Note: This problem requires the use of Excel or similar spreadsheet software. It is potentially more timeconsuming than other problems worth similar point values. I urge you not to attempt it unless you have completed the rest of the exam. Consider the health insurance market. Suppose health insurance plans are denoted by j and are differentiated along two dimensions: price, pj, and generosity, gj. In each period consumers may purchase one of J plans or no plan at all (we call this the \"outside good\"). The utility that consumer i gets from purchasing plan j is given by: Uij = 04R; + gj + Eij' In this equation, a is the price sensitivity (and so must be a negative number) and B repre- sents how much consumers care about the generosity of the plan. eij is an individualplan Specic unobservable term. Note that EU is the only term in the equation with an i subscript, which means that individuals only differ in their 6,-3- draw. Suppose that the utility of the outside good is normalized to zero and that 6,]- is indepen dently and identically drawn across individuals and plans according to the TypeI Emtreme Value Distribution. The details of this distribution aren't important for the purposes of this question. What is important is that under that assumption, it can be shown that the mar ket share of plan j, sj is given by the legit share fmriction1 (thus this differentiated products demand system is often called \"logit demand\"), is given by S_ _ exmapj + 691-) .7 _ J ' 1 + 21m exp(apk + 69k) '1-4521 a\\ 1w - n d'"! On the supply side, there are two rms, each of which offers one plan, so j E {1, 2}. The marginal cost of each plan is constant and is a quadratic function of the generosity,2 given by mcj = 703' + 7133? + 723992-- In this problem, we will be solving this model numerically. We will use the following parameters: 05 = 1,5 = 1:701 = 0,711 = 0-75,\"le = {15,702 = 025,712 = 5,722 = 25- a. (1 point) By visual examination, it looks like rm 2 has to pay less for generosity than rm 1 does. Let's suppose that rm 1 sets g1 : 1 and rm 2 sets gg : 2. First, calculate the marginal costs for each rm. b. (4 points) Suppose that rm 2 sets a price equal to twice their marginal cost. What is the protmaximizing price for rm 1? What are the prots for both rms? Hint: This is very difcult to determine analytically. Instead, I recommend putting the prot function into an Excel worksheet, and using the \"Solver\" tool under the \"Data\" menu (you may have to rst add it in to your installation of Excel, see https : / / support .microsoft . com/ en-us/ office/load-the-solver-add-in-in-excel-61292613c-d53b-46b4872c-e24772f078ca). This is a tool that allows you to numerically maximize or minimize functions of one or more variables. Similar tools exist for Google Sheets and other spreadsheet software. To make sure that you have the spreadsheet programmed correctly, given the parameters above, when p1 = 2, 91 = 1,p2 = 4, 92 = 2, you should get an a 0.183546 and 7T2 z 0.157554. One way to nd an equilibrium is by repeatedly iterating backandforth between best reSponses i.e. maximize prots for one rm taking the other rm's price as given, then maximize prots for the other rm taking its competitor's price as given, and so on. In general, with numerical algorithms such as this, it is necessary to dene a \"convergence criterion.\" In other words, we can't ever get the best responses to exactly line up because there will always be a small approximation error. Let pt1 be the price we nd for rm 1 at step t of the algorithm and pi\" be the price we nd for rm 1 in the next step. We will say this algorithm converges if hog1 pgl