Edited to add sub-questions. Please help with (e) and (f). Thank you!

Suppose that the following utility function u(x1, x2) = 21nxi +31nx2 represents well the preferences of a typical household from a given population, where good 1 and good 2 represent two different categories of goods. (a) Find the marginal rate of substitution between good 1 and good 2. Is the marginal rate of substitution increasing or decreasing in X? How do we interpret this? (b) Find the demand functions for good 1 and good 2 in terms of the prices of these two goods and the income of the household. (C) Suppose that the price of good 1 is Pi = 2, the price of good 2 is P2 = 3, and the household income is m = 10. Find the consumption bundle that maximizes the household's utility. (d) Suppose the price of good 1 increases to Pi 4. What consumption bundle does the household demand after the price increase? = (e) Given the price increase, how much income does the typical household need to remain as happy (have the same utility) as they were before the price change? What commodity bundle would the household acquire if they had that additional income, given the new prices? (f) Decompose the total change in the consumption of good 1 and good 2 into substitution and income effects. In a clearly-labeled diagram with software on the horizontal axis, show the income and substitution effects of the increase in the price of software. (Hint: use the Slutsky equation.) Suppose that the following utility function u(x1, x2) = 21nxi +31nx2 represents well the preferences of a typical household from a given population, where good 1 and good 2 represent two different categories of goods. (a) Find the marginal rate of substitution between good 1 and good 2. Is the marginal rate of substitution increasing or decreasing in X? How do we interpret this? (b) Find the demand functions for good 1 and good 2 in terms of the prices of these two goods and the income of the household. (C) Suppose that the price of good 1 is Pi = 2, the price of good 2 is P2 = 3, and the household income is m = 10. Find the consumption bundle that maximizes the household's utility. (d) Suppose the price of good 1 increases to Pi 4. What consumption bundle does the household demand after the price increase? = (e) Given the price increase, how much income does the typical household need to remain as happy (have the same utility) as they were before the price change? What commodity bundle would the household acquire if they had that additional income, given the new prices? (f) Decompose the total change in the consumption of good 1 and good 2 into substitution and income effects. In a clearly-labeled diagram with software on the horizontal axis, show the income and substitution effects of the increase in the price of software. (Hint: use the Slutsky equation.)