1. A consumer's preferences are described by the utility function U = Ax1 x22, where U refers to utility units in "utils", A = 2, a = 2, a2 = 3. The prices of x1 and x2 are denoted as p and p2 and income is denoted as m.. (a) (b) (c) (d) (f) Write down the Lagrange function for this consumer's problem. Derive the first order conditions. Solve for the Lagrange multiplier, A, and give an economic interpretation of its meaning. Assuming that the optimal bundle is such that x > 0 and x > 0, under what conditions are the the first order conditions derived in part (a), sufficient for an optimum? Show that these conditions are met. Derive the consumer's demand functions x (P1, P2, m) and xz (P1, P2, m). Suppose p = 9 and P2 = 3 in dollars per unit, and m = 36 dollars. De- termine the optimal quantities, x and x2, demanded by this consumer. Determine her utility in utils. Plot a graph showing the consumer's budget line and indifference curve. It is recommended that you use Excel or other graphing software. Put x on the horizontal axis with a range of 0 to 4. Put x2 on the vertical axis with a range of 0 to 25. Make sure you label the curves as BL1 and U1 for the budget line and indifference curve, respectively, and label the axes. Now suppose the government wants to raise revenue and imposes a commodity tax of $9 per unit of x1. Determine the optimal consumption bundle (x, x) when the tax is applied. What is the consumer's new utility level? How much revenue is raised by this tax? We will denote this tax revenue as R.