The homework is due on Wednesday, September 7. Each questions is worth 0.8 points. No partial credits.

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1 Homework : Econ500 Fall, 0 The homework is due on Wednesday, September 7. Each questions is worth 0. points. No partial credits. For the graphic arguments, use the graphing paper that is attached. Clearly indicate the feasible set by shading it. Always graph at least three lines representing the objective, and draw an arrow in the direction in which the objective increases. For the computer exercises, you need to attach a printout of each Excel worksheet. In addition, you need to write down all the formulae that you use. Question 1 Suppose an oil company has supplies of five crude products. In the refinery the crude products can be used to make two refined products x 1 and x, which the company can sell at a price of Dollars each. The company has a fixed supply of the crude products. In particular supplies of the crude products are,,,, and 0, respectively. For technological reasons, the company must produces at least 1 unit of good for every 3.75 units of good 1. The company wants to maximize the total revenue from selling the product. As a consequence, the company solves the following optimization problem. maxx 1 + x subject to (i) x 1 + x (ii) x 1 + x (iii) x 1 + x (iv) x 1 + x (v) 3x 1 + x 0 (vi) x 1 15x 0 (vii) x 1 0 (viii) x Determine the optimum graphically.. Do there exist prices for the output such that the firm will only produce good? 3. Do there exist prices for the output such that the firm will only produce good 1? 1

2 Question A firm wishes to produce 0 units of a product at the lowest possible cost. Two inputs are needed. The costs of the inputs are and Dollars, respectively. In order to produce 0 units of output, the inputs must fulfill 5x 1 + 3x 0. Thus, the firm solves: min x 1 + x subject to (i) 5x 1 + 3x 0 (ii) x 1 0 (iii) x Determine the optimal choice of x 1 and x graphically. Note: This is a minimization problem, i.e., moving down and to the left and your graph decreases costs.. Determine prices for the inputs such that you do not just get a solution that is on the corner of the feasible set. Are the prices unique? Question 3 A utility function is given by u(x 1, x )=x 1 + x. Suppose prices are p 1 = 3 and p =. Income is I= 0. The utility maximization problem is therefore given by max x 1 + x subject to (i) 3x 1 + x 0 (ii) x 1 0 (iii) x 0. Solve the optimization problem graphically. Question A person with utility function u(x 1, x )=x 1 + 3x wants to find the least costly consumption that provides a utility of 0 when prices are p 1 =, p =. To do this, you need to solve the following minimization problem. min x 1 + x subject to (i) x 1 + 3x 0 (ii) x 1 0 (iii) x 0. Solve the optimization problem graphically. How much money does the person need to afford this consumption bundle. Question 5 A firm produces three outputs using 5 inputs which are in fixed supply. The prices of the outputs are 3, 1, and 5, respectively. The input amounts are 50, 5,

3 0, 00, and 0. Thus, the firms solves max 3x 1 + x + 5x 3 subject to (i) x 1 + x + x 3 50 (ii) x 1 + x + x 3 5 (iii) 5x 1 + x + x 3 0 (iv) x 1 + 3x + x 3 00 (v) 5x 1 + x + 3x 3 0 (vi) x 1 0 (vii) x 0 (viii) x 3 0 Solve the optimization problem using Excel. Determine which constraints bind and which are slack. Question A automobile parts supplier uses different inputs to produce 5 outputs. The firm has already signed contracts to deliver 0 units of the first output, of the second, 15 of the third, and 0 of the fourth and 0 of the fifth good. The production of outputs is linear. The prices of the inputs 1,,3,,11,3,, 9,, and 15, respectively. Thus, the firm faces a linear programming problem, which we now specify. min 1x 1 + x + 3x 3 + x + 11x 5 + 3x + x 7 + 9x + x x subject to x 1,...,x (i) x 1 + x + x 3 + 5x + x 5 + x + 1.x 7 + x + 3x 9 + x 0 (ii) 5x 1 + 5x + x 3 + x + 3x 5 + 3x + x 7 + x + x 9 + 0x (iii) 0.5x 1 + x + x 3 + 3x + x 5 + 7x + 3x 7 + 3x + x 9 + x 15 (iv).5x 1 +.x + 5x 3 + x + x 5 + 3x + x 7 + x + x 9 + x 0 (v) 1.5x 1 + x + 3x 3 + x + 0.x 5 + x + x 7 + x + 3x 9 + x 0 (vi) x 1 0,... x Solve for the optimal values of x i, i=1,..., x by using Excel. Also determine which constraints bind and which are slack. 3

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