Akerlof's Lemons Market

🚗 Car quality $q$ is uniform on $[0, 1]$ ; the seller values a car at $q$ and the buyer at $mq$ , with $m = 1.5$ .

Find the buyer's willingness to pay $WTP(p)$ at price $p$ , and show that the market fully unravels so that $p^* = 0$ is the unique equilibrium.

WTP(p) = m \cdot \frac{p}{2} = 0.75\,p

p_{n+1} = 0.75\,p_n, \qquad p_0 = 1

m = 1.5

— willingness to pay lies below the 45-degree line; the market unravels to

p = 0

Show proof

At price $p$ only sellers with $q \le p$ offer, so the average quality offered is $p/2$ .
Hence $WTP(p) = 1.5 \cdot p/2 = 0.75p$ .
For every $p > 0$ , $WTP(p) = 0.75p < p$ , so $p^* = 0$ is the only solution of $WTP(p) = p$ .
The unraveling iterates are $p_{n+1} = 0.75\,p_n$ : $1 \to 0.75 \to 0.5625 \to 0.421875 \to 0.316406 \to \cdots \to 0$ . The traded fraction is $p^* = 0$ , so nothing trades.
Full-information total surplus is $E[(m-1)q] = 0.5/2 = 0.25$ . With no trade in equilibrium the welfare loss is also $0.25$ — the entire gains from trade vanish.
$\text{loss} = \frac{m-1}{2} - \frac{(m-1)(p^*)^2}{2} = 0.25 - 0 = 0.25$

In the same model with $m = 2$ , verify the full-trade equilibrium $p^* = 1$

and compute the traded fraction and the welfare loss.

WTP(p) = 2 \cdot \frac{p}{2} = p

m = 2

— willingness to pay coincides with the 45-degree line; full trade holds at

p^* = 1

Show proof

With $m = 2$ , $WTP(p) = p$ , so the unraveling process stalls: $p_{n+1} = p_n$ , i.e. $1 \to 1 \to 1 \to \cdots$ . The equilibrium price is $p^* = 1$ .
Every seller ( $q \le 1$ ) offers, so the traded fraction is $1$ . Traded surplus is $(m-1)(p^*)^2/2 = 0.5$ , equal to the full-information surplus $(m-1)/2 = 0.5$ , so the welfare loss is $0$ — adverse selection disappears.

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