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Economic Dynamics
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In 「Economic Dynamics」we study the change through time, or the absence of change,in variables (economic indicators) representing economic conditions. Let me explain what kind of problems we are going to deal with.
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I think that the students who took microeconomics at undergraduate level is familiar with the following optimization problem. \[
+\begin{aligned}
+ &\max_{c\in \mathbb{R}^N_+} u(c) \\
+ &\text {subject to}\quad p \cdot c \le I
+\end{aligned}
+\]
+
The problem concerns the choice of consumption \(c = (c_1, \dots, c_N)\) that maximixes utility \(u(c)\) given prices \(p = (p_1, \dots, p_N)\) and income \(I\). The standar assumption for this consumer model is that the economy has different types of goods \(N\), whose the price of good \(n\), \(n = 1, \dots, N\), is set at \(p_n\). Consumers wil purchase it in \(c_n\) units. To purchase all goods, consumer need to pay \(p \cdot c = \sum_{n=1}^N p_n x_n\) subject to its budget constraint.
+
The order of lining up each goods \(n = 1, \dots, N\) is completely arbitrary. There is also no reason for lining up \(c_n\) and \(c_{n + 1}\) in this order. Therefore, for example, even if monotonicity, \(c_1 < c_2 < c_3,\dots\), is obtained, it can not be interpreted as if it happened by chance.
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Let’s try to change the interpretation without changing the model. Essentially, there is only one good in the economy. For example, \(c_1\) is consumed in 2001, \(c_2\) in 2002, and each \(c_n\) is interpreted as a variable representing economic activity at a certain point in time. Such change in the interpretation changes the interpretation of the result without changing the model to be solved. In other words, the above-mentioned monotonicity means that consumption will increase year by year. For the sake of explanation, \(N\) is finite, but we often consider cases where \(N = \infty\). When \(N\) is finite, this model is called finite horizon model. Otherwise, it is called infinite horizon model.
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Each variable solves a time-labelled model. The economic dynamics has the main task to investigate the temporal nature that the solution satisfies. We are interested in the following propositions.
-- Theoretical analysis of linear rational expectation model
-- Computer simulation
-- Introduction of Markov-switching rational expectations model
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I think there are many students who use by themselves or just heard the name “Dynare”“, that is a standard tool for macro quantitative analysis. On the other hand, I think that there are many students who do not know how calculations are made on the back side. […] The first part of this lecture is to understand the theory and open the black box of Dynare, that is no other than an application with an excellent interface.
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The second part is to get the students to acquire basic techniques of computer simulation. Since the linear rational expectation models have been playing a major role in quantitative analysis of macroeconomics in recent years, I believe it will be of great use to each research project. Unless you work very hard you cannot master mathematics and programming. I want you to develop comprehensive problem solving skills while coming back and forth between theory and its implementation. In this lecture we will mainly use R language.
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Finally, we introduce the Markov-switching rational expectations model, a non linear model that has become very popular in the last years. Like linear models (or some kinds of), there is a characteristic to obtain a necessary and sufficient condition for the stability / instability which is theoretically easy to handle. It is known that Taylor’s condition related to monetary policy rules can be relaxed, and research is actively underway from both theory and empirical research.
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Organization
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These notes are made in accordance with the system of quarter that consists of 15 lectures (twice a week, 90 minutes × 8 weeks each time). In principle, we are going to read through each chapter in order. Since I set up 「exercises」to try to build a comprehensive understanding, I would like you tackle all of them.
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The first few chapters summarize the definitions and results that you learnt in the first year of undergraduate, which are necessary for the analysis of dynamic systems, so it should be a list of things that most of students already know. Therefore, you can skip reading providing that you solve practice problems. However, since each chapter is arranged to be increasingly difficult, I recommend you to read it if you are a little worry. Furthermore, I think that it is not useful to read only the chapters of interest if you do not have a preious knowledge of this field.
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@(intro)Chapter: Introduction
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-- What kind of model do you want to solve?
-- Building a programming environment
+- Is it constant throughout the time, or will it settle to a convergence state after a sufficient amount of time? Or, does we observe convergence and constant oscillation path?
+- Is the convergence path monotonically increasing (or decreasing) or oscillating?
+- In the case we observe constant oscillation, the oscillation is regular (periodic) or irregular (caos)?
