Macroeconomic Theory II Homework 4 - olution Professor Gianluca Violante, TA: Diego Daruich New York University pring 2014 Problem 1 Time is discrete and indexed by t {0, 1,..., }. The economy is populated by a continuum of measure one of infinitely-lived households indexed by i on [0, 1]. Preferences for individual i are given by U ( c i 0, c i 1,... ) = E 0 β t u ( c i t), where β (0, 1) is the discount factor, c t 0 denotes consumption at time t, u > 0 and u < 0. Each household i in period t receives a stochastic idiosyncratic endowment of the consumption good ε i t {ε h, ε l } which follows a Markov chain where π ε hh = Pr { ε i t = ε h ε i t 1 = ε h}, π ε ll = Pr { ε i t = ε l ε i t 1 = ε l}, π ε hl = 1 π ε hh, and πε lh = 1 πε ll. Households can trade a risk-free asset a in zero-net supply with return R t, and they can accumulate debt up to the exogenous borrowing limit b. t=0 Part a Define a stationary recursive competitive equilibrium (RCE) for this economy, and discuss its existence and uniqueness. First, let us define some objects. Let = A E denote the state space for an individual agent, where A = [ b, ā] (for some ā) 1 and E = {ε h, ε l }. Also, let B() be the Borel σ-algebra induced by. It is helpful to define the transition function Q : B() [0, 1], as the function that gives us, for any arbitrary Borel sets A E, the probability that an agent with current states (a, ε) will be in those Borel sets next period. We can define Q as Q[A E, (a, ε)] = 1[a (a, ε) A] ε E π(ε, ε) (1) The solution is based on an earlier version by Dan Greenwald and Miguel de Faria e Castro, NYU. Comments and errors should be addressed to ddaruich@nyu.edu. 1 How do we know such ā exists? ee proof of existence and uniqueness of stationary distribution given prices. Otherwise, we could also write the set A as being non-compact for now. 1
where 1 is the indicator function, a (a, ε) is the policy function for savings of an agent with states (aε) and π(ε, ε) = Pr(ε t+1 = ε ε t = ε). Also note that there is no aggregate uncertainty in this economy: in a stationary equilibrium, all aggregate variables will be constant, including the value of aggregate production. Let (π, 1 π ) denote the stationary distribution of ε. Then, aggregate output is constant at Y = π ε h + (1 π )ε l and it is easy to check that π 1 π ll = 2 π ll π hh ince there is no aggregate uncertainty, I shall assume that R t is time-invariant when defining a stationary equilibrium. That is, R t = R. Before defining the equilibrium, it is also useful to write down the problem of an agent with states (a, ε). We can write the recursive formulation of that agent s problem as V (a, ε) = max c,a { u(c) + β ε E π(ε, ε)v (a, ε ) } (2) subject to c + a = Ra + ε c 0 a b A stationary recursive competitive equilibrium consists of a value function V : R, policy functions (c, a ) : R + A, a price R and a stationary measure λ : B() [0, 1] such that 1. Given the price R, V is the solution to the Bellman Equation in (2), and (c, a )(a, ε) are the associated policy functions. 2. Markets clear: goods : assets : c(a, ε)dλ(a, ε) = A E a (a, ε)dλ(a, ε) = 0 A E A E εdλ(a, ε) = Y 3. For all (A E) B(), the invariant measure λ satisfies λ(a E) = Q[A E, (a, ε)]dλ(a, ε) where Q is defined in (1). A E Existence and uniqueness of stationary distribution given prices 2
The major difference between this exchange economy and the production economy discussed in class is that assets need to be in zero net supply. This is equivalent to defining the demand of assets as 0 r. The supply of capital behaves in a similar way to the discussion in class. To prove the existence of a stationary equilibrium we need to verify that there exists a unique invariant distribution λ. To satisfy theorem 12.13 in LP for the existence of the equilibrium we need the existence and uniqueness of the invariant distribution, which (using theorem 12.12 in LP) follows from the following assumptions 2 : For compactness of the state space we need to assume DARA preference. Then compactness is guaranteed for every r that satisfies β (1 + r) < 1. 