When only the seller knows the quality

In the previous two chapters the private information was “how much I am willing to pay” or “what my cost is”, which affects only the division of gains. Another kind of private information bears directly on the value of what is being traded: the condition of a used car, a job applicant’s ability, whether a supplier can deliver on time. Here asymmetric information does not merely complicate the division; it can make the whole market disappear.

# Lemons: quality q ~ U[0, 1], known only to the seller, who wants at least q; a buyer
# values q at k*q. At price p only q <= p is offered (mean p/2), so buyers then pay k*p/2.
for k in (1.5, 1.9, 2.5):
    path = [1.0]
    for _ in range(200):
        path.append(min(1.0, k * path[-1] / 2))
    n = next((i for i, x in enumerate(path) if x < 0.01), None)
    end = f"under 1 % left after {n} rounds" if n else "no unravelling"
    print(f"k = {k}: " + " -> ".join(f"{x:.2f}" for x in path[:6]) + f" ...; {end}")
# Certification at cost kappa reveals q; certified cars sell at k*q. Uncertified cars
# still spiral (k < 2) and do not trade, so a seller certifies iff k*q - kappa >= q.
k = 1.5
print(f"k = {k}: full-information gains {(k - 1) / 2:.3f} per car; hidden quality: 0")
print(" kappa | certify if q >= | traded | net gains | spent on certificates")
for kappa in (0.02, 0.05, 0.10, 0.20):
    qs = kappa / (k - 1)
    spent = kappa * (1 - qs)
    net = (k - 1) * (1 - qs ** 2) / 2 - spent
    print(f"  {kappa:.2f} |      {qs:.2f}       |  {1 - qs:4.0%}  |   {net:.3f}   |"
          f"        {spent:.3f}")
# k = 1.5: 1.00 -> 0.75 -> 0.56 -> 0.42 -> 0.32 -> 0.24 ...; under 1 % left after 17 rounds
# k = 1.9: 1.00 -> 0.95 -> 0.90 -> 0.86 -> 0.81 -> 0.77 ...; under 1 % left after 90 rounds
# k = 2.5: 1.00 -> 1.00 -> 1.00 -> 1.00 -> 1.00 -> 1.00 ...; no unravelling
# k = 1.5: full-information gains 0.250 per car; hidden quality: 0
#  kappa | certify if q >= | traded | net gains | spent on certificates
#   0.02 |      0.04       |   96%  |   0.230   |        0.019
#   0.05 |      0.10       |   90%  |   0.202   |        0.045
#   0.10 |      0.20       |   80%  |   0.160   |        0.080
#   0.20 |      0.40       |   60%  |   0.090   |        0.120

With $k = 1.5$ the price is multiplied by 0.75 each round: from 1.00 to 0.75, 0.56, 0.42…; after 17 rounds fewer than 1 % of cars remain on the market. With $k = 1.9$ the shrinking is much slower, but it still takes about 90 rounds to fall below 1 %: as long as $k < 2$, the spiral ends at zero. With $k = 2.5$ the buyers’ premium for quality is high enough: at a price of 1 the average quality on offer is 0.5, and 0.5 times 2.5 is 1.25, still above the top quality, so the market does not shrink at all.

The certification part shows the economics of “costly signals”. With $k = 1.5$, full information would create an average gain of 0.25 per car; with hidden information, 0. Once certification at cost $\kappa$ is available, owners of cars with $q \ge \kappa/(k-1)$ choose to certify. At a certification cost of 0.02, 96 % of cars trade and the net gain is 0.230, of which 0.019 is spent on certification. The dearer certification is, the fewer cars are certified, yet the more is spent on certificates: at a cost of 0.20 only 60 % of cars trade, the net gain is 0.090, and the 0.120 spent on certificates already exceeds the net gain.

Spence’s signals: affordable for the good, unaffordable for the bad

Certification means a third party verifies quality. Another remedy is for the informed side itself to send a signal: an action that may be useless in itself but costs different types different amounts.

# Spence signalling: productivity H = 2 or L = 1; a signal e costs 0.5e (high type) or
# 1.0e (low type) and adds nothing to output. Separation at e*: low type does not mimic,
# L >= H - cL e*; high type signals, H - cH e* >= L. Pooling pays the average.
H, L = 2.0, 1.0

def separating(e, cH, cL):
    return L >= H - cL * e and H - cH * e >= L

grid = [k / 100 for k in range(301)]
ok = [e for e in grid if separating(e, 0.5, 1.0)]
print(f"separating e* in [{min(ok):.2f}, {max(ok):.2f}] = [(H-L)/cL, (H-L)/cH]")
e = min(ok)
print(f"least-cost e* = {e:.2f}: high type nets {H - 0.5 * e:.2f}, low type {L:.2f}")
print(" share high | pooling wage | high type separate vs pool | waste per worker")
for lam in (0.2, 0.5, 0.8):
    pool = lam * H + (1 - lam) * L
    sep = H - 0.5 * e
    print(f"    {lam:.1f}     |     {pool:.2f}     |       {sep:.2f} vs {pool:.2f}"
          "         |"
          f"      {lam * 0.5 * e:.2f}")
n_eq = sum(separating(x, 1.0, 1.0) for x in grid)
print(f"equal costs for both types: separating levels = {n_eq} (knife-edge e = 1 only)")
# separating e* in [1.00, 2.00] = [(H-L)/cL, (H-L)/cH]
# least-cost e* = 1.00: high type nets 1.50, low type 1.00
#  share high | pooling wage | high type separate vs pool | waste per worker
#     0.2     |     1.20     |       1.50 vs 1.20         |      0.10
#     0.5     |     1.50     |       1.50 vs 1.50         |      0.25
#     0.8     |     1.80     |       1.50 vs 1.80         |      0.40
# equal costs for both types: separating levels = 1 (knife-edge e = 1 only)

Separating equilibria exist for every $e^*$ in the interval [1, 2]. At the lower end, $e^* = (H – L)/c_L = 1$, the low type is just indifferent between imitating and not; at the upper end, $e^* = (H – L)/c_H = 2$, the high type is just indifferent between signalling and not. So there is more than one separating equilibrium; under a commonly used equilibrium refinement only the cheapest one ($e^* = 1$) survives (Cho and Kreps 1987).

The cost of signalling is real. At $e^* = 1$ the high type nets 1.50, 0.5 less than the 2 she would earn under full information, and those 0.5 are pure waste: the signal produces nothing. With 80 % high types, each worker spends 0.40 on signalling on average, and the high type would herself be better off under pooling (1.80 against 1.50); yet in the separating equilibrium nobody can stop signalling unilaterally. The last line shows that if the signal costs both types the same, only a knife-edge point remains at which both types are exactly indifferent, and the signal cannot stably separate them.

Chapter summary

When only the seller knows quality, buyers can offer only a price based on average quality, good sellers withdraw and the price falls round after round; in the model with uniform quality and a buyer premium below 100 %, the market eventually collapses entirely. But a direct test in the used pickup-truck market found no lemons effect, suggesting that inspection, reputation and warranties often offset asymmetric information. Third-party certification and self-chosen signals can both separate the good from the bad. A signal works only under the single-crossing condition, with a cost in the range where imitation does not pay but the genuine side still wants to pay; separating equilibria are usually multiple, and pure signals are a social waste. The next chapter turns inside organisations: what happens when a metric is used to reward and punish.