Market for lemons derivation

Question

Akerlof's 1970 paper 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 quality is $\mu = p/2$. The paper states that to drive the expressions for supply and quality, the uniform distribution of car quality is used. How is this done exactly?

Answer 1

A type-1 seller will trade her car only if the car's measure of quality $x$ (privately known) is less than or equal to the average car quality in the market, $\mu$. $x \in [0,2]$ since no buyer values a car more than $2$ and a car cannot be sold for less than $0$. To find the probability of a car having quality less than or equal to $\mu$, we consider the CDF of $x$, $F(x)$, assuming $x$ is uniformly distributed over the support $[0,2]$:

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

So the probability that a car has $x \leq \mu$ is $P(x \leq \mu) = F(\mu) = \frac{\mu}{2}$. A seller facing a price $p$ for her car becomes indifferent towards selling at $\mu = p \implies F = \frac{p}{2}$. Scaling up by a total of $N$ cars possessed by type-1 sellers implies that the market supply from type-1 sellers is $S_1(p) = \frac{p}{2}N$.