# 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 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?

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 $$\mu = \frac{p}{2}$$.

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 stablish 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 stablished $$p$$ 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 $$p/2$$

• 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. Commented Feb 11 at 1:25
• 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! Commented Feb 12 at 19:44