1. Tutorial 4: Asymmetric Information
Matthew Robson

2. 4.1 Asymmetric Information
A particular good is either high quality (type A) or low quality (type B). Sellers know which type they are selling. Buyers cannot distinguish type at the time of purchase. So there can be only a single price p, and in effect a single market. Suppliers are price-takers, with market supply functions for Types A and B respectively:
𝑠 𝐴 =𝑚𝑎𝑥 5𝑝−30 , 𝑠 𝐵 =20
Relative to supply, there is a large number of potential buyers, each of whom would be willing to pay up to £16 (per unit) for known Type A, and £4 for known Type B. Buyers are risk-neutral. 1

3. 4.1
Let 𝜑 denote the probability, as estimated/perceived by buyers, that any given unit of the good is Type A rather than Type B. Define an equilibrium in this market as a situation in which supply equals demand, and also 𝜑 is correct, i.e.:
𝜑= 𝑠 𝐴 𝑠 𝐴 + 𝑠 𝐵
Market for Lemons, George Akerlof
-Second Hand Car Sales: Good Cars and Bad Cars (lemons)
-Asymmetry of Information between buyers and sellers.
-Up to a price of £6 the only cars in market are lemons 2

4. 4.1 (a)
Explain why, for a unit of unknown type, buyers are willing to pay up to 𝑣=4+12𝜑 .
We know buyers are risk neutral.
Willing to pay £16 for Type A and £4 for Type B.
The probability that the good is Type A is: 𝜑= 𝑠 𝐴 𝑠 𝐴 + 𝑠 𝐵
Therefore, buyers willingness to pay for an unknown type is:
𝑣=φ16+ 1−φ 4
𝑣=4+12𝜑 3

5. 4.1 (b)
By calculating in each case the correct 𝜑, and the corresponding v, explain why the market would not be in equilibrium at any of the following prices:
𝑝= 3 𝑝= 𝑝= 𝑝= 12
If 𝑣<𝑝 then there is zero demand, and so excess supply. If 𝑣>𝑝 then there is excess demand, given the assumption of a large number of buyers relative to supply. 4

6. 4.1 (b) If 𝑝=3 𝑠 𝐴 =𝑚𝑎𝑥 5𝑝−30 , 0 𝑠 𝐵 =20 𝑠 𝐴 =0 𝑠 𝐵 =20
𝑠 𝐴 =𝑚𝑎𝑥 5𝑝−30 , 𝑠 𝐵 =20
𝑠 𝐴 = 𝑠 𝐵 =20
𝜑= 𝑠 𝐴 𝑠 𝐴 + 𝑠 𝐵
𝜑=0
𝑣=4+12𝜑
𝑣=4
𝑝=3, 𝜑=0, 𝑣=4, 𝑠𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑑𝑒𝑚𝑎𝑛𝑑 5

7. So what are the 𝜑’s and 𝑣’s for each of the following prices:
𝑝= 3 𝑝= 𝑝= 𝑝= 12
𝑝=3, 𝜑=0, 𝑣=4, 𝑠𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑑𝑒𝑚𝑎𝑛𝑑
𝑝=6, 𝜑=0, 𝑣=4, 𝑠𝑜 𝑧𝑒𝑟𝑜 𝑑𝑒𝑚𝑎𝑛𝑑
𝑝=9, 𝜑=3/7, 𝑣≈9.14, 𝑠𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑑𝑒𝑚𝑎𝑛𝑑
𝑝=12, 𝜑=3/5, 𝑣=11.2, 𝑠𝑜 𝑧𝑒𝑟𝑜 𝑑𝑒𝑚𝑎𝑛𝑑 6

8. 4.1 (c)
Similarly, explain why the market would be in equilibrium at any of the following prices:
𝑝= 4 𝑝= 𝑝= 10
 𝑝=4, 𝜑=0, 𝑣=4, 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚
𝑝=8, 𝜑=1/3, 𝑣=8, 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚
𝑝=10, 𝜑=1/2, 𝑣=10, 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 7

9. Supply:
-There is a single market as buyers cannot distinguish between Type A and Type B
-So for any given p the ‘correct’ 𝜑 can be found
Demand:
-A function of p and 𝜑, so different curves for different 𝜑
-For the ‘correct’ 𝜑 from the supply curve, the demand curve can be traced out 8

