%----------new chapter \chapter{The Lucas Model} \label{ch:lucas} \index{Lucas model (begin 1)} The present-value interpretation of the monetary model underscores the idea that we should expect the exchange rate to behave like the prices of other assets---such as stocks and bonds. This is one of that model's attractive features. One of its unattractive features is that the model is ad hoc in the sense that the money demand functions upon which it rests were not shown to arise explicitly from decisions of optimizing agents. Lucas's~[\ref{bib:Lucas.JME}] \index{Lucas, R.E.} neoclassical model of exchange rate determination gives a rigorous theoretical framework for pricing foreign exchange and other assets in a flexible price environment and is not subject to this criticism. It is a dynamic general equilibrium model of an endowment economy with complete markets where the fundamental determinants of the exchange rate are the same as those in the monetary model. The economic environment for dynamic general equilibrium analysis needs to be specified in some detail. To make this task manageable, we will begin by modeling the real part of the economy that operates under a barter system. We will obtain a solution for the real exchange rate and real stock-pricing formulae. This perfect-markets real general equilibrium model is sometimes referred to as an Arrow~[\ref{bib:Arrow}]--Debreu~[\ref{bib:Debreu}] \index{Arrow, K.J.}\index{Debreu, G.} model because it can be mapped into their static general equilibrium framework. We know that the Arrow--Debreu competitive equilibrium yields a Pareto \index{Arrow-Debreu model} \index{Social Optimum (Pareto optimum see social optimum)} Optimum. Why is this connection useful? Because it tells us that we can understand the behavior of the market economy by solving for the social optimum and it is typically more straightforward to obtain the social optimum than to directly solve for the market equilibrium. In order to study the exchange rate, we need to have a monetary economy. The problem is that there is no role for fiat money in the Arrow--Debreu environment. The way that Lucas gets around this problem is to require people to use money when they buy goods. This \marginpar [(76)$\Rightarrow$]{$\Leftarrow$(76)} requirement is called a `cash-in-advance' constraint and is a popular strategy for introducing money in general equilibrium along the lines of the transactions motive for holding money. A second popular strategy that puts money in the utility function will be developed in chapter \ref{ch:redux}. The models we will study in this chapter and in chapter \ref{ch.rbcers} have no market imperfections and exhibit no nominal rigidities. Market participants have complete information and rational expectations. Why study such a perfect world? First, we have a better idea for solving frictionless and perfect-markets models so it is a good idea to start in familiar territory. Naturally, these models of idealized economies will not fully explain the real world. So we want to view these models as providing a benchmark against which to measure progress. If and when the data `reject' these models, take one should note the manner in which they are rejected to guide the appropriate extensions and refinements to the theory. There is a good deal of notation for the model which is summarized in Table \ref{tab:lucas.notation}. \section{The Barter Economy} \label{sec:Lucas.barter} Consider two countries each inhabited by a large number of individuals who have identical utility functions and identical wealth. People may believe that they are individuals but the respond in the same way to changes in incentives. Because people are so similar you can normalize the constant populations of each country to 1 and model the people in each country by the actions of a single {\it representative agent (household) in Lucas model}. \index{Representative agent} This is the simplest way to aggregate across individuals so that we can model macroeconomic behavior. `Firms' in each country are pure endowment streams that generate a homogeneous nonstorable country-specific good using no labor or capital inputs. Some people like to think of these firms as fruit trees. You can also normalize the number of firms in each country to 1. $x_t$ is the exogenous domestic output and $y_t$ is the exogenous foreign output. The evolution of output is given by $x_t=g_t x_{t-1}$ at home and by $y_t= g^*_t y_{t-1}$ abroad where $g_t$ and $g^*_t$ are {\it random} gross rates of change that evolve according to a stochastic process that is known by agents. Each firm issues one perfectly divisible share of common stock which is traded in a competitive stock market. The firms pay out all of their output as dividends to shareholders. Dividends form the sole source of support for individuals. We will let $x_t$ be the {\it numeraire} good and $q_t$ be the price of $y_t$ in terms of $x_t$. $e_t$ is the ex-dividend market value of the domestic firm and $e^*_t$ is the ex-dividend market value of the foreign firm. The domestic agent consumes $c_{xt}$ units of the home good, $c_{yt}$ units of the foreign good and holds $\omega_{xt}$ shares of the domestic firm and $\omega_{yt}$ shares of the foreign firm. Similarly, the foreign agent consumes $c^*_{xt}$, units of the home good, $c^*_{yt}$ units of the foreign good and holds $\omega^*_{xt}$ shares of the domestic firm and $\omega^*_{yt}$ shares of the foreign firm. The domestic agent brings into period $t$ wealth valued at \begin{equation} W_t = \omega_{xt-1}(x_t+e_t)+\omega_{yt-1}(q_t y_t + e^*_t), \label{eq:lucas.barter.bc1} \end{equation} where $x_t+e_t$ and $q_t y_t + e^*_t$ are the with-dividend value of the home and foreign firms. The individual then allocates current wealth towards new share purchases $e_t \omega_{xt} + e^*_t \omega_{y_t},$ and consumption $c_{xt} + q_t c_{y_t}$ \begin{equation} W_t=e_t \omega_{xt} + e^*_t \omega_{y_t} + c_{xt} + q_t c_{y_t}. \label{eq:lucas.barter.bc2} \end{equation} Equating (\ref{eq:lucas.barter.bc1}) to (\ref{eq:lucas.barter.bc2}) gives the consolidated budget constraint \begin{equation} c_{xt} + q_t c_{y_t} + e_t \omega_{xt} + e^*_t \omega_{y_t} = \omega_{xt-1}(x_t+e_t)+\omega_{yt-1}(q_t y_t + e^*_t). \label{eq:lucas.bc1} \end{equation} Let $u(c_{xt},c_{yt})$ be current period utility and $0 <\beta < 1$ be the subjective discount factor. The domestic agent's problem then is to choose sequences of consumption and stock purchases, $\{c_{xt+j},c_{y_t+j},\omega_{xt+j},\omega_{yt+j}\}_{j=0}^{\infty}$, to maximize expected lifetime utility \begin{equation} E_t\left(\sum_{j=0}^{\infty} \beta^j u(c_{xt+j},c_{yt+j})\right), \label{eq:lucas-maxu.1} \end{equation} subject to (\ref{eq:lucas.bc1}). You can transform the constrained optimum problem into an unconstrained optimum problem by substituting $c_{xt}$ from (\ref{eq:lucas.bc1}) into (\ref{eq:lucas-maxu.1}). The objective function becomes \begin{equation} \begin{array}{l} u(\omega_{xt-1}(x_t+e_t) + \omega_{yt-1}(q_t y_t + e^*_t) -e_t \omega_{xt} - e^*_t \omega_{y_t} - q_t c_{y_t}, c_{y_t}) \\ [2mm] \ \ \ + E_t[\beta u(\omega_{xt}(x_{t+1}+e_{t+1})+\omega_{yt}(q_{t+1} y_{t+1} + e^*_{t+1}) \\ [2mm] \ \ \ \ -e_{t+1} \omega_{xt+1} - e^*_{t+1} \omega_{y_{t+1}} - q_{t+1} c_{y_{t+1}}, c_{y_{t+1}})] \ \ + \cdots \label{eq:unconstrained} \end{array} \end{equation} Let $u_1(c_{xt},c_{yt})=\partial u(c_{xt},c_{yt})/\partial c_{xt}$ be the marginal utility of $x$-consumption and $u_2(c_{xt},c_{yt})=\partial u(c_{xt},c_{yt})/\partial c_{yt}$ be the marginal utility of $y$-consumption. Differentiating (\ref{eq:unconstrained}) with respect to $c_{yt}, \omega_{xt}$, and $\omega_{yt}$, setting the result to zero and rearranging yields the Euler equations \index{Euler equations, Lucas model} \marginpar [(77)$\Rightarrow$]{$\Leftarrow$(77)} \begin{eqnarray} c_{yt}: & & q_t u_1(c_{xt},c_{yt}) = u_2(c_{xt},c_{yt}), \label{eq:lucas.euler.1}\\ \omega_{xt}: & & e_t u_1(c_{xt},c_{yt})=\beta E_t[u_{1}(c_{xt+1},c_{yt+1})(x_{t+1}+e_{t+1})],\label{eq:lucas.euler.2}\\ \omega_{yt}: & & e^*_t u_1(c_{xt},c_{yt})=\beta E_t[u_{1}(c_{xt+1},c_{yt+1})(q_{t+1}y_{t+1}+e^*_{t+1})].\label{eq:lucas.euler.3} \end{eqnarray} These equations must hold if the agent is behaving optimally. (\ref{eq:lucas.euler.1}) is the standard intratemporal optimality condition that equates the relative price between $x$ and $y$ to their marginal rate of substitution. Reallocating consumption by adding a unit of $c_{y}$ increases utility by $u_2(\cdot)$. This is financed by giving up $q_t$ units of $c_{x}$, each unit of which costs $u_1(\cdot)$ units of utility for a total utility cost of $q_t u_1(\cdot)$. If the individual is behaving optimally, no such reallocations of the consumption plan yields a net gain in utility. (\ref{eq:lucas.euler.2}) is the intertemporal Euler equation for purchases of the domestic equity. The left side is the utility cost of the marginal purchase of domestic equity. To buy incremental shares of the domestic firm, it costs the individual $e_t$ units of $c_x$, each unit of which lowers utility by $u_1(c_{xt},c_{yt})$. The right hand side of (\ref{eq:lucas.euler.2}) is the utility expected to be derived from the payoff of the marginal investment. If the individual is behaving optimally, no such reallocations between consumption and saving can yield a net increase in utility. An analogous interpretation holds for intertemporal reallocations of consumption and purchases of the foreign equity in (\ref{eq:lucas.euler.3}). The foreign agent has the same utility function and faces the analogous problem to maximize \begin{equation} E_t\left(\sum_{j=0}^{\infty} \beta^j u(c^*_{xt+j},c^*_{yt+j})\right), \label{eq:lucas-maxu.2} \end{equation} subject to \begin{equation} c^*_{xt} + q_t c^*_{y_t} + e_t \omega^*_{xt} + e^*_t \omega^*_{y_t} = \omega^*_{xt-1}(x_t+e_t)+\omega^*_{yt-1}(q_t y_t + e^*_t). \nonumber \end{equation} The analogous set of Euler equations for the foreign individual are \index{Euler equations, Lucas model} \begin{eqnarray} c^*_{yt}: & & q_t u_1(c^*_{xt},c^*_{yt})=u_2(c^*_{xt},c^*_{yt}), \label{eq:lucas.euler.4}\\ \omega^*_{xt}: & & e_t u_1(c^*_{xt},c^*_{yt})=\beta E_t[u_{1}(c^*_{xt+1},c^*_{yt+1})(x_{t+1}+e_{t+1})],\label{eq:lucas.euler.5}\\ \omega^*_{yt}: & & e^*_t u_1(c^*_{xt},c^*_{yt})=\beta E_t[u_{1}(c^*_{xt+1},c^*_{yt+1})(q_{t+1}y_{t+1}+e^*_{t+1})].