Slides III - Complete Markets Julio Garín University of Georgia Macroeconomic Theory II (Ph.D.) Spring 2017 Macroeconomic Theory II Slides III - Complete Markets Spring 2017 1 / 33
Outline 1. Risk, Uncertainty, and Neumann-Morganstern preferences 2. Two-Period Arrow-Debreu Model 2.1 Actuarially Fair Prices 2.2 Risk Sharing 2.3 Equilibrium Price Determination 3. Infinite Horizon 3.1 Time-0 Trading 3.1.1 Risk Sharing 3.1.2 No aggregate uncertainty 3.2 Asset Pricing 3.3 Sequential Trading 3.4 Equivalence of Two Equilibria 3.5 History, Recursivity Wealth Macroeconomic Theory II Slides III - Complete Markets Spring 2017 2 / 33
Uncertainty and Expected Utility Theory We usually don t have all information available about the future. We typically use Expected Utility Theory. Von Neumann-Morgenstern preferences. Let s focus first in a case with two states of the world. It s all about trade-offs. Intertemporal elasticity of substitution (IES). Coefficient of relative risk aversion (RRA). Macroeconomic Theory II Slides III - Complete Markets Spring 2017 3 / 33
Arrow-Debreu Model of State-Contingent Securities The baseline model of consumption under uncertainty. What will it teach us? Agents are expected-utility maximizers. We will start with a simple two-period model. Why? Endowment economy. Income at t + 1, y t+1, is uncertain. State-contingent (Arrow-Debreu securities) are available. Pay one unit of output if state s occurs, otherwise pay zero. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 4 / 33
Two-Period Problem With Only Two States Definitions: Number of state-s securities that the agent holds. Price paid for those securities. Probability that state s occurs. There are complete markets : There are A-D securities for every state of the world. This doesn t imply perfect consumption smoothing. We will also allow a non-contingent bond, b t+1. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 5 / 33
We have to solve: max c t, c t+1 (1), c t+1 (2), B t+1 (1), B t+1 (2), b t+1 u(c t ) + β [π 1 u(c t+1 (1)) + π 2 u(c t+1 (2))] subject to: c t = y t b t+1 P t (1)B t+1 (1) P t (2)B t+1 (2) c t+1 (1) = y t+1 (1) + B t+1 (1) + (1 + r)b t+1 c t+1 (2) = y t+1 (2) + B t+1 (2) + (1 + r)b t+1 Macroeconomic Theory II Slides III - Complete Markets Spring 2017 6 / 33
Analyzing The First-Order Conditions 1 1 + r = P t(1) + P t (2) (1) What if this condition is not satisfied? This would represent an arbitrage opportunity. Market must drive prices such that the condition holds. Non-contingent bonds are redundant. They can be replicated by holding A-D securities. Euler equation under uncertainty. u (c t ) = (1 + r)β E t u (c t+1 ) (2) Macroeconomic Theory II Slides III - Complete Markets Spring 2017 7 / 33
We can also show: u [c t+1 (1)] u [c t+1 (2)] = P t(1) π 2 (3) P t (2) π 1 If prices are actuarially fair, two marginal utilities are equal. Consumption would be perfectly smoothed across states. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 8 / 33
Economy With Heterogeneous Agents Two agents: Igor, i, and James, j. Receive different income streams. Idiosyncratic risk. For some state s: βπ s u [ c k t+i (s)] λ k t = P t (s) for k = i, j (4) Since they face the same price... Macroeconomic Theory II Slides III - Complete Markets Spring 2017 9 / 33
Economy With Heterogeneous Agents u [ c j t+i (s) ] u [ c i t+i (s)] = λj t λ i t (5) Lambdas are not state-dependent. MUs are perfectly correlated across states. State-contingent bonds allows for smoothing. Insurance against idiosyncratic risk. MUs can only change due to aggregate risk. How this related to the representative agent assumption? If markets are complete, idiosyncratic risk will be insured. Only aggregate risk remains. Individuals will move up and down together. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 10 / 33
Equilibrium Price Determination Assume that there are many agents, but they are all identical. Then prices must be such that in equilibrium agents consume their endowment. In that case: u [y t+1 (1)] u [y t+1 (2)] = P t(1) π 2 P t (2) π 1 If y t+1 (1) = y t+1 (2) and utility isn t linear, prices are not actuarially fair. State-contingent bonds that deliver output when output is scarce are more expensive (relative to actuarially fair prices). Idea at the foundation of all modern asset-pricing theory. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 11 / 33
