There are three types of individuals who dier in their marginal valuations of leisure. Individuals have utility functions that can be described as follows: U" =1nxn+nxln(1F), where x is consumption, 1 is labor supply, and 11 indicates type. A third of all individuals falls into each of the three following categories of types: :1 = 1.12, n = 1, n = 2. Labor supply is bounded between 0 and 1. The price of private consumption is l, and the wage rate is also equal to l for all individuals. All individuals have non-labor income equal to 1/4- The following exercises explore the potential work disincentives of welfare programs in this economy. In each of the following parts of the question, please explain your results graphically as well as algebraically. a) In the absence of a welfare program, how much labor will individuals of each type supply? b) The government has realized the distribution of consumption across individuals is very unequal. It decides to establish a welfare program to achieve more equity in this economy. The program guarantees all individuals who choose to take up welfare transfers a grant of U4. However, this grant is taxed away at a rate of one-forone with labor earnings; that is, the benet reduction rate (or the implicit tax rate) is l (100%). How much labor does each type choose to supply under this program? c) After introducing the welfare pro gram, the government is surprised by how high welfare outlays are. The administration decides to improve the labor supply incentives of the welfare program by reducing the implicit tax rate on earnings to U2 (i-e., by allowing individuals to keep half of labor earnings and still participate in welfare). How much labor does each type of individual choose to supply under this program? What happens to total government outlays? (In your answer you can assume there is one person of each type-)