## International Trade with Monopolistic Competition and Heterogenous Firms

This problem is from Tim Kehoe. It showed up on the trade prelims in 2019 and 2015

### 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.

# LaTeX Dump

\section*{Problem 3 (HW4)} Consider the economy where consumers solve the following: $\max\left[(1-\alpha)\log c_{0}+\frac{\alpha}{\rho}\log\int_{0}^{m}c(z)^{\rho}dz\right]$ $\text{s.t. }p_{0}c_{0}+\int_{0}^{m}p(z)x(z)dz=wl+\pi$ $c(z)\geq0$ Here $\alpha,\rho\in(0,1)$. Furthermore, $m$ is the measure of firms, which is dtermined in equilbvirum. The measure of potential firms is fixed at $\mu>0$ . Suppose that good zero is produced with production function $y_{0}=l_{0}$. \subsection*{a) Suppose producer of good z takes the prices of other producers as given. Then this producer has the production fucntion $y(z)=\max\left[x(z)\left(l(z)-f\right),0\right]$ where $x(z)>0$ is the firms prductivity and $f>0$. Solve the firms profit maximizing problem and derive a rule for optimal pricing. } The solution to the consumer's problem in reponse to prices is $c_{0}=(1-\alpha)\frac{w\bar{l}+\pi}{p_{0}}$ $c(z)=\frac{\alpha(w\bar{l}+\pi)}{p(z)^{\frac{1}{1-\rho}}\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}$ Good 1 is produced in a competitive market with $y_{0}=l_{0}$ so $w=p_{0}$ and we can normalize prices so that $p_{0}=w=1$. The firm that makes good $z$wants problem is to set prices to maximize profit. If they produce at all, revenue is $p(z)c(z)$ and cost is $f+\frac{c(z)}{x(z)}$ because $w=1.$ Thus the firms problem is given the prices of other firms, choose $p(z)$ to solve $\max_{p(z)}\left(p(z)\frac{\alpha(w\bar{l}+\pi)}{p(z)^{\frac{1}{1-\rho}}\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}-\frac{\alpha(w\bar{l}+\pi)}{x(z)p(z)^{\frac{1}{1-\rho}}\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}-\frac{f}{x(z)}\right)$ Setting the derivative equal to zero yields \begin{align*} \frac{\rho}{\rho-1}p(z)^{\frac{1}{1-\rho}}\frac{\alpha(w\bar{l}+\pi)}{\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)} & =\frac{1}{\rho-1}p(z)^{\frac{1}{1-\rho}-1}\frac{\alpha(w\bar{l}+\pi)}{x(z)\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}\\ p(z) & =\frac{1}{\rho x(z)} \end{align*} Because $\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)$ does not change with respect to an individual firm's behavior. \subsection*{b) Suppose firm productivities are distributed on the interval $x\protect\geq1$ according to the Pareto distribution with distribution function $F(x)=1-x^{-\gamma}$ where $\gamma>2$ and $\gamma>\frac{\rho}{1-\rho}.$ Define an equilibirum for this economy.} An equiliibrum for this economy consists of the number of manufacturing firms $m$ in the market, prices $p(z)$, wage $w$ consumption allocations $c(z)$ and production plans $y(z)$ ,$l(z)$ such that \begin{itemize} \item Given prices and wages, the consumer's allocation solves \end{itemize} $\max\left[(1-\alpha)\log c_{0}+\frac{\alpha}{\rho}\log\int_{0}^{m}c(z)^{\rho}dz\right]$ $\text{s.t. }p_{0}c_{0}+\int_{0}^{m}p(z)c(z)dz=w\bar{l}+\pi$ $c(z)\geq0$ \begin{itemize} \item $p_{0}=w$ \item Given the indirect demand function $c(z)=\frac{\alpha(w\bar{l}+\pi)}{p(z)^{\frac{1}{1-\rho}}\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}$ firm j chooses $p(z)$ to solve $\max_{p(z)}\left(p(z)\frac{\alpha(w\bar{l}+\pi)}{p(z)^{\frac{1}{1-\rho}}\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}-\frac{\alpha(w\bar{l}+\pi)}{x(z)p(z)^{\frac{1}{1-\rho}}\left(\int_{0}^{m}p(z')^{\frac{-\rho}{1-\rho}}dz'\right)}-f\right)$ \item $p(z)c(z)-wc(z)/x(z)-f\leq0$  with equality if $c(z)>0$ \item $y_{0}=l_{0}$ \item $c(z)=y(z)=\max\left[x(z)\left(l(z)-f\right),0\right]$ \item $l_{0}+\int_{0}^{m}l(z)=\bar{l}$ \end{itemize} \subsection*{c) Suppose in equilibirum that not all firms