Information, Risk and Economic Policy: A Dynamic Contracting Approach

Transcription

1 Information, Risk and Economic Policy: A Dynamic Contracting Approach Noah University of Wisconsin-Madison

2 Or: What I ve Learned from LPH As a student, RA, and co-author Much of my current work builds directly on tools and lessons I learned from Lars, especially in the 2002 RFS paper with Cagetti and Sargent (CHSW) and the 2006 JET paper with Sargent and Turmuhambetova (HSTW). Today: overview of two recent applications of continuous time dynamic contracting: Optimal unemployment insurance with cyclical fluctuations Risk taking with delegated investment

3 Some lessons from LPH Lesson 1: Study a problem of true economic interest. Lesson 2: Learn the appropriate tools. Don t settle for a fixed stock. Lesson 3: Get the theory right and formulate the problem in the right way. Lesson 4: Be generous with your students and get them involved.

4 Applying lessons from LPH Lesson 1: Study a problem of true economic interest. How does private information distort allocations? How should unemployment benefits vary over time and over the business cycle? How should contracts be structured to limit excess risk taking by managers? Lesson 2: Learn the appropriate tools. Martingale representations, Girsanov, stochastic maximum principle and BSDEs, jump process control Lesson 3: Formulate the problem in the right way. Formulate contracting problem as a change of measure Lesson 4: Get your students involved. My recent work is with Rui Li, UW PhD (2012), now at UMass-Boston.

5 A Warning from Sargent Sargent [to me]: In seminars, Lars makes the audience work. But they know that they can trust him to deliver and make their effort worthwhile. They don t have that trust in you.

6 : Overview Agent consumes, puts forth effort. Principal owns capital, pays agent. dk t = [f (k t, a t ) c t ] dt + σ(k t )dw t k t capital, a t effort, c t agent payment, W t std Brownian motion. Principal observes only k t, not a t (or noise W t ). Contract specifies c(t, k), so capital evolution depends on its whole past. Difficult to handle directly. Key idea: change focus from evolution of state to distribution of outcomes: k C[0, T]. Effort a changes distribution. Let a 0 : f (k, a 0 ) c(t, k) 0. dk t = σ(k t )dw 0 t Any other effort changes measure: ā Pā on C[0, T].

7 Agent s Problem: Optimality Conditions Formulate agent s problem as choosing effort to influence density of change of measure. Apply stochastic maximum principle to derive optimality conditions. Agent s promised utility is the co-state associated with the density process. [ ] q t = ρe t e ρ(t s) u(c s, a s )ds t dq t = ρ [q t u(c t, a t )] dt + γ t σ(k t )dw t = ρ [q t u(c t, a t )] dt + γ t (dk t [f (k t, a t ) c t ] dt) γ t gives local volatility of agent s utility its loading on new information. Will be key for incentives. The optimal effort a solves: max {ρu(c t, a) + γ t [f (k t, a) c t ]} a

8 Application: Optimal Unemployment Insurance with Cyclical Fluctuations In normal times, unemployment benefits provide replacement rate (47% average) for 26 weeks. In recessions, federal extended benefits provide an additional 13 weeks of benefits. In severe recessions, these are extended further. Most recent recession: 99 weeks in high unemployment states. What should be the optimal pattern (level, duration) of unemployment insurance over the cycle when workers put forth unobservable search effort? How would this affect outcomes? Level and duration of unemployment in booms and recessions. Tradeoff increased insurance with less information in a recession.

9 The Model Continuous time version of Hopenhayn-Nicolini (1996), with business cycles and multiple unemployment spells. Extension of previous work to setting with controlled (employment) & uncontrolled (aggregate) jumps All jobs pay wage ω. Workers are risk averse, put forth search effort a, consume c. No outside consumption when unemployed. Unemployment agency risk-neutral, minimizes transfers subject to providing a given level of utility. Business cycle: boom is a period of high job finding rates, low unemployment rates. s = G, B is state of economy, arrival intensity of a job is: q s (a t ) = q s0 + q s1 a t, q B (a) < q G (a) Exogenous separation intensity: p B > p G Aggregate state intensity: λ B > λ G.

