## (Solved) Econ 450 Problem Set #5 Due: April 21, 2017 Professor Rasmus Lentz Department of Economics University of Wisconsin-Madison Problem 1 - The Shirking...

Consider the following wage search model. Determine steady state unemployment rate using logic of bathrub model and then use answer to calculate unemployment rate. Question is attached

Econ 450 Problem Set #5
Due: April 21, 2017
Professor Rasmus Lentz
Department of Economics
University of Wisconsinâ€“Madison Problem 1 â€“ The Shirking Model
Consider a worker with utility function over wages and effort, u(w, e) = w âˆ’ e. A firm is considering
how to optimally set wages so as to induce the worker to deliver a given effort level e = eÂ¯. Effort
is normally unobserved but with monitoring probability Î³ âˆˆ (0, 1), the firm can directly observe
the workerâ€™s effort. If the firm monitors the worker and it finds that the worker is shirking, it
is committed to firing the worker. The firmâ€™s revenue is directly tied to the workerâ€™s effort in a
non-verifiable way (that is, it is deterministic and observable to the firm, but it cannot use revenue
Â¯
observations to make assertions about the workerâ€™s effort in a court of law). Specifically, R(Â¯
e) = R,
and R(0) = 0. The firmâ€™s profits are given by Î (Î³, w|e) = R(e) âˆ’ w âˆ’ c(Î³), where c(Â·) is the
monitoring cost. Define c(Î³) = Î³.
Define wâˆ— as the market clearing wage for which unemployment is zero. Unemployment in the
economy is going to be a positive function of the difference between the actual market wage w
Ë† and
âˆ—
Ë†
wâˆ— , u(w)
Ë† = wâˆ’w
.
w
Ë†
The workerâ€™s expected utility can be written up contingent on the effort choice and the firmâ€™s
wage offer w,
u(w, eÂ¯) = w âˆ’ eÂ¯

 
u(w, 0) = (1 âˆ’ Î³)w + Î³ u(w)b
Ë† + 1 âˆ’ u(w)
Ë† w
Ë† ,
where the last equation reflects the probability of being fired in case of being caught shirking in
which case we are stylistically capturing the threat of unemployment by saying that the worker will
then be unemployed with probability u and receive benefits b, or find a new job with probability
(1 âˆ’ u) and then receive the market wage w.
Ë†
1. Determine the minimum firm wage requirement w
Ëœ such that the workerâ€™s incentive compatibility constraint (IC) is satisfied, u(w,
Ëœ eÂ¯) â‰¥ u(w,
Ëœ 0).
2. Determine the optimal monitoring choice given that the firm sets the wage equal to w,
Ëœ Î³ âˆ— (w).
Ëœ
3. Now, impose the condition that in equilibrium it must be that all firms set the same wage.
Consequently, the market wage must equal the firmâ€™s optimal wage choice, w
Ë† = w.
Ëœ Use the
binding (IC) constraint imposing w
Ë†=w
Ëœ and the optimal monitoring choice to characterize
the equilibrium efficiency wage w.
Ë†
Â¯ so that the above solution is indeed an equilibrium (that
4. Determine a basic condition on R
âˆ—
is, Î (Î³ (w),
Ë† w|Â¯
Ë† e) â‰¥ Î (0, 0|0).)
5. How is unemployment affected by an increase in the benefit level, b?
1 Problem 2 â€“ Search
Jane is searching for a summer job. She has surveyed the market and has determined that the
offer distribution for summer jobs is uniform with a lower bound of \$5, 000 and an upper bound of
\$20, 000. Hence, any offer in the range between \$5, 000 and \$20, 000 is equally likely, so Pr(of f er =
x) = 1/15000, for any x âˆˆ [5000, 20000]. Obtaining offers requires resources and effort, and Jane
has determined that the cost of obtaining an offer is equivalent to a monetary payment of c = 500
dollars. Jane can draw as many offers as she pleases. Each one costs her 500 dollars. After each
draw she decides to accept or reject the current offer. If she rejects the offer, she draws again. If
she accepts, she is done searching. Define the reservation level, R, such that all offers below R are
rejected and all offers at or above R are accepted.
1. Determine the average value of a draw from the offer distribution.
2. For a given reservation level, R, determine the expected search cost.
3. For a given reservation level, R, determine the expected value of a job conditional on it being
accepted.
4. The value of search for a given reservation level strategy is the expected value of an acceptable
job minus expected search cost. Determine the value of searching as a function of the given
reservation level, U (R). Draw U (R) in a figure with R on the horizontal axis and U (R) on
the vertical axis. Include the 45â—¦ line in the graph.
5. Determine the optimal choice of R.
6. What is the average value of a job that Jane accepts? What is the average number of offers
that Jane will draw before an acceptable offer is made?
7. Suppose that the offer distribution changes so that the lower bound is 0 dollars and the upper
bound is \$25, 000. The distribution remains uniform so all offers between 0 and 25000 are
equally likely. What is the value of the average realization from the distribution? What is
Janeâ€™s new optimal choice of R? What is the average number of offers Jane will draw before
accepting? Explain why R changes and the direction of change. Problem 3 â€“ Unemployment
Consider the following wage search model: During a period an unemployed worker receives one
wage offer with probability Î». With probability 1 âˆ’ Î», the unemployed worker does not receive
an offer. The wage offer is drawn from the wage offer distribution F (w) which is assumed to be
uniform over the interval [0,100]. Specifically this implies that the probability of receiving an offer
w
Ëœ less than or equal to some level w is Pr(w
Ëœ â‰¤ w) = w/100. During any period, an employed
worker loses her job with probability Î´. For any given reservation wage R such that unemployed
workers reject offers below R and accept offers above R, determine the steady state unemployment
rate using the logic of the bathtub model. Use your answer to calculate the unemployment rate for
Î´ = 0.017, Î» = 0.5, and R = 35. 2

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