3 The Feller property is immediate and does not require additional assumptions. For the monotonicity of the transition function we need to assume that π hh > π ll. Then monotonicity follows from the fact that policy function a is increasing in (a, ε). MMC is satisfied if the Markov chain is ergodic. Existence and uniqueness of the recursive competitive equilibrium Given aggregate demand and supply are continuous functions of interest rate, we can show existence by checking that demand is greater than supply as interest rate R is close to 0 and supply is greater than demand as R is close to 1 β. Unfortunately as usual, we cannot show uniqueness as there could be more than one interest rates that correspond to zero net asset. There are several reasons why the choice of aggregate asset holdings might be nonmonotonic in the interest rate. First, income effect and substitutional effect of interest rate work in opposite directions. Which one dominates the other is determined by the specific set of preference parameters. econd, even we consider only the income effect, the effect interest rate on aggregate asset allocation is still ambiguous, because agents are allowed to hold negative asset balances (borrow). If a < 0, then higher interest rate will make the household more constrained and so a might decrease with interest rate. Part b uppose now that b = 0, and that u (c) = c1 γ 1 γ. What is so special about this case? Determine the equilibrium risk-free return as a function of the model s parameters and prove that the return is lower than 1/β. Explain how the risk-free rate changes with the ratio ε h /ε l, with the persistence parameter π hh, and with risk aversion γ. Now we have that b = 0 (so that agents cannot borrow) and CRRA utility. Note that if agents cannot borrow, they cannot lend either - since the asset market must clear at zero net supply, whoever wants to lend needs someone who is willing to borrow in order to undertake the asset trade - but agents cannot borrow in the first place. Therefore, this economy will be autarkic and agents will simply consume their own current income, c t = ε t, t 0. 2 ee notes at the end of the document. 3 Using Chamberlain and Wilson (2000) result for βrgeq1, since we have enough stochasticity. And that R = 0 would lead to too much borrowing in an equilibrium (since no one has to pay, they all borrow up to the limit). Then, we use standard dynamic programming results to argue that savings policy function a is increasing (using the concavity of the utility function) and continuos (theorem of the maximum). 3
What would agents want to do? Assume for a moment that (π hh, π ll ) are both strictly less than one (so that no state is absorbing). Then agents will have a consumption smoothing motive, as high agents will want to save for when they get a low income shock, and low agents will want to borrow. The latter are technologically constrained by b. High endowment agents, however, do not face any sort of technological constraint. Therefore, aggregate conditions will have to be such that high endowment agents find it optimal not to save. ince high endowment agents determine the interest rate, what this tells us is that R must be such that, in equilibrium, the Euler Equation for agents with high endowments reads: Applying our specific functional form yields Rearranging: u (ε h ) βr[π hh u (ε h ) + (1 π hh )u (ε l )] (ε h ) γ βr[π hh (ε h ) γ + (1 π hh )(ε l ) γ ] R 1 β π hh ε γ h ε γ h + (1 π hh)ε γ l Any interest rate that is lower than the RH is consistent with an equilibrium in which high endowment agents find it optimal not to save and simply consume their entire endowment - this is fairly intuitive: for high agents not to save, the interest rate must be extremely low. ince γ > 0, x γ is a decreasing function of x. Therefore, ε γ h < ε γ l. Thus we have that π hh ε γ h ε γ h + (1 π hh)ε γ l Once again, this holds assuming that π hh < 1. But, then, we immediately obtain that < 1 as we wanted to