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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Checkpoint 5 Worked-out solution available at LarsonAppliedCalculus.com

The demand and supply functions for a product are modeled by

Demand :   p =  – 0.2 x +   8  and  S u p p l y :   p =   0.1   x   +   2 where p is the price (in dollars) and x is the number of units (in millions). Find the consumer and producer surpluses for this product.

To determine

To calculate: The consumer and producer surpluses for a product whose demand and supply functions are modelled by Demand:p=0.2x+8 and Supply:p=0.1x+2

Explanation

Given Information:

The demand and supply functions for a product are modeled by

Demand:p=0.2x+8 and Supply:p=0.1x+2

Here, p is the price (in dollars) and x is the number of units (in millions).

Formula used:

The consumer surplus is the area of the region bounded by the graph of the demand function, the horizontal line p=p0 and the vertical line x=0.

Consumer surplus=0x0[demand functionprice]dx

The producer surplus is the area of the region bounded by the graph of the supply function, the horizontal line p=p0 and the vertical line x=0.

Producer surplus=0x0[pricesupply function]dx

Here (x0,p0) is the equilibrium point at which the demand function and supply function intersect.

Calculation:

Consider the demand and supply functions,

Demand:p=0.2x+8 and Supply:p=0.1x+2

Compute the points of intersection of two graphs by setting the functions equal to each other and solving for x,

Demand=Supply0.2x+8=0.1x+20.2x0.1x=280.3x=6

Divide both sides by 0.3,

(0.3x)(0.3)=(6)(0.3)x=20

Substitute x=20 in the demand function p=0.2x+8 and compute the price,

p0=0.2×20+8=4+8=4

So, the equilibrium point occurs at x=20 million and the price is $ 4 per unit.

Use the formula for the consumer surplus and calculated it for the provided product as follows:

Consumer surplus=0x0[demand functionprice]dx=020[(0.2x+8)4]dx=020[0

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