# Holmstrom (1999) - Managerial Incentive Problems

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Has article title | Managerial Incentive Problems |

Has author | Holmstrom |

Has year | 1999 |

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- This page is referenced in BPP Field Exam Papers

## Contents

## Reference(s)

Holmstrom B., (1999) "Managerial Incentive Problems: A Dynamic Perspective," Review of Economic Studies, 66(1): 169-182 pdf

## Abstract

The paper studies how a person's concern for a future career may influence his or her incentives to put in effort or make decisions on the job. In the model, the person's productive abilities are revealed over time through observations of performance. There are no explicit output-contingent contracts, but since the wage in each period is based on expected output and expected output depends on assessed ability, an "implicit contract" links today's performance to future wages. An incentive problem arises from the person's ability and desire to influence the learning process, and therefore the wage process, by taking unobserved actions that affect today's performance. The fundamental incongruity in preferences is between the individual's concern for human capital returns and the firm's concern for financial returns. The two need be only weakly related. It is shown that career motives can be beneficial as well as detrimental, depending on how well the two kinds of capital returns are aligned.

## The Holmstrom Career Concerns Model

This model features:

- Moral Hazard
- Infinite periods
- Rational Expectation Equilibrium (i.e. Perfect Bayesian Nash)

### Quick Summary

Holmstrom (1999/1983) models a claim from Fama (1980) that market forces will provide implicit incentives for agents to work (through their 'career concerns'), and that explicit incentives, through contracts, may not be needed.

The Holmstrom model assumes:

- Additively and time seperable utility
- No contracts (no way to link past performance to wages directly); wages are based on the prior period's output and paid in advance
- Private costs with Inada conditions
- Randomly drawn managerial ability is private, leads to asymmetric information
- Output is observed and is the sum of ability, effort and noise
- Infinately lived agents
- A competitive labour market

How to solve:

- Competitive labour market bids up to the expected output that a manager will provide
- In equilibrium the beliefs of the labour market are correct
- Use the conditional normal distribution equation to solve the Bayesian updating

Key results:

- The labour market acts as an indirect mechanism for linking past-performance to wages
- Career concerns generally do not provide first best incentives, they over and/or under shoot
- Career concerns (i.e. the incentive effects arising from career concerns) are strongest at the start of the career and weakest at the end

### Set-up

Utility of the Manager is given by:

[math]U(w,a)=\sum_{t=1}^{\infty }\beta ^{t-1}[w_{t}-g(a_{t})]\,[/math]

where: [math]\beta\,[/math] is the discount factor, [math]w_{t}\,[/math] are wages, [math]g(\cdot)\,[/math] is the cost of effort, and [math]a_{t}\,[/math] is the effort.

Output is given by:

[math]y_{t}=\eta + a_{t} +\varepsilon_{t}\,[/math]

where [math]\varepsilon\,[/math] is noise and [math]\eta\,[/math] is the ability of the manager, such that [math]\varepsilon _{t}\sim N(0,\frac{1}{h_{\varepsilon }})\,[/math], and [math]\eta \sim N(m_{1},\frac{1}{h_{1}})\,[/math].

The market will set wages:

[math]w_{t}(y^{t-1})=\mathbb{E}[y_{t}|y^{t-1}]=\mathbb{E}[\eta |y^{t-1}]+a_{t}(y^{t-1})\,[/math]

The manager will best respond by choice an effort sequence:

[math]\underset{\{a_{t}(y^{t-1})\}_{t=1}^{\infty }}{\max }\;\sum_{t=1}^{\infty}\beta ^{t-1}[\mathbb{E}w_{t}(y^{t-1})-\mathbb{E}g(a_{t}(y^{t-1}))]\,[/math]

### Two Period Model

Wages are paid in advance so in the second period the agent exerts no effort. In equilbrium [math]a_{1}^{*}\,[/math] is correctly anticipated.

The market observes [math]z_{1}\equiv \eta +\varepsilon _{1}=y_{1}-a_{1}^{*}\,[/math] and uses this to form its conditional expectation [math]\mathbb{E}[\eta|z_{1}]\,[/math]. Given the assumption of normality we can use the conditional normal equation to give:

- [math][\eta |z_{1}]\sim N(m_{2},\frac{1}{h_{2}})\,[/math]

where

- [math]m_{2}(z_{1})=\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot z_{1}\,[/math]

- [math]h_{2}=h_{1}+h_{\varepsilon }\,[/math]

In equilibrium the beliefs are correct and the wage is bid up to the expected output:

- [math]w_{1}=Ey_{1}=E[\eta +a_{1}^{*}+\varepsilon _{1}]=m_{1}+a_{1}^{*}\,[/math]:

Likewise:

- [math]w_{2}(y_{1}) = m_{2}(z_{1})+a_{2}^{*}\,[/math]

where [math]a_{2}^{*}=0\,[/math]

From the manager's perspective the second period wage is in expectation:

- [math]\mathbb{E}[w_{2}(y_{1})] = \mathbb{E}[m_{2}(z_{1})] = \frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot\underset{\mathbb{E}z_{1}}{\underbrace{(\overset{\mathbb{E}y_{1}}{\overbrace{m_{1}+a_{1}}}-a_{1}^{*})}}\,[/math]

Putting this into the utility function:

- [math]\underset{a_{1}}{\max }\;w_{1}-g(a_{1})+\beta (w_{2}-g(a_{2}))\,[/math]