show. The comparative statics are as follows: R < 1 β 1. Endowment Ratio - Rewrite the condition on the interest rate to read R 1 (ε h /ε l ) γ β π hh (ε h /ε l ) γ + 1 π hh where I simply multiply and divide the RH by ε γ l. Clearly, when this ratio increases, the RH term decreases. Therefore, the maximum possible value for the interest rate that is consistent with equilibrium decreases, R. The intuition is that an increase in this ratio raises the variance of income and thus increases the incentives to smooth consumption (since the agent is risk-averse), which can only be done through saving. ince the incentives to save increase, the interest rate must decrease further so as to ensure the autarkic market clearing for assets (i.e. to ensure that rich agents do not save). Note that if the consumer were risk-neutral, γ = 0, this effect would not exist. 2. Persistence - ince ε γ h < ε γ l, as π hh we observe the RH term increasing, so that the maximum interest rate compatible with equilibrium increases R. An increase in the persistence of high income makes income less volatile for rich agents (i.e. the likelihood of remaining rich increases), thereby decreasing the incentives to save. This allows R to increase slightly while still ensuring autarky in the asset market. 4
3. Risk Aversion - This effect is less obvious. The easiest way to think about this is to take the inverse of the RH π hh + (1 π hh )(ε h /ε l ) γ When γ, since ε l < ε h the above expression increases. Thus our expression must decrease, meaning that R.This is in fact intuitive: if the agent becomes more risk-averse, he or she will have more incentives to save, forcing the interest rate down to ensure equilibrium. Problem 2 Consider a stationary economy populated by a continuum of measure one of infinitely lived, ex-ante equal agents with preferences over sequences of consumption and leisure given by E 0 t=0 [ ] β t u (c it ) + e ψ it v (1 h it ). Agents face individual shocks to their preference for leisure ψ it, which follow the stochastic process ψ it = ρψ i,t 1 + η it, with η it N (0, σ η ). Agents can save but cannot borrow. Production takes place through the aggregate technology C t + K t+1 (1 δ) K t = AKt α Ht 1 α where C t, K t and H t are, respectively, aggregate consumption, aggregate capital, and aggregate hours at time t. Labor and asset markets are competitive and clear, every period, with prices w t and r t, respectively. The government taxes capital income at a fixed flat rate τ. Tax revenues are returned to agents as tax-exempt lump-sum transfers b. Households can evade taxes by deciding every period t the fraction of capital income φ it to declare in their tax return, i.e., the fraction of capital income on which they pay taxes. Let x it be the total undeclared taxes at time t. The government, knowing that agents may have evaded taxes at time t 1, at time t can monitor and perfectly verify the past period individual tax returns. Let π be the probability that, at time t, the time t 1 tax returned of a household is subject to monitoring. The household finds out whether her t 1 period tax return is monitored at the beginning of period t, i.e., before consumption decisions are taken. In the event the household is caught, at time t the tax agency gives her a fine equal to z (x i,t 1 ), where x i,t 1 is the tax amount due from the past period, with z (0) = 0 and z (x i,t 1 ) > 1. Part a Write down the problem of the household in recursive form, making explicit the individual and the aggregate state variables. There are several ways to set-up this problem. The individual state variables at time t will be (a it, ψ it, x it 1, m it ), where a it is current asset holdings, ψ it is the preference shock (you could had written ψ it 1, η it instead), x it 1 is the amount of taxes that went undeclared in the previous period and m it {0, 1} denotes whether the agent is being monitored by the government this period or not. 5