10. Equilibria:
-Here the intersection of the demand and supply shows the three equilibria
-Each has a different value of 𝝋
p = 10
p = 8
p = 4 9

11. 4.1 (c)
Do your calculations in (b) suggest anything about the stability of any of these equilibria? What is distinctive about the equilibrium at 𝑝= 4
Answers from (b) suggest that 𝒑=𝟖 is unstable, in that for p slightly higher than 8 there is excess demand (presumably causing p to rise), while for p slightly lower than 8 there is excess supply (presumably causing p to fall).
The equilibrium at 𝑝=4 is a separating equilibrium, in that only Type B sell at that price.
So if we don’t just want lemons in the market the higher price of 𝑝=10 may have to be the price to pay. 10

12. 4.2 The Labour Market
In a particular field of employment there are two types of applicants: high quality (Type A) and low quality (Type B). An applicant (seller) knows what type they are. Employers (buyers) cannot distinguish type at the time of offering employment contracts, although they know that a fixed proportion 𝝋 of applicants are of Type A.
Relative to the number of applicants, there is a large number of potential employers. Each would be willing to pay a wage of up to 70 to a known Type A, and 30 to a known Type B, and each is risk neutral in their valuation of an unknown type. 11

13. 4.2 (a)
In each case identifying the crucial assumption(s), explain why on the assumptions so far:
the equilibrium wage cannot be more than 30+40𝜑
(ii) the equilibrium wage cannot be less than 𝜑
(iii) hence, the equilibrium wage must be 𝜑
Why is the equilibrium here unique, in contrast to exercise 4.1? 12

14. 4.2 (a)
Employers are assumed to be risk-neutral, which implies that they will offer no more than 𝜑70+ 1−𝜑 30=30+40𝜑 to an applicant of unknown type.
Employers are risk-neutral, and also numerous relative to applicants. So competition between them pushes the wage up to the maximum they will offer, i.e 𝜑
follows from (i) and (ii).
The equilibrium is unique because 𝜑 is assumed fixed. In 4.1, by contrast, 𝜑 varies with price. 13

15. 4.2 (b)
(b) Now assume that there is a qualification that only Type A applicants are capable of obtaining, at a cost to them of 𝑐 𝐴 . Employers can offer a wage w either unconditionally (w,0), or conditional (w,1) on the applicant having obtained that qualification.
In a pooling equilibrium there is a uniform offer made by all employers, and accepted by both types of applicant. Explain why, on the assumptions so far, in a pooling equilibrium:
the offer must be unconditional
the wage offered must be 30+40𝜑 14

16. 4.2 (b)
(i) By assumption Type B are unable to obtain the qualification. So, if the equilibrium offer was conditional on obtaining the qualification then only Type A could accept, in which case it would not be a pooling equilibrium.
(ii) Given that both types accept the offer, the explanation is exactly as in (a). 15

17. 4.2 (b)
In a separating equilibrium employers make two distinct offers: a conditional offer ( 𝑤 𝐴 ,1) accepted only by Type A applicants, and an unconditional offer ( 𝑤 𝐵 ,0) accepted only by Type B. Explain why,
on the assumptions so far, in a separating equilibrium:
(iii) 𝑤 𝐵 =30
(iv) 𝑤 𝐴 =70
Hence, explain why there can be no separating equilibrium if 𝑐 𝐴 >40 16

18. 4.2 (b)
(iii) Given that only Type B accept the ( 𝑤 𝐵 ,0) offer, competition between employers pushes 𝑤 𝐵 up to the maximum they will pay for known Type B. 𝑤 𝐵 =30.
(iv) Given that only Type A accept the ( 𝑤 𝐴 ,1) offer, competition between employers pushes 𝑤 𝐴 up to the maximum they will pay for known Type A. 𝑤 𝐴 =70.
If 𝑐 𝐴 >40 then Type A applicants prefer to accept (30,0) rather than (70,1). 17

19. 4.2 (c) Assume from now on that 𝑐 𝐴 =18.
Assume that 𝜑=0.5. If all other employers were offering only (50,0), find a profitable alternative offer that would attract only Type A applicants. Hence explain why a pooling equilibrium does not exist here. Explain why a pooling equilibrium would exist if instead 𝜑=0.6. 18

