# Dump of LaTeX solution

\section*{Consider an economy in which the consumption space is the set of functoins $c:R_{+}\times R_{+}\to R_{+}$. 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, $a(x,t)$>$\bar{a}(x,t)=e^{-x}$. At $t=0$, there is a $z(0)>0$ such that $a(x,0)=e^{-x}$ for all $x\bar{a}(x,t)\\ 0 & \text{if }a(x,t)=\bar{a}(x,t) \end{cases} \] $b(v,t)=\begin{cases} b>0 & a(v,t)>\bar{a}(v,t)\\ 0 & a(v,t)=\bar{a}(v,t) \end{cases}$ \section*{(a) Provide a motivation for both the utility function and the production technology described above.} The utility function's inner integral reflects consumers who like having a variety of goods but are content with (don't get infinitely negative utility from) consuming 0 of some non-zero measure of goods. This reflects that some goods are too advanced to be worth the cost, while others become outdated. The motivation for the prodcution function is that goods vary in the level of technological sopistication needed to make them. Goods beyond the technological 'frontier' of a society are increasingly difficult to make. A neolithic society could in principle make a large mechanical computer, but doing so would take so much time and effort that everyone would starve. But these same technologies that are initially difficult to make also tend to be less labor intensive once fully researched. And the more people working on advanced technology, the faster the frontier of technology is moved outward. Population growth feedback loop? \section*{(b) Define the equilibrium and characterize as much as possible.} Note that there is no storage, or intertemporal trade, so consumption cannot be smoothed over time. An equilbirum consists of prices$w(t),P(x,t)$and consumption allocations$c(x,t)$and labor allocations$l(x,t)$such that \begin{itemize} \item Given prices, consumers choose$c(x,t)$so that at each time$c(x,t)$maximizes $\int_{0}^{\infty}e^{-\rho t}\left[\int_{0}^{\infty}\log(c(x,t)+1)dx\right]dt$ $\ni\int_{0}^{\infty}P(x,t)c(x,t)dx\leq w(t)\ \forall t$ $c(x,t)\geq0$ \item Given prices, firms choose labor to maximize $\max_{l(x,t)}\left[p(x,t)\frac{l(x,t)}{a(x,t)}-w(t)l(x,t)\right]$ \item Perfectly competitive firms: $P(x,t)=w(t)a(x,t)$ \item Non-negativity in prices \item Markets clear. For all x,t: $y(x,t)a(x,t)=l(x,t)\ \ \ \forall x,t$ $\bar{l}c(x,t)=y(x,t)=\frac{l(x,t)}{a(x,t)}$ $\int_{0}^{\infty}l(x,t)dx=\bar{l}$ \item The production function evolves as a fucntional of$l(x,t)$as described: $\bar{a}(x,t)\equiv e^{-x}$ $\frac{\frac{\partial}{\partial t}a(x,t)}{a(x,t)}=\begin{cases} -\int_{0}^{\infty}b(v,t)l(v,t)dv & \text{if }a(x,t)>\bar{a}(x,t)\\ 0 & \text{if }a(x,t)=\bar{a}(x,t) \end{cases}$ $b(v,t)=\begin{cases} b>0 & a(v,t)>\bar{a}(v,t)\\ 0 & a(v,t)=\bar{a}(v,t) \end{cases}$ \end{itemize} \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash \textemdash{} \subsubsection*{There is some cutoff productivity.} Because there is no storage, and consumers are identical, budget constraint is period by period. And with perfect competition, the budget becomes $\int_{0}^{\infty}a(x,t)c(x,t)dx\leq1$ FOC for any$c(x,t)>0$is thus: $\frac{\partial\mathcal{L}}{\partial c(x,t)}=\frac{1}{c(x,t)+1}-\lambda_{t}a(x,t)$ $c(x,t)=\frac{1}{\lambda_{t}a(x,t)}-1$ $\frac{1}{a(x,t)\left(c(x,t)+1\right)}=\lambda_{t}$ Define$\tilde{a_{t}}\equiv\frac{1}{\lambda_{t}}$. Then for all$c(x,t)>0,\begin{align*} c(x,t) & =\frac{\tilde{a_{t}}}{a(x,t)}-1 \end{align*} $c(x,t)a(x,t)=\tilde{a_{t}}-a(x,t)$ And ifa(x,t)>\tilde{a_{t}}$,$c(x,t)$must then be zero. Thus$\tilde{a_{t}}$is a cutoff value, and goods are produced iff they have better productivity than the cutoff. \subsubsection*{The shape is always symetric around some point.