Full-Blown Infinite Horizon: Preliminaries History s t = [s 0, s 1,..., s t ] keeps track of the sequence of states that have been realized. Households have von Neumann-Morganstern preferences. There are two alternative trading arrangements: 1. Arrow-Debreu structure. 2. Sequential trading (Radner). Under some conditions, both are equivalent. Consumption allocation at time t will depend only on aggregate endowment. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 12 / 33
Time-0 Trading Macroeconomic Theory II Slides III - Complete Markets Spring 2017 13 / 33
Time-0 Trading Agents buy and sell, at the beginning of time, a full set of state-contingent securities for each possible history and for each point in time. Probabilities: π t (s t ) denotes unconditional probability of history s t being realized. Note that π t varies through time. π t (s t s τ ) for t > τ, denotes conditional probability. I households named i = 1,..., I. Each receives its own endowment, yt i (s t ). qt 0 (s t ) is the time-0 price of a security that pays one unit of the endowment good at time t if history s t is realized. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 14 / 33
Time-0 Trading: Household Problem In equilibrium, prices are such that markets clear. The problem for household i is then subject to: max {{ct i (s t )} s t } t=0 t=0 s t β t u[c i t(s t )]π t (s t ) t=0 qt 0 (s t )ct(s i t ) s t qt 0 (s t )yt i (s t ) t=0 s t Note that: Single lifetime budget constraint. We don t keep track of how many state-contingent claims the household purchases. Consumption and income don t need to be equal to each other in every state. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 15 / 33
Equilibrium Definition A price system is a sequence of functions { qt 0 (s t ) } t=0. An allocation is a list of sequences of functions c i = { ct(s i t ) } t=0, one for each i. Definition (Competitive Equilibrium) A competitive equilibrium is a feasible allocation and price system such that, given the price system, the allocation solves each household s problem. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 16 / 33
F.O.C. Let µ i be the Lagrange multiplier on household s i s lifetime budget constraint. β t u [ c i t(s t ) ] π t (s t ) = µ i q 0 t (s t ) We can show that, consistent with what we derived before, different household s MU are perfectly correlated. After inverting the MU and plugging into the aggregate feasibility constraint, u {u [ ] } 1 ct j (s t µi ) = i µ j yt i (s t ) i ct j (s t ) only depends on the current aggregate endowment. All idiosyncratic risk is traded away. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 17 / 33
Example 1: Risk Sharing Assume u(c) = c1 γ 1 1 γ. Consumption of household i will be given by ( ) 1 ct(s i t ) = ct j (s t µi γ ) µ j And using the feasibility constraint we can derive ct j (s t ) = χj 1 yt i (s t ) i where χ j = µ 1/γ j i µ 1/γ i gives the fraction of aggregate endowment that agent j consumes in each period. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 18 / 33
Example 2: No Aggregate Uncertainty Two households. Receive yt 1 (s t ) = s t and yt 2 (s t ) = 1 s t, where s t is a r.v. taking on values in [0, 1] Since aggregate endowment is constant c i t(s t ) = c i. From the F.O.C. and the lifetime budget constraint we can show that: Is this familiar? c i = (1 β) What if β 1 = 1 + r? t=0 s t β t π t (s t )y i t (s t ) Individual consumptions will depend on the distribution of s t. Note that c 1 + c 2 = 1. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 19 / 33
Asset Pricing Macroeconomic Theory II Slides III - Complete Markets Spring 2017 20 / 33
Asset Pricing Rearranging the F.O.C. Does it make sense? q 0 t (s t ) = β t u [ c i t(s t ) ] π t (s t ) µ i As before, negative relationship between price and equilibrium consumption. People are willing to pay more for deliveries when aggregate endowment is scarce. Expression in terms of household i is without loss of generality. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 21 / 33
Applications We can use these prices to price all sort of redundant assets. Just apply the price to the dividend/payment that the asset pays. Let {d t (s t )} t=0 be the stream of claims on time t, history st consumption. No arbitrage implies: Examples: 1. Riskless consol. 2. Riskless strips. p 0 0 (s 0) = qt 0 (s t )d t (s t ) t=0 s t Macroeconomic Theory II Slides III - Complete Markets Spring 2017 22 / 33
