Modern aggregation theory and index number theory were introduced into monetary aggregation by Barnett (1980). The widely used Divisia monetary aggregates were based upon that paper. A key result upon which the rest of the theory depended was Barnett's derivation of the user-cost price of monetary assets. To make that critical part of Barnett''s results available prior to publication of that paper in the Journal of Econometrics, Barnett repeated that proof two years earlier in Economics Letters. Both papers have become seminal to the subsequent literature on monetary asset quantity and user cost aggregation. The extension of that literature to risk with intertemporally non-separable preferences now has become available in a working paper by Barnett and Wu (2004), and that paper will appear in volume 1, number 1 of the new journal, Annals of Finance. We are making available the key results from that paper below, without the proofs, which will be available in the longer paper.

Shu Wu

William Barnett

English

Research Papers in Economics

Intertemporally non−separable monetary−asset risk
adjustment and aggregation 
William Barnett Shu Wu
University of Kansas University of Kansas
Abstract
Modern aggregation theory and index number theory were introduced into monetary
aggregation by Barnett (1980). The widely used Divisia monetary aggregates were based
upon that paper. A key result upon which the rest of the theory depended was Barnett’s
derivation of the user−cost price of monetary assets. To make that critical part of Barnett’s
results available prior to publication of that paper in the Journal of Econometrics, Barnett
repeated that proof two years earlier in Economics Letters. Both papers have become seminal
to the subsequent literature on monetary asset quantity and user cost aggregation. The
extension of that literature to risk with intertemporally non−separable preferences now has
become available in a working paper by Barnett and Wu (2004), and that paper will appear in
volume 1, number 1 of the new journal, Annals of Finance. We are making available the key
results from that paper below, without the proofs, which will be available in the longer paper.
We thank participants at the 11th Global Finance Annual Conference and Yuqing Huang for helpful comments.
Citation: Barnett, William and Shu Wu, (2004) "Intertemporally non−separable monetary−asset risk adjustment and
aggregation." Economics Bulletin, Vol. 5, No. 13 pp. 1−9
Submitted: June 27, 2004.  Accepted: July 6, 2004.
URL: http://www.economicsbulletin.com/2004/volume5/EB−04E00005A.pdf 
Introduction 
 
  Modern aggregation theory and index number theory were introduced into monetary 
aggregation by Barnett (1980).  The widely used Divisia monetary aggregates were based upon 
that paper.  A key result upon which the rest of the theory depended was Barnett’s derivation of 
the user-cost price of monetary assets.  To make that critical part of Barnett’s (1980) results 
available prior to publication of the full paper in the Journal of Econometrics, Barnett (1978) 
repeated that part of the theory two years earlier in Economics Letters.  Both papers have become 
seminal to the subsequent literature on monetary asset quantity and user cost aggregation.   
The extension of that literature to risk with intertemporally non-separable preferences 
now has become available in a working paper by Barnett and Wu (2004), and that paper will 
appear in volume 1, number 1 of the new journal, Annals of Finance.  That complete paper, in 
working paper form, is online in the Economics Working Paper Archive, at 
http://econwpa.wustl.edu/eprints/mac/papers/0406/0406009.abs.  In this paper, we are making 
available the key results from the working paper, without the proofs, which will be available in 
the longer paper. We extend the monetary-asset user-cost risk adjustment of Barnett, Liu, and 
Jensen (1997) and their risk-adjusted Divisia monetary aggregates to the case of multiple non-
monetary assets and intertemporal non-separability. Our model can generate potentially larger 
and more accurate CCAPM user-cost risk adjustments than those found in Barnett, Liu, and 
Jensen (1997).  
Barnett (1978, 1980, 1997) produced the microeconomic theory of monetary aggregation 
under perfect certainty, derived the formula for the user cost of monetary assets, and originated 
the Divisia monetary aggregates to track the theory’s quