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Tutorial 1.31 Determine the market equilibrium for a given scenario
Market equilibrium is the point of compromise between the suppliers and the demanders of a product.
In other words, the market equilibrium is the point of intersection when the supply function is equal to the demand function: S(x) = D(x). The point at which the market equilibrium occurs is sometimes called the equilibrium point (Q, P).
example 1.31a Suppose that the owner of a local shoe store will buy only 20 pairs of a certain shoe if the unit price is $80 but 30 pairs if the unit price is $55. The manufacturer of the shoes is willing to provide 60 pairs if the unit price is $70 but only 50 pairs if the unit price is $50. If both of the supply and demand functions are linear, find the market equilibrium for this scenario.
Just follow these steps to answer this word problem…
• Determine which sentence refers to the supply function: manufacturer = supplier
• Determine which sentence refers to the demand function: owner = retailer
• Use the given information to create ordered pairs (Q, P)…two ordered pairs for the supply function and two ordered pairs for the demand function:
supply function: P1 = (60, 70); P2 = (50, 50) demand function: P1 = (20, 80); P2 = (30, 55)
87
• Use the points to determine the slope (m) of each function:
supply function: 2 1
2 1
50 70 20 2 50 60 10 y y
m x x
− − −
= = = =
− − −
demand function: 2 1
2 1
55 80 25 2.5 30 20 10
y y m x x
− − −
= = = = −
− −
• Use either of the S(x) points to determine its y-intercept:
m = 2, P1 = (60, 70) → y = mx + b → b = y – mx → b = 70 – 2(60) = –50 m = 2, P2 = (50, 50) → y = mx + b → b = y – mx → b = 50 – 2(50) = –50
• Use either of the D(x) points to determine its y- intercept:
m = –2.5, P1 = (20, 80) → y = mx + b → b = y – mx → b = 80 – (–2.5)(20) = 130 m = –2.5, P2 = (30, 55) → y = mx + b → b = y – mx → b = 55 – (–2.5)(30) = 130
• Use the calculated values of m and b to write the supply function in its slope-intercept format:
m = 2, b = –50 → S(x) = 2x – 50
• Use the calculated values of m and b to write the demand function in its slope-intercept format:
m = –2.5, b = 130 → D(x) = 130 – 2.5x
• To find the market equilibrium, set the two functions equal to each other and solve for the variable x (which represents the quantity agreed upon):
2x – 50 = 130 – 2.5x → 2x + 2.5x = 130 + 50 → 4.5x = 180 → 4.5 180 4.5 4.5
x= → x = 40
• Finally, substitute the value of x into either the supply function of the demand function to determine the agreed upon price:
supply function: S(x) = 2x – 50 → S(40) = 2(40) – 50 → S(x) = 80 – 50 = 30
demand function: D(x) = 130 – 2.5x → D(40) = 130 – (2.5)(40) → D(x) = 130 – 100 = 30
• Use the derived values to answer the word problem:
The equilibrium point (Q, P) = (40, 30)…which means that the agreed upon quantity was 40 pairs of shoes at the unit price of $30.
example 1.31b Suppose a group of retailers will buy only 74 televisions from a wholesaler if the unit price is $370 but 130 televisions if the unit price is $300. The wholesaler is willing to supply only 82 televisions if the unit price is $436 and 154 televisions if the unit price is $517. If both of the supply and demand functions are linear, find the market equilibrium for this scenario.
• Determine which sentence refers to the supply function: wholesaler = supplier
• Determine which sentence refers to the demand function: retailers = retailer
• Use the given information to create ordered pairs (Q, P)…two ordered pairs for the supply function and two ordered pairs for the demand function:
supply function: P1 = (82, 436); P2 = (154, 517) demand function: P1 = (74, 370); P2 = (130, 300)
• Use these points to determine the slope (m) of each function:
supply function: 2 1
2 1
517 436 81 9 1.125 154 82 72 8
y y m x x
− −
= = = = =
− −
demand function: 2 1
2 1
300 370 70 5 1.25 130 74 56 4
y y m x x
− − − −
= = = = = −
− −
• Use either of the S(x) points to determine its y-intercept:
m = 1.125, P1 = (82, 436) → y = mx + b → b = y – mx → b = 436 – (1.125)(82) = 343.75 m = 1.125, P2 = (154, 517) → y = mx + b → b = y – mx → b = 517 – (1.125)(154) = 343.75
• Use either of the D(x) points to determine its y- intercept:
m = –1.25, P1 = (74, 370) → y = mx + b → b = y – mx → b = 370 – (–1.25)(74) = 462.5 m = –1.25, P2 = (130, 300) → y = mx + b → b = y – mx → b = 300 – (–1.25)(130) = 462.5
• Use the calculated values of m and b to write the supply function in its slope-intercept format:
m = 1.125, b = 343.75 → S(x) = 1.125x + 343.75
• Use the calculated values of m and b to write the demand function in its slope-intercept format:
m = –1.25, b = 462.50 → D(x) = 462.50 – 1.25x
• To find the market equilibrium, set the two functions equal to each other and solve for the variable x (which represents the quantity agreed upon):
1.125x + 343.75 = 462.50 – 1.25x → 1.125x + 1.25x = 462.50 – 343.75 2.375x = 118.75 → 2.375 118.75
2.375 2.375
x = → x = 50
• Finally, substitute the value of x into either the supply function of the demand function to determine the agreed upon price:
supply function:
S(x) = 1.125x + 343.75 S(50) = 1.125(50) + 343.75 S(x) = 56.25 + 343.75 = 400
demand function:
D(x) = 462.50 – 1.25x D(50) = 462.50 – 1.25(50) D(x) = 462.50 – 62.50 = 400
• Use the derived values to answer the word problem:
The equilibrium point (Q, P) = (50, 400)…which means that the agreed upon quantity was 50 televisions at the unit price of $400.
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Recall that the slope of all linear demand functions will be negative while the slope of all linear supply functions will be positive. If we are not sure which given information should be used for the supply function or which given information should be used for the demand function, just determine the slope for each of the sentences…the negative slope will be the demand function.
NOTE: Although the same slope values were used for the supply functions in each of these examples that may not always be the case.
NOTE: Although the same slope values were used for the demand functions in each of these examples that may not always be the case.
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