This question is very similar to Question #1 but recasts the problem in terms of a single type of good consumed over time instead of two different types of goods within the same period of time. Consider the problem of an individual that has Y dollars to spend on consuming over two periods. Let cl denote the amount of consumption that the individual would like to purchase in period 1 and ()2 denote the amount of consumption that the individual would like to consume in period 2. The individual begins period 1 with Y dollars and can purchase cl units of the consumption good at a price P1 and can save any unspent wealth. Use 51 to denote the amount of savings the individual chooses to hold at the end of period 1. Any wealth that is saved earns interest at rate r so that the amount of wealth the individual has at his / her diSposal to purchase consumption goods in period 2 is (1 +T)81. This principal and interest on savings is used to nance period 2 consumption. Again, for simplicity, we can assume that it costs P2 dollars to buy a unit of the consumption good in period 2. The individual's total happiness is measured by the sum of period utility across time, u(cl) + u(cg). Let u(c) be an increasing function that is strictly concave in the amount of consumption c enjoyed by the individual. Also assume that the function u(c) satises the Inada condition limc_,0 u'(c) = 00 where u'(c) 2 die) is the rst-derivative of the utility function 15(0) with respect to c. 1. The individual faces a budget constraint in period 1 of P1c1 + 31 = Y and a period 2 budget constraint of P262 2 (1 + r)31. Interpret each of these two constraints in words. 2. The individual's problem is to choose the consumption bundle (01,62) optimally. Specically, max {u(C1) + u(C2)} C1362381 subject to the two budget constraints above. Using the Method of Lagrange, let A1 be the Lagrange multiplier on the period 1 budget constraint and A2 be the Lagrange multiplier attached to the period 2 budget constraint. The Lagrangean can be written as 5(01, C2, 81, A1, A2) = 1:5(61) + \"(C2) + A1[Y - P101 81] + A2 [(1 + T)81 P202]- Take the rst orderderivative of the Lagrangean function with respect to cl, (:2, 81, A1 and A2. Set each of them equal to zero in order provide the equations that can be used to identify the Lagrangean functions \"critical points\". 3. Take the rst-order conditions for cl and c2 and obtain expressions for A1 and A2. Use these expressions for A1 and A2 in the rst-order condition with respect to 31. This will give you a trade-off between purchasing more c1 at the expense of c2 or vice versa. 4. In words, interpret the economic trade-off between good one and good two that you obtained in the previous step