Public sector economics question

Consider an economy with one (type of) household and one (type of) firm. The household sells labour, L, to the firm and buys the consumer good (x) produced by the firm. The utility function of the household is U=2x1/2-L. The firm's production function is x=L. Assume that labour is the numeraire good and that the wage rate is 1. The government taxes labour income to satisfy a total revenue requirement R=1/4. Assume first that the government uses a strictly proportional tax, i.e. a household with total pre-tax income Z pays the income tax T(Z)=tZ where t20. The government seeks to maximise the representative household's utility while collecting total income tax T(Z)=R=1/4. (Assume that government spends the tax proceeds in a way that does not impact the household's utility, i.e. the tax proceeds are wasted.) a) Show that the price of the consumer good is p = 1 when the firm takes prices as parametrically given. b) Find the household's labour supply and consumer good demand as a function of the tax rate t. c) Find the household's (indirect) utility as a function of the income tax rate t. d) Set up the government's problem and show that the income tax rate maximising social welfare while raising the required tax revenue is * =1/2. e) Show that when t*=1/2, the household's labour supply is L*=1/2 and the consumption is x*=1/4. Calculate the household's utility level U* in this case. Now assume that the government modifies the income tax system. When taxing incomes (= labour supply) up until L*=1/2 by the tax rate t*=1/2, the government collects the required revenue. It therefore decides to allow the part of household income that is above 1/2 to be untaxed, while the tax rate remains t=t*=1/2 for the part of income below 1/2 To summarize, the tax rate according to the new regressive tax system is: t=1/2 for L S L*=1/2 and t=0 for L > L*=1/2. Denote by AL the extra labour the household sells in addition to L', i.e. the total labour supply is now L*+AL. Likewise, denote by Ax the extra consumption the household buys in addition to x*, i.e. total consumption is now x*+ Ax. As the tax rate on income exceeding L' is zero, all extra income can be spent on consumption, i.e. AL= Ax. f) Explain briefly why the household's utility can be written as U=2(x*+ Ax)1/2+(L*+ AL). when Ax2 0 and AL20. g) Show that Ax= AL=3/4 maximises the household's utility given that "extra income" is untaxed. I.e. maximise with respect to Ax and AL the household's utilityU=2(x*+ Ax)1/2+(L*+ AL). subject to Ax = AL , taking into account that L'=1/2 and x*=1/4 are constants. h) Calculate the representative household's utility under the new regressive tax system where the marginal rate of high incomes is equal to zero. i) Compare the household's utility under the strictly proportional income tax (cf. e)) with the utility under the regressive tax system (cf. h)). Explain the economic intuition behind the result