Liquidity, Productivity and Efficiency

Transcription

1 Liquidity, Productivity and Efficiency Ehsan Ebrahimy University of Chicago August 9, 2011 Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 1

2 Introduction Efficiency of private liquidity provision: Liquidity Pledgeability Existing literature abstracts from: - Investment heterogeneity - Endogeneity of the assets liquidity choice Investment heterogeneity Endogenous liquidity choice Farhi and Tirole (2011) Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 2

3 Introduction Key features: Limited pledgeability: - Limited commitment - Limited enforcement - Moral hazard Higher returns Lower pledgeability Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 3

4 Outline Model Setup Competitive Equilibrium Steady States Efficiency and Welfare Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 4

5 Model Setup: Preferences and Technology OLG economy of entrepreneurs Young, middle aged and old Unit measure of each for t 0 Young receives e > 0 perishable consumption goods Middle aged invest in: - Type i, return and pledgeability (R i,θ i R i ) - Type l, return and pledgeability (R l,θ l R l ) Consume only when old Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 5

6 Model Setup: Problem of the Middle Aged Middle aged at t > 0 solves: max i t,x it,x lt 0 R i x it +R l x lt (1+r t )i t s.t. x it +x lt (1+r t 1 )e+i t (1+r t )i t θ i R i x it +θ l R l x lt x it and x lt investment in type i and l i t, funds borrowed from young at t (1+r t 1 )e is return to past investment Assumption. R i > R l > 1 and θ i R i < θ l R l < 1. Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 6

7 Competitive Equilibrium Lemma. If 1+r t < R i, borrowing constraint of the middle aged entrepreneurs binds at t. If borrowing constraint binds at t > 0, middle aged solves: max i t Λ(θ,R;r t )i t +Φ(θ,R;r t 1 )e s.t. ( ) ( ) θi R i (1+r t 1 ) θl R l (1+r t 1 ) e i t e. 1+r t θ i R i 1+r t θ l R l Original problem: max i t,x it,x lt 0 R i x it +R l x lt (1+r t)i t s.t. x it +x lt (1 +r t 1 )e+i t (1+r t)i t θ i R i x it +θ l R l x lt Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 7

8 Competitive Equilibrium Λ(θ,R;r t ) is net return of increase in i t while FC binds: ( ) ( ) (θl θ i )R i R l (1 θi )R i (1 θ l )R l Λ(θ,R;r t ) (1+r t ) θ l R l θ i R i θ l R l θ i R i To increase i t by ɛ > 0 Borrowing constraint binds: (1+r t)i t = θ ir ix it +θ l R l x lt - x it by δ > 0 - x lt by ɛ+δ Investment size, return net gain Λ(θ,R;r t)ɛ Λ(θ,R;r t) can be positive or negative Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 8

9 Demand and Supply of Fund i s t = e rt i d t = ψ(w t 1;r t ) i t Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 9

10 Demand and Supply of Fund i s t = e rt i d t = ψ(w t 1;r t ) i t Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 10

11 Demand and Supply of Fund i s t = e rt i d t = ψ(w t 1;r t ) i t Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 11

12 Competitive Equilibrium Definition. Let 1+r Λ (θ,r) be the gross interest rate that makes Λ(θ,R;r t ) zero: 1+r Λ (θ,r) (θ l θ i )R i R l (1 θ i )R i (1 θ l )R l Market clearing i t = e Market clearing + optimal policy of middle aged: 1+r t = θ l R l (2+r t 1 ) If θ l R l (2+r t 1 ) < 1+r Λ (θ,r) 1+r t = θ i R i (2+r t 1 ) If θ i R i (2+r t 1 ) > 1+r Λ (θ,r) 1+r t = 1+r Λ (θ,r) Otherwise Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 12

13 Competitive Equilibrium Definition. A competitive equilibrium is a sequence of {i t,x it,x lt,r t } t=0 and initial wealth w 1 = (1+r 1 )e that maximize consumption for the old entrepreneur, satisfy the interest rate conditions above and 1+r t < R i for all t > 0. Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 13

14 Steady State F set of (θ,r) that: R i > R l > 1 and θ ir i < θ l R l < 1 Financing constraint binds at SS F = F l F m F i such that at SS: Only liquid in F l Mix of both in F m Only illiquid in F i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 14

15 Steady State Lemma. For any (θ,r) F, there is a unique and stable steady state equilibrium where: 1+r ss l = θ lr l 1 θ l R l if (θ,r) F l 1+r ss m = 1+r Λ (θ,r) if (θ,r) F m 1+r ss i = θ ir i 1 θ i R i if (θ,r) F i Proposition. Given any (θ,r) F, and an initial condition 1+r 1 < R i, there exists a unique competitive equilibrium that converges to the steady state corresponding to (θ, R) which is given in the above lemma. Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 15

16 Steady State R i = 4, R l = θl F l F m F i θ i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 16

17 Steady State: Properties of Equilibria 1+r ss : non-monotone in θ i / monotone in θ l Investment demand Substitution effect Decline in the real interest rates after 90s θ i x ss l x ss i +xss l : monotone in θ i / non-monotone in θ l Partial Eqm: θ l xss l x ss i +xss l θ i xss l x ss i +xss l General Eqm: θ l r ss θ i r ss Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 17

18 Steady State: Interest Rate Contour R i = 4,R l = θl θ i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 18

19 Steady State: High and Low θ i x ss l x ss i +x ss l R i = 4, R l = x ss l x ss i +x ss l θ l Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 19