- [math]\therefore \underset{a_{1}}{\max }\;m_{1}+a_{1}^{*}-g(a_{1})+\beta \cdot \left[\underset{\mathbb{E}[w_{2}(y_{1})]}{\underbrace{\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot(m_{1}+a_{1}-a_{1}^{*})}}\right] \,[/math]

- [math]g^{\prime }(a_{1}^{*})=\beta \cdot \frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\in (0,1)\,[/math]

As [math]g^{\prime }(a_{1}^{*}) \lt 1\,[/math], it must be the case that [math]a_{1}^{*} \lt a^{FB}\,[/math]. Likewise if the agent lives for [math]T\,[/math] periods then [math]a_{T}^{*} = 0\,[/math], so effort declines from the first period onwards, but some effort is exerted in the first period and it is increasing in [math]\beta\,[/math] (as the future becomes more important). The manager exerts a higher effort in the first period because he knows that this be attributed to ability in the second period and so result in a higher wage, but in a rational expectations equilibrium this effort is anticipated by the market and the manager is forced into making it because otherwise the market will downgrade his ability.

### Infinite Horizon with fixed ability

First we define:

- [math]z_{t}\equiv \eta +\varepsilon _{t}=y_{t}-a_{t}^{*}(y^{t-1})\,[/math]

Then we apply the Bayesian updating as before:

- [math][\eta |z^{t-1}]\sim N(m_{t},\frac{1}{h_{t}})\,[/math]

To give:

- [math]m_{t}(z^{t-1})=\frac{h_{1}}{h_{1}+(t-1)h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+(t-1)h_{\varepsilon }}\cdot \sum_{s=1}^{t-1}z_{s}\,[/math]

and

- [math]h_{t}=h_{1}+(t-1)h_{\varepsilon}\,[/math]

Noting that [math]h_{t} \rightarrow \infty\,[/math] which implies that as time progresses the market gets an ever better estimate of the manager's ability and so the manager's wages [math]a_{t}^{*}(y^{t-1}) \rightarrow 0\,[/math] (see below).

The manager's ex-ante expected wage is:

- [math]\mathbb{E}[w_{t}(y^{t-1})]=\frac{h_{1}}{h_{t}}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{t}}\cdot \sum_{s=1}^{t-1}\underset{\mathbb{E}z_{s}}{\underbrace{[\overset{\mathbb{E}y_{s}}{\overbrace{m_{1}+a_{s}}}-\mathbb{E}a_{s}^{*}(y^{s-1})]}+\mathbb{E}a_{t}^{*}(y^{t-1})}\,[/math]

Which can be substituted into the manager's utility function:

- [math]\underset{\{a_{t}(y^{t-1})\}_{t=1}^{\infty }}{\max }\;\sum_{t=1}^{\infty}\beta ^{t-1}[\mathbb{E}w_{t}(y^{t-1})-\mathbb{E}g(a_{t}(y^{t-1}))]\,[/math]

Which is solved by a first order condition to give:

- [math]g^{\prime }(a_{t}^{*})=\sum_{s=t}^{\infty }\beta ^{s-t}\cdot \alpha_{s}\equiv \gamma _{t}\,[/math]

where [math]\alpha _{s}\equiv \frac{h_{\varepsilon }}{h_{s}}\,[/math]

So early in the manager's career he will work hard (though possibly still below first best, depending on the parameterization), and this work 'ethic' will tend to zero as his career progresses.

### Infinite Horizon with varying ability

For incentives not to disappear there must always be some uncertainty about the manager's ability, so now suppose:

- [math]\eta _{t+1}=\eta _{t}+\delta _{t}\,[/math]

where [math]\delta _{t}\sim N(0,\frac{1}{h_{\delta}})\,[/math]

Bayesian updating on the mean is as before:

- [math]m_{t+1}=\mu _{t}m_{t}+(1-\mu _{t})z_{t}\,[/math]

where [math]\mu _{t}=\frac{h_{t}}{h_{t}+h_{\varepsilon}}\,[/math]

However, Bayesian updating on the variance (precision) is different:

- [math]h_{t+1}=\frac{(h_{t}+h_{\varepsilon })h_{\delta }}{h_{t}+h_{\varepsilon}+h_{\delta}}\,[/math]

Essentially the shocks prevent the market from learning the true ability.

Looking at how the variance changes over time we take:

- [math]\frac{\partial h_{t+1}}{\partial h_{t}} = \frac{h_{\delta }^{2}}{(h_{t}+h_{\varepsilon }+h_{\delta })^{2}}\in (0,1)\,[/math]

So must conclude that the variance tends to a steady state and not to zero. This in turn leads to steady state effort, which we can solve for by equating the marginal benefit to the marginal cost of a change:

- [math]g^{\prime }(a^{*})=\beta (1-\mu ^{*})+\beta ^{2}\mu ^{*}(1-\mu^{*})+\beta ^{3}(\mu ^{*})^{2}(1-\mu ^{*})+\cdot \cdot \cdot=\frac{\beta (1-\mu ^{*})}{1-\beta \mu ^{*}}\,[/math]

Which in turn leads to Holmstrom's proposition 1 that the stationary effort is [math]a^{*}\leq a^{FB}\,[/math] and only equal to first best if [math]\beta =1,\,\frac{1}{h_{\varepsilon }}\gt 0\,[/math], and [math]\frac{1}{h_{\delta }}\gt 0\,[/math]