Lecture notes for Macroeconomics I 2004 Yale University

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Introduction, These lecture notes cover a one semester course The overriding goal of the course is. to begin provide methodological tools for advanced research in macroeconomics The. emphasis is on theory although data guides the theoretical explorations We build en. tirely on models with microfoundations i e models where behavior is derived from basic. assumptions on consumers preferences production technologies information and so on. Behavior is always assumed to be rational given the restrictions imposed by the primi. tives all actors in the economic models are assumed to maximize their objectives. Macroeconomic studies emphasize decisions with a time dimension such as various. forms of investments Moreover it is often useful to assume that the time horizon is. infinite This makes dynamic optimization a necessary part of the tools we need to. cover and the first significant fraction of the course goes through in turn sequential. maximization and dynamic programming We assume throughout that time is discrete. since it leads to simpler and more intuitive mathematics. The baseline macroeconomic model we use is based on the assumption of perfect com. petition Current research often departs from this assumption in various ways but it is. important to understand the baseline in order to fully understand the extensions There. fore we also spend significant time on the concepts of dynamic competitive equilibrium. both expressed in the sequence form and recursively using dynamic programming In. this context the welfare properties of our dynamic equilibria are studied. Infinite horizon models can employ different assumptions about the time horizon of. each economic actor We study two extreme cases i all consumers really dynasties live. forever the infinitely lived agent model and ii consumers have finite and deterministic. lifetimes but there are consumers of different generations living at any point in time. the overlapping generations model These two cases share many features but also have. important differences Most of the course material is built on infinitely lived agents but. we also study the overlapping generations model in some depth. Finally many macroeconomic issues involve uncertainty Therefore we spend some. time on how to introduce it into our models both mathematically and in terms of eco. nomic concepts, The second part of the course notes goes over some important macroeconomic topics. These involve growth and business cycle analysis asset pricing fiscal policy monetary. economics unemployment and inequality Here few new tools are introduced we instead. simply apply the tools from the first part of the course. Motivation Solow s growth model, Most modern dynamic models of macroeconomics build on the framework described in. Solow s 1956 paper 1 To motivate what is to follow we start with a brief description of. the Solow model This model was set up to study a closed economy and we will assume. that there is a constant population,2 1 The model,The model consists of some simple equations. Ct It Yt F Kt L 2 1,It Kt 1 1 Kt 2 2,It sF Kt L 2 3.
The equalities in 2 1 are accounting identities saying that total resources are either. consumed or invested and that total resources are given by the output of a production. function with capital and labor as inputs We take labor input to be constant at this point. whereas the other variables are allowed to vary over time The accounting identity can also. be interpreted in terms of technology this is a one good or one sector economy where. the only good can be used both for consumption and as capital investment Equation. 2 2 describes capital accumulation the output good in the form of investment is. used to accumulate the capital input and capital depreciates geometrically a constant. fraction 0 1 disintegrates every period, Equation 2 3 is a behavioral equation Unlike in the rest of the course behavior. here is assumed directly a constant fraction s 0 1 of output is saved independently. of what the level of output is, These equations together form a complete dynamic system an equation system defin. ing how its variables evolve over time for some given F That is we know in principle. t 0 and Yt Ct It t 0 will be given any initial capital value K0. In order to analyze the dynamics we now make some assumptions. No attempt is made here to properly assign credit to the inventors of each model For example the. Solow model could also be called the Swan model although usually it is not. lim sFK K L 1 1, F is strictly concave in K and strictly increasing in K. An example of a function satisfying these assumptions and that will be used repeat. edly in the course is F K L AK L1 with 0 1 This production function. is called Cobb Douglas function Here A is a productivity parameter and and 1. denote the capital and labor share respectively Why they are called shares will be the. subject of the discussion later on, The law of motion equation for capital may be rewritten as. Kt 1 1 Kt sF Kt L, Mapping Kt into Kt 1 graphically this can be pictured as in Figure 2 1.
Figure 2 1 Convergence in the Solow model, The intersection of the 45o line with the savings function determines the stationary. point It can be verified that the system exhibits global convergence to the unique. strictly positive steady state K that satisfies,K 1 K sF K L or. K sF K L there is a unique positive solution,Given this information we have. Theorem 2 1 K 0 K0 0 Kt K,Proof outline,1 Find a K candidate show it is unique. 2 If K0 K show that K Kt 1 Kt t 0 using Kt 1 Kt sF Kt L. Kt If K0 K show that K Kt 1 Kt t 0, 3 We have concluded that Kt is a monotonic sequence and that it is also bounded.
