Class 36 (answers) Asymmetric information

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Microeconomics II 
Undergraduate degree in Economics 
 
Class nr. 36 
 
Subject: 5. Market failures 
 5.3. Asymmetric information (practice classes) 
 
 
Exercise 37.1. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 441-442 
 
There are two types of electric pencil-sharpener producers. “High-quality” 
manufacturers produce very good electric pencil-sharpeners that consumers value at 
$14. “Low-quality” manufacturers produce less good electric pencil-sharpeners that 
consumers value at $8. At the time of purchase, customers cannot distinguish between a 
high-quality and a low-quality product; nor can they identify the manufacturer. 
However, they can determine the quality of the product after purchase. The consumers 
are risk neutral or indifferent. If they have probability “q” of buying a high-quality 
product and “1 – q” probability of getting a low-quality product, the consumers’ 
valuation of the product is given by the expected value. Each type of manufacturer can 
manufacture the product at a constant unit cost of $11.50. All manufacturers behave 
competitively. 
 
a) Suppose that the sale of low-quality electric pencil-sharpeners is illegal, so 
that the only items allowed to appear on the market are of high quality. What 
will be the equilibrium price? 
b) Suppose that there were no high-quality manufacturers. How many low-
quality pencil-sharpeners would be sold in equilibrium? 
c) Could there be an equilibrium in which equal (positive) quantities of the two 
types of pencil sharpeners appear in the market? 
d) Now we change our assumptions about the technology. Suppose that each 
producer can choose to manufacture either a high-quality or a low-quality 
pencil-sharpener, with a unit cost of $11.50 for the former and $11 for the 
latter. What would we expected to happen in equilibrium? 
e) Assuming the conditions described in question d), what good would it do if 
the government banned the production of low-quality electric pencil-
sharpeners? 
 
 
 
 
 
 
Microeconomics II 
Undergraduate degree in Economics 
 
 
Answers: 
a) 
Because both manufactures behave as if they were in a perfect competition market, by 
assumption, they will fix the price at the marginal cost level. The equilibrium price will 
be $11.50. The manufactures that chose a higher price would not be able to sell all their 
production. The difference between the $14 from consumer’s valuation and the $11.50 
from the production cost contributes for the consumer surplus. 
b) 
For the consumer the low-quality electric pencil sharpeners have a value of $8, however 
they cost $11.50 to produce, therefore the manufacturers will not produce any low-
quality of pencil-sharpeners under this conditions. 
c) 
If there is competition, the market price will be equal to the marginal cost, which is 
$11.50. In order to sell pencil-sharpeners it’s necessary that the consumer’s valuation of 
the average (p) pencil-sharpener is, at least, equal to that cost value. 
( )
( )
11.5 14 1 8 11.5
11.5 8 0.58 3
6
p q q
q q
≥ ⇔ × + − × ≥ ⇔
−⇔ ≥ ⇔ ≥ 
 
 
 
 
 
 
 
 
 
At least 58.(3)% of the pencil-sharpeners produced and sold should be of “high-quality” 
in order for a market to exist, therefore is not possible to have an equilibrium in which 
we would have equal quantities of both types of pencil sharpeners in the market. 
 
d) 
The incentive of each manufacturer is to produce “low-quality” pencil sharpeners, 
because that’s what allows him to minimize his costs, without any significant impact on 
the revenue (therefore maximizing the profits), since customers cannot distinguish 
between both products, and they value all the pencil-sharpeners equally, by their 
expected value, at the time of purchasing. 
p 
q 
11.5 
8 
0 0.58(3) 1 
p 
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Undergraduate degree in Economics 
 
All manufacturers face the same incentives; therefore everyone will want to produce 
"low-quality" pencil sharpeners. This will have an impact on consumer perceptions 
about the quality of electric pencil sharpeners that are being produced and placed in the 
market, lowering the value they assign to the probability that the pencil sharpener that 
they buy is of "high quality" ("q") to zero in the limit. With q = 0, the expected 
valuation of the pencil sharpener to be produced is: 
( ) ( ) 88011408114 =×−+×=×−+× qq 
which is lower than the production cost of either type of electric pencil sharpener. 
Consumers will not be willing to pay for the average quality of the electric pencil 
sharpeners that they find on the market, so the market disappears. There is no 
transaction of electric pencil sharpeners. 
e) 
The prohibition on the sale of electric pencil sharpeners would put us back in the 
conditions of question a). The gain from the governmental measure can be measured by 
the consumer surplus generated in this case, a value that is zero in d), since there would 
be no market. 
 
