Investment Decisions under Market Imperfections
Microeconomics for Finance
Juan F. Imbet
Master in Finance, 1st Year - Paris Dauphine University - PSL
Introduction to Market Imperfections
Real-world markets deviate from the perfect competition assumptions.
Market imperfections include:
- Asymmetric information
- Transaction costs
- Taxes and regulations
- Market power
- Behavioral biases
These imperfections affect investment decisions and asset pricing.
Asymmetric Information
One party has more or better information than the other.
Types:
- Adverse selection: occurs before transaction
- Moral hazard: occurs after transaction
In financial markets: lenders vs. borrowers, insurers vs. insured, etc.
The Principal-Agent Problem
A Simple Principal-Agent Model
Two periods: contract written at \(t=0\), production realized at \(t=1\).
Output \(y\) combines effort and noise: \(y = e + \varepsilon\), with
- Effort \(e \in [0, \bar e]\) chosen by the manager (agent) and not observed.
- Shock \(\varepsilon\) has mean zero and variance \(\sigma^2\); realizations are observed.
Manager preferences: \(U = \mathbb{E}[u(w) - c(e)]\) with risk aversion \(u(w) = -\exp(-\rho w)\) and effort cost \(c(e) = \tfrac{k}{2} e^2\), \(k>0\).
Owner (principal) is risk neutral and wants to maximize expected profit \(\Pi = \mathbb{E}[y - w] = e - \mathbb{E}[w]\).
Contracting with Hidden Effort
Contract offers wage schedule \(w(y)\) contingent on output.
Feasible contracts must satisfy:
- Participation: expected utility at least outside option \(\bar U\): \[ \mathbb{E}[u(w(y)) - c(e)] \ge \bar U. \]
- Incentive compatibility: chosen effort solves \[ e \in \arg\max_{\tilde e} \mathbb{E}[u(w(\tilde e + \varepsilon)) - c(\tilde e)]. \]
Without hidden effort, principal would enforce \(e^{FB} = \arg\max_e (e - c(e)) = 1/k\).
Hidden effort creates a wedge between first-best and second-best outcomes.
Optimal Linear Contract (1): Agent Side
- Consider linear wage: \(w(y) = a + b y = a + b(e+\varepsilon)\).
- Distribution: \(w \sim \mathcal{N}\big(a + b e,\; b^2 \sigma^2\big)\) since \(\varepsilon\) is mean 0, variance \(\sigma^2\).
- CARA utility with normal pay implies certainty equivalent (CE): \[ CE = \underbrace{a + b e}_{\text{mean wage}} - \underbrace{\tfrac{1}{2} \rho b^2 \sigma^2}_{\text{risk premium}} - \underbrace{\tfrac{k}{2} e^2}_{\text{effort cost}}. \]
- Agent chooses \(e\) to maximize \(CE\). \[ \frac{\partial CE}{\partial e} = b - k e = 0 \;\Rightarrow\; e^{SB} = \frac{b}{k}. \]
- Effort increases one-for-one with incentive intensity \(b\) and decreases with effort cost parameter \(k\).
Optimal Linear Contract (1b): CARA-Normal Certainty Equivalent
- If \(w \sim \mathcal{N}(\mu_w, \sigma_w^2)\) and \(u(w) = -e^{-\rho w}\) (CARA), then \[ \mathbb{E}[u(w)] = -\mathbb{E}[e^{-\rho w}] = - e^{-\rho \mu_w + \tfrac{1}{2} \rho^2 \sigma_w^2}. \]
- The certainty equivalent \(CE\) solves \(u(CE) = \mathbb{E}[u(w)]\): \[ -e^{-\rho CE} = - e^{-\rho \mu_w + \tfrac{1}{2} \rho^2 \sigma_w^2} \;\Rightarrow\; CE = \mu_w - \tfrac{1}{2} \rho \sigma_w^2. \]
- In our contract: \(\mu_w = a + b e\), \(\sigma_w^2 = b^2 \sigma^2\) giving \[ CE = a + b e - \tfrac{1}{2} \rho b^2 \sigma^2 - \tfrac{k}{2} e^2. \]
- Effort cost \(\tfrac{k}{2}e^2\) subtracts from CE because utility is \(u(w) - c(e)\).
- This derivation justifies the earlier expression used for agent optimization.
