2 Scenario 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 % 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. 2.1 Your task Your task is to estimate the impact of the proposed tax on low-income households, and determine the level of compensation necessary to ensure that they are no worse off as a result of the tax. You are also to assess the claims regarding the cost of compensation, and effect of compensation on lowincome household demand for electricity. 2.2 Lowincome households Using data from household surveys, you have determined that the 'typical' lowincome household has a monthly income of $240!]. The household's preferences are represented by the utility function, U = fr. 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 MU, = x2. The price of electricity is currently P, = $10!] per Mwh, and the price of the composite good is normalised to P, = 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 F; to represent the price of electricity. You should substitute for P}. and if using the values in the scenario. {15 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? [5 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? [3 marks} Step 5: What level of compensation must the government provide to the lowincome household to restore iE welfare to the pretax level. Hint: The decomposition consumption basket delivers the initial level of utility at the final prices. [15 marks]