According to Vela Velupillai, Orthodox Economic Theory is not Turing Complete, because it is "replete with uncomputabilites". On this 2009's article he states:
Economic theory, at every level and at almost all frontiers – be it microeconomics or macroeconomics, game theory or IO
– is now almost irreversibly dominated by computational, numerical and experimental considerations. Curiously, though, none of the frontier emphasis from any one of these three points of view – computational, numerical or experimental – is underpinned by the natural algorithmic mathematics of either computability theory or constructive analysis. In particular, the much
vaunted field of computable general equilibrium theory, with explicit claims that it is based on constructive and computable
foundations is neither the one, nor the other. Similarly, Newclassical Economics, the dominant strand in Macroeconomics,
has as its formal core so-called Recursive Macroeconomic Theory. The dominance of computational and numerical analysis,
powerfully underpinned by serious approximation theory, is totally devoid of computable or constructive foundations.
The reasons for this paradoxical lack of interest in computability or constructivity considerations, even while almost the
whole of economic theory is almost completely dominated by numerical, computational and experimental considerations,
are quite easy to discern: the reliance of every kind of mathematical economics on real analysis for formalization. [...]
Anyone with a modicum of expertise in recursion theory, constructive analysis or even nonstandard analysis in its constructive modes, would find, in any reading from these more algorithmically oriented perspectives, the citadel of economic theory, game theory and IO replete with uncomputabilities, undecidabilities and non-constructivities – even elements of incompleteness.
Section 2 of that paper explores some examples of uncomputability in economic theory.
This author is one of the leading figures in the field called "Computable Economics", which is an attempt to produce Turing Complete economic theory.
In 2000 he compiled a book on the subject. There is also a 2009 handbook of Computable Economics. The 2000's book opens with a quote that hints why Game Theory is not Turing Complete:
[T]here are games in which the player who in theory can always win cannot do so in practice because it is impossible to supply him with effective instructions regarding how he should play in order to win. Rabin (1957: 148; emphasis added)
Then, the authors state:
The key word is “effective,” referring to a procedure whose execution is specified in a finite series of instructions, each of which is finite in length and where all the details of the execution are specified exactly, thereby leaving no room for magic, miracles, or other such metaphysical entities. The exact meaning of “effecitivity” is mathematically equivalent, under the Church–Turing thesis, to “computability.”
So, as I understand this, there are some games for which there is no algorithm that could help us "compute" such optimal wining strategy in practice. Hence, Game Theory is uncomputable, ergo Turing incomplete.
Herbert Simon is perhaps the highest profile, relatively mainstream economist contributing toward Computable Economics. Actually, he got both the Nobel Prize in Economics and the ACM Turing Award (regarded as the "nobel prize for computing").