Part 3 Demographic Changes and Pension System Consider a two-period household optimization problem with children. A household has one parent and n children. For simplicity, 11 may not be an integer. Children do not make any decisions. In period 1, the parent chooses her consumption 83', her children's total consumption my, and savings 51 (interest rate is r). In period 2, the parent chooses her consumption (33, and her children will be independent and form it new households. The parent earns a wage as , which is subject to a social security tax rate T, in period 1, and receives a pay-as-you-go pension payment p in period 2. The utility function ofa household is U = high?) + yn7log[cy) + log{cg), where y and \"r are parameters controlling the importance the parent places on children's utility, and ,6 is a discount factor (log denotes natural logarithm}. I} (a) Suppose a large number of identical households live in an economy. What is the expression of pension payment p implied by the government' 5 budget balance condition? ['lpt] D {b} Set up a household maximization problem. [lpt] I} (c) What is the expression of c3 and 51 when a household hiaximizes its utility? Lay out rst order conditions before solving the problerrL [3pt] I} (d) China has experienced a substantial decline in children to parents ratio. Discuss how this situation affect household savings in the economy described above. [2pt] I:> (e) Let :0 = 1, r = [102, ,3 = [198, ,u = 0.5, qr = 0.5, T = 0.2, n = 1. Calculate cg. [lpt] I} (f) Change 11 to [15. If the government intends to keep 03 unchanged from (e), how much will it adjusts the social security tax rate T? Note that other parameters does not change. [lpt] I:> {g} China has adopted the "two-child policy" in 2016. Explain why it has a very small effect on the number of newborns. [1 pt]