For a given amount of power, higher voltage equals lower current. Lower current means you can use smaller conductors.
The real problem with DC distribution is the inability to readily convert from one voltage to another. In an AC system, power is distributed at a high voltage and transformed to a lower voltage at the point of use. That's the purpose of the transformer mounted on the pole feeding your house.
If we had to distribute power at 240 volts, the cables would be prohibitively large, which was part of the problem with Edison's system. For example, 100 amps at 240 volts is the same amount of power as 1.8 amps at 13,000 volts (a common distribution voltage). There's a very significant difference in the size of wire necessary to carry 1.8 amps as compared to the size necessary to carry 100 amps.
The problem with DC distribution is the difficulties in converting from one voltage to another, as I pointed out earlier. In an AC system, a transformer is all that's needed to convert from 13 kV to 240 volts at your house.
Because of the skin effect at 60 Hz, DC power can use a slightly smaller wire size than an equivalent amount of power at AC. However, only when very high power and long distances are involved do the lower losses in the cables offset the difficulties of converting DC voltages.
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