## 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)?

## Symmetric Melitz-Ottaviano Model

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

### Setup

Different types of goods: There's a continuum of types of goods indexed from $0$ to $m$. Good $0$, the numeraire good, is homogenous. Think of it like grain. Each other good with index in $(0,m]$ is somewhat interchangeable, and produced by a monopoly. Call these the differentiated goods. You can think of these goods as different songs or different brands of clothing or something else along those lines.

Consumer's problem: Consumers have Dixit Stiglitz utility functions. They want some of the numeraire good, and some diversified bundle of the differentiated goods. The consumer's utility maximization problem is to choose $c_z$ for each $z$ in the range $[0,m]$ to solve:

$\max \; \left[[1-\alpha]\ln c_0 + {\alpha \over \rho} \ln \left( \int_0^m c_z^\rho \, dz \right)\right]$ subject to the constraints: \begin{gather} c_z\geq 0 \quad \forall z \in [0,m] \tag{NonNeg}\\ p_0 c_0 + \int_0^m p_z c_z \, dz \leq w\bar l + \pi \tag{Budget}\\ \end{gather}

In the above consumer's problem, $0 < \alpha < 1$, and $0 < \rho < 1$. $m$ represents the measure of firms, which is determined in equilibrium, and taken as given by the consumer. The measure of potential firms is fixed at $\mu>0$. As you might expect, $p_z$ represents the price of good $z$, $w$ represents the real wage, and $\pi$ represents the total dividend profits from the firms, all of which the consumer takes as given.

Firm production: Different amounts of labor $l_z$ can be used in the production of each good $z$. Labor is the only input. Good $0$ is produced using the CRS production function $\bbox[#eee8d5, 5px]{y_0=l_0}$ ,while each of the differentiated goods is produced according to the production function $\bbox[#eee8d5, 5px]{y_z(l_z) = \max \left\{ x_z \cdot [l_z-f] , 0 \right\}}$ where $x_z > 0$ represents the individual firm's productivity level, while $f > 0$ represents a fixed cost in terms of units of labor that is the same across all the monopolistic firms.

### Problem (a): Optimal Pricing Rule

Suppose that the producer of differentiated good $z$ takes the prices $p_{z^\prime}$, for $z^\prime \neq z$, as given. Solve the firm's profit maximization problem to derive an optimal pricing rule.

##### Solution:

First, we need to figure out how much of good $z$ the consumer will be willing to buy as a function of the prices. Once we know the consumer's demand function, we can use it to describe the firm's profits as a function of the prices. And once we have that information, we can find our optimal pricing rule.

