than a hard-and-fast rule
Given his pragmatic stance on regulation, and that according to our best modern understanding banking regulation is essential, Tarullo, 2019) he would likely not oppose it, although without inventing time machine we can only conjecture what he would say. In addition, central bank does not fulfill the same role in present day fiat monetary system as it did under gold standard. Again, save of having time machine we can just conjecture how he would respond to that. Smith in his writing does not directly address fiat money systems, these became commonplace only during 20th century (there are arguably few historical exceptions (see Peter Bernholz (2003). Monetary Regimes and Inflation: History, Economic and Political Relationships) but Smith does not discuss these, and he might not have been even aware of such systems.
Does ... Capitalism support a Central Bank/Federal Reserve?
- Capitalism is name for economic/social system. A system has no will it cannot support anything. That is like asking if solar system supports banning smoking in public places.
- If you are interested in knowing if having central bank is consistent with capitalism that depends on your definition of capitalism. The word capitalism is typically not used in economics as it is very poorly defined and it can mean anything (there are scholars that consider USSR capitalist country). For example, Mankiws principles of economics have word capitalism printed only three times (in a textbook of over 850 pages), and all of those three times, two are quotes from newspapers and one is from a quote from Churchill used in introduction to one chapter.
However, if you wish to use the dictionary definition of capitalism (from Merriam-Webster):
: an economic system characterized by private or corporate ownership of capital goods, by investments that are determined by private decision, and by prices, production, and the distribution of goods that are determined mainly by competition in a free market
then central banks or Fed are fully consistent with capitalist system. The description above talks about system that is just characterized by private ownership, and where the distribution of goods is mainly determined by competition in free market. Thus, having central bank or even whole sectors like let's say healthcare, controlled by government is fully consistent with capitalism provided the economy mainly relies on free market.
However, under some different specific definitions of capitalism the above may not hold. Since capitalism is poorly defined and value laden term it is best to just avoid it as done throughout most of modern economic literature (or if you have to use it it is best to refer to some specific definition).