Advanced Macroeconomics 5. Rational Expectations and Asset Prices - PowerPoint PPT Presentation

Advanced Macroeconomics 5. Rational Expectations and Asset Prices Karl Whelan School of Economics, UCD Spring 2020 Karl Whelan (UCD) Asset Prices Spring 2020 1 / 43

A New Topic We are now going to switch gear and leave the IS-MP-PC model behind us. One of the things we’ve focused on is how people formulate expectations about inflation. We put forward one model of how these expectations were formulated, an adaptive expectations model in which people formulated their expectations by looking at past values for a series. Over the next few weeks, we will look at an alternative approach that macroeconomists call “rational expectations”. This approach is widely used in macroeconomics and we will cover its application to models of ◮ Asset prices, particularly stock prices. ◮ Household consumption and fiscal policy. ◮ Exchange rates Karl Whelan (UCD) Asset Prices Spring 2020 2 / 43

Rational Expectations Almost all economic transactions rely crucially on the fact that the economy is not a “one-period game.” Economic decisions have an intertemporal element to them. A key issue in macroeconomics is how people formulate expectations about the in the presence of uncertainty. Prior to the 1970s, this aspect of macro theory was largely ad hoc relying on approaches like adaptive expectations. This approach criticised in the 1970s by economists such as Robert Lucas and Thomas Sargent who instead promoted the use of an alternative approach which they called “rational expectations.” In economics, saying people have “rational expectations” usually means two things: They use publicly available information in an efficient manner. Thus, 1 they do not make systematic mistakes when formulating expectations. They understand the structure of the model economy and base their 2 expectations of variables on this knowledge. Karl Whelan (UCD) Asset Prices Spring 2020 3 / 43

How We Will Describe Expectations We will write E t Z t +2 to mean the expected value the agents in the economy have at time t for what Z is going to be at time t + 2. We assume people have a distribution of potential outcomes for Z t +2 and E t Z t +2 is mean of this distribution. So E t is not a number that is multiplying Z t +2 . Instead, it is a qualifier explaining that we are dealing with people’s prior expectations of a Z t +2 rather than the actual realised value of Z t +2 itself. We will use some basic properties of the expected value of distributions. Specifically, the fact that expected values of distributions is what is known as a linear operator meaning E t ( α X t + k + β Y t + k ) = α E t X t + k + β E t Y t + k For example, E t (5 X t + k ) = 5 E t ( X t + k ) And E t ( X t + k + Y t + k ) = E t X t + k + E t Y t + k Karl Whelan (UCD) Asset Prices Spring 2020 4 / 43

Example: Asset Prices Asset prices are an increasingly important topic in macroeconomics. The most recent two global recessions—the “dot com” recession of 2000/01 and the “great recession” of 2008/09—were triggered by big declines in asset prices following earlier large increases. A framework for discussing these movements is thus a necessary part of any training in macroeconomics. Consider an asset that can be purchased today for price P t and which yields a dividend of D t . The asset could be a share of equity in a firm with D t being the dividend payment but it could also be a house and D t could be the net return from renting this house out If this asset is sold tomorrow for price P t +1 , then it generates a rate of return on this investment of r t +1 = D t + ∆ P t +1 P t This rate of return has two components, the first reflects the dividend received during the period the asset was held, and the second reflects the capital gain (or loss) due to the price of the asset changing from period t to period t + 1. Karl Whelan (UCD) Asset Prices Spring 2020 5 / 43

A Different Form for the Rate of Return Equation The gross return on the asset, i.e. one plus the rate of return, is 1 + r t +1 = D t + P t +1 P t A useful re-arrangement of this equation that we will work with is: D t P t +1 P t = + 1 + r t +1 1 + r t +1 In this context, rational expectations means investors understand this equation and that all expectations of future variables must be consistent with it. This implies that � D t P t +1 � E t P t = E t + 1 + r t +1 1 + r t +1 where E t means the expectation of a variable formulated at time t . The stock price at time t is observable to the agent so E t P t = P t , implying � � D t P t +1 P t = E t + 1 + r t +1 1 + r t +1 Karl Whelan (UCD) Asset Prices Spring 2020 6 / 43

Constant Expected Returns Assume the expected return on assets is constant. E t r t + k = r k = 1 , 2 , 3 , ..... Can think of this as a “required return”, determined perhaps the rate of return available on some other asset. Last equation on the previous slide becomes 1 + r + E t P t +1 D t P t = 1 + r This is an example of a first-order stochastic difference equation . Stochastic means random or incorporating uncertainty. Because such equations occur commonly in macroeconomics, we will discuss a general approach to solving them. Karl Whelan (UCD) Asset Prices Spring 2020 7 / 43