The aggregate state for this economy is the distribution of individual states λ(a, ψ, x, m), from which other aggregate states such as K or b can be derived. I always implicitly assume that the agent takes λ as a given when solving the recursive problem. I am going to write the problem in a way that it is not needed to include the additional decision variable φ, and the agent simply chooses over x (the amount of taxes that go undeclared from this period to the next). We can then set up the recursive problem as subject to { } V (a, x, ψ, n) = max u(c) + e ψ v(1 h) + βe ψ [πv (a, x, ψ, 1) + (1 π)v (a, x, ψ, 0)] c,a,h,x c + a = [1 + r(1 τ)]a + x + wh + b nz(x) x τra x, c, a 0 h [0, 1] A few comments: first, the expectation in the value function is taken with respect to the two independent random variables (ψ, n ). econd, the choice of φ is subsumed in the first and second constraints. Part b Write down the individual first-order necessary condition that characterizes the optimal tax evasion choice. Let (λ 1, λ 2, µ 1, µ 2, µ 3 ) denote the Lagrange multipliers for each constraint (and let us just assume the agent chooses interior leisure, for simplicity). Then, we can write the FOC with respect to x as u (c) + βe ψ [πv 2 (a, x, ψ, 1) + (1 π)v 2 (a, x, ψ, 0)] λ 2 + µ 1 = 0 where I am also implicitly assuming that c > 0. Now, we need to apply the envelope condition to the two value functions: when the agent is monitored and when he or she is not. For the monitoring case, with n = 1, the agent will pay an extra tax z(x) for each unit of x that was not reported. Then V 2 (a, x, ψ, 1) = u (c)z (x) While for the no-monitoring case, n = 0, the previous choice of x does not affect the consumer in any way V 2 (a, x, ψ, 0) = 0 Iterating forward and plugging back in the FOC with respect to x, we see that u (c) = βπe ψ [u (c )z (x )] + λ 2 µ 1 Clearly, we will have that x geq0, hence µ 1 = 0 (since no agent would ever overreport the amount of taxes it needs to pay). We cannot, however, say anything about λ 2 - some agents may prefer to report losses, which is not allowed -. Thus the FOC for tax evasion is However, if the solution were interior, λ 2 = 0. u (c) = βπe ψ [u (c )z (x )] + λ 2. 6
Part c Define a stationary recursive competitive equilibrium for this economy. Let the state space be = A X Ψ N = R 2 + R {0, 1}. Let B() denote the product σ-algebra over. A tationary Recursive Competitive Equilibrium for this economy is a value function V : R, policy functions for the household (c, a, x, h) : R 3 + [0, 1], policies for the firm (H, K), prices (r, w), government policies (τ, b), and a stationary measure λ : [0, 1] such that 1. Given prices and government policies, V solves the Bellman Equation in Part 1, and (c, a, x, h) are the associated policy functions. 2. Given prices (r, w), the firm chooses (H, K) optimally F K (K, H) = r + δ F H (K, H) = w 3. Markets clear labor : assets : goods : h(a, x, ψ, n)dλ = H a (a, x, ψ, n)dλ = K c(a, x, ψ, n)dλ + K = F (K, H) + (1 δ)k 4. The government budget constraint is balanced τradλ x (a, x, ψ, n)dλ + π z(x)dλ = b 5. For all Borel sets (A, X, P, N ) B(), the invariant measure satisfies λ(a, X, P, N ) = Q[(A, X, P, N ), (a, x, ψ, n)]dλ(a, x, ψ, n) where Q : B() [0, 1] is a transition function defined as Q[(A, X, P, N ), (a, x, ψ, n)] =1[a (a, x, ψ, n) A x (a, x, ψ, n) X ] [π1[{1} N ] + (1 π)1[{0} N ]] ψ df (ψ, ψ) where F (ψ, ψ) is the transition function implied by the AR structure ψ = ρψ + η, with η N (0, σ η ). P Problem 3 Time is discrete and indexed by t {0, 1,..., }. The economy is populated by a continuum of measure one of infinitely-lived households indexed by i on [0, 1]. Preferences for individual i are given by U ( c i 0, c i 1,...; h i 0, h i 1,... ) = E 0 β t u ( c i t, 1 h i t), 7 t=0