\label{eq:lucas.euler.6} \end{eqnarray} A set of four adding up constraints on outstanding equity shares and the exhaustion of output in home and foreign consumption complete the specification of the barter model \begin{eqnarray} \omega_{xt}+\omega^*_{xt} & = & 1, \label{eq:lucas.addingup.1}\\ \omega_{yt}+\omega^*_{yt} & = & 1, \label{eq:lucas.addingup.2}\\ c_{xt} + c^*_{xt} & = & x_t, \label{eq:lucas.addingup.3}\\ c_{yt} + c^*_{yt} & = & y_t. \label{eq:lucas.addingup.4} \end{eqnarray} \ \\ \index{Social Optimum (Pareto optimum see social optimum) (begin)} {\it Digression on the social optimum.} You can solve the model by grinding out the equilibrium, but the complete markets and competitive setting makes available a `backdoor' solution strategy of solving the problem confronting a fictitious social planner. The stochastic dynamic barter economy can conceptually be reformulated in terms of a static competitive general equilibrium model--the properties of which are well known. The reformulation goes like this. We want to narrow the definition of a `good' so that it is defined precisely by its characteristics (whether it is an $x-$good or a $y-$good), the date of its delivery ($t$), and the state of the world when it is delivered ($x_t, y_t$). Suppose that there are only two possible values for $x_t$ ($y_t$) in each period---a high value $x_h (y_h)$ and a low value $x_{\ell} (y_{\ell})$. Then there are 4 possible states of the world $(x_h,y_h), (x_h,y_{\ell}), (x_{\ell},y_h)$, and $(x_{\ell},y_{\ell})$. `Good 1' is $x$ delivered at $t=0$ in state 1. `Good 2' is $x$ delivered at $t=0$ in state 2, `good 8' is $y$ delivered at $t=1$ in state 4, and so on. In this way, all possible future outcomes are completely spelled out. The reformulation of what constitutes a good corresponds to a complete system of forward markets. Instead of waiting for nature to reveal itself over time, we can have people meet and contract for all future trades today (Domestic agents agree to sell so many units of $x$ to foreign agents at $t=2$ if state 3 occurs in exchange for $q_2$ units of $y$, and so on.) After trades in future contingencies have been contracted, we allow time to evolve. People in the economy simply fulfill their contractual obligations and make no further decisions. The point is that the dynamic economy has been reformulated as a static general equilibrium model. Since the solution to the social planner's problem is a Pareto optimal allocation and you know by the fundamental theorems of welfare economics that the Pareto Optimum supports a competitive equilibrium, it follows that the solution to the planner's problem will also describe the equilibrium for the market economy.\footnote{Under certain regularity conditions that are satisfied in the relatively simple environments considered here, the results from welfare economics that we need are, i)~A competitive equilibrium yields a Pareto Optimum, and ii)~Any Pareto Optimum can be replicated by a competitive equilibrium.} We will let the social planner attach a weight of $\phi$ to the home individual and $1-\phi$ to the foreign individual. The planner's problem is to allocate the $x$ and $y$ endowments optimally between the domestic and foreign individuals each period by maximizing \begin{equation} \mbox{E}_t\sum_{j=0}^{\infty}\beta^j\left[ \phi u(c_{xt+j},c_{yt+j}) + (1-\phi)u(c^*_{xt+j},c^*_{yt+j})\right], \label{eq:lucas.e1} \index{Social planner's problem, Lucas model} \end{equation} subject to the resource constraints (\ref{eq:lucas.addingup.3}) and (\ref{eq:lucas.addingup.4}). Since the goods are not storable, the planner's problem reduces to the timeless problem of maximizing \begin{displaymath} \phi u(c_{xt},c_{yt})+(1-\phi)u(c^*_{xt},c^*_{yt}), \end{displaymath} subject to (\ref{eq:lucas.addingup.3}) and (\ref{eq:lucas.addingup.4}). The Euler equations for this problem are \index{Euler equations, Lucas model} \begin{eqnarray} \phi u_1(c_{xt},c_{yt}) &=&(1-\phi)u_1(c^*_{xt},c^*_{yt}),\label{eq:lucas.e2}\\ \phi u_2(c_{xt},c_{yt}) &=& (1-\phi) u_2(c^*_{xt},c^*_{yt}).\label{eq:lucas.e3} \end{eqnarray} (\ref{eq:lucas.e2}) and (\ref{eq:lucas.e3}) are the optimal \index{Risk sharing, efficient conditions} or efficient risk-sharing conditions. Risk-sharing is efficient when consumption is allocated so that the marginal utility of the home individual is proportional, and therefore perfectly correlated, to the marginal utility of the foreign individual. Because individuals enjoy consuming both goods and the utility function is concave, it is optimal for the planner to split the available $x$ and $y$ between the home and foreign individuals according to the relative importance of the individuals to the planner. The weight $\phi$ can be interpreted as a measure of the size of the home country in the market version of the world economy. Since we assumed at the outset that agents have equal wealth, we will let both agents be equally important to the planner and set $\phi=1/2$. Then the Pareto optimal allocation is to split the available output of $x$ and $y$ equally \begin{displaymath} c_{xt} = c^*_{xt}=\frac{x_t}{2}, \ \ \ \mbox{and} \ \ \ c_{yt} = c^*_{yt}=\frac{y_t}{2}. \end{displaymath} Having determined the optimal quantities, to get the market solution we look for the competitive equilibrium that supports this Pareto optimum. \index{Social Optimum (Pareto optimum see social optimum) (begin)} \ \\ {\it The market equilibrium}. If agents owned only their own country's firms, individuals would be exposed to idiosyncratic country-specific risk that they would prefer to avoid. The risk facing the home agent is that the home firm experiences a bad year with low output of $x$ when the foreign firm experiences a good year with high output of $y$. One way to insure against this risk is to hold a diversified portfolio of assets. A diversification plan that perfectly insures against country-specific risk and which replicates the social optimum is for each agent to hold stock in half of each country's output.\footnote{Agents cannot insure against world-wide macroeconomic risk (simultaneously low $x_t$ and $y_t$).} The stock portfolio that achieves complete insurance of idiosyncratic risk is for each individual to own half of the domestic firm and half of the foreign firm\footnote{Actually, Cole and Obstfeld~[\ref{bib:Cole.Obstfeld}]) \index{Cole, H.L.}\index{Obstfeld, M.} showed that trade in goods alone are sufficient to achieve efficient risk sharing in the present model. These issues are dealt with in the end-of-chapter problems.} \begin{equation} \omega_{xt}=\omega^*_{xt} = \omega_{yt} = \omega^*_{yt} = \frac{1}{2}. \index{Risk pooling equilbrium} \label{eq:lucas.shares} \end{equation} We call this a `pooling' equilibrium because the implicit insurance scheme at work is that agents agree in advance that they will pool their risk by sharing the realized output equally. \ \\ {\it The solution under constant relative-risk aversion utility}. Let's adopt a particular functional form for the utility function to get explicit solutions. We'll let the period utility function be constant relative-risk aversion in $C_t=c^{\theta}_{xt} c^{1-\theta}_{yt}$, a Cobb-Douglas index of the two goods \index{Cobb-Douglas, consumption index} \begin{equation} u(c_x,c_{y}) = \frac{C_t^{1-\gamma}}{1-\gamma}. \label{eq:lucas.utility.1} \index{Constant relative risk aversion utility} \end{equation} Then \begin{displaymath} u_1(c_{xt},c_{yt})=\frac{\theta C_t^{1-\gamma}}{c_{xt}}, \end{displaymath} \begin{displaymath} u_2(c_{xt},c_{yt})=\frac{(1-\theta)C_t^{1-\gamma}}{c_{yt}}. \end{displaymath} and the Euler equations \index{Euler equations, Lucas model} (\ref{eq:lucas.euler.1})--(\ref{eq:lucas.euler.6}) become \begin{eqnarray} q_t &=& \frac{1-\theta}{\theta}\frac{x_t}{y_t}, \label{eq:lucas.barter.1}\\ \frac{e_t}{x_t} & = & \beta E_t \left[\left(\frac{C_{t+1}}{C_t}\right)^{(1-\gamma)} \left(1+\frac{e_{t+1}}{x_{t+1}}\right)\right], \label{eq:lucas.barter.2}\\ \frac{e^*_t}{q_ty_t} & = & \beta E_t \left[\left(\frac{C_{t+1}}{C_t}\right)^{(1-\gamma)} \left(1+\frac{e^*_{t+1}}{q_{t+1}y_{t+1}}\right)\right]. \label{eq:lucas.barter.3} \end{eqnarray} From (\ref{eq:lucas.barter.1}) the real exchange rate $q_t$ is determined by relative output levels. (\ref{eq:lucas.barter.2}) and (\ref{eq:lucas.barter.3}) are stochastic difference equations in the `price-dividend' ratios $e_t/x_t$ and \marginpar [(79)$\Rightarrow$]{$\Leftarrow$(79)} $e^*_t/(q_ty_t)$. If you iterate forward on them as you did in (\ref{eq:mm1-sde.1}) for the monetary model, the equity price--dividend ratio can be expressed as the present discounted value of future consumption growth raised to the power $1-\gamma$. You can then get an explicit solution once you make an assumption about the stochastic process governing output. This will be covered in section \ref{sec:calibrate.lucas} below. An important point to note is that there is no actual asset trading in the Lucas model. Agents hold their investments forever and never rebalance their portfolios. The asset prices produced by the model are {\it shadow prices} that \index{Shadow prices} must be respected in order for agents to willingly to hold the outstanding equity shares according to (\ref{eq:lucas.shares}). \section{The One-Money Monetary Economy} \label{sec:lucas.onemoney} In this section we introduce a single world currency. The economic environment can be thought of as a two-sector closed economy. The idea is to introduce money without changing the real equilibrium that we characterized above. One of the difficulties in getting money into the model is that the people in the barter economy get along just fine without it. An unbacked currency in the Arrow--Debreu world that generates no consumption payoffs will not have any value in equilibrium. To get around this problem, Lucas prohibits barter in the monetary economy and imposes a `cash-in-advance' constraint that requires people to use money to buy goods. As we enter period $t$ the following specific cash-in-advance transactions technology must be adhered to. \index{Cash-in-advance transactions technology} \begin{enumerate} \item $x_t$ and $y_t$ are revealed. \item $\lambda_t$, the exogenous stochastic gross rate of change in money is revealed. The total money supply $M_t$, evolves according to \linebreak $M_t = \lambda_t M_{t-1}$. The economy-wide increment $\Delta M_t = (\lambda_{t}-1)M_{t-1}$, is distributed evenly to the home and foreign individuals where each agent receives the lump-sum transfer $\frac{\Delta M_t}{2}=(\lambda_t-1)\frac{M_{t-1}}{2}$. \item A centralized securities market opens where agents allocate their wealth towards stock purchases and the cash that they will need to purchase goods for consumption. To distinguish between the aggregate money stock $M_t$ and the cash holdings selected by agents, denote individual's choice variables by lower case letters, $m_t$ and $m^*_t$. Securities market closes. \item Decentralized goods trading now takes place in the `shopping mall.' Each household is split into `worker--shopper' pairs. The shopper takes the cash from security markets trading and buys $x$ and $y-$goods from {\it other} stores in the mall (shoppers are not allowed to buy from their own stores). The home-country worker collects the $x-$ endowment and offers it for sale in an $x-$good store in the `mall.' The $y-$goods come from the foreign country `worker' in the foreign country who collects and sells the $y-$endowment in the mall. The goods market closes. \item The cash value of goods sales are distributed to stockholders as dividends. Stockholders carry these nominal dividend payments into the next period. \end{enumerate} The state of the world is the gross growth rate of home output, foreign output, and money $(g_t,g^*_t, \lambda_t)$, and is revealed {\it prior} to trading. Because the within-period uncertainty is revealed before any trading takes place, the household can determine the precise amount of money it needs to finance the current period consumption plan. As a result, it is not necessary to carry extra cash from one period to the next. If the (shadow) nominal interest rate is always positive, households will make sure that all the cash is spent each period.\footnote{It may seem strange to talk about the interest rate and bonds since individuals do not hold nor trade bonds. That is because bonds are redundant assets in the current environment and consequently are in zero net supply. But we can compute the shadow interest rate to keep the bonds in zero net supply. The equilibrium interest rate is such that individuals have no incentive either to issue or to buy nominal debt contracts. We will use the model to price nominal bonds at the end of this section.} To formally derive the domestic agent's problem, let $P_t$ be the {\it nominal} price of $x_t$. Current-period wealth is comprised of dividends from last period's goods sales, the market value of ex-dividend equity shares and the lump-sum monetary transfer \begin{eqnarray} W_t &=& \underbrace{\frac{ P_{t-1} (\omega_{xt-1}x_{t-1}+\omega_{yt-1}q_{t-1}y_{t-1}) }{P_t}}_{\mbox{Dividends}} \nonumber\\ && + \underbrace{\omega_{xt-1}e_t + \omega_{y_t-1}e^*_t}_{\mbox{Ex-dividend share values}} + \underbrace{\frac{\Delta M_t}{2P_t}}_{\mbox{Money transfer}}. \label{eq:lucas.b1} \end{eqnarray} In the securities market, the domestic household allocates $W_t$ towards cash $m_t$ to finance shopping plans and to equities \begin{equation} W_t = \frac{m_t}{P_t} + \omega_{xt}e_t + \omega_{y_t}e^*_t. \label{eq:lucas.b2} \end{equation} The household knows that the amount of cash required to finance the current period consumption plan is \begin{equation} m_t = P_t(c_{xt}+q_tc_{yt}). \index{Cash-in-advance, Constraint} \label{eq:lucas.cia.1} \end{equation} The cash-in-advance constraint is said to bind. Substituting (\ref{eq:lucas.cia.1}) into (\ref{eq:lucas.b2}), and equating the result to (\ref{eq:lucas.b1}) eliminates $m_t$ and gives the simpler consolidated budget constraint \begin{eqnarray} &&c_{xt}+q_t c_{yt}+\omega_{xt}e_t+\omega_{yt}e^*_t = \frac{P_{t-1}}{P_t}[\omega_{xt-1}x_{t-1}+\omega_{yt-1}q_{t-1}y_{t-1}] \nonumber \\ &&\hspace*{12mm} + \frac{\Delta M_t}{2P_t}+\omega_{xt-1}e_t+\omega_{yt-1}e^*_t. \label{eq:lucas.consolid.bc} \end{eqnarray} The domestic household's problem is therefore to maximize \begin{equation} \mbox{E}_t\left(\sum_{j=0}^{\infty} \beta^j u(c_{xt+j},c_{yt+j})\right), \label{eq:lucas-maxu.3} \end{equation} subject to (\ref{eq:lucas.consolid.bc}). As before, the terms that matter at date $t$ are \begin{displaymath} u(c_{xt},c_{yt}) + \beta E_t u(c_{xt+1},c_{yt+1}), \end{displaymath} so you can substitute (\ref{eq:lucas.consolid.bc}) into the utility function to eliminate $c_{xt}$ and $c_{xt+1}$ and to transform the problem into one of unconstrained optimization. The Euler equations characterizing optimal household behavior are \marginpar [(81-83)$\Rightarrow$]{$\Leftarrow$(81-83)} \begin{eqnarray} \hspace*{-5mm}c_{yt}: & &\hspace*{-5mm} q_t u_1(c_{xt},c_{yt}) = u_2(c_{xt},c_{yt}), \\ \hspace*{-5mm}\omega_{xt}: & &\hspace*{-5mm} e_t u_1(c_{xt},c_{yt}) = \beta E_t \left[u_1(c_{xt+1},c_{yt+1})\left(\frac{P_{t}}{P_{t+1}} x_t + e_{t+1} \right)\right],\\ \hspace*{-5mm}\omega_{yt}: & &\hspace*{-5mm} e^*_t u_1(c_{xt},c_{yt}) =\beta E_t \left[u_1(c_{xt+1},c_{yt+1})\left( \frac{P_{t}}{P_{t+1}} q_t y_t + e^*_{t+1} \right)\right]. \end{eqnarray} The foreign household solves an analogous problem. Using the foreign cash-in-advance constraint \begin{equation} m^*_t = P_t(c^*_t+q_tc^*_{yt}). \end{equation} the consolidated budget constraint for the foreign household is \begin{eqnarray} &&c^*_{xt}+q_t c^*_{yt}+\omega^*_{xt}e_t+\omega^*_{yt}e^*_t = \frac{P_{t-1}}{P_t}[\omega^*_{xt-1}x_{t-1}+\omega^*_{yt-1}q_{t-1}y_{t-1}] \nonumber \\ &&\hspace*{12mm} + \frac{\Delta M_t}{2P_t}+\omega^*_{xt-1}e_t+\omega^*_{yt-1}e^*_t. \label{eq:lucas.4nhhbc} \end{eqnarray} The job is to maximize \begin{displaymath} \mbox{E}_t\left(\sum_{j=0}^{\infty} \beta^j u(c^*_{xt+j},c^*_{yt+j})\right), \end{displaymath} subject to (\ref{eq:lucas.4nhhbc}). The foreign household's problem generates a symmetric set of Euler equations \index{Euler equations, Lucas model} \marginpar [(84-86)$\Rightarrow$]{$\Leftarrow$(84-86)} \begin{eqnarray} \hspace*{-5mm}c^*_{yt}:& & q_t u_1(c^*_{xt},c^*_{yt}) = u_2(c^*_{xt},c^*_{yt}), \nonumber\\ \hspace*{-5mm}\omega^*_{xt}:& & e_t u_1(c^*_{xt},c^*_{yt}) = \beta \mbox{E}_t \left[u_1(c^*_{xt+1},c^*_{yt+1})\left(\frac{P_{t}}{P_{t+1}} x_t + e_{t+1} \right)\right],\nonumber\\ \hspace*{-5mm}\omega^*_{yt}: & & e^*_t u_1(c^*_{xt},c^*_{yt}) =\beta \mbox{E}_t \left[u_1(c^*_{xt+1},c^*_{yt+1})\left( \frac{P_{t}}{P_{t+1}} q_t y_t + e^*_{t+1} \right)\right].\nonumber \end{eqnarray} The adding-up constraints that complete the model are \begin{eqnarray} 1&=&\omega_{xt}+\omega^*_{xt},\nonumber\\ 1&=&\omega_{yt}+\omega^*_{yt},\nonumber\\ M_t &=& m_t + m^*_t, \nonumber \\ x_t &=& c_{xt}+c^*_{xt},\nonumber \\ y_t &=& c_{yt}+c^*_{yt}.