Question 2 (50 marks): The following graph is found in most economics texts and it illustrates how

an economy can get outside its PPF through free trade. Solving this problem is a straight-forward

application of Lagrange.

1

Skippy lives on a small island full of resources that lets her produce two goods: x and y. Her production

possibility frontier (constraint) is given by x2 + y2 = 300. Skippy works alone to maximize her utility

which is given by U(x; y) = x2y.

(1) Set up the Lagrangian function for Skippy and solve for x and y. Use your solution to calculate U.

(10 marks)

(2) Carefully graph and label the PPF and Indifference Curve at the optimum. (10 marks)

Now Skippy has the opportunity of trading in the world market. Skippy can trade whatever x and y

she has at prices px and py. To maximize her gains from trade, Skippy needs to follow a 2-step process.

Step 1. First, she maximizes her endowment position (maximize trading line) subject to her PPF, where

the solution to this Lagrange problem (xE, yE) is her production decision and are her endowments she

will use for trade.

Step 2. Second, she maximizes utility subject to the constraint, pxxE + pyyE = pxx + pyy, where x

and y are the quantities she trades for that maximize her utility.

(3) If px = py = $1, please solve the Lagrangian that maximizes her endowments. Then use the endow-

ment value and prices are given in the trading constraint to maximize her utility by the Lagrangian method.

Calculate her utility number. (20 marks)

(4) Give an economic interpretation to each Lagrangian multiplier in the endowment-maximization

problem and the utility-maximization problem in (3). (10 marks)

Question 2 (50 marks): The following graph is found in most economics texts and it illustrates how an economy can get outside its PPF through free trade. Solving this problem is a straight-forward application of Lagrange. The Gains From Trade YE B u* PPF "Trading Line X Figure 1: 1 Skippy lives on a samll island full of resources that lets her produce two goods: r and y. Her production possibility frontier (constraint) is given by x2 + y2 = 300. Skippy works alone to maximize her utility which is given by U(x, y) = x+y. (1) Set up the Lagrangian function for Skippy and solve for 7 and y. Use your solution to calculate U. (10 marks) (2) Carefully graph and label the PPF and Indifference Curve at the optimum. (10 marks) Now Skippy has the opportunity of trading in the world market. Skippy can trade whatever r and y she has at prices P2 and Py. To maximize her gains from trade, Skippy needs to follow a 2-step process. Step 1. First, she maximizes her endowment position (maximize trading line) subject to her PPF, where the solution to this lagrange problem (XE, YE) is her production decision and are her endowments she will use for trade. Step 2. Second, she maximizes utility subject to the constraint, Prle + PyYE = Pzx* +Pyy", where r* and y* are the quantities she trades for that maximize her utility. (3) If Pa = Py = $1, please solve the Lagrangian that maximizes her endowments. Then use the endow- ment value and prices given in the trading constraint to maximize her utility by the Lagrangian method. Calculate her utility number. (20 marks) (4) Give an economic intrepretation to each Lagrangian multiplier in the endowment-maximization problem and the utility-maximization problem in (3). (10 marks)