Revise notation for insurance – lectures give an informal treatment
Adverse Selection, Signalling, and Screening
Asymmetric Information
One party is more informed than the other
Leads to inefficiencies
o Adverse Selection: neither party can take action in advance to reduce asymmetry
o Signalling: informed party can take action in advance: choose signals, and
uninformed can act on those signals
o Screening: uninformed party can take action in advance: make offers, and informed
parties choose among those offers
Adverse Selection
Half used cars are bad, half are good
Sellers know the quality of car they own, buyers don’t know the quality of used cars
Sellers value bad cars at 4, good cars at 12 (average = 8), buyers value bad cars at 6, good
cars at 12 (average = 11)
No information
Neither sellers no buyers know quality of any car, all cars sell for 8 ≤ p ≤ 11
Full information
Both parties know quality of each car
o Bad cars sell for 4 ≤ p B ≤ 6
o Good cars sell for 12 ≤ pG ≤ 16
Sellers know but Buyers Don’t
All used cars sell at same price:
o If p ≥12, both cars offered for sale, buy buyers know half bad half good, so won't
pay more than 11
o If p ≤11, then owners of good cars aren’t willing to sell
Market unravels from top: good cars withdrawn
o Only bad cars are offered for sale at 4 ≤ p B ≤ 6
o Equilibrium is inefficient – gains from trade in good cars aren’t realised
Signalling
Two types of worker: θ H >θ L >0 , λ=Pr ( θ=θ H )
Worker of type θ has productivity θ , and no outside opportunities ie unemployed get 0:
¿
therefore w = E [ θ ] =λ θ H + (1−λ ) θ L in absence of signalling – Pareto efficient
Education has no value other than signalling (pure signal) – how realistic is this?
Worker’s utility: w−c ( e , θ ), where c ( e , θ ) is cost of education to worker: eg c=e2 / θ
o c ( 0 ,θ )=0 – its free not to have any education
o c 'e ( e , θ ) >0 – education is costly
o c 'ee' ( e , θ ) >0 – education is costly at an increasing rate
o c 'θ ( e , θ ) <0 – education more costly for θ L than θ H
o c eθ ( e ,θ )< 0 – indifference curves cross just once
(Spence-Mirrelees condition) – means MC lower for
highly productive workers
Two firms, CRTS, one worker
Any eqbm with signalling is not welfare maximising, expected per capita output reduced by
expected cost of signalling
Timing
1. Nature selects θ , privately observed by worker
2. Worker chooses e
3. Firms observe e , and simultaneously make wage offers – Bertrand competition
4. Worker decides which offer (if any) to accept
, Revise notation for insurance – lectures give an informal treatment
If at 1. different types choose same e, say e , then signal is uninformative, so at 2. both firms
offer w ( e ) =E [ θ ]
If at 1. Different types choose different e’s, say e L , e H , then signal is informative, so at stage
2. Both firms offer w ( e L ) =θ L and w ( e H )=θ H
¿ ¿
If θ were publicly observable, then choices would be efficient ( e ( θ H ) =e ( θ L )=0 ) and firms
¿ ¿
earn zero profit ( w ( θ H ) =θH , w ( θ L )=θ L )
If θ not observable and firms made this offer, then all workers would choose e=0 , claim to
be H, and firms would lose money
Pooling Equilibrium
e ¿ ( θ L ) =e ¿ ( θ H )=e ¿ , w ¿ ( e¿ )=E [ θ ]
w ¿ ( e ) fairly arbitrary for e ≠ e ¿
¿
Two extremes for e
Any e ¿ ∈ [ 0 , e ' ] can be sustained
¿
LHS with e =0 is Pareto dominant – has same outcome as no-signalling equilibrium
RHS is both types get education, so get w=E [ θ ] but suffer education costs
If all types choose the same signalling level, then the signal is uninformative and both types
earn the wage under no information: E ( θ )
Example
Suppose employers have beliefs over worker productivity as follows
¿
Pr (θ H )= p if e ≥ e¿
{0 if e< e
Ie if workers do not educate then they are definitely low productivity, if they do then they are
high productivity with probability p ⇒ uneducated workers earn θ L and educated workers
earn E ( θ )
e¿
Equilibrium is a pooling one: at 0 if θ H − < E ( θ ) ie productive types prefer the expected
θH
wage to signalling and earning the high wage (and by construction so do low types)
e¿
If θ H − > E ( θ ) then we pool on e*, even though both types would prefer to not signal at
θH
¿
e
all, ie E ( θ ) > E ( θ )− . Why do they pool?
