Discussion Class Notes

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1 Department of Economics University of Minnesota Macroeconomic Theory Varadarajan V. Chari Spring 2015 Discussion Class Notes Keyvan Eslami The following notes are prepared as a complement to V.V. Chari s Macroeconomic Theory course, taught as the third part of Macroeconomics Sequence to first year graduate students, based on the requirements of the course, and under his supervision. Stokey et al. 1989) and Ljungqvist and Sargent 2012) has been used as the major references.

2 1 Stochastic Dynamic Programming We extensively considered sequential problems in the context of neoclassical growth models, in the deterministic settings, previous mini. We also considered the recursive counterparts of these problems, and established the relationship between these two formulations; this was done under the topic of principle of optimality. We start this mini by moving into the world of uncertainty; consider the same old neoclassical growth model, with a single type of infinitely lived consumer, and a single composite good, that can be consumed or invested. The only difference is that, now, instead of a certain amount of output for a given amount of capital stock, output is affected by an exogenous random shock. In this new setting, the functional equation we have to consider takes the following form: v k, z) = { } sup u [zf k) k ] + β v k, z ) Q z, dz ). 1) k [0,zfk)] Z Here, k is an endogenous state variable, and z is an exogenous state variable, that follows a known stochastic process, namely, a stationary Markov process. As you might expect, our first step would be, like the deterministic case, to establish the relation between the solution to this functional equation and the policy associated with it) and the corresponding value of a sequence problem. To do so, we need a little bit of preliminaries. 1.1 Measure Spaces and Measure Functions Let X be a set and X a σ-algebra over X; X is simply a collection of subsets of X that is closed under countable complement and intersection operations. The pair, X, X ) is called a measurable space. A measure is a function, µ : X R +, that satisfies the following conditions: null set has measure zero; µ ) = 0, 1

3 µ ) is countably additive; ) µ E i = i I i I µ E i ), for every disjoint countable collection of sets, I, in X. It is important to bear in mind that µ ) is not defined on a general sub-collection of X. The triple X, X, µ) is called a measure space. Given two measurable spaces, X 1, X 1 ) and X 2, X 2 ), X = X 1 X 2, together with X, the smallest σ-algebra containing all the rectangles of the form E 1 E 2 where E 1 X 1 and E 2 X 2 ), constitutes a measurable space. X, X ) is called the product space of X 1, X 1 ) and X 2, X 2 ). µ ) is called a probability measure if µ X) = 1. Then, we call the triple X, X, µ) a probability space. The introduction of measure theory in real analysis opened a whole new door to the study of probability; what became modern probability theory. Intuitively, X can be thought of as a set of possible outcomes. Then, elements of X can be viewed as events. Then, µ E), for any E X gives the probability of an event. Note that, naturally, if {E i } i I is a countable collection of disjoint events in X, we expect the probability of one of these events happening to be the sum of the probability of each of these events.) Given this concise introduction into measure spaces, we can now define a transition function, as follows: Definition 1 Let X, X ) be a measurable space. A transition function is a function Q : X X [0, 1], such that: 1. for each x X, Q x, ) is a probability measure on X, X ), 2. and, for each E X, Q, E) is a X -measurable function. Using the definition of transition functions, we can describe a stationary Markov process by means of a transition function, Q ); Q x, E) gives the probability that next period s state lies in E, given that this period s state is x; i.e. Q x, E) = P r {x t+1 E x t = x}. 2

4 Now, consider an exogenous random shock, z, that can take values in a set Z. Suppose Z, Z ) is the corresponding measurable space. Moreover, assume the transition of z is given by the transition function Q ), and g : Z R is a Z -measurable function. Then, we can write the expectation of g ), conditioned on this periods realization of z as: E [g z ) z] = g z ) Q z, dz ), where the integral is Lebesgue integral. Z 1.2 Principle of Optimality Let us get back to our stochastic dynamic program, and define each element precisely. 1 To do so, we consider the general case of 1, in which k and z need not be scalars, but general vectors. To this end, suppose X, X ) and Z, Z ) are measurable spaces, and S, S ) = X Z, X Z ) is the product space. S is the set of states for this system, whose elements consist of an exogenous state, z Z, and an endogenous state, x X. We will assume that the exogenous state follows a stationary Markov process, whit transition function Q ), defined on Z, Z ). Given today s state, s, let us denote the set of feasible next period states by Γ s); Γ : S X is the feasibility correspondence. Let A be the graph of Γ ): A = {x, y, z) X X Z y Γ x, z)}. Suppose F : A R is one-period payoff function, and β 0 is the discount factor. Recall from the last mini that, in a sequential setting, a planner s problem in each period is to choose a sequence of contingent plans that maximize the discounted expected value of utility. These contingent plans can be viewed as functions of the history of shocks up to a 1 What follows is based on Chapter 9 of Stokey et al. 1989). 3

5 given date t, namely, z t : max {k t+1 z t )} zt Z t t 0 E 0 t=0 β t u [ z t f [ k t z t 1 )] k t+1 zt, z t 1)] 2) s.t. 0 k t+1 zt, z t 1) z t f [ k t z t 1 )] for all z t 1 and z t, given k 0 and z 0. To generalize this problem, to a setting where z is a Markov process instead of a Markov chain, let Z t, Z t ) be the product space up to period t 1, and z t Z t be the partial history of shocks up to date t. Then, we can define a plan as follows: Definition 2 A plan is a value π 0 X and a sequence of measurable functions π t : Z t X, for t = 1, 2,.... A plan π is feasible from s 0 S, if: 1. π 0 Γ s 0 ), 2. and, π t z t ) Γ [π t 1 z t 1 ), z t ], for all z t Z t and t = 1, 2,.... Let us denote the set of all feasible plans from s 0 by Π s 0 ). Recall from the deterministic case that, our first requirement for the sequence problem to be well defined was for Γ ) to be a non-empty correspondence. Here, we need a stronger assumption that ensures the existence of measurable functions. This is done in the following assumptions. Assumption 1 Γ ) is non-empty valued, and A is X X Z )-measurable. Moreover, Γ ) has a measurable selection; i.e. there exists a measurable function h : S X such that h s) Γ s) for all s S. It is straightforward to show that, under Assumption 1, Π s 0 ) is non-empty for all s 0 S. Next, we have to specify how the planner calculates the expectations in 2. To do so, given the transition function of the Markov process governing z, it is straightforward at least intuitively) to construct a probability measure explaining the evolution of history up to date t, namely µ t z 0, ) : Z t [0, 1]. Now, if we define a σ-algebra, A as: A = {C X X Z C A}, 4

