Question

Suppose that the waiter in a restaurant can either work hard (e = 3) or neglect his duties (e = 0). In addition, suppose that the revenue of the restaurant depends on the \"states of nature\" because they are beyond the control of either the agent or the principal. Formally: $\begin{aligned} R(3) &= \begin{pmatrix} H \\ L \end{pmatrix} \begin{pmatrix} probability \\ 0.7 \\ 0.3 \end{pmatrix} \\ R(0) &= \begin{pmatrix} H \\ L \end{pmatrix} \begin{pmatrix} probability \\ 0.4 \\ 0.6 \end{pmatrix} \end{aligned}$ And the utility of the waiter is denoted by: $U^* = \begin{cases} Ew - e & \text{if he devotes an e level of effort} \\ 20 & \text{if he works at another place} \end{cases}$ where Ew is the expected wage, and e is the effort exerted by the worker. a) Set up the following: i. expected wage equation of the worker if e = 3, and if e = 0. ii. Participation constraint iii. Incentive constraint b) Using the equations for participation and incentive constraints, solve for the wage if the waiter works hard ($w^H$) and if he shirks ($w^L$).

          Suppose that the waiter in a restaurant can either work hard (e = 3) or neglect his duties (e = 0).
In addition, suppose that the revenue of the restaurant depends on the \"states of nature\" because
they are beyond the control of either the agent or the principal. Formally:
$\begin{aligned}
R(3) &= \begin{pmatrix} H \\ L \end{pmatrix} \begin{pmatrix} probability \\ 0.7 \\ 0.3 \end{pmatrix} \\
R(0) &= \begin{pmatrix} H \\ L \end{pmatrix} \begin{pmatrix} probability \\ 0.4 \\ 0.6 \end{pmatrix}
\end{aligned}$
And the utility of the waiter is denoted by:
$U^* = \begin{cases} Ew - e & \text{if he devotes an e level of effort} \\ 20 & \text{if he works at another place} \end{cases}$
where Ew is the expected wage, and e is the effort exerted by the worker.
a) Set up the following:
i. expected wage equation of the worker if e = 3, and if e = 0.
ii. Participation constraint
iii. Incentive constraint
b) Using the equations for participation and incentive constraints, solve for the wage if the
waiter works hard ($w^H$) and if he shirks ($w^L$).

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Suppose that the waiter in a restaurant can either work hard (e = 3) or neglect his duties (e = 0).
In addition, suppose that the revenue of the restaurant depends on the s̈tates of natureb̈ecause
they are beyond the control of either the agent or the principal. Formally:
R(3) =
< p m a t r i x >

< p m a t r i x >

R(0) =
< p m a t r i x >

< p m a t r i x >
And the utility of the waiter is denoted by:
U^* = Ew - e if he devotes an e level of effort
20 if he works at another place
where Ew is the expected wage, and e is the effort exerted by the worker.
a) Set up the following:
i. expected wage equation of the worker if e = 3, and if e = 0.
ii. Participation constraint
iii. Incentive constraint
b) Using the equations for participation and incentive constraints, solve for the wage if the
waiter works hard (w^H) and if he shirks (w^L).

Added by Mary S.

Principles of Economics

Gregory Mankiw 8th Edition

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Suppose that the waiter in a restaurant can either work hard (e = 3) or neglect his duties (e = 0). In addition, suppose that the revenue of the restaurant depends on the states of nature because they are beyond the control of either the agent or the principal. Formally: H probability 0.7 R(3) = L probability 0.3 H probability 0.4 R(0) = L probability 0.6 And the utility of the waiter is denoted by: (Ew - e) if he devotes any level of effort U* = 20 if he works at another place, where Ew is the expected wage, and e is the effort exerted by the worker. a) Set up the following: i. Expected wage equation of the worker if e = 3 and if e = 0 ii. Participation constraint iii. Incentive constraint b) Using the equations for participation and incentive constraints, solve for the wage if the waiter works hard (w) and if he shirks (w).

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