θH
o Suppose there is separating eqbm: H types earn high wage, L types earn L; if
¿
e
profitable for L to signal as an H it will do so and earn θ H − >θ
θL L
o But then all L types will do this as they strictly prefer it, so we pool on e* and
everyone incurs a signalling cost ie is worse off
Separating Equilibrium
e ¿ ( θ L ) ≠ e ¿ ( θ H ) , w ¿ ( e ¿ ( θ L ) )=θ L , w ¿ ( e ¿ ( θ H ) ) =θ H
Adverse Selection, Signalling, and Screening
Asymmetric Information
One party is more informed than the other
Leads to inefficiencies
o Adverse Selection: neither party can take action in advance to reduce asymmetry
o Signalling: informed party can take action in advance: choose signals, and
uninformed can act on those signals
o Screening: uninformed party can take action in advance: make offers, and informed
parties choose among those offers
Adverse Selection
Half used cars are bad, half are good
Sellers know the quality of car they own, buyers don’t know the quality of used cars
Sellers value bad cars at 4, good cars at 12 (average = 8), buyers value bad cars at 6, good
cars at 12 (average = 11)
No information
Neither sellers no buyers know quality of any car, all cars sell for 8 ≤ p ≤ 11
Full information
Both parties know quality of each car
o Bad cars sell for 4 ≤ p B ≤ 6
o Good cars sell for 12 ≤ pG ≤ 16
Sellers know but Buyers Don’t
All used cars sell at same price:
o If p ≥12, both cars offered for sale, buy buyers know half bad half good, so won't
pay more than 11
o If p ≤11, then owners of good cars aren’t willing to sell
Market unravels from top: good cars withdrawn
o Only bad cars are offered for sale at 4 ≤ p B ≤ 6
o Equilibrium is inefficient – gains from trade in good cars aren’t realised
Signalling
Two types of worker: θ H >θ L >0 , λ=Pr ( θ=θ H )
Worker of type θ has productivity θ , and no outside opportunities ie unemployed get 0:
¿
therefore w = E [ θ ] =λ θ H + (1−λ ) θ L in absence of signalling – Pareto efficient
Education has no value other than signalling (pure signal) – how realistic is this?
Worker’s utility: w−c ( e , θ ), where c ( e , θ ) is cost of education to worker: eg c=e2 / θ
o c ( 0 ,θ )=0 – its free not to have any education
o c 'e ( e , θ ) >0 – education is costly
o c 'ee' ( e , θ ) >0 – education is costly at an increasing rate
o c 'θ ( e , θ ) <0 – education more costly for θ L than θ H
o c eθ ( e ,θ )< 0 – indifference curves cross just once
(Spence-Mirrelees condition) – means MC lower for
highly productive workers
Two firms, CRTS, one worker
Any eqbm with signalling is not welfare maximising, expected per capita output reduced by
expected cost of signalling
Timing
1. Nature selects θ , privately observed by worker
2. Worker chooses e
3. Firms observe e , and simultaneously make wage offers – Bertrand competition
4. Worker decides which offer (if any) to accept
, Revise notation for insurance – lectures give an informal treatment
If at 1. different types choose same e, say e , then signal is uninformative, so at 2. both firms
offer w ( e ) =E [ θ ]
If at 1. Different types choose different e’s, say e L , e H , then signal is informative, so at stage
2. Both firms offer w ( e L ) =θ L and w ( e H )=θ H
¿ ¿
If θ were publicly observable, then choices would be efficient ( e ( θ H ) =e ( θ L )=0 ) and firms
¿ ¿
earn zero profit ( w ( θ H ) =θH , w ( θ L )=θ L )
If θ not observable and firms made this offer, then all workers would choose e=0 , claim to
be H, and firms would lose money
Pooling Equilibrium
e ¿ ( θ L ) =e ¿ ( θ H )=e ¿ , w ¿ ( e¿ )=E [ θ ]
w ¿ ( e ) fairly arbitrary for e ≠ e ¿
¿
Two extremes for e
Any e ¿ ∈ [ 0 , e ' ] can be sustained
¿
LHS with e =0 is Pareto dominant – has same outcome as no-signalling equilibrium
RHS is both types get education, so get w=E [ θ ] but suffer education costs
If all types choose the same signalling level, then the signal is uninformative and both types
earn the wage under no information: E ( θ )
Example
Suppose employers have beliefs over worker productivity as follows
¿
Pr (θ H )= p if e ≥ e¿
{0 if e< e
Ie if workers do not educate then they are definitely low productivity, if they do then they are
high productivity with probability p ⇒ uneducated workers earn θ L and educated workers
earn E ( θ )
e¿
Equilibrium is a pooling one: at 0 if θ H − < E ( θ ) ie productive types prefer the expected
θH
wage to signalling and earning the high wage (and by construction so do low types)
e¿
If θ H − > E ( θ ) then we pool on e*, even though both types would prefer to not signal at
θH
¿
e
all, ie E ( θ ) > E ( θ )− . Why do they pool?
θH
o Suppose there is separating eqbm: H types earn high wage, L types earn L; if
¿
e
profitable for L to signal as an H it will do so and earn θ H − >θ
θL L
o But then all L types will do this as they strictly prefer it, so we pool on e* and
everyone incurs a signalling cost ie is worse off
Separating Equilibrium
e ¿ ( θ L ) ≠ e ¿ ( θ H ) , w ¿ ( e ¿ ( θ L ) )=θ L , w ¿ ( e ¿ ( θ H ) ) =θ H
