My advisor told me not to send Duke my 20-page essay but only its introduction section. The reason was that the graduate committee will not have enough time to read a lengthy paper. What their most concern was that whether or not students are able to use what the knowledge they learnt from undergraduate economics courses to conduct graduate level research, a combination of economics, english, and mathematics or statistics knowledge. If you just put down several math equations without detailed analysis, that would not work for the writing sample requirement. good luck.