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)?1 d2(y1)?2d3(y1)?3
• c(y2)d1(y2)?1 d2(y2)?2d3(y2)?3
• c(y3)d1(y3)?1 d2(y3)?2d3(y3)?3

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The Economy

• H agents, S states, JL long lived assets paying
  d(y) in state y JSS-JL short lived assets
  paying d(y) just next period (payoffs are
  linearly independent)
• Agent i has endowment ei(y), agents wealth is
  wi(y)ei(y)S?ij(y)qj(y) where qj is price of
  asset j
• Short lived assets in zero net supply (think of
  bonds)
• Utility is time separable

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Arrow-Debreu Equilibrium

• A set of prices p(t,y) and consumption policies
  ci(t,y) s.t.
• Sci(t,y)Sei(y)Sdj(y) for all y,t (markets
  clear)
• ci(t,y) argmax ESU(ci(t,y)) s.t.
  Sp(t,y)ci(t,y)Sp(t,y)wi(t,y)

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Financial Market Equilibrium

• A process for portfolio holdings ?ji(t,y) and
  asset prices qj(t,y)
• S?ji(t,y) S?ji(t-1,y) for all j (markets clear)
• ci(t,y) , ?ji(t,y) argmax ESU(ci(t,y)) s.t.
• ci(t,y)ei(y)S?ji(t-1,y)(qj(t-1,y)dj(y))-S?ji(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

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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.

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Solving the model

• We want to solve for following quantities
• ci(y), (SH) (each agents state contingent
  consumption)
• qj(y) (SS) (each assets state contingent
  price)
• ?ij(y) (SSH) (each agents state contingent
  asset holdings)
• Define p(y)U1(c1(y)) (from FOC of A-D
  equilibrium) and wi(y)ei(y)S?ij(y)dj(y)

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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)ßEV(y)pcß?V
• W(y)p(y)w(y)ßEW(y)pwß?W
• V(y)I- ß?-1(p(y)c(y))I- ß?-1(p(y)w(y))W(y
  )
• I- ß?-1(p(y)(ci(y)-wi(y)))0 (S(H-1)
  equations)
• Market Clearing Sci(y)Swi(y) (S equations)

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• Step 2
• Euler equations qj(y)p(y)ßEp(y)(qj(y)dj(y
  )) ?
• qj(y)p(y)I-ß?-1ß?(p(y)dj(y))
  (SS equations)
• Step 3
• Budget constraint
• S?ij(z)(qj(y)dj(y))ci(y)-ei(y)S?ij(y)qj(y)
  (SSH equations)
• Now solve SH nonlinear and SSSSH linear
  equations, easy if S,H are small

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Zero Trading Volume

• S?ij(y)(qj(z)dj(z))S?ij(x)(qj(z)dj(z)) ?
• S(?ij(y)-?ij(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
  ?(x)?(y) for all x and y (if D has full rank and
  Dx0 than x0)
• 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