Holmstrom (1999) - Managerial Incentive Problems - Revision history

Ed at 23:14, 29 September 2020

2020-09-29T23:14:57Z

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Ed at 16:47, 29 September 2020

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imported>Ed at 01:00, 28 April 2010

2010-04-28T01:00:59Z

imported>Ed at 01:00, 28 April 2010

2010-04-28T01:00:50Z

imported>Ed: /* Infinite Horizon with varying ability */

2010-04-28T00:59:21Z

Infinite Horizon with varying ability

imported>Ed at 00:58, 28 April 2010

2010-04-28T00:58:40Z

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 {{Article
 |Has page=Holmstrom (1999) - Managerial Incentive Problems
-|Has title=Managerial Incentive Problems
+|Has bibtex key=
 |Has author=Holmstrom
 |Has year=1999

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 |Has page=Holmstrom (1999) - Managerial Incentive Problems
 |Has title=Managerial Incentive Problems
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+|Has author=Holmstrom
 |Has year=1999
 |In journal=

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 |Has title=Managerial Incentive Problems
 |Has author=
-|Has year=
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 |In journal=
 |In volume=

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 {{Article
 |Has page=Holmstrom (1999) - Managerial Incentive Problems
-|Has title=
+|Has title=Managerial Incentive Problems
 |Has author=
 |Has year=

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 {{Article
-|Has page=
+|Has page=Holmstrom (1999) - Managerial Incentive Problems
 |Has title=
 |Has author=

← Older revision		Revision as of 16:47, 29 September 2020
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		+	{{Article
		+	\|Has page=
		+	\|Has title=
		+	\|Has author=
		+	\|Has year=
		+	\|In journal=
		+	\|In volume=
		+	\|In number=
		+	\|Has pages=
		+	\|Has publisher=
		+	}}
	*This page is referenced in [[BPP Field Exam Papers]]		*This page is referenced in [[BPP Field Exam Papers]]

← Older revision		Revision as of 01:00, 28 April 2010
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	*This page is referenced in [[BPP Field Exam Papers]]		*This page is referenced in [[BPP Field Exam Papers]]
		+

	==Reference(s)==		==Reference(s)==

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 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>
+:<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 }}>0\,</math>, and <math>$\frac{1}{h_{\delta }}>0\,</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 }}>0\,</math>, and <math>\frac{1}{h_{\delta }}>0\,</math>

← Older revision		Revision as of 00:58, 28 April 2010
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	:<math>g^{\prime }(a_{1}^{*})=\beta \cdot \frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\in (0,1)\,</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}^{}) < 1\,</math>, it must be the case that <math>a_{1}^{} < 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.	+	As <math>g^{\prime }(a_{1}^{}) < 1\,</math>, it must be the case that <math>a_{1}^{} < 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 }}>0\,</math>, and <math>$\frac{1}{h_{\delta }}>0\,</math>

← Older revision		Revision as of 01:00, 28 April 2010
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		+	*This page is referenced in [[BPP Field Exam Papers]]
		+
	==Reference(s)==		==Reference(s)==
	Holmstrom B., (1999) "Managerial Incentive Problems: A Dynamic Perspective," Review of Economic Studies, 66(1): 169-182 [http://www.edegan.com/pdfs/Holmstrom%20(1999)%20-%20Managerial%20Incentive%20Problems.pdf pdf]		Holmstrom B., (1999) "Managerial Incentive Problems: A Dynamic Perspective," Review of Economic Studies, 66(1): 169-182 [http://www.edegan.com/pdfs/Holmstrom%20(1999)%20-%20Managerial%20Incentive%20Problems.pdf pdf]