Indeterminacy and Sunspots in Macroeconomics

Transcription

1 Indeterminacy and Sunspots in Macroeconomics Wednesday September 6 th : Lecture 5 Gerzensee, September 2017 Roger E. A. Farmer Warwick University and NIESR

2 Topics for Lecture 5 Sunspots (Cass-Shell paper) Incomplete participation Incomplete markets The Farmer-Woodford Model and self-fulfilling prophecies 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 2

3 Reading Cass and Shell Do Sunspots Matter Journal of Political Economy 1983 Farmer and Woodford, Self-Fulfilling Prophecies and the Business Cycle Macroeconomic Dynamics 1997 Farmer Macroeconomics of Self-Fulfilling Prophecies MIT Press Chapter 10 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 3

4 Sunspots: the Cass-Shell paper Intrinsic uncertainty: affects preferences endowment or technology Extrinsic uncertainty: does not affect any of the fundamentals 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 4

5 Sunspots: the Cass-Shell paper Question: Can there be an equilibrium in an economy with complete financial markets where allocations are different across states that differ only in extrinsic uncertainty? Answer: Yes. If there is incomplete participation in financial markets This is ALWAYS true in an overlapping generations model 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 5

6 Sunspots: the Cass-Shell paper Do sunspots matter? JPE (2) Two generations Complete markets Two dates Two commodities at date 2 Two states at date 2, α and β No commodities at date 1 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 6

7 Two Market Structures Cass and Shell consider two market structures: (A)Securities Markets As in Arrow s formulation of GE (B)Contingent Commodities As in Debreu s formulation of GE 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 7

8 The Timing of Events Timing G1 born Securities traded Sunspot G2 born Commodities traded 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 8

9 (A) Arrow Securities 1) The securities market opens. Agents of generation 1 trade securities contingent on realization of sunspot. 2) A sunspot is realized 3) Generation 2 is born 4) Trade takes place in the commodities markets 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 9

10 (B) Contingent Commodities 1) Market opens at the beginning of time. Agents trade the following commodities, x ' α, x ( α, x ' β, x ( β 2) In this structure Generation 2 is present, but is not allowed to trade across states of nature 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 10

11 Securities and Arrow Securities Suppose that there are 2 periods, l goods and n states in period 2, Suppose further that there are m securities and the security a - pays a vector b = (b ' -, b ( -, b 3 - ) of dollars in each state s = 1,, n For example, the payoff of the j th Arrow security b 8 = (0,, 1, 0) ',...,8,..,3 pays one dollar in state j and 0 otherwise 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 11

12 Securities and Arrow Securities If the matrix B = ' b ' b' 3 b? ' b? 3 Has full row rank, we say that markets are complete A complete set of Arrow securities is the special case when 1 0 B = I /3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 12

13 Securities and Arrow Securities When markets are complete, the household faces a single budget constraint and can transfer income between the present and all future states When markets are incomplete, the household faces multiple budget constraints (Can you explain why?) 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 13

14 Incomplete Markets and Incomplete Participation Equilibria are Pareto Inefficient when markets are incomplete Some economists argue that markets are obviously incomplete because we do not observe securities designed to be contingent on every observable contingency Other economists argue that this doesn't matter. If a market is important: it will be created 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 14

15 Incomplete Markets and Incomplete Participation Although it seems unlikely (to me) that markets are complete: Cass and Shell assume complete markets in order to make a point. Even if every conceivable security is traded in financial markets: those markets will still lead to Pareto Inefficient outcomes if there is incomplete participation 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 15

16 Incomplete Participation Is incomplete participation a reasonable assumption? YES: because we cannot trade securities in markets that open before we are born 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 16

17 An Equivalence Result If there are complete markets preferences are time consistent (Von-Neumann Morgenstern preferences satisfy this assumption) Then Representation (A) with complete financial markets is equivalent to Representation (B) with complete contingent commodities 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 17

18 Sunspots Since (A) and (B) are equivalent, Cass and Shell proceed with the complete contingent commodity representation 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 18

19 Some Definitions x B α x ' B α, x ( B α x B β x ' B β, x ( B β h α and β Superscripts 1 and 2 vectors of contingent commodities indexes households index the state index commodities 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 19

20 Generation 1 The problem of G 1 max U H π J u B x B s JMN,O p x B p ω B where and x B x B ' α, x B ( α, x B ' β, x B ( β p p ' α, p ( α, p ' β, p ( β 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 20

21 Generation 2 The two problems of G 2 max U u B x B s, h G (, s = α, β p(s) x B (s) p(s) ω B (s) where and x B (s) x B ' s, x B ( s p(s) p ' s, p ( s 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 21

22 Types of Uncertainty Definition 1: Uncertainty is intrinsic if it affects preferences or endowments (e.g. the weather, taste shocks) Uncertainty is extrinsic if it does not affect preferences or endowments (e.g. opinion piece in the Wall Street Journal with no basis in fact. Fake news?) 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 22

23 An Assumption Assumption 1: All uncertainty is extrinsic This implies that ω B α = ω B β for h G ' G ( 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 23

24 A Definition (Sunspots) Definition 2: Sunspots matter if there is an equilibrium where for at least one h x B α x B β 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 24

25 Two Definitions of Pareto Efficiency If there are two possible future states: state α and state β Is the person born tomorrow in state α the same person as if she were born in state β? Why is that an interesting question? 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 25

26 Ex Post Pareto Efficiency If we consider Mr. α to be a different person from Mr. β then sunspot allocations are Pareto Efficient. Mr. α is treated differently from Mr. β but that s ok because they are different people. We say that a sunspot equilibrium, in this case, is Ex Post Pareto Efficient 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 26

