Home Production, Consumption Tax, and the La'er Curve: Consider the following oneperiod representative household model. There is a representative household and she decides how to allocate her time between market and nonmarket market activity. Let n denote the time allocated to market activity! and (1 7 n) denote the time allocated to nonmarket activity so that her total time endowment adds up to 1. Her preferences are: u.(cm._ ch) : (cfn + c (1) where p g 1 is a parameten Cm. denotes the consumption of market-produced good and ch denotes the consumption of homeproduced good, egg cooking, babysitting. l\\-larketproduced good can only be purchased in the market and subject to a budget constraint as follows: (1 + T)cm : "Lu-n... (2) Here, T is the consumption tax on the marketproduced good, and u: is the constant wage rate. Note that 71 corresponds to the labor supply in our consumption-leisure framework. Home- produced good cannot be purchased from the market and is produced (at home) according to a linear technology: ch : a(1 it). Note that there is no direct utility from leisure._ but indirectly from the production of home good. Moreoven there is no tax on home-produced good: e. g. the government cannot collect tax from the food you cook and eat. at home. Per the purposes of this question= there is no labor income tax either. (a) Write down the Lagrangian for the utility maximization problem and obtain the rst order conditions. (h) Solve for optimal n. Holding everything else constant5 how does an increase in the produc- tivity of home production5 i.e.5 larger a, aect RE the labor supply? Explain intuitively. (c) For the rest of the questiom assume that ,0 : 0.5, o: : 1.0. and w : 1.0. Solve for optimal cm and ch. (d) Calculate government tax revenue when T : 1. (e) \"-Yhat value of T maximizes the government tax revenue