The inverse demand function views price as a function of quantity. The inverse demand function of the firm will be:
[tex]\rm P = 9 - 0.25Q[/tex]
The profit-maximizing price and level of production is the point where MR = MC. The profit-maximizing price is $8.50 and the production level is 2 units.
What is inverse demand function?
Inverse demand function is the function that represents price as the function of quantity demanded. The graph of inverse demand puts price on the y axis and quantity demanded on the x axis.
a. The inverse demand function of the firm will be calculated as follows:
Given:
Demand function is:
[tex]\rm Q = 36 - 4P[/tex]
The inverse demand function is the demand function in terms of price. Therefore inverse demand function will be:
[tex]\rm Q = 36 - 4P\\\\4P = 36 - Q\\\\P = \dfrac{36}{4} - \dfrac{Q}{4}\\\\P = 9 - 0.25Q[/tex]
b. The profit-maximization point is the point at which MR=MC.
MR refers to the marginal revenue and MC is the marginal cost.
MC can be calculated as the first derivative of the cost function:
[tex]\rm C(Q) = 4 + 4Q + Q^2\\\\MC = \dfrac{d}{dQ}(4 + 4Q + Q^2)\\\\MC = 4 + 2Q[/tex]
The total revenue can be calculate as a product of price and quantity.
[tex]\rm Total\:revenue = Price \: \times Quantity\\\\\rm Total\:revenue = (9-0.25Q) \times Q\\\\\rm Total\:revenue = 9Q - 0.25Q^2[/tex]
The Marginal revenue can be calculated as the first derivative of the total revenue function:
[tex]\rm TR = 9Q - 0.25Q^2\\\\MR = \dfrac{d}{dQ}(9Q - 0.25Q^2)\\\\MR = 9 - 0.50Q[/tex]
The profit maximization point will be:
[tex]\begin{aligned}\rm MR &= MC\\\\\rm 9 - 0.50Q &= 4 + 2Q\\\\\rm 9 - 4 &= \rm 2Q + 0.50Q\\\\\rm 5 &= \rm 2.50Q \\\\\rm Q &= \dfrac{5}{2.50}\\\\\rm Q&= 2\end[/tex]
The profit maximization quantity therefore is 2 units.
Substituting the value of Q in inverse demand function:
[tex]\rm P = 9 - 0.25Q\\\\\rm P = 9 - 0.25(2)\\\\\rm P = 9 - 0.50\\\\\rm P = \$8.50[/tex]
Therefore the profit profit-maximizing price is $8.50.
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