   Chapter 11, Problem 37RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Taxes Suppose the supply and demand functions for a product are p = 40 + 20 q and p = 5000 q + 1 respectively, where p is in dollars and q is the number of units. Find the tax t that maximizes the tax revenue T and find the maximum tax revenue.

To determine

To calculate: The tax per item that will maximize the total tax revenue if the demand function for a fixed period of time is given by p=5000q+1 and the supply function before taxation is p=40+20q.

Explanation

Given Information:

The provided expression is demand function for a fixed period of time is given by p=5000q+1 and the supply function before taxation is p=40+20q.

Where t is the tax per unit.

The total tax T is equal to T=tq, where t is the tax per unit and q is the number of units.

The tax revenue is maximized by finding the value of the variable at first derivative is equal to 0. Also checking by verifying that T(q)0 or not.

Formula Used:

The following procedure are used to maximizing total tax revenue,

Step-1 Write the supply function after taxation,

Step-2 Equate the demand function and new supply function to get number of units and tax.

Step-3 Calculate the total revue function T that will be product of number of units and tax.

Step-4 Calculate the derivative of revenue function and set equal to zero and solve.

Step-5 Calculate second derivative test to verify the result.

Calculation:

The provided expression is p=40+20q,

After the taxation, the supply function is equal to p=40+20q+t. The demand function will meet the new supply function is:

Put the new supply(after taxation) equal to the demand.

40+20q+t=5000q+1

Solve to equation further to get the value of t,

Add (5000q+1) on both the sides,

40+20q+t(5000q+1)=5000q+1(5000q+1)40+20q+t(5000q+1)=0

Deduct t from both the sides,

40+20q+t(5000q+1)t=0t

Multiply by -1 on both the sides,

t=(5000q+1)4020q

Then, compute the total tax by the formula T=tq,

T=q((5000q+1)4020q)=((5000qq+1)40q20q2)

Tax is maximized as follows:

The first derivative of T=((5000qq+1)40q20q2) is as follows:

T(q)=ddq

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