###### Step 1: Finding the consumer's demand function:

Note that the budget constraint will be binding because utility is strictly increasing. Let's assume that the nonegativity constraints are nonbinding. That is, assume $c_z > 0 \, \forall z \in [0,m]$. With this assumption, we can set up a Lagrangian for the consumer's optimization problem like so: $\mathcal{L} = [1-\alpha]\ln c_0 + {\alpha \over \rho} \ln \left( \int_0^m c_z^\rho \, dz \right) - \lambda \left[ p_0 c_0 + \int_0^m p_z c_z \, dz - w\bar l - \pi \right]$ First order Conditions are: ${\partial \mathcal{L} \over \partial c_0} = 0 = {1-\alpha \over c_0} - \lambda p_0$ ${\partial \mathcal{L} \over \partial c_h} = 0 = {\alpha \over \rho}{1 \over \int_0^m c_z^\rho \, dz } \rho c_h^{\rho-1} - \lambda p_h$ $-{\partial \mathcal{L} \over \partial \lambda} = 0 = p_0 c_0 + \int_0^m p_z c_z \, dz - w\bar l - \pi$ where the FOC for $c_h$ applies for any differentiated good $h\in(0,m]$. Rearrange to get: ${1-\alpha \over c_0} = \lambda p_0$ ${\alpha c_h^{\rho-1} \over \int_0^m c_z^\rho \, dz } = \lambda p_h$ $p_0 c_0 + \int_0^m p_z c_z \, dz = w\bar l + \pi$ Further rearrange the first order conditions for consumption to get $p_0 c_0 = {1-\alpha \over \lambda}$ $p_h c_h = {\alpha c_h^\rho \over \lambda \int_0^m c_z^\rho \, dz }$ Plug these into the budget constraint: ${1-\alpha \over \lambda} + \int_0^m {\alpha c_h^\rho \over \lambda \int_0^m c_z^\rho \, dz } \, dh = w\bar l + \pi$ ${1-\alpha \over \lambda} + {\alpha \over \lambda \int_0^m c_z^\rho \, dz } \int_0^m { c_h^\rho } \, dh = w\bar l + \pi$ ${1-\alpha \over \lambda} + {\alpha \over \lambda } = w\bar l + \pi$ ${1 \over \lambda} = w \bar l + \pi$ Now that we have an explicit expression for $\lambda$, we can substitute it back into the FOC for undifferentiated consumption: $\bbox[#eee8d5, 5px]{p_0 c_0 = [1-\alpha]\cdot [ w\bar l + \pi ]}$ And this, along with the budget constraint, implies that $\int_0^m p_z c_z \, dz = \alpha \cdot [ w\bar l + \pi ]$ One last piece of the puzzle: If we divide the FOCs for two distinct differentiated goods, $h$ and $k$, we get ${\alpha c_h^{\rho-1} \int_0^m c_z^\rho \, dz \over \alpha c_k^{\rho-1} \int_0^m c_z^\rho \, dz } = {\lambda p_h \over \lambda p_k}$ ${ c_h^{\rho-1} \over c_k^{\rho-1} } = { p_h \over p_k}$ $c_k = c_h \left({p_k \over p_h} \right)^{1\over \rho - 1}$ Combine these two results to get: \begin{align} \alpha \cdot [ w\bar l + \pi ] &= \int_0^m p_z c_z \, dz \\ &= \int_0^m p_z c_h \left({p_z \over p_h} \right)^{1\over \rho - 1} \, dz \\ &= c_h \left({1 \over p_h} \right)^{1\over \rho - 1} \int_0^m p_z p_z^{1\over \rho - 1} \, dz \\ &= c_h \left({1 \over p_h} \right)^{1\over \rho - 1} \int_0^m p_z^{\rho\over \rho - 1} \, dz \\ \end{align} So we get the following equation for the consumption demand of differentiated good $h$: $\bbox[#eee8d5, 5px]{c_h = {\alpha \cdot [ w\bar l + \pi ]\cdot p_h^{1\over \rho - 1} \over \int_0^m p_z^{\rho\over \rho - 1} \, dz }}$ Or more succintly, if we denote $P\equiv \int_0^m p_z^{\rho\over \rho - 1} \, dz$, $\bbox[#eee8d5, 5px]{c_h = {\alpha \cdot [ w\bar l + \pi ]\cdot p_h^{1\over \rho - 1} \cdot {1\over P} }}$

###### Step 2: Firm's optimization problem:

From step 1, we've learned that If differentiated firm $h$ sets their price at $p_h$, then they'll sell $c_h = y_h = \alpha [ w\bar l + \pi ] p_h^{1\over \rho - 1} {1\over P}$ units of goods.

From the setup, we are told that when firm $h$ is producing non-negative quantities of goods, their output as a function of labor is given by $y_h (l_h) = x_h l_h - f$ Rearranging this, we get that the amount of labor needed to produce $y_h$ units of output is $l_h (y_h) = {y_h \over x_h} + f$

Put these together and we get that the profit for firm $h$ will be \begin{align} \pi_h &= p_h y_h - w l_h \\ &= p_h y_h - w \left[{y_h \over x_h} + f\right] \\ &= y_h \left[ p_h - {w \over x_h} \right] - f\\ &= \left[ \alpha [ w\bar l + \pi ] p_h^{1\over \rho - 1} {1\over P} \right] \cdot \left[ p_h - {w \over x_h} \right] - f\\ \end{align} The derivative of profit with respect to price is then: ${\partial \pi_h \over \partial p_h} = \alpha [ w\bar l + \pi ] {1\over P} \left[ {\rho \over \rho - 1} p_h^{1\over \rho - 1} - {w \over x_h} {1\over \rho - 1}{p_h^{{1\over \rho - 1}-1} } \right]$ Set this equal to zero and solve for p_h to get: ${\rho \over \rho - 1} p_h^{1\over \rho - 1} = {w \over x_h} {1\over \rho - 1}{p_h^{{1\over \rho - 1}-1} }$ $\rho p_h^{1\over \rho - 1} = {w \over x_h}{p_h^{{1\over \rho - 1}-1} }$ $\bbox[#eee8d5, 5px]{p_h = {w \over \rho x_h }}$ This is the optimal pricing rule for differentiated firm $h$.

