# 14.282 Fall 2025 Midterm 1 --- Dear Yifan and or Bob, I'm sorry I didn't have enough capacity to fully absorb module 1,2,3. I got LLM's help to write midterm solution. I tried not to "rubber-stamp" and hope my LLM cares less about its reputation (problem 2). Entrepreneurship setting is my reservoir of imagination, so I made loose analogies, marked with cute 🐣. Thanks for bearing it. # 1. Solution ## Problem 1: Career Concerns and Incentive Contracts ### Part (a): Pure Career Concerns (p observable, not contractible) **Equilibrium:** **Wages:** $\{w_1, w_2\}$ **Efforts:** $\{a_1, a_2\}$ where $a_t = (a_{1t}, a_{2t})$ **Beliefs:** $E[\eta | p_1]$ Period 2: $a_2^* = 0$, $w_2^* = E[\eta|p_1]$ Period 1: Competition implies $w_1^* = \bar{u}$ Agent maximizes: $w_1 - c(a_1) + \delta E[\eta|p_1]$ Bayesian updating: $E[\eta|p_1] = \varphi(p_1 - g \cdot \hat{a}_1)$ where $\varphi = \frac{h_\phi}{h+h_\phi}$ FOC yields: $a_1^* = \delta\varphi g$ **Why $a_1 \neq 0$:** Career concerns create implicit incentive $\delta\varphi$. Higher effort increases $p_1$, which increases perceived ability and future wage. (e.g. Early-stage founders working without salary—building reputation for future rounds.) **Dependence on $\cos\theta$:** Let $\cos\theta = \frac{f \cdot g}{\|f\|\|g\|}$. Effort level $\|a_1^*\|$ is independent of $\cos\theta$ since agent optimizes along $g$. However, social value $f \cdot a_1^* = \delta\varphi\|f\|\|g\|\cos\theta$ depends on alignment. **Dependence on $\varphi$:** $\frac{\partial a_1^*}{\partial\varphi} = \delta g > 0$. Higher signal precision strengthens career concerns, increasing effort. At $\varphi = 0$ (pure noise), career concerns vanish and $a_1^* = 0$. **Summary table:** | $\varphi$ 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 | 🐣 *Entrepreneurial Analogy:* in extremely volatile sectors (e.g., crypto ventures during bubble periods) where market timing dominates skill—investors cannot distinguish ability from luck, eliminating reputation incentives. --- ### Part (b) **Period 2 equilibrium:** $b_2^* = \frac{f \cdot g}{2\|g\|^2} = \frac{\|f\|}{2\|g\|}\cos\theta$ (from solving ∂/∂b₂: (f·g) - 2b₂|g|² = 0) $a_2^* = b_2^* g$ **Period 1 equilibrium:** Total incentive: explicit bonus $b_1$ plus implicit career concern $\delta\varphi$ FOC: $a_1^* = (b_1 + \delta\varphi)g$ Optimal contract sets $b_1^* = \max\{0, b_2^* - \delta\varphi\}$ **Why $b_1^* \neq b_2^*$:** Explicit and implicit incentives are substitutes. Firms reduce explicit bonus by exactly the career concern component: $b_1^* = b_2^* - \delta\varphi \quad \text{(if interior)}$ **Summary table:** | Component | Period 1 | Period 2 | |-----------|----------|----------| | Explicit ($b_t$) | $b_2^* - \delta \varphi$ | $b_2^*$ | | Implicit (career) | $\delta \varphi$ | $0$ | | **Total** | $b_2^*$ | $b_2^*$ | **Wisdom of Weak Incentives:** When $\delta \varphi \geq b_2^*$: - Career concerns alone sufficient - $b_1^* = 0$ (corner solution) - Effort: $a_1^* = \delta \varphi g < a_2^* = b_2^* g$ **How $a_1^*$ differs from $a_2^*$:** Interior solution: $a_1^* = a_2^*$ (total incentive constant) Corner solution ($\delta\varphi > b_2^*$): $b_1^* = 0$, so $a_1^* = \delta\varphi g \neq b_2^* g = a_2^*$ **Public observability:** Public contracts weaken implicit incentives by enabling market decomposition: observing $b_1$ allows inference of expected effort, isolating ability from $p_1$. This reduces the marginal reputation gain per unit effort (smaller $\delta\varphi$), requiring compensating explicit incentives (higher $b_1^*$). Private contracts prevent decomposition—market attributes entire $p_1$ to ability, amplifying reputation returns to effort (larger $\delta\varphi$), allowing firms to reduce $b_1^*$. **Summary table:** | Condition | $b_1^*$ | $a_1^*$ | Relationship | | ------------------------ | ---------------------------- | ------------------------------------- | --------------- | | $\delta \varphi < b_2^*$ | $b_2^* - \delta \varphi > 0$ | $(b_1^* + \delta \varphi)g = b_2^* g$ | $a_1^* = a_2^*$ | | $\delta \varphi = b_2^*$ | $0$ | $\delta \varphi g = b_2^* g$ | $a_1^* = a_2^*$ | | $\delta \varphi > b_2^*$ | $0$ | $\delta \varphi g > b_2^* g$ | $a_1^* > a_2^*$ | 🐣 *Entrepreneurial