27 Rawls and the Veil of Ignorance Imagine putting yourself in a state, before you are born. The Philosopher John Rawls calls this experiment, the veil of ignorance Behind the veil of ignorance, you face the following gamble. With probability p you can be born as a rich person in Switzerland in 2017 or with probability 1 p you can be born as a slave in Sparta in 1BC Alternatively, you can avoid this gamble and be born as a middle class citizen of a European country in /3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 27

28 Ex-Ante Pareto Efficiency Many of us would pay quite a lot to avoid a probability of the slavery option Under the veil of ignorance, we consider Mr. α to be the same person as Mr. β. We say that a sunspot equilibrium in this case is Ex Ante Pareto Inefficient because it randomizes over the good and the bad outcomes even when the mean gamble is feasible 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 28

29 The Sunspot Propositions: Proposition 1: If G ( (unrestricted participation) then sunspots do not matter Proposition 2: Sunspot equilibria are Ex Ante Pareto Inefficient Proposition 3: If G ( there may be equilibria where sunspots matter 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 29

30 Sunspots and Indeterminacy The proof of the existence of sunspots is constructive Cass and Shell set up a model with three equilibria and they construct a new equilibrium that randomizes over two of the equilibria in the world with certainty That would not be possible if there was complete participation. (Can you explain why?) 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 30

31 Sunspots and Indeterminacy We have studied models with dynamic indeterminacy. And we have learned about sunspots. Next: I will show how to put the two together Sunspot equilibria are constructed by randomizing over the equilibria of models with multiple certainty equilibria It is easy to construct sunspot equilibria in models with an indeterminate set of equilibria because there are so many equilibria to randomize over 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 31

32 The Farmer-Woodford Example The FW example is a two-period overlapping generations model with production y [ Output c [ Consumption n [ Labor supply M [ Money p [ $ Price of a good g Govt. Spending 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 32

33 Households max U = n [ ( + E [ c [a' utility function y [ = n [ p [ n [ = M [ p [a' c [a' = M [ p [a' c [a' = p [ n [ technology first period budget constraint second period budget constraint life-cycle budget constraint 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 33

34 Solution max 3 b U = n [ ( + E [ p [ p [a' n [ n [ = E [ c b c bde Labor supply y [ = E [ c b c bde Output supply 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 34

35 Money Demand n [ = y [ = M [ p [ m [ m [ = E [ p [ p [a' Output equals labor supply equals real value of money balances 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 35

36 Government M [ M [f' = p [ g Government prints enough money to purchase g goods m [ m [f' p [f' p [ = g p [f' p [ = m [ g m [f' 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 36

37 Market Clearing p [ = m [a' g p [a' m [ Endogenous money supply m [ = E [ p [ p [a' Household optimization 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 37

38 Equilibrium p [ = m [a' g p [a' m [ Endogenous money supply m [ = E [ p [ p [a' Household optimization m [ = E [ m [a' g m [ Equilibrium 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 38

39 Equilibrium m [ = E [ m [a' g m [ Any stochastic process that satisfies this equation is an equilibrium m ' = M ' p ' M ' is the initial dollar value of the money supply. There is no economic condition pinning down p ' 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 39

40 Non-stochastic Equilibria m [a' = m [ ( + g Non-stationary nonstochastic equilibria m ( m + g = 0 mg = 1 ± 1 4g 2 Stationary nonstochastic equilibria 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 40

41 Characterizing Equilibria m [a' m [a' = m [ ( + g 45 k line Two stationary equilibria g mg ' = g mg ' mg ( mg ( = g m [ 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 41

42 Characterizing Equilibria m [a' m [a' = m [ ( + g 45 k line g A nonstationary equilibrium mg ' m l m ( m ' mg ( m [ 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 42

43 Sunspots and Indeterminacy Let s recap I have provided a simple example of a rational expectations model with no fundamental uncertainty where there are two stationary equilibria and there is a continuum of non-stationary equilibria Check for yourself that and m ' mg ', mg ( element of an equilibrium sequence is the first All of these sequences converge, asymptotically, to mg ' 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 43

44 Sunspots and Indeterminacy There is also an equilibrium in which m n [ mm' = {mg ( n } mm' Which equilibrium will occur? The answer depends on the initial price level 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 44

45 Sunspots and Indeterminacy If p ' = M ' mg ( the economy will begin at, and remain at, the upper steady state forever. If instead, p ' = M ' mg ' the economy will begin at, and remain at, the lower steady state forever 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 45

46 Sunspots and Indeterminacy If p ' = M ' mg ', M ' mg (, There will be a self-fulfilling sequence of beliefs abut future prices that will cause real balances to converge to the lower steady state Al of these sequences are Pareto Inefficient 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 46

47 Sunspots and Indeterminacy We can also add a random variable (a sunspot) to the difference equation that describes equilibria m [a' = m [ ( + g + ε [a' And as long as E [ ε [ = 0 Sequences generated by the equation obey all of the conditions that define an equilibrium 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 47

48 Stochastic Equilibria m [a' invariant distribution m [a' = m [ ( + g + ε [a' g + ε r g + ε s m [ 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 48

49 Summary Sunspot equilibria occur if allocations differ in the face of extrinsic uncertainty They are constructed as randomizations across the multiple equilibria of a non-stochastic model Monetary models always have indeterminate sets of multiple equilibria and are therefore good candidates to generate sunspot equilibria 9/3/17 (c) Roger E. A. Farmer, Gerzensee Lectures 49