(VCG) I have this problem:

Anna, Bjorn, and Christian are students living in the same kollegium, and they consider buying a smoke machine to make their kitchen parties more fun (they are the only ones in their kollegium willing to pay for it). The machine costs 600kr; if the decision is made to purchase it, then the cost is split equally between the three. Anna, Bjorn and Christian all have quasi-linear preferences over the smoke machine and private consumption (measured in kr). Both Bjorn and Christian value the smoke machine at 250kr, while Anna values the smoke machine at 80kr.. 1. Is it (Pareto) efficient for them to buy the smoke machine? 2. Suppose they decide to use a VCG mechanism and everyone reports truthfully. Do they end up buying the smoke machine? What are the total payments that each student has to make? Now, instead of being 80kr., assume that Anna's valuation is some number vA, which could be anything between 50 and 300kr 3. For which values of vA is it (Pareto) efficient to buy the smoke machine? 4. Suppose again that they decide to use a VCG mechanism and everyone reports truthfully. Derive the total payments that each students have to make, as a function of Anna's valuation VA.4. The smoke machine gets purchased when vA 2 100. Anna is pivotal when vA 0). Bjorn and Christian are pivotal when A E (100, 150) and otherwise not. Their payments are: 200 if vA > 150, Ti(VA) = 350 - VA if vA E (100, 150), O if vA