Exercise 37.4. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 444-445 
 
Old MacDonald produces hay. He has a single employee, Jack. If Jack works for x 
hours he can produce x bales of hay. Each bale of hay sells for $1. The cost of Jack of 
working x hours is ( )
10
2xxc = . 
a) What is the efficient number of bales of hay for Jack to cut? 
b) If Jack didn’t have another choice to get a work (thus the maximum amount 
Jack could earn elsewhere is zero), how much would MacDonald have to 
pay him to get him to work the efficient amount of hours? 
c) What is MacDonald’s net profit? 
d) Suppose that Jack has an alternative where he would receive $1 for passing 
out leaflets, an activity that involves no effort whatsoever. How much would 
he have to receive from MacDonald for producing the efficient number of 
bales of hay? 
e) Suppose that the opportunity for passing out leaflets is no longer available, 
but that MacDonald decides to rent his hayfield out to Jack for a flat fee. 
How much would MacDonald rent it for? 
 
Answers: 
a) 
Production function: ( ) xxf = 
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Undergraduate degree in Economics 
 
Profit: ( ) ( )( )xfsxf −=Π 
Jack’s participation constraint: ( )( ) ( ) 0≥− xcxfs 
Owner’s optimization problem: 
( ) ( )( )
( )( ) ( ) ⇔⎪⎩
⎪⎨
⎧
≥−
−=Π
0..
max
xcxfsts
xfsxf
x 
(using the participation constraint in equality) 
( ) ( )xcxf
x
−=Π⇔ max 
( ) ( )[ ] ( ) ( ) ( ) ( ) ⇔∂
∂=∂
∂⇔=∂
∂−∂
∂⇔=−∂
∂
x
xc
x
xf
x
xc
x
xfxcxf
x
00 
(marginal productivity = marginal cost) 
5
10
21 =⇔=⇔ xx 
5 is the efficient number of bales of hay that Jack should cut, which would imply 5 
hours of work. 
 
b) 
Jack’s participation constraint: ( )( ) ( ) 0** ≥− xcxfs 
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2* * 5* * 0 * 0 * 5 5 2.5
10 10 10
x x
s f x c x s x s x s s− ≥ ⇔ − ≥ ⇔ ≥ ⇔ ≥ ⇔ ≥
 
The value of $2.5 is enough to make Jack indifferent between working 5 hours and not 
working at all. If he works 5 hours, he earns $2.5 and incurs a cost equivalent to $2.5 for 
the effort (given by Jack’s cost function). If he doesn't work he will not earn any money 
or have any effort cost. If old MacDonald will only pay him if he cuts 5 bales of hay, 
any other effort level, would imply a negative net gain to Jack. Nevertheless, there are 
other efficient ways to determine the payment function ( )( )xfs . 
 
c) 
MacDonald’s net profit: 
( ) ( ) ( )( )xfsxfx −=Π 
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )* * * 5 5 5 5 5 2.5 2.5x f x s f x f s f f sΠ = − = − = − = − = 
 
d) 
Jack’s new participation constraint: ( )( ) ( ) 1** ≥− xcxfs 
Microeconomics II 
Undergraduate degree in Economics( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2* * 5* * 1 * 1 * 1 5 1 5 3.5
10 10 10
x x
s f x c x s x s x s s− ≥ ⇔ − ≥ ⇔ ≥ + ⇔ ≥ + ⇔ ≥
 
e) 
Now, Jack keeps the quantity produced to himself: 
( )( ) ( ) Rxfxfs −= 
R ≡ rent 
Jack maximizes the following objective function: 
( )( ) ( ) ( ) ( )xcRxfxcxfs −−=− 
This assures that ( ) ( )
x
xc
x
xf
∂
∂=∂
∂ 
How can we determine the rent (R)? 
Old MacDonald will want the maximum R subject to the participation constraint: 
( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
2 2
max max max
. . * * 0 . . * * 0 . . * *
* 5* * * 5 2.5
10 10
R R R
R R R
s t s f x c x s t f x R c x s t f x c x R
x
R f x c x R x R R
⎧ ⎧ ⎧⎪ ⎪ ⎪⇔ ⇔ ⇒⎨ ⎨ ⎨− ≥ − − ≥ − ≥⎪ ⎪ ⎪⎩ ⎩⎩
⇒ = − ⇔ = − ⇔ = − ⇔ =
 
Old MacDonald would charge a rent of $ 2.5 to Jack, in order to rent him the hay field.