Questions on microeconomics, provide solutions for the same.
(a) In the following pair of games, check whether the players' preferences over lotteries on the strategy profiles are identical (i.e. row player's preferences on the left to the row player's preferences on the right and column player's preferences on the left to the column player's preferences on the right). L M R. L M R. 2,-2 1,1 -3,7 12.-1 5.0 -3,2 b 1,10 0,4 0,4 b 5.3 3,1 3.1 -2,1 1,7 -1,-5 C -1.0 5.2 1.-2 (b) Under Postulates P1-5 of Savage, let D1, D2. . ... D,, be disjoint non-null events such that Dj~D,~ ~Do, where > and ~ are the at least as likely as and as likely as relations between events, derived from betting preferences as in the class. Given any subsets / and N' of {1, 2, ...,n}, show that UD UD - IN| 2 IN'. iEN iEN'Consider an economy with two types of individuals: skilled and unskilled workers. The only differ- ence between the two is that the skilled have a higher hourly wage w. = 40 than the unskilled do, wu = 10. Suppose that there are 400 unskilled and 100 skilled workers in this economy. Suppose that each individual has a utility function over consumption (c) and leisure (1) of the following form: U(c, 1) = In(c) + 2In(1) where I E [0, 24]. 1. (a) (3 minutes) Write down the individual's budget constraint in terms of consumption and leisure. Draw the budget constraints for the skilled and unskilled workers in the same graph with leisure on the r-axis. (b) (7 minutes) Solve for each individual's optimal leisure, labor, and consumption choice. 2. (10 minutes) Now suppose that the government wants to redistribute from the skilled to the unskilled workers. It levies an income tax which collects 20% of each skilled worker's earnings and then uses the tax revenue to give an equal amount (lump-sum transfer) T to each unskilled worker. So only the skilled workers are taxed and only the unskilled workers receive the transfer. (a) (4 minutes) On the same set of axes, draw the new budget constraints faced by the two types of individuals. (b) (4 minutes) Solve for each skilled individual's new optimal leisure, labor, and consump- tion. Does labor supply change? What is the intuition behind this result? (c) (2 minutes) Compute the total tax revenue collected by the government from taxing the skilled individuals. 3. (a) (4 minutes) Suppose that for every tax dollar collected, 6.25 cents are lost due to admin- istrative costs. Suppose that the government sets 7 so that it spends what it collects (the government balances its budget). How large is T"?An economy has two agents, Bill and Bob. Bill has $110, and Bob has $200. Utility of agents in this economy is characterized by the following function of income: U = u(y) = [ log(y - 60) if y 160 The minimum level of income possible in this economy is 60. Each agent is about to choose a new business venture, and has a choice between project A and project B. Neither project requires any investment up front. Project A yields revenues of 20 with probability = and revenues of -10 with probability -. Project B yields revenues of 4 with probability one-half and revenues of 5 with probability one-half. Throughout this problem, assume that fractional income is possible. (a) (5 points) Which project would each agent choose? Provide intuition for your answer. (b) (5 points ) If Bill and Bob each choose an investment project each year and receive the associated income for 20 years, will the expected gap in their incomes be larger or smaller at the end of this period than it was initially ? How does this relate to attitudes toward risk? You do not need to calculate income over 20 years, just provide intuition. (c) (10 points) Now, assume that there is a job available that provides fixed wage income. What salary would the job have to provide in order to induce Bill to take the job rather than entering a new business venture? What salary would the job have to provide in order to induce Bob to take the job? Which is higher, and why? Algebraic expressions are acceptable as answers
