<p>It doesnt seem completely right to me Kenneth's anwser. The main reason is because he's assuming that people will sell up to the quality equal to the mean quality. Nevertheless, this wouldnt hold firstly, if type 2 consumers had cars, secondly given the price <span class="math-container">$p=2$</span> they supply <span class="math-container">$N$</span> cars not the mean value and thirdly he says that quality equals price when the paper says quality equals price divided by two.</p>
<p>I think what you need to take into account is that sellers know the quality of their car. And if they were able to establish a price according to their valuation it would be:</p>
<p><span class="math-container">$$p_i=x_i$$</span></p>
<p>(This is given by their utility function form)</p>
<p>However, they have a fix price established <span class="math-container">$p$</span> for all qualities, then they will offer their cars up to that price (not up to the mean quality), whenever the price is <span class="math-container">$p\leq 2$</span>.</p>
<p>Then, you arise to with a similar mathematical procedure as Kenneth's:</p>
<p><span class="math-container">$$S=pN/2$$</span></p>
<p>(If <span class="math-container">$p\geq 2$</span> supply will be simply equal to <span class="math-container">$N$</span>)</p>
<p>Then it's easy to conclude that mean quality would be <span class="math-container">$p/2$</span></p>

CC BY-SA 4.0
2024-02-11T01:25:20.500Z
Thanks for following up on this. I've reviewed this question with fresh eyes after nearly six years later, and I believe I have greatly improved my answer so that it gets to the heart of the supply-side derivation. I think it fills in the holes that your response above perhaps alludes to, but otherwise in my view leaves somewhat unclear.
Kenneth Rios
CC BY-SA 4.0
2024-02-12T19:44:16.667Z
Now your response is complete. It took me quite long to solve this paper (the supply part), and when I saw your first response it helped me a lot to arrive to my result. So thank you too!
MG123
CC BY-SA 4.0
2024-01-10T03:48:04.737Z
2024-12-31T03:16:09.387Z
2024-12-31T03:16:09.387Z
MG123
<p>Given type-1 traders' marginal utility for numeraire <span class="math-container">$M$</span> and car quality <span class="math-container">$x$</span>, a type-1 trader will supply a car <em>only if</em> the car's measure of quality <span class="math-container">$x$</span> (privately known) is less than or equal to the dollar price <span class="math-container">$p$</span> that she would receive for it. The model assumes that car quality <span class="math-container">$x$</span> is uniformly distributed over <span class="math-container">$[0,2]$</span>. To determine the probability of a car having quality less than or equal to <span class="math-container">$p$</span>, we consider the CDF of <span class="math-container">$x$</span>, <span class="math-container">$F(x)$</span>:</p>
<p><span class="math-container">$$
F(x)=
\begin{cases}
0 &amp;&amp; x \leq 0 \\ 
\frac{x}{2} &amp;&amp; x \in  [0,2] \\ 
1 &amp;&amp; x \geq 2.
\end{cases}
$$</span></p>
<p>The probability that a given car has <span class="math-container">$x \leq p$</span> is <span class="math-container">$P(x \leq p) = F(p) = \frac{p}{2}$</span> (for <span class="math-container">$p \leq 2$</span>). Scaling up by a mass of <span class="math-container">$N$</span> cars possessed across all type-1 traders implies that the aggregate market supply of cars from type-1 traders is <span class="math-container">$S_1(p) = \frac{p}{2}N$</span>. Since only cars with <span class="math-container">$x \in [0, p]$</span> are supplied, it follows once more from <span class="math-container">$F(x)$</span> that the average supplied car quality is <span class="math-container">$\mu = \frac{p}{2}$</span>.</p>

CC BY-SA 4.0
2018-10-10T20:39:44.837Z
2024-02-11T01:48:03.910Z
2024-02-11T01:48:03.910Z
Kenneth Rios
<p>Akerlof's 1970 <a href="https://www.sas.upenn.edu/%7Ehfang/teaching/socialinsurance/readings/fudan_hsbc/Akerlof70(2.1).pdf" rel="nofollow noreferrer">paper</a> models the utility of two trading groups as</p>
<p><span class="math-container">$$
U_1 = M + \sum_{i=1}^n x_i  \\
U_2 = M + \sum_{i=1}^n \frac{3}{2} x_i
$$</span>
where <span class="math-container">$M$</span> is the consumption of good other than cars, <span class="math-container">$x_i$</span> is the quality of the <span class="math-container">$i$</span>th car, and <span class="math-container">$n$</span> is the number of cars.</p>
<p>The quality of the cars held by group one have a uniformly distributed quality <span class="math-container">$0 \leq x \leq 2$</span> and the price of goods other than the cars is unitary.</p>
<p>Income for the two groups is denoted <span class="math-container">$Y_1$</span> and <span class="math-container">$Y_2$</span>.</p>
<p>The paper goes on to give the demand for cars for type one traders:
<span class="math-container">$$
D_1 = Y_1/p \quad \quad  \mu/p &gt; 1  \\
D_1 = 0 \quad \quad  \mu/p &lt; 1
$$</span>
The supply of cars from type one is
<span class="math-container">$$
S_1 = pN/2 \quad \quad p \leq 2
$$</span>
and their average quality is <span class="math-container">$\mu = p/2$</span>.  The paper states that to derive the expressions for supply and average quality, the uniform distribution of car quality is used.  How is this done exactly?</p>

