2 Scenario Your client. lnsureCorp. 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 IUDUD households in Autarka interested in purchasing income insurance. Your client expects that the fixed cost of launching the income insurance product will be $2.DDD.DDU per year. and that each policy issued to a customer will cost the company an additional $1.50D in sales commissions. 2.1 Your task Your client wants you to analyse the potential market for income insurance and report on the following: a 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 $16. per year. which they spend on food {good x) and clothing (good y]. Their preferences oyer consumption baskets are represented by the utility function. U = le4y1r4_ The associate marginal utilities are. 1:4 1:4 y x MU). = 4x334 and MU}. = 41'3\". The price of food is PJr = $8 per meal. and the price of clothing is P}. = 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 $5.4m] 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: in Type all equations using the 'lnsert Equation\" function (or equivalent}. o 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 wordfpaqe limit for the analysis section. Step 1: Deriye 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 $l.. 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 $5,4DD. What is the household's associated level of utility? (It) 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} Step 6: What is a household's maximum willingness to pay for insurance? (Hint: Use the utility function U = .Hfr-l for this step only.) [8 marks} Date : _ _ Page XTRA EDGE Step 1 Given that, Autarka has an income = $ 160000 per year Shend on food = x ( goods ) spend on clothing = y ( goods ) And utility function will be given by U = X y Ad the associate mu are given by mux = y y 4 * MUy = X 4 4 4 4 also given that income reduced to $6 400 hey yearstep - 2 (1) and MRS XY The marginal rate of substitution is represented by Now expression for lyrical household's Marginal rate of subsititution will be MRS NY - MUX MUy equation ( 1) here the MU y = 4 x 34 now hut values of MU x and MUy in eq, (1) we get , MR S x y = 4 4 * 4 4 4 4 x 3/4 x by y ly x y y x 3/4 x Vy x (3+4 ) y MRS XY = y X This the expression for typical household MRS.Date : _ XTRA Page EDGE Ams - 2 to find myhical household's optimal consumption basket when its income is $ 160000 here income I = $160000 force of good - Rx = $ 8 freeice of down = Py = $128 we know the equation of budget line I = ( PX ) X + ( RX ) y now put the values 160000 = 8 ( X) + 128 ( 4) - 1 Now Indifference alive and budget line are langent they have same slope. slope of I C - MRS xy = - MUX Muy MUX = Y & MUY = X Slope of IC - - y X therefore the slope of budget line will be PX PY- 8 EDGE 128 16 omflion - Y = X 16 16 1 = X (2) 160000 = 8* + 128y 160000 - 8 X 16Y + 128 4 160000 128 4 + 1284 16000 0 = 256 4 y = 625 and 16Y = X 10,000 = X therefore optimal household's consumption for goods x q y will 10,000 and 625 3 Explanat Same as And - 2 I = P X / X + Py (Y ) 6400 = 8 (x ) + 128 ( ) ) Slope of IC = MRS Xy = - MUX MUY " slop of IC = - YPage Date : 1 - - XTRA EDGE scope of Budget line - Px py = - 8 128 = -L 16 16 Y = X 6400 = 8x+ 128 Y 6400 = 1287+ 1284 6400 _y 95 6 Y = 25 X = 400