\nonumber \end{eqnarray} To solve the model, aggregate the cash-in-advance constraints over the home and foreign agents and use the adding-up constraints to get \begin{equation} M_t= P_t(x_t+q_t y_t). \index{Quantity equation} \label{eq:lucas.quantity-eq1} \end{equation} This is the quantity equation for the world economy where velocity is always 1. The single money generates no new idiosyncratic country-specific risk. The equilibrium established for the barter economy (constant and equal portfolio shares) is still the perfect risk-pooling equilibrium \index{Risk pooling equilibrium} \begin{displaymath} \omega_{xt}=\omega^*_{xt}=\omega_{yt}=\omega^*_{yt} = \frac{1}{2}, \end{displaymath} \begin{displaymath} c_{xt}=c^*_{xt}=\frac{x_t}{2}, \end{displaymath} \begin{displaymath} c_{yt}=c^*_{yt}=\frac{y_t}{2}. \end{displaymath} The only thing that has changed are the equity pricing formulae, which now incorporate an `inflation premium.' \index{Inflation premium} The inflation premium arises because the nominal dividends of the current period must be carried over into the next period at which time their real value can potentially be eroded by an inflation shock. \ \\ \index{Constant relative risk aversion utility } {\it Solution under constant relative risk aversion utility}. Under the utility function (\ref{eq:lucas.utility.1}), the real exchange rate is $q_t = \left[\frac{1-\theta}{\theta}\right]\left(\frac{x_t}{y_t}\right)$. \marginpar [(87)$\Rightarrow$]{$\Leftarrow$(87)} Substituting this into (\ref{eq:lucas.quantity-eq1}), the inverse of the gross inflation rate is $\frac{P_t}{P_{t+1}}=\frac{M_t}{M_{t+1}}\frac{x_{t+1}}{x_t}$. Together, these expressions can be used to rewrite the equity pricing equations as \begin{eqnarray} \frac{e_t}{x_t} &= & \beta E_t \left[\left( \frac{C_{t+1}}{C_t} \right)^{(1-\gamma)} \left(\frac{M_{t}}{M_{t+1}} + \frac{e_{t+1}}{x_{t+1}}\right)\right], \label{eq:lucas.monetary.equity.1} \\ \frac{e^*_t}{q_ty_t} &= & \beta E_t \left[ \left( \frac{C_{t+1}}{C_t} \right)^{(1-\gamma)} \left( \frac{M_t}{M_{t+1}} + \frac{e^*_{t+1}}{q_{t+1}y_{t+1}} \right) \right].\label{eq:lucas.monetary.equity.2} \end{eqnarray} To price nominal bonds, you are looking for the shadow price of a hypothetical nominal bond such that the public willingly keeps it in zero net supply. Let $b_t$ be the nominal price of a bond that pays one dollar at the end of the period. The utility cost of buying the bond is $u_1(c_{xt},c_{yt})b_t/P_t$. In equilibrium, this is offset by the discounted expected marginal utility of the one-dollar payoff, $\beta\mbox{E}_t[u_1(c_{xt+1},c_{yt+1})/P_{t+1}]$. Under the constant relative risk aversion utility function (\ref{eq:lucas.utility.1}) we have \begin{equation} b_t = \beta E_t \left[ \left(\frac{C_{t+1}}{C_t}\right)^{(1-\gamma)} \frac{M_t}{M_{t+1}}\right]. \label{eq:bond.price} \index{Nominal bond price, Lucas model} \end{equation} If $i_t$ is the nominal interest rate, then $b_t=(1+i_t)^{-1}$. Nominal interest rates will be positive in all states of nature if $b_t<1$ and is likely to be true when the endowment growth rate and monetary growth rates are positive. \section{The Two-Money Monetary Economy} \label{sec:lucas.two.monies} To address exchange rate issues, you need to introduce a second national currency. Let the home country money be the `dollar' and the foreign country money be the `euro.' We now amend the transactions technology to require that the home country's $x$--goods can only be purchased with dollars and the foreign country's $y$--goods can only be purchased with euros. In addition, $x-$dividends are paid out in dollars and $y-$dividends are paid out in euros. Agents can acquire the foreign currency required to finance consumption plans during securities market trading. Let $P_t$ be the dollar price of $x$, $P^*_t$ be the euro price of $y$, and $S_t$ be the exchange rate expressed as the dollar price of euros. $M_t$ is the outstanding stock of dollars, $N_t$ is the outstanding stock of euros and they evolve over time according to \begin{displaymath} M_t= \lambda_t M_{t-1}, \ \ \ \mbox{and} \ \ \ N_t = \lambda^*_t N_{t-1}, \end{displaymath} where $(\lambda_t, \lambda^*_t)$ are exogenous random gross rates of change in $M$ and $N$. If the domestic household received transfers only of $M$, it faces foreign purchasing-power risk because it it also needs $N$ to buy $y$-goods. Introducing the second currency creates a new country-specific risk that households will want to hedge. The complete markets paradigm allows \index{Complete markets} markets to develop whenever there is a demand for a product. The products that individuals desire are claims to future dollar and euro transfers.\footnote{In the real world, this type of hedge might be constructed by taking appropriate positions in futures contracts for foreign currencies.} So to develop this idea, let $r_{t} $ be the price of a claim to all future dollar transfers in terms of $x$ and $r^*_{t}$ be the price to all future euro transfers in terms of $x$. Let there be one perfectly divisible claim outstanding for each of these monetary transfer streams. Let the domestic agent hold $\psi_{Mt}$ claims on the dollar streams and $\psi_{N_t}$ claims on the euro streams whereas the foreign agent holds $\psi^*_{Mt}$ claims on the dollar stream and $\psi^*_{Nt}$ claims on the euro stream. Initially, the home agent is endowed with $\psi_M=1, \psi_N=0$ and the foreign agent has $\psi^*_N=1, \psi^*_M=0$ which they are free to trade. Now to develop the problem confronting the domestic household, note that current-period wealth consists of nominal dividends paid from equity ownership carried over from last period, current period monetary transfers the market value of equity and monetary transfer claims \begin{eqnarray} W_{t}& =& \underbrace{ \frac{P_{t-1}}{P_{t}}\omega_{xt-1}x_{t-1} + \frac{S_tP^*_{t-1}}{P_{t}} \omega_{yt-1}y_{t-1}}_{\mbox{Dividends}}\nonumber\\ & + & \underbrace{ \frac{\psi_{Mt-1}\Delta M_{t}}{P_{t}} + \frac{\psi_{Nt-1}S_t \Delta N_{t}}{P_{t}}}_{\mbox{Monetary Transfers}} \nonumber\\ & + & \underbrace{ \omega_{xt-1}e_{t} + \omega_{yt-1}e^*_{t} + \psi_{Mt-1}r_{t} + \psi_{Nt-1}r^*_{t}}_{\mbox{Market value of securities}}. \label{eq:twocountbc} \end{eqnarray} This wealth is then allocated to stocks, claims to future monetary transfers, dollars and euros for shopping in securities market trading according to \begin{equation} W_t = \omega_{xt}e_t + \omega_{yt}e^*_t + \psi_{Mt}r_t + \psi_{Nt}r^*_t + \frac{m_t}{P_t} + \frac{n_tS_t}{P_t}. \label{eq:lucas.wealth.allo} \end{equation} The current values of $x_t, y_t, M_t, $ and $N_t$ are revealed before trading occurs so domestic households acquire the exact amount of dollars and euros required to finance current period consumption plans. In equilibrium, we have the binding cash-in-advance constraints \begin{equation} m_t = P_t c_{xt}, \label{eq:lucas.home.cia} \end{equation} \begin{equation} n_t= P^*_t c_{yt}, \end{equation} which you can use to eliminate $m_t$ and $n_t$ from the allocation of current period wealth to rewrite (\ref{eq:lucas.wealth.allo}) as \begin{equation} W_t = \underbrace{c_{xt} + \frac{S_tP^*_t}{P_t} c_{yt}}_{\mbox{Goods}} + \underbrace{\omega_{xt}e_t + \omega_{yt}e^*_t}_{\mbox{Equity}} + \underbrace{\psi_{Mt}r_t + \psi_{Nt}r^*_t}_{\mbox{Money transfers}}. \label{eq:lucas.two.co.bc.1} \end{equation} The consolidated budget constraint of the home individual is therefore \begin{eqnarray} c_{xt} & + & \frac{S_tP^*_t}{P_t} c_{yt} + \omega_{xt}e_t + \omega_{yt}e^*_t +\psi_{Mt}r_t + \psi_{Nt}r^*_t = \frac{P_{t-1}}{P_{t}}\omega_{xt-1}x_{t-1} \nonumber \\ &&+ \frac{S_tP^*_{t-1}}{P_{t}} \omega_{yt-1}y_{t-1} + \frac{\psi_{Mt-1}\Delta M_{t}}{P_{t}} +\frac{\psi_{Nt-1}S_t \Delta N_{t}}{P_{t}}\nonumber \\ &&+ \omega_{xt-1}e_{t} + \omega_{yt-1}e^*_{t} + \psi_{xt-1}r_{t} + \psi_{yt-1}r^*_{t}. \label{eq:lucas.bc.3} \end{eqnarray} The domestic household's problem is to maximize \begin{equation} E_t\left(\sum_{j=0}^{\infty} \beta^j u(c_{xt+j},c_{yt+j})\right) \end{equation} subject to (\ref{eq:lucas.bc.3}). The associated Euler equations are \index{Euler equations, Lucas model} \marginpar [(88-92)$\Rightarrow$]{$\Leftarrow$(88-92)} \begin{eqnarray} \hspace*{-13mm}c_{yt}: & & \frac{S_tP^*_t}{P_t} u_1(c_{xt},c_{yt}) = u_2(c_{xt},c_{yt}), \label{eq:lucas.euler.2.1}\\ \hspace*{-13mm}\omega_{xt}: & & e_t u_1(c_{xt},c_{yt}) = \beta E_t\left[u_{1}(c_{xt+1},c_{yt+1})\left(\frac{P_t}{P_{t+1}}x_t + e_{t+1}\right)\right],\label{eq:lucas.euler.2.2}\\ \hspace*{-13mm}\omega_{yt}: & & e^*_t u_1(c_{xt},c_{yt}) = \beta E_t \left[u_{1}(c_{xt+1},c_{yt+1})\left(\frac{S_{t+1}P^*_t}{P_{t+1}}y_t+ e^*_{t+1}\right)\right],\label{eq:lucas.euler.2.3}\\ \hspace*{-13mm}\psi_{Mt}: & & r_t u_1(c_{xt},c_{yt})= \beta E_t \left[u_{1}(c_{xt+1},c_{yt+1})\left(\frac{\Delta M_{t+1}}{P_{t+1}} + r_{t+1}\right)\right],\label{eq:lucas.euler.2.4}\\ \hspace*{-13mm}\psi_{Nt}: & & r^*_t u_1(c_{xt},c_{yt}) = \beta E_t\left[ u_{1}(c_{xt+1},c_{yt+1})\left(\frac{\Delta N_{t+1}S_{t+1}}{P_{t+1}}+r^*_{t+1}\right)\right].