### Problem (b): Define equilibrium

Suppose that good $0$ is is produced with the constant-returns production function $y_0 = l_0$. Suppose that firm productivities are distributed on the interval $x\geq 1$ according to the Pareto distribution with distribution function $F(x)=1-x^{-\gamma}$ where $\gamma > 2$ and $\gamma > {\rho \over 1-\rho}$. Also suppose the measure of potential firms is fixed at $\mu$. Define an equilibrium for this economy.

##### Solution:
An equilibrium in this economy consists of
Such that the following conditions are satisfied:
###### Markets Clear:
$\bar{l} = l_0 + \int_z=0^m l_0 \, dz$ $y_0 = c_0, \;\; y_z = c_z \;\forall z\in (0,m]$

### Problem (c): Productivity Threshold

Suppose that, in equilibrium, not all potential firms actually produce. Find an expression for the productivity of the least productive firm that produces. That is, find a productivity $\bar{x} > 1$ such that no firm with $x(z) < \bar{x}$ produces and all firms with $x(z) \geq \bar{x}$ produce. Relate the measure of firms that produce, $m$ to the measure of potential firms $\mu$ and to the cutoff $\bar{x}$.

##### Solution:

The latter part is easy. $F(\bar{x})=1-\bar{x}^{-\gamma}$ is the portion of potential firms with productivity at or below $\bar{x}$. Thus $\mu F(\bar{x})$ is the total measure of such firms. And if only firms at or above this threshold produce, then the measure $m$ or producing firms will be $\bbox[#eee8d5, 5px]{m = \mu \cdot [1- F(\bar{x})] = \mu \bar{x}^{-\gamma}}$

Actually finding this threshold $\bar{x}$ will be a bit trickier. A firm will produce iff it can make a non-negative profit by doing so. We need to find a firm's profits in equilibrium as a function of their productivity level, then the productivity threshold $\bar{x}$ will be the productivity at which the firm makes exactly 0 profit.

Recall from part (a) the following facts about a differentiated firm $h$ which produces non-zero output: $\pi_h = p_h y_h - w l_h \tag{profit}$ $l_h = {y_h \over x_h} + f \tag{labor required}$ $c_h = {\alpha \cdot [ w\bar l + \pi ]\cdot p_h^{1\over \rho - 1} \over \int_0^m p_z^{\rho\over \rho - 1} \, dz } \tag{demand}$ $p_h = {w \over \rho x_h } \tag{optimal pricing}$ And also note that $y_h = c_h$ in equilibrium for every firm which produces.

###### Step 1: Convert the above to functions of productivity.

Optimal pricing as a function of productivity is fairly straightforward: $p(x) = {w \over \rho x}$

