# 14.282 Midterm 1 - FINAL ANSWERS (Iteration 2) **Student:** 思 중군 함대🐅 **Course:** 14.282 Organizational Economics, Fall 2025 --- [[iter(eval)]] ## Problem 1: Career Concerns and Incentive Contracts ### Part (a): Pure Career Concerns (p observable, not contractible) #### I. Equilibrium Specification **Wages:** $\{w_1, w_2\}$ **Efforts:** $\{a_1, a_2\}$ where $a_t = (a_{1t}, a_{2t})$ **Beliefs:** $E[\eta | p_1]$ #### II. Solution by Backward Induction **Period 2:** Since no future exists, the agent has no career concern motive. The agent chooses effort to maximize current payoff, but with no explicit incentive contract and no future benefit: $ \max_{a_2} \quad w_2 - c(a_2) = w_2 - \frac{1}{2}(a_{12}^2 + a_{22}^2) $ **Result:** $a_2^* = (0, 0)$ **Wage determination:** With competitive principals and rational expectations: $ w_2^* = E[y_2 | p_1] = E[f_1 a_{12} + f_2 a_{22} + \eta + \varepsilon_2 | p_1] = E[\eta | p_1] = \varphi p_1 $ where in equilibrium, $\hat{a}_1 = a_1^*$, so $E[\eta | p_1] = \varphi(p_1 - g \cdot a_1^*) = \varphi p_1$ after substituting equilibrium belief. Actually, more precisely: $E[\eta | p_1] = \varphi(p_1 - g_1 a_{11}^* - g_2 a_{21}^*)$. In equilibrium with correct beliefs, market knows effort, so: $ w_2^* = \varphi p_1 \quad \text{(simplified, as market correctly anticipates } a_1^* \text{)} $ **Period 1:** The agent maximizes: $ \max_{a_1} \quad w_1 - c(a_1) + \delta[w_2^* - c(a_2^*)] $ Substituting $w_2^* = E[\eta | p_1]$ and $a_2^* = 0$: $ \max_{a_1} \quad w_1 - \frac{1}{2}(a_{11}^2 + a_{21}^2) + \delta \varphi(p_1 - g \cdot \hat{a}_1) $ where $p_1 = g_1 a_{11} + g_2 a_{21} + \eta + \phi_1$. In equilibrium, $\hat{a}_1 = a_1$, so: $ \max_{a_1} \quad w_1 - \frac{1}{2}(a_{11}^2 + a_{21}^2) + \delta \varphi(g_1 a_{11} + g_2 a_{21} + \eta + \phi_1 - g_1 a_{11} - g_2 a_{21}) $ $ = w_1 - \frac{1}{2}(a_{11}^2 + a_{21}^2) + \delta \varphi(\eta + \phi_1) $ Wait, this doesn't look right. Let me reconsider. Actually, the market updates belief about η based on observed p₁. The agent's marginal benefit from effort is the effect on market's belief. Let me redo this properly. **Correct approach:** Market observes $p_1 = g \cdot a_1 + \eta + \phi_1$. Market's belief: $E[\eta | p_1] = \varphi(p_1 - g \cdot \hat{a}_1)$ where $\hat{a}_1$ is the market's conjecture of agent's effort. Agent's payoff from period 1: $ w_1 - c(a_1) + \delta E[\eta | p_1] $ Agent chooses $a_1$ taking $\hat{a}_1$ as given (since agent is atomistic). Taking derivative: $ \frac{\partial}{\partial a_{i1}}\left[- \frac{1}{2}(a_{11}^2 + a_{21}^2) + \delta \varphi(g_1 a_{11} + g_2 a_{21} + \eta + \phi_1 - g \cdot \hat{a}_1)\right] = 0 $ $ -a_{i1} + \delta \varphi g_i = 0 $ **First-order condition:** $ a_{i1}^* = \delta \varphi g_i \quad \text{for } i = 1, 2 $ Or in vector form: $ a_1^* = \delta \varphi g $ where $g = (g_1, g_2)$. **Wage:** Competition among principals drives profits to zero: $ w_1^* = \bar{u} $ The agent's participation constraint binds in period 1. #### III. Why is $(a_{11}, a_{21}) \neq (0, 0)$? **Intuition:** Even without an explicit performance contract, the