Answer the following questions giving explanations where necessary.

1. [8 points] Let the production function be Q (t) = f (t, K(t), L(t)) = exp (gt) K(t)" L(t) 1 -, (1) where Q is total output, f (.) is the production function, K is capital input, L is labor input, and t is the time which indicates that the production function can shift over time due to technological changes. The notation exp (.) denotes the exponential function, g > 0 is a constant, and a E (0, 1) is the share of capital in the production function. Derive the expression of the growth rate of Q (t), i.e., do(t) Q(t) (2) in terms of the growth rates of capital and labor, dk (t) dL(D) and K (t) L(t) and two model parameters, o and g.At the end of each year an independent organisation ranks the performance of the unit trusts invested in a particular sector, and classifies them into four quartiles (with quartile 1 being the best performer). Past experience has shown that, at the end of each year, a fund will either remain in the same quartile or will move to a neighbouring quartile. In fact, there is a probability 1-20 that a fund will remain in the same quartile and, where upward or downward movements are both possible, these are equally likely. However, it has been found that a fund that has remained in the top or bottom quartile for two consecutive years has a probability of 1- (/