where p,- denotes the price of good 1', and I denotes income. The country is populated by L agents, and in equilibrium total consumption of a good must be equal to total production of that good: 11:,- = Leg. (8%) (b) (60 By solving the consumer's utility maximization problem, show that the (inverse) residual demand that the rm producing good i faces is described by the following equation: He\") m where A is the Lagrange multiplier associated to the budget constraint. Plot the residual demand function (1) on the ($1,111,) plane. Does the residual demand function depend on the number of rms? And on the prices charged by other rms? To answer this question, you can solve for A by plugging (1) into the budget constraint. Labor is the only factor of production, hence production costs are expressed in units of labor. Each rm 3' produces with the same technology, described by the following cost function: 5(a) = a + 512:,- where 1(a) are the units of labor rm 1' needs to produce 3:,- units of good 3', ,6 > I] denotes the marginal cost, and a > 0 denotes the xed cost of production. Both a and ,6 are 60113111011 across rms. Write down the rm's prot maximization problem. Solve the rm's prot maximization problem, 229., compute the optimal quantity produced I: and the optimal selling price p:. Does the optimal price depend on the number of rms N? Does the optimal price diifer across rms? Why? In equilibrium full employment must hold, \"5.6. all the labor force of the country must be employed by the rms. Compute the longrun equilibrium number of rms N *. Describe how N * depends on the parameters of the model. Provide economic intuition for your