# Market for lemons derivation

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?

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 $$S_1(p) = \frac{p}{2}N$$.