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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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Consumer and Producer Surpluses In Exercises 43-48, find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). See Example 5.

Demand Function Supply Function

p = 50 0.5 x                          p = 0.125 x

To determine

To calculate: The consumer and producer surpluses for a product whose demand and supply functions are modeled by Demand:p=500.5x and Supply:p=0.125x

Explanation

Given Information:

The demand and supply functions for a product are modeled by

Demand:p=500.5x and Supply:p=0.125x

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=500.5x and Supply:p=0.125x

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

Demand=Supply500.5x=0.125x0.5x0.125x=0500.625x=50

Divide both sides by 0.625,

(0.625x)(0.625)=(50)(0.625)x=80

Substitute x=80 in the demand function p=500.5x and compute the price,

p0=500.5×80=5040=10

So, the equilibrium point occurs at x=80 million and the price is $ 10 per unit.

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

Consumer surplus=0x0[demand functionprice]dx=080[(500

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