Freakonomics co-author Steven Levitt blogs an idea  which, frankly, demonstrates why we're pretty justified in keeping a sharp divide between economists and politicians:
In Glen’s voting mechanism , every voter can vote as many times as he or she likes. The catch, however, is that you have to pay each time you vote, and the amount you have to pay is a function of the square of the number of votes you cast. As a consequence, each extra vote you cast costs more than the previous vote. Just for the sake of argument, let’s say the first vote costs you $1. Then to vote a second time would cost $4. The third vote would be $9, the fourth $16, and so on. A person who cast four votes would have to pay a total of $30 (1+4+9+16=30). Twenty votes would cost $2,870. One hundred votes would cost you more than $300,000. Five hundred votes would cost more than $40 million. So eventually, no matter how much you like a candidate, you choose to vote a finite number of times.
What is so special about this voting scheme? People end up voting in proportion to how much they care about the election outcome. The system captures not just which candidate you prefer, but how strong your preferences are. Given Glen’s assumptions, this turns out to be Pareto efficient  — i.e., no person in society can be made better off without making someone else worse off.
Levitt deals with some potential criticisms on his blog, but only in passing; and while he argues that there is support for the idea in a laboratory, the laboratory experiments didn't deal with the major problem with the idea in the real world, which is that when the difference in wealth spreads several orders of magnitude, it couldn't fail to give more voice to those with more wealth, especially when it comes to issues where the rich speak as one (like, perhaps, taxation of the wealthy).
In addition, the proposal is only examined from an economists point of view, when it is an area also well studied by political scientists. An important aspect of voting, for instance, is that while we may talk of "wasted" votes in majoritarian systems, very little has been actually wasted. If you have to buy votes, then "safe" constituencies would basically never change hands, as the minority party's turnout would collapse. That, in turn, would likely see the majority party's turnout also collapse, which could set up frankly strange chaotic cycles, especially in a three+ party system.
Levitt also mentions the prospect of fraud, but focuses on a strange aspect; the problem seems less to be that people would sell their votes, and more that a system set up to take multiple votes per person removes one hurdle to voter fraud that we have now.
Add to those problems the fact that the system as designed locks anyone out of the electoral process who doesn't have enough money to spare on it; that one-person-one-vote was always defended for philosophical, rather than practical, reasons; and that a far more serious problem with elections from the point of view of an economist is that being forced to communicate acceptance of a broad set of policies with only a yes or no answer to a question every five years is a stupidly inefficient way to gauge public preferences.
So: be glad economists don't run countries, only their money.