6 for the discounted expected payoff to be well-defined a the general version of 2, we can impose the following assumption: Assumption 2 F : A R is A -measurable, and one of the followings holds: 1. F 0 or F For each s 0 = x 0, z 0 ) S and each plan π Π s 0 ), F [π t 1 z t 1 ), π t z t ), z t ] is µ t z 0, )-integrable, for t = 1, 2,..., and the limit F x 0, π 0, z 0 ) + lim n n t=1 exists although it might not be bounded). Z t β t F [ π t 1 z t 1 ), π t z t ), z t ] µ t z 0, dz t) Notice that, under Assumptions 1 and 2, discounted expected payoff in a planner s problem) is well-defined, and we can define the value of a planner as: v s) = sup {F x 0, π 0, z 0 ) π Πs) + lim n n β t F [ π ) t 1 z t 1, π ) ] t z t, z } t µ t z 0, dz t). 3) Z t t=1 The recursive counterpart of this optimization problem can be written in therms of the following functional equation: { } v s) = v x, z) = sup F x, y, z) + β v y, z ) Q z, dz ). 4) y Γs) Z If there exists a function v ) that solves this functional equation, then, we can define the associated policy correspondence as: G x, z) = { } y Γ x, z) v x, z) = F x, y, z) + β v y, z ) Q z, dz ). 5) Z How can we generate a plan from this correspondence? Notice that, now, G x, z) being non-empty is not enough to ensure the existence of a feasible plan; now we need existence 5

7 of a measurable selection from G ). If such a measurable selection exists, say g 0, g 1,... is a sequence of measurable selections from G ), then we can generate a plane as: π 0 = g 0 s 0 ), π t z t ) = g t [ πt 1 z t 1 ), z t ], for all z t Z t and t 1. Now, we are ready to present a partial counterpart of the principle of optimality for the stochastic setting: Theorem 1 Principle of Optimality) Suppose X, X ), Z, Z ), Q, Γ, F, and β satisfy Assumptions 1 and 2. Let v ) be the solution to 3, and v ) the solution to 4, so that: lim β t v [ π ) ] t 1 z t 1, z t µ t z 0, dz t) = 0, t Z t for all π Π s 0 ), and all s 0 S. Let G ) be the correspondence defined by 5, which is non-empty and permits a measurable selection. Then v = v, and the plan generated by G ) attains the supremum in Solution to the Functional Equation Next step, is to ensure the existence of a solution to the functional equation. Like the deterministic case, when the return function is bounded, there is a good chance that this is the case. In this section, we consider the fairly general assumptions under which, this is the case, by focusing on the case of bounded returns. Assumption 3 X is a convex Borel set in R l, and X is its Borel subsets. Assumption 4 One of the followings holds: 1. Z is a countable set, and Z is the σ-algebra containing all of its subsets. 2. Z is a compact Borel set in R k, with its Borel subsets Z, and the transition function Q ) has the Feller property. 6

8 The key role of Assumption 4 is to ensure that the integral in 4, Mf y, z) = Z v y, z ) Q z, dz ), for all y, z) X Z, 6) maps a bounded continuous function v : X Z R into the space of bounded continuous functions over X Z. Moreover, this assumption ensures that, if v ) is increasing or concave, then the integral would be an increasing or concave function of y, z). Given this property, the rest is quite similar to the stochastic case; first, we may use Blackwell s sufficient conditions to ensure that the mapping defined by 4 is a contraction, and then use the Contraction Mapping Theorem to ensure the existence of a fixed point. First we need the following two assumptions: Assumption 5 The correspondence Γ : X Z X is non-empty, compact-valued, and continuous. Assumption 6 The function F : A R is bounded and continuous, and β 0, 1). Now, we have the following theorem: Theorem 2 Under Assumptions 3-6, the operator T, defined by { } T f x, z) = sup F x, y, z) + β f y, z ) Q z, dz ), 7) y Γs) Z maps the set of bounded continuous functions, C S), into itself, and has a unique fixed point in this set, v ) C S). Moreover, for all v 0 ) C S): T n v 0 v β n v 0 v, n = 1, 2,.... In addition, the correspondence G ) defined by 5 is non-empty, compact-valued, and upper hemi-continuous. 1.4 Inheritance Properties of the Value Function If the operator M in 6 preserves the monotonicity and concavity of the integrand, it is natural to expect that the value function inherits these properties from the payoff function; 7

9 what we had in the deterministic case, as well. To formalize this idea, let us introduce the following assumptions. Note that, A i denotes the i-section of the set A, in what follows. Assumption 7 For each y, z) X Z, F, y, z) : A yz R is strictly increasing. Assumption 8 For each z Z, x x implies Γ x, z) Γ x, z). Now, we have our first inheritance property of the value function: Theorem 3 Under Assumptions 3-8, the fixed point of operator T in 7 is strictly increasing in x, for each z Z. The value function inherits the concavity of the payoff function as well: Assumption 9 For each z Z, F,, z) : A z R is strictly concave in x, y). Assumption 10 The set A z is convex. Theorem 4 Under Assumptions 3-6 and 9-10, the fixed point of operator T in 7 is strictly concave in x, for each z Z, and the corresponding policy correspondence is a continuous function. Finally, v ) inherits the differentiability of the payoff function, too: Assumption 11 For a fixed z Z, F,, z) is continuously differentiable in x, y), in the interior of A z. Theorem 5 Suppose Assumptions 3-6 and 9-11 hold, v ) C S) is the fixed point of operator T in 7, and g : S X is the corresponding value function. If x 0 intx, and g x 0, z 0 ) intγ x 0, z 0 ), then v, z 0 ) is continuously differentiable in x at x 0, and we have: v i x 0, z 0 ) = F i [x 0, g x 0, z 0 ), z 0 ], for i = 1, 2,..., l. 2 Weak Convergence of Markov Processes When dealing with competitive equilibrium concept in the deterministic case, the allocations are usually thought of sequences of real numbers in the appropriate commodity space). We often are interested in the asymptotic properties of these allocations; if they converge to a finite limit, and, if so, what is this limit. 8

