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public sector economics. the above question is on the topic commodity taxation.

Consider an economy with a representative firm and a representative household. The household sells labour, L, and buys the consumer good, x, produced by the firm. The utility function of the household is U=2x /2-L . The firm has the production function x=L. The consumer good is the numeraire good with price 1. The wage rate paid by the firm is labelled w. The government imposes a linear tax on wage income. In specific, the total income tax paid by a household with income Z is T=-T+tZ, where 1 20 and t20 are parameters set by the government. (Notice: If T 20, then the income tax system is progressive as the average tax rate increases with income.) a) Solve the firm's problem to find the wage rate w paid by the firm. b) Find the household's consumer good demand and labour supply as functions of t and T. c) Find the household's (indirect) utility as a function of t and T. d) The government balances its budget, i.e. 7 = 0. Find the household's (indirect] utility as a function of t under the assumption that T = 0. e) Find the tax rate t* maximising the household's utility (under the assumption of a balanced government budget). f) Explain the result obtained in e)