Analogy:* Kickstarter (public funding) vs. VC side deals (private)—the former exposes raised amount and terms, enabling backers to isolate founder ability from resources, weakening reputation incentives; the latter hides deal structure, so startup success appears fully attributable to founder skill, intensifying reputation-driven effort while allowing VCs to offer lower valuations. --- ## Problem 2: Cheap Talk with Reputation **Entrepreneurial Context Mapping:** | Variable | Model Definition | Venture Interpretation | |----------|-----------------|------------------------| | **Principal** | Decision-maker (uninformed) | Investor | | **Agent** | Advisor (informed, observes s) | Founder | | **s** | State ∈ {0,1}, P[s=1]=p | Market condition (0=weak, 1=strong) | | **d** | Principal's decision ∈ [0,1] | Funding level (0=low, 1=high) | | **m** | Agent's message ∈ {0,1} | Pitch tone (0=conservative, 1=aggressive) | | **φ(m)** | Posterior: P[unbiased\|m] | Investor belief: "founder is aligned" | | **q** | Prior: P[unbiased] | Prior: aligned founder probability | | **λ** | Weight on reputation | Value of appearing aligned | | **x** | Biased weight on d | Growth-biased preference for funding | | **y** | Unbiased weight on (s-d)² | Aligned preference for strategy-market fit | **Agent Types:** - Biased: $u^b = xd + \lambda\phi(m)$ (prefers high d regardless of s) - Unbiased: $u^u = -y(s-d)^2 + \lambda\phi(m)$ (prefers d=s) --- ### Part (a): No Full Separation **Claim:** No equilibrium where $m(s) = s$ for both types. **Proof by contradiction:** Suppose $m^u(s)=m^b(s)=s$. Principal: $\phi(0)=\phi(1)=1 \Rightarrow d_0=0, d_1=1$ Biased at $s=0$: - Truth: $u^b = 0 + \lambda q = \lambda q$ - Lie ($m=1$): $u^b = x + \lambda q$ Since $x>0$, deviation profitable. $\square$ 🐣 *Entrepreneurial Analogy:* If aligned founders truthfully reveal market conditions, growth-biased founders facing weak markets (s=0) deviate to aggressive pitches (m=1) to maximize funding (xd). This breaks truth-telling—exactly why pitch tone rarely maps to actual traction in seed rounds. --- ### Part (b): No Equilibrium where $m^u = 0, m^b = 1$ **Claim:** No equilibrium with $m^u(0)=m^u(1)=0, m^b(0)=m^b(1)=1$. Principal: $\phi(0)=1, \phi(1)=0 \Rightarrow d_0=d_1=p$ Biased type: - Stay ($m=1$): $u^b = xp$ - Deviate ($m=0$): $u^b = xp + \lambda$ Deviation profitable. $\square$ 🐣 *Entrepreneurial Analogy:* Suppose aligned founders signal conservatively (m=0) while growth-biased founders signal aggressively (m=1). _Growth-biased types costlessly mimic conservative signals to capture reputation premium λ (ANGIE IS NOT SURE)._ Venture markets cannot segregate by preference alignment through cheap talk alone. --- ### Part (c): No Equilibrium where $m^u = 1, m^b = 0$ **Claim:** No equilibrium with $m^u(0)=m^u(1)=1, m^b(0)=m^b(1)=0$. Principal: $\phi(0)=0, \phi(1)=1 \Rightarrow d_0=d_1=p$ Biased type: - Stay ($m=0$): $u^b = xp$ - Deviate ($m=1$): $u^b = xp + \lambda$ Deviation profitable. $\square$ 🐣 *Entrepreneurial Analogy:* Reverse signaling—asking growth-biased founders to appear disciplined while aligned founders claim growth—is equally unstable. When reputation rewards favor visionary narratives (λ>0), directionally-biased types always deviate to aggressive messages, flipping intended sorting. --- ### Part (d): Pooling on $m=1$ Equilibrium **Equilibrium:** $m^u(s)=m^b(s)=1$ for all $s$. Principal: uninformative signal, $d^*=p$ Off-path: $\phi(0)=0 \Rightarrow d_0=1$ Unbiased at $s=0$: - Pool (m=1): $-yp^2 + \lambda q$ - Deviate ($m=0$): $-y$ **IC:** $\lambda q \geq y(1-p^2)$ **Condition:** $\lambda \geq \frac{y(1-p^2)}{q}$ 🐣 *Entrepreneurial Analogy:* When reputation stakes exceed honesty costs—$\lambda > y(1-p^2)/q$—even aligned founders with weak traction (s=0) adopt "moonshot" narratives. This threshold formalizes Y Combinator demo days: competitive funding markets converge to uniformly optimistic pitches. Founders prefer reputational safety (λq) to truthful precision. --- ### Part (e): Pooling on $m=0$ Equilibrium **Equilibrium:** $m^u(s)=m^b(s)=0$ for all $s$. Principal: $d^*=p$ Off-path: $\phi(1)=0 \Rightarrow d_1=1$ Unbiased at $s=1$: - Pool: $-y(1-p)^2 + \lambda q$ - Deviate ($m=1$): $0$ **IC:** $\lambda q \geq y(1-p)^2$ When Part (d) condition fails, this equilibrium exists. 🐣 Entrepreneurial Analogy: Post-bubble markets where exaggeration cost > reputation reward: founders coordinate on conservative guidance. Condition $\lambda q < y(1-p)^2$ says "appearing delusional is costlier than appearing unambitious." Post-2001 dotcom era exemplifies this—after [Pets.com](http://Pets.com) and Webvan collapsed, VCs penalized optimistic projections. Modest guidance became the shared safe strategy. Pooling on low message sandbags equilibrium. --- **Equilibrium Regions:** | λ Range | Equilibrium | |---------|-------------| | $\lambda \geq \frac{y(1-p^2)}{q}$ | Pool on m=1 | | $\lambda < \frac{y(1-p)^2}{q}$ | Neither pooling | | Between | Pool on m=0 | **Summary Table:** More information → decisions can be better tailored to reality. Less information → decisions rely on averages or priors → welfare falls. | Equilibrium | Information Revealed | Decision d | Efficiency | | ------------------ | -------------------- | ---------------------------------------- | ------------------- | | Full separation | Exact s | d(s)=s on each state | Fully efficient | | Partial separation | State interval | d=E[s\|interval] on average state in bin | Partially efficient | | Pooling | None | d=p=E[s] on prior | Highly inefficient | ![[Pasted image 20251027174419.png|400]] --- # 2. Synthesis #### **1. Two Faces of Reputation** This midterm contrasts reputation's role across two canonical agency problems: **Problem 1** (Career concerns): Performance (p) affects market beliefs about ability (η). Future wage competition creates implicit incentives. Principal adjusts explicit contracts accordingly (b₁* < b₂*). **Problem 2** (Cheap talk): Messages (m) affect receiver beliefs about sender type (biased/unbiased). Crawford & Sobel (1982) showed preference misalignment yields partition equilibria. Adding reputation concerns (λ·φ(m)) generates pooling even with aligned outcome preferences. --- #### **2. Reputation Across Information Structures** | Dimension | Hidden Action (P1) | Hidden Information (P2) | | -------------------------- | -------------------------- | ------------------------ | | **Posterior belief about** | Agent's ability (η) | Agent's type (θ) | | **Updated via** | Performance traces (p) | Message choice (m) | | **Induced behavior** | Effort on measurable tasks | Pooling on safe messages | | **Result** | a₁* = δφg₁ > 0 | $m^u(s) = m^b(s)$ | **Problem 1:** Market observes p = f·a + η + ε. Even without explicit contract on p, agent internalizes future wage gains E[w₂|p]. Competition ensures wages track beliefs: w₂ = E[η|p]. Result: implicit incentives δφg₁. **Problem 2:** Receiver observes only m, which sender controls. Crawford & Sobel: preference misalignment (x,y > 0) limits information transmission. Adding reputation (λ) creates second obstacle: unbiased type at s=0 prefers pooling on m=1 when λq ≥ y(1-p²). Even truthful types lie to avoid appearing biased. --- #### **3. Literature to visit after exam curated by LLM Holmström, B. (1999). "Managerial Incentive Problems: A Dynamic Perspective." _Review of Economic Studies_, 66(1), 169–182. → Career concerns as implicit incentives; market competition generates effort without explicit contracts. Crawford, V., & Sobel, J. (1982). "Strategic Information Transmission." _Econometrica_, 50(6), 1431–1451. → Cheap talk with preference misalignment yields partition equilibria; full separation impossible. Morris, S. (2001). "Political Correctness." _Journal of Political Economy_, 109(2), 231–265. → With identical outcome preferences, reputational concerns alone generate pooling. Advisor lies to avoid appearing biased; if λ sufficiently large, no information transmitted. Gibbons, R., & Murphy, K. (1992). "Optimal Incentive Contracts in the Presence of Career Concerns." _Journal of Political Economy_, 100(3), 468–505. → Empirical evidence: CEO compensation shows b₁ < b₂—explicit incentives weaken when career concerns strongest. Gibbons, R. (1998). "Incentives in Organizations." _Journal of Economic Perspectives_, 12(4), 115–132. → Implicit incentives from market forces substitute for explicit contracts when contracting incomplete.