10 As we saw last class, the competitive equilibrium allocations in the stochastic environment are not in general sequences of numbers, but sequences of measurable) functions; in other words, random variables. So, the asymptotic properties of these allocations can not be stated in terms of convergence of a real sequence in Euclidean norm. Instead, we have to consider the asymptotic behavior of the random variables that represent the allocations. This, first of all, calls for some appropriate definition of convergence of random variables, and, next, theorems that ensure the convergence in this new sense. When the random shocks in the economy follow a Markov process, we are able to write the sequence problem of a planner in particular, and the equilibrium in general, as a stationary dynamic programming problem. Moreover, as you have seen in the class, when the stochastic process governing the exogenous shocks is Markov, we can describe the transition of the endogenous states as a Markov process as well. This, in turns, enables us to focus on the definition of convergence of random variables that are defined over a Markov space. More importantly, there are theorems that guarantee this type of convergence for Markov processes, under justifiable assumptions. In this session, we are going to define a notion of convergence for Markov processes, and state the theorems that ensure their convergence. 2.1 Some Definitions Weak Convergence of Probability Measures There are different notions of convergence of probability measures. But, what we really are interested in is weak convergence. Let S, ρ) be a metric space, and S be its Borel subsets; i.e. S is the σ-algebra generated by the family of open sets in S. We usually think of S as a subset of k-dimensional Euclidean space.) Let C S) be the set of continuous functions defined on S, and Λ S, S ) the set of probability measures defined on S, S ). Definition 1 A sequence of probability measures {λ n } n 1 Λ S, S ) is said to converge weakly to λ Λ S, S ), written λ n λ, if lim f s) λ n ds) = f s) λ ds), f C S). n S S 9

11 Note that, when f ) is a continuous function on S, and λ ) is a probability measure on S, S ), fdλ is the expectation of f ) with respect to λ ). Therefore, we say a sequence of probability measures converges to a probability measure if the expectation of a all the continuous functions converges. A more intuitive definition of the weak convergence of probability measures is provided in the following theorem. Theorem 1 Suppose {λ n } n 1 and λ are probability measures in Λ S, S ). Then λ n λ if and only if for all A S with λ A) = 0. lim λ n A) = λ A), n This theorem says a sequence of probability measures converges weakly to a probability measure if and only if, in the limit, they assign the same probability to the sets that do not have dense boundaries i.e. boundaries with positive probability under the latter) Transition Functions, and Markov and Adjoint Operators We talked about a Markov process on S, S ) last class, and how we can characterize it by means of transition function, Q ). For any s S and A S, Q s, A) gives the probability of event A happening in next period, given this period s shock is s. For a transition function Q ), the probability of event A happening after N periods, given this period s realization of shock is s, can be derived as: Q N s, A) = = where Q 1 s, A) = Q s, A). S S S... Q s N 1, A) Q s N 2, ds N 1 )... Q s, ds 1 ) S Q N 1 s, A) Q s, ds ), There are two important operators associated with every Markov process and transition function, in general). Definition 2 Given Q ), a transition function on S, S ), the Markov operator associated 10

12 with Q ) is defined as: T f s) = f s ) Q s, ds ), s S, S where f ) is a S -measurable function defined on S. The adjoint operator of T is defined as: T λ A) = Q s, A) λ ds), A S, S where λ ) is a probability measure over S, S ). Intuitively, given a Markov process, Markov operator determines the expected value of function f ) in the next period, given the realization of shock in this period. On the other hand, the adjoint operator of T given the unconditional) probability of an event next period, given the unconditional probability of the shock this period. Question What can you say about the relation between T and T? The following equation gives the relation between the two: T f s) λ ds) = f s ) T λ ds ). S S Each side of this equation gives the unconditional expectation of f ) next period, given the probability measure of shocks this period. Given a Markov process, characterized by a transition function, we can use the associated Markov operator to calculate next period s expected value of a measurable) function, as a function of this period s realization of shock. Feller property and monotonicity of the transition function determine if the Markov operator preserves properties of the function whose expected value we are interested in. Definition 3 A transition function Q ) is said to have Feller property on S, S ) if the associated Markov operator maps the space of bounded continuous functions into itself. It is said to be monotone if for every non-decreasing function f ) on S, T f ) is also non-decreasing. Feller property and monotonicity of the Markov operator have counterparts in terms of adjoint operators. The following theorem states these counterparts. 11

13 Theorem 2 A transition function Q ) on S, S ) is monotone if and only if for all λ, µ Λ S, S ), T µ first order stochastically dominates T µ, if µfirst order stochastically dominates λ. Moreover, Q ) has the Feller property if and only if λ n λ implies T λ n T λ. Remark Now you can observe why we talked about monotonicity in class in terms of stochastic dominance of probability measures. 2.2 Monotone Markov Processes As we mentioned before, when the exogenous shocks follow a Markov process, solving a stationary dynamic program and finding the associated policy functions enable us to write the endogenous states themselves as Markov processes, by finding the appropriate transition function. Suppose P ) represents such a transition function on the space S, S ); here, S is the set of possible states of the economy, not just endogenous states. Assume T and T are the associated Markov and adjoint operators for this transition function, respectively. We can use these operators to construct a probability measure that characterizes the state of the economy endogenous and exogenous states) in each period. Recall again, that the states in this setting are random variables.) One question of particular interest is what are the asymptotic properties of the state? Given the probability measure over the state in a given period, say period t = 0, λ 0 ), according to the definition of adjoint operator, the probability measure for the next period s state is given by λ 1 = T λ. Similarly, the probability measure of the state in t = 2 is given by λ 2 = T λ 1 = T 2 λ 0. This way, we can construct a sequence of probability measures, {λ n } n 0. The question is, under what conditions, does this sequence converges to a limit in the weak convergence sense? A probability measure λ ) is called the invariant measure of the Markov process P ) if T λ = λ ; i.e. if λ ) is the unconditional probability of the state in period t, it will be the unconditional probability of the next period s state, as well. If we let λ 0 = λ, it is clear that λ n = λ for all n, and hence the sequence {λ n } n 0 converges to a limiting probability measure, λ. The important question is, if such an invariant measure exists, will {λ n } converge to it for 12

