1. Tutorial 3: Market Failures
Matthew Robson

2. Q 3.1
Consider a market with inverse demand function 𝑝=20−0.1𝑞. Each firm i has a linear cost function 𝑚 𝑖 + 𝑞 𝑖 𝑐 𝑖 (fixed cost and variable cost), and thus profit given by 𝜋 𝑖 = 𝑝− 𝑐 𝑖 𝑞 𝑖 − 𝑚 𝑖 . Assume that 𝑐 1 =2.
(a) Sketch a graph of the inverse demand function, and hence explain diagrammatically why Consumer Surplus in this market is equal to −𝑝 𝑞 .
Sketch a graph of firm i’s marginal cost function, and hence explain diagrammatically why Producer Surplus for firm i is given by 𝑝− 𝑐 𝑖 𝑞 𝑖 . 1

3. Q 3.1 (a)
Consumer Surplus
Producer Surplus 2

4. Q 3.1
(b) Assume that Firm 1 is a monopolist, so that 𝑞=𝑞 1 . Find Firm 1’s profit-maximising output 𝑞 1 , and the corresponding market price 𝑝. Evaluate Consumer and Producer Surplus at that point. What is the value of deadweight loss? 3

5. Q 3.1 (b) 𝜋 1 = 𝑝− 𝑐 1 𝑞 1 − 𝑚 1 𝑐 1 =2 𝑝=20−0.1𝑞 𝜋 1 = 𝑝−2 𝑞 1 − 𝑚 1
𝜋 1 = 𝑝− 𝑐 1 𝑞 1 − 𝑚 1
𝑐 1 =2
𝑝=20−0.1𝑞
𝜋 1 = 𝑝−2 𝑞 1 − 𝑚 1
𝜋 1 =18 𝑞 1 −0.1 𝑞 1 2 − 𝑚 1
𝜕 𝜋 1 𝜕 𝑞 1 =18−0.2 𝑞 1 =0
𝑞 1 =90, 𝑝=11, 𝜋 1 =810− 𝑚 1 4

6. Note: Sum of Consumer Surplus and Profit is: 1215− 𝑚 1
Q 3.1 (b)
Consumer Surplus:
0.5 20−𝑝 𝑞= −11 90=405
Producer Surplus:
𝑝− 𝑐 𝑖 𝑞 𝑖 = 11−2 90 = 810
Deadweight Loss:
(0.5(20−2)180)−810−405 = 405
Note: Sum of Consumer Surplus and Profit is: 1215− 𝑚 1 5

7. Q 3.1 (b)
Monopolist 6

8. Q 3.1
(c) A Government Regulator requires Firm 1 to set price 𝑝=2. Evaluate Consumer and Producer Surplus at that point. By how much is the sum of Consumer and Producer Surplus greater than in (b)? What is the maximum value of 𝑚 1 such that Firm 1 makes a non-negative profit here? 7

9. Q 3.1 (c) From the Demand Function 𝑝=20−0.1𝑞 p=2 implies q=180 So:
Consumer Surplus:
0.5 20−𝑝 𝑞= −2 180=1620
Producer Surplus:
𝑝− 𝑐 𝑖 𝑞 𝑖 = 2−2 180 =0
Sum of the Surpluses: 1620 is 405 greater than in (b)
Firm 1’s profit is 0− 𝑚 1 .
So 𝑚 1 =0. 8

10. Govt. Regulation, p=2
Q 3.1 (c) 9

11. Q 3.1
(d) The Regulator instead wants to enforce the lowest price p that allows Firm 1 to make a non-negative profit. If 𝑚 1 =450, what is that price? [Hint: use the demand function to substitute for 𝑞 1 in the profit function, and then solve 𝜋 1 =0 for 𝑝.]
At that price, what is the sum of Consumer and Producer Surpluses? 10

12. Q 3.1 (d)
𝜋 1 = 𝑝− 𝑐 1 𝑞 1 − 𝑚 1 , 𝑐 1 =2, 𝑝=20−0.1𝑞
𝜋 1 = 𝑝−2 𝑞 1 − 𝑚 1 = 𝑝−2 200−10𝑝 − 𝑚 1 =0
𝑝−2 200−10𝑝 −450=0
220𝑝−10 𝑝 2 −850=0
𝑝 2 −22𝑝+85=0
𝑝−5 𝑝−17 =0
𝒑=𝟓, 𝑝=17
If 𝑝=5 implies q=200−10 5 =150 . So:
From (a) Consumer Surplus = 1125, Producer Surplus = 450
So the Sum of Surpluses is 360 greater than in (b). 11

