PCP MicroeconomicsSession 12

Takako Fujiwara-Greve

Homework 10

Real-world Externalities Noise, pirating music CDs, blocking sunlight of n

eighbors, land price near a new railway Recycling may be positive externalities to future g

eneration 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

Kreps, 15.2

Jo Jack Jim

u(40k) 1 212.1320344 300

u(0) 0.49 70.71067812 223.6067977

EU 0.6175 106.0660172 242.7050983

7500 for sure 0.65 111.8033989 239.7915762

CE around 6000 6250 8905.764747

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

quit100

loaf 0.95 √9500 + 0.05 √ 245,000 = 100.421

0.75 √9500 + 0.25 √ 245,000 -10= 102.232try

hard

kills0.6√9500 + 0.4 √ 245,000 - 20= 101.091

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 contr

act) maximizes your profit?

Problem 19.1: Let’s solve! Find the cheapest (wage, bonus) to induce “try but not ha

rd” level of effort b= utility from base wage x= utility from base wage + bonus

quit100

loaf 0.95 b + 0.05 x

0.75 b + 0.25 x -10try

hard

kills0.6 b + 0.4 x - 20

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 (- 95

06.25) = 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…

induced action base wage bonus profit

loaf 10000 0 -7,000

try but not hard 9506.25 12,250 2431.25

hard work 8711.05 16,889 8533.37

killing level 1600 56,000 400

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?