In 1973, Nobel Prize Winner, Professor Wassily Leontief opened the door to a new
ID: 1202530 • Letter: I
Question
In 1973, Nobel Prize Winner, Professor Wassily Leontief opened the door to a new era of mathematical modeling in economics. While building the bridge between mathematics and economics, he proved various results. For example: Suppose a nation's economy is divided into several sectors (e.g. manufacturing, communication, entertaninment, etc.). Suppose that for each sector, we know its total output for one year and we know exactly how this output is divided or "exchanged" among the other sectors of the economy. Let the total dollar value of a sector's output be called the price of that output. Leontief proved that there exist a specific values, called equilibrium prices, that can be assigned to the total outputs of the various sections in such a way that the income of each sector exactly balances its expenses. Suppose an economy has 4 sectors: Mining, Lumber, Entertainment, and Services. Mining sells 10% of its output to Lumber, 60% to Entertainment, and retains/keeps the rest. Lumber sells 15% of its output to Mining, 50% to Entertainment, 20% to Services, and retains/keeps the rest. Entertainment seIIs 20% of its output to Mining, 15% to Lumber, 20% to Services, and retains/keeps the rest. Services sells 20% of its output to Mining, 10% to Lumber, 50% to Entertainment, and retains/keeps the rest. (a) Construct the exchange table for this economy. (b) Find the equilibrium prices for the economy. Tip: See Section 1.6, Example 1, for an Example.Explanation / Answer
Let the output of the four sectors be X,Y,Z,N
Hence
Mining Lumber, entertainment service
0 10 60 0
15 0 50 20 X x y z n
20 15 0 20
20 10 50 0
10y + 60z=100--------------------1
15x+50 z+20n =100-------------2
20x+15y+20n=100--------------3
20x+10y+50z= 100------------4
Taking equation 2 and 3
3x+15y+50z= 0
20x+10y+50z=100
Hence, 15x-5y=100
10y +60z=100
15x+5y+60z=200
5x+15y+50z=100
Hence, -40y-90z= -100
or 40y+90z=100
Hence 40y+90z=100
10y+60z= 100
therefore,
40y+90z=100
40y+240z= 400
--------------------------
150z=300
Hence , z=20
substituting,
40y+1800=100
y= 1700/40=42.5
and,
15x-5y=100
15x- 212.5=100
Hence, equilibrium prices are -
Mining =20.7
Lumber=42.5
Entertainment= 20
Services= 60
Hence, solved.
x=20.7
substituting,
15 * 20.7+50 *20+20n=100
300 +1000+20n = 100
1300 + 20n=100
Hence, n= 60