# Econ Analysis Paper

Topics: Game theory, Optimization, Wage Pages: 7 (1281 words) Published: April 24, 2013
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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...