14 every initial distribution of the state, λ 0, in at least the weak convergence sense? If S is a compact subset of k-dimensional Euclidean space, by a powerful theorem in probability theory, Helly s Theorem, we can be sure that any transition function defined on S, S ) that has the Feller property, has an invariant measure. This invariant measure looks like a good candidate for the asymptotic distribution of the sequence of probability measures {λ n } n 0, constructed above. However, there is a major problem; the invariant measure might not be unique, and, hence, we cannot be sure in which one of these distributions we might end up in the limit. Figure 1, from Stokey et al. 1989), illustrates this problem; let S = [a, b] a compact subset of R), and let the transition function be defined as follows: for a given state s in period t, next period s state is a draw from the uniform distribution on the interval [h s), H s)]. You can check and see that the transition function corresponding to this process satisfies the Feller property. However, if we start at s = a with probability one, we will get stuck in the set E 1 forever, and, if we start from s = b with probability one, we end up in set E 2. To avoid this type of multiplicity of the asymptotic distribution, we impose the following assumption. Assumption 1 There exists c S, ε > 0, and N 1 such that P N a, [c, b]) ε and P N b, [a, c]) ε. Assumption 1 is know as mixing property; it says, if we give the process enough time, there is always a positive possibility of going from one tail of distribution to the other. Now, we are ready to state the main result of this class. Theorem 3 Let S = [a, b] R l. If P ) is monotone, has the Feller property, and satisfies Assumption 1, then P ) has a unique invariant probability measure λ ), and for all λ 0 Λ S, S ), T n λ 0 λ. 3 The Risk-Free Rate in Incomplete Economies You saw that, in the framework of Arrow-Debreu economies with a full set of securities, agents can perfectly insure themselves against idiosyncratic shocks. Huggett 1993) describes an economy in which the set of insurance instruments is not complete. Therefore, 13

15 Figure 1. Multiple Ergodic Sets 14

16 agents face the risk of fluctuations in their consumption. In the absence of complete insurance instruments, agents have an incentive to over-accumulate assets. Thus, to discourage this over-saving and attain the equilibrium, the interest rate on the risk-free bonds must be lower than what it would be in a complete market. Our purpose is not to discuss the importance of the model, its relevance, or the intuition behind this; you would see and hear more about this in class. Instead, we want to prove a technical result that will have an important role in its implications. To this end, we will try to give a rough outline of the model. After that, we will jump immediately to the result we want to prove. 3.1 The Model Consider an exchange economy with a continuum of agents of measure one. In each period, agents receive an endowment of a non-storable consumption good, in the set E = {e l, e h }, where e h > e l. The stochastic process governing the endowments of each agent follows a Markov chain, that is characterized by transition function π ), and is independent across agents. We will assume π e e) > 0, for all e, e E. Agents maximize the expected value of their lifetime utility, which is represented by the following utility function: [ E t=0 β t c1 σ) 1 σ) where β 0, 1), and σ > 1. Agents budget constrains is given by c + qa a + e, ], where c 0, and a ā. Here, q is the constant price of the assets, and ā is the lower bound on agents credit balances. If we let X = A E be the state-space, for A = [ā, ), for an agent with individual state x X, the functional equation describing agent s decision problem is then given by v x; q) = max a Γx;q) { c 1 σ) 1 σ) + β } v a, e ; q) π e e), 8) e 15

17 where Γ x; q) = {c, a ) c + qa a + e, c 0, a ā}. We will denote the policy functions associated with this functional equation by c x; q) and a x; q). Naturally, this equation describes a mapping of the form T v x; q) = max a Γx;q) { c 1 σ) 1 σ) + β } v a, e ; q) π e e). e It is straightforward to infer, from what we have seen so far, that if q > 0 and ā+el āq > 0, then, T ) defines a contraction mapping and there is a unique bounded continuous function defined over X, v ) C X), that solves the functional equation in 8). Moreover, v ) is strictly increasing, strictly concave, and continuously differentiable in a. Moreover, c ) and a ) are continuous functions, and a ) is non-decreasing in a, and strictly increasing in a for x; q) so that a x; q) > ā. In addition, c a, e) is strictly increasing in a. Exercise 1 Prove the statements of last paragraph. 3.2 Theorem 2 in Huggett 1993) We do not intend to discuss the importance of Huggett 1993) s model, or its implications. Our goal is to prove a rather technical theorem that drives these implications. To be able to state and quite understand) Theorem 2 in Huggett 1993), we first have to prove the following lemma. Lemma 1 Suppose q > 0, ā + el āq > 0, β < q, and π eh e h ) π π h e l ). Then, there is a subset of the state-space, S = [ā, ā] E X, with the property that, if an agent starts out in S, then he in fact, his state) will remain in S. To prove this result, we will follow the following steps. Lemma 2 Under the conditions of Lemma 1, a a, e l ) < a, for a > ā. Proof. To prove this, we follow a variation argument as follows; suppose this was not the case, and there exists some a A so that a a, e l ) a. In particular, assume a a, e l ) = a > ā the argument for a a, e l ) > a follows a similar argument, so, this assumption will not 16

