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This homework explores real-world externalities, including noise pollution, music piracy, and how nearby infrastructure like railways affects land prices. It examines positive externalities from recycling and the impact of studying with motivated peers. The assignment also discusses the marginal cost of commuting via bridges or tunnels along with utility analyses in uncertain scenarios involving moral hazard. Key concepts include incentives for workers, optimal contract structures, and how to balance motivation and risk-sharing for maximum employer profits.
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PCP MicroeconomicsSession 12 Takako Fujiwara-Greve
Homework 10 • Real-world Externalities • Noise, pirating music CDs, blocking sunlight of neighbors, land price near a new railway • Recycling may be positive externalities to future generation • Studying with highly motivated friends
Kreps 14.1 • (a) commute time via the bridge = 30 + nB/20k • = commute time via the tunnel = 40 + (400k - nB)/5000 • <=> nB = 360,000 nT = 40,000 • 30 + nB/20k = 48 minutes • (b) total commute time = nB ·48 + nT · 48 = 400 k · 48 =19.2 million minutes
14.1 (c ) • Minimize nB (30 +nB/20k) + (400k - nB) (40 + (400k - nB)/5000) • By differentiation -170 + nB/2000 = 0 • n* B = 340,000 • n*T = 60,000 • Commute time • Bridge 30 + 340k/20k =47 min. • Tunnel 40 + 60k/5k =52 min. • (d) impose toll of 0.5 for the bridge
EU approach • Jo: EU from the gamble = 0.25 ·1 + 0.75 · 0.49 = 0.615 • Utility from sure $7500 ≈ 0.65 • Jack: EU =0.25 √45,000 + 0.74 √5000 ≈ 106.066 • Utility from sure $7500 = √12500 ≈111.68 • Jim: EU = 0.25 √90,000 + 0.75 √50,000 ≈242.705 • Utility from sure $7500 = √57500 ≈239.79
Solving for Certainty Equivalent • Jack • x + 5000 = (106.066) => CE ≈ 6250 < 7500 • Jim • x + 50,000 = (242.705) => CE ≈ 8905>7500 2 2
Kreps, 15.3 (a) • 40,000 - 0.05· 750,000 = 2500
Moral Hazard • Insured factory owner: will she take care of dangerous things well? • Workers: will they work hard when they are not supervised? • Partnership: will partners work hard to increase the joint profit? • Actions of one party affects the welfare of others, where the interests of the parties are not the same
Incentives • Direct financial incentives • If you work hard, I pay you more • Reciprocity, reputation • If you are nice to others, others will be nice to you later • Intrinsic motivation, social norm • Feel proud or feel guilty for being bad
Solution 1: write a contract • Determine the “correct” action • Write it in the contract • Problem • How can we monitor/measure/prove?
Solution 2: Put all responsibility on the person who takes actions • No insurance • Tie payment to the worker’s sales • Problem • No risk sharing • Simultaneous moral hazard
Fundamental problems with incentives • The desired actions cannot be specified contractually • Measurement, monitoring, enforceability • Even if the desired action is taken, there is uncertainty about the consequences • Loading the full consequences on the party taking actions is undesirable • Could share the risk and improve all
Salesperson Compensation • Sale --> you (employer) get $60,000 • No sale -->you get $0 • Salesperson’s choices disutility • Kills himself: sale with prob. 0.5 40 • Works hard: sale with prob. 0.4 20 • Not hard: sale with prob. 0.25 10 • Loafs: sale with prob. 0.05 0 • S’s utility √wage - disutility • Outside opportunity: $10,000 w/ no disutility • Employer must give at least utility 100 • Employer: risk neutral
