Dilip Mookherjee SOLUTIONS TO MIDTERM EXAMINATION
نویسنده
چکیده
Household i’s utility maximization problem is to maximize min{xi1 λi1 , xi2 λi2 , . . .} subject to Σlpl[xil − ωil] ≤ 0. It is optimal for i to equalize xil λil across all goods (first note it is strictly optimal to equate this ratio across all goods with a positive price, and then adjust the consumption of the free goods so that their ratio is at least as high as for those goods that are expensive). Hence without loss of generality, household i’s demand for good l is proportional to its demand for good 1: i.e., xil = λil λi1xi1. The household must also exhaust its budget, otherwise it can increase the consumption of all goods in step and be better off. Hence total expenditure Σlplxil = xi1Σlpl λil λi1 must equal total income Σlplωil, implying that it is optimal for i to consume the first good in quantity xi1 = λi1 Σlplωil Σplλil , and correspondingly the other goods in quantity xik = λik Σlplωil Σplλil . This corresponds to an excess demand function for i: Zik = λik Σlplωil Σplλil − ωil
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