This is the end of part I and II. This is often also the end of a course. This is a good moment to look back at what has been accomplished. After 14 or 15 lectures and the same number of exercise classes, the amount of material covered is fairly impressive.
In terms of maximization tools, this …rst part has covered Solving by substitution
Lagrange methods in discrete and continuous time
Dynamic programming in discrete time and continuous time Hamiltonian
With respect to model building components, we have learnt how to build budget constraints
how to structure the presentation of a model how to derive reduced forms
From an economic perspective, the …rst part presented the two-period OLG model
the optimal saving central planner model in discrete and continuous time the matching approach to unemployment
the decentralized optimal growth model and an optimal growth model with money
Most importantly, however, the tools presented here allow students to “become in-dependent”. A very large part of the Economics literature (acknowledging that game theoretic approaches have not been covered here at all) is now open and accessible and the basis for understanding a paper in detail (and not just the overall argument) and for presenting their own arguments in a scienti…c language are laid out.
Clearly, models with uncertainty present additional challenges. They will be presented and overcome in part III and part IV.
Stochastic models in discrete time
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In part III, the world becomes stochastic. Parts I and II provided many optimization methods for deterministic setups, both in discrete and continuous time. All economic questions that were analyzed were viewed as “su¢ ciently deterministic”. If there was any uncertainty in the setup of the problem, we simply ignored it or argued that it is of no importance for understanding the basic properties and relationships of the economic question. This is a good approach to many economic questions.
Generally speaking, however, real life has few certain components. Death is certain, but when? Taxes are certain, but how high are they? We know that we all exist - but don’t ask philosophers. Part III (and part IV later) will take uncertainty in life seriously and incorporate it explicitly in the analysis of economic problems. We follow the same distinction as in part I and II - we …rst analyse the e¤ects of uncertainty on economic behaviour in discrete time setups in part III and then move to continuous time setups in part IV.
Chapter 7 and 8 are an extended version of chapter 2. As we are in a stochastic world, however, chapter 7 will …rst spend some time reviewing some basics of random variables, their moments and distributions. Chapter 7 also looks at di¤erence equations. As they are now stochastic, they allow us to understand how distributions change over time and how a distribution converges - in the example we look at - to a limiting distribution.
The limiting distribution is the stochastic equivalent to a …x point or steady state in deterministic setups.
Chapter 8 looks at maximization problems in this stochastic framework and focuses on the simplest case of two-period models. A general equilibrium analysis with an overlapping generations setup will allow us to look at the new aspects introduced by uncertainty for an intertemporal consumption and saving problem. We will also see how one can easily understand dynamic behaviour of various variables and derive properties of long-run distributions in general equilibrium by graphical analysis. One can for example easily obtain the range of the long-run distribution for capital, output and consumption. This increases intuitive understanding of the processes at hand tremendously and helps a lot as a guide to numerical analysis. Further examples include borrowing and lending between risk-averse and risk-neutral households, the pricing of assets in a stochastic world and a
…rst look at ’natural volatility’, a view of business cycles which stresses the link between jointly endogenously determined short-run ‡uctuations and long-run growth.
Chapter 9 is then similar to chapter 3 and looks at multi-period, i.e. in…nite horizon, problems. As in each chapter, we start with the classic intertemporal utility maximization problem. We then move on to various important applications. The …rst is a central planner stochastic growth model, the second is capital asset pricing in general equilibrium and how it relates to utility maximization. We continue with endogenous labour supply and the matching model of unemployment. The next section then covers how many maximization problems can be solved without using dynamic programming or the Lagrangian. In fact, many problems can be solved simply by inserting, despite uncertainty. This will be illustrated with many further applications. A …nal section on …nite horizons concludes.
Stochastic di¤erence equations and moments
Before we look at di¤erence equations in section 7.4, we will …rst spend a few sections reviewing basic concepts related to uncertain environments. These concepts will be useful at later stages.
7.1 Basics on random variables
Let us …rst have a look at some basics of random variables. This follows Evans, Hastings and Peacock (2000).
7.1.1 Some concepts
A probabilistic experiment is an occurrence where a complex natural background leads to a chance outcome. The set of possible outcomes of a probabilistic experiment is called the possibility space. A random variable (RV) X is a function which maps from the possibility space into a set of numbers. The set of numbers this RV can take is called the range of this variable X.
The distribution function F associated with the RV X is a function which maps from the range into the probability domain [0,1],
F (x) =Prob (X x) :
The probability that X has a realization of x or smaller is given by F (x) :
We now need to make a distinction between discrete and continuous RVs. When the RV X has a discrete range then f (x) gives …nite probabilities and is called the probability function or probability mass function. The probability that X has the realization of x is given by f (x) :
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When the RV X is continuous, the …rst derivative of the distribution function F f (x) = dF (x)
dx
is called the probability density function f . The probability that the realization of X lies between, say, a and b > a is given by F (b) F (a) = Rb
a f (x) dx: Hence the probability that X equals a is zero.
7.1.2 An illustration
Discrete random variable
Consider the probabilistic experiment ’tossing a coin twice’. The possibility space is given by fHH; HT; T H; T T g. De…ne the RV ’Number of heads’. The range of this variable is given by f0; 1; 2g : Assuming that the coin falls on either side with the same probability, the probability function of this RV is given by
f (x) =
Think of next weekend. You might consider going to a pub to meet friends. Before you go there, you do not know how much time you will spend there. If you meet a lot of friends, you will stay longer; if you drink just one beer, you will leave soon. Hence, going to a pub on a weekend is a probabilistic experiment with a chance outcome.
The set of possible outcomes with respect to the amount of time spent in a pub is the possibility space. Our random variable T maps from this possibility space into a set of numbers with a range from 0 to, let’s say, 4 hours (as the pub closes at 1 am and you never go there before 9 p.m.). As time is continuous, T 2 [0; 4] is a continuous random variable.
The distribution function F (t) gives you the probability that you spend a period of length t or shorter in the pub. The probability that you spend between 1.5 and two hours in the pub is given by R2
1:5f (t) dt;where f (t) is the density function f (t) = dF (t) =dt.