where β (0, 1) is the time discount factor, c 0 denotes consumption and h 0 hours worked. Household i at time t supplies the optimal amount of hours worked h i t to a competitive labor market, taking the hourly wage ω t as given. Each household owns a single private firm, i.e. firm i is identified with household i. Each firm produces the final good y (which can be used for conumption or investment) through y i t = F ( z i t, k i t, n i t), where F is a production technology displaying constant-returns in capital and labor. The variable z i t represents a firm-specific productivity shock which follows the Markov process π (z, z) = Pr { z i t+1 = z z i t = z } with z min = 0. Households can freely hire labor on the market for their firm, but here is no market to trade claims on physical capital, thus households can only invest the capital they own in their firm. Moreover, k i t 0. Capital depreciates at rate δ. Let π i t denote the profits accruing to individual i from operating her firm at time t. Households can trade a risk-free bond b in zero-net supply with return R t and they can accumulate debt up to the natural borrowing constraint. Assume that the economy is in a stationary equilibrium. Part a What is the natural borrowing constraint faced by each household? This question highlights some of the difficulties in establishing what a natural borrowing constraint should look like. Basically, the natural borrowing limit will depend on whether we assume that the agent is able to commit the repayment of its own debts or if the agent is allowed to default. It is first helpful to look at the budget constraint for an individual agent i. We can define it sequentially as c i t + k i t+1 + b i t+1 = F (z i t, k i t, n i t) ωn i t + ωh i t + Rb i t + (1 δ)k i t on the LH we say that the agent can spend money in consumption, capital investment or bond investment. The first two terms on the RH are the net profit of the private firm (production minus wage bill), the third term is wage income from the labor market and the last two terms are bond return and undepreciated capital (that can be sold in the goods market). Note that I dropped the time indices on prices (R, ω), since we are focusing on a stationary equilibrium in which there is no aggregate uncertainty. The household derives income from two sources: capital and labor. ince we are studying a natural borrowing limit, we have to look at what can the agent afford to pay in a worst case scenario, when z min = 0. Imposing c i t + kt+1 i 0, we find the following: Iterate forward to obtain that b i t 1 R j=0 b i t bi t+1 [F (zi t, k i t, n i t) ωn i t] ωh i t (1 δ)k i t R F (z i t+j, ki t+j, ni t+j ) ωni t+j + ωhi t+j + (1 δ)ki t+j R j b i T + lim T R T 8
Now, the natural debt limit should be such that the agent should be able to repay its debts in every state of the world. In particular, the agent should be able to repay its debts in a worst-case scenario where z t = z min = 0 forever. Clearly, in such an event, the agent would optimally set n i t = 0 (the agent would not hire any worker, since the marginal gain from doing so is zero) and kt+j i = 0, j 1 (the logic is the same, the agent would not purchase any capital). 4 This, and imposing a transversality condition on bond holdings, allows us to slightly simplify the expression to yield Iterating one period ahead leaves us with b i t 1 R (1 δ)k t 1 R b i t+1 1 R (1 δ)ki t+1 1 R j=0 j=0 ωh i t+j R j ωh i t+1+j R j That is, the agent can borrow against the capital stock he or she will hold in the next period (adjusted for depreciation), and future human capital income. There is a fundamental problem with this, however: both k t+1 and {h t+1+j } j=0 are endogenous variables, chosen by the agent! Therefore, it is as if the agent can influence its own natural borrowing limit - so this seems to be a very complicated problem! There are two approaches to this. The first is just to assume that the agent will do whatever it takes to repay its debts, even if this includes setting h t = 1 forever and not investing in capital today (regardless of the states). This leaves us with b t+1 1 ω 1 R R j = 1 R 1 ω j=0 The other approach is to assume that the agent will choose optimally, and will do so in order to repay its debts. In this case, the natural debt limit will be a function of the current states, Φ(z t, k t, b t ). ince we are