produce. Find an expression fo the productivity of the least productive firm that produces. Relate the measure of firms that produce $m$ to the measure of potential firms $\mu$ and the cutoof $\bar{x}$. } $f(x)=\gamma x^{-\gamma-1}$ If a firm has producitivty at the cutoff, then it must be that they make zero profits in equilibrium. This will make it so that any firm with lower productivity does not produce. First let us convert the integral of prices over index to one of prices over productiivty: \begin{align*} \left(\int_{0}^{m}p(z)^{\frac{-\rho}{1-\rho}}dz\right) & =\mu\int_{\bar{x}}^{\infty}p(z)^{\frac{-\rho}{1-\rho}}dF(x)\\ & =\mu\int_{\bar{x}}^{\infty}p(z)^{\frac{-\rho}{1-\rho}}\gamma x^{-\gamma-1}dx\\ & =\frac{\mu\rho^{\frac{\rho}{1-\rho}}(1-\rho)\gamma\bar{x}^{\frac{\rho-\gamma(1-\rho)}{1-\rho}}}{\gamma(1-\rho)-\rho} \end{align*} Thus indirect demand for a firm with producitivyt $x$ is $c(x)=\frac{\alpha(w\bar{l}+\pi)}{p(z)^{\frac{1}{1-\rho}}\left(\frac{\mu\rho^{\frac{\rho}{1-\rho}}(1-\rho)\gamma\bar{x}^{\frac{\rho-\gamma(1-\rho)}{1-\rho}}}{\gamma(1-\rho)-\rho}\right)}=\frac{\rho\left(\gamma(1-\rho)-\rho\right)\alpha(w\bar{l}+\pi)x^{\frac{1}{1-\rho}}}{\mu(1-\rho)\bar{x}^{\frac{\rho-\gamma(1-\rho)}{1-\rho}}}$ Thus when we set profit of firm with productivity $\bar{x}$ to zero: $0=\left(\frac{\rho\left(\gamma(1-\rho)-\rho\right)\alpha(w\bar{l}+\pi)x^{\frac{1}{1-\rho}}}{\mu(1-\rho)\bar{x}^{\frac{\rho-\gamma(1-\rho)}{1-\rho}}}\right)\left(\frac{1}{\rho\bar{x}}-\frac{1}{\bar{x}}\right)-f$ which implies $\bar{x}=\left(\frac{(\gamma-\rho\alpha)\mu f}{\gamma(1-\rho)-\rho\alpha\bar{l}}\right)^{\frac{1}{\gamma}}$ And the portion of firms which produce is equal to the portion of firms with productivity at least $\bar{x}$. $m/\mu=1-F(\bar{x})$ \subsection*{d) Suppose 2 countries are engaged in trade. Each country $i$ has population $\bar{l_{i}}$ and potential firms $\mu_{i}$ . A firm in country i faces a fixed cost of exporting to country $j,$ $j\protect\neq i$ of $f_{e}$ where $f_{e}>f_{d}=f$. Each country also imposes an ad valorem tariff $\tau$ on imports of differentiated goods form the other county. The revenue is given in a lump sum to the consumer in that country. Define an equilibrium fo this world economy.} INCOMPLETE An equilibrium is the number of manufacturing firms, consumption allocaations for the consumers in each country, the prices of goods in their home countries, wages in each country, and production plans for the firms in each nation whereby \begin{itemize} \item Given prices and wages, the consumer in country $i$ has an allocation which solves \end{itemize} $\max\left[(1-\alpha)\log c_{0}+\frac{\alpha}{\rho}\log\int_{0}^{m}\left(c_{i}^{i}(z)+c_{i}^{j}(z)\right)dz\right]$ $\text{s.t. }p_{0}c_{0}+\int_{0}^{m}\left(c_{i}^{i}(z)p_{i}^{i}(z)+c_{i}^{j}(z)\left(p_{j}^{i}(z)\right)\right)dz=w\bar{l}+\pi$ $c(z)\equiv\left(c_{i}^{i}(z)+c_{i}^{j}(z)\right)\geq0$ $p_{j}^{i}=p_{i}^{i}+f_{e}-f$ ? ... INCOMPLETE \subsection*{e) Suppose that the two countries have the same population and measure of potential firms. Explain how to characterize the equilibrium production pattersn with a cutoff value as in part c. } By symettry, both countries are producing the same measure of goods People will buy from whichever producer of $z$ can provide it most cheaply, so the only trade will occur with goods where luck of the draw has made one producer so much more effective that the productivity gains outweigh the transport costs. This is uncorelated to $z$. So this will be largely random and just be a certain poriton of goods. \subsection*{f) Disuss the strengths and limitations of this sort of model for accounting for firm level data on exports.} Good: Most firms don't export, just like in this model. More productive firms are more likely to export, like in the model. Bad: ..