} Because$\frac{\frac{\partial}{\partial t}a(x,t)}{a(x,t)}$is the same for all$x$such that$a(x,t)>e^{-x}$,$a(x,t)=\max\left\{ e^{-x},\alpha(t)e^{x-2z(0)}\right\} $for some scaling factor$\alpha(t)$Also note that$\alpha(t)e^{x-2z(0)}=e^{x-2z(t)}$where$z(t)\equiv z(0)-\frac{1}{2}\log\alpha(t)$. And$e^{-x}=\alpha(t)e^{x-2z(0)}$at$x=z(t).$So at any time$t$, there is a$z(t)>0$such that$a(x,t)=e^{-x}$for all$xz^{2}(0).$There is no borrowing or lending. Define an equilibrium. } (no transportation costs so prices are the same between countries.) A competitive equilibirum consists of prices$w_{1}(t),w_{2}(t),P(x,t)$; consumption functions$c_{1}(x,t),c_{2}(x,t)$, labor allocation functions$l_{1}(x,t)$,$l_{2}(x,t)$; and output such that$y_{1}(x,t)$,$y_{2}(x,t)$\begin{itemize} \item Given prices, in each country$i$, consumers choose$c_{i}$so that at each time$c_{i}$maximizes $\int_{0}^{\infty}e^{-\rho t}\left[\int_{0}^{\infty}\log(c_{i}(x,t)+1)dx\right]dt$ $\ni\int_{0}^{\infty}P(x,t)c_{i}(x,t)dx\leq w_{i}(t)\ \forall t$ $c_{i}(x,t)\geq0$ \item Given prices, each firm chooses labor allocation$l_{i}(x,t)\geq0$to solve $\max_{l_{i}(x,t)}P(x,t)\frac{l_{i}(x,t)}{a_{i}(x,t)}-w_{i}(t)l_{i}(x,t)$ \item Market Clears. For all x,i,t,: $y_{i}(x,t)=\frac{l_{i}(x,t)}{a_{i}(x,t)}$ $\bar{l_{i}}=\int_{0}^{\infty}l_{i}(x,t)dx$ $\bar{l_{1}}c_{1}(x,t)+\bar{l_{2}}c_{2}(x,t)=\frac{l_{1}(x,t)}{a_{1}(x,t)}+\frac{l_{2}(x,t)}{a_{2}(x,t)}$ Perfectly competitive firms: $P(x,t)=\min_{i}\left[w_{i}(t)a_{i}(x,t)\right]$ \item Non-negativity in prices \item Evolution of production function by LearningByDoing: $\frac{\frac{\partial}{\partial t}a_{i}(x,t)}{a_{i}(x,t)}=\begin{cases} -\int_{0}^{\infty}b_{i}(v,t)l_{i}(v,t)dv & \text{if }a_{i}(x,t)>\bar{a}(x,t)\\ 0 & \text{if }a_{i}(x,t)=\bar{a}(x,t) \end{cases}$ $b_{i}(v,t)=\begin{cases} b>0 & a_{i}(v,t)>\bar{a}(v,t)\\ 0 & a_{i}(v,t)=\bar{a}(v,t) \end{cases}$ \end{itemize} \section*{(d)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$z(t)$s are sufficiently large such that good$0$is not produced in equilibirum.} Consumers will only buy a good from the nation that produces it most cheaply. Let$w_{2}(t)$be the numeraire, and let$W\equiv\frac{w_{1}}{w_{2}}=w_{1}$. Then the price of a good produced in country 1 is$Wa_{1}(x,t)$and the price of a good produced in country 2 is$a_{2}(x,t)$. In each graph below, which plots good index vs price,$Wa_{1}(x,t)$is the solid blue line,$a_{2}(x,t)$is the dotted orange line, The solid green line represents the maximum price that consumers from country 1 will pay for a good, and the dotted red line is the maximum price for consumers from country 2. For each graph, there is a region shaded with yellow horizontal lines, cooresponding to the consumption patterns of country 2 (area=$1$), and blue vertical lines for country 1 (area=$W$). Whenever a good is produced by both countries, it really means that that region of goods is produced by some combination of the two countries which is not specified in equilibirum.