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