Under fairly weak assumptions, the most a standard rational economic agent is willing to pay for an item they don't own (WTP) and the least they're willing to accept in exchange for that item if they already own it (WTA) should be identical. In experiments with humans, psychologists and economists have repeatedly found WTP-WTA gaps suggesting that humans aren't rational in at least this specific way. This has been interpreted as the endowment effect* and evidence for prospect theory. According to prospect theory, people are loss averse. Roughly this means that that, given their current ownership set, people value not losing stuff more highly than gaining stuff. Thus once someone gains ownership of something they suddenly value it much more highly. This "endowment effect"* on one's valuation of an item has been put forth as an explanation for the observed WTP - WTA gaps.

*Wikipedia confusingly defines the endowment effect as the gap itself, i.e. as the phenomena to be explained instead of the explanation. I suspect this is a difference in terminology among economists and psychologists, where psychologists use the wiki definition and economists use the definition I give here. However, calling the WTP-WTA gap an "endowment effect" is a bit misleading because a priori the gap may not have anything to endowments at all.

A paper (pdf) by Charlie Plott and Kathryn Zeiler investigates WTP-WTA gaps and it turns out that they may just be due to subjects not quite understanding the experimental protocols, particularly in the value elicitation process. Here's an important quote from their conclusion, but do read the paper for details: 

The issue explored here is not whether a WTP-WTA gap can be observed. Clearly, the experiments of KKT and others show not only that gaps can be observed, but also that they are replicable. Instead, our interest lies in the interpretation of observed gaps. The primary conclusion derived from the data reported here is that observed WTP-WTA gaps do not reflect a fundamental feature of human preferences. That is, endowment effect theory does not seem to explain observed gaps. In addition, our results suggest that observed gaps should not be interpreted as support for prospect theory.

A review of the literature reveals that WTP-WTA gaps are not reliably observed across experimental designs. Given the nature of reported experimental designs, we posited that differences in experimental procedures might account for the differences across reported results. This conjecture prompted us to develop procedures to test for the robustness of the phenomenon. We conducted comparative experiments using procedures commonly used in studies that report observed gaps (i.e., KKT). We also employed a "revealed theory" methodology to identify procedures reported in the literature that provide clues about experimenter notions regarding subject misconceptions. We then conducted experiments that implemented the union of procedures used by experimentalists to control for subject misconceptions. The comparative experiments demonstrate that WTP-WTA gaps are indeed sensitive to experimental procedures. By implementing different procedures, the phenomenon can be turned on and off. When procedures used in studies that report the gap are employed, the gap is readily observed. When a full set of controls is implemented, the gap is not observed.

The fact that the gap can be turned on and off demonstrates that interpreting gaps as support for endowment effect theory is problematic. The mere observation of the phenomenon does not support loss aversion-a very special form of preferences in which gains are valued less than losses. That the phenomenon can be turned on and off while holding the good constant supports a strong rejection of the claim that WTP-WTA gaps support a particular theory of preferences posited by prospect theory. Loss aversion might in some sense characterize preferences, but such a theory most likely does not explain observed WTP-WTA gaps. Exactly what accounts for observed WTP-WTA gaps? The thesis of this paper is that observed gaps are symptomatic of subjects' misconceptions about the nature of the experimental task. The differences reported in the literature reflect differences in experimental controls for misconceptions as opposed to differences in the nature of the commodity (e.g., candy, money, mugs, lotteries, etc.) under study.