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Math  /  Algebra

QuestionThe price-demand equation for a particular flashlight is given by p=119−0.004xp=119-0.004 x, where xx is the number of flashlights demanded when the price is pp dollars each. The flashlight manufacturers will produce no flashlights if the price is $71\$ 71 or less, and they will market 5,500 flashlights when the price is $93\$ 93 per flashlight. (Assume the price-supply equation is linear.) (a) Find the consumers' surplus for this commodity. \ \square(b)Findtheproducers′surplusforthiscommodity.$ (b) Find the producers' surplus for this commodity. \$ \square$

Studdy Solution

STEP 1

1. The price-demand equation is given by p=119−0.004x p = 119 - 0.004x .
2. The price-supply equation is linear.
3. Consumers' surplus is the area between the demand curve and the equilibrium price, up to the equilibrium quantity.
4. Producers' surplus is the area between the supply curve and the equilibrium price, up to the equilibrium quantity.
5. The equilibrium point is where the quantity demanded equals the quantity supplied.

STEP 2

1. Determine the equilibrium price and quantity.
2. Find the consumers' surplus.
3. Find the producers' surplus.

STEP 3

To find the equilibrium point, we need to determine the price-supply equation. We know: - At p=71 p = 71 , x=0 x = 0 . - At p=93 p = 93 , x=5500 x = 5500 .
The supply equation is linear, so we can use the point-slope form of a line to find it. First, calculate the slope m m :
m=93−715500−0=225500=0.004 m = \frac{93 - 71}{5500 - 0} = \frac{22}{5500} = 0.004
Now, use the point-slope form with the point (0,71) (0, 71) :
p−71=0.004(x−0) p - 71 = 0.004(x - 0) p=0.004x+71 p = 0.004x + 71

STEP 4

Set the demand equation equal to the supply equation to find the equilibrium quantity xe x_e and price pe p_e :
119−0.004x=0.004x+71 119 - 0.004x = 0.004x + 71
Combine like terms:
119−71=0.004x+0.004x 119 - 71 = 0.004x + 0.004x 48=0.008x 48 = 0.008x
Solve for x x :
x=480.008=6000 x = \frac{48}{0.008} = 6000
Substitute x=6000 x = 6000 back into either the demand or supply equation to find p p :
p=119−0.004(6000)=119−24=95 p = 119 - 0.004(6000) = 119 - 24 = 95
The equilibrium point is (6000,95) (6000, 95) .

STEP 5

Consumers' surplus is the area between the demand curve and the equilibrium price, up to the equilibrium quantity. The formula for consumers' surplus is:
Consumers’ Surplus=∫06000(119−0.004x) dx−95×6000 \text{Consumers' Surplus} = \int_{0}^{6000} (119 - 0.004x) \, dx - 95 \times 6000
Calculate the integral:
∫(119−0.004x) dx=119x−0.002x2+C \int (119 - 0.004x) \, dx = 119x - 0.002x^2 + C
Evaluate from 0 to 6000:
[119(6000)−0.002(6000)2]−[119(0)−0.002(0)2] [119(6000) - 0.002(6000)^2] - [119(0) - 0.002(0)^2] =714000−72000 = 714000 - 72000 =642000 = 642000
Subtract the rectangle area:
642000−570000=72000 642000 - 570000 = 72000
Consumers' surplus is 72000 \boxed{72000} .

STEP 6

Producers' surplus is the area between the supply curve and the equilibrium price, up to the equilibrium quantity. The formula for producers' surplus is:
Producers’ Surplus=95×6000−∫06000(0.004x+71) dx \text{Producers' Surplus} = 95 \times 6000 - \int_{0}^{6000} (0.004x + 71) \, dx
Calculate the integral:
∫(0.004x+71) dx=0.002x2+71x+C \int (0.004x + 71) \, dx = 0.002x^2 + 71x + C
Evaluate from 0 to 6000:
[0.002(6000)2+71(6000)]−[0.002(0)2+71(0)] [0.002(6000)^2 + 71(6000)] - [0.002(0)^2 + 71(0)] =72000+426000 = 72000 + 426000 =498000 = 498000
Subtract from the rectangle area:
570000−498000=72000 570000 - 498000 = 72000
Producers' surplus is 72000 \boxed{72000} .

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