18 cause any loss in generality). Suppose the agent increases this period s consumption by qɛ, by decreasing her savings by ɛ. This change will increase this periods utility by q e l + a qa) σ ɛ = q e l + 1 q) a) σ ɛ, while having a discounted expected utility cost of β [π e h e l ) v a, e h ; q) + π e l e l ) v a, e l ; q)] ɛ. By envelope condition, we can write this as β [ π e h e l ) e h + a qa ) σ + π e l e l ) e l + a qa) σ] ɛ, where a = a a, e h ). Note that, e + a qa ) σ = c a, e) σ. So, if we could show c a, e) is non-decreasing in its second argument, then β [ π e h e l ) e h + a qa ) σ + π e l e l ) e l + a qa) σ] β e l + a qa) σ. Then, since q/β > 1, by assumption, we could write: q e l + a qa) σ ɛ β [π e h e l ) v a, e h ; q) + π e l e l ) v a, e l ; q)] ɛ. This means, a decrease of savings by ɛ would increase the expected utility of the agent, and hence a a, e l ) = a would not be an optimal decision. It remains to show c a, e) is non-decreasing in e. Exercise 2 Prove that c a, e) is non-decreasing in e. Lemma 3 If v a, e) > β/q) E [v a, e ) e] for a a > ā, then a a, e) < a for a a. Proof. If a a, e) = ā, we are done. The first order condition for an agent s problem, when a a, e) > ā, is q e + a qa a, e)) σ = β [π e h e) v a a, e), e h ; q) + π e l e) v a a, e), e l ; q)]. 9) 17

19 By envelope condition: e + a qa a, e)) σ = v a, e). Since v ) is concave, for the equality in 9) to hold when a a, we must have a a, e) < a. Lemma 4 Under the conditions of Lemma 1, there exists a A so that a a, e h ) = a. Proof. Suppose this was not the case, so that a a, e h ) > a, for all a A. By Lemma 2, we know that a a, e l ) < a. Therefore, a a, e h ) > a a, e l ), for all a. Thus: a + e l qa a, e h ) a + e l qa a, e l ), c a, e h ) e h e l ) c a, e l ), c a, e l) c a, e h ) 1 e h e l ) c a, e h ). Concavity and monotonicity of v implies that v a 1, e) v a 2, e) > 0, for all a 2 a 1. As mentioned above, c a, e) is strictly increasing in a. But, v ) is bounded and strictly increasing. Therefore, lim a v a, e) must be zero. By envelope condition, v a, e) = u c a, e)) = c a, e) σ. Therefore, as a, c a, e) must tend to infinity. As a result, as a, we have: v a, e h ) v a, e l ) = [ ] σ [ c a, el ) 1 e ] σ h e l ) = 1. c a, e h ) c a, e h ) This means, since β/q < 1, there is some a, large enough, so that v a, e h ) v a, e l ) β q, for a a. As we proved in Lemma 2, v a, e h ) v a, e l ). Therefore, the hypothesis of Lemma 3 will hold, and this is a contradiction. Lemma 2 and 4 imply that the policy function a a, e), looks like the one depicted in Figure 2. As it is apparent in this figure, there is an interval in A, which we denote by S = [ā, ā], so that, when the agent s asset holdings is in S, it will stay in the same set. This proves our claim in Lemma 1. 18

20 Figure 2. Existence of an Ergodic Set 19

21 Now that we know we can restrict our attention to a compact subset of R, as the space of endogenous state, we can check as see whether the hypothesis of convergence theorem of the Markov processes that we stated last class are satisfies or not. 4 A Shopping Time Monetary Economy In a complete market model, fiat currency i.e. money that has no intrinsic value and is not convertible into goods that do) would be valueless. To see why this is the case, consider an agent that holds positive amount of money. A necessary condition for this decision to be optimal is transversality: lim t [ t 1 s=0 ] 1 m t+1 = r s ) p t A no arbitrage argument implies that the real return on money, p t /p t+1, has to be equal to 1 + r t in order for the agents to hold money. Thus: t 1 s=0 t r s ) = s=0 = p t p 0. p s+1 p s If we substitute this into the transversality condition, we have: m t+1 lim = 0. t p 0 Thus, if we consider the inconvertible money one for which lim t m t+1 > 0), p 0 = ; i.e. money must have no value. Therefore, to assign positive value to money, we have to alter the complete market structure in some way. Today, we do this by means of introducing a shopping technology to the economy, so that buying consumption goods requires time and effort, and this effort is decreasing in the real balances an agent holds. You are going to see how we can do this in the context of cash-in-advance models, in class. 20

22 4.1 The Model Consider an endowment economy with a single type of representative households. Household is endowed with one unit of time, and y units of consumption good in each period. Its preferences are represented by the following utility function over consumption and leisure: β t u c t, l t ). t=0 To acquire the consumption good, household has to spend some shopping time; specifically, purchasing c t units of consumption good, when household is holding m t+1 /p t units of real balances, calls for s t units of shopping time. s t, c t, and m t+1 /p t are related according to the following shopping technology: < 0. An example of such a func- where we assume H, H c, H c,c, H m p, m p tion is s t = H c t, m ) t+1, p t > 0, and H m p, H c, m p s t = c t m t+1 /p t ɛ. Even though this is not quite intuitive, you can think of s t as the effort, when an individual pays for his purchases in cash, and he does not want to carry around more than m t+1 of it. Therefore, to purchase c t, he has to go p t c t /m t+1 times to the bank. If each time takes ɛ of effort, the total effort would be s t. Given that an individual has to allocate his time between leisure and shopping, he faces a constraint of the form l t + s t = 1, every period. The government, in this economy, has to finance a stream of exogenous expenditures, {g t } t=0, through fiscal and monetary policy, characterized by lump-sum taxation, τ, issuing debt, B, and printing money, M. 4.2 General Equilibrium The notion of competitive equilibrium in this economy is exactly what you would expect it to be. 21

