2 Scenario Your client. InsureCorp, is an insurance company considering launching an 'income insur- ance' product in the island nation of Autarka. Income insurance is a product that fully insures a household against changes in income caused by a major injury or illness. At present, no businesses are selling income insurance products in Autarka. Initial mar- ket research suggests that there are 10.000 households in Autarka interested in purchasing income insurance. Your client expects that the fixed cost of launching the income insurance product will be $20.000.000 per year. and that each policy issued to a customer will cost the company an additional $1.500 in sales commissions. 2.1 Your task Your client wants you to analyse the potential market for income insurance and report on the following: o What is the maximum price the company can charge a household for an income insurance policy? a What is the expected profit (or loss) for the company if it becomes a monopoly provider of income insurance? a Is there a risk that rival insurance companies will also enter the market. selling identical income insurance products? If so. what would be the expected profit of your client? (You should assume that any competitors would face the same costs as your client.) 2.2 Household welfare A typical household in Autarka has an income of $160000 per year. which they spend on food (good x) and clothing (good y). Their preferences over consumption baskets are represented by the utility function, U = X114 lid- The associate marginal utilities are. 311/4 Km. MUX = and MUy= 4y3/4' 4x3\" The price of food is P = $8 per meal. and the price of clothing is F'y = 128 per item. Each household has a 10% probability of experiencing a major injury or illness in any given year. If a household experiences a major injury or illness. its income is reduced to $6.400 per year. 3.2 Analysis In the analysis section you must complete each of the steps detailed below. When com- pleting the steps you must: a Type all equations using the \"Insert Equation' function (or equivalent). a Show all of your working and include sufficient written description for the reader to follow your process. Note that hand draw figures and equations are not acceptable. There is no worprage limit for the analysis section. Step 1: Derive an expression for the typical household's marginal rate of substitution. (4 marks) Step 2: Find the typical household's optimal consumption basket when its income is $160,000. What is the household's associated level of utility? (10 marks) Step 3: Find the typical household's optimal consumption basket when its income is $6.400. What is the household's associated level of utility? (10 marks) Step 4: What is the typical household's expected utility if it does not purchase insurance? (4 marks) Step 5: What is the expected payout to the typical household if it does purchase insurance? (4 marks)