This problem is from Tim Kehoe. It showed up on the trade prelims in 2019 and 2015
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:
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.
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.
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.
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 \(P\equiv \int_0^m p_z^{\rho\over \rho - 1} \, dz \), \[\bbox[#eee8d5, 5px]{c_h = {\alpha \cdot [ w\bar l + \pi ]\cdot p_h^{1\over \rho - 1} \cdot {1\over P} }}\]
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 \(h\).
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.
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 \(\bar{x}\).
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.
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}
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}
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.
Discuss the strengths and limitations of this sort of model for account for firm-level data on exports.