Hi tutor, can you assist me in solving the following questions(updated, utility function was missing on the previous post) below ?

In the island nation ofAutarka, electricity is primarily generated by burning fossil fuels, a significant contributor to climate change. The government has proposed introducing a 50% 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.

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 low-income household demand for electricity.

Using data from household surveys, you have determined that the 'typical' low-income household has a monthly income of$1800. The household's preferences are represented by the utility function,

U=xy ,

wherexrepresents the quantity of electricity consumed, measure in Megawatt hours (Mwh), andyrepresents the quantity of the composite good consumed. The associate marginal utilities are,

MUx=y2andMUy= 2xy.

The price of electricity is currentlyPx= $100 per Mwh, and the price of the composite good is normalised toPy= 1.

The first section in your report is the introduction. The introduction provides the reader with the background to the report. You should proved a summary of the purpose of your research, the methodology employed, and summarise your key findings. The introduction has a one hundred limit.

In this section you must complete each of the steps detailed below. When completing the steps you must:

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.

There is no word/page limit for the analysis section.

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. UsePxto represent the price of electricity. You should substitute forPyandIusing 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? (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?

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.

conclusion

Here you must summarise your findings, and discuss the implications on these finding for the scenario. Be sure to base your conclusion on the results in your analysis section and avoid assertions or speculation. The conclusion has a one hundred limit.