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Given the demand function for movie tickets for adults and children: PA = 100 ...
Nov 13, 2024
Given the demand function for movie tickets for adults and children: PA = 100 – 2qA ----- adult DD function PC = 60 – 2qC ------ child's DD function The marginal and average cost to run a theatre is MC = AC = RM20. Based on the information above: (a). What are the price and quantity of tickets if the owner implement third degree price discrimination. (b) Find the marked demand function. Calculate the price and quantity of tickets without price discrimination. (c) Calculate the consumer surplus and producer surplus for the situation at (a) and (b). Explain the effect of price discrimination towards the society’s welfare.
Answer
a
To implement third-degree price discrimination, we need to find the optimal price and quantity for adults and children separately. For adults, set the marginal revenue (MR) equal to marginal cost (MC): Demand function: PA=100−2qA P_A = 100 - 2q_A Total Revenue (TR) for adults: TRA=PA⋅qA=(100−2qA)qA=100qA−2qA2 TR_A = P_A \cdot q_A = (100 - 2q_A)q_A = 100q_A - 2q_A^2 Marginal Revenue (MR): MRA=d(TRA)dqA=100−4qA MR_A = \frac{d(TR_A)}{dq_A} = 100 - 4q_A Setting MRA=MC MR_A = MC : 100−4qA=20 100 - 4q_A = 20 4qA=80 4q_A = 80 qA=20 q_A = 20 Price for adults: PA=100−2(20)=60 P_A = 100 - 2(20) = 60 For children, similarly: Demand function: PC=60−2qC P_C = 60 - 2q_C Total Revenue (TR) for children: TRC=(60−2qC)qC=60qC−2qC2 TR_C = (60 - 2q_C)q_C = 60q_C - 2q_C^2 Marginal Revenue (MR): MRC=60−4qC MR_C = 60 - 4q_C Setting MRC=MC MR_C = MC : 60−4qC=20 60 - 4q_C = 20 4qC=40 4q_C = 40 qC=10 q_C = 10 Price for children: PC=60−2(10)=40 P_C = 60 - 2(10) = 40
b
The market demand function combines both adult and child demand. The total quantity demanded at each price can be found by adding the quantities from both groups. For adults: qA=20 q_A = 20 at PA=60 P_A = 60 For children: qC=10 q_C = 10 at PC=40 P_C = 40 Total quantity: Q=qA+qC=20+10=30 Q = q_A + q_C = 20 + 10 = 30 To find the market demand function, we can express it as: P=100−2qA P = 100 - 2q_A for adults and P=60−2qC P = 60 - 2q_C for children. At Q=30 Q = 30 , we can find the price by substituting back into either demand function. Using the adult function: P=100−2(20)=60 P = 100 - 2(20) = 60 Using the child function: P=60−2(10)=40 P = 60 - 2(10) = 40 The market price without discrimination is determined by the higher price, which is P=60 P = 60 and total quantity Q=30 Q = 30
c
Consumer surplus (CS) and producer surplus (PS) can be calculated for both scenarios. For (a): CS for adults: 12×(100−60)×20=400 \frac{1}{2} \times (100 - 60) \times 20 = 400 CS for children: 12×(60−40)×10=100 \frac{1}{2} \times (60 - 40) \times 10 = 100 Total CS = 400+100=500 400 + 100 = 500 PS = Total Revenue - Total Cost = (60×20+40×10)−(30×20)=1200−600=600 (60 \times 20 + 40 \times 10) - (30 \times 20) = 1200 - 600 = 600 For (b): CS = 12×(100−60)×30=600 \frac{1}{2} \times (100 - 60) \times 30 = 600 PS = (60×30)−(30×20)=1800−600=1200 (60 \times 30) - (30 \times 20) = 1800 - 600 = 1200 Price discrimination increases producer surplus but may reduce consumer surplus, affecting overall welfare
Answer
Price for adults: RM60, quantity: 20; Price for children: RM40, quantity: 10; Market price without discrimination: RM60, quantity: 30.
Key Concept
Price discrimination allows firms to maximize profits by charging different prices to different consumer groups based on their willingness to pay.
Explanation
Price discrimination can lead to higher producer surplus and lower consumer surplus, impacting overall societal welfare.
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