Brown Inc. has the following weekly demand and cost relationships: q = 2000 - 25

Question

Brown Inc. has the following weekly demand and cost relationships:

                        q = 2000 - 250P

                        TC = 0.006q2 + q

1. Determine the total revenue maximizing levels of output (q) and price (P).

2. Derive the value for price elasticity of demand at the total revenue maximizing levels of output and price.

3. Determine the profit maximizing levels of output (q) and price (P).

4. Derive and interpret the value for price elasticity of demand at the total profit maximizing levels of output and price.

5. What is maximum weekly profit?

6. On the same graph sketch marginal cost (MC), marginal revenue (MR), and the inverse demand curve (= average revenue). On the this graph, denote the profit-maximizing levels of q and P .

Explanation / Answer

Demand function: q = 2000 - 250P

250P = 2000 - q

P = 5 - 0.004q

Cost function: TC = 0.006q2 + q

Marginal cost (MC) = dTC/dq = 0.012q + 1

(1) Total revenue (TR) = P x q = 5q - 0.004q2

Revenue is maximized when dTR/dq (= MR) = 0.

MR = 5 - 0.008q = 0

0.008q = 5

q = 625

p = 5 - (0.004 x 625) = 5 - 2.5 = 2.5

(2) Price elasticity of demand = (dq/dP) x (P/q) = -250 x (2.5 / 625) = -1

(3) Profit is maximized when MR equals MC.

5 - 0.008q = 0.012q + 1

0.02q = 4

q = 200

p = 5 - (0.004 x 200) = 5 - 0.8 = 4.2

(4) Price elasticity of demand = (dq/dP) x (P/q) = -250 x (4.2 / 200) = -5.25

This means that if price is increased (decreased) by 1%, quantity demanded will decrease (increase) by 5.25%.

NOTE: As per Answering Policy, first 4 parts have been answered.

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