dealing with worst case scenarios, we just have to look at the policies h(z t, k t, b t ) when z t = 0, k t = 0 and b t = Φ. Thus we get that b t+1 (z t, k t, b t ) Φ(z t, k t, b t ) = 1 R (1 δ)k (z t, k t, b t ) 1 ωh(0, 0, Φ) R R j Note that finding this alternative debt limit requires computing the optimal policies (as function of the states), and then solving a fixed point problem that determines Φ: Φ(z t, k t, b t ) = 1 R (1 1 δ)k (z t, k t, b t ) + h[0, 0, Φ(z t, k t, b t )] R 1 If we have the policy functions, we can compute the function Φ, hence the natural borrowing limit will fluctuate over time. For simplicity, I assume that Φ = 1 R 1ω in the remainder of the exercise. j=0 Part b List explicitly the individual state variables and write down the problem of the household in recursive formulation. The discussion in the previous exercise already spoiled some of the work we develop here: the individual state variables will be the current productivity level z t, the current stock of capital k t and bond holdings b t. The state space is = Z K B, where Z is the state space for the Markov process π, K = R + and B = [ Φ, + ), 4 We are assuming here that F (0, n, k) = 0 for all (n, k). Otherwise, the statement of z min = 0 makes no difference. 9
where Φ is the natural borrowing limit computed before. We can express the agent s problem recursively as { V (z, k, b) = max c,k,b,h,n u(c, 1 h) + β z Z π(z, z)v (z, k, b ) } (3) subject to c + k + b = F (z, k, n) ωn + ωh + Rb + (1 δ)k c, n, k 0 b Φ h [0, 1] Part c Define a Recursive Competitive Equilibrium (RCE) for this economy. A RCE for this economy is a value function V : R, policies (c, k, n, b, h) R 3 + [ Φ, + ) [0, 1], prices (R, ω) and a stationary distribution λ : B() [0, 1] such that: 1. Given prices, V solves the Bellman Equation (3) and (c, k, n, b, h) are the associated policy functions. 2. Markets clear: (labor) : (bonds) : (goods) : = h(z, k, b)dλ(z, k, b) = n(z, k, b)dλ(z, k, b) b (z, k, b)dλ(z, k, b) = 0 c(z, k, b)dλ(z, k, b) + k (z, k, b)dλ(z, k, b) F [z, k, n(z, k, b)]dλ(z, k, b) + (1 δ) kdλ(z, k, b) 3. For all (Z K B) B(), the invariant distribution satisfies λ(z K B) = Q[(Z K B), (z, k, b)]dλ(z, k, b) where the transition function Q : B() [0, 1] is defined as Q[(Z K B), (z, k, b)] = 1[k (z, k, b) K b (z, k, b) B] π(z, z) z Z 10
Part d Discuss the existence and uniqueness of the RCE for this economy. The existence and uniqueness of the stationary invariant distribution given prices (w, r) can be shown in a similar way to problem 1. o is the existence of a stationary equilibrium. As for uniqueness, the argument in problem 1 still holds here since natural borrowing constraint is negative. What is worse, the actual effect of wage increase on labor supply is also ambiguous because of the income affect (less leisure) and the substitution effect (more leisure). On top of this, is the effect coming from the interconnected effects of the wage and the interest rate. Therefore uniqueness cannot be guaranteed. Part e Imagine now that a new financial market opens allowing agents to invest their own k in other households firms. What is a simple way to represent the new economy with this additional market? In particular, list what the individual state variables become in this case. A simple way to model this would be to add an additional state variable κ, corresponding to capital invested in other firms. Correspondingly, there would be a new control κ. The budget constraint would now look like c + k + b + κ = F (z, k, n) ωn + ωh + Rb + (1 δ)k + (1 + r δ)κ where r is the equilibrium return on capital in the economy (constant in a stationary equilibrium). ince capital would now become freely mobile across firms, it does not matter in which firm capital is invested, since returns will have to be equalized so as to prevent arbitrage opportunities. In equilibrium, all capital and labor will concentrate in the firms with the highest productivity level z max due to constant returns to scale. ince capital becomes concentrated in those