agent exerts positive effort in period 1 due to **career concerns**. The mechanism: 1. Higher effort → higher observed performance $p_1$ 2. Market interprets high $p_1$ as signal of high ability $\eta$ 3. Higher inferred $\eta$ → higher period 2 wage $w_2$ The implicit incentive is $\delta \varphi$, which acts like an endogenous bonus rate. This is the "career concern" effect identified by Holmström (1982/1999). **Mathematical insight:** $ \frac{\partial w_2}{\partial a_{i1}} = \varphi g_i \delta $ This marginal return to effort equals marginal cost at equilibrium: $ a_{i1}^* = \delta \varphi g_i $ #### IV. How does $\{a_t\}_{t=1}^2$ depend on $\cos(\theta)$? **Define alignment:** $\cos(\theta) = \frac{f \cdot g}{\|f\| \|g\|}$ **Key insight:** In part (a), where $p$ is not contractible, $\cos(\theta)$ does NOT affect effort levels directly. **Why?** - Agent always exerts effort along direction $g$ (the observable metric) - $\cos(\theta)$ measures alignment between firm value ($f$) and performance metric ($g$) - Effort level: $\|a_1^*\| = \delta \varphi \|g\|$ (independent of $\theta$) **What does $\cos(\theta)$ affect?** The **social value** of effort: $ \text{Value created} = f \cdot a_1^* = f \cdot (\delta \varphi g) = \delta \varphi \|f\| \|g\| \cos(\theta) $ **Interpretation:** - High $\cos(\theta)$ → effort well-aligned with firm value → efficient - Low $\cos(\theta)$ → effort creates "window dressing" → inefficient - $\cos(\theta) = 0$ → effort orthogonal to value → complete waste **Summary:** $\cos(\theta)$ affects EFFICIENCY, not effort level, in pure career concerns. #### V. How does $\{a_t\}_{t=1}^2$ depend on $\varphi = \frac{h_\phi}{h + h_\phi}$? **Definition:** $\varphi$ is the **precision weight** in Bayesian updating, measuring how accurately performance $p_1$ reflects ability $\eta$. **Bayesian interpretation:** $ \varphi = \frac{h_\phi}{h + h_\phi} = \frac{\text{signal precision}}{\text{signal precision} + \text{prior precision}} $ **Effect on effort:** **Period 1:** $ \frac{\partial a_1^*}{\partial \varphi} = \delta g > 0 $ Higher signal precision → stronger career concerns → more effort. **Period 2:** $ a_2^* = 0 \quad \text{(independent of } \varphi \text{)} $ **Intuition:** | $\varphi$ Level | Market Interpretation | Period 1 Effort | Logic | |-----------------|----------------------|-----------------|-------| | $\varphi \to 1$ | "Performance = Ability" | $a_1^* \to \delta g$ | Effort strongly affects reputation | | $\varphi \to 0$ | "Performance = Noise" | $a_1^* \to 0$ | "It's all luck anyway" | | $\varphi = 0$ | No learning | $a_1^* = 0$ | Career concerns vanish completely | **Extreme cases:** - $\varphi = 0$: No information in $p_1$ about $\eta$ → no career concerns → $a_1^* = 0$ - $\varphi = 1$: Perfect signal → maximum career concerns → $a_1^* = \delta g$ --- ### Part (b): Contractible Performance ($w_t = s_t + b_t p_t$) #### I. Equilibrium Contracts **Period 2: Pure Moral Hazard** Agent's problem: $ \max_{a_2} \quad b_2 p_2 - c(a_2) = b_2(g \cdot a_2 + \eta + \phi_2) - \frac{1}{2}\|a_2\|^2 $ **FOC:** $ b_2 g = a_2 $ Thus: $ a_2^* = b_2 