Optimal Linear Contract (2a): Principal Optimization Setup
- Profit given contract and agent response: \[ \Pi = \mathbb{E}[y - w] = e - (a + b e). \]
- Substitute agent effort \(e = b/k\): \[ \Pi = \frac{b}{k} - a - \frac{b^2}{k}. \]
- Participation (binding): \(CE = \bar U\) implies \[ a = \bar U + \tfrac{k}{2} e^2 + \tfrac{1}{2} \rho b^2 \sigma^2 - b e. \]
- Replace \(e = b/k\) to eliminate \(a\) and \(e\) in \(\Pi\): \[ \Pi = e - \bar U - \tfrac{1}{2} k e^2 - \tfrac{1}{2} \rho b^2 \sigma^2, \quad e=\frac{b}{k}. \]
- Express purely in \(b\), choose \(b\) to maximize \(\Pi(b)\). \[ \Pi(b) = \frac{b}{k} - \bar U - \frac{b^2}{2k} - \frac{1}{2} \rho b^2 \sigma^2. \]
Optimal Linear Contract (2b): Solving for \(b^{\star}\) and Effort
- First-order condition: \[ \frac{\partial \Pi}{\partial b} = \frac{1}{k} - \frac{b}{k} - \rho b \sigma^2 = 0. \]
- Solve for incentive slope: \[ b^{\star} = \frac{k}{1 + k \rho \sigma^2}. \]
- Implied second-best effort: \[ e^{SB} = \frac{b^{\star}}{k} = \frac{1}{1 + k \rho \sigma^2}. \]
- Comparison with first-best (\(e^{FB} = 1/k\)): \[ \frac{e^{SB}}{e^{FB}} = \frac{k}{1 + k \rho \sigma^2} < 1. \]
- Gap arises from need to insure the risk-averse agent against output noise.
Optimal Linear Contract (3): Interpretation & Limits
- Incentive vs. insurance trade-off:
- Benefit of higher \(b\): raises effort \(e = b/k\) linearly.
- Cost of higher \(b\): raises agent’s risk premium \(\tfrac{1}{2} \rho b^2 \sigma^2\) quadratically.
- Limiting cases:
- \(\rho \to 0\) (risk-neutral agent) or \(\sigma^2 \to 0\) (no noise): \(b^{\star} \to 1\), \(e^{SB} \to e^{FB}\).
- Large \(\rho\) or \(\sigma^2\): \(b^{\star} \approx 1/(\rho \sigma^2)\), \(e^{SB} \approx 1/(k \rho \sigma^2) \to 0\).
- Comparative statics: \[ \frac{\partial b^{\star}}{\partial \rho} < 0, \quad \frac{\partial b^{\star}}{\partial \sigma^2} < 0, \quad \frac{\partial b^{\star}}{\partial k} = \frac{1}{(1 + k \rho \sigma^2)^2} > 0. \]
- Although \(b^{\star}\) rises with \(k\), effort \(e^{SB} = b^{\star}/k\) falls as \(k\) increases.
- Welfare loss (effort gap): \[ e^{FB} - e^{SB} = \frac{1}{k} - \frac{1}{1 + k \rho \sigma^2} = \frac{\rho \sigma^2}{1 + k \rho \sigma^2}. \]
- Smaller \(\sigma^2\) (better performance metric) improves incentives.
Optimal Linear Contract (4): Decomposition & Implementation
- Fixed part \(a\) provides insurance and adjusts for participation: \[ a = \bar U + \tfrac{k}{2} (e^{SB})^2 + \tfrac{1}{2} (b^{\star})^2 \rho \sigma^2 - b^{\star} e^{SB}. \]
- Expected wage: \[ \mathbb{E}[w] = a + b^{\star} e^{SB} = a + \frac{(b^{\star})^2}{k}. \]
- Agent rents: zero if participation binds (all surplus beyond reservation extracted).
- Practical implementation:
- \(b\) via performance bonus or equity/stock grants.
- \(a\) as base salary ensuring risk sharing.
- Reduce \(\sigma^2\) using better KPIs/auditing to raise feasible \(b\).
- Performance measurement innovations substitute for costly incentives by lowering noise.
Implementing the Optimal Contract
(2b) - Step 1: calibrate parameters \((k, \rho, \sigma^2, \bar U)\) from data or managerial judgment. - Step 2: compute \(b^{\star}\) and implied effort \(e^{SB}\). - Step 3: determine fixed component \(a\) from the participation constraint: \[ a = \bar U + \tfrac{k}{2} (e^{SB})^2 + \tfrac{1}{2} (b^{\star})^2 \rho \sigma^2 - b^{\star} e^{SB}. \] - Step 4: communicate performance metrics and ensure verifiability of output \(y\).
- Contract balances incentives (through \(b\)) and insurance (through \(a\)).
Economic Interpretation of Constraints
Incentive compatibility ensures the manager prefers intended effort; otherwise, hidden actions would undermine performance.
Participation guarantees willingness to sign; often benchmarked to outside salary offers.
Comparative statics:
- Higher risk aversion \((\rho)\) or noise \((\sigma^2)\) lowers \(b^{\star}\), weakening incentives.
- Lower effort cost \((k)\) allows stronger incentives and higher effort.
Real-world contracts blend salary, bonus, and stock grants to approximate the optimal mix of incentives and insurance.
Moral Hazard in Financial Contracts