firms, the distribution of capital per se becomes irrelevant. To understand why, note that the single source of heterogeneity in this economy were the productivity shocks. However, agents no longer face these: if zt i < z max, the agent simply sets k = 0 and invests everything in κ, in the highest productivity firm. Thus all agents face the same return on capital in the economy. This means that heterogeneity is completely eliminated and the problem now admits aggregation as a representative agent problem, with resource constraint given by C + K = F (z max, K, N) + (1 δ)k Notes on FP, Monotonicity and MMC Conditions Feller Property Alternative 1 For this property we need operator T to preserve continuity and boudedness. Boundedness is straightforward from compactness of A E. For continuity argument we need to provide some details of the proof. Lets take an arbitrary sequence {(a n, ɛ n )} n=1 such that lim n (a n, ɛ n ) = (a, ɛ). We know that Q ((a n, ɛ n ), s) weakly Q ((a, ɛ), s) as n 11
iff for any f C 5 By Aleksandrov Theorem iff lim n f (s) Q ((a n, ɛ n ), ds) = f (s) Q ((a, ɛ), ds) Q ((a n, ɛ n ), s) weakly Q ((a, ɛ), s) as n Q ((a n, ɛ n ), ) Q ((a, ɛ), ) for all such that Q ((a, ɛ), ) = 0 Note that since a is continuos then lim n a (a n, ɛ n ) = a (a, ɛ). Also note that Q ((a n, ɛ n ), (A E)) Q ((a, ɛ), (A E)) is violated only if a (a, ɛ) A. Hence if a (a, ɛ) A then Q ((a n, ɛ n ), ) Q ((a, ɛ), ). One can show that Q ((a, ɛ), (A E)) = 0 iff (a) or (b) hold: a a (a, ɛ) A b ɛ E π (ɛ, ɛ) = 0 In case (a) we just have shown that Q ((a n, ɛ n ), (A E)) Q ((a, ɛ), (A E)). In case of (b) Q ((a n, ɛ n ), ) Q ((a, ɛ), ) obviously holds since both Q ((a, ɛ), ) and Q ((a n, ɛ n ), ) are zero starting from some n. Combining this altogether we get that T preserves continuity. Alternative 2 Using theorem 9.14 in LP. Our space is measurable and the transition function satisfies that 1. A is a convex Borel set in R. 2. E is countable and we are using the Borel sigma-algebra. 3. avings policy function is continuous (using theorem of the maximum). Hence, all assumptions are satisfied, so the transition function has the Feller property. Monotonicity Monotonicity has two components: monotonicity in a and in ɛ. T preserves monotonicity iff for any f that is increasing in both arguments T f is also increasing. First lets prove monotonicity in a. We need to show that for all a, â : â a f (s) Q ((â, ɛ), ds) f (s) Q ((a, ɛ), ds) for all increasing f We know that this is equivalent to Q ((â, ɛ), ds) F D Q ((a, ɛ), ds). We also know that there is a criterion for FD relation which is g F D h iff G (x) H (x) for all x 5 Here convergence is in terms of r.v. that are associated with probability measures Q ((a n, ɛ n ), ) and Q ((a, ɛ), ) 12
where G ( ) is cdf for g and H ( ) is cdf for h. In our case cdf is Q ((â, ɛ), (α, e)) where (α, e) = {(a, ɛ) (a, ɛ) (A E) and (a, ɛ) (α, e)} Note that since a (a, ɛ) is monotonic then Q ((â, ɛ), (α, e)) looks like a step and threshold for Q ((â, ɛ), (α, e)) is higher then for Q ((a, ɛ), (α, e)) hence Q ((â, ɛ), (α, e)) Q ((a, ɛ), (α, e)).from this we can conclude that Q ((â, ɛ), (α, e)) F D Q ((a, ɛ), (α, e)) and hence f (s) Q ((â, ɛ), ds) f (s) Q ((a, ɛ), ds) for all increasing f Now lets prove monotonicity in ɛ.we need to show that for all ɛ, ˆɛ : ˆɛ ɛ f (s) Q ((a, ˆɛ), ds) f (s) Q ((a, ɛ), ds) for all increasing f We will show it for the case of E = {ɛ H, ɛ L }. The argument that we need to use is also based on FD relation. However we need to impose more conditions. ince Q ((a, ɛ), A (α, e) E (α, e)) = I { a (a, ɛ) A (α, e) } π ( ɛ, ɛ ) ɛ E(α,e) then I { a (a, ˆɛ) A (α, e) } I { a (a, ˆɛ) A (α, e) } for all (α, e) (4) and π ( ɛ, ˆɛ ) π ( ɛ, ɛ ) for all (α, e) (5) ɛ E(α,e) ɛ E(α,e) implies that Q ((a, ˆɛ), A (α, e) E (α, e)) Q ((a, ɛ), A (α, e) E (α, e)) for all (α, e) It is easy to check that increasing a is sufficient for 4 to hold and π (ɛ H, ɛ H ) π (ɛ L, ɛ L ) implies 5. QED MMC Note that necessary condition for MMC to hold is ergodicity of Markov chain. 13