Growth, Technological Asset Pricing and Consumption Risk
Stavros Panageas and Jianfeng Yu Prof. Sargent's Reading Group
Technological Growth, Asset Pricing and Consumption Risk p.1/16
Introduction
Contemporaneous consumption growth shows little correlation with excess returns Empirical failure of CAPM Consumption over longer horizons (3 years) shows high correlation with excess return (Parker and Julliard, JPE 2005) long horizon versions of CAPM perform better This paper tries to rationalize the new well documented fact in a GE framework with a non standard production sector
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The model: Timing
There is a sequence of different technological epochs N , N (-..., -1, 0, 1... + ), that arrive at the Poisson rate >0 At the beginning of epoch N a large scale technological change occurs and each firm draws a tree specific productivity (ij,N ) U (0, 1) i.i.d. across epochs and (i) = 0 (1 - i) , i [0, 1] heterogeneity across firms Within each epoch, a firm decides if to adopt the new technology, paying a fixed cost, or not. Once the epoch changes the firm loses the option to plant a tree of any previous epochs At each time t a "small", disembodied aggregate shock t hits the entire economy, dt = dt + dBt t
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The model: Technology
There is a continuum of firms indexed by j [0, 1] Earnings stream of a tree planted in N : YN,i,t = (A)N (iN )t , Total earnings:
N
(A)N : vintage effect, A > 1
(1)
Yj,t =
n=-
An (ij,n )1(Xj,n =1) t = Xj,t t ~
(2)
Total earnings (output) in the economy at t:
1 1
Yt =
0
Yj,t dj =
0
Xj,t djt = Xt t
(3)
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The model: Technology
f o A Share price: PN,j,t = Pj,t + PN,j,t + PN,t A Pj,t o PN,j,t f PN,t
= Xj,t Et
t
Hs s ds Ht AN (ij,N )
= sup Et 1( <N +1 )
H Hs s ds - q Ht Ht
= Et
n=N +1
Hn o PN,j,n Ht
The problem of the firm is to choose the optimal stopping time within each epoch
Technological Growth, Asset Pricing and Consumption Risk p.5/16
Optimal Stopping Time
j,N
=
N tN +1
inf
t t: MN (iN,j )
>0
(4)
No firm will find it optimal to install a tree at the beginning of the new epoch The firms with the more productive trees will start planting first Strong correlation effect due to the continuity of and
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Preferences
There is a continuum of consumers/gardeners aggregated in a single representative agent that owns all the firms Ut = U (Ct , MtC ) MtC = maxst {Cs }
UC > 0, UCC < 0, UM C < 0 (envy), UCM C > 0 (catching up with the Joneses) Consumer's problem
Cs ,dls C e-(s-t) U (Cs , Ms )ds-
max Et
t
e-(s-t) UC (s)(s) dls
t
fixed disutility per tree planted
s.t.
Et
t
Hs Cs ds Ht
1
Nt
PN,j,t dj + Et
0 n=- t
Hs qs dls Ht
Technological Growth, Asset Pricing and Risk Consumption p.7/16
Equilibrium
A competitive equilibrium for this economy is a set of stochastic processes {Ct , Xt , Kn,t , dlt , qt } s.t.
Ct and dlt solve the optimization problem of the household
Firms determine the optimal time to plant a tree by solving the problem described before The goods market clears : Ct = Yt for all t 0 The gardening market clears
1
dlt = dKn,t
Kn,t =
0
1(Xj,n =1) dj ~
The markets for all assets clear
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Parametrization
U (Ct , MtC ) = (MtC )
1- Ct
(
1-
=
Ct - Ct C) Mt
1-
>1
SDF unaffected by investment decisions: C Mt = maxst s UC = ( M t )- = ( Mtt )- C
t
qt = q AN M,N M,N =historical maximum at the start of new epoch
the compensation for gardening must share the same trend as aggregate output the amount of gardening services must be stationary gardening fee must be constant within each epoch the cost of planting a tree must be such that at the beginning of each epoch no firms has incentive to adopt the new technology
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Consumption-Output Dynamics
Yt = Ct = Xt t =
Xt N A t AN
log Ct = xt + N log A + log t
predictable part of consumption growth
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Consumption-Output Dynamics
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Calibration and Unconditional Moments
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Countercyclical Variation in Expected Returns
Growth options value over aggregate stock market value
o+f wt = 1 KNt 1 A 0 PN,j,t dj f o PN,j,t dj + PN,t 1 KNt f o PN,j,t dj + PN,t
+
Lemma 1:
o+f wt
=
e-xt
t f( M , M t Mt N
)+e-xt
wx < 0
Expected exces...