Adverse Selection and the Lemons Problem
When quality is private information, markets can unravel. Akerlof (1970) showed that if sellers know car quality $q \sim F[q_L, q_H]$ but buyers do not, buyers offer the expected value $\mathbb{E}[q]$. High-quality sellers exit if $\mathbb{E}[q] < q_H$, lowering the expectation further and triggering a cascade until only lemons remain. Formally, equilibrium exists at quality $q^*$ satisfying:
\[\mathbb{E}[q \mid q \leq q^*] = q^*\]Moral Hazard and the Principal-Agent Model
A principal offers a contract to an agent who takes unobservable effort $e \in {e_L, e_H}$. Output $y$ is stochastic: $y = e + \epsilon$, $\epsilon \sim \mathcal{N}(0, \sigma^2)$. With risk-averse agent (CARA utility $-e^{-r w}$) and linear contract $w(y) = \alpha + \beta y$, the optimal incentive power $\beta^*$ balances effort provision against risk-bearing:
\[\beta^* = \frac{1}{1 + r \sigma^2 c''}\]where $c’’$ is the curvature of the effort cost function. Higher agent risk aversion $r$ or output variance $\sigma^2$ reduces optimal incentives.
Spence Signalling Equilibrium
In Spence (1973), workers have productivity $\theta \in {\theta_L, \theta_H}$ and choose education $e \geq 0$ at cost $c(e, \theta) = e/\theta$. A separating equilibrium exists at education levels $(e_L^, e_H^)$ satisfying the single-crossing incentive constraints:
\(\theta_H - e_H^*/\theta_H \geq \theta_H - e_L^*/\theta_H \quad \text{(high type prefers } e_H^*)\) \(\theta_L - e_L^*/\theta_L \geq \theta_L - e_H^*/\theta_L \quad \text{(low type prefers } e_L^*)\)
The least-cost separating equilibrium sets $e_L^* = 0$ and $e_H^* = \theta_L$. Education conveys information purely through its differential cost — it need not raise productivity to be valuable as a signal.