Answer the following questions below.

A life insurance company issues a whole life assurance to a life aged x exact. The benefit is 1 if death occurs in the first year, 2 if death occurs in the second year, 3 if death occurs in the third year, and so on. The death benefit is payable at the end of the year of death. Level premiums are payable annually in advance. There is an initial expense of / and a renewal expense of e at the start of each year, including the first. (i) Write down: (a) the equation of value (b) an expression for the gross premium prospective reserve at integer time r (c) an expression for the gross premium retrospective reserve at integer time 1. [3] (ii) Show that the expressions in (i)(b) and (i)(c) are equal if the premium, the prospective reserve and the retrospective reserve are calculated using the same basis. [4] [Total 7]2. Consider the following CES function for K > 0 and L > 0: Q = -SK-P - (1-8) L-P (0 p2 -1