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1. Consumer's optimization. Dewi's preferences over the consumption of goods 1 and 2 are represented by the utility function u(X1, X2) = a log(x1) + (1 - a) log(x2), where x1 and x2 are the quantities of the two goods, a E (0, 1) and log denotes the natural logarithm. Assume further that Dewi's disposable income is R > 0 and the prices of the goods are denoted as pi > 0 for i = 1, 2. (a) Write down Dewi's optimization problem (20 points) (b) Write down the complete set of necessary conditions for optimization (15 points) (c) What are the amounts x1 and x2 purchased by Dewi? (15 points)