g $ Principal wants to maximize firm value. If principal could choose any $a_2$, optimal would be: $ a_2^{FB} = f $ Since agent responds along $g$, principal chooses $b_2$ to induce effort in that direction: $ b_2^* g = \text{projection of } f \text{ onto } g = \frac{f \cdot g}{\|g\|^2} g $ Therefore: $ b_2^* = \frac{f \cdot g}{\|g\|^2} = \frac{\|f\|}{\|g\|} \cos(\theta) $ **Resulting effort:** $ a_2^* = \frac{f \cdot g}{\|g\|^2} g $ **Salary:** Competition implies zero profit: $ s_2^* = E[y_2 | p_1] - b_2^* E[p_2 | p_1] = E[\eta | p_1] - b_2^* (g \cdot a_2^* + E[\eta | p_1]) $ Simplifying (since $E[\phi_2] = 0$): $ s_2^* = -b_2^* g \cdot a_2^* $ **Period 1: Explicit + Implicit Incentives** Agent's problem: $ \max_{a_1} \quad b_1 p_1 - c(a_1) + \delta[E[w_2 | p_1] - c(a_2^*)] $ With $w_2 = s_2 + b_2 p_2$ and $E[w_2 | p_1] = s_2^* + b_2^* E[p_2 | p_1]$, but the key is: $ E[w_2 | p_1] = E[\eta | p_1] + \text{contract terms} $ The career concern component is: $ \frac{\partial E[w_2 | p_1]}{\partial a_1} = \frac{\partial E[\eta | p_1]}{\partial p_1} \cdot \frac{\partial p_1}{\partial a_1} $ With public contracts, market knows $b_1$, so: $ E[\eta | p_1, b_1] = \varphi(p_1 - g \cdot a_1^{exp}(b_1)) $ where $a_1^{exp}(b_1) = b_1 g$ (expected effort given $b_1$). The implicit incentive from updating is: $ \frac{\partial E[\eta | p_1, b_1]}{\partial a_1} = \varphi g $ **Total marginal return to effort:** $ (b_1 + \delta \varphi) g $ **FOC:** $ a_1^* = (b_1 + \delta \varphi) g $ **Optimal contract:** To induce same effort as period 2 (if interior solution): $ b_1 + \delta \varphi = b_2^* $ Thus: $ \boxed{b_1^* = b_2^* - \delta \varphi} $ **Constraint:** $b_1^* \geq 0$. If $\delta \varphi > b_2^*$, then: $ b_1^* = 0 \quad \text{(corner solution)} $ This is **Holmström's "Wisdom of Weak Incentives"**: when implicit incentives are strong enough, no explicit incentive needed. **Salary:** $ s_1^* = \bar{u} - b_1^* E[p_1] $ #### II. What makes $b_1^*$ different from $b_2^*$? **Key difference:** Presence of implicit career-concern incentive $\delta \varphi$ in period 1. **Substitutability:** The relationship $b_1^* = b_2^* - \delta \varphi$ shows that: - Explicit incentive ($b_1^*$) and implicit incentive ($\delta \varphi$) are **perfect substitutes** - One unit increase in implicit → exactly one unit decrease in explicit - Total incentive remains constant: $b_1^* + \delta \varphi = b_2^*$ **Economic interpretation:** | Component | Period 1 | Period 2 | |-----------|----------|----------| | Explicit ($b_t$) | $b_2^* - \delta \varphi$ | $b_2^*$ | | Implicit (career) | $\delta \varphi$ | $0$ | | **Total** | $b_2^*$ | $b_2^*$ | The firm "gets free incentives" from career concerns, so reduces explicit bonus accordingly. **Wisdom of Weak Incentives:** When $\delta \varphi \geq b_2^*$: - Career concerns alone sufficient - $b_1^* = 0$ (corner solution) - Effort: $a_1^* = \delta \varphi g < a_2^* = b_2^* g$ #### III. How does $a_1^*$ differ from $a_2^*$? Why? **Case 1: Interior Solution** ($\delta \varphi < b_2^*$) $ a_1^* = (b_1^* + \delta \varphi) g = b_2^* g = a_2^* $ **Result:** $a_1^* = a_2^*$ (same effort level) **Why?