duction to Financial Mathematics - Semester 1, 2017 Assignment 7 Algebra To be handed in by 2pm, Monday 8th of May Examples: A. Let the technology (input-output) matrix for an open economy comprised of fishing, agriculture and mining industries be 0.5 0.1 0.1 A = 0.3 0.5 0.2 . 0.1 0.3 0.4 If the surpluses of 110 units of fishing output and 50 units of each of agriculture and mining goods are desired, find the output of each industry. B. Use your knowledge of closed Leontief economic models to complete the entries of the input-output matrix A and then find the output for the industries in the model. G I H 0.4 0.2 Government A= 0.2 0.3 Industry 0.5 0.5 Households Questions \u0014 \u0015 0 5/6 1. The input-output matrix of a closed economic model is A = . Solve 1 1/6 the closed Leontief economic model equation (I A)x = 0 for the production vector x. 2. In village of 90 people, each person either goes fishing or gathers nuts. All food is shared between the villagers and none is traded outside the village. The production of 3 units of fish requires two people and one unit of fish. The production of 1 unit of nuts requires one person. Each person in the village requires 1/3 units of fish and 2/3 units of nuts. (a) Write the input-output matrix A for the closed Leontief economic model which describes this village. Hint: the matrix is 3 3 and the sectors are fish, nuts and people (labour). (b) Solve the closed Leontief economic model equation (I A)x = 0 by Gauss-Jordan elimination to find the production of fish and nuts. Hint: There are 90 people in the village so set the people component of the production vector x to 90. (c) In this village, how many people go fishing and how many people gather nuts? 0 2 2 3. Consider matrix A = 3 0 1 . 2 1 1 (a) Calculate det(A). (b) Find all cofactors of A and find A1 . 4. Three neighbours have backyard vegetable gardens. Neighbour T grows tomatoes, neighbour C grows corn and neighbour L grows lettuce. The neighbours only sell their produce to each other. Neighbour T buys 1/5 of the tomatoes, 1/5 of the corn and 3/5 of the lettuce. Neighbour C buys 2/5 of the tomatoes, 1/5 of the corn and 1/5 of the lettuce. (a) What portion of each crop does neighbour L get? (b) What is the production vector x? (c) If the highest priced crop sells for $42, what prices should be assigned to each of the crops? 5. The matrix 0.2 0 0.2 0.3 0.1 B = 0 0.4 0.5 0.5 0.5 0.3 0 0.2 0 0 0.2 0.2 0.1 0.2 0 is the input-output matrix of a closed Leontief economic model. (a) Complete the missing row. (b) If industry four (that is, the industry described by the fourth column) produces 360 units, find the production vector x. Give your final answers correct to one decimal place. (Hint: MATLAB might be helpful here.)