13. Govt. Regulation; Non-Negative π
Q 3.1 (d) 12

14. Q 3.1
(e) Now assume that Firms 1 and 2 are duopolists, so that 𝑞=𝑞 1 +𝑞 2 and assume that the two firms’ cost functions are identical, i.e. that 𝑐 2 =2 and 𝑚 1 = 𝑚 2 =𝑚. In the Cournot Equilibrium:
-What is the sum total of Consumer and Producer Surpluses?
-What is the maximum value of m such that each firm makes a non- negative profit?
-What is the maximum value of m such that the sum total of Consumer Surplus and profits is no less than in (b)? 13

15. Q 3.1 (e)
𝜋 1 = 18−0.1 𝑞 1 + 𝑞 2 𝑞 1 −𝑚
𝜕 𝜋 1 𝜕 𝑞 1 =18−0.2 𝑞 1 −0.1 𝑞 2 =0
Profit Max at: 𝑞 1 =90−0.5 𝑞 2
Symmetrically: 𝑞 2 =90−0.5 𝑞 1
So the equilibrium is : 𝑞 1 = 𝑞 2 =60, 𝑝=8,
𝜋 1 = 𝜋 2 =360−𝑚
From (a): Consumer Surplus = 720, Producer Surplus = 720
The sum of surpluses is The sum of total consumer surpluses and profits is – 2m
Non-Negative Profit: m =360, No Less than (b): m=225 14

16. Q 3.1 (e)
Duopoly; Equal m 15

17. Q 3.1
(f) Assume instead that 𝑐 2 =8 . In the Cournot equilibrium, what is the sum total of Consumer and Producer Surpluses? 16

18. Q 3.1 (f) As in (e) for Firm 1 : 𝑞 1 =90−0.5 𝑞 2 . But for Firm 2:
𝜋 2 = 12−0.1 𝑞 1 + 𝑞 2 𝑞 2 −𝑚
𝜕 𝜋 2 𝜕 𝑞 2 =12−0.2 𝑞 2 −0.1 𝑞 1 =0
Profit Max at: 𝑞 2 =60−0.5 𝑞 1
So the equilibrium is : 𝑞 1 = 80, 𝑞 2 =20, 𝑝=10, 𝜋 1 =640− 𝑚 1 , 𝜋 2 =40− 𝑚 2
From (a): Consumer Surplus = 500, Producer Surplus = 680
The sum of surpluses is 1180. 17

19. Q 3.1 (f)
Duopoly; Different m 18

20. Q 3.1
(g) What is the point of all this? 19

21. Govt. Regulation; p=2 Monopolist Govt. Regulation; Non-Negative π
Duopoly; Equal m
Duopoly; Different m 20

22. Q 3.2
Individuals A and B have incomes 𝑚 𝐴 =50 and 𝑚 𝐵 =100. They each spend a part of their incomes, 𝑥 𝐴 and 𝑥 𝐵 respectively, on a good with a mutual negative externality, their preferences being represented by the following utility functions:
𝑈 𝐴 = 𝑥 𝐴 𝑥 𝐵 − 𝑥 𝐴 𝑈 𝐵 = 𝑥 𝐵 𝑥 𝐴 − 𝑥 𝐵 0.8
 In a Nash equilibrium, A chooses 𝑥 𝐴 to maximise 𝑈 𝐴 , taking 𝑥 𝐵 as given, while B similarly chooses 𝑥 𝐵 to maximise 𝑈 𝐵 , taking 𝑥 𝐴 as given. So the equilibrium can be found by solving each individual’s maximisation problem, giving a solution equation for each of 𝑥 𝐴 and 𝑥 𝐵 , and then (if necessary) simultaneously solving these two equations. 21

23. (Product Rule: 𝑓𝑔 ′ = 𝑓 ′ 𝑔+𝑓𝑔′)
Q 3.2
𝑈 𝐴 = 𝑥 𝐴 𝑥 𝐵 − 𝑥 𝐴 0.6
(Product Rule: 𝑓𝑔 ′ = 𝑓 ′ 𝑔+𝑓𝑔′)
𝜕 𝑈 𝐴 𝜕 𝑥 𝐴 = 𝑥 𝐵 𝑥 𝐴 𝑥 𝐵 − − 𝑥 𝐴 𝑥 𝐴 𝑥 𝐵 − 𝑥 𝐴 −0.4
𝑥 𝐵 𝑥 𝐴 𝑥 𝐵 − − 𝑥 𝐴 = 𝑥 𝐴 𝑥 𝐵 − 𝑥 𝐴 0.4
𝑥 𝐵 50− 𝑥 𝐴 = 𝑥 𝐴 𝑥 𝐵 0.6
0.4 50− 𝑥 𝐴 =0.6 𝑥 𝐴
𝑥 𝐴 =20 22

