1. Suppose all individuals in the economy have the following preferences over two con- sumption goods, x1 and 72, given by: U(21, 12) = In1 + 12 and fixed income /. (You can think of a representative consumer here.) Assume the producer prices of both goods are fixed and normalized to unity, so p1 = pz = 1. Suppose the government needs to raise a given amount of revenue R. and can only raise this revenue using commodity taxes, 71 and 12, so consumer prices are given by q; = (1 + T;) for j = 1, 2. (a) Set up the household's maximization problem. Solve for the optimal commodity demands. How does the demand for both goods depend on its own price? The price of the other good? (b) Write out the household's indirect utility function v(71, 72, /) and determine the expressions for Ov/071, Ov/072 and Ov/Of. (c) Set up the government's optimal tax problem. Write out the first-order condi- tions for n and 72 and provide some brief interpretation for each. (d) Suppose now individuals misperceive the tax rate on good 1 and choose their optimal consumption of good 1 based on this subjective price, that is, of = 1 + min where my E (0, 1]. Solve for the optimal demand of good 1 as a function of qi. Determine the tax price elasticity of demand for good 1, that is, the percentage change in demand for good 1 with a one percentage change in the tax rate on good 1. How does an increase in my affect this elasticity? What implications does this have for how the government would optimally tax good 1 relative to good 2? (Just a brief explanation is needed here.)