## 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

### Setup:

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.