Explanation Given Information: The provided expression is demand function for a fixed period of time is given by p = 5000 q + 1 and the supply function before taxation is p = 40 + 20 q . Where t is the tax per unit. The total tax T is equal to T = t q , 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 + 20 q , After the taxation, the supply function is equal to p = 40 + 20 q + t . The demand function will meet the new supply function is: Put the new supply(after taxation) equal to the demand. 40 + 20 q + t = 5000 q + 1 Solve to equation further to get the value of t, Add − ( 5000 q + 1 ) on both the sides, 40 + 20 q + t − ( 5000 q + 1 ) = 5000 q + 1 − ( 5000 q + 1 ) 40 + 20 q + t − ( 5000 q + 1 ) = 0 Deduct t from both the sides, 40 + 20 q + t − ( 5000 q + 1 ) − t = 0 − t Multiply by -1 on both the sides, t = ( 5000 q + 1 ) − 40 − 20 q Then, compute the total tax by the formula T = t q , T = q ( ( 5000 q + 1 ) − 40 − 20 q ) = ( ( 5000 q q + 1 ) − 40 q − 20 q 2 ) Tax is maximized as follows: The first derivative of T = ( ( 5000 q q + 1 ) − 40 q − 20 q 2 ) is as follows: T ′ ( q ) = d d q

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 11, Problem 41RE

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 40RE

Chapter 11, Problem 1T

Students have asked these similar questions

Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows. From To Hamilton Butler Clermont Southern Gas 10 20 15 Northwest Gas 12 15 18 (a) Develop a network representation of this problem. (Submit a file with a maximum size of 1 MB.) Choose File No file chosen Assignment 3 graph.docx Score: 1 out of 1 Comment: (b) Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs (in thousands of…

Use the method of undetermined coefficients to solve the given nonhomogeneous system. dx dt = 2x + 3y − 8 dy dt = −x − 2y + 6 X(t) =

As discussed in Section 8.3, the Markowitz model uses the variance of the portfolio as the measure of risk. However, variance includes deviations both below and above the mean return. Semivariance includes only deviations below the mean and is considered by many to be a better measure of risk. (a) Develop a model that minimizes semivariance for the Hauck Financial data given in the file HauckData with a required return of 10%. Assume that the five planning scenarios in the Hauck Financial Services model are equally likely to occur. Hint: Modify model (8.10)–(8.19). Define a variable ds for each scenario and let ds ≥ R − Rs with ds ≥ 0. Then make the objective function: Min 1 5 5 s = 1 ds2. Let FS = proportion of portfolio invested in the foreign stock mutual fund IB = proportion of portfolio invested in the intermediate-term bond fund LG = proportion of portfolio invested in the large-cap growth fund LV = proportion of portfolio invested in the large-cap value fund…

Chapter 11 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 11.1 - 1. Ch. 11.1 - 2. If Ch. 11.1 - Prob. 3CP Ch. 11.1 - 4. Find . Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Find the derivatives of the functions in Problems...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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 = 5000 q + 1 and the supply function before taxation is p = 40 + 20 q .

Solution Summary: The author calculates 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 tax

Concept explainers

Videos

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.

Want to see the full answer?

Chapter 11 Solutions