Question

A marketing firm is faced with fixed cost of 22500 and it's total variable cost is given as TVC=2000Q+0.2Q^2. If the demand function is P=2500-0.8Q. find:
1) the total revenue function in terms of Q
2) the total cost function in terms of Q
3) the profit function in terms of Q
4) the level of output that maximizes profit and the profit level.
5) the value of marginal cost and marginal revenue at this profit.
6) the level of output for which average cost is minimized.

Answer 1

1) The total revenue function in terms of Q

P=2500-0.8Q

$TR = P\times Q$

TR = $(2500-0.8Q)\times Q$

TR = $2500Q - 0.8Q^2$

2) the total cost function in terms of Q

Total cost (TC) = Fixed Cost + Variable cost

Fixed cost = 22,500

Variable cost = $2000Q+0.2Q^2$

TC = $22,500 + 2000Q+0.2Q^2$

3) The profit function in terms of Q

Profit = Total revenue – Total cost

Profit = $2500Q - 0.8Q^2 – (22,500 + 2000Q+0.2Q^2)$

Profit = $2500Q - 0.8Q^2 – 22,500 - 2000Q - 0.2Q^2$

Profit = $500Q- Q^2 – 22,500$

4) the level of output that maximizes profit and the profit level.

Profit maximizing level of output:

Getting the first order conditions by differentiating profit with respect to quantity:

dProfit/dQuantity = 0

dProfit/dQuantity = $\frac{d(500Q- Q2 – 22,500)}{dQ}$

dP/dQ = 500 – 2Q = 0

2Q = 500

Q = $\frac{500}{2} = 250$

5) The value of marginal cost and marginal revenue at this profit.

Marginal cost (MC) = dTotal Cost/dQuantity

TC = $22,500 + 2000Q+0.2Q^2$

MC = $\frac{d(22,500 + 2000Q+0.2Q2)}{dQ}$

MC = $2000 + 2\times0.2Q$

MC = 2,000 + 0.4Q

When Q = 250

MC = $2,000 + 0.4\times250$

MC = 2,000 + 100 = 2,100

Marginal Revenue (MR) = dTotal Revenue/dQuantity

TR = 2500Q - 0.8Q²

MR = $\frac{d(TR = 2500Q - 0.8Q^2)}{dQ}$

MR = 2500 – 1.6Q

When Q = 250

MR = $2500 – 1.6\times250$

MR = 2500 – 400 = 2,100

6) the level of output for which average cost is minimized.

Average cost (AC) = $\frac{Total Cost}{Q}$

TC =$22,500 + 2000Q+0.2Q^2$

AC = $\frac{(22,500 + 2000Q+0.2Q^2)}{Q}$

AC = $\frac{22,500}{Q} + 2000 + 0.2Q$

$\frac{dAC}{Dq} = 0$

dAC /dQ =$\frac{d(22,500/Q + 2000 + 0.2Q)}{dQ} = 0$

= $- 22,500Q^-2 + 0.2 = 0$

22,500Q^-2 = 0.2

Q² = $\frac{22,500}{0.2} = 112500$

Q = $\sqrt{112,500} = 335.4102$