Now use a math theorem a monotone bounded sequence has a limit. The proof of this theorem establishes not only global convergence but also that conver. gence is monotonic The result is rather special in that it holds only under quite restrictive. circumstances for example a one sector model is a key part of the restriction. 2 2 Applications,2 2 1 Growth, The Solow growth model is an important part of many more complicated models setups. in modern macroeconomic analysis Its first and main use is that of understanding. why output grows in the long run and what forms that growth takes We will spend. considerable time with that topic later This involves discussing what features of the. production technology are important for long run growth and analyzing the endogenous. determination of productivity in a technological sense. Consider for example a simple Cobb Douglas case In that case the capital share. determines the shape of the law of motion function for capital accumulation If is. close to one the law of motion is close to being linear in capital if it is close to zero but. not exactly zero the law of motion is quite nonlinear in capital In terms of Figure 2 1. an close to zero will make the steady state lower and the convergence to the steady. state will be quite rapid from a given initial capital stock few periods are necessary to. get close to the steady state If on the other hand is close to one the steady state is. far to the right in the figure and convergence will be slow. When the production function is linear in capital when equals one we have no. positive steady state 2 Suppose that sA 1 exceeds one Then over time output would. keep growing and it would grow at precisely rate sA 1 Output and consumption. would grow at that rate too The Ak production technology is the simplest tech. nology allowing endogenous growth i e the growth rate in the model is nontrivially. determined at least in the sense that different types of behavior correspond to different. growth rates Savings rates that are very low will even make the economy shrink if. sA 1 goes below one Keeping in mind that savings rates are probably influenced. by government policy such as taxation this means that there would be a choice both. by individuals and government of whether or not to grow. The Ak model of growth emphasizes physical capital accumulation as the driving. force of prosperity It is not the only way to think about growth however For example. This statement is true unless sA 1 happens to equal 1. k 1 k 2 kt,Figure 2 2 Random productivity in the Solow model. one could model A more carefully and be specific about how productivity is enhanced. over time via explicit decisions to accumulate R D capital or human capital learning. We will return to these different alternatives later. In the context of understanding the growth of output Solow also developed the. methodology of growth accounting which is a way of breaking down the total growth of. an economy into components input growth and technology growth We will discuss this. later too growth accounting remains a central tool for analyzing output and productivity. growth over time and also for understanding differences between different economies in. the cross section,2 2 2 Business Cycles, Many modern studies of business cycles also rely fundamentally on the Solow model. This includes real as well as monetary models How can Solow s framework turn into a. business cycle setup Assume that the production technology will exhibit a stochastic. component affecting the productivity of factors For example assume it is of the form. F At F Kt L, where At is stochastic for instance taking on two values AH AL Retaining the assump. tion that savings rates are constant we have what is depicted in Figure 2 2. It is clear from studying this graph that as productivity realizations are high or low. output and total savings fluctuate Will there be convergence to a steady state In the. sense of constancy of capital and other variables steady states will clearly not be feasible. here However another aspect of the convergence in deterministic model is inherited. here over time initial conditions the initial capital stock lose influence and eventually. after an infinite number of time periods the stochastic process for the endogenous. variables will settle down and become stationary Stationarity here is a statistical term. one that we will not develop in great detail in this course although we will define it and. use it for much simpler stochastic processes in the context of asset pricing One element. of stationarity in this case is that there will be a smallest compact set of capital stocks. such that once the capital stock is in this set it never leaves the set the ergodic set. In the figure this set is determined by the two intersections with the 45o line. 2 2 3 Other topics, In other macroeconomic topics such as monetary economics labor fiscal policy and.
asset pricing the Solow model is also commonly used Then other aspects need to be. added to the framework but Solow s one sector approach is still very useful for talking. about the macroeconomic aggregates,2 3 Where next, The model presented has the problem of relying on an exogenously determined savings. rate We saw that the savings rate in particular did not depend on the level of capital. or output nor on the productivity level As stated in the introduction this course. aims to develop microfoundations We would therefore like the savings behavior to be. an outcome rather than an input into the model To this end the following chapters. will introduce decision making consumers into our economy We will first cover decision. making with a finite time horizon and then decision making when the time horizon is. infinite The decision problems will be phrased generally as well as applied to the Solow. growth environment and other environments that will be of interest later. Dynamic optimization, There are two common approaches to modelling real life individuals i they live a finite. number of periods and ii they live forever The latter is the most common approach. but the former requires less mathematical sophistication in the decision problem We will. start with finite life models and then consider infinite horizons. We will also study two alternative ways of solving dynamic optimization problems. using sequential methods and using recursive methods Sequential methods involve maxi. mizing over sequences Recursive methods also labelled dynamic programming methods. involve functional equations We begin with sequential methods and then move to re. cursive methods,3 1 Sequential methods,3 1 1 A finite horizon. Consider a consumer having to decide on a consumption stream for T periods Con. sumer s preference ordering of the consumption streams can be represented with the. utility function,U c0 c1 cT, A standard assumption is that this function exhibits additive separability with. stationary discounting weights,U c0 c1 cT t u ct, Notice that the per period or instantaneous utility index u does not depend on.
time Nevertheless if instead we had ut the utility function U c0 c1 cT would still. be additively separable, The powers of are the discounting weights They are called stationary because the. ratio between the weights of any two different dates t i and t j i only depends on. the number of periods elapsed between i and j and not on the values of i or j. The standard assumption is 0 1 which corresponds to the observations that hu. man beings seem to deem consumption at an early time more valuable than consumption. further off in the future, We now state the dynamic optimization problem associated with the neoclassical. Lecture notes for Macroeconomics I 2004 Per Krusell Please do NOT distribute without permission Comments and suggestions are welcome 1 2 Chapter 1 Introduction These lecture notes cover a one semester course The overriding goal of the course is to begin provide methodological tools for advanced research in macroeconomics The emphasis is on theory although data guides the theoretical

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