First-Order Stochastic Difference Equations Lots of models in economics take the form y t = ax t + bE t y t +1 The equation just says that y today is determined by x and by tomorrow’s expected value of y . But what determines this expected value? Rational expectations implies a very specific answer. Under rational expectations, the agents in the economy understand the equation and formulate their expectation in a way that is consistent with it: E t y t +1 = aE t x t +1 + bE t E t +1 y t +2 This last term can be simplified to E t y t +1 = aE t x t +1 + bE t y t +2 because E t E t +1 y t +2 = E t y t +2 . This is known as the Law of Iterated Expectations: It is not rational for me to expect to have a different expectation next period for y t +2 than the one that I have today. Karl Whelan (UCD) Asset Prices Spring 2020 8 / 43

Repeated Substitution Substituting this into the previous equation, we get y t = ax t + abE t x t +1 + b 2 E t y t +2 Repeating this by substituting for E t y t +2 , and then E t y t +3 and so on gives y t = ax t + abE t x t +1 + ab 2 E t x t +2 + .... + ab N − 1 E t x t + N − 1 + b N E t y t + N Which can be written in more compact form as N − 1 � b k E t x t + k + b N E t y t + N y t = a k =0 Usually, it is assumed that N →∞ b N E t y t + N = 0 lim So the solution is ∞ � b k E t x t + k y t = a k =0 This solution underlies the logic of a very large amount of modern macroeconomics. Karl Whelan (UCD) Asset Prices Spring 2020 9 / 43

Repeated Substitution Substituting this into the previous equation, we get y t = ax t + abE t x t +1 + b 2 E t y t +2 Repeating this by substituting for E t y t +2 , and then E t y t +3 and so on gives y t = ax t + abE t x t +1 + ab 2 E t x t +2 + .... + ab N − 1 E t x t + N − 1 + b N E t y t + N Which can be written in more compact form as N − 1 � b k E t x t + k + b N E t y t + N y t = a k =0 Usually, it is assumed that N →∞ b N E t y t + N = 0 lim So the solution is ∞ � b k E t x t + k y t = a k =0 This solution underlies the logic of a very large amount of modern macroeconomics. Karl Whelan (UCD) Asset Prices Spring 2020 9 / 43

Repeated Substitution Substituting this into the previous equation, we get y t = ax t + abE t x t +1 + b 2 E t y t +2 Repeating this by substituting for E t y t +2 , and then E t y t +3 and so on gives y t = ax t + abE t x t +1 + ab 2 E t x t +2 + .... + ab N − 1 E t x t + N − 1 + b N E t y t + N Which can be written in more compact form as N − 1 � b k E t x t + k + b N E t y t + N y t = a k =0 Usually, it is assumed that N →∞ b N E t y t + N = 0 lim So the solution is ∞ � b k E t x t + k y t = a k =0 This solution underlies the logic of a very large amount of modern macroeconomics. Karl Whelan (UCD) Asset Prices Spring 2020 9 / 43

Repeated Substitution Substituting this into the previous equation, we get y t = ax t + abE t x t +1 + b 2 E t y t +2 Repeating this by substituting for E t y t +2 , and then E t y t +3 and so on gives y t = ax t + abE t x t +1 + ab 2 E t x t +2 + .... + ab N − 1 E t x t + N − 1 + b N E t y t + N Which can be written in more compact form as N − 1 � b k E t x t + k + b N E t y t + N y t = a k =0 Usually, it is assumed that N →∞ b N E t y t + N = 0 lim So the solution is ∞ � b k E t x t + k y t = a k =0 This solution underlies the logic of a very large amount of modern macroeconomics. Karl Whelan (UCD) Asset Prices Spring 2020 9 / 43

Summation Signs For those of you unfamiliar with the summation sign terminology, summation signs work like this 2 � z k = z 0 + z 1 + z 2 k =0 3 � z k = z 0 + z 1 + z 2 + z 3 k =0 and so on. The term ∞ � b k E t x t + k k =0 is just a compact way of writing x t + bE t x t +1 + b 2 E t x t +2 + ... Karl Whelan (UCD) Asset Prices Spring 2020 10 / 43

Applying Solution to Asset Prices Our asset price equation 1 + r + E t P t +1 D t P t = 1 + r is a specific case of the first-order stochastic difference equation with y t = P t x t = D t 1 a = 1 + r 1 b = 1 + r This implies that the asset price can be expressed as follows N − 1 � k +1 � N � � 1 1 � P t = E t D t + k + E t P t + N 1 + r 1 + r k =0 Karl Whelan (UCD) Asset Prices Spring 2020 11 / 43