Exercise 2 Suppose there are N countries. In this exercise we extend the Krugman (1980) model to N countries, derive a gravity equation, and solve for the number of products produced in equilibrium. There are iceberg trade costs 7;; in order to ship from country to country j (i.e. one needs to ship 7;; > 1 units of the good from to j in order for one unit to arrive in j). Preferences are given by N Uj = E Bijcis ( w ) "odw where Oj is a parameter such that E- So bij (w) dw = 1. Technology is such that li = dit Bi> Tijlij, 1=1 where ij = CijLj. Utility maximization yields the following demand function for the individual con- sumer (note that there are L; consumers in country j): Pij wj Cij Oij Pi where Pi = Eil So eqp ; dw denotes the aggregate price index. 1. Consider a firm located in country i. Derive its profit maximizing price in order to sell to country j. Label this price pij. Note that each firm is small compared to the overall economy, hence the firm's choice does not affect the price index, Pj 2. Plug this optimal price into the demand function and then both the demand function and the optimal price into this expression for the aggregate trade flows from i to j, Xij: Xij = MiPij Cij Lj, where Mi denotes the number of firms producing in country i. 3. Good market clearing and zero profits imply that wiLi = E,=1 Xij holds. Plug in your solution for Xij derived above and solve for Mi (.", Biwi). Plug this expression into Xij. This will lead to a gravity equation