10 Promise Keeping and Incentives Define two indicator states: st J = 1 if employed, 0 unemployed, st S = 1 if recession, 0 boom. Construct associated martingales mt J, mt S, i.e. if st J = 0, st S = 1: dm J t = s J t q B (a t )dt. Promised utility [ ] q t Et a ρ e ρ(t s) u(c s, a s )ds t dq t = ρ[q t u(c t, a t )]dt + gt J dmt J + gt S dmt S = ρ[q t u(c t, a t )]dt + g J t [ s J t q B (a t )dt] + g S t dm S t Incentive constraint = agent s optimality condition: { } a t arg max ρu(c t, a) + g J a t q s (a)

11 Optimal Contract Unemployment agency is risk neutral, minimizes expected discounted transfers to agent, both when unemployed and employed (allow tax on wages), subject to delivering a given level of promised utility. Also consider benchmark contract: fixed benefit for 26 weeks in booms, 39 weeks in recessions. Estimate switching process on Shimer (2012) finding rate data (H-P filtered) to get λ s. Read off separations p s. Calibrate model (simulate large population of workers) under benchmark contract to match: average finding rates in boom, recession, elasticity of unemployment duration w.r.t. benefit ( 0.7, range 0.5-1).

12 Summary Statistics Benchmark Optimal Boom Recess Boom Recess Unemp Rate (%) Unemp Duration (weeks) Finding Rate (month) Separation Rate (month) Net Cost/Worker (% of ω)

13 Consumption Over Unemployment Spell: Recession Consumption (Replacement Ratio) Unobservable Effort Observable Effort Benchmark Weeks Unemployed

14 Effort Over Unemployment Spell: Recession Effort Unobservable Effort Observable Effort Benchmark Weeks Unemployed

15 Consumption Over Unemployment Spell Boom Recession Consumption (Replacement Ratio) Weeks Unemployed

16 Effort Over Unemployment Spell Boom Recession Effort Weeks Unemployed

17 Recessions and Extended Benefits Simulate recession and compare benefits extension. Benchmark: 5.3% 6.7%, 99-Week: 5.3% 6.8%. Optimal: 3.6% 4.0% Unemployment Rate Benchmark 99 week Optimal Time (Weeks)

18 Application: Hidden Risk Taking in Delegated Investment New type of agency friction common recently: managers and traders taking actions subjecting their firms to large potential losses. Rajan (2011): After all the profits from such [risk taking] activities would look a lot healthier if no one new the risks they were taking. Accordingly, Citibank s off-balance sheet conduits, holding an enormous quantity of asset-backed securities funded with short-term debt were hidden from all but the most careful analysis. Other example: AIG selling credit default swaps. Earn premia in good times, large losses in bad. Pay-for-performance contracts alleviate other agency frictions, but give incentive to take on excess risk.

19 The Model Risk neutral owner hires risk averse manager to operate firm, instantaneous cash flow AK t. Evolution of K t : dk t = (I t + µk t ) dt + K t (σ o dw ot + σ u dw ut ) + (1 ϕ) K t (λ t dt dn t ). I t investment by manager, W ot observable, W ut unobservable (to owner): sources of std moral hazard Disasters: Poisson N t, arrival λ t controlled by manager, destroy fraction 1 ϕ (0, 1), of capital. Martingale. Resource constraint: C t + I t + D t = AK t. Manger power utility over C t, owner risk neutral over D t. Total payment to manager: p t AK t C t + I t.

20 Promise Keeping and Incentives Promised utility evolution: dq t = ρ (W t u(c t)) dt + H tσ odw ot + G tσ udw ut + J t( λ tdt + dn t) = ρ (W t u(c t)) dt + H tσ odw ot + J t( λ tdt + dn t) + Gt K t [dk t (I t + µk t) dt K tσ odw ot (1 ϕ) K t(λ tdt dn t)] Incentive constraints for i t = I t /K t, λ t : i t λ t arg max {ρu(p t AK t ik t ) + G t i} i arg max λ [G t(1 ϕ) + J t ] λ We analyze the simple special case i {i, i}, λ {λ, λ}. Exploit homogeneity to write owner s value function as: 1 ((1 γ) W ) 1 γ V (K, W ) = Kv = Kv(w). K

21 Some Results Benefits of partnerships: If the manager is always paid a fixed fraction of cash flows (p t = p) then he will never choose the high risk λ. Hazards of pay-for-performance: If the owner tries to implement a standard moral hazard pay-for-performance contract ignoring the manager s control over λ, the manager will always choose the high risk λ. Costs of incentives: In the optimal contract accounting for moral hazard and hidden risk, it can become too costly to provide incentives and so is optimal for the manager to choose the high risk λ.

22 Risk Taking in the Optimal Contract High Unobservable Risk Taking Risk Taking Level Low w_l Normalized Continuation Utility w_r High Observable Risk Taking Risk Taking Level Low w_l Normalized Continuation Utility w_r

23 Conclusion Wide scope of applications for dynamic contracting models in economics. Continuous time methods make wider classes of models amenable to analysis. Current applications extend baseline models to include jumps, are suggest information frictions qualitatively and quantitatively important. None of my research would have been possible without what I ve learned from LPH.