Carlos enjoys drinking Soda and Juice.Let x denote thequantityofSoda,and y denote the quantity of Juice (x and y are measured in liters) that Carlos consumes every month.Carlos's preferences are represented by the utility function (x,y) = lnx+ lny.Carlos's monthly income is I.The price of one liter Soda is px, and the price of one liter of Juice is py.

(a)Find the Marshallian demand functions for Soda and Juice, x(px,py,I) and y(px,py,I), by writing down Carlos's optimization problem and solving it with the Lagrange method.

(b)What is Carlos's demand for each of the goods at px= $20, py= $30, and I= $240? What is the highest utility level, U, that Carlos can achieve if px= $20, py= $30, and I= $240?

(c)The government wants to support Soda makers, there-fore it start taxing Juice at the rate = 2, so that Carlos will pay py = $32 per liter of Juice.Will such taxpolicyre-ally support Soda makers?How much of each beverage will Carlos consume with the tax policy?What will be the tax revenue collected by the government?

(d)What income will Carlos need in order to keep him as happy as be-fore the tax?How much of each beverage will Carlos consume with this income?

(e)What are the income and substitution effects of the tax?