<microeconomics><asymmetric-information>
Market for lemons derivation
Victor Maxwell
# Market for lemons derivation

(Asked by: username_0 on 2018-10-10)

Akerlof's 1970 [paper](https://www.sas.upenn.edu/%7Ehfang/teaching/socialinsurance/readings/fudan_hsbc/Akerlof70(2.1).pdf) models the utility of two trading groups as

$$
U\_1 = M + \sum\_{i=1}^n x\_i \\
U\_2 = M + \sum\_{i=1}^n \frac{3}{2} x\_i
$$
where $M$ is the consumption of good other than cars, $x\_i$ is the quality of the $i$th car, and $n$ is the number of cars.

The quality of the cars held by group one have a uniformly distributed quality $0 \leq x \leq 2$ and the price of goods other than the cars is unitary.

Income for the two groups is denoted $Y\_1$ and $Y\_2$.

The paper goes on to give the demand for cars for type one traders:
$$
D\_1 = Y\_1/p \quad \quad \mu/p > 1 \\
D\_1 = 0 \quad \quad \mu/p < 1
$$
The supply of cars from type one is
$$
S\_1 = pN/2 \quad \quad p \leq 2
$$
and their average quality is $\mu = p/2$. The paper states that to derive the expressions for supply and average quality, the uniform distribution of car quality is used. How is this done exactly?

**Tags:** microeconomics, asymmetric-information


---

## 2 answers

### Answer by username_2 on 2018-10-10. Score: 5 | Accepted answer

Given type-1 traders' marginal utility for numeraire $M$ and car quality $x$, a type-1 trader will supply a car *only if* the car's measure of quality $x$ (privately known) is less than or equal to the dollar price $p$ that she would receive for it. The model assumes that car quality $x$ is uniformly distributed over $[0,2]$. To determine the probability of a car having quality less than or equal to $p$, we consider the CDF of $x$, $F(x)$:

$$
F(x)=
\begin{cases}
0 && x \leq 0 \\
\frac{x}{2} && x \in [0,2] \\
1 && x \geq 2.
\end{cases}
$$

The probability that a given car has $x \leq p$ is $P(x \leq p) = F(p) = \frac{p}{2}$ (for $p \leq 2$). Scaling up by a mass of $N$ cars possessed across all type-1 traders implies that the aggregate market supply of cars from type-1 traders is $S\_1(p) = \frac{p}{2}N$. Since only cars with $x \in [0, p]$ are supplied, it follows once more from $F(x)$ that the average supplied car quality is $\mu = \frac{p}{2}$.


---

### Answer by username_1 on 2024-01-10. Score: 0

It doesnt seem completely right to me Kenneth's anwser. The main reason is because he's assuming that people will sell up to the quality equal to the mean quality. Nevertheless, this wouldnt hold firstly, if type 2 consumers had cars, secondly given the price $p=2$ they supply $N$ cars not the mean value and thirdly he says that quality equals price when the paper says quality equals price divided by two.

I think what you need to take into account is that sellers know the quality of their car. And if they were able to establish a price according to their valuation it would be:

$$p\_i=x\_i$$

(This is given by their utility function form)

However, they have a fix price established $p$ for all qualities, then they will offer their cars up to that price (not up to the mean quality), whenever the price is $p\leq 2$.

Then, you arise to with a similar mathematical procedure as Kenneth's:

$$S=pN/2$$

(If $p\geq 2$ supply will be simply equal to $N$)

Then it's easy to conclude that mean quality would be $p/2$

> **Comments:**
> **username_2**: Thanks for following up on this. I've reviewed this question with fresh eyes after nearly six years later, and I believe I have greatly improved my answer so that it gets to the heart of the supply-side derivation. I think it fills in the holes that your response above perhaps alludes to, but otherwise in my view leaves somewhat unclear.  
> **username_1**: Now your response is complete. It took me quite long to solve this paper (the supply part), and when I saw your first response it helped me a lot to arrive to my result. So thank you too!  

---

