Question 3. Joe Elder is an elderly single man living by himself. He receives M = $3600 per month from Social Security. He has no other income. Joe lives in a cold climate and uses natural gas to heat his home. Let X be the quantity of natural gas that Joe uses each month. The price of natural gas is Px = $4/per gallon. Joe's utility function is U(X,Y)= X¹/2Y¹/2, where Y is the amount of money that Joe has available to spend on all other goods besides natural gas. Therefore, the Price of Y is Py = $1. a) How much Natural Gas will Joe buy each month if he maximizes his utility? b) How much money will Joe have left to spend on all other goods each month if he maximizes his utility? Now suppose the price of natural gas rises to Px = $9/per gallon. Since you have education in Economics, you have been assigned to a committee to determine how much to increase Joe's Social Security payment to account for the increase in the price of natural gas. c) How much of Joe's $3600/month income would he be willing to pay to avoid the price increase of Natural Gas? d) (1 point) Is your answer to part (c) the Compensating Variation or the Equivalent Variation? e) (4 points) How much additional money would Joe need in order to have the same utility level after the price change as before the price change? f) (1 point) Is your answer to part (e) the Compensating Variation or the Equivalent Variation? g) of the total change in the quantity demanded of X, how much is due to the substitution effect and how much is due to the income effect? Note: since there is an increase in the price of good X, these values will be negative.

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