Now use this to convert the expression for the price index: \begin{align} P &\equiv \int_0^m p_z^{\rho\over \rho - 1} \, dz \\ & = \mu \int_{\bar{x}}^\infty \bbox[border:2px dashed #eee8d5,3px]{p(x)}^{\rho\over \rho - 1} \, \bbox[border:2px dashed #eee8d5,3px]{dF(x)}\\ & = \mu \int_{\bar{x}}^\infty \bbox[border:2px dashed #eee8d5,1px]{{\left[w \over \rho x\right]}}^{\rho\over \rho - 1} \, \bbox[border:2px dashed #eee8d5,3px]{\gamma {x}^{-\gamma - 1} dx} \\ & = \mu \gamma {\left[w \over \rho \right]}^{\rho\over \rho - 1} \cdot \int_{\bar{x}}^\infty {\left[1 \over x\right]}^{\rho\over \rho - 1} \, { {x}^{-\gamma - 1} \, dx} \\ & = \mu \gamma {\left[w \over \rho \right]}^{\rho\over \rho - 1} \cdot \int_{\bar{x}}^\infty x^{\left[ {\rho \over 1-\rho} - \gamma - 1\right]} \, dx \\ & = {\mu \gamma \over \left[ {\rho \over 1-\rho} - \gamma \right]} {\left[w \over \rho \right]}^{\rho\over \rho - 1} \cdot x^{\left[ {\rho \over 1-\rho} - \gamma \right]} \Big|_{\bar{x}}^\infty \\ & = {\mu \gamma \over \left[ {\rho \over 1-\rho} - \gamma \right]} {\left[w \over \rho \right]}^{\rho\over \rho - 1} \cdot \left[-\bar{x}^{\left[ {\rho \over 1-\rho} - \gamma \right]} \right] \\ & = - {\mu \gamma \over \left[ {\rho \over 1-\rho} - \gamma \right]} {\left[w \over \rho \right]}^{\rho\over \rho - 1} \bar{x}^{\left[ {\rho \over 1-\rho} - \gamma \right]} \\ & = - \mu \gamma {1-\rho \over \rho -\gamma + \gamma \rho } {\left[w \over \rho \right]}^{\rho\over \rho - 1} \bar{x}^{\left[ {\rho \over 1-\rho} - \gamma \right]} \\ \end{align} Note that the evalutation of the integral requires use of the fact that $\gamma > {\rho \over 1-\rho}$. We can pretty this up a bit by rearranging things to get $P = \mu \gamma {1-\rho \over \gamma - \gamma \rho - \rho } {\left[w \over \rho \right]}^{\rho\over \rho - 1} \bar{x}^{\left[ {\rho \over 1-\rho} - \gamma \right]}$

And then we can plug this into the demand function to get \begin{align} y(x) = c(x) &= {\alpha \cdot [ w\bar l + \pi ]\cdot \dashbox{#eee8d5}{p(x)}^{1\over \rho - 1} \over \dashbox{#eee8d5}{\int_0^m p_z^{\rho\over \rho - 1} \, dz} } \\ \newcommand{\PExpression}{\mu \gamma {1-\rho \over \gamma - \gamma \rho - \rho } {\left[w \over \rho \right]}^{\rho\over \rho - 1} \bar{x}^{\left[ {\rho \over 1-\rho} - \gamma \right]} } & = {\alpha \cdot [ w\bar l + \pi ]\cdot \dashbox{#eee8d5}{\left[{w \over \rho x}\right]}^{1\over \rho - 1} \over \dashbox{#eee8d5}{\PExpression} } \\ & = \alpha [w\bar l + \pi] {\gamma -\gamma\rho - \rho \over \mu \gamma [1-\rho] } {\rho \over w} x^{1 \over 1 - \rho} \bar{x}^{\left[\gamma - { \rho \over 1-\rho} \right]} \end{align}

Next, we can get the expression for the profits of a firm with productivity $x > \bar{x}$: \begin{align} \pi(x) &= p(x) y(x) - w l(x) \\ &= p(x) y(x) - w {y(x) \over x} - w f \\ & = \left[\dashbox{#eee8d5}{ p(x)} - {w \over x} \right] \dashbox{#eee8d5}{y(x)} -wf \\ \newcommand{YExpression}{{\alpha [w\bar l + \pi] {\gamma -\gamma\rho - \rho \over \mu \gamma [1-\rho] } {\rho \over w} x^{1 \over 1 - \rho} \bar{x}^{\left[\gamma - { \rho \over 1-\rho} \right]} }} & = \left[ \dashbox{#eee8d5}{{w \over \rho x}} - {w \over x} \right] \dashbox{#eee8d5}{\left[\YExpression\right]} -wf \\ & = \cancel{\left[ {1-\rho \over \rho}{w \over x} \right]} \alpha [w\bar l + \pi] {\gamma -\gamma\rho - \rho \over \mu \gamma \cancel{[1-\rho]} } {\cancel{\rho} \over \cancel{w}} {x^{1 \over 1 - \rho} \over \color{red}{x}} \bar{x}^{\left[\gamma - { \rho \over 1-\rho} \right]} -wf \\ &= \alpha[w\bar{l}+\pi]\frac{\gamma-\gamma\rho-\rho}{\mu\gamma}x^{\frac{\rho}{1-\rho}}\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}-wf \end{align}

