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Y Files X + 6 C' & yaleinstructurecom 2021 Problem Set 5.pdf o I: e so utions, : e sure your responses are concise an- to t e point. 1 Optimal Consumption and Labor Supply Ms. Williams's preferences over consumption of food and leisure are represented by the utility function U(C, L) : lnC+ ln(16 , L) C 2 0 is the number of pounds of food she eats in a day, 16 L is the number of hours of leisure per day, and L is the number of hours she works in a clay.1 Note that In is the is the natural logarithm. Ms. Williams earns $111 per hour spent working. A pound of food costs $1). Ms. Williams also receives $1r in dividends from stocks she owns, and pays a lump-sum tax $T (a lump-sum tax is a xed amount due irrespective of whether a person earns income or not). Ms. Williams works L hours, earns her labor income, and then purchases C pounds of food for consumption of her labor and dividend income, net of taxes. She chooses C and L to maximize her utility U(C, L), taking all prices as given. 1. Write Ms. Williams's optimization problem, inclusive of all constraints. Make sure that all magnitudes in the budget constraint, both incomes and expenditures, are exprased in $, which in this model is the unit of account. Remember that neither labor supply nor leisure can be too low nor too large, and that pounds of food cannot be negative.2 (I expect you to write each of these constraints in mathematical terms). '1 in assuming here that Ms. Williams sleeps eight hours a day. I am assuming the only activities are sleep, leisure and work, 2Assume that the 8 hours of sleep cannot be modied