f01mt_a - GS/ECON 5300 Public Economics I October 2001...

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GS/ECON 5300 Public Economics I October 2001 Answers to Mid–term 1. The best answer is to take the definition of the excess burden L ( p, t, u ) E ( p + t, u ) - E ( p, u ) - t · X c ( p + t, u ) where p is the vector of pre–tax prices, t is the vector of unit taxes, u is the original ( pre– tax ) level of utility, and X c ( · , u ) is the vector of the person’s compensated demands. E ( p, u ) is the person’s expenditure function, defined as the cost of the lowest–cost consumption bundle which gives her a utility level of u , when the price vector is p . From this definition of the expenditure function E ( p, u ) = p · X c ( p, u ) and E ( p + t, u ) = ( p + t ) · X c ( p + t, u ) since the compensated demand function is the vector of quantities of goods which minimize the cost of the given level of utility. Therefore L ( p, t, u ) =( p + t ) · X c ( p + t, u ) - p · X c ( p, u ) - t · X c ( p + t, u ) = p · X c ( p + t, u ) - p · X c ( p, u ) X c ( p, u ) is the bundle which minimizes the cost of attaining a utility level u , when prices are p . Therefore any other consumption bundle which achieves the same level of utility must cost more than p · X c ( p, u ). In particular, the bundle X c ( p + t, u ) achieves the same level of utility u . So — at prices p X c ( p + t, u ) costs more than the bundle X c ( p, u ), so that p · X c ( p + t, u ) > p · X c ( p, u )
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