\label{eq:lucas.euler.2.5} \end{eqnarray} The foreign agent solves the analogous problem which generate a set of symmetric Euler equations, do not need to be stated here. We know that in equilibrium, the cash-in-advance constraints bind. The cash-in-advance constraints for the foreign agent are \begin{equation} m^*_t = P_tc^*_{xt}, \end{equation} \begin{equation} n^*_t = P^*_t c^*_{yt} \label{eq:lucas.foreign.cia} \end{equation} In addition, we have the adding-up constraints \begin{eqnarray} 1 &=& \psi_{Mt}+\psi^*_{Mt},\nonumber \\ 1 &=& \psi_{Nt}+\psi^*_{Nt},\nonumber \\ x_t & = & c_{xt}+c^*_{xt}, \nonumber \\ y_t &=& c_{yt} + c^*_{yt}, \nonumber \\ M_t &=& m_t + m^*_t, \nonumber \\ N_t &=& n_t + n^*_t. \nonumber \end{eqnarray} Together, the adding-up constraints and the cash-in-advance constraints give a unit-velocity quantity equation for each country \begin{displaymath} M_t = P_t x_t \index{Quantity equation, Lucas model} \end{displaymath} \begin{displaymath} N_t = P^*_t y_t, \end{displaymath} which can be used to eliminate the endogenous nominal price levels from the Euler equations. The equilibrium where people are able to pool and insure against their country-specific risks is given by \marginpar [(93)$\Rightarrow$]{$\Leftarrow$(93)} \begin{displaymath} \omega_{xt}=\omega^*_{xt} = \omega_{yt}=\omega^*_{yt} = \psi_{Mt}=\psi^*_{Mt} = \psi_{Nt}=\psi^*_{Nt} = \frac{1}{2}. \end{displaymath} Both the domestic and foreign representative households own half of the domestic endowment stream, half of the foreign endowment stream, half of all future domestic monetary transfers and half of all future foreign monetary transfers. In short, they split the world's resources in half so the pooling equilibrium supports the symmetric allocation $c_{xt}=c^*_{xt} = \frac{x_t}{2}$ and $c_{yt}=c^*_{yt} = \frac{y_t}{2}$. To solve for the nominal exchange rate $S_t$, we know by (\ref{eq:lucas.euler.2.1}) that the real exchange rate is \begin{equation} \frac{u_2(c_{xt},c_{yt})}{u_1(c_{xt},c_{yt})} = \frac{S_{t} P^*_{t}}{P_{t}} = \frac{S_{t}N_{t} x_{t}}{M_{t}y_{t}}. \label{eq:lucas.e6} \end{equation} Rearranging (\ref{eq:lucas.e6}) gives the nominal exchange rate \begin{equation} S_{t}= \frac{u_2(c_{xt},c_{yt})}{u_1(c_{xt},c_{yt})}\frac{M_t}{N_t}\frac{y_t}{x_t}. \label{eq:lucas.e7} \index{Nominal exchange rate, Lucas model} \end{equation} As in the monetary approach, the fundamental determinants of the nominal exchange rate are relative money supplies and relative GDPs. The two major differences are first that in the Lucas model the exchange rate depends on preferences (utility), and second that it does not depend explicitly on expectations of the future. \ \\ \index{Constant relative risk aversion utility} {\it The solution under constant relative risk aversion utility}. Using the utility function (\ref{eq:lucas.utility.1}), the equilibrium real exchange rate is $q_t =((1-\theta)/\theta)(x_t/y_t)$. The income terms cancel out and the exchange rate is \marginpar [(94)$\Rightarrow$]{$\Leftarrow$(94)} \begin{equation} S_t = \frac{(1-\theta)}{\theta}\frac{M_t}{N_t}. \label{eq:lucas.nomexra} \end{equation} The Euler equations are \index{Euler equations, Lucas model} \begin{eqnarray} \frac{e_t}{x_t} & = & \beta E_t \left[ \left(\frac{C_{t+1}}{C_{t}} \right)^{(1-\gamma)} \left(\frac{M_t}{M_{t+1}}+\frac{e_{t+1}}{x_{t+1}} \right) \right], \\ \frac{e^*_t}{q_ty_t} & = & \beta E_t \left[ \left(\frac{C_{t+1}}{C_t} \right)^{(1-\gamma)} \left(\frac{N_t}{N_{t+1}}+\frac{e^*_{t+1}}{q_{t+1}y_{t+1}} \right) \right], \\ \frac{r_t}{x_t} & = & \beta E_t \left[ \left(\frac{C_{t+1}}{C_{t}} \right)^{(1-\gamma)} \left( \frac{\Delta M_{t+1}}{M_{t+1}} + \frac{r_{t+1}}{x_{t+1}} \right) \right], \\ \frac{r^*_t}{x_t} & = & \beta E_t \left[ \left(\frac{C_{t+1}}{C_{t}} \right)^{(1-\gamma)} \left(\frac{1-\theta}{\theta} \frac{\Delta N_{t+1}}{N_{t+1}} + \frac{r^*_{t+1}}{x_{t+1}} \right) \right]. \end{eqnarray} Just as you can calculate the equilibrium price of nominal bonds even though they are not traded in equilibrium, you can compute the equilibrium forward exchange rate even though there is no explicit forward market. To do this, let $b_{t}$ be the date $t$ dollar price of a 1-period nominal discount bond that pays one dollar at the beginning of period $t$+1, and let $b^*_{t}$ be the date $t$ euro price of a 1-period nominal discount bond that pays one euro at the beginning of period $t$+1. By covered interest parity (\ref{ch1eq:CIA} ), the one-period ahead forward exchange rate is, \begin{equation} F_t = S_t \frac{b^*_{t}}{b_{t}}. \index{Forward exchange rate, Lucas model} \end{equation} The equilibrium bond prices are \begin{equation} b_{t} = \beta E_t\left[\left(\frac{C_{t+1}}{C_t}\right)^{1-\gamma}\frac{M_t}{M_{t+1}}\right], \label{eq:lucas.bondprice} \index{Nominal bond price, Lucas model} \end{equation} \begin{equation} b^*_{t} = \beta E_t \left[ \left(\frac{C_{t+1}}{C_{t}}\right)^{1-\gamma} \frac{N_t}{N_{t+1}} \right]. \end{equation} \clearpage \begin{table} \protect\caption{Notation for the Lucas Model} \label{tab:lucas.notation} \begin{center} \vspace*{-5mm} \begin{tabular}{|lp{4.5in}|}\hline\hline %\multicolumn{2}{|l|}{Lucas Model Notation} \\ \hline $x$& The domestic good.\\ $y$& The foreign good.\\ $q$& Relative price of $y$ in terms of $x$.\\ $c_x$& Home consumption of home good.\\ $c_y$& Home consumption of foreign good.\\ $C$ & Domestic Cobb-Douglas consumption index, $c_x^{\theta}c_y^{(1-\theta)}$.\\ $C^*$ &Foreign Cobb-Douglas consumption index, $c^{*\theta}_x c^{*(1-\theta)}_y$.\\ $c^*_x$ & Foreign consumption of home good.\\ $c^*_y$ & Foreign consumption of foreign good.\\ $\omega_x$ & Shares of home firm held by home agent.\\ $\omega_y$ & Shares of foreign firm held by home agent.\\ $\omega^*_x$ & Shares of home firm held by foreign agent.\\ $\omega^*_y$ & Shares of foreign firm held by foreign agent.\\ $s$ & Nominal exchange rate. Dollar price of euro.\\ $e$ & Price of home firm equity in terms of $x$.\\ $e^*$ & Price of foreign firm equity in terms of $x$.\\ $P$ & Nominal Price of $x$ in dollars.\\ $P^*$& Nominal Price of $y$ in euros.\\ $M$ & Dollars in circulation.\\ $N$ & Euros in circulation.\\ $\lambda_t$& Rate of growth of $M$.\\ $\lambda^*_t$& Rate of growth of $N$.\\ $m$& Dollars held by domestic household.\\ $m^*$& Dollars held by foreign household.\\ $n$& Euros held by domestic household.\\ $n^*$& Euros held by foreign household.\\ $r_t$ & Price of claim to future dollar transfers in terms of $x$.\\ $r^*_t$&Price of claim to future euro transfers in terms of $x$.\\ $\psi_{Mt}$& Shares of dollar transfer stream held by home agent.\\ $\psi_{Nt}$& Shares of euro transfer stream held by home agent.\\ $\psi^*_{Mt}$& Shares of dollar transfer stream held by foreign agent.\\ $\psi^*_{Nt}$& Shares of euro transfer stream held by foreign agent.\\ $b_t$&Price of one-period nominal bond with one-dollar payoff.\\ \hline \end{tabular} \end{center} \end{table} \clearpage \section{Introduction to the Calibration Method} \label{sec:lucas.the.calibration} \index{Calibration method} The Lucas model plays a central role in asset-pricing research. Chapter \ref{ch:fin.puzzles} covers some tests of its predictions using time-series econometric methods. At this point we introduce an alternative and popular methodology called {\it calibration}. In the calibration method, the researcher simulates the model given `reasonable' values to the underlying taste and technology parameters and looks to see whether the simulated observations match various features of the real-world data. Because there is no capital accumulation or production, the technology in the Lucas model is a stochastic process governing the evolution of $x_t$ and $y_t$. The reasonably simple mechanics underlying the model makes its calibration relatively straightforward. Our work here will set the stage for the next chapter as real business cycle researchers rely heavily on the calibration method to evaluate the performance of their models. Cooley and Prescott~[\ref{bib:Cooley-Prescott}] set out the \index{Cooley T.F.}\index{Prescott, E.C.} ingredients of the calibration method proceeds as follows. \begin{enumerate} \item Obtain a set of measurements from real-world data that we want to explain. These are typically a set of sample moments such as the mean, the standard deviation, and autocorrelations of a time-series. Special emphasis is often placed on the cross-correlations between two series which measure the extent of their {\it co-movements}. \item Solve and calibrate a candidate model. That is, assign values to the {\it deep parameters} of tastes (the utility function) and technology (the production function) that make sense or that have been estimated by others. \index{Deep parameters} \item {\it Run} ({\it simulate}) the model by computer and generate time-series of the variables that we want to explain. \item Decide whether the computer generated time-series implied by the model `look like' the observations that you want to explain.\footnote{The standard analysis is not based on classical statistical inference, although Cecchetti {\it et.al.}~[\ref{bib:CLM.2}], \index{Cecchetti, S.G.}\index{Burnside, C.} Burnside~[\ref{bib:Burnside}], Gregory and Smith~[\ref{bib:Greg_Smith}] \index{Gregory, A.W.}\index{Smith, G.W.} show how calibration methods can be combined with classical statistical inference, but the practice has not caught on.} \end{enumerate} \section{Calibrating the Lucas Model} \index{Calibration method, applied to Lucas model (begin range)} \label{sec:calibrate.lucas} % Gauss program: \lucas-w\lucas.pg1 and \lucas-w\measure.pgm \index{Calibration method, measurement in Lucas model} {\it Measurement.} The measurements that we ask the Lucas model to match are the volatility (standard deviation) and first-order autocorrelation of the gross rate of depreciation, $S_{t+1}/S_t$, the forward premium $F_t/S_t$, the realized forward profit $(F_t-S_{t+1})/S_t$, and the slope coefficient from regressing the gross depreciation on the forward premium. Using quarterly data for the U.S. and Germany from 1973.1 to 1997.1, the measurements are given in the row labeled `data' in Table \ref{tab:lucas.measure}. \begin{table}[h] \protect\caption{Measured and Implied Moments, US-Germany} \label{tab:lucas.measure} \begin{center} \vspace*{0mm} \begin{tabular}{||c|c|ccc|ccc||}\hline\hline & & \multicolumn{3}{c|}{Volatility} &\multicolumn{3}{c||}{Autocorrelation} \\ \hline & Slope & $\frac{S_{t+1}}{S_t}$ & $\frac{F_t}{S_t}$ & $\frac{(F_t-S_{t+1})}{S_t}$ & $\frac{S_{t+1}}{S_t}$ & $\frac{F_t}{S_t}$ & $\frac{(F_t-S_{t+1})}{S_t}$ \\\hline Data & -0.293& 0.060 & 0.008 & 0.061 & 0.007 & 0.888 & 0.026 \\\hline Model& -1.444& 0.014 & 0.006 & 0.029 & 0.105 & 0.006 & 0.628 \\\hline\hline \end{tabular} \end{center} \vspace*{0mm} \footnotesize Note: Model values generated with $\gamma=10$, $\theta=0.5$. \normalsize \end{table} The implied forward and spot exchange rates exhibit the so-called forward premium puzzle---that the forward premium predicts the future depreciation, but with a negative sign. Recall that the uncovered interest parity condition implies that the forward premium predicts the future depreciation with a coefficient of 1. The depreciation and the realized profit exhibit volatility of similar magnitude which is much larger than the volatility of the forward premium. All three series exhibit substantial serial dependence. \ \\ {\it Calibration.