## Symmetric Melitz-Ottaviano Model

This problem is from Doireann Fitzgerald. It showed up on the trade prelims in 2016, 2017 (fall), and 2019

### Setup

Start with a closed economy population by $L$ identical households, each of which supplies $1$ unit of labhor inelastically. Household $i$'s preferences are: $U_i = c_{i0} + \alpha \sum_{j=1}^N c_{ij} - \frac{\beta}{2} \sum_{j=1}^N (c_{ij})^2 - \frac{\gamma}{2} \left( \sum_{j=1}^N c_{ij} \right)^2$ where good $0$ is a homogenous numeraire good and $j=1,...,N$ are differentiated goods. The number $N$ of differentiated goods in this economy is endogenous. The labor market is perfectly competitive. The numeraire good is produced under constant returns to scale at unit cost and sold in a perfectly competitive market. This implies $w=1$. In order to produce a differentiated good, the firm must incur and upfront cost given by $Sw = s$. After that, production is constant returns to scale, and each unit of labor produces $1 / \sigma$ units of the good, so marginal cost is given by $\sigma$ for all firms. Assume monopolistic competition in the differentiated goods sector. That is, each firm chooses its price to maximize profit taking as given the behavior or all other firms.

### Problem (b): Consumer maximization problem and FOC.

Set up the maximization problem of the consumer. What is the first order condition for household $i$'s consumption of differentiated good $h$ (ie FOC with respect to $c_{ih}$? You can assume there is positive consumption of the numeraire, and $N > 0$.

### Problem (c): Sum over differentiated goods.

Sum this condition across all $h = 1,...,N$ and use the notation $\overline{p} = (1/N) \sum_{h=1}^N p_h$ to obtain an expression for $\sum_{h=1}^N c_{ih}$ in terms of $N$, $\overline{p}$ and preference parameters.

### Problem (d): Derive demand function.

Use the solution to part (c) to express $c_{ih}$, household $i$'s demand for good $h$ as a function of $p_h$, $N$, $\overline{p}$ and preference parameters. What is the market demand $q_h$ (remember that there are $L$ agents)?

### Problem (e): Firm's profit and pricing rule.

Now write down the profits of differentiated goods firm $h$ as a function of $p_h$, marginal cost $\sigma$, $L$, $N$, $\overline{p}$ and preference parameters. Assuming that firm $h$ takes $\overline{p}$ as given (ie monopolistic competition), what is the first order condition with respect to $p_h$? Use this to write the firm's optimal $p_h$ as a function of $\sigma$, $N$, $\overline{p}$, and preference parameters.

### Problem (f): Price index and profit.

Now use the fact that all differentiated goods are identical so $p_h=\overline{p}$ to write $\overline{p}$ as a function of $\sigma$, $N$ and preference parameters. Use this to write maximized profit as a function of $\sigma$, $L$, $N$, and preference parameters.

### Problem (g): Number of firms with free entry.

What is the free entry condition? Rearrange it to obtain an expression for $N$ as a fucntion of $\sigma$, $L$, $S$ and preference parameters. You can assume $\alpha - \sigma > 0$.

### Problem (h): Solve for quantities produced.

Use the market demand together with teh expressions for $\overline{p}$ and $N$ to solve for $q_h$, production of good $h$, as a function of $\sigma$, $L$, $S$ and preference parameters.

### Problem (i): Pareto Optimality

Is the competitive equilibrium in this closed economy Pareto optimal?

### Problem (j): Opening to trade.

Suppose the competitive economy opens up to (costless) trade with an identical economy with $L^*$ households. What happens to the set of varieties the consumer has access to? What happens to $q_h$? What happens to the set of varieties produced in the home country? Give some intuition for what happens. What happens to welfare?

### Problem (k): Alternate Preferences.

How would you answers to (j) differ if preferences were CES as in Krugman(1980)?

Solution