Answer Happy

Please, help me to solve this problem, the marked questions 6, 7, 8, 10, 11, 12, 13, 14. Thank you! Luxembourgers consume only coffee and cigarettes. Each Luxembourger has the same utility function u(x, y) = ^y1-4, where r and y are the amounts of coffee and cigarettes consumed, and a € (0,1). Luxembourger i has income mi, and the aggregate national income is M = Limi. 1. Which of the following functions does not represent the same preferences as the utility function above? 10. Suppose that instead the government simply wants to collect in taxes amount S > 0, where S < (1 - a)M. First, the government decides to introduce an income tax in order to collect S. Income tax works the following way: each consumer pays fraction de (0,1) of her income. That is income of consumer i after tax is (1 - d)mi, and dm; goes to the government. Find d such that the government collects S in taxes (as a function of Pa, P, M,,S). 1. (Continued) Alternatively suppose the government decides to tax cigarettes. It collects fraction ty € (0,1) of the sale price of cigarettes. That is each consumer pays (1 +typ, for cigarettes, and typy goes to the government. Find ty such that the government collects S in taxes (as a function of Pr, Py, M,,S). 12. (Continued) How much additional income should we give each person i in order to compen- sate i for the introduction of the cigarette tax in question 11 (as a function of pr.py, M, a, S)? 13. (Continued) Which one from the above two policies is better for consumers, income tax or cigarette tax? Explain your answer. (a) Income tax (b) Cigarette tax 14. (Continued) Suppose the government cannot introduce an income tax, but can tax both coffee and cigarettes instead. It chooses a tax rate for coffee, t, (0,1), and a tax rate for cigarettes, ty € (0,1). Find t, and ty such that the government collects S in taxes and this is the most efficient way to do so as a function of pr. P, M,,S). That is there is no other pair of tax rates which collects S or more in taxes and leaves each consumer with a higher utility. (a) a Inc + (1 - a) In y (b) 100.222(1-4) (c) 241-a)zy (d) xyl-a + (1 - a) 2. Find an absolute value of MRS at a point r = 10, y = 20 when a = 0.5. 3. Find an absolute value of slope of a budget line of i when p, = 1 and Py = 2. 4. Find coffee and cigarettes demand function of consumer i. 5. Find a total demand (demand aggregated across all consumers) on cigarettes in Luxem- bourg as a function of (P1.py, M). The government wants to reduce the cigarette consumption by d percents using a tax. Suppose that after the introduction of the tax the price of cigarettes is simply (1 + t Py; where t is a tax rate. Find tax rate t that implements the d percents reduction in the nationwide cigarettes consumption. That is if c is the initial amount of cigarettes consumed, then after the introduction of the tax the new cigarette consumption is (1 - d).. (Continued) The introduction of the tax appears to be very unpopular among Luxembourg- ers. The government is afraid of loosing the support of voters, and decides to compensate them by distributing subsidies, such that the income with subsidy is m:(1+si). Find the subsidy s; for each Luxembourger i, such that after the introduction of both, the tax from (a) and the subsidy, the utility of each i is the same as before the changes. Show that, to the annoyance of the government, after imposing the tax computed in (a) and compensat- ing the people using subsidies si, it does not achieve the goal of d percents reduction in smoking (Continued) Find a tax rate and a subsidy such that the government achieves d percents reduction in the cigarette consumption and the consumers receive the same utility as before the tax.