24. (Product Rule: 𝑓𝑔 ′ = 𝑓 ′ 𝑔+𝑓𝑔′)
Q 3.2
𝑈 𝐵 = 𝑥 𝐵 𝑥 𝐴 − 𝑥 𝐵 0.8
(Product Rule: 𝑓𝑔 ′ = 𝑓 ′ 𝑔+𝑓𝑔′)
𝜕 𝑈 𝐵 𝜕 𝑥 𝐵 = 𝑥 𝐴 𝑥 𝐵 𝑥 𝐴 − − 𝑥 𝐵 𝑥 𝐵 𝑥 𝐴 − 𝑥 𝐵 −0.2
𝑥 𝐴 𝑥 𝐵 𝑥 𝐴 − − 𝑥 𝐵 = 𝑥 𝐵 𝑥 𝐴 − 𝑥 𝐵 −0.2
𝑥 𝐴 100− 𝑥 𝐵 = 𝑥 𝐵 𝑥 𝐴 0.8
− 𝑥 𝐵 =0.8 𝑥 𝐵
𝑥 𝐵 =20 23

25. Q 3.2
(a) In the equilibrium, to one decimal place what are the values of the above utility functions?
𝐴 𝑈 𝐴 = 𝑈 𝐵 =31.9
𝐵 𝑈 𝐴 = 𝑈 𝐵 =9.4
𝐶 𝑈 𝐴 = 𝑈 𝐵 =34.9
𝐷 𝑈 𝐴 = 𝑈 𝐵 =21.7
𝐸 𝑈 𝐴 = 𝑈 𝐵 =33.3 24

26. Q 3.2 (a)
𝑥 𝐴 =20, 𝑥 𝐵 =20
𝑈 𝐴 = 𝑥 𝐴 𝑥 𝐵 − 𝑥 𝐴 = − =7.7
𝑈 𝐵 = 𝑥 𝐵 𝑥 𝐴 − 𝑥 𝐵 = − =33.3
Answer:
𝐸 𝑈 𝐴 = 𝑈 𝐵 =33.3 25

27. Q 3.2
b) Which of the following is preferred by both A and B to the equilibrium? 
𝐴 𝑥 𝐴 = 𝑥 𝐵 =17
𝐵 𝑥 𝐴 = 𝑥 𝐵 =18
𝐶 𝑥 𝐴 = 𝑥 𝐵 =10
𝐷 𝑥 𝐴 = 𝑥 𝐵 =13
𝐸 𝑁𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 26

28. Q 3.2 (b) Calculate the utilities for each scenario:
𝐴 𝑥 𝐴 =10, 𝑥 𝐵 =17, 𝑈 𝐴 =7.40, 𝑈 𝐵 =𝟑𝟖.𝟏𝟒
𝐵 𝑥 𝐴 =13, 𝑥 𝐵 =18, 𝑈 𝐴 =7.66, 𝑈 𝐵 =𝟑𝟔.𝟐𝟓
𝐶 𝑥 𝐴 =17, 𝑥 𝐵 =10, 𝑈 𝐴 =𝟏𝟎.𝟎𝟖, 𝑈 𝐵 =32.91
𝐷 𝑥 𝐴 =18, 𝑥 𝐵 =13, 𝑈 𝐴 =𝟗.𝟏𝟏, 𝑈 𝐵 =𝟑𝟑.𝟑𝟕
𝑶 𝒙 𝑨 =𝟐𝟎, 𝒙 𝑩 =𝟐𝟎, 𝑼 𝑨 =𝟕.𝟕𝟎, 𝑼 𝑩 =𝟑𝟑.𝟑𝟎
Answer:
𝐷 𝑥 𝐴 = 𝑥 𝐵 =13 27

29. Q 3.2
(c) Explain why there is no Pareto-efficient outcome with 𝑥 𝐴 >0 and 𝑥 𝐵 >0. [Hint: what happens to 𝑈 𝐴 and 𝑈 𝐵 , as 𝑥 𝐴 and 𝑥 𝐵 change but with 𝑥 𝐴 𝑥 𝐵 remaining constant?]
If 𝑥 𝐴 and 𝑥 𝐵 are positive, then (say) halving the value of each will keep 𝑥 𝐴 𝑥 𝐵 and 𝑥 𝐵 𝑥 𝐴 unchanged. So in each utility function the first element remains unchanged, while the second increases. 28