} Let random variables be denoted with a `tilde.' The `technology' that underlies the model are the exogenous monetary growth rates $\tilde{\lambda}, \tilde{\lambda}^*$, and the exogenous output growth rates $\tilde{g},\tilde{g}^*$. Let the state vector be $\tilde{\underline{\phi}} =(\tilde{\lambda},\tilde{\lambda}^*,\tilde{g},\tilde{g}^*)$. The process governing the state vector is a finite-state Markov chain \index{Markov chain} with stationary probabilities (see the chapter appendix). Each element of the state vector is allowed to be in either of one of two possible states--high and low. A `1' subscript indicates that the variable is in the high growth state and a `2' subscript indicates that the variable is in the low growth state. Therefore, $\lambda=\lambda_1$ indicates high domestic money growth, $\lambda=\lambda_2$ indicates low domestic money growth. Analogous designations hold for the other variables. The 16 possible states of the world are \ \\ \begin{tabular}{cc} $\underline{\phi}_1 =(\lambda_1,\lambda^*_1,g_1,g^*_1)$&\hspace*{20mm}$\underline{\phi}_9 =(\lambda_2,\lambda^*_1,g_1,g^*_1)$\\ $\underline{\phi}_2 =(\lambda_1,\lambda^*_1,g_1,g^*_2)$&\hspace*{20mm}$\underline{\phi}_{10}=(\lambda_2,\lambda^*_1,g_1,g^*_2)$\\ $\underline{\phi}_3 =(\lambda_1,\lambda^*_1,g_2,g^*_1)$&\hspace*{20mm}$\underline{\phi}_{11}=(\lambda_2,\lambda^*_1,g_2,g^*_1)$\\ $\underline{\phi}_4 =(\lambda_1,\lambda^*_1,g_2,g^*_2)$&\hspace*{20mm}$\underline{\phi}_{12}=(\lambda_2,\lambda^*_1,g_2,g^*_2)$\\ $\underline{\phi}_5 =(\lambda_1,\lambda^*_2,g_1,g^*_1)$&\hspace*{20mm}$\underline{\phi}_{13}=(\lambda_2,\lambda^*_2,g_1,g^*_1)$\\ $\underline{\phi}_6 =(\lambda_1,\lambda^*_2,g_1,g^*_2)$&\hspace*{20mm}$\underline{\phi}_{14}=(\lambda_2,\lambda^*_2,g_1,g^*_2)$\\ $\underline{\phi}_7 =(\lambda_1,\lambda^*_2,g_2,g^*_1)$&\hspace*{20mm}$\underline{\phi}_{15}=(\lambda_2,\lambda^*_2,g_2,g^*_1)$\\ $\underline{\phi}_8 =(\lambda_1,\lambda^*_2,g_2,g^*_2)$&\hspace*{20mm}$\underline{\phi}_{16}=(\lambda_2,\lambda^*_2,g_2,g^*_2)$.\\ \end{tabular} \ \\ \index{Markov chain, Transition matrix} We will denote the $16 \times 16$ probability transition matrix for the state by ${\bf P}$, where $p_{ij}=\mbox{P}[\tilde{\underline{\phi}}_{t+1}=\underline{\phi}_j | \tilde{\underline{\phi}}_{t}=\underline{\phi}_i]$ the $ij-$th element. The price of the domestic and foreign currency bonds are, \linebreak $ b_t=\beta\mbox{E}_t[(g^{\theta}_{t+1}g^{*(1-\theta)}_{t+1})^{1-\gamma}]/\lambda_{t+1},$ and $ b^*_t=\beta\mbox{E}_t[(g^{\theta}_{t+1}g^{*(1-\theta)}_{t+1})^{1-\gamma}]/\lambda^*_{t+1},$ under the constant relative risk aversion utility function (\ref{eq:lucas.utility.1}). Since their values depend on the state of the world, we say that these are {\it state-contingent} bond prices. Next, define $G=[(g^{\theta}g^{*(1-\theta)})^{1-\gamma}]/\lambda$ and $G^*=[(g^{\theta}g^{*(1-\theta)})^{1-\gamma}]/\lambda^*$, and let $d=\lambda/\lambda^*$ be the gross rate of depreciation of the home currency. The possible values of $G$ and $G^*$ and $d$ are given in Table \ref{tab:calib.lucas.states}, %------------------------------------------------------------------- \begin{table}[t] \protect\caption{Possible State Values} \label{tab:calib.lucas.states} \vspace*{3mm} \begin{tabular}{||l|l|l||}\hline\hline $G_{1} =[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{1} =[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{1}= \lambda_1/\lambda^*_1 $\\ $G_{2} =[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{2} =[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{2}= \lambda_1/\lambda^*_1 $\\ $G_{3} =[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{3} =[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{3}= \lambda_1/\lambda^*_1 $\\ $G_{4} =[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{4} =[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{4}= \lambda_1/\lambda^*_1 $\\ $G_{5} =[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{5} =[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{5}= \lambda_1/\lambda^*_2 $\\ $G_{6} =[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{6} =[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{6}= \lambda_1/\lambda^*_2 $\\ $G_{7} =[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{7} =[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{7}= \lambda_1/\lambda^*_2 $\\ $G_{8} =[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_1$ &$G^*_{8} =[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{8}= \lambda_1/\lambda^*_2 $\\ $G_{9} =[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{9} =[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{9}= \lambda_2/\lambda^*_1 $\\ $G_{10}=[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{10}=[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{10}=\lambda_2/\lambda^*_1 $ \\ $G_{11}=[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{11}=[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{11}=\lambda_2/\lambda^*_1 $ \\ $G_{12}=[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{12}=[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_1$ &$d_{12}=\lambda_2/\lambda^*_1 $ \\ $G_{13}=[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{13}=[(g_1^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{13}=\lambda_2/\lambda^*_2 $ \\ $G_{14}=[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{14}=[(g_1^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{14}=\lambda_2/\lambda^*_2 $ \\ $G_{15}=[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{15}=[(g_2^{\theta}g_1^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{15}=\lambda_2/\lambda^*_2 $ \\ $G_{16}=[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda_2$ &$G^*_{16}=[(g_2^{\theta}g_2^{*(1-\theta)})^{1-\gamma}]/\lambda^*_2$ &$d_{16}=\lambda_2/\lambda^*_2 $ \\ \hline\hline \end{tabular} \end{table} Suppose the current state is $\underline{\phi}_k$. By (\ref{eq:lucas.nomexra}), the spot exchange rate is given by $(1-\theta)d_k/\theta$. The domestic bond price is \\ $b_k=\beta \sum_{i=1}^{16}p_{k,i}G_i,$ the foreign bond price is $b^*_k=\beta \sum_{i=1}^{16}p_{k,i}G^*_i,$ the expected gross change in the nominal exchange rate is $\sum_{i=1}^{16}p_{k,i}d_i$, and the state-$k$ contingent risk premium is \index{Risk premium, Lucas state contingent} \begin{displaymath} rp_k=\sum_{i=1}^{16}p_{k,i}d_i- \frac{(\sum_{i=1}^{16}p_{k,i}G^*_i)}{(\sum_{i=1}^{16}p_{k,i}G_i)}. \end{displaymath} Next, we must estimate the probability transition matrix. The first question is whether we should use consumption data or GDP? In the Lucas model, consumption equals GDP so there is no theoretical presumption as to which series we should use. Since prices depend on utility which depends on consumption. From this perspective, it makes sense to use consumption data which is what we do. The consumption and money data are from the {\it International Financial Statistics} and are in {\it per capita} terms. The next question is what estimation technique to use? Using generalized method of moments or simulated method of moments (see chapter \ref{chapter.emetrics}.\ref{sec:emetrics.gmm} and chapter \ref{chapter.emetrics}.\ref{ssec:emetrics.smm}) to estimate the transition matrix might be good choices if the dimensionality of the problem were smaller. Since we don't have a very long time span of data, it turns out that estimating the transition probability matrix ${\bf P}$ by GMM or by the SMM does not work well. Instead, we `estimate' the transition probabilities by counting the relative frequency of the transition events. Let's classify the growth rate of a variable as being high-growth whenever it lies above its sample mean and in the low-growth state otherwise. Then set high-growth states $\lambda_1$, $\lambda^*_1$, $g_1$, and $g^*_1$ to the average of the high-growth rates found in the data. Similarly, assign the low-growth states $\lambda_2$, $\lambda^*_2$, $g_2$, and $g^*_2$ to the average of the low-growth rates found in the data. Using per capita consumption and money data for the US and Germany, and viewing the US as the home country, the estimates of the high and low state values are \begin{tabular}{l} $\lambda_1= 1.010$--average US money growth good state,\\ $\lambda_2= 0.990$--average US money growth bad state, \\ $\lambda^*_1= 1.011$--average German money growth good state,\\ $\lambda^*_2= 0.991$--average German money growth bad state, \\ $g_1=1.009$--average US consumption growth good state, \\ $g_2=0.998$--average US consumption growth bad state, \\ $ g^*_1= 1.012$--average German consumption growth good state,\\ $ g^*_2= 0.993$--average German consumption growth bad state.