** Firm adjusts explicit bonus to offset career concerns, maintaining total incentive at optimal level $b_2^*$. **Case 2: Corner Solution** ($\delta \varphi \geq b_2^*$) $ b_1^* = 0 \implies a_1^* = \delta \varphi g $ While: $ a_2^* = b_2^* g $ **Result:** $a_1^* \neq a_2^*$ Specifically: - If $\delta \varphi = b_2^*$: $a_1^* = a_2^*$ - If $\delta \varphi > b_2^*$: $a_1^* > a_2^*$ (!) **Why different?** When career concerns are very strong ($\delta \varphi > b_2^*$), the firm cannot reduce explicit bonus below zero. The agent "over-exerts" effort in period 1 relative to first-best, driven by excessive career concerns. **Summary table:** | Condition | $b_1^*$ | $a_1^*$ | Relationship | |-----------|---------|---------|--------------| | $\delta \varphi < b_2^*$ | $b_2^* - \delta \varphi > 0$ | $(b_1^* + \delta \varphi)g = b_2^* g$ | $a_1^* = a_2^*$ | | $\delta \varphi = b_2^*$ | $0$ | $\delta \varphi g = b_2^* g$ | $a_1^* = a_2^*$ | | $\delta \varphi > b_2^*$ | $0$ | $\delta \varphi g > b_2^* g$ | $a_1^* > a_2^*$ | #### IV. Why is public observability of contracts important? **Mechanism:** Public observability affects the **strength of implicit incentives** through its impact on market inference. **Private Contracts (not observable):** Market observes only $p_1$, not $b_1$. Belief updating: $ E[\eta | p_1] = \varphi(p_1 - g \cdot \hat{a}_1^{private}) $ where $\hat{a}_1^{private}$ is market's conjecture without knowing $b_1$. Market must guess effort level, leading to: - **Higher implicit incentive** (market attributes more of high $p_1$ to ability) - **Lower optimal $b_1^*$** (firm exploits this) **Public Contracts (observable):** Market observes both $p_1$ and $b_1$. Belief updating: $ E[\eta | p_1, b_1] = \varphi(p_1 - g \cdot a_1^{exp}(b_1)) $ where $a_1^{exp}(b_1) = (b_1 + \delta \varphi)g$ is the expected effort given known contract. Market can **decompose** performance: $ p_1 = \underbrace{g \cdot a_1^{exp}(b_1)}_{\text{effort component}} + \underbrace{\eta}_{\text{ability}} + \phi_1 $ Result: - **More accurate inference** about ability - **Lower implicit incentive** (effort component factored out) - **Higher optimal $b_1^*$** (firm must compensate) **Formal comparison:** Let $\delta \varphi^{private}$ and $\delta \varphi^{public}$ denote implicit incentives under each regime. - Private: $\varphi^{private}$ reflects uncertainty about effort → larger - Public: $\varphi^{public}$ reflects only ability uncertainty → smaller Thus: $ b_1^{private*} = b_2^* - \delta \varphi^{private} < b_2^* - \delta \varphi^{public} = b_1^{public*} $ **Economic insight:** Public contracts act as **hard evidence** (à la Grossman-Milgrom), preventing the firm from "free-riding" on excessive career concerns. This: 1. Reduces career distortions 2. Makes reputation system more efficient 3. Increases explicit incentive pay **Connection to Problem 2:** This parallels the hard evidence vs. cheap talk distinction. Public contracts = verifiable information → better information transmission → more efficient outcomes. --- ## Problem 2: Cheap Talk with Reputation ### Setup Summary **Players:** Agent (advisor), Principal (decision-maker) **State:** $s \in \{0, 1\}$ with $P(s=1) = p$ **Types:** - Unbiased (u): $u^u = -y(s-d)^2 + \lambda \phi(m)$ - Biased (b): $u^b = xd + \lambda \phi(m)$ - Prior: $P(\text{unbiased}) = q$ **Message:** $m \in \{0, 1\}$ (cheap talk) **Decision:** Principal chooses $d$ after observing $m$ **Reputation:** $\phi(m) = P(\text{unbiased} | m)$ --- ### Part (a): No Full Separation **Claim:** There exists no equilibrium where $m(s) = s$ for both types. **Proof by contradiction:** Suppose separating equilibrium exists: - Unbiased: $m^u(0) = 0$, $m^u(1) = 1$ - Biased: $m^b(0) = 0$, $m^b(1) = 1$ **Principal's response:** Using Bayes' rule on path: - $m=0$ could come from either type when $s=0$ - $m=1$ could come from either type when $s=1$ - Both messages equally informative about type: $\phi(0) = \phi(1) = q$ Principal's optimal decisions: - $m=0$: $d_0 = E[s | m=0] = 0$ (both types report truthfully) - $m=1$: $d_1 = E[s | m=1] = 1$ **Check biased type's incentive when $s=0$:** **Truth-telling** ($m=0$): $ u^b(d_0, m=0, s=0) = x \cdot 0 + \lambda \cdot q = \lambda q $ **Lying** ($m=1$): $ u^b(d_1, m=1, s=0) = x \cdot 1 + \lambda \cdot q = x + \lambda q $ **Comparison:** Since $x > 0$: $ x + \lambda q > \lambda q $ The biased type strictly prefers to lie. The reputation term $\lambda q$ is identical in both cases (both on equilibrium path), so the biased type optimizes only the decision payoff $xd$. **Conclusion:** Biased type deviates when $s=0$. Equilibrium breaks. $\square$ **Crawford-Sobel (1982) insight:** With cheap talk (costless messages) and preference misalignment, full separation impossible. The biased type can always mimic unbiased to get preferred decision. --- ### Part (b): No Equilibrium where $m^u = 0, m^b = 1$ **Claim:** No equilibrium exists where unbiased type always sends $m=0$ and biased type always sends $m=1$. **Proposed equilibrium:** - Unbiased: $m^u(0) = 0$, $m^u(1) = 0$ - Biased: $m^b(0) = 1$, $m^b(1) = 1$ **Principal's beliefs:** - $m=0 \implies$ unbiased type $\implies \phi(0) = 1$ - $m=1 \implies$ biased type $\implies \phi(1) = 0$ **Principal's responses:** - $m=0$: $d_0 = E[s | \text{unbiased}] = P(s=1) \cdot 1 + P(s=0) \cdot 0 = p$ - $m=1$: $d_1 = \arg\max_d E[u_p | \text{biased}] = 1$ (since biased always wants high $d$, principal infers they're advocating for it) Actually, let me reconsider. If principal knows sender is biased, principal should optimize own payoff: - $u_p = -(s-d)^2$ - With $P(s=1) = p$: $d_1^* = p$ Hmm, but the problem says "biased always prefers higher decisions no matter what is s", so principal knowing sender is biased doesn't help much. Let me use the off-equilibrium belief that supports this: principal believes biased would send $m=1$ regardless of $s$, so $m=1$ is uninformative. **Better approach:** After $m=1$, principal believes sender is biased. Since biased type's message is independent of $s$, the message is uninformative about state. Principal uses prior: $ d_1 = p $ After $m=0$, principal believes sender is