You can achieve the desired effect by modifying the .bst file, with a small caveat as pointed out by the 'comments' in the aea.bst file:
Copying of this file is authorized only if either
(1) you make absolutely no changes to your copy, including name, or
(2) if you do make changes, you name it something other than
btxbst.doc, plain.bst, unsrt.bst, alpha.bst, abbrv.bst, agsm.bst,
dcu.bst, cje.bst, aer.bst, or kluwer.bst.
This restriction helps ensure that all standard styles are identical.
If you make the changes as I show below, please save it as a new .bst file and name it something else, like robinho.bst perhaps.
In the .bst file, find (Ctrl+F) the FUNCTION {output.month.year} entry. This controls the formatting of the month and year output. So to add the desired brackets (note that this will change all of the formatting of the year field to include brackets), replace
FUNCTION {output.month.year}
{
space month plain.space.output
space year plain.comma "year" output.check
}
with
FUNCTION {output.month.year}
{
space month plain.space.output
space "(" year * ")" * plain.comma "year" output.check
}
To influence the location of the year field, go ahead and find (Ctrl+F) in the .bst file, FUNCTION {article}. Replace
FUNCTION {article}
{ output.bibitem
author.item.check
format.title.if.not.sortkey.check
crossref missing$
{ space journal italic comma "journal" output.check
output.month.year
output.vol.num.pages
}
{ space format.article.crossref plain.space output.nonnull
comma format.pages plain.space.output
}
if$
fin.entry
}
with
FUNCTION {article}
{ output.bibitem
author.item.check
output.month.year
format.title.if.not.sortkey.check
crossref missing$
{ space journal italic comma "journal" output.check
output.vol.num.pages
}
{ space format.article.crossref plain.space output.nonnull
comma format.pages plain.space.output
}
if$
fin.entry
}
Save as robinho.bst. (Do so, for the reasons I mentioned at the beginning of this answer.)
Then consider the following MWEB:
\begin{filecontents*}{\jobname.bib}
@Article{abeler2010gift,
author = {Abeler, Johannes and Altmann, Steffen and Kube, Sebastian and Wibral, Matthias},
title = {Gift exchange and workers' fairness concerns: When equality is unfair},
year = {2010},
volume = {8},
number = {6},
pages = {1299--1324},
journal = {Journal of the European Economic Association},
publisher = {Wiley Online Library},
}
\end{filecontents*}
\documentclass{article}
\usepackage{natbib}
\usepackage{filecontents}
\begin{document}
\section{Test}
Here is an example to cite \citet{abeler2010gift} in text.
\bibliographystyle{robinho}% <-----------
\bibliography{\jobname}% <-----------
% \nocite{*}
\end{document}
which gives the output:

bstfile. To give good answers it is important to have a minimal but working example with bibliography (MWEB) not only to reproduce the problem but also to test our suggestions. And we need as much information as possible what is wrong with the example and what should be different. So please help us to help you. – Schweinebacke Jul 26 '17 at 08:04biblatex+biber. It can be configured withoutbsthacking and has several other advantages. – Schweinebacke Jul 26 '17 at 08:06bibfile and cannot replace it, e.g., by thebiblatex-examples.bibofbiblatexwe need abibfile with the entries that are valid for the example. But please have a look at the link in my first comment! – Schweinebacke Jul 26 '17 at 08:42filecontentsas shown in the community answer from the MWEB link above. If it is too large use a file hoster and add a link. – Schweinebacke Jul 26 '17 at 09:24biblatex+biberinstead obbibtex+natbib-file also be welcome? – Schweinebacke Jul 26 '17 at 09:45article? (What aboutbook,incollection,inbooketc.?) – Troy Jul 26 '17 at 15:54filecontentsas Schweinebacke mentioned in earlier comments.filecontentsis especially useful when constructing an MWEB, which you can now copy-paste as a whole into your system to compile (unlike the code in your example, which is fragmented). All we ask is for you to extend the courtesy in doing the same for us when you are asking a question relating to bibliography etc in the future. It saves us a little trouble! :) – Troy Jul 26 '17 at 17:55