###### Step 2: Finding the threshold productivity.

A firm will produce iff they can make positive profits iff $x = \bar{x}$. Thus \begin{align} 0 &= \pi(\bar{x}) \\ &= \alpha[w\bar{l}+\pi]\frac{\gamma-\gamma\rho-\rho}{\mu\gamma}\bar{x}^{\frac{\rho}{1-\rho}}\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}-wf \\ &= \alpha[w\bar{l}+\pi]\frac{\gamma-\gamma\rho-\rho}{\mu\gamma}\bar{x}^\gamma-wf \end{align} $\bar{x}^\gamma = {wf \over \alpha[w\bar{l}+\pi]} \frac{\mu\gamma}{\gamma-\gamma\rho-\rho}$ $\bar{x} = \left[{w f \mu \gamma \over \alpha[w\bar{l}+\pi][\gamma-\gamma\rho-\rho]}\right]^{1\over\gamma}$

But we aren't quite done, because this expression includes two endogenous parameters $w$ and $\pi$. We can deal with $w$ by just setting $p_0 = w$ as the numeraire. But we need to find the total industry profit: \begin{align} \pi &= \mu \int_{\bar{x}}^\infty \dashbox{#eee8d5}{\pi(x)} \, \dashbox{#eee8d5}{ dF(x)} \\ &= \mu \int_{\bar{x}}^\infty \dashbox{#eee8d5}{\left[\alpha[w\bar{l}+\pi]\frac{\gamma-\gamma\rho-\rho}{\mu\gamma}x^{\frac{\rho}{1-\rho}}\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}-wf\right]} \, \dashbox{#eee8d5}{ \gamma {x}^{-\gamma - 1} \, dx}\\ &= \left[\mu\alpha[w\bar{l}+\pi]\frac{\gamma-\gamma\rho-\rho}{\mu\gamma}\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}\gamma \int_{\bar{x}}^{\infty}x^{\left[\frac{\rho}{1-\rho}-\gamma-1\right]}\,dx\right]-\left[\mu wf \gamma\int_{\bar{x}}^{\infty}{x}^{-\gamma-1}\,dx\right] \\ &= \left[\cancel{\mu}\alpha[w\bar{l}+\pi]\frac{\cancel{\gamma-\gamma\rho-\rho}}{\cancel{\mu}\cancel{\gamma}}\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}\frac{1-\rho}{\cancel{\rho-\gamma+\gamma\rho}}\cancel{\gamma}x^{\left[\frac{\rho}{1-\rho}-\gamma\right]}\Big|_{\bar{x}}^{\infty}\right]-\left[\frac{\mu w f \cancel{\gamma}}{-\cancel{\gamma}}{x}^{-\gamma}\Big|_{\bar{x}}^{\infty}\right] \\ &= \left[\alpha[w\bar{l}+\pi]\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}{1-\rho \over -1}\left[0-\bar{x}^{\left[\frac{\rho}{1-\rho}-\gamma\right]}\right]\right]-\left[-\mu w f \left[0-{\bar{x}}^{-\gamma}\right]\right] \\ &= \left[\alpha[w\bar{l}+\pi]\cancel{\bar{x}^{\left[\gamma-\frac{\rho}{1-\rho}\right]}}[1-\rho]\cancel{\bar{x}^{\left[\frac{\rho}{1-\rho}-\gamma\right]}}\right]-\mu wf{\bar{x}}^{-\gamma}\\ &= \alpha[w\bar{l}+\pi][1-\rho]-\mu wf{\bar{x}}^{-\gamma} \end{align} $\pi - \alpha[1-\rho]\pi = \alpha w\bar{l}[1-\rho]-\mu wf{\bar{x}}^{-\gamma}$ $\pi = {\alpha w\bar{l}[1-\rho]-\mu wf{\bar{x}}^{-\gamma} \over 1 - \alpha[1-\rho]}$

