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In the island nation of Autarka, electricity is primarily generated by burning fossil fuels, a significant contributor to climate change

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In the island nation of Autarka, electricity is primarily generated by burning fossil fuels, a significant contributor to climate change. The government has proposed introducing a 60% tax on retail electricity prices to provide consumers with an incentive to save electricity and purchase more energy efficient appliances. Community groups in Autarka have expressed concern that the proposed tax will dis proportionately hurt low income households. These households spend a larger proportion of their income on energy, and may not be able to afford to purchase new appliances. In response, the government has suggested providing compensation to low income households to ensure that they are no worse off. However, some commentators have criticised this suggestion, claiming that compensation would be too expensive, and drive up demand for electricity amongst low income households.
Low-income households
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 = x^2 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,
MUx = 2xy and MUy =x^2
The price of electricity is currently Px = $100 per Mwh, and the price of the composite good is normalised to Py = 1.
Step 1: Derive the marginal rate of substitution for the typical low-income household. (4 marks)
Step 2: Find the household's electricity demand function, and composite good demand function. Use Px to represent the price of electricity. You should substitute for Py and I using the values in the scenario. (16 marks)
Step 3: Find the household's optimal consumption basket under the initial prices (no tax). What utility does the household derive from this basket? (6 marks)
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? (8 marks)
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.