1. This problem will show how to choose some parameters of a BBC model to match the data, a process known as calibration. Suppose preferences are given by: 00 B to + n)t[(1 a) log q + alog(1 N0] i=0 here n > 0 is the population growth rate and ct and Ni are per capita consumption and hours. Suppose labor~augn1enting technology grows at rate 9 so At = (1 + 9):. Thus the aggregate resource constraint is: Ct + It 2 (1 + g)(1a}tk: Ntla: where It is per capita investment an is: is the per capita capital stock. Finally the law of motion for the capital in per capita terms is: (1 +9)\" + \")kHl : (1 (\"kt + It (a) Working directly with the social planner's problem, find the first order con dition for hours worked and also find the Euler equation for the optimal con sumption allocation. (b) This model has a balanced growth path (BGP) in which hours worked N! is constant and all other per capita variables grow at the constant rate Q, i.e. kt+1 : (l + gcg and so on. Using the two relations derived in part (a) and the law of motion for capital1 find three equations relating the hours N , the capital/output ratio k/y, the consumption/output ratio (2/3}, and the investment/capital ratio I/k to each other and the parameters of the model. (c) Suppose a = 0.4, n = 0.01 and g = 0.016, which are estimated from US data. 1. Given a value of I/k = 0.08 in the data, find a value of (5 consistent with this in the BGP. ii. Given a value of 19/3; : 3.3 and your value of 5 find a value of? from the BGP relations. iii. Given a value of N : 0.33 and y/c : 1.33 nd a value of a from the BGP relations. 2. Suppose that a household has preferences given by: log(Cm) + log(CC) l75 log(Cm') -l- log(CC')