Government-Mandated Hiring Freeze. Echoing events in the U.S. from early 2017, suppose the government has set an economy-wide "hiring freeze." Operationally, the "hiring freeze" means that there is an upper limit V of job vacancies that firms can advertise in their recruiting process. More precisely, because the upper limit is much smaller than the profit-maximizing value of job vacancies, firms will be subject to a "hiring freeze constraint" when determining profit-maximizing values for vacancies v and labor demand nº. The Lagrangian for the firm's profit-maximization is A-f(n)-w-n-o-v+μ (qFIND .v-n°) +2.(v-v), hiring freeze constaint in which stands for the Lagrange multiplier for the "hiring freeze constraint." All other notation is exactly the same as in Chapters 27, 28, and 29. For the sake of simplicity, there are no "long- lasting jobs" throughout this question - i.e., the job turnover parameter is fixed at p=1. a. Based on the Lagrange function stated above, compute the first-order conditions with respect to nº and v. Clearly display each first-order condition by drawing a box around it. b. Based on the pair of first-order conditions from part a, rewrite them in a way so that your final solution is @= in which the term in ellipsis ("...") on the right-hand side is left to you to determine. Clearly display the final solution by drawing a box around it, and clearly and carefully provide important algebraic steps/logic that lead to the solution. (Note: in the final solution, the term in ellipsis ("...") on the right-hand side should not include the Lagrange multiplier , but may include the Lagrange multiplier 2.)
Government-Mandated Hiring Freeze. Echoing events in the U.S. from early 2017, suppose the government has set an economy-wide "hiring freeze." Operationally, the "hiring freeze" means that there is an upper limit V of job vacancies that firms can advertise in their recruiting process. More precisely, because the upper limit is much smaller than the profit-maximizing value of job vacancies, firms will be subject to a "hiring freeze constraint" when determining profit-maximizing values for vacancies v and labor demand nº. The Lagrangian for the firm's profit-maximization is A-f(n)-w-n-o-v+μ (qFIND .v-n°) +2.(v-v), hiring freeze constaint in which stands for the Lagrange multiplier for the "hiring freeze constraint." All other notation is exactly the same as in Chapters 27, 28, and 29. For the sake of simplicity, there are no "long- lasting jobs" throughout this question - i.e., the job turnover parameter is fixed at p=1. a. Based on the Lagrange function stated above, compute the first-order conditions with respect to nº and v. Clearly display each first-order condition by drawing a box around it. b. Based on the pair of first-order conditions from part a, rewrite them in a way so that your final solution is @= in which the term in ellipsis ("...") on the right-hand side is left to you to determine. Clearly display the final solution by drawing a box around it, and clearly and carefully provide important algebraic steps/logic that lead to the solution. (Note: in the final solution, the term in ellipsis ("...") on the right-hand side should not include the Lagrange multiplier , but may include the Lagrange multiplier 2.)
Chapter10: Cost Functions
Section: Chapter Questions
Problem 10.10P
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Transcribed Image Text:Government-Mandated Hiring Freeze. Echoing events in the U.S. from early 2017, suppose the
government has set an economy-wide "hiring freeze." Operationally, the "hiring freeze" means that
there is an upper limit V of job vacancies that firms can advertise in their recruiting process. More
precisely, because the upper limit v is much smaller than the profit-maximizing value of job
vacancies, firms will be subject to a “hiring freeze constraint" when determining profit-maximizing
values for vacancies v and labor demand nº.
The Lagrangian for the firm's profit-maximization is
A f(n°)-w•n° –@•v+µ' (q°
FIND
v-nº)+2' · (V-v),
hiring
freeze
constaint
in which 2' stands for the Lagrange multiplier for the "hiring freeze constraint." All other notation
is exactly the same as in Chapters 27, 28, and 29. For the sake of simplicity, there are no "long-
lasting jobs" throughout this question – i.e., the job turnover parameter is fixed at p=1.
a. Based on the Lagrange function stated above, compute the first-order conditions with respect to
nº and v. Clearly display each first-order condition by drawing a box around it.
b. Based on the pair of first-order conditions from part a, rewrite them in a way so that your final
solution is
0 =...
in which the term in ellipsis (“...") on the right-hand side is left to you to determine. Clearly
display the final solution by drawing a box around it, and clearly and carefully provide
important algebraic steps/logic that lead to the solution. (Note: in the final solution, the term in
ellipsis (“...") on the right-hand side should not include the Lagrange multiplier u', but may
include the Lagrange multiplier a' .)
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