Using data from household surveys, you have determined that the typical low-income household has a monthly income of $2400. The household's preferences are represented by the utility function,

U = x² y,

where x represents the quantity of electricity consumed, measure in Megawatt hours (Mwh), and y represents the quantity of the composite good consumed. The associate marginal utilities are,

MU_x = 2xy and MU_y = x²

The price of electricity is currently P_x = $100 per Mwh, and the price of the composite good is normalised to P_y = 1.

Step 1: Derive the marginal rate of substitution for the typical low-income household.

Step 2: Find the household's electricity demand function, and composite good demand function. Use P_x to represent the price of electricity. You should substitute for P_y and I using the values in the scenario.

Step 3: Find the household's optimal consumption basket under the initial prices (no tax). What utility does the household derive from this basket?

Step 4: Find the household's optimal consumption basket if a carbon tax is applied to electricity. What utility does the household derive from this basket?

Step 5: What level of compensation must the government provide to the low-income household to restore its welfare to the pre-tax level. Hint: The decomposition consumption basket delivers the initial level of utility at the final prices.

Thanks!