$M_{i}^{j}\equiv$the leftmost point at which the price curve for country$j$is equal to the maximum price paid by consumers in country$i$. Likewise,$N_{i}^{j}$is the rightmost such point. First some impossibilities: \subsubsection*{Impossibility 1:$W<1$} \includegraphics{LbdCase-1} In this case, all goods are cheaper to produce in country 1, so none will be produced in country 2. But without lending, there is no way for the balance of trade to work out. This would mean, for example, that the total labor income of country 2 is$0$. But then country 2 cannot possible inmport goods at nonnegative price. So this example contradicts the setup of the problem. \subsubsection*{Impossiblity 2:$W>e^{2(z_{1}(t)-z_{2}(t))}$} \includegraphics{LbdCase-2} This results in all goods being cheaper in country 2, which also makes balanced trade impossible. \subsubsection*{Case 1: Equal wages} \includegraphics{LbdCase1annotated}$W=1$is the minimum possible$W$with trade. Because wages are the same, the budget constraint is the same for each country, and consumption patterns will be identical. Both countries consume goods in the range$[M,N^{1}]$Goods with$N^{1}\geq x>z_{2}(t)$will be exclusively produced by country 1. Goods with$M\leq x\leq z_{2}(t)$will be produced and by some combination of the two countries, which is not uniquely specified in equilibirum. \subsubsection*{Case 2: Maximum possible wage ratio} \includegraphics{LbdCase2annotated} Here$W=e^{2(z_{1}(t)-z_{2}(t))}$. All goods with$x\in[M_{1}^{2},M_{2}^{2})$will be produced exclusively by country 2, and consumed only by country 1. Goods with$x\in[M_{2}^{2},z_{1}(t))$will be produced by country 2, and consumed by both countries. Goods with$x\in\left[z_{1}(t),N_{2}\right]$will be produced and consumed by both countries. Goods with$x\in(N_{2},N_{1}]$will be produced by both countries and consumed only by country 1. \subsubsection*{Case 3-5: Intermediate wages} If$1v$. \subsubsection*{In case 3,}$P(v,t)<\tilde{P_{2}}$, and so this crossover good will be produced and consumed by both countries. \includegraphics{LbdCase3annotated} Country 1 produces in the range$\left[M_{1}^{2},v\right]$Country 1 produces in the range$\left[v,N_{1}^{1}\right]$Country one consumes the range$\left[M_{1}^{2},N_{1}^{1}\right]$Country one consumes the range$\left[M_{2}^{2},N_{2}^{1}\right]$\subsubsection*{In Case 4,}$\tilde{P_{1}}>P(v,t)>\tilde{P_{2}}$and so good$v$will be produced by both countries, but consumed only by country 1. \includegraphics{LbdCase4annotated} In fact, there is now a gap around$v$in the middle of country 2's range of consumed goods. Only country 2 consumes goods in this gap. \subsubsection*{In Case 5,}$P(v,t)>\tilde{P_{1}}$and so neither country consumes good$v$\includegraphics{LbdCase5annotated} Consumption and production are like Case 3, except that there is now a region of goods around$v$such that neither country consumes or produces in this region. \emph{(Note: All the other cases had graphs generated using actual parameters for$z_{1},z_{2},w_{1},w_{2}$, but I couldn't find nice looking parameters for case 5, so I just faked the graph.)} \section*{(e) 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 does$\bar{l_{2}}$need to be so that$z_{2}(t)$eventually passes$z_{1}(t)$? Describe the economic significance of this result.