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Math  /  Data & Statistics

Question7. [-/1.48 Points] DETAILS MY NOTES LARAPCALC10 5.5.058. ASK YOUR TEACHER
PRACTICE ANOTHER Factory orders for an air conditioner are about 4000 units per week when the price is $350\$ 350 and about 6000 units per week when the price is $325\$ 325. The supply function is given by ρ=0.0275x\rho=0.0275 x. Find the consumer and producer surpluses. (Assume the demand function is linear.) consumer surplus \ \square$ producer surplus \$ Need Help? Read It Watch It

Studdy Solution

STEP 1

What is this asking? We need to find the consumer and producer surpluses for air conditioners, given how many are bought at two different price points, and knowing how the number supplied changes with price. Watch out! Don't mix up supply and demand!
Also, remember the area of a triangle is 12⋅base⋅height\frac{1}{2} \cdot \text{base} \cdot \text{height}.

STEP 2

1. Find the demand function
2. Find the equilibrium point
3. Calculate the consumer surplus
4. Calculate the producer surplus

STEP 3

We're given two points on the demand curve: (4000 units,$350)(\text{4000 units}, \$\text{350}) and (6000 units,$325)(\text{6000 units}, \$\text{325}).
Since we're assuming the demand function is linear, we can **find the slope**!

STEP 4

Remember, slope is change in pricechange in quantity\frac{\text{change in price}}{\text{change in quantity}}.
So, our slope is $325−$3506000−4000=−$252000=−$0.0125 per unit.\frac{\text{\(\$325\)} - \text{\(\$350\)}}{\text{6000} - \text{4000}} = \frac{-\$25}{2000} = - \$\text{0.0125 per unit}. This means for every extra unit demanded, the price goes down by $\$0.0125.
Makes sense, right?

STEP 5

Now we can **use the point-slope form** of a linear equation: p−p1=m(x−x1)p - p_1 = m(x - x_1), where (x1,p1)(x_1, p_1) is one of our points and mm is the slope.
Let's use the point (4000,350)(\text{4000}, \text{350}): p−350=−0.0125(x−4000).p - 350 = -0.0125(x - 4000).

STEP 6

**Simplify** to get our demand function: p=−0.0125x+50+350p = -0.0125x + 50 + 350 p=−0.0125x+400.p = -0.0125x + 400.

STEP 7

The **equilibrium point** is where supply equals demand.
We have our demand function, p=−0.0125x+400p = -0.0125x + 400, and our supply function, p=0.0275xp = 0.0275x.
Let's set them equal!

STEP 8

−0.0125x+400=0.0275x.-0.0125x + 400 = 0.0275x. **Add** 0.0125x0.0125x to both sides to get 400=0.04x.400 = 0.04x.

STEP 9

Now, **divide** both sides by 0.04 to find the equilibrium quantity: x=4000.04=10000.x = \frac{400}{0.04} = 10000. So, the equilibrium quantity is **10,000 units**.

STEP 10

Plug this back into either the supply or demand function to find the equilibrium price.
Let's use the supply function because it's simpler: p=0.0275⋅10000=$275.p = 0.0275 \cdot 10000 = \$275. The equilibrium price is $\$275.

STEP 11

The **consumer surplus** is the area of the triangle formed by the demand curve, the horizontal line at the equilibrium price (p=275p = 275), and the vertical axis.

STEP 12

The **height** of this triangle is the difference between the y-intercept of the demand function (400) and the equilibrium price (275), which is 400−275=125400 - 275 = 125.

STEP 13

The **base** of the triangle is the equilibrium quantity, which is 10000.

STEP 14

So, the consumer surplus is 12⋅10000⋅125=$625,000.\frac{1}{2} \cdot 10000 \cdot 125 = \$625,000.

STEP 15

The **producer surplus** is the area of the triangle formed by the supply curve, the horizontal line at the equilibrium price (p=275p = 275), and the vertical axis.

STEP 16

The **height** of this triangle is the equilibrium price, which is 275.

STEP 17

The **base** of this triangle is the equilibrium quantity, which is 10000.

STEP 18

So, the producer surplus is 12⋅10000⋅275=$1,375,000.\frac{1}{2} \cdot 10000 \cdot 275 = \$1,375,000.

STEP 19

Consumer Surplus: $625,000\$625,000 Producer Surplus: $1,375,000\$1,375,000

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