We present a theory of endogenous policy preferences and electoral competition with boundedly rational voters who find it costly to recall detailed information. Voters are otherwise fully rational, and they strategically choose how much    memory to devote to processing political information. We find that even if all voters start with a common prior such that given this prior they all prefer a moderate policy over either a left or a right alternative, and even if they only observe common signals that in the limit would make a perfectly rational observer certain that the moderate policy is indeed best for everyone, voters with costly memory capacity will eventually prefer extreme policies, and the electorate polarizes: some voters support the left policy, and some support the right policy. Two fully rational political parties polarize as well, one pandering to the voters on the left, and the other pandering to the voters on the right.

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