\\ \end{tabular} \ \\ Now classify the data into the $\underline{\phi}$ states according to whether the observations lie above or below the mean then set the transition probabilities $p_{jk}$ equal to the relative frequency of transitions from state $\phi_j$ to $\phi_k$ found in the data. The ${\bf P}$ estimated in this fashion, rounded to 2 significant digits, is \index{Markov chain, Transition matrix} \footnotesize \begin{displaymath} \left[\begin{array}{cccccccccccccccc} .00&.00&.20&.00&.40 &.00& .00 &.00 &.20 &.00 &.00& .00& .20& .00& .00&.00\\ .20&.20&.20&.20&.00 &.20& .00 &.00 &.00 &.00 &.00& .00& .00& .00& .00&.00 \\ .17&.17&.00&.17&.17 &.00& .00 &.00 &.00 &.00 &.00& .00& .00& .00& .17&.17 \\ .00&.00&.00&.00&.17 &.00& .00 &.00 &.00 &.17 &.33& .17& .00& .00& .17&.00 \\ .08&.08&.08&.08&.15 &.08& .08 &.08 &.15 &.08 &.08& .00& .00& .00& .00&.00 \\ .20&.00&.00&.00&.20 &.00& .00 &.00 &.00 &.00 &.20& .00& .00& .20& .20&.00 \\ .00&.00&.00&.20&.40 &.00& .00 &.20 &.00 &.00 &.00& .00& .20& .00& .00&.00 \\ .25&.00&.00&.00&.00 &.50& .00 &.00 &.00 &.00 &.00& .00& .00& .00& .00&.25 \\ .00&.14&.00&.00&.00 &.00& .14 &.00 &.14 &.14 &.00& .00& .00& .14& .14&.14 \\ .00&.00&.00&.00&.00 &.00& .25 &.00 &.25 &.00 &.00& .25& .25& .00& .00&.00 \\ .00&.00&.20&.00&.20 &.00& .00 &.00 &.20 &.20 &.00& .20& .00& .00& .00&.00 \\ .00&.25&.00&.25&.25 &.00& .00 &.00 &.00 &.00 &.00& .00& .00& .25& .00&.00 \\ .00&.00&.00&.00&.13 &.00& .00 &.13 &.13 &.00 &.13& .13& .25& .00& .13&.00 \\ .00&.00&.20&.00&.00 &.00& .00 &.00 &.00 &.00 &.00& .00& .20& .00& .40&.20 \\ .00&.00&.00&.00&.25 &.00& .25 &.13 &.00 &.00 &.00& .00& .13& .13& .00&.13 \\ .00&.00&.00&.20&.00 &.20& .00 &.00 &.00 &.00 &.00& .00& .20& .20& .20&.00 \\ \end{array} \right] \end{displaymath} \normalsize \ \\ {\it Results}. We set the share of home goods in consumption to be $\theta=1/2$, the coefficient of relative risk aversion to be $\gamma=10$, and the subjective discount factor to be $\beta=0.99$ and simulate the model as follows. Draw a sequence of $T$ realizations of the gross change in the exchange rate, the forward premium, and the risk premium with the initial state vector drawn from probabilities of the initial probability vector, $\underline{v}$. Let $u_t$ be a $iid$ uniform random variable on $[0,1]$. The rule for determining the initial state is, \ \\ \begin{tabular}{ll} $\underline{\phi}_1 $ if & $u_t < v_1$\\ $\underline{\phi}_2 $ if & $v_1 < u_t < \sum_{j=1}^2 v_j$\\ $\underline{\phi}_3 $ if & $\sum_{j=1}^2 v_j < u_t < \sum_{j=1}^3 v_j$\\ \vdots & \vdots \\ $\underline{\phi}_{16} $ if & $\sum_{j=1}^{15} v_j < u_t < 1$\\ \end{tabular} \ \\ For subsequent observations, suppose that at $t=1$ we are in state $k$. Then the state at $t=2$ is determined by \ \\ \begin{tabular}{ll} $\underline{\phi}_1 $ if & $u_t < p_{k1}$\\ $\underline{\phi}_2 $ if & $p_{k1} < u_t < \sum_{j=1}^2 p_{kj}$\\ $\underline{\phi}_3 $ if & $\sum_{j=1}^2 p_{kj} < u_t < \sum_{j=1}^3 p_{kj}$\\ \vdots & \vdots \\ $\underline{\phi}_{16} $ if & $\sum_{j=1}^{15} p_{kj} < u_t < 1$\\ \end{tabular} \ \\ Figure \ref{fig:lucas.1}.A shows 97 simulated values of $S_{t+1}/S_t$ and $F_t/S_t$ generated from the model. Notice that these two series appear to be negatively correlated. This certainly is not what you would expect to see if uncovered interest parity held. But we know from chapter \ref{chapter.markets} that market participation of risk-averse agents is potentially a key reason behind the failure of UIP. Figure \ref{fig:lucas.1}.B shows the simulated values of the predicted forward payoff $\mbox{E}_t(S_{t+1}-F_t)/S_t$ and the realized payoff $(S_{t+1}-F_t)/S_t.$ The thing to notice here is that the predicted payoff or risk premium seems too small to explain the data. The largest predicted state contingent risk premium is actually only 0.14 percent on a quarterly basis. \index{Risk premium, Lucas state contingent} \marginpar [(96)$\Rightarrow$]{$\Leftarrow$(96)} %\begin{center} %\begin{tabular}{l|ccccc}\hline %State & 1 & 2 & 3 & 4 \\ %Risk Premium &0.011 & -0.040 & 0.043 & 0.094 \\\hline %State & 5 & 6 & 7 & 8\\ %Risk Premium &-0.011& 0.007 & 0.023 & -0.054 \\\hline %State & 9 & 10 & 11 & 12 \\ %Risk Premium & 0.039 & 0.113 & 0.137 & 0.091 \\\hline %State & 13 & 14 & 15 & 16\\ %Risk Premium & -0.007 & 0.001 &0.069 & -0.037\\\hline %\end{tabular} %\end{center} Now we generate 10000 time-series observations from the model and use them to calculate slope coefficient, volatility, and autocorrelation coefficients shown in the row labeled `model' in Table \ref{tab:lucas.measure}. As can be seen, the implied volatility of the depreciation and of the realized profit is much too small. The implied persistence of the depreciation and the forward premium is also too low to be consistent with the data. The model does predict that the forward rate is a biased predictor of the future spot rate due to the presence of a risk premium. However, the size of the implied risk premium appears to be too small to provide an adequate explanation for the data. We study the forward premium puzzle in greater detail in chapter \ref{ch:fin.puzzles}. \index{Lucas model (end 1)} \index{Calibration method, applied to Lucas model (end range)} \clearpage %------------------------------------------------------------ \begin{figure}[ht] \begin{flushleft} \vspace*{4.5in} \hspace*{0in}\special{wmf:lucas1.wmf x=5.08in y=4.5in} \caption{From the Lucas Model. A: Implied gross one-period ahead change in nominal exchange rate $S_{t+1}/S_t$ and current forward premium $F_t/S_t$ (in boxes). B. Implied ex post forward payoff $(S_{t+1}-F_t)/S_t$ (jagged line) and risk premium $\mbox{E}_t(S_{t+1}-F_t)/S_t$ (smooth line).} \label{fig:lucas.1} \end{flushleft} \end{figure} %------------------------------------------------------------ \clearpage \ \\ \begin{tabular}{|lp{4.5in}|}\hline \multicolumn{2}{|l|}{\underline{Lucas Model Summary}} \\ & \\ 1.& It is a flexible-price, complete markets, dynamic general equilibrium model with optimizing agents. It is logically consistent and provides the micro-foundations for international asset pricing.\\ 2.& The Lucas model provides a framework for pricing assets, including the exchange rate, in an international setting. The exchange rate depends on the same set of fundamental variables as predicted by the monetary model. The empirical predictions of the model will be developed more fully in chapter \ref{ch:fin.puzzles}. \\ 3. & There is no trading volume for any of the assets. The prices derived in the model are shadow values under which the existing stock of assets are willingly held by the agents.\\ 4. & Output is taken to be exogenous so the model not well equipped to explain quantities such as the current account.\\ 5. &The Lucas model is designed to help us understand the determination of the prices of assets---exchange rates, bonds, and stocks---that are consistent with equilibrium choices of consumption. Because it is an endowment model, the dynamics of consumption (or alternatively output) are taken exogeneously. This is actually a virtue of the model since a model with production, while perhaps more `realistic,' does not change the underlying asset pricing formulae which are based on the Euler equations for the consumer's problem but complicates the job by forcing us to write down a model where equilibrium decisions of the firm generate not only realistic asset price movements but also realistic output dynamics. It is therefore not necessary or even desirable to introduce production in order to understand equilibrium asset pricing issues. \\ \hline \end{tabular} \clearpage \small \index{Markov chain (begin range)} \section*{Appendix--Markov Chains} Let $X_t$ be a random variable and $x_t$ be a particular realization of $X_t$. A Markov chain is a stochastic process $\{X_t\}_{t=0}^{\infty}$ with the property that the information in the current realized value of $X_t=x_t$ summarizes the entire past history of the process. That is, \begin{equation} \mbox{P}[X_{t+1}=x_{t+1}|X_{t}=x_t,X_{t-1}=x_{t-1},\ldots,X_0=x_0] = \mbox{P}[X_{t+1}=x_{t+1}|X_t=x_t]. \label{eq:lucas.markov.1} \end{equation} A key result that simplifies probability calculations of Markov chains is, \begin{result} If $\{X_t\}_{t=0}^{\infty}$ is a Markov chain, then \marginpar [(98)$\Rightarrow$]{$\Leftarrow$(98)} \begin{displaymath} \mbox{P}[X_t=x_t \cap X_{t-1}=x_{t-1}\cap \cdots \cap X_0=x_0] = \nonumber \end{displaymath} \begin{equation} \mbox{P}[X_t=x_t|X_{t-1}=x_{t-1}]\cdots \mbox{P}[X_1=x_1|X_0=x_0]\mbox{P}[X_0=x_0]. \label{eq:lucas.markov.2} \end{equation} \label{result:markov.1} \end{result} {\bf Proof}: Let $A_j$ be the event $(X_j=x_j)$. You can write the left side of (\ref{eq:lucas.markov.2}) as, \begin{eqnarray} \mbox{P}(A_t \cap A_{t-1}\cap \cdots \cap A_0) & = & \mbox{P}(A_t|\bigcap_{j=0}^{t-1}A_j) \mbox{P}(\bigcap_{j=0}^{t-1}A_j) \ \ \mbox{(multiplication rule)} \nonumber \\ &=& \mbox{P}(A_t|A_{t-1})\mbox{P}(\bigcap_{j=0}^{t-1}A_j) \ \ \mbox{(Markov chain property)} \nonumber \\ &=& \mbox{P}(A_t|A_{t-1})\mbox{P}(A_{t-1}|\bigcap_{j=0}^{t-2}A_j) \mbox{P}(\bigcap_{j=0}^{t-2}A_j) \ \ \mbox{(mult. rule)} \nonumber \\ &=& \mbox{P}(A_t|A_{t-1})\mbox{P}(A_{t-1}|A_{t-2}) \mbox{P}(\bigcap_{j=0}^{t-2}) \ \ \mbox{(Markov chain)} \nonumber \\ & \vdots & \nonumber \\ &=& \mbox{P}(A_t|A_{t-1})\mbox{P}(A_{t-1}|A_{t-2})\cdots \mbox{P}(A_{1}|A_{0})\mbox{P}(A_0) \nonumber \end{eqnarray} \hspace*{4in}\rule{3mm}{3mm} Let $\lambda_j$, $j=1,\ldots,N$ denote the possible states for $X_t$. A Markov chain has stationary probabilities if the transition probabilities from state $\lambda_i$ to $\lambda_j$ are time-invariant. That is, \begin{displaymath} \mbox{P}[X_{t+1}=\lambda_j|X_t=\lambda_i] = p_{ij} \index{Markov chain, Transition matrix} \end{displaymath} Notice that in Markov chain analysis the first subscript denotes the state that you condition on. For concreteness, consider a Markov chain with two possible states, $\lambda_1$ and $\lambda_2$, with transition matrix, \begin{displaymath} {\bf P}=\left[\begin{array}{cc}p_{11}& p_{12} \\ p_{21}&p_{22} \end{array} \right], \end{displaymath} where the rows of ${\bf P}$ sum to 1. \begin{result} The transition matrix over $k$ steps is \begin{displaymath} {\bf P}^k=\underbrace{{\bf P}{\bf P}\cdots{\bf P}}_{\mbox{k}} \end{displaymath} \end{result} {\bf Proof.