The Leontief Input-Output Model

.. Discrete Math Name KEY Lesson 07.1-The Leontief Input-Output Model Exercise 1 Period The Leontief Input-Output Model was developed by Harvard economist, Wassily Leontief, in the 1960s. He began his study by constructing input-output tables that described the flow of goods and services among various sectors of the economy in the United States. Today, his models can be applied to complex economies with hundreds of production sectors or to a situation as small as a single company that produces only one product. He was awarded the Nobel Prize in economics in 1973 for his work in this area. Power Source is a company that manufactures batteries used to power electric motors. However, not all the batteries produced by the company are available for sale outside the company. For every 100 batteries produced, three (3%) are used within the company. 203 In general, for a total production of P batteries by this company, 0.03P batteries will be used internally and P-0.03P will be available for external sales to customers. If D represents demand, the number of batteries available for external sales, then the number of batteries available for external sales is D=P-0.03P Example. Suppose the company receives an order for 5,000 batteries. What must the total production be to satisfy this external demand for batteries? To find the total production necessary, substitute 5,000 for D in the previous equation and solve for P. D-P-0.03P 5,000P-0.03P 5,000=P(1-0.03) factor P 5,155 batterics The company must produce 5,155 batteries to satisfy an external demand for 5,000 batteries. 1. The High-Tech Computer Company produces computer chips. For every dollar's worth of chips it produces for external sales, it uses 2 cents worth in its manufacturing process. a. Write an equation for High-Tech that represents external demand D in terms of total production P. D=P .02 P b. What must the total production of chips be to meet an external demand of $20,000 worth of chips? 20.000 = P(1-.02) - P=20408.16 Now take a look at a two-sector economy. Suppose Power Source buys an electric motor company and begins producing motors as well as batteries. The company's primary reason for this merger is that electric motors are needed to produce batteries. The production requirements for each division are described on the following page. Battery Division 1. For the battery division to produce 100 batteries, it uses 3% of its own batteries. 2. For every 100 batteries produced, I motor is required from the motor division. Motor Division 1. For the motor division to produce 100 motors, it must use 4% of its own motors. 2. For every 100 motors produced, 8 batteries are required from the battery division. 0.08 The production needs within this two- sector economy can be illustrated in the weighted digraph to the right. 0.03 0.01 0.04 The battery division requires 0.036 batteries from its own division and 0.01b motors from the motor division. The motor division requires 0.04m motors from its own division and 0.08m batteries from the battery division. Another way to view the company's production needs is in a consumption matrix for the economy. To B M ०६ Battery Motor Battery 0.03 0.08 Motor 0.01 0.04 From M 10 A 04 As an example, to produce 200 batteries, the battery division will need (0.03) (200) 6 batteries from the battery division and (0.01) (200) = 2 motors from the motor division. To produce 50 motors, the motor division will need (0.08)(50) = 4 batteries from the battery division and (0.04)(50) 2 motors from the motor division. 2. Suppose the High-Tech Computer Company adds another division that produces computers. Each division within the expanded company uses some of the other division's products. For every dollar's worth of chips it produces, the Chip Division requires 2 cents' worth of computer chips and 1 cent's worth of computers. For every dollar's worth of computers it produces, the Computer Division requires 20 cents' worth of computer chips and 3 cent's worth of computers. a. Construct and label a digraph and a consumption matrix for this economy. .02 2 01 03 chip 2 이 Comp 5.03 and from the chip camp Computer Division. chip chip 1000[0] 100 30 c. Suppose the Computer Division produces $5000 worth of computers. How much input does it need from itself and from the Chip Division. Chb. Suppose the Chip Division produces $1000 worth of chips. How much input does it need from itself $20 chips $10 computers Lomp comp chip comp 20 5000[0][0] $ 1000 chips, $so comps Discrete Maih Total (1-05) 1394 Name KEY Lesson 07.1-The Leontief Input-Output Model Exercise 2 Period 1. A utility company produces electric energy. Suppose that 5% of the total production of electricity is used up within the company to operate equipment needed to produce the electricity. Complete the following production table for this one-sector economy. Total Production Units Units Used Internally Unit for External Sales 500 0.05(500) = 25 500-0.05(500) = 475 900 105 (900) = 45 900-45 = 2000 100 1900 5000 05) 250 4750 2500 05(1500) 125 2,375 7500 375 7,125 P P-P(r) P-.06.959 For questions 2-4, use the following consumption matrix for a company with two departments, Service and Production. T Service Production Service [0.05 0.20 Production 0.04 0.01 2. Draw a weighted digraph to show the flow of goods and services within this company. 05 S 20 P .01 3. Complete the following 5 a. For every dollar's worth of output, the service department requires from its own department and cents' worth of input cents' worth of input from the production department b. For every dollar's worth of output, the production department requires input from its own department and 0 cents' worth of cents' worth of input from the service department. c. Use the information from parts a and b and the matrix equation DP-CP to compute the amount of computer chips-and computers available for sales outside the company (external-demand). D= 40 000 10 800 1900 29 200 48 100 Service Production 2. Recall the consumption matrix from Lesson 07.1. C- Service [0.05 0.20] Production 0.04 0.01 a. Suppose that the total output for the service department of the company is $20 million and the total output for the production department is $40 million. Write a production matrix P. P =[ 20 000 000 b. Compute the matrix product CP to find the output from cach department that the company uses internally. CP = [ 1 200 000 c. Compute the output that is available for sales demands outside the company (external demand). D= P-CP 38 800 000 d. The company must meet external demands of $25 million in service and $50 million in products over a period of time. What must be the total production in service and products to meet this demand? USE EQUATION ON FRUNT D=P-CP 10 0 05.2 04.01 -1 25 000 000 50 000 ROD 38 00000. 53 000 00 3. Recall the consumption matrix for the two-sector economy involving agriculture and transportation. C = Agriculture Agriculture Transportation Transportation 0.30 0.10 0.35 0.15 Suppose the economy must meet external demands of $10 million in agricultural products and $15 million in transportation services. Find the total production of agriculture products and transportation services necessary to satisfy these demands.