Equilibrium with Complete Markets

Transcription

1 Equilibrium with Complete Markets Jesús Fernández-Villaverde University of Pennsylvania February 12, 2016 Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

2 Arrow-Debreu versus Sequential Markets n previous lecture, we discussed the preferences of agents in a situation with uncertainty. Now, we will discuss how markets operate in a simple endowment economy. Two approaches: Arrow-Debreu set-up and sequential markets. Under some technical conditions both approaches are equivalent. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

3 Environment We have agents, i = 1,...,. Endowment: (e 1,..., e ) = {e 1 t (s t ),..., e t (s t )} t=0,s t S t Tradition in macro of looking at endowment economies. Why? Consumption, risk-sharing, asset pricing. Advantages and shortcomings. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

4 Allocation Definition An allocation is a sequence of consumption in each period and event for each individual: (c 1,..., c ) = {c 1 t (s t ),..., c t (s t )} t=0,s t S t Definition Feasible allocation: an allocation such that: c i t (s t ) 0 for all t, all s t S t, for i = 1, 2 ct i (s t ) et(s i t ) for all t, all s t S t Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

5 Pareto Effi ciency Definition An allocation {(ct 1 (s t ),..., ct (s t ))} t=0,s t S is Pareto effi cient if it is t feasible and if there is no other feasible allocation {( c t 1 (s t ),..., c t (s t ))} t=0,s t S t such that u( c i ) u(c i ) for all i u( c i ) > u(c i ) for at least one i Ex ante versus ex post effi ciency. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

6 Arrow-Debreu Market Structure Trade takes place at period 0, before any uncertainty has been realized (in particular, before s 0 has been realized). As for allocation and endowment, Arrow-Debreu prices have to be indexed by event histories in addition to time. Let p t (s t ) denote the price of one unit of consumption, quoted at period 0, delivered at period t if (and only if) event history s t has been realized. We need to normalize one price to 1 and use it as numeraire. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

7 Arrow-Debreu Equilibrium Definition An Arrow-Debreu equilibrium are prices {ˆp t (s t )} t=0,s t S and allocations t ({ĉt i (s t )} t=0,s t S ) t,.., such that: 1 Given {ˆp t (s t )} t=0,s t S, for i = 1,..,, {ĉ i t t (s t )} t=0,s t S solves: t s.t. max {c i t (s t )} t=0,s t S t t=0 t=0 s t S t p t (s t )c i t (s t ) s t S t β t π(s t )u(c i t (s t )) t=0 c i t (s t ) 0 for all t s t S t p t (s t )e i t(s t ) 2 Markets clear: ĉ i t (s t ) = e i t(s t ) for all t, all s t S t Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

8 Welfare Theorems Theorem Let ({ĉ i t (s t )} t=0,s t S t ),.., be a competitive equilibrium allocation. Then, ({ĉ i t (s t )} t=0,s t S t ),.., is Pareto effi cient. Theorem Let ({ĉt i (s t )} t=0,s t S ) t,.., be Pareto effi cient. Then, there is a an A-D equilibrium with price {ˆp t (s t )} t=0,s t S that decentralizes the allocation t ({ĉt i (s t )} t=0,s t S ) t,..,. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

9 Sequential Markets Market Structure Now we will let trade take place sequentially in spot markets in each period, event-history pair. One period contingent OU s: financial contracts bought in period t that pay out one unit of the consumption good in t + 1 only for a particular realization of s t+1 = j tomorrow. Q t (s t, s t+1 ): price at period t of a contract that pays out one unit of consumption in period t + 1 if and only if tomorrow s event is s t+1 (zero-coupon bonds). a i t+1 (st, s t+1 ): quantities of these Arrow securities bought (or sold) at period t by agent i. These contracts are often called Arrow securities, contingent claims or one-period insurance contracts. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

10 Period-by-period Budget Constraint The period t, event history s t budget constraint of agent i is given by c i t (s t ) + s t+1 s t Q t (s t, s t+1 )a i t+1(s t, s t+1 ) e i t(s t ) + a i t(s t ) Note: we only have prices and quantities. Many economists use expectations in the budget constraint. We will later see why. However, this is bad practice. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

11 Natural Debt Limit We need to rule out Ponzi schemes. Tail of endowment distribution: A i t(s t ) = τ=t p τ (sτ ) p s τ s t t (s t ) ei τ(s τ ) A i t(s t ) is known as the natural debt limit. Then: a i t+1(s t+1 ) A i t+1(s t+1 ) Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

12 Sequential Markets Equilibrium Definition A SME is prices for Arrow securities { Q t (s t, s t+1 )} t=0,s { t S t,s t+1 S and ( allocations ĉt i (s t ), { ât+1 i (st, s t+1 ) } ) } s t+1 such that: S,.., t=0,s t S t 1 For i = 1,..,, given { Q t (s t, s t+1 )} t=0,s t S t,s t+1 S, for all i, {ĉt i (s t ), { ât+1 i (st, s t+1 ) } s t+1 S } t=0,s t S solves: t max {c i t (s t ),{a i t+1 (s t,s t+1 )} st+1 S } t=0,s t S t s.t. c i t (s t ) + t=0 s t S t β t π(s t )u(c i t (s t )) s t+1 s t Q t (s t, s t+1 )a i t+1(s t, s t+1 ) e i t(s t ) + a i t(s t ) c i t (s t ) 0 for all t, s t S t a i t+1(s t, s t+1 ) A i t+1(s t+1 ) for all t, s t S t Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

