Question

Consider a company that can hire N workers. Each of them may work for the company or remain unemployed, depending on the number of jobs offered by the company. All workers are identical. The company sells its product in a competitive market. The production function of the firm is ? = ? (?, ?), where ? is the number of workers and ? is random shock. The production ? takes the values {?1 , ?2 , . . . , ??} and the probability that the production takes the value ?? is ?? (?). At the moment of signing the contract, neither the workers nor the company know the state of nature, but ex-post everybody knows the result. In general, the wage of each worker depends on the number of workers that are employed by the company and on the result of the random shock: if ? workers are employed and the result obtained is ?? we denote by ?? (?) the salary obtained by each worker. The company is risk neutral and maximizes expected net benefit ?(?) − ??(?), where ?(?) and ?(?) are (respectively) the expected result and the expected wage, whenever n workers are hired. The utility function of the workers is of the form ?(?), with ?′(?) > 0, ?′′(?) < 0, and where ? is the salary of the worker if he is hired or is the unemployment subsidy ? if the worker does not work.

(a) Find the wage offered by the company, depending on the number of employed workers. (That is, fix the number of workers employed and determine the optimum salary for that number.) (b) Using the result obtained in (a), determine the optimum level of employment offered by the company and how does it change, depending on the unemployment subsidy.

Answer 1

xpected net benefit conperinon) will he geuo, (pecefect fon, &) - ns = 0 fonie) - u s Hence, the no. of workers that will be employed will depend upon s, ie. the unemployment subsidy It is inveersely related with s le as s decreases employment wall so increase.

Expected oueput will be , p, ch) + р, си) 3, 4 - - - - -(СИ) XR Expected net benefit will be руси) –ир, си) у си) ] - , pcИ) — и расизм - - - - - - | Ак руси) – ирси) wе си ] Now , but x = fcn, E) den, t) руси) – ирүси) , (0)] - - - - - - - Jenit) per) - мрк си) да си | chit) Lруси) + раз си ) + - -- РЕСИ) - и руси) w1 СИ) + руси) у си) + - - + - - - pk си) w k(и) | from we know that, P2ch) w2ch) + - + Pkih) w kla)=s Pich) w,ch) + щемсе, som of all probabilities is ! ] - Jenie) (1) - ns som of all mobabilties is

Given the from Weres N workers. The production function, of the firm is xa denie) n px, 2, 3 , - - - - x k} the probability, the produchon funchon will take value xi is Pach) of n workers one employed, and the result obtained is ni, salany miu be denoted my wich) Given the urinly function now), with uw) o and u" (W220 unemployment subsidy is sy of the workers as uneniployed The worked will be indifferent between working and not working, if the following condition holds ie, wech) s – equanon o since initially workers don't know the exact nature of production result, so expected wages. nâu be, pich Wilh) + P₂ Uh) W211) + - - - - + pk Wp Ch) As wich) wau be the wage, it is the output and se nau be the output, only with I probability piens similarly we can deduce for the othen so from equanon we can say that, Pichs wiens + Ren) w2ch) + - - - & Pk wach) s -

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