Alice runs a restaurant and it is only open on Wednesdays and Saturdays. Her tot

Question

Alice runs a restaurant and it is only open on Wednesdays and Saturdays. Her total cost is TC=10+q2 where q is number of meals served per day. Her restaurant is a price taker. A meal is sold at $12 on Wednesday and $20 on Saturday. a. Get her average cost per day and marginal cost per day. b. How many meals does she serve on Wednesday to maximize profit? And Saturday? How much is her profit each day? Per week? c. A newspaper noticed the price difference on different days and reported it. Now to make consumers happy all restaurants in the town have to sell meals for $16 regardless of the days. Calculate Alice’s optimal number of meals to serve and her profit each day and per week. d. Still assume the meals are sold at $16 each day. Now Alice has to hand in 10% of her profit as income tax to the government. How many meals should she serve each day? How much is her profit after tax? e. Still assume the meals are sold at $16 each day. Now suppose that due to competition all restaurants serve a free dessert with each meal. This costs Alice $2 for each meal she serves. Calculate her problem and profit each day and per week

Explanation / Answer

TC = 10 + q2

(a)

Average cost, AC = TC / q = (10 / q) + q

Marginal cost, MC = dTC / dq = 2q

(b)

For a competitive price taker, profit is maximized by equating price with MC:

(i) Wednesday:

2q = 12

q = 6

ATC = (10 / q) + q = (10 / 6) + 6 = 7.67

Profit = q x (P - ATC) = 6 x (12 - 7.67) = 25.98

(ii) Saturday:

2q = 20

q = 10

ATC = (10 / q) + q = (10 / 10) + 10 = 11

Profit = q x (P - ATC) = 10 x (20 - 11) = 90

(iii)

Weekly profit = 25.98 + 90 = 115.98

(c)

Let optimal number of meals = Q (per day)

Equating MC with price:

2Q = 16

Q = 8

So, daily average cost = (10 / Q) + Q = (10 / 8) + 8 = 9.25

daily profit = Q x (P - ATC) = 8 x (16 - 9.25) = 54

Weekly profit = 2 x 54 = 108

(d)

Let optimal number of meals = Q (per day)

Pre-tax Profit, Z = Q x (16 - ATC) = Q x [16 - (10 / Q) + Q] = 16Q - 10 + Q2

Post-tax profit, Z1 = 0.90 x (16Q - 10 + Q2)

Profit is maximized when dZ1 / dQ = 0

0.90 x (16 - 2Q) = 0

Q = 16 / 2 = 8

So, Revenue = P x Q = 16 x 8 = 128

TC = 10 + Q2 = 10 + 64 = 74

Profit = 128 - 74 = 54

Profit after tax = 54 x 0.90 = 48.6

NOTE: Out of 5 questions, the first 4 are answered.

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