Trade Prelim Notes

Learning by Doing - Prelim Problem

This problem is from Tim Kehoe. It showed up on the trade prelims in Fall of 2017 and Spring of 2014


Consider an economy in which the consumption space is the set of functions $c:R_{+}\times R_{+}\to R_{+}$. In $c(x,t)$ the index $x$ denotes the type of good and the index $t$ denotes the date at which it is consumed. An individual consumer has preferences given by the functional \[u(c)=\int_{0}^{\infty}e^{-\rho t}\left[\int_{0}^{\infty}\log(c(x,t)+1)dx\right]dt\] Goods are produced using a single factor of production, labor: \[y(x,t)=\frac{l(x,t)}{a(x,t)}\] Each consumer has an endowment of labor equal to $l$, and the total number of consumers is fixed at $\bar{l}$. The unit labor requirement $a(x,t)$ is bounded from below, $a(x,t) > \bar{a}(x)$ where $$\bar{a}(x)=e^{-x}$$ At $t=0$, there is a $z(0)>0$ such that $a(x,0)=e^{-x}$ for all $x < z(0)$ and that $a(x,0) = e^{x-2 z(0)}$ for all $x\geq z(0)$. There is learning by doing of the form $$\frac{\dot{a}(x,t)}{a(x,t)}= \begin{cases} -\int_{0}^{\infty}b(v,t)l(v,t)\thinspace dv & \text{if }a(x,t)>\bar{a}(x)\\ 0 & \text{if }a(x,t)=\bar{a}(x) \end{cases}$$ Here $\dot{a}(x,t)$ denotes the partial derivative of $a(x,t)$ with respect to $t$ and $$b(v,t)= \begin{cases} b>0 & a(v,t)>\bar{a}(v)\\ 0 & a(v,t)=\bar{a}(v) \end{cases}$$ There is no storage.

Problem (a): Motivation

Provide a motivation for both the utility function and the production technology described above.

Problem (b): Characterize Equilibrium

Define an equilibrium for this economy. Characterize the equilibrium as much as possible.

Problem (c): Two-Country Equilibrium

Consider now a two country world in which the two countries are identical except in their endowments of labor and their initial technology levels. In particular, $z_1 (0) > z_2 (0)$. There is no borrowing or lending. Define an equilibrium.

Problem (d): Different possibilities for equilibrium

Suppose that $z^1 (t) > z^2 (t)$. Explain carefully and illustrate two of the five qualitatively different possible equiblirbium configurations for production, consumption and trade at time $t$. (To make things easier, assume the $z^1 (t)$ and $z^2 (t)$ are sufficiently large such that good $x=0$ is not produced in equilibirum.)

Problem (e): Explain Dynamics

Briefly describe the dynamics of the model, explaining the crucial role played by the sizes of the two countries, $\bar{l}^1$ and $\bar{l}^2$. In particular, how much larger than $\bar{l}^1$ does $\bar{l}^2$ have to be so that $z^2 (t)$ eventually passes $z^1 (t)$? Discuss the economic significance of this result.