Consider the Cobb Douglass utility function for two privately produced and consumed normal goods, 2 and...

  

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Consider the Cobb Douglass utility function for two privately produced and consumed normal goods, 2 and 2: (1) U = f(1, 12) = 112. Assume a # 5, and a +8+1. The consumer is subject to the budget constraint P1 +=1 We can express the marginal utilities as MU1 = art-1 (2) (3) MU2 = 3x2- 2a. Express the optimality condition in terms of the MRS and the price ratio. Simplify your expression as much as possible. Be sure your notation indicates that consumers are price-takers in this competitive market. 2b. Rearrange your expression until 2 is on the left hand side by itself with all other vari- ables on the right hand side. 2c. Use substitution and insert your answer for part 2b into the budget constraint (equation 2) above. Solve for the demand zi as a function of p, P2, I, a and 3. Simplify as much as possible. 2d. Consider a per-unit (excise) tax on 22, such that the new price is p2 +7. Show how this tax affects consumer demand for r, by drawing the indifference curves and budget constraints with an without the tax (on the same graph). Also show how the after-tax price of 12 affects the optimality condition as well as rj Give yourself plenty of space and make the details of the graph visable. Consider the Cobb Douglass utility function for two privately produced and consumed normal goods, 2 and 2: (1) U = f(1, 12) = 112. Assume a # 5, and a +8+1. The consumer is subject to the budget constraint P1 +=1 We can express the marginal utilities as MU1 = art-1 (2) (3) MU2 = 3x2- 2a. Express the optimality condition in terms of the MRS and the price ratio. Simplify your expression as much as possible. Be sure your notation indicates that consumers are price-takers in this competitive market. 2b. Rearrange your expression until 2 is on the left hand side by itself with all other vari- ables on the right hand side. 2c. Use substitution and insert your answer for part 2b into the budget constraint (equation 2) above. Solve for the demand zi as a function of p, P2, I, a and 3. Simplify as much as possible. 2d. Consider a per-unit (excise) tax on 22, such that the new price is p2 +7. Show how this tax affects consumer demand for r, by drawing the indifference curves and budget constraints with an without the tax (on the same graph). Also show how the after-tax price of 12 affects the optimality condition as well as rj Give yourself plenty of space and make the details of the graph visable.

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