CAN SOMEONE PLEASE HELP WITH QUESTION SIX? The Utility Function is U=XY7!

Consider individuals that have a utility function U = X Y7for X, trips to the gym per fortnight and Y, dollars of spending on all other goods.Assume the individuals' income per fortnight is M=$320 and theprice per gym trip is Px=$10. A $2 per trip subsidy has been introduced.

Question 2 50 marks

Consider individuals that have a utility function U = X Y7for X, trips to the gym per fortnight and Y, dollars of spending on all other goods.Assume the individuals' income per fortnight is M and theprice per gym trip is of PX.

1. Show that the Marginal Rate of Substitution (MRS) of a gym trip in terms of $ of all other goods is equal Y/7X.
2. Show that individual demand function for gym trips per fortnight X is X* = M / 7PX
3. Suppose that price per gym trip PX=$10. Consider Jenny who has a utility function U = X Y7and fortnightly income of $320. How many trips does to the gym does Jenny make?
4. Suppose the government introduces a subsidy of $2 per trip to the gym. How many trips to the Gym does Jenny make after the subsidy? How much does it cost the subsidy the government?
5. If Jenny was instead given a lumpsum subsidy equivalent to the cost of the $2 per trip subsidy so long as she was gym member, how many trips to the gym would Jenny make? Use Jenny'sutility function to explain which scheme Jenny prefers.Note you will get very large numbers for utility even after dividing by a billion.
6. Calculate the size of the income and substitution effect for the $2/trip subsidy and the lump- sum subsidy. Why does the $ subsidy encourage more demand than the lumpsum subsidy?