23 Definition 1 Given {g t, τ t } t=0, B 0 = b 0, and M 0 = m 0 > 0, an equilibrium in this economy is a price system,{p t, R t } t=0, an allocation for the households, {c t, l t, s t, b t, m t } t=0, a government policy, {M t, B t } t=1, so that: 1. given the price system and taxes, {c t, l t, s t, b t, m t } t=0 solves household s problem, given by max {c t,l t,s t,b t,m t} β t u c t, l t ) 10) t=0 s.t. c t + m t+1 p t + b t+1 R t s t + l t = 1 s t = H c t, m ) t+1, p t y + m t p t + b t τ t 2. government runs a balanced budget; g t + B t = τ t + B t+1 R t + M t+1 M t p t, t 0, 11) 3. and, markets clear in each period t 0; a) c t + g t = y, b) M t = m t, c) and B t = b t. Note that, B t in the definition of equilibrium is the government s debt to the private sector, maturing at the beginning of period t, and M t is the total supply of money. R t is the gross rate of return, therefore 1/R t is the face value of government or private) debt Money Demand Given this solution concept, for the usual functional forms for the utility function that we have seen so far, an equilibrium allocation must satisfy the following first order conditions: 1. u c c t, l t ) = λ t + µ t H c c t, m t+1 p t ), 22

24 2. u c c t, l t ) = µ t, 3. λ t R t = βλ t+1, 4. and, λt p t = β λ t+1 p t+1 µt p t H m p c t, m t+1 p t ), where β t λ t is the Lagrange multiplier on the household s budget constraint, and β t µ t is the multiplier on the shopping technology. If we multiply 4 by p t, we get: Denoting p t /p t+1 by Rt m write this equality as: λ t = βλ t+1 p t p t+1 µ t H m p c t, m ) t+1. p t the real return of money), as we did in the introduction, we can λ t = βλ t+1 R m t After substituting for βλ t+1 from 3, we get: µ t H m p c t, m ) t+1. p t ) 1 Rm t λ t = µ t H m c p t, m ) t+1. 12) R t p t Now, we can substitute for λ t from 1, and for µ t from 2, to get: ) 1 Rm t [u c c t, l t ) u c c t, l t ) H c c t, m )] t+1 = u c c t, l t ) H m c p t, m ) t+1, R t p t p t which can be simplified more into: ) [ 1 Rm t uc c t, l t ) R t u c c t, l t ) H c c t, m )] t+1 + H m c p t, m ) t+1 = 0. 13) p t p t Noting that l t = 1 H c t, m t+1 /p t ), this equation defines a money demand function implicitly of the form m t+1 p t = F ) c t, Rm t. 14) R t 23

25 The important remark about this function is that it is increasing in both its arguments, if we note that the coefficient on the left hand side of 13) is positive. To see why this is the case, consider the equality in 12) once more; since H m < 0, for the agents to hold positive p amounts of both money and government bonds in any given period, we must have ) 1 Rm t 0. R t This, in fact, is a no arbitrage condition; if the real return on money was greater than that of bonds, agents could make arbitrary large amount of profit by accumulating debt and holding money. Not only would this give them the opportunity to decrease their shopping time, but also would give them unbounded profit. Question Why the inverse can still be true? Why can we have Rt m R t in equilibrium? This is due the transaction technology; households want real balances to carry out their transactions. Therefore, even when the real return on money is less than that of bonds, they would choose to hold money in equilibrium, though, money demand is increasing in the return of money. This feature of this economy breaks down the no arbitrage relation between money and bonds, and creates value for money, unlike the complete market economy, discussed at the beginning of this section Equilibrium Determination We will focus on the government policies with g t = g, t 0, τ t = τ, t 1, B t = B, t 1, and τ 0 and B 0 that are potentially different from τ and B. This enables us to study the equilibria in which the economy moves from an initial state at t = 0 short-run equilibrium) to a stationary path from date t = 1 onward long-run equilibrium). Notice that, when g t = g for all t 0, from the good s market clearing condition in the equilibrium, c t = c = y g for all t 0. Seeking a stationary equilibrium from date t = 1, 24

26 we also need I don t know why wee need these in the initial period as well!) p t p t+1 = R m, t 0, R t = R, t 0, s t = s, t 0. Since l t = 1 s t = l for all t 0 in this equilibrium, from conditions 1 and 2 of household s optimization problem: u c c t, l t ) = λ t + u c c t, l t ) H c c t, m ) t+1, p t λ t+1 λ t = 1. If we substitute this result into condition 3, we get: R t = 1 β. Moreover, since c t is constant in all dates, we can write 14) as m t+1 p t = f R m ). 15) As we noted earlier, f R m ) 0. Now, if we substitute this into government s budget constraint for t 1, we have: R 1) g τ + B R = 1 R m ) f R m ). 16) The right hand side of this equation is government s revenue from printing currency know as seigniorage). Since we require R m to be constant at all dates, from the money demand equation, M 1 /p 0 = 25

27 f R m ). Therefore, government s budget constraint at date t = 0 can be written as: g + B 0 = τ 0 + B R + M 1 p 0 M 0 p 0, g + B 0 = τ 0 + B R + f Rm ) M 0 p 0. 17) Equations 15), 16), and 17) completely characterize equilibrium; from 16), given R = β 1, we can calculate R m, and use it to determine p 0. Given p 0, we can pin down the sequence of prices, and money supply that supports this equilibrium through 15) and definition of R m. 4.3 A Few Monetary Doctrines Having a monetary model at our disposal, we can use it to study different government policies. In what follows, we give examples of some monetary doctrines, and address the model implications regarding them Quantity Theory of Money Suppose we increase the initial money supply, M 0, by some factor γ, while leaving all the other variables unchanged. From Equations 16) and 17), we can see that such a change would only increase the initial price level and sequence of money supplies by the same factor, γ, while leaving everything else intact Sustained Deficit Causes Inflation If we draw the graph of 1 R m ) f R m ) for a utility function of the form u c, l) = and a shopping technology of the form c1 δ 1 δ) + l1 α 1 α), H c, m) = c 1 + m, 26