20. 4.2 (c) From (b), if 𝜑=0.5 then a pooling equilibrium would be when:
𝑤=30+40𝜑=50
So: (50,0). But an alternative offer of (69,1) would be accepted by Type A applicants, in preference to (50,0), and at a wage less than the maximum the employer is willing to offer a known Type A. So (50,0) is not an equilibrium.
If instead 𝜑=0.6 then a pooling equilibrium would when:
𝑤=30+40𝜑=54
at (54,0). For an alternative offer (𝑤,1) to be accepted by Type A would now require 𝑤>72 (as 𝑐 𝐴 =18), which is more than an employer is willing to offer a known Type A. So (54,0) is an equilibrium. 19

21. 4.2 (c)
Assume that 𝜑=0.6. If all other employers were offering only (30,0) and (70,1), find a profitable alternative offer that would attract both types. Hence explain why a separating equilibrium does not exist here. Explain why a separating equilibrium would exist if instead 𝜑=0.5.
So, in equilibrium, what proportion of applicants obtain the qualification if 𝜑=0.5.? And what proportion if 𝜑=0.6.? 20

22. 4.2 (c)
From (b), a separating equilibrium would be at (30,0) and (70,1). If 𝜑=0.6 then an employer is willing to offer up to 54 to an unknown type. An alternative offer of (53,0) would be accepted by both types. So (30,0) and (70,1) is not an equilibrium.
If instead 𝜑=0.5 then an employer is willing to offer no more than 50 to a unknown type. For an alternative offer (w,0) to attract both types would require 𝑤> 52 . So (30,0) and (70,1) is an equilibrium.
So, if 𝜑=0.5 then in equilibrium 50% of applicants (i.e. all Type A) obtain the qualification. But if 𝜑=0.6 then in equilibrium 0% of applicants obtain the qualification. 21

23. 4.2 (d)
 (d) Now assume that Type B applicants are also capable of obtaining the same qualification, but at a higher cost of 𝑐 𝐵 =34 . Explain why, in a separating equilibrium:
(i) 𝑤 𝐵 =30
(ii) 𝑤 𝐴 =64
Hence explain why there is no equilibrium at all if 𝜑=0.5.
Given the assumed values of 𝑐 𝐴 and 𝑐 𝐵 , what is the largest 𝜑 at which there would be a separating equilibrium? And what is the smallest value of 𝜑 at which there would be a pooling equilibrium? 22

24. 4.2 (d)
(i) Same as (b)(iii). Given that only Type B accepts the ( 𝑤 𝐵 ,0) offer, competition between employers pushes 𝑤 𝐵 up to the maximum they will pay for known Type B. 𝑤 𝐵 =30.
(ii) As in (b)(iv), competition between risk neutral employers pushes 𝑤 𝐴 up to the maximum they will pay for known Type A, so long as 𝑤 𝐴 ,1 is accepted only by Type A. But if 𝑤 𝐴 >64 then Type B applicants will also prefer to accept ( 𝑤 𝐴 ,1) rather than (30,0). 23

25. 4.2 (d)
A separating equilibrium would be at (30,0) and (64,1). If 𝜑=0.5 then an employer is willing to offer up to 50 to an unknown type. An alternative offer of (47,0) would attract both types. So (30,0) and (64,1) is not an equilibrium. We already know from (c) that if 𝜑=0.5 then there is no pooling equilibrium. So there is no equilibrium at all, if 𝜑=0.5.
(30,0) and (64,1) is an equilibrium iff:
30+40𝜑≤46 =64−18 𝑖.𝑒. 𝜑≤0.4
(30+40𝜑, 0) is an equilibrium iff:
30+40𝜑≥52 =70−18 𝑖.𝑒. 𝜑≥0.55 24

26. Signalling
One important assumption that was changed was related to signalling.
In (a) we assume that the individuals type in unknown.
But in (b) we assume that due to their qualification they can signal their type.
A good signal must be costly-to-fake and ensure full-disclosure occurs.
Toad’s Croak
-Male toads croak at night to attract mates. The female toads prefer the larger toads, and so need to have some signal in the dark as to which toads are the largest.
-So the toads have a signal, their croak. The larger the toads are the deeper their croak will be, while smaller toads can only emit higher pitched croaks.
-The signal is costly-to-fake, the smaller toads cannot croak deeply, and ensures full-disclosure occurs, as individual toads must disclose even unfavourable information about themselves (a high pitched croak) as silence would mean they had more to hide (they were even smaller). 25