The answer to this question will depend on how you construe 'cognitive abilities.' For instance, for certain formulations the answer is trivially yes: an economist who learns psychology may come away with psychological knowledge that could help his economics, like how to space out his studying to maximally improve retention; or how to deal with the cognitive dissonance of causing widespread financial collapse. Demonstrations of transfer in the other direction -- from economics to psychology -- are even easier to demonstrate. Our department, for example, has in recent years seen an influx of psychology graduate students whose backgrounds are in much more mathematically sophisticated disciplines. These students are able to leverage their mathematical skills to great advantage in what has, in past years, sometimes been a field with excessively 'fluffy' methodology. But in both these instances -- psych --> economics and economics --> psych -- we're talking not so much about 'transfer' of knowledge as of literal re-use. If you buy paper plates for a picnic, can you also use them inside your house? Of course you can.
The question gets harder when we zoom in further. If we accept as given that the techniques of economics would be useful in the field of psychology, what about the atomic cognitive units which collectively constitute the edifice of economics knowledge? Does the way of viewing the world one adopts when one has knowledge of economics, or the process of abstract thinking and symbolic manipulation one must acquire in the course of it, help with other disciplines in ways that do not directly relate to full-blown techniques one learns in economics?
The short answer is that nobody knows, so far as I'm aware. Which doesn't mean that people don't have strongly held beliefs. For many years students spent years learning Latin because it was thought to have various beneficial effects, though these have never been demonstrated to my knowledge. Coming back to the economics example, the issue is that nobody seems to know what it even means to know economics. Aside from the specific ability to solve economics problems, what kind of cognitive capacity can an economics expert demonstrate? We have only begun to be able to answer this question with regards to mathematics (c.f. Ansari et al, 2005.) or to much of anything else. Our sense of what mental representations are -- of what knowledge is built from -- is extraordinarily primitive.
However, there are substantive reasons to continue to believe in the possibility of non-trivial transfer. The most compelling of these is the study of metaphor, for which the canonical work is probably Lakoff & Johnson's (1980) phenomenal and very readable book (see also Bowdle & Gentner, 2005; Holyoak & Thagard 1997 for more technical treatments.) The idea is that knowledge structures representing one thing can sometimes be mapped onto knowledge structures representing another thing. When such a mapping can be found, one is able to make additional inferences on the second thing that correspond to inferences based on the first thing. According to these theorists, the ability to make inferences about the world that flow from these kinds of metaphorical transfer account for a large part of our intellectual ability.
An example might be useful. A prominant metaphor, according to Lakoff and Johnson, is "An argument is a war." According to this metaphor, we might understand a destination domain (argument) in terms of a source domain (war.) This mapping between the two domains allows us to structure the former in terms of the latter. Wars have winners and losers, for instance; wars end when one side surrenders or is destroyed; one succeeds in war by hurting one's enemy; and so on. By extension, then, an argument has a winner and loser; an argument ends in surrender or destruction; and in an argument the goal is to hurt your opponent.
This construction of what it means to have argument, and of how one should conduct onself in the course of an argument, would lead to very different results than, say, the metaphor "An argument is a dance." It's clear, from this formulation, how consequential is the choice of metaphor in constructing one's attitude to and behaviors during an argument. Metaphors are not simply colorful language (the argument goes) they are instructions for how we might leverage certain knowledge structures and behaviors (those involving war, in this case) for application in a completely unrelated domain (an argument, in this case.)
The most powerful idea latent in this metaphorical construction of cognition is that metaphorical extension is not only (or even mostly) about language -- any knowledge representation can potentially be aligned with another, and the tools and circumstances governing the use of the former could (potentially) be used to gain 'experience' in the latter automatically, for free. (See Durso et al., (1994) for a nice example of how this might culminate in an 'insight' experience.) And in fact, DARPA has been very interested in this kind of 'bootstrap learning' looking for ways where experience in a relatively cheap (in dollars or risk) domain might be leveraged to benefit a more expensive domain, although the results of this work have not yet lit the world on fire.
Concerning the automaticity of this process, the answer seems to be both yes and no. Insofar as people do use some kind of metaphorical extension to understand the world, and to apply knowledge across situations, then the process occurs automatically, at least to some extent. (Note that the details of this process are still very much in question.) But even if it is automatic, Chrysikou (2006) showed that the cross-pollination between disparate types of knowledge can be improved by conscious effort. In her study, subjects who spent ten minutes performing the "creative uses" task (in which they generated novel and unusual applications for various objects; for example, a creative use for a toothpick might be as a javelin for a grasshopper) solved significantly more insight problems than subjects who did not perform the CUT beforehand. One theory for these results is that the process of coming up with creative uses energized the machinery used to create ad hoc categories (see Barsalou 1983), and that the ability to blend disparate forms of knowledge in these dynamic bundles is an important component in problem-solving.
This seems relevant to the original question in that whatever benefit is automatically gained simply by possessing a varied store of cross-discipline knowledge, seems to be enhanced by purposeful engagement and cross-linkage between these cognitive components, an enhancement that manifests concretely in increased problem-solving performance.
All that being said, in my opinion the extent and mechanisms of cross-domain transfer are still largely undetermined. Despite the considerable theoretical literature in support of it, at this point the proponents of deep transfer are running on faith as much as anything.
EDIT: Note that an earlier question on this site explores some of these same ideas.
Ansari, D., Garcia, N., Lucas, E., Hamon, K., & Dhital, B. (2005). Neural correlates of symbolic number processing in children and adults. NeuroReport, 16(16), 1769.
Barsalou, L. W. (1983). Ad hoc categories. Memory & Cognition, 11(3), 211–227.
Bowdle, B. F., & Gentner, D. (2005). The career of metaphor. Psychological Review, 112(1), 193–216.
Chrysikou, E. G. (2006). When Shoes Become Hammers: Goal-Derived Categorization Training Enhances Problem-Solving Performance. Journal of Experimental Psychology: Learning, Memory, and Cognition, 32(4), 935–942.
Durso, F. T., Rea, C. B., & Dayton, T. (1994). Graph-Theoretic Confirmation of Restructuring during Insight. Psychological science, 5(2), 94–98. Association for Psychological Science.
Holyoak, K. J., & Thagard, P. (1997). The analogical mind. The American psychologist, 52(1), 35–44.
Lakoff, G., & Johnson, M. (1980). Metaphors We Live By. The University of Chicago Press.