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Lecture 2

May 24, 2011

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Other ways to provide incentives
Distortions Self Enforcing Contracts Promotions Efficiency Wages Career Concerns Non-Monetary Incentives Teams- Free Riding

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Distortions
So adding risk is one way to get b < 1. However, empirical evidence about this seems mixed. Let’s consider another story-distortions in performance measures. An implicit assumption so far. Firm can write a contract on y (output in our simple model but can be thought of as benefit to the firm). In other words when y is realized there is a court that can verify y and make sure that the firm pays the worker what the wage contract specifies. Reasonable for stock price. Courts can verify stock prices for publicly traded companies at any point in time, making contracts easy to verify. e.g. a stock based compensation plan.

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Distortions

But how about privately held firms? The financial value of the firm is not known and even if it is it may not be the only factor that affects the owners benefit. Similarly consider non-profit organizations.

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Distortions-Model
Basic Elements. A risk neutral firm and a risk neutral worker. Worker can take two actions (or put in effort in two areas) a1 and a2 . Objective function for firm: E (y − w ). 1 2 1 2 Objective function for worker: E (w ) − a1 − a2 2 2 Production (Benefit) Function: y = f1 a1 + f2 a2 + y with E ( y ) = 0. This cannot be verified by a court. Performance measure: p = g1 a1 + g2 a2 + p with E ( p ) = 0. This can be verified by a court. Firms offers a take it or leave it contract of the form w = s + bp where s is the salary component and b measures the strength of incentives. Timing. Firm offers contract. Worker decides whether to participate (if not his outside option is u0 ). Worker chooses effort. Output realized. Contract enforced.

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Distortions- Efficient actions
To compute efficient levels of a1 and a2 , first compute total surplus. 1 2 1 2 1 2 1 2 Total Surplus = E (y )−E (w )+E (w )− a1 − a2 = f1 a1 +f2 a2 − a1 − a2 2 2 2 2 The efficient levels are a1 = f1 and a2 = f2 .

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Optimization- Firm
Substituting E (y ) = f1 a1 + f2 a2 and E (w ) = s + b(g1 a1 + g2 a2 ), we can write down the main elements of the firm’s optimization problem. Firm’s Objective Function : f1 a1 + f2 a2 − s − b(g1 a1 + g2 a2 ) Constraints a1 = bg1 (Incentive Compatibility Condition for action 1) a2 = bg2 (Incentive Compatibility Condition for action 2) 1 2 1 2 s + b(g1 a1 + g2 a2 ) − a1 − a2 ≥ u0 (Participation 2 2 Constraint)

The choice variables for the firm are b, s and a1 , a2 .

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Optimization- Firm
So you can rewrite the problem as
s,b,a1 ,a2

Max

f1 a1 + f2 a2 − s − b(g1 a1 + g2 a2 )

Subject to a1 = bg1 a2 = bg2 1 2 1 2 s + b(g1 a1 + g2 a2 ) − a1 − a2 ≥ u0 2 2

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Optimization- Firm

Solving this problem involves a series of substitutions.
1

2

From the participation constraint, we have 1 2 1 2 s + b(g1 a1 + g2 a2 ) = a1 + a2 + u0 . Substitute this in 2 2 the objective function. Substitute the incentive compatibility conditions into the objective function

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Optimization- Firm

So you can rewrite the problem as 1 1 2 2 Max f1 bg1 + f2 bg2 − b 2 g1 − b 2 g2 − u0 b 2 2

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Optimization- Firm
The first order conditions give us
2 2 f1 g1 + f2 g2 − b(g1 + g2 ) = 0

This implies b= f1 g1 + f2 g2 2 2 g1 + g2

Also using the formula for inner products this can be rewritten as b= f12 + f22 2 2 g1 + g2

cos(θ)

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Measuring Distortions

Distortions depend on two things. Alignment Scale

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Applications

Do correlations between y and p matter? Accounting measures and stock price. Job training programmes. Tradeoff between risk and distortion

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Trust Game
Consider the following setting. A firm and a worker The worker can trust that the firm will compensate him and exert effort (TRUST) or he can not trust the firm to compensate...
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