+- Does the economic state depend or not on the past path?
+- Is the economic state uniquely determined …
+- etc.
+
Equations that characterize the solution of the model are called dynamic equation or dynamic system. In economic models, as an equilibrium condition satisfied by optimization problem composed of various constraints, utility maximization of consumer ・profit maximization of firms, it is possible to get equations that satisfy adjacent variables. \[
+F(x_t, x_{t + 1}) = 0, \quad t = 1, 2, \dots
+\] However, \(x_t\) is a vector of economic variables of interest. We changed the index representing timeto \(t\). When dealing with cases where the economic environment changes due to external factors (shocks) such as economic policies \[
+ F(x_t, x_{t+1}, z_t) = 0
+\] we will obtain a dynamic equation. These systems are implicit systems in the sense that variables are determined as solutions of implicit function.
+
In order to know the change over time of the economy, time series of \(x_1, x_2, \dots\) are necessary. As consequence, ideally a sequential calculation \[
+x_{n + 1} = G_t (x_t, z_t)
+\] is necessary. Or, if there is no time dependency of $ G $, \[
+x_{n+1} = G(x_t, z_t)
+\] it could be possible. The dynamic system as exemplified above can not be solved.
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@(complex numbers)Chapter: Complex numbers
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+
Intro target model
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In this lecture, we consider the following dynamic system.
+
\[\begin{equation}
+ A\mathbb{E}_{t}x_{t+1}=Bx_{t}+Cz_{t} (\#eq:lre)
+\end{equation}\]
+
\(x_t\) is a vector of variables (endogenous variables) determined to solve the model, while \(z_t\) is a variable determined outside the model (exogenous variable). \(A\), \(B\), \(C\) are matrices of appropriate size. \(\mathbb E_t\) is a conditional expectation.
+A system, as the above-mentioned, consisting only of a matrix product and addition is called linear system. If we rewrite the above linear equation as
+
\[\begin{equation}
+ Bx_{t} = A\mathbb{E}_{t}x_{t+1} - Cz_{t} (\#eq:lre2)
+\end{equation}\]
+
we can read it as the current economic variable is determined by expectation for the future. Systems such as @ref(eq:lre) and @ref(eq:lre2) are called linear rational expectations model.
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Although we do not go deeper into model derivation in these notes, it would be sufficient if you were aware that variables such as price and consumption are determined to reflect forward-looking expectations. The relationship where the price is determined reflecting the expectation can be understood through the following example.
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The basic formula of economic theory is that the stock price of a company is determined so that the value that the company produces in future will be discounted with consideration of the interest rate (or discount rate).
+Moreover, 「discount」means to adjust depreciation for the amount of time you have to wait before you get profit.
+
+The reason for establishing such a relationship can be understood relatively easily.
+Let's say that the discounted present value of the expected earnings of a company is temporarily higher than the total stock price of the company.
+If you purchase such stocks you can earn profits higher than the purchase price over a long period of time,
+so buyers collect such stocks and market prices rise. On the other hand,
+let's suppose the expected earnings are temporarily lower than the total stock price.
+Shareholders have incentives to sell shares, but at such market prices, buyers are unwilling to buy.
+Stock prices will fall until no shareholder wants to sell them even if the selling price would be lower.
+It is the previous price formula that is achieved as a result of eliminating the discrepancy between the anticipated value and the market price.
+
+If there is a chance to earn margins (arbitrage opportunity), the price will adjust quickly through the market, and,
+as consequence, arbitrage opportunities will be lost.
+Based on the idea above, economic theory often makes analysis by focusing on the economic environment (equilibrium)
+after the arbitrage opportunity has been eliminated.
+
Usually, economic models are described by non linear systems, but if you limit the analysis around \(z_t = 0\) (\(t > 0\)), the non-stochastic equilibrium point \(x^* = x_t = x_{t+1}\) is known to be well approximated by a linear system.
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The analysis of linear systems can be incorporated into the analysis of coefficients matrices, so it is very easy to handle both theoretically and numerically. As consequence, The analytical method we learn from now is the first step to analyze what kind of change in economic variables will occur when small shocks are added to the economic equilibrium. In practice it is difficult to derive quantitative implications from only linear approximation, so to make useful analysis it is necessary to learn higher order approximation techniques. However, you cannot understand nonlinear theory without understanding linear theory. Let’s go steadily step by step.