28 it would look like an inverse U-curve for standard parameter values). Thus, for a given level of government deficit, g τ + B R 1) in Equation 16), there are two levels of real R return on money that can be sustained in equilibrium. If we choose the higher level as the favorable policy note that, higher rate of return on money corresponds to lower levels of inflation), then an increase in the government deficit would result in a decrease in R m ; this is in accordance with the classical point of view that sustained deficit would increase inflation Optimality of Friedman Rule What is the optimal level of real balances in this economy? The social value of real balances in this economy is to decrease the shopping time, and leave the households with more leisure time. By increasing the rate of return on money, government can increase the real balances, as shown by Equation 15). On the other hand, government can run a surplus, by financing it trough τ. This implies, through Equation 16), that it can achieve any level of R m 1, β 1 ). This would lead to an increase in welfare, and is in accordance with Friedman s idea of satiating the economy with real balances, as much as possible. 5 A New-Keynesian Model You were introduced to a monopolistic competition model with price rigidities in class where only a constant fraction of firms, the so-called flexible-price firms, could change their prices after the realization of a shock; the rest had to set their prices at the beginning of each period and keep their promise of delivering the demand after the realization of exogenous shocks. Today, we are going to present a similar basic New-Keynesian model in which monopolistic firms can adjust their prices in each period with a constant probability of 1 θ. This is based on the staggered price-setting model of Calvo 1983). Thus, after the price is set, the expected duration of this price being effective is 1/ 1 θ). This creates a so-called!!) micro-founded rigidity in aggregate price level. To make this interesting, imagine there is a fairy in this economy, which we refer to as Calvo fairy! This fictitious being appears at the beginning of each period t, chooses fraction 1 θ) of firms 27

29 randomly and taps them on the shoulder, giving them the permission to change their prices. This story occurs in every period. A more realistic story is that there are small costs of changing prices in this economy, referred to as menu costs. So that, in each period, only a fraction θ of firms find it too costly to adjust their prices to unanticipated shocks in economy. We will invite you to think about the difference of these two settings, in terms of persistence of unanticipated nominal shocks in the economy! For a firm that sets its price in period t, there is a 1 θ) chance of resetting price next period. This means there is a 1 θ) probability of the price in t being in effect only in period t, without experiencing any rigidity in price. With probability θ, the firm has to keep its price fixed in period t + 1. So, the chance of a one-period price rigidity is θ 1 θ). Therefore, the expected length of price rigidity is θ k 1 θ) k, k=0 which is a geometric power series converging to 1 θ) 1. We are going to study the implications of such assumptions for the neutrality of monetary policy and nominal shocks in such an economy. 5.1 Households The economy is inhabited by an infinitely-lived representative household whose preferences over streams of consumption and labor are represented by the following utility function with expected form: E 0 t=0 where C t is a consumption index, given by [ 1 C t := 0 β t U C t, N t ), C t i) ɛ 1 ɛ ] ɛ ɛ 1 di. Here, C t i) is the consumption of a differentiated good indexed by i. We will assume there is an exogenous continuum of such goods in the economy, of unit mass. 28

30 The budget constraint of the household is given by 1 0 P t i) C t i) di + Q t B t B t 1 + W t N t + T t, where p t i) is the price of variety i, B t is a one-period risk free bond with face value price Q t, W t is the wage rate, and T t is the lump-sum transfers to the households possibly in terms of profits, etc.). Thus, we may write the problem of the household as: max E 0 β t U C t, N t ) t=0 [ 1 s.t. C t = C t i) ɛ 1 ɛ ] ɛ ɛ 1 di P t i) C t i) di + Q t B t B t 1 + W t N t + T t lim E t B t ) 0, T where the last condition rules out the possibility of Ponzi schemes in the economy. Exercise 1 What is the difference between condition lim T E t B t ) 0, and the no-ponzi scheme condition you have seen previously e.g. in Tim s course)? Which one is more strict? We can think of a household s problem in period t as consisting of two parts: deciding about the fraction of wealth household wants to allocate to consumption, and dividing this fraction between different varieties of consumption goods. We begin with the latter; i.e. given amount W of wealth dedicated to consumption and price of each differentiated good, how would household divide W between varieties. Given a constant amount of wealth, W, spent on consumption in period t, 1 0 P t i) C t i) di = W, 29

31 we can write the problem of the household as choosing between the different varieties as: max Ci) U C, N) [ 1 s.t. C = C i) ɛ 1 ɛ ] ɛ ɛ 1 di P i) C i) di = W. The first order condition for the optimal consumption of variety i is: U C C, N) C i) ɛ 1 ɛ ɛ [ 1 0 C k) ɛ 1 ɛ ] ɛ ɛ+1 ɛ 1 dk = λp i), i [0, 1], where λ is the Lagrange multiplier on the constraint of the problem. By writing the same condition for variety j, and dividing the two equations, we have: C i) 1/ɛ P i) = 1/ɛ C j) P j), [ ] ɛ P j) C i) = C j). P i) If we substitute this result into the definition of consumption index, we get: and, hence: C = C = [ 1 0 [ 1 0 [[ ] ɛ ɛ 1 ] ɛ ɛ 1 P j) ɛ C j)] di, P i) [ ] ɛ 1 P j) C j) ɛ 1 ɛ di P i) ] ɛ ɛ 1 P j) ɛ C j) = [ ] ɛ C, 1 P 0 i)1 ɛ ɛ 1 di [ ] ɛ P j) C j) = C, 18) P, 30

32 where: [ 1 P := 0 ] 1 1 ɛ P i) 1 ɛ di. 19) We will refer to Equation 18) as the demand function for variety j, as a function of its price P j), aggregate consumption index C, and aggregate price index P. To see the intuition behind the aggregate price index, let us substitute this result back into the constraint of the optimization problem above, to get: W = = P i) C i) di [ ] ɛ P i) P i) Cdi P 0 = P i)1 ɛ di [ 1 P 0 i)1 ɛ di [ 1 = P i) 1 ɛ di 0 =P.C. ] ɛ 1 ɛ C ] 1+ ɛ 1 ɛ C Therefore, we may write the first part of a household s problem simply as: max E 0 β t U C t, N t ) s.t. t=0 P t C t + Q t B t B t 1 + W t N t + T t lim E t B t ) 0, T where P t is defined in 19). The first order conditions for this problem are: 1. β t U C C t, N t ) = P t λ t, 2. β t U N C t, N t ) = W t λ t, 3. and Q t λ t = s t+1 λ t+1 s t+1 ) where s t+1 is an index for next period s state). If we combine conditions 1 and 2, we get the intratemporal optimality condition for labor 31

