Demand Equation and Revenue Graphing

1. Monthly demand function is highlighted below:

Supply function before taxation is highlighted below

         

Tax per item that will maximize the total tax revenue =?

Let the supply function after taxation is given below:

       

At equilibrium

    

The total tax would be T that is given below:

       

In order to maximize the total tax, take first derivative of function and put equal to zero.

   

Tax per item

     

Therefore, the tax per item would be 4800.

2. Demand function 

       

To find the revenue of function take first derivative of R.

         

Hence, the revenue function is

        

                                                                

3. The given function

           

Elasticity of demand function, region of elasticity, unitary and inelastic demand =?

Now,

Elasticity of demand

             

In case of unitary elasticity,

             

4. It can be concluded from the above that for the values of p between 0 < p < (20/3) demand would be elastic. Similarly, for the values of p between (20/3) <p<20, demand would be              inelastic.

5. The given cost is shown below:

            

   a) The cost of producing 25 units

           

Now,

            

Hence, the cost of producing 25 units would be $572.43.

   b) Number of unit produced when the total cost $250

             

Now,

               

6. Growth of number of employees of company is highlighted below:

              

   a) Number of employee in company after 1 year

             

Hence, the number of employee in company after 1 year would be 240.

   b) Maximum number of employees of the company

                                                       

Based on the above shown graph, it can be seen that the maximum number of employee of the company would be 250.

Cost Function and Elasticity Analysis

7. The function for the same of a product is highlighted below:

               

   a) Sale of the company after 6 months

                 

   b) The month after which the sales be below 100,000

                 

It can be said that sale will take 10 months to be below 100,000.

     c) Graph S(x)

                                                                        

8. Retention by students of material learned earlier is highlighted below:

                

Retention rate after 3 weeks =?

Rate of retention would be first derivation of given function.

              

Now,

              

Therefore, the retention rate after 3 weeks would be 4.787 per week.

9. Monopoly market revenue from selling the x number of units is represented below:

              

    a) Marginal revenue when the number of units is 5

        The marginal revenue can be determined after taking the first derivative of the revenue function.

                          

The marginal revenue function is

                         

Now, marginal revenue for 5 units

                       

  b) Value of x for maximum revenue

 For this, the marginal revenue function would be taken as equal to zero.  

                       

Hence, at maximum revenue the number of units would be 2.5.

c) Revenue function

                                                                          

10. The given demand and supply function for a product is highlighted below:

     Demand function

     

Supply function

     

Let the tax per item is t.

The supply function after taxation is highlighted below:

   

At equilibrium the demand and supply function would be same.

    

Now,

Let the total tax revenue is T.

The T would a function of t and q.

        

In order to maximize the total tax revenue, put the first derivative of the function equal to zero.

       

Now,

Tax per item from equation 1

      

Therefore, the tax per item would be 168.

11.“When price changes cause significant changes in demand, the demand is said to be _____________elastic ______ for that product and if price changes cause relatively little change in           demand, the demand is said to be _____inelastic _____________ for that product.” 

12.  If the point of elasticity of demand is greater than 1, the demand is __________elastic______. If the point of elasticity of demand is less than 1, the demand is                                           ____________inelastic____.

13. Demand function of a product is highlighted below:

         

Now,

Point price elasticity of demand

        

        

 It can be seen from the above that point price elasticity of demand is lower than 1. This indicates that the demand is inelastic in nature. Hence, the inelastic demand nature would indicate that increase in price will also increase the total revenue.