20 Steady State: Properties of Equilibria R i = 4, R l = arg max θ l x ss l x ss i +x ss l 0.20 θl F l F m F i θ i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 20

21 Efficiency and Welfare Definition. A competitive equilibrium is constrained Pareto efficient if a social planner cannot make at least someone strictly better off while keeping all others at least as well off by a reallocation that respects the pledgeability constraint. Constrained Pareto efficient {c t,x it,x lt } t=0 solves: max {c t,x it,x lt } t=0 λ t c t t=0 λ t > 0 are Pareto weights. c t +x it +x lt R i x it 1 +R l x lt 1 +e x it +x lt θ i R i x it 1 +θ l R l x lt 1 +e Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 21

22 Efficiency and Welfare: A Reallocation Let (θ,r) F l F m x ss l > 0 Planner reduces (1+r ss )e, by δ > 0: FC slack x i, by ɛ > 0/ x l, by ɛ+δ Maximum ɛ when FC binds: δ =(θ l R l θ ir i)ɛ+θ l R l δ ɛ = 1 θ lr l θ l R l θ ir i δ Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 22

23 Efficiency and Welfare: A Reallocation Change in utility: ( V ss 1 θl R l = R i (1+ 1 θ ) lr l )R l +1 δ θ l R l θ i R i θ l R l θ i R i Proposition. For any (θ,r) F l F m, one has V ss 0, and consequently the steady state is constrained Pareto inefficient, if and only if r Λ (θ,r) 0. Outside steady state: Lemma. Given (θ,r) F l F m, if r Λ (θ,r) 0 any competitive equilibrium is constrained Pareto inefficient. Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 23

24 Inefficient Equilibria R i = 4, R l = r Λ = 0 θl Ineff. liquid Ineff. illiquid θ i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 24

25 Efficiency and Welfare: A Reallocation Proposition. All competitive equilibria in F l are constrained Pareto inefficient if and only if: R i R l R l 1 > 1 Pareto frontier of F? Proposition. Given (θ,r) F, if r Λ (θ,r) > 0 any allocation satisfying resource and pledgeability constraints with equality for t 0, including all competitive equilibria, is constrained Pareto efficient. Moreover, any allocation satisfying resource and pledgeability constraints with equality for t 0 such that x lt = 0,t T for some T 0, including all equilibria in F i, is constrained Pareto efficient. Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 25

26 Inefficient Equilibria R i = 4, R l = r Λ = θl Ineff. liquid Ineff. illiquid θ i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 26

27 Efficiency and Welfare: A Reallocation Reinterpret OLG 3 infinitely lived agents: (y, m, o), (y, m,... (i) m, o), (y, m, o),... (ii) o), (y, m, o), (y,... (iii) Discount factor β (0,1) Proposition. A competitive equilibrium for (θ,r) F m F i such that r Λ (θ,r) > 0 reinterpreted as above is constrained Pareto inefficient, if β(θ,r) β < 1 for some threshold β(θ,r) (0,1). Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 27

28 Inefficient Equilibria R i = 4, R l = r Λ = 0 θl Ineff. liquid Ineff. illiquid θ i Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 28

29 Efficiency and Welfare: Regulated Economy Pareto reallocation regulating α t = x lt x it+x lt : Inefficiently liquid α t Inefficiently illiquid α t Proposition. A Pareto improving reallocation for a small enough δ > 0 can be implemented by a regulation that sets α t = α t for t T where T 0 for the OLG as well as the reinterpreted economy. For inefficiently liquid equilibria, this regulation can result in an allocation on the Pareto frontier. Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 29

30 Efficiency and Welfare: Regulated Economy (θ,r) F l and w t 1 = (1+r t 1 )e low: max i t,x it,x lt 0 R i x it +R l x lt (1+r t )i t s.t. x it +x lt w t 1 +i t (1+r t )i t θ i R i x it +θ l R l x lt Unregulated Eqm: i t = e 1+r t = θ l R l (2+r t 1 ) V = (1 θ l )R l (w t 1 +e) Regulated Eqm (α t = 0): ĩ t = e 1+ r t = θ i R i (2+r t 1 ) Ṽ = (1 θ i )R i (w t 1 +e) Ṽ > V (1 θ i)r i > (1 θ l )R l Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 30

31 Efficiency and Welfare: Regulated Economy r Λ (θ,r) 0 (1 θi)ri 1 θ ir i (1 θ l)r l 1 θ l R l Note that V ss (θ,r) = (1 θ i)r i 1 θ i R i, investing only in type i (1 θ l)r l 1 θ l R l, investing only in type l Low w t 1 = (1+r t 1 )e low θ Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 31

32 Efficiency and Welfare: Regulated Economy Inefficiently illiquid α r CE t > 0, Pareto reallocation r PO t More traditional Inefficiently liquid α r CE t 0, Pareto reallocation r PO t Overinvestment r CE t Sign of r CE t < 0, r CE t > r CE t > 0 < r CE t < 0 < r PO t < 0 can be a misleading indicator Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 32

33 Conclusion Endogenous liquidity choice Investment heterogeneity Positive implications: Share of liquid type non-monotone in θ l Interest rate non-monotone in θ i Normative implications: Endogenous liquidity pecuniary externality inefficiency Inefficiently liquid / Inefficiently illiquid (more traditional) Pareto reallocation regulating share of liquid investment Sign of interest rate misleading indicator of inefficiency Effect of bubbles or public liquidity? Ehsan Ebrahimy Liquidity, Productivity and Efficiency -p. 33