Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents

1. Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents Kenneth Judd, Felix Kubler, Karl Schmedders Presented by Jack Favilukis

2. In a nutshell • Standard portfolio theory predicts that agents will need to trade assets in order to rebalance portfolios in response to shocks, new information, etc. (in partial equilibrium) • This paper: in general equilibrium with dynamically complete markets there will be no trade of infinite horizon assets • Consumption and prices move together in general equilibrium to negate any need for trading

3. Intuition • If everything is state dependent, in state y agent i will always want to consume ci(y) and asset j will always pay dj(y) • If markets are dynamically complete there are as many independent assets as states c(y1)=d1(y1)*θ1+d2(y1)*θ2+d3(y1)*θ3 c(y2)=d1(y2)*θ1+d2(y2)*θ2+d3(y2)*θ3 c(y3)=d1(y3)*θ1+d2(y3)*θ2+d3(y3)*θ3

4. The Economy • H agents, S states, JL long lived assets paying d(y) in state y; JS=S-JL short lived assets paying d(y) just next period (payoffs are linearly independent) • Agent i has endowment ei(y), agent’s wealth is wi(y)=ei(y)+Σθij(y)qj(y) where qj is price of asset j • Short lived assets in zero net supply (think of bonds) • Utility is time separable

5. Arrow-Debreu Equilibrium A set of prices p(t,y) and consumption policies ci(t,y) s.t. • Σci(t,y)=Σei(y)+Σdj(y) for all y,t (markets clear) • ci(t,y) = argmax E[ΣU(ci(t,y))] s.t. Σp(t,y)ci(t,y)≤Σp(t,y)wi(t,y)

6. Financial Market Equilibrium A process for portfolio holdings θji(t,y) and asset prices qj(t,y) • Σθji(t,y)=Σθji(t-1,y) for all j (markets clear) • [ci(t,y) , θji(t,y)] = argmax E[ΣU(ci(t,y))] s.t. ci(t,y)=ei(y)+Σθji(t-1,y)(qj(t-1,y)+dj(y))-Σθji(t,y)qj(t,y) • There is a one to one correspondence between the two types of equilibria if markets are complete and there are no bubbles

7. Time Homogeneity • Every efficient equilibrium exhibits time-homogeneous Markovian consumption processes for all agents • Step 1: Solve for c(y) and use above as justification • Step 2: Use c(y) from Step 1 and Euler equations to show that asset prices must also be time homogenous, solve for q(y) • Step 3: Show that asset holdings must also be Markovian (concavity of utility function implies they are a function of wealth, which is Markovian). Solve for asset holdings.

8. Solving the model • We want to solve for following quantities: • ci(y), (S*H) (each agent’s state contingent consumption) • qj(y) (S*S) (each asset’s state contingent price) • θij(y) (S*S*H) (each agent’s state contingent asset holdings) • Define p(y)=U1’(c1(y)) (from FOC of A-D equilibrium) and wi(y)=ei(y)+Σθij(y)dj(y)

9. Step 1 • We can compute present value of consumption V and the present value of endowments and portfolio holdings W • V(y)=p(y)c(y)+βE[V(y+)]=pc+βΠV+ • W(y)=p(y)w(y)+βE[W(y+)]=pw+βΠW+ • V(y)=[I- βΠ]-1(p(y)*c(y))=[I- βΠ]-1(p(y)*w(y))=W(y) • [I- βΠ]-1(p(y)*(ci(y)-wi(y)))=0 (S*(H-1) equations) • Market Clearing: Σci(y)=Σwi(y) (S equations)

10. Step 2 • Euler equations: qj(y)p(y)=βE[p(y+)(qj(y+)+dj(y+))]  qj(y)*p(y)=[I-βΠ]-1βΠ(p(y)*dj(y)) (S*S equations) Step 3 • Budget constraint: Σθij(z)(qj(y)+dj(y))=ci(y)-ei(y)+Σθij(y)qj(y) (S*S*H equations) • Now solve S*H nonlinear and S*S+S*S*H linear equations, easy if S,H are small

11. Zero Trading Volume • Σθij(y)(qj(z)+dj(z))=Σθij(x)(qj(z)+dj(z))  Σ(θij(y)-θij(x))(qj(z)+dj(z))=0 for all y,x,z • Kubler and Schmedders (2003) showed that qj(z)+dj(z) has rank S, therefore it must be that θ(x)=θ(y) for all x and y (if D has full rank and Dx=0 than x=0) • There is trade in short lived securities: i.e. always want $100 of 1 year bonds, $200 of 2 year bonds, next year would need to sell $100 of 1 year bonds