Problem 3 As part of a marketing campaign, a firm decides to hire 100 promoters who will have to give away advertising fliers from morning till evening to people passing by. Promoters will be assigned various spots I the city for the whole day. In order to make sure that promoters are not cheating (not giving away fliers in their assigned locations), the firm will send an incognito controller whose task will be to visit some of the spots (randomly chosen) in which promoters should be giving away fliers during the day. Promoters are not familiar with the controller's personality, so there is no way for them to determine whether they are being monitored or not. Assume also that the controller works with 100% precision, so that if the controller visits one of the designated promoter locations, she will know with complete confidence whether the promoter in that location is actually working or shirking in other words, there are no ways for the promoters to outsmart the controller). Additionally, to make the calculations simpler, we will be assuming that each promoter has only two options: either to work as expected through the whole day or abstain from work completely (i.e., we will be assuming that the promoter cannot choose to give fliers away during only some part of the day, while being absent from the designated location during the rest of the day). The firm's utility depends on promoters' efforts and on the remuneration payable to them: Vr = x-w, where Vi is the utility the firm receives from a promoter, x is the promoter's actual level of effort (taking only the values of 0 or 1), w is the fixed amount of remuneration a promoter will get to receive on the next day after the campaign. Each promoter's utility is given by Vo = vw, and their disutility from work is Cp = x-. For simplicity, we will be assuming that the promoters' alternative utility is zero (which is more than realistic since it is usually implies that they use their spare time to the campaign job). Task: write down the Lagrangian for the problem.

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