33 supply: U N C t, N t ) U C C t, N t ) = W t P t. 20) Moreover, we can derive the Euler equation by combining 2 and 3: Q t = βe t UC C t+1, N t+1 ) U C C t, N t ) P t P t+1 ). 21) Remark To see how we can derive this, assume for the moment that next period s state is given by s t. Then: β t U C C t, N t ) Q t = β t+1 U C C t+1 s t+1 ), N t+1 s t+1 )) P t P s t+1 s t+1 ) t+1 21) can be derived using this equality. =β t+1 E t U C C t+1, N t+1 ) P t+1. Now, if we assume that the period utility takes the following separable form, then, we can write 21) as: U C, N) = 1 = E t β 1 Q t C σ t+1 C σ t which can be written in natural log terms as: C1 σ 1 σ) N 1+φ 1 + φ), P t P t+1 ), 1 = E t exp log β + i t σ c t+1 π t+1 )). 22) The lower case letters indicate the logarithm of the variables. i t is defined as log 1/Q t ). Remark i t is the net nominal rate of return. To see why, note that the gross nominal rate of 32

34 return is 1/Q t. If we denote this by 1 + i t, then: ) 1 log 1 + i t ) = log Q t i t. On a balanced growth path, where consumption grows at a constant rate γ, we have: i = log β + σγ + π. A first order Taylor expansion of 22) around the balanced growth path yields: exp log β + i t σ c t+1 π t+1 ) 1 + log β + i t σ c t+1 π t+1. If we substitute this result into Equation 22), we get a first order log-linearization of this condition: log β + i t σe t c t+1 c t ) E t π t+1 ) = 0, c t = E t c t+1 ) 1 σ [log β + i t E t π t+1 )]. 23) On the other hand, under the assumption of separable utility form, we may write Equation 20) as: N φ t C σ t In logarithmic terms, we can rewrite this as: = W t P t. w t p t = σc t + φn t. 24) To introduce money explicitly in the model, we add an ad-hoc log-linear money demand equation to the household side of the economy as well: m t p t = y t ηi t. This equation, determines the path of nominal interest rate i t as a function of exogenous 33

35 money supply M t in the equilibrium. To see the intuition behind this money demand function, consider the following problem faced by a household that values money holdings we have seen how such a utility form may emerge from a monetary economy): max s.t. [ E 0 β t t=0 Ct 1 σ 1 σ) + M t/p t ) 1 ν 1 ν) ] N 1+φ t 1 + φ) P t C t + Q t B t + M t B t 1 + M t 1 + W t N t + T t lim E t B t ) 0. T The first order conditions for the household are: 1. β t C σ t = P t λ t, 2. β t 1 P t Mt P t ) ν = λt λ t+1, 3. and Q t λ t = λ t+1. If we substitute from 3 and 1 into 2, we get: β t 1 ) ν Mt = 1 Q t ) β t 1 P t P t Mt P t ) ν = 1 Q t ) C σ t. P t C σ t, Using our definition of nominal rate of return, we may write this condition as: M t /P t ) ν C σ t = 1 e it i t. Now, if we take the natural logarithm, we may write this equation as follows: m t p t = σ ν c t 1 ν log 1 e it) σ ν c t 1 ν e it 1) i t. 34

36 Assuming unit elasticity of real) money demand with respect to consumption i.e. σ/ν = 1) and incorporating market clearing condition, then, imply: m t p t = y t ηi t. 5.2 Firms There is a continuum of monopolistically competitive firms in the economy that produce a differentiated good according to the same technology, given by: Y t i) = A t N t i). Firm i faces a demand function of the form in 18), and takes the aggregate price index P t and aggregate consumption index C t as given. In each period, fraction 1 θ of the firms have the chance to adjust their prices while fraction θ has to keep their price fixed and meet the demand in the market). Therefore, the average interval between price changes for a firm is 1/ 1 θ). In this sense, θ is an index of price stickiness in this economy. The problem faced by a firm that is chosen in period t to adjust its price can be written as: max p t k=0 s.t. Y t+k t = θ k E t Qt,t+k P t Y t+k t Ψ t+k Yt+k t ))) ) P ɛ t C t+k, P t+k where Ψ is the cost function. Q t,t+k is the stochastic discount factor; it determines how 35

37 future nominal profits are discounted in date t s utility terms, and can be evaluated as: k 1 ) Q t,t+k =E t Q t+s s=0 k 1 UC C t+s+1, N t+s+1 ) =E t βe t+s U s=0 C C t+s, N t+s ) ) =β k UC C t+k, N t+k ) P t E t, U C C t, N t ) P t+k P t+s P t+s+1 where the second equality follows from the Law of Iterated Expectations. It is clear that any firm tapped by the Calvo fairy to adjust its price would face the same problem in period t. Therefore, all the firms adjusting prices in period t would choose If we substitute for Y t+k t into the objective function, we can rewrite the problem of the monopolist as: ) ) max P t k=0 θ k E t Qt,t+k P t ) 1 ɛ P ɛ t+kc t+k Ψ t+k P t ) ɛ P ɛ t+kc t+k ))) The first order condition for this problem, with respect to the control Pt yields:: ))) θ k E t Q t,t+k 1 ɛ) Pt ) ɛ Pt+kC ɛ t+k + ɛ Pt ) ɛ 1 Pt+kC ɛ Ψ t+k Yt+k t t+k = 0. Y t+k t k=0 If we substitute back Y t+k t, we get: ))) θ k E t Q t,t+k 1 ɛ) Y t+k t + ɛ Pt ) 1 Ψ t+k Yt+k t Y t+k t = 0. Y t+k t k=0 If we multiply both sides of the equality by P t / 1 ɛ), we have: k=0 )) θ k E t Q t,t+k Y t+k t Pt ɛ ɛ 1) ψ t+k t = 0, where ψ t+k t := Ψ t+k Yt+k t ) / Yt+k t is the nominal marginal cost of the firm in period t + k. 36