Answer to Question #192417 in Accounting for robel

Question

A firm’s demand curve in period 1 is Q=25 - P. Fixed costs are 20 and marginal costs per unit are 5. (5 marks) a. Derive equations for total revenue and marginal revenue. b. At what output will marginal revenue be zero? c. At what price will total revenue be maximized? d. At what price and output will profit be maximized? e. Calculate the maximum profits the firm makes.

Answer 1

Solution:

a.). Equations for total revenue and marginal revenue:

Total Revenue (TR) = P $\times$ Q

First, derive the inverse function of the demand function

Q = 25 – P

P = 25 – Q

TR = (25 – Q) Q = 25Q – Q²

TR = 25Q – Q²

Marginal revenue = change in TR divided by change in Q

MR = $\frac{\partial TR} {\partial Q} = 25 - 2Q$

MR = 25 – 2Q

 

b.). Profit maximizing quantity and price can be determined by setting MR equal to zero, which occurs at the maximum level of output.

Set MR = 0 to determine the level of output

MR = 25 – 2Q

25 – 2Q = 0

25 = 2Q

Q = $\frac{25}{2} = 12.5$

Q = 12.5

Marginal revenue will be zero when output is 12.5 units

 

c.). Total revenue will be maximized at the quantity at which marginal revenue is equal to zero.

When MR = 0, quantity is equal to 12.5 units

Substitute for price in the inverse demand function:

P = 25 – Q

P = 25 – 12.5

P = 12.5

Total revenue will be maximized when the price is 12.5.

 

d.). Profit will be maximized when MR = MC

MC = 5

MR = 25 – 2Q

25 – 2Q = 5

25 – 5 = 2Q

20 = 2Q

Q = $\frac{20}{2} = 10$

Substitute to derive P:

P = 25 – Q

P = 25 – 10

P = 15

Profit maximizing price = 15

Profit maximizing quantity = 10

 

e.). Maximum profits = TR – TC

TR = P x Q = 15 x 10 = 150

TC = VC + FC

FC = 20

VC = MC per unit x Number of units = 5 x 10 = 50

TC = 50 + 20 = 70

Maximum profits = 150 – 70 = 80

Maximum profits = 70