Great! We now have an expression for $\pi$ in terms of $\bar{x}$ and vice versa. Plug one into the other, and we can solve. First note that $\bar{x}^{-\gamma} = \left[{w f \mu \gamma \over \alpha[w\bar{l}+\pi][\gamma-\gamma\rho-\rho]}\right]^{-1}$ $\bar{x}^{-\gamma} = \left[{\alpha[w\bar{l}+\pi][\gamma-\gamma\rho-\rho] \over w f \mu \gamma }\right]$ So \begin{align} \pi &= \alpha[w\bar{l}+\pi][1-\rho]-\mu wf\left[{\alpha[w\bar{l}+\pi][\gamma-\gamma\rho-\rho] \over w f \mu \gamma }\right] \\ &= \alpha[w\bar{l}+\pi][1-\rho]-\cancel{\mu wf}\left[{\alpha[w\bar{l}+\pi][\gamma-\gamma\rho-\rho] \over\cancel{ w f \mu} \gamma }\right] \\ &= \alpha[w\bar{l}+\pi][1-\rho]-\frac{\alpha[w\bar{l}+\pi][\gamma-\gamma\rho-\rho]}{\gamma}\\ &= \alpha[w\bar{l}+\pi][1-\rho-1+\rho+\frac{\rho}{\gamma}] \\ &= \alpha[w\bar{l}+\pi]\frac{\rho}{\gamma} \end{align} $\pi-\alpha\frac{\rho}{\gamma}\pi=\alpha w\bar{l}\frac{\rho}{\gamma}$ $\pi=\frac{\alpha w\bar{l}\frac{\rho}{\gamma}}{1-\alpha\frac{\rho}{\gamma}}=\frac{\alpha w\bar{l}\frac{\rho}{\gamma}}{\left[\frac{\gamma-\alpha\rho}{\gamma}\right]}=\frac{\alpha w\bar{l}\rho}{\gamma-\alpha\rho}$ And plugging this back into the formula for $\bar{x}$, we get \begin{align} \bar{x} &=\left[\frac{wf\mu\gamma}{\alpha[w\bar{l}+\dashbox{#eee8d5}{\pi}][\gamma-\gamma\rho-\rho]}\right]^{\frac{1}{\gamma}}\\ &=\left[\frac{wf\mu\gamma}{\alpha[w\bar{l}+\dashbox{#eee8d5}{\frac{\alpha w\bar{l}\rho}{\gamma-\alpha\rho}}][\gamma-\gamma\rho-\rho]}\right]^{\frac{1}{\gamma}}\\ &= \left[\frac{wf\mu\gamma}{\alpha w\bar{l}[1+\frac{\alpha\rho}{\gamma-\alpha\rho}][\gamma-\gamma\rho-\rho]}\right]^{\frac{1}{\gamma}}\\ &= \left[\frac{wf\mu\gamma}{\alpha w\bar{l}[\frac{\gamma}{\gamma-\alpha\rho}][\gamma-\gamma\rho-\rho]}\right]^{\frac{1}{\gamma}}\\ &= \left[\frac{f\mu[\gamma-\alpha\rho]}{\alpha\bar{l}[\gamma-\gamma\rho-\rho]}\right]^{\frac{1}{\gamma}} \end{align} $\bbox[#eee8d5, 5px]{\bar{x}=\left[\frac{f\mu[\gamma-\alpha\rho]}{\alpha\bar{l}[\gamma-\gamma\rho-\rho]}\right]^{\frac{1}{\gamma}}}$ \dashbox{#eee8d5}{test}

TODO: Maybe skip right to $\pi_h = \left[ \alpha [ w\bar l + \pi ] p_h^{1\over \rho - 1} {1\over P} \right] \cdot \left[ p_h - {w \over x_h} \right] - f$

### Problem (e): Costly Entry Definition

Suppose that the set of potential firms $\mu$ is not fixed but instead is determined by a costly entry condition. Potential firms pay an entry cost $\phi > 0$ in terms of their country’s labor. Consider the closed economy model in part b and explain how the definition of equilibrium changes.

### Problem (e): The Real World

Discuss the strengths and limitations of this sort of model for account for firm-level data on exports.