Question

The annual demand function for a particular motor car is
estimated as:
D = 16000-10P/3+ Y2/1000
where D =annual demand, P =price in £'s and Y =average
disposable income.
(i) Given that the retail price next year will be £12 000, whilst
average disposable income is expected to be £8000, estimate
next year's annual demand. If the manufacturer
receives 80% of the retail price for each car sold, estimate
the manufacturer's revenue next year.
(ii) Find the retail price to maximise manufacturer's revenue
next year.
(iii) If the marginal cost per car is estimated to be £6000, find
the price to maximise profit next year.
(iv) In the subsequent year the retail price is expected to rise to
£13 000, whilst incomes should increase by 5%. Estimate
demand and manufacturer's revenue for that year, and use
this information to estimate the price and income demand
elasticities

Answer 1

(i) The annual demand function for the particular motor car is given as D=16000-10P/3+2Y/1000 where D, P, and Y denote the annual demand of the car, the retail price of the car in pounds, and the average disposable income of the consumers. The  P and Y in next year are given as 12000 and 8000 pounds respectively. The manufacturer obtains 80% or 0.8 of the P.

Therefore, lugging the respective values of the variables given into the annual demand function of the car, we get:-

D=16000-10P/3+Y^2/1000

D=16000-10*(12000)/3+(8000)^2/1000

D=16000-120,000/3+64,000,000/1000

D=16000-40000+64000

D=40,000

Hence, next year's annual demand for the manufacturing car would be 40,000 cars.

The price received by the manufacturer=(0.8*12000)=9600 pounds for each car sold and thus, the total sales revenue of the manufacturer during next year=(9600 pounds*40,000 cars)=384,000,000 pounds

(ii) Based on the respective values of the variables given in the question, the annual demand function of the car can be written or expressed as D=16000-10P/3+64000

Hence, the inverse annual demand function of the car can be stated or derived as:-

D=16000-10P/3+64000

D=80000-10P/3

10P/3=80000-D

10P=240,000-3D

P=(240,000-3D)/10

P=24000-0.3D

Thus, the total revenue or TR of the manufacturer during the next year would be=P*D=(24000-0.3D)*D=24000D-0.3D^2. Therefore, the marginal revenue or the MR of the manufacturer would be: MR= dTR/dD=24000-0.6D. Now, based on the profit-maximizing condition or principle, the manufacturer would maximize its TR where the MR is equal to 0.

Therefore, based on the revenue-maximizing condition or principle, it can be stated:-

24000-0.6D=0

-0.6D=-24000

D=-24000/-0.6

D=40,000

Hence, when the MR of the car manufacturer is 0 the overall annual demand for the car becomes 40,000 cars.

Now, plugging the value of the revenue-maximizing annual demand of the car into the inverse annual demand function, we can get:

P=24000-0.3D

P=24000-0.3*(40000)

P=24000-12000

P=12000

Therefore, the revenue-maximizing price of the car manufacturer would be 12000 pounds.

(iii) The marginal cost per car or MC is given as 6000 pounds and based on the profit-maximizing condition or principle of any competitive firm, the output produced by the firm would correspond to the equality between the price of the concerned good or service and the marginal cost of production of the good or service. The inverse demand function is P=24000-0.3D

Therefore, based on the profit-maximizing condition of any competitive firm, it can be stated:-

P=MC

24000-0.3D=6000

-0.3D=6000-24000

-0.3D=-18000

D=-18000/-0.3

D=60,000

Thus, the profit-maximizing number of cars in the market would be 60,000 cars.

Now, plugging the value of the profit-maximizing number of cars into the inverse demand function, we get:-

P=24000-0.3D

P=24000-0.3*(60,000)

P=24000-18000

P=6000

Hence, the profit-maximizing price of the car manufacturer would be 6000 pounds.

(iv) The initial average disposable income or Y of the consumers is 8000 pounds and it increases by 5% or 0.05. Therefore, the new or subsequent Y would be=8000 pounds+(0.05*8000 pounds)=8400 pounds. The P increases to 13000 pounds from 12000 pounds in the subsequent year. Therefore, the percentage in the P, in this case, =(13000-12000)/12000=0.083 or 8.3%.

Therefore, plugging the new values of P and Y into the annual demand function of the car, we get:-

D=16000-10P/3+Y^2/1000

D=16000-10*(13000)/3+(8400)^2/1000

D=16000-130,000/3+70560

D=16000-43,333.33+70560

D=70560-27333.33

D=43226.67

Hence, as a result of the subsequent changes in P and Y the D changed from 40000 cars to 43227 cars approximately. Thus, the percentage change in D would be=(43227-40000)/40000=0.08 or 8% approximately. Now, based on the mathematical formula to calculate the price and income elasticity of demand of any commodity or product, the price elasticity of demand of any food or product=Percentage change in quantity demanded of the product or good divided by the percentage change in price of the product or good and income elasticity of demand of any good or product would be Percentage change in the quantity demanded of any product or good divided by the percentage change in disposable income of the consumers.

Thus, based on the mathematical formula to calculate the price elasticity of demand of any good, we can state:-

Percentage change in D/Percentage change in P

=0.08/0.083

=0.96 approximately

Therefore, the price elasticity of demand of the car would be 0.96 approximately.

Thus, based on the mathematical formula to calculate the income elasticity of demand of any good, we can state:-

Percentage change in D/Percentage change in Y

=0.08/0.05

=1.6

Hence, the income elasticity of demand of the car would be 1.96 in this instance.

The total revenue or TR of the car manufacturer, in this case, would be=(13000 pounds*43227 cars)=561,951,000 pounds approximately.