} For country 2 to eventually surpass country 1, it must be that$\bar{l_{2}}\geq\left(2+e^{2(z_{1}-z_{2})}\right)\bar{l}_{1}$. - The percentage change of$a_{i}$for technology which isn't fully researched is$b\bar{l}_{i}\cdot$(portion of labor in country$i$allocated to producing hi tech goods.)$\frac{\partial}{\partial t}z_{i}(t)$is half of this rate. Thus the rate of change for$z_{i}$is proportional to the amount of$\bar{l_{i}}$being used to produce goods$x>z_{i}$Note that in all 5 cases, Country 1 has a weakly greater share of its labor contributing to hi-tech fields. So for country 2 to catch up, it must have a larger population. Also note that Case 1 can only happen if country 1 is larger, because the majority of the share of each country's consumption is produced in country 1. (The majority of two things adds up to more than the smaller of the two things.) For similar reasons, \textbf{Case 2 can only happen if country 2 is larger.} - Case 1 is the only case in which a country does not have increasing technology. In case 1, country 2 devotes all of their labor to fully-research technology, and so$z_{2}(t)$remains static while$z_{1}(t)$increases, until eventually the economy shifts into case 3 as country 1 exhausts their potential to shift labor into higher-tech industries without driving up wages. In all other cases,$\frac{\partial}{\partial t}z_{i}(t)$is positive for both countries. - In Case 3-5, both countries technologically develop. When the relative size of the gap between tech levels in countries increase, it drives the equilibrium from case 3 to 4 to 5. And a decreasing gap drives things in the other direction. \textbf{If Country 1 is bigger}, country 1 grows faster, and so the economy tends towards Case E over time. \textbf{If the countries are of equal size,} then Case 3 has country 1 growing relatively faster, driving the equilibrium to case 5. In case 5, each country devotes precisely half of its labor to producing hi-tech goods and thus both$z_{i}$increase the same. With the gap between tech levels constant, but tech levels inceasing, the relative size of the gap shrinks, driving the equilbirum to Case 4. \textbf{If country 2 is bigger, }then Case 5 is quickly driven down into case 4. Cases 3 and 4 both exhibit widening gaps in technology. - So the only way for the gap between$z_{i}$s to shrink is for country 2 to be bigger and the equilibrium to be in Case 2. Even in this case, for the tech gap to shrink, it must be that Country 2 has a greater amount of labor in high tech industries. Country 1 uses all of it's labor on goods with$x>z_{2}$. This labor is equivalent to$w_{1}\bar{l_{1}}$units of country 2's labor. When figuring out the share of labor used above the tech thresholds, we can, without loss of generality, presume that there is some$H$such that country 1 does all production for$x>z_{2}+H$and country two does all prodcution for$x$\frac{\partial}{\partial t}z_{1}(t)$, it must be that $\frac{\bar{l_{2}}-e^{2(z_{1}(t)-z_{2}(t))}\bar{l}_{1}}{2}>\bar{l_{1}}$ $\bar{l_{2}}>\left(2+e^{2(z_{1}(t)-z_{2}(t))}\right)\bar{l_{1}}$ - The economic implication here is that if developing nations want to technologically advance they should become very large. Beacuse a nation in this model is defined as a region in which labor can freely move around, one strategy for development is to join unions with other nations to allow free movement of labor.