A Household welfare
A typical household in Autarka has an income of $160,000 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 = x 1/4y y 1/4.
The associate marginal utilities are, MUx = y 1/4 / 4x 3/4 and MUy = x 1/4 /4y 3/4 . The price of food is Px = $8 per meal, and the price of clothing is Py = 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. In the analysis section you must complete each of the steps detailed below. When com- pleting the steps you must: • Type all equations using the 'Insert Equation' function (or equivalent). • 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 word/page 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)
Step 6: What is a household's maximum willingness to pay for insurance? (Hint: Use the utility function U = √I/64 for this step only.) (8 marks)
