The showkeys
package might do just what you're looking for:
\documentclass[12pt]{article}
\usepackage[style=verbose,citestyle=alphabetic,sorting=nty,natbib=true]{biblatex}
\begin{filecontents}{MCMC.bib}
@ARTICLE{Ait04,
author = {Y. A\"{i}t-Sahalia},
title = {Disentangling Diffusion from Jumps},
journal = {Journal of Financial Economics },
year = {2004},
volume = {74},
pages = {487-528}
}
@ARTICLE{Ait02,
author = {Y. A\"{i}t-Sahalia},
title = {Maximum-Likelihood Estimation of Discretely-Sampled Diffusions: A Closed-Form Approximation Approach},
journal = {Econometrica },
year = {2002},
volume = {70},
pages = {223-262}
}
\end{filecontents}
%\bibliography{MCMC}
\addbibresource{MCMC.bib}% is recommended for biblatex
\usepackage{showkeys}
\begin{document}
\nocite{*}
\printbibliography
\end{document}

I solved my issues of compatibility between the BibTex file exported from Zotero and BibLatex. I recap the solution here.
Apparently BibLatex is much more sensible than BibTex when it comes to reading a BibTex file...
To export correctly the bibliography from Zotero and then into a Tex document via BibLatex I used this BibTeX Export Translator (modified by Robin Wilson) with two important tweaks:
1) On line 11 of the translator script you need to set "exportNotes": true,
to "exportNotes": false,
(of course just in case you have added notes to your Zotero items, e.g. you extract annotations from PDFs). This will avoid BibLatex crashing on too long fields.
2) On line 2256 you need to tell the translator to avoid adding a comma after each bibliography item (the comma will result in BibLatex giving a warning on the first line of each entry excluding the first...) changing this line: Zotero.write((first ? "" : ",\n\n") + "@"+type+"{"+citekey);
with this line: Zotero.write((first ? "" : "\n\n") + "@"+type+"{"+citekey);
Another possible source of problems of the BibTeX Export Translator could be the character encoding. In my translator I kept as in source file "exportCharset": "ISO-8859-1",
on line 10. I tried to change it to UTF-8 but it created another class of issues, not with BibLatex (it run without errors) but with Latex,
[1{/usr/local/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
(./bib/price2012.tex [2]
! Undefined control sequence.
<to be read again> \edef \blx@tempa {193\x
{FFFD}\x {FFFD}\x {FFFD}219}
l.8 \printbibliography[heading=subbibliography]
for each refsection. It also messed with the "pages" field of some items, outputting:
Andrea B. Hollingshead. “Information suppression and status persistence in group decision making the effects of communication media”. In: Human Communication Research 23.2 (1996), 193fffdfffdfffd219.
I then switched back to "exportCharset": "ISO-8859-1",
and both BibLatex and Latex run without warnings or errors. The PDF output was correct:
Andrea B. Hollingshead. “Information suppression and status persistence in group decision making the effects of communication media”. In: Human Communication Research 23.2 (1996), 193–219.
and I also get correct output with the umlaut of Habermas:

EDIT: The problem with the page field was probably due by a odd "-" character that you sometimes get to separate the 2 page numbers when you download the citation from the Internet.
Best Answer
should be what you are looking for. It prints all of your entries, including the uncited ones into your bibliography. You enter it somewhere between
\begin{document} ... \end{document}
I'm including locksteps comment into the answer: