Pooling equilibrium in akerlof's lemon market with certification cost
We are looking for a pooling equilibrium where both types choose to certify their cars.
The certification has two outcomes $\{success,\;failure\}$.
The buyer faces three possible types of histories: (1) (type $h_1$) a successful certification, (2) (type $h_2$) an unsuccessful certification, (3) (type $h_3$) no certification. (each of these histories are followed by a standard lemons market probem with a price offer from the seller and a buy/not buy decision from the buyer).  Then, by Bayes rule (wherever possible), the interim beliefs following history $h$ are:

$Pr(good|successful\; test) = \frac{2}{3}$
$Pr(good|unsuccessful\; test) = 0$
Off path Belief: $Pr(good|no\; test) = 0$.

Game following history $h_1$:
Buyer's maximum willingness to pay is $\frac{2}{3}100+\frac{1}{3}20 = \frac{220}{3}&gt;50$. Hence the equilibrium price in this game $p(h_1)=\frac{220}{3}$ and both types sell their cars.
Gross Payoff for either type: $\frac{220}{3}-C$

Game following history $h_2$:
Buyer's maximum willingness to pay = 20 &lt;50. In this case, the equilibrium price is $p(h_2)=20$ and hence only the lemons sell.
Gross payoff to lemons: 20-C
Gross payoff to good cars: 50 - C

Thus the expected payoff to Lemons if they choose to certify: $\frac{1}{4}\frac{220}{3}+\frac{3}{4}20 -C = \frac{400}{12}-C$.
The expected payoff to Good cars following certification: $\frac{220}{3}-C$.

Off-path choices (history $h_3$):
To sustain this equilibrium, neither type should deviate to no testing. Given off path beliefs, the buyer's maximum willingness to pay is 20&lt;50. Hence only lemons are sold in this scenario, yielding payoffs 20 and 50 for lemons and good cars respectively.

Pooling can be sustained when:

$\frac{220}{3}-C&gt;50 \implies C&lt;\frac{70}{3}$, and
$\frac{400}{12} - C &gt; 20 \implies C&lt;\frac{160}{12}$

Hence pooling is an equilibrium as long as $C&lt;\frac{40}{3}$