unbiased. Unbiased sends $m=0$ regardless of $s$ (both when $s=0$ and $s=1$), so again uninformative: $ d_0 = p $ Wait, this doesn't work either. Let me reconsider the structure. **Actually, the equilibrium specifies:** - Unbiased always sends 0 (both states) - Biased always sends 1 (both states) So: - $m=0 \implies$ unbiased $\implies d_0 = p$ (uninformative about $s$) - $m=1 \implies$ biased $\implies d_1 = p$ (uninformative about $s$) This gives same decision both ways, which seems odd. Let me re-examine. Perhaps the specification is: - $d_0$ should be chosen to maximize principal's payoff given belief sender is unbiased - Since unbiased has same preferences as principal, principal would want to follow unbiased's implicit advice - But unbiased sends $m=0$ for both states, so no advice conveyed - Thus $d_0 = p$ Similarly, $d_1 = p$. **Check unbiased type when $s=1$:** **Stay** ($m=0$): $ u^u = -y(p-1)^2 + \lambda \cdot 1 = -y(1-p)^2 + \lambda $ **Deviate** ($m=1$): $ u^u = -y(p-1)^2 + \lambda \cdot 0 = -y(1-p)^2 $ **Comparison:** Stay payoff: $-y(1-p)^2 + \lambda$ Deviate payoff: $-y(1-p)^2$ Since $\lambda > 0$, staying is better. No profitable deviation. Hmm, this suggests equilibrium might exist. Let me check other types. **Check unbiased type when $s=0$:** **Stay** ($m=0$): $ u^u = -y(p-0)^2 + \lambda \cdot 1 = -yp^2 + \lambda $ **Deviate** ($m=1$): $ u^u = -y(p-0)^2 + \lambda \cdot 0 = -yp^2 $ Again, staying better due to reputation. **Check biased type:** **Stay** ($m=1$): $ u^b = xp + \lambda \cdot 0 = xp $ **Deviate** ($m=0$): $ u^b = xp + \lambda \cdot 1 = xp + \lambda $ **Profitable deviation!** Biased type prefers to deviate to $m=0$ to gain reputation, since decision is same either way. **Conclusion:** Equilibrium fails because biased type deviates. $\square$ --- ### Part (c): No Equilibrium where $m^u = 1, m^b = 0$ **Claim:** No equilibrium exists where unbiased always sends $m=1$ and biased always sends $m=0$. **Proposed equilibrium:** - Unbiased: $m^u(0) = 1$, $m^u(1) = 1$ - Biased: $m^b(0) = 0$, $m^b(1) = 0$ **Principal's beliefs:** - $m=0 \implies \phi(0) = 0$ (biased) - $m=1 \implies \phi(1) = 1$ (unbiased) **Principal's responses:** - $m=0$: $d_0 = p$ (biased uninformative) - $m=1$: $d_1 = p$ (unbiased uninformative) **Check biased type:** **Stay** ($m=0$): $ u^b = xp + \lambda \cdot 0 = xp $ **Deviate** ($m=1$): $ u^b = xp + \lambda \cdot 1 = xp + \lambda $ **Profitable deviation!** Since decisions are identical ($d_0 = d_1 = p$), biased type strictly prefers to deviate to $m=1$ to gain reputation $\lambda$. **Conclusion:** Equilibrium fails. $\square$ **General insight:** When both types pool on different messages but decisions are same, the type assigned worse reputation will deviate to improve reputation costlessly. --- ### Part (d): Pooling on $m=1$ Equilibrium **Equilibrium candidate:** $ m^u(0) = m^u(1) = m^b(0) = m^b(1) = 1 $ **Principal's response:** Message $m=1$ is uninformative (all types send it). Using prior: $ d^* = E[s] = p $ **Off-equilibrium belief:** For $m=0$ (off path), use pessimistic belief: $ \phi(0) = 0 $ This implies