13 Definition (cont.) 2. For all t 0 ĉt i (s t ) = et(s i t ) for all t, s t S t ât+1(s i t, s t+1 ) = 0 for all t, s t S t and all s t+1 S Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

14 Equivalence of Arrow-Debreu and Sequential Markets Equilibria A full set of one-period Arrow securities is suffi cient to make markets sequentially complete. Any (nonnegative) consumption allocation is attainable with an appropriate sequence of Arrow security holdings {a t+1 (s t, s t+1 )} satisfying all sequential markets budget constraints. Later, when we talk about asset pricing, we will discuss how to use Q t (s t, s t+1 = j) to price any other security. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

15 Pareto Problem We will extensively exploit the two welfare theorems. Negishi s (1960) method to compute competitive equilibria: 1 We fix some Pareto weights. 2 We solve the Pareto problem associated to those weights. 3 We decentralize the resulting allocation using the second welfare theorem. All competitive equilibria correspond to some Pareto weights. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

16 Social Planner s Problem We solve the social planners problem: max ({c i t (s t )} t=0,s t S t ),.., s.t. where α i are the Pareto weights. α i β t π(s t )u(ct i (s t )) t=0 s t S t ct i (s t ) = et(s i t ) for all t, s t S t ct i (s t ) 0 for all t, s t S t Definition An allocation ({c i t (s t )} t=0,s t S t ),.., is Pareto effi cient if and only if it solves the social planners problem for some (α i ),..., [0, 1]. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

17 Perfect nsurance We write the lagrangian for the problem: { α i β t } max ( π(s t )u(ct i (s t )) ({ct i (s t )} t=0,s t S t ),.., +λ t (s t ) [ e i t (s t ) ct i (s t ) ]) t=0 s t S t where λ t (s t ) is the state-dependent lagrangian multiplier. We forget about the non-negativity constraints and take FOCs: α i β t π(s t )u (c i t (s t )) = λ t ( s t ) for all i, t, s t S t Then, by dividing the condition for two different agents: Definition u (ct i (s t )) u (ct j (s t )) = α j α i An allocation ({c i t (s t )} t=0,s t S t ),.., has perfect consumption insurance if the ratio of marginal utilities between two agents is constant across time and states. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

18 rrelevance of History From previous equation, and making j = 1: ( ) ct i (s t ) = u 1 α1 u (ct 1 (s t )) α i Summing over individuals and using aggregate resource constraint: et(s i t ( ) ) = u 1 α1 u (ct 1 (s t )) α i which is one equation on one unknown, c 1 t (s t ). Then, the pareto-effi cient allocation ({c i t (s t )} t=0,s t S t ),.., only depends on aggregate endowment and not on s t. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

19 Theory Confronts Data Perfect insurance implies proportional changes in marginal utilities as a response to aggregate shocks. Do we see perfect risk-sharing in the data? Surprasingly more diffi cult to answer than you would think. Let us suppose we have CRRA utility function. Then, perfect insurance implies: ct i (s t ( ) 1 ) ct j (s t ) = αi γ α j Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

20 ndividual Consumption Now, c i t (s t ) = c j t (s t ) 1 γ αj 1 γ αi ct i (s t ) = and we sum over i e i t(s t ) = cj t (s t ) α 1 γ j 1 γ αi Then: 1 ct j (s t ) = α γ j α 1 γ i et(s i t ) = θ j y t (s t ) i.e., each agent consumes a constant fraction of the aggregate endowment. Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

21 ndividual Level Regressions Take logs: log c j t (s t ) = log θ j + log y t (s t ) f we take first differences, log c j t (s t ) = log y t (s t ) Equation we can estimate: log c j t (s t ) = α 1 log y t (s t ) + α 2 log e i t(s t ) + ε i t Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

22 Estimating the Equation How do we estimate? log c j t (s t ) = α 1 log y t (s t ) + α 2 log e i t(s t ) + ε i t CEX data. We get α 2 is different from zero (despite measurement error). Excess sensitivity of consumption by another name! Possible explanation? Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

23 Permanent ncome Hypothesis Build the Lagrangian of the problem of the household i: t=0 s t S t β t π(s t )u(c i t (s t )) µ i t=0 Note: we have only one multiplier µ i. p t (s t ) ( et(s i t ) ct i (s t ) ) s t S t Then, first order conditions are β t π(s t )u (c i t (s t )) = µ i p t (s t )for all t, s t S t Substituting into the budget constraint: t=0 1 s t S t µ i β t π(s t )u (c i t (s t )) ( e i t(s t ) c i t (s t ) ) = 0 Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24

24 No Aggregate Shocks Assume that e i t(s t ) is constant over time. From perfect insurance, we know then that c i t (s t ) is also constant. Let s call it ĉ i. Then (cancelling constants) t=0 s t S t β t π(s t ) ( e i t(s t ) ĉ i ) = 0 ĉ i = (1 β) t=0 s t S t β t π(s t )e i t(s t ) Later, we will see that with no aggregate shocks, β 1 = 1 + r. Then, ĉ i = r 1 + r t=0 s t S t ( ) 1 t π(s t )e i 1 + r t(s t ) Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, / 24