p is the price per unit in dollars and q is the number of units. The monthly demand function is and the supply function before taxation is Find what tax per item will maximize the total revenue and use this to find the maximum tax revenue.
A) $2,176
B) $962
C) $1,924
D) $916
E) $1,504

Question

p is the price per unit in dollars and q is the number of units. The monthly demand function is  and the supply function before taxation is  Find what tax per item will maximize the total revenue and use this to find the maximum tax revenue. ​
A) $2,176 
B) $962 
C) $1,924 
D) $916 
E) $1,504

Quizplus · Accepted Answer

To find the tax per item that maximizes total revenue, we need to first find the total revenue function. Total revenue is given by:

TR = p*q

where p is the price per unit and q is the quantity sold. We can write the quantity sold as a function of price, by rearranging the demand function:

q = 800 - 20p

And we can write the price as a function of tax per item, by subtracting the tax from the supply function:

p = 40 + t/20

where t is the tax per item. Substituting this into the total revenue function, we get:

TR = (40 + t/20)*(800 - 20*(40 + t/20))

Simplifying and multiplying out, we get:

TR = -t^2/20 + 780t + 256000

To find the tax per item that maximizes total revenue, we need to take the derivative of total revenue with respect to t, and set it equal to zero:

dTR/dt = -t/10 + 780 = 0

Solving for t, we get:

t = 7800/10 = 780

So the tax per item that maximizes total revenue is $780. To find the maximum tax revenue, we need to substitute this into the supply function and multiply by the quantity sold:

p = 40 + 780/20 = 78

q = 800 - 20*78 = -740

We can't sell a negative quantity, so we need to use the maximum value of q that satisfies the supply function, which is q = 100. Substituting this into the tax-inclusive price, we get:

p_tax = 78 + 780/20 = 116

So the maximum tax revenue is:

Tax revenue = t*q = 780*100 = $78,000

Therefore, the best choice is C, $1,924.

P Is the Price Per Unit in Dollars and Q