principal believes only biased would deviate, so: $ d_0 = p \quad \text{or} \quad d_0 = 1 $ (Biased type wants high decision regardless of state; if principal knows sender is biased and uninformative, best response is still $d_0 = p$. But for strongest deterrent, use $d_0 = 1$.) Let's use $d_0 = 1$ for clearer analysis. **Incentive compatibility: Unbiased type, $s=0$** **Stay** ($m=1$): $ u^u = -y(p-0)^2 + \lambda q = -yp^2 + \lambda q $ **Deviate** ($m=0$): $ u^u = -y(1-0)^2 + \lambda \cdot 0 = -y $ **IC condition:** $ -yp^2 + \lambda q \geq -y $ $ \lambda q \geq y(1-p^2) = y(1-p)(1+p) $ **Equilibrium condition:** $ \boxed{\lambda \geq \frac{y(1-p^2)}{q}} $ **Interpretation:** For equilibrium to exist, reputation value $\lambda$ must be large enough that unbiased type (who sees $s=0$) prefers to: - Accept wrong decision ($d=p$ instead of $d=0$) - Maintain reputation ($\phi = q$) Rather than: - Reveal truth (send $m=0$) - Get correct decision ($d=0$) but lose reputation ($\phi=0$) **Key insight:** High reputation concerns ($\lambda$) **destroy information transmission**. Even honest types lie to protect reputation. --- ### Part (e): Pooling on $m=0$ Equilibrium **Equilibrium candidate:** $ m^u(0) = m^u(1) = m^b(0) = m^b(1) = 0 $ **Principal's response:** Message $m=0$ uninformative: $ d^* = p $ **Off-equilibrium belief:** For $m=1$, use: $ \phi(1) = 0 \implies d_1 = 1 $ **IC: Unbiased type, $s=1$** **Stay** ($m=0$): $ u^u = -y(p-1)^2 + \lambda q = -y(1-p)^2 + \lambda q $ **Deviate** ($m=1$): $ u^u = -y(1-1)^2 + \lambda \cdot 0 = 0 $ **IC condition:** $ -y(1-p)^2 + \lambda q \geq 0 $ $ \lambda q \geq y(1-p)^2 $ **Equilibrium condition:** $ \boxed{\lambda \geq \frac{y(1-p)^2}{q}} $ **Relationship to part (d):** Condition (d): $\lambda \geq \frac{y(1-p^2)}{q} = \frac{y(1-p)(1+p)}{q}$ Condition (e): $\lambda \geq \frac{y(1-p)^2}{q}$ Since $(1-p)^2 < (1-p)(1+p)$ for $p > 0$: $ \frac{y(1-p)^2}{q} < \frac{y(1-p)(1+p)}{q} $ **Result:** - High $\lambda$: Both equilibria exist, but (d) has tighter condition - Intermediate $\lambda$: Only (e) exists - Low $\lambda$: Neither exists (no pooling equilibrium) **Summary:** | $\lambda$ Range | Equilibrium | |-----------------|-------------| | $\lambda \geq \frac{y(1-p)(1+p)}{q}$ | Both pooling equilibria exist | | $\frac{y(1-p)^2}{q} \leq \lambda < \frac{y(1-p)(1+p)}{q}$ | Only pool on $m=0$ | | $\lambda < \frac{y(1-p)^2}{q}$ | Neither pooling equilibrium | **Economic lesson:** Reputation concerns create **babbling equilibria** where all types send same message → principal learns nothing → decisions suboptimal. The paradox: caring about reputation (wanting to appear unbiased) makes agents unable to transmit information, even when they have valuable information to share. This connects to **Crawford & Sobel (1982)**: cheap talk + preference misalignment + reputation concerns → information transmission fails. **Connection to Problem 1:** Just as public contracts act as hard evidence to restore information transmission, here we'd need verifiable messages (hard evidence) rather than cheap talk to enable truthful communication.