Consumer has quadratic preferences and cares about consumption over two periods: U(c0,c1)=21(c0c)221(c1c)2. Assume that the real interest rate, r, is zero, and the time discount factor, , equals 1. (Note (!) that consumption can be negative if preferences are quadratic.) (a) Consumer's disposable income in period 0 equals 10 , and in period 1 equals 20. There's no uncertainty. Write down the Euler equation and find the optimal consumption levels in periods 0 and 1 , and the optimal savings. (b) Assume now that period 0 income stays at 10 , while period 1 income is uncertain. There're two possible states of nature that might realize in period 1 with probability =31, income will equal 0 in period 1 if state 0 occurs whereas with probability 1=32 income will equal 30 in period 1 if state 1 occurs. Consumer has to make decision about her consumption and saving for period 0 before uncertainty is resolved. Consumer now maximizes expected utility EU(c0,c1)=21(c0c)221(c1(0)c)2(1)21(c1(1)c)2 where c1(k) is consumption in period 1 , state k=0,1. (i) variance of income in period 1. (ii) Find the optimal consumption and saving in period 0 , and consumption in period 1 in both states of nature. (iii) Does your answer for the optimal consumption in period 0 and savings differ from the answer to (5a), and why it does or why it doesn't? Assume now that income in period 1 state 0 equals 0 with probability =0.99 and income in period 1 state 1 equals 2000 with probability 1=0.01. (i) Write down the Euler equation and find the expected value and variance of income in period 1. (ii) Find the optimal consumption and saving in period 0 , and consumption in period 1 in both states of nature. (iii) Does your answer for the optimal consumption in period 0 and savings differ from the answer to (5b), and why it does or why it doesn't? Assume now that each period's utility function is u(c)=ln(c). Continue assuming that the real interest rate, r, is zero, and the time discount factor, , equals 1. Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (5a). Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (5b). Compare the optimal saving to the value you found in (5b) and argue why they are different (if different at all). Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (5c). Compare the optimal saving to the value you found in (5c) and argue why they are different (if different at all). Consumer has quadratic preferences and cares about consumption over two periods: U(c0,c1)=21(c0c)221(c1c)2. Assume that the real interest rate, r, is zero, and the time discount factor, , equals 1. (Note (!) that consumption can be negative if preferences are quadratic.) (a) Consumer's disposable income in period 0 equals 10 , and in period 1 equals 20. There's no uncertainty. Write down the Euler equation and find the optimal consumption levels in periods 0 and 1 , and the optimal savings. (b) Assume now that period 0 income stays at 10 , while period 1 income is uncertain. There're two possible states of nature that might realize in period 1 with probability =31, income will equal 0 in period 1 if state 0 occurs whereas with probability 1=32 income will equal 30 in period 1 if state 1 occurs. Consumer has to make decision about her consumption and saving for period 0 before uncertainty is resolved. Consumer now maximizes expected utility EU(c0,c1)=21(c0c)221(c1(0)c)2(1)21(c1(1)c)2 where c1(k) is consumption in period 1 , state k=0,1. (i) variance of income in period 1. (ii) Find the optimal consumption and saving in period 0 , and consumption in period 1 in both states of nature. (iii) Does your answer for the optimal consumption in period 0 and savings differ from the answer to (5a), and why it does or why it doesn't? Assume now that income in period 1 state 0 equals 0 with probability =0.99 and income in period 1 state 1 equals 2000 with probability 1=0.01. (i) Write down the Euler equation and find the expected value and variance of income in period 1. (ii) Find the optimal consumption and saving in period 0 , and consumption in period 1 in both states of nature. (iii) Does your answer for the optimal consumption in period 0 and savings differ from the answer to (5b), and why it does or why it doesn't? Assume now that each period's utility function is u(c)=ln(c). Continue assuming that the real interest rate, r, is zero, and the time discount factor, , equals 1. Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (5a). Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (5b). Compare the optimal saving to the value you found in (5b) and argue why they are different (if different at all). Write down the Euler equation and find the optimal consumption in periods 0 and 1 and optimal saving in period 0 given the data in (5c). Compare the optimal saving to the value you found in (5c) and argue why they are different (if different at all)