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+
Deterministic model
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Actually, even if probabilistic factors disappear, the fundamental policy of analysis does not change. In other words, by establishing a method to analyze non-stochastic (deterministic) system
+
\[\begin{equation}
+ Ax_{t+1} = Bx_t + Cz_t (\#eq:lsys)
+\end{equation}\]
+
you can make a simulation/analysis of @ref(eq:lre). First, after talking about analysis of @ref(eq: lsys), let’s proceed with theoretical analysis introducing probabilistic factors.
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Furthermore, non-deterministic analysis will be described in several stages.
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+- \(A\) is a regular case
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-- Review of complex numbers
-- Why are complex numbers necessary?
+- Analytical approach
+- Numerical approach
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@(matrix)Chapter: Review of matrices
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+- \(A\) is an irregular case
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-- Review of matrices
-- Similarity between matrix product and complex numbers product
+- Analytical approach
+- Numerical approach
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In the case where \(A\) is regular, since @ref(eq: lsys) is equivalent to
+
\[\begin{equation}
+ x_{t+1} = A^{-1}Bx_t + A^{-1}Cz_t (\#eq:lsys2)
+\end{equation}\]
+
(\(A^{-1}\) is the inverse matrix of \(A\)), it is formally the same as a standard linear system (state space equation).
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Then, let’s deal with \(A\) as irregular case. The above-mentioned @ref(eq:lsys), in the field of control theory the subject is the descriptor system, also known as implicit system. Those who are going to conduct research in this field should probably be aware that the same concept is being used with different names in different fields.
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By the way, the descriptor system is used to represent a 「non-causal」 system. In other words, it is a system where information of the future influences the present. Is not it a story you have heard somewhere? In economics, “forward-looking” is called non-causal in the control theory.
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@(eigen)Chapter: Eigenvalues of matrices
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-- Review of eigenvalues of the matrices
-- Clarify the relationship of complex eigenvalues of linear system
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Difference from control theory
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Economics and control theory deal with slightly different subjects. In control theory, we deal with objects which are operating arbitrarily without introducing control variables. On the other hand, in economic theory, the control variable (eg. consumption) is a part of the elements constituting the model of the analysis target. As consequence, the economic model is designed to represent a forward-looking phenomenon even if \(A\) is regular.
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In dynamic system theory and control theory, when dealing with a system like @ref(eq: lsys), we usually give as many initial conditions as \(x\). In such a case, since it is possible to sequentially obtain the solution of the system starting from the initial condition, there is no particular hardship in solving the solution. On the other hand, the expression of forward-looking in economics is independent of the system equation @ref(eq: lsys). That is, forward-looking is handled in such a manner that initial conditions are given to some elements of the variable (vector) \(x\) and no initial condition is given to the remaining elements. Components with initial conditions are called predetermined component or predetermined variable. Components without initial conditions are called non-predetermined component or non-predetermined variable.
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For example, let’s say that you have created a model where the share \(a\), the price of the stock \(p\), the external influence on the stock price \(u\), and the vector \((a, p)\) satisfy the following dynamic equation.
+
\[
+\begin{bmatrix}
+ a_{t+1} \\
+ p_{t+1}
+\end{bmatrix}
+=
+B
+\begin{bmatrix}
+ a_{t} \\
+ p_{t}
+\end{bmatrix}
++
+\begin{bmatrix}
+ 0 \\
+ u_{t}
+\end{bmatrix}
+\]
+
In this case, \(a_0\) is given as an initial value, but it is a matter of typical macroeconomics that \(p_0\) is determined based on expectations for $a_0 $ and \(u\).
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Chapter 5: Eigenspace
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Stability and determinacy
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Now, how can we solve a problem where initial condition is not given to some variables? Of course, we can not solve such problems unless we place any restrictions on \(a_1, a_2, \dots\), \(p_0, p_1, \dots\).
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@BlanchardKahn1980 proposed the following conditions. Economic agents do not act based on expectations that diverge at a speed faster than geometric series. In other words, if the initial value can be set to 1 as a result of excluding initial values that diverge faster than the geometric series, that point is a unique balance. However, in general there are numerous routes that satisfy the stability (non-geometric divergence) while satisfying the dynamic equation. We call the cases that are unique determinate, and the non-determinate cases indeterminate.
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