Tail Assets After stripping the first τ 1 periods, what is the time 0 value of the remaining dividend stream? pτ(s 0 τ ) = qt 0 (s t )d t (s t ) t τ s t s τ We can convert this to time τ, history s τ units: p τ τ(s τ ) p0 τ(s τ ) q 0 τ(s τ ) = q0 t (st ) t τ s t s τ q 0 τ(s τ ) d t(s t ) Defining qt τ q t (s t )/qτ(s 0 τ ) we obtain pτ(s τ τ ) = qt τ (s t )d t (s t ) t τ s t s τ Note what q τ t (s t ) really is. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 23 / 33
One Period Returns and Stochastic Discount Factor What is the time τ price at history s τ of a claim to a random payoff, ω(s τ+1 )? p τ τ(s τ ) = s τ+1 q τ τ+1(s τ+1 )ω(s τ+1 ) Using our definition of q τ t (s t ) we have that for any individual i [ ] 1 = E τ β u (c τ+1 ) u (c τ ) R τ+1 = E τ [m τ+1 R τ+1 ] where R τ+1 is the one-period gross return on the asset and m τ+1 is a stochastic discount factor. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 24 / 33
Sequential Trading Macroeconomic Theory II Slides III - Complete Markets Spring 2017 25 / 33
Sequential Trading Arrow securities. Tildes denote equilibrium variables from this environment. New definitions: Claims to time-t endowment households bring to t. Price of an Arrow security. The budget constraint is given by: c i t(s t ) + s t+1 ã i t+1(s t+1 ) Q i t(s t+1 s t ) ỹ i t (s t ) + ã i t(s t ) Macroeconomic Theory II Slides III - Complete Markets Spring 2017 26 / 33
Sequential Trading: Borrowing Limit Before there was a payment clearance system. We still need to make sure that everyone lives within their means. We impose the no-ponzi scheme: ã i t+1 (st+1 ) qτ t+1 (s τ )yτ(s i τ ) A i t(s t ) τ=t+1 s τ s t Note that prices are not in terms of Qs. This is a natural borrowing limit. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 27 / 33
Sequential Trading: Household Problem The problem for household i is then subject to: max ct i (s t ),{ãt+1 i (s t+1 s t )} t=0 s t β t u[c i t(s t )]π t (s t ) c i t(s t ) + s t+1 ã i t+1(s t+1 ) Q i t(s t+1 s t ) ỹ i t (s t ) + ã i t(s t ) (6) ã i t+1(s t+1 ) A i t(s t ) (7) Let η i t(s t ) and ν i t(s t ) be the multipliers on (6) and (7). Macroeconomic Theory II Slides III - Complete Markets Spring 2017 28 / 33
F.O.C. Combing the F.O.C.s and using the Inada condition on utility, we obtain [ Q t(s i t+1 s t ) = β u c t+1 i (st+1 ) ] u [ c t(s i t ) ] π t+1 (s t+1 ) π t (s t ) From time-0 endowment: q 0 t (s t ) = β u [ c i t(s t ) ] π(s t ) µ i and q t+1 t (s t ) q0 t+1 (st+1 ) q 0 t (s t ) What if Q i t(s t+1 s t ) = q t+1 t (s t+1 )? Macroeconomic Theory II Slides III - Complete Markets Spring 2017 29 / 33
Equivalence Define the household i s current and future net claims under time-0 equilibrium: Υ i t(s t ) = τ=t qτ(s t τ ) [ cτ(s i τ ) yτ(s i τ ) ] s τ s t We can show that if i, a0 i (s 0) = 0, then the lifetime budget constraints are the same. Also, ãt(s i t ) = Υ i t(s t ). Therefore, equilibrium allocation are the same. Welfare theorems apply, so Pareto optimum is achieved. Intuition? Macroeconomic Theory II Slides III - Complete Markets Spring 2017 30 / 33
Markov and Recursivity So far, probabilities could depend on entire past history. Pricing kernels and wealth distributions both depend on history s t. Both are time-varying functions. Now let s assume that states follow a Markov process. If the same is true for endowment, we will have yt i (s t ) = yt i (s t ). The problem takes a recursive structure. We will have a new equilibrium concept. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 31 / 33
History Independence and Equilibrium Outcomes We have u {u [ ] } 1 ct j (s t µi ) i µ j = yt i (s t ) i Given the Markov property, we can express ct(s i t ) = c i (s t ) c() is a time-invariant mapping that depends only on current state. Therefore, we can rewrite the Euler: [ Q t(s i t+1 s t ) = β u c i (s t+1 ) ] u [ c i π(s t+1 s t ) (s t )] Macroeconomic Theory II Slides III - Complete Markets Spring 2017 32 / 33
This history independence extends to the household s wealth level Υ i t(s t ) = Ῡ i t(s t ) Recall what Υ i was. Why doesn t it depend on past endowment realizations? Household already insured himself against previous endowment realizations. Macroeconomic Theory II Slides III - Complete Markets Spring 2017 33 / 33