} For the two state process, define \begin{eqnarray} p^{(2)}_{ij} &=& \mbox{P}[X_{t+2}=\lambda_j|X_t=\lambda_i]\nonumber \\ &=& \mbox{P}[X_{t+2}=\lambda_j \cap X_{t+1}=\lambda_1|X_t=\lambda_i] + \mbox{P}[X_{t+1}=\lambda_j \cap X_{t+1}=\lambda_2|X_t=\lambda_i] \nonumber \\ &=& \sum_{k=1}^{2}\mbox{P}[X_{t+1}=\lambda_j \cap X_{t+1}=\lambda_k|X_t=\lambda_i] \nonumber \\ & = & \frac{\mbox{P}[X_{t+1}=\lambda_j \cap X_{t+1}=\lambda_k \cap X_t=\lambda_i]}{\mbox{P}(X_t=\lambda_i)} \label{eq:lucas.app.1} \end{eqnarray} Now by property \ref{result:markov.1}, the numerator in last equality can be decomposed as, \begin{equation} \mbox{P}[X_{t+2}=\lambda_j|X_{t+1}=\lambda_k] \mbox{P}[X_{t+1}=\lambda_k|X_t=\lambda_i] \mbox{P}[X_t=\lambda_i] \label{eq:lucas.app.2} \end{equation} Substituting (\ref{eq:lucas.app.2}) into (\ref{eq:lucas.app.1}) gives, \begin{eqnarray} p^{(2)}_{ij}&=& \sum_{k=1}^{2}\mbox{P}[X_{t+1}=\lambda_j|X_{t+1}=\lambda_k] \mbox{P}[X_{t+1}=\lambda_k|X_t=\lambda_i] \nonumber \\ &=& \sum_{k=1}^{2}p_{kj}p_{ik} \nonumber \end{eqnarray} which is seen to be the $ij-$th element of the matrix ${\bf P}{\bf P}$. The extension to any arbitrary number of steps forward is straightforward. \hspace*{2mm}\rule{3mm}{3mm} \marginpar [(99)$\Rightarrow$]{$\Leftarrow$(99)} \index{Markov chain (end range)} \clearpage \small \section*{Problems} \begin{enumerate} \item {\it Risk sharing in the Lucas model } [Cole-Obstfeld~(1991)]. \index{Cole, H.L.}\index{Obstfeld, M.} Let the period utility function be $u(c_{x},c_{y}) = \theta \ln c_x + (1-\theta) \ln c_y$ for the home agent and $u(c^*_{x},c^*_{y}) = \theta \ln c^*_x + (1-\theta) \ln c^*_y$ for the foreign agent. Suppose That capital is internationally immobile. The home agent owns all of the $x-$endowment $(\phi_x=1)$, the foreign agent owns all of the $y-$endowment $(\phi^*_y=1)$. Show that in the equilibrium under portfolio autarchy, trade in goods alone is sufficient to achieve efficient risk sharing. %{\sf Solution: The planner's problem is to maximize %\begin{displaymath} %\phi[\theta \ln c_x+(1-\theta)\ln c_y]+(1-\phi)[\theta \ln %c^*_x+(1-\theta)\ln c^*_y] %\end{displaymath} %subject to %\begin{displaymath} %c_x + c^*_x = x, \ \ \ \mbox{and} \ \ \ c_y + c^*_y = y. %\end{displaymath} %With $\phi=1/2$, the efficient risk sharing conditions are, %\begin{displaymath} %(1-\phi)c_{xt}=\phi c^*_{xt}, \ \ \ (1-\phi) c_{yt}=\phi c^*_{yt}. %\end{displaymath} % %If the home agent owns all of the $x-$ firm, the foreign agent owns %all of the $y-$ firm, under zero capital mobility with goods trade, %the home agent's problem is to maximize %\begin{displaymath} %\theta \ln c_{xt} + (1-\theta)\ln c_{yt} %\end{displaymath} %subject to %\begin{displaymath} %c_{xt} + q_t c_{yt} = x_t. %\end{displaymath} %Taking $q_t$ as given, we get $q_t = %[(1-\theta)/\theta](c_{xt}/c_{yt})$. Substitute this into the budget %constraint to get the demand for $x$, $c_{xt}=\theta x_t$. %The foreign agent maximizes %\begin{displaymath} %\theta \ln c^*_{xt} + (1-\theta)\ln c^*_{yt} %\end{displaymath} %subject to %\begin{displaymath} %c^*_{xt} + q_t c^*_{yt} = q_ty_t. %\end{displaymath} %The foreign demand for $y$ is $c^*_{yt}=(1-\theta)y_t$. The %equilibrium under goods trade with zero capital mobility is %therefore, %\begin{eqnarray} %c_{xt} = \theta x_t, & c^*_{xt}=(1-\theta)x_t \nonumber \\ %c_{yt} = \theta y_t, & c^*_{yt}=(1-\theta)y_t \nonumber %\end{eqnarray} %We get, % %\begin{tabular}{ccc} % & \multicolumn{2}{c}{Marginal Utility} \\ % & $x_t $ & $y_t$ \\ \hline %Home & $\theta/x_t$ & $(1-\theta)/y_t$ \\ %Foreign &$ \theta/x_t$ &$ (1-\theta)/y_t$ \\\hline %\end{tabular} % %The equilibrium attains the efficient risk-sharing condition with %$\phi=\theta$. %} \item Consider now the single-good model. Let $x_t$ be the home endowment and $x^*_t$ be the foreign endowment of the same good. The planner's problem is to maximize \begin{displaymath} \phi \ln c_t + (1-\phi) \ln c^*_t \end{displaymath} subject to $c_t + c^*_t = x_t + x^*_t$. Under zero capital mobility, the home agent's problem is to maximize $\ln(c_t)$ subject to $c_t = x_t$. The foreign agent maximizes $\ln(c^*_t)$ subject to $c^*_t=x^*_t$. Show that asset trade is necessary in this case to achieve efficient risk sharing. %{\sf Solution. The efficient risk-sharing condition is %$\phi/c_t = (1-\phi)/c^*_t$. %The implied allocations are %$c_t = \phi(x_t+x^*_t)$ %and %$c^*_t = (1-\phi)(x_t+x^*_t)$. %Under zero capital mobility, no goods trade can occur. The %equilibrium is that the agents consume their endowments and live in %complete autarchy %$c_t = x_t$ %and %$c^*_t=x^*_t$. If $x_t$ and $x^*_t$ %are imperfectly correlated, the marginal utility of the home and %foreign agents will also be imperfectly correlated. %} \item {\it Nontraded goods}. \index{Nontraded goods} Let $x$ and $y$ be traded as in the model of this chapter. In addition, let $N$ be a nonstorable nontraded domestic good generated by an exogenous endowment, and let $N^*$ be a nonstorable nontraded foreign good also generated by exogenous \marginpar [(100)$\Rightarrow$]{$\Leftarrow$(100)} endowment. Let the domestic agent's utility function be $u(c_{xt},c_{yt},c_N)=(C^{1-\gamma})/(1-\gamma)$ where $C=c_x^{\theta_1}c_y^{\theta_2}c_N^{\theta_3}$ with $\theta_1+\theta_2+\theta_3=1$. The foreign agent has the same utility function. Show that trade in goods under zero capital mobility does not achieve efficient risk sharing. %{\sf Solution. The planner's problem is to maximize %\begin{displaymath} %\phi u(c_x,c_y,c_N) + (1-\phi)u(c^*_x,c^*_y,c^{*}_{N^*}) %\end{displaymath} %subject to %\begin{eqnarray} %x &= & c_x + c^*_x \nonumber \\ %y &= & c_y + c^*_y \nonumber \\ %c_N&=& N \nonumber \\ %c^*_N{^*}&=& N^* \nonumber %\end{eqnarray} %We have, %\begin{eqnarray} %u_1=\theta_1\frac{C^{1-\gamma}}{c_x}, & u^*_1 = \theta_1 %\frac{C^{*(1-\gamma)}}{c^*_x} \nonumber \\ %u_2=\theta_2\frac{C^{1-\gamma}}{c_y}, & u^*_2 = \theta_2 %\frac{C^{*(1-\gamma)}}{c^*_y} \nonumber \\ %u_3=\theta_3\frac{C^{1-\gamma}}{c_N}, & u^*_3 = \theta_3 %\frac{C^{*(1-\gamma)}}{c^*_{N^*}} \nonumber %\end{eqnarray} %Efficient risk-sharing requires, %\begin{eqnarray} %\phi u_1 &=& (1-\phi)u^*_1 \nonumber \\ %\phi u_2 &=& (1-\phi)u^*_2 \nonumber \\ %C_N &=& N \nonumber \\ %C^*_{N^*} &=& N^* \nonumber %\end{eqnarray} %The first two conditions for this utility function give, %\begin{eqnarray} %\phi %c_x^{\theta_1(1-\gamma)-1}c_y^{\theta_2(1-\gamma)}N^{\theta_3(1-\gamma)} %&=& %(1-\phi)[x-c_x]^{\theta_1(1-\gamma)-1}[y-c_y]^{\theta_2(1-\gamma)} %N^{*\theta_3(1-\gamma)}\nonumber \\ %\phi %c_x^{\theta_1(1-\gamma)}c_y^{\theta_2(1-\gamma)-1}N^{\theta_3(1-\gamma)} %&=& %(1-\phi)[x-c_x]^{\theta_1(1-\gamma)}[y-c_y]^{\theta_2(1-\gamma)-1} %N^{*\theta_3(1-\gamma)}\nonumber %\end{eqnarray} %The planner's allocate rule of $x$ and $y$ to the domestic and foreign %agents is contingent on $N $ and $N^*$. % %In the zero-capital mobility but trade in goods problem, domestic guy %maximizes $u(c_x,c_y,c_N)$ subject to %\begin{displaymath} %c_x + q_y c_y + q_Nc_N = x + q_N N %\end{displaymath} %where $ q_y$ is the relative price of $y$ in terms of $x$ and $q_N$ %is the relative price of $N$ in terms of $x$. From the first-order %conditions, we get, %\begin{eqnarray} %q_y & = & \frac{\theta_2 c_x}{\theta_1 c_y}\nonumber \\ %q_N &=& \frac{\theta_3c_x}{\theta_1 N} \nonumber %\end{eqnarray} %Substitute the first Euler equation into the budget constraint to get %the domestic demand function for $x$, %\begin{displaymath} %c_x = \frac{\theta_1}{\theta_1+\theta_2} x %\end{displaymath} % %Foreign guy maximizes $u(c^*_x,c^*_y, c^*_N)$ subject to %\begin{displaymath} %c^*_x+q_y c^*_y + q^*_{N^*} c^*_{N^*} = q_y y + q^*_{N^*} N^* %\end{displaymath} %where $q^*_{N^*}$ is the relative price of $N^*$ in terms of $x$. %} \item Derive the exchange rate in the Lucas model under log utility, $U(c_{xt},c_{yt})=\theta \ln(c_{xt})+(1-\theta)\ln(c_{yt})$ and \marginpar [(101)$\Rightarrow$]{$\Leftarrow$(101)} compare it with the solution under constant relative risk aversion utility. \item Use the high and low growth states and the transition matrix given in section 4.5 to solve for the price-dividend ratios for \marginpar [(102)$\Rightarrow$]{$\Leftarrow$(102)} equities. What does the Lucas model have to say about the volatility of stock prices? How does the behavior of equity prices in the monetary economy differ from the behavior of equity prices in the barter economy? \end{enumerate} \normalsize