If you can specify an effort level in a contract • “S chooses effort level A and be paid X if a sale is made and Y if not” • Efficient risk sharing: X = Y • No effort is ok: √w - 0 ≥100 <=> w = $10,000 • Employer’s profit = (0.05)(60,000) - 10,000 = - 7000 • Let him try: √w - 10 ≥100 <=> w = $12,100 • Profit = (0.25)(60,000) - 12,100 = 2900
Work hard: √w - 20≥100 <=> w = 14,400 • Profit = (0.4) (60,000) - 14,400 = 9600 • Kill himself: √w - 40 ≥100 <=> w = 19,600 • Profit = (0.5)(60,000) - 19,600 = 10,400 • Optimal action for the employer = killing level
If effort is not contractable • If S is risk neutral, put all risk on him • But S is risk averse… • Shall the employer take all risk? • No! Then S will not make any effort • C.f: pride, reputation, promotion…
Try a bonus contract • Base wage = 9500 • Bonus = 15,000 if and only if a sale is made 100 quit loaf 0.95 √9500 + 0.05 √ 245,000 = 100.421 try 0.75 √9500 + 0.25 √ 245,000 -10= 102.232 hard 0.6√9500 + 0.4 √ 245,000 - 20= 101.091 kills 0.5 √9500 + 0.5 √ 245,000 - 40= 86.996
For this bonus contract • Salesperson will try but not hard • Profit = 0.25 · 60,000 - 9500 - 0.25 · 15,000 = 1750 • Trade-off • Efficient risk sharing => put risk on the employer • Motivation => tie the wage to outcomes (risky) • Optimal solution: compromise of these
To find “optimal” contract • That maximizes the profit • Step 1: For each of possible effort level (action), what is the cheapest way to motivate him to do? • Step 2: Which effort level (with the cheapest contract) maximizes your profit?
Problem 19.1: Let’s solve! • Find the cheapest (wage, bonus) to induce “try but not hard” level of effort • b= utility from base wage • x= utility from base wage + bonus 100 quit loaf 0.95 b + 0.05 x try 0.75 b + 0.25 x -10 hard 0.6 b + 0.4 x - 20 kills 0.5 b + 0.5 x - 40
Only two binding constraints • To maximize your profit we only need to satisfy • Participation constraint • 0.75 b + 0.25 x - 10 = 100 • > would also work, but why pay more? • Incentive constraint (not to choose easier action) • 0.75 b + 0.25 x - 10 = 0.95 b + 0.05 x • Check later that other effort levels are not better
Solution b=97.5, x=147.5 => base wage =9506.25, bonus = 12,250 • 0.5 b + 0.5 x - 40 < 0.6 b + 0.4 x - 20 < 0.75 b + 0.25 x - 10 ok • Your profit 0.25 (60,000 -21,756.25) + 0.75 (- 9506.25) = 2431.25
To “induce”? loafing • Participation constraint only • No need to give him a bonus • Base wage = 10,000 bonus =0 • Your profit = 0.05· 60000 - 10000 = -7000
To induce hard work • Participation constraint • 0.6 b + 0.4 x - 20 = 100 • Incentive constraint • 0.6 b + 0.4 x - 20 = 0.75 b + 0.25 x - 10 • Check others of course • Solution: b = 93.33.., x = 160 • Base wage = 8711.05, bonus = 16,889 • Your profit = 8533.37
To induce killing level • Participation constraint • 0.5 b + 0.5 x - 40 = 100 • Incentive constraint • 0.5 b + 0.5 x - 40 = 0.6 b + 0.4 x - 20 • =>b = 40, x = 240 • Base wage = 1600, bonus = 56,000 • Your profit = 400
Therefore… If the action can be specified in a contract, killing level was optimal and your profit = 10,400
Suggested Exercises • Kreps, Problem 19.3 • Answers to the above will be posted in my website on Friday 15th • For a limited time, probably until early August • You can pick up your homework 11 anytime at my office from 15th (I hope). Answers are posted asap.
Summary of the Course • Optimization • Marginal X, supply, demand, price discrimination … • Market equilibrium • Efficiency, surplus • Externalities • Risk and expected utility • Hidden information (Adverse selection) • Moral hazard
Towards the final… • 1 sheet of A4-size paper, EJ, JE dictionary (incl. electronic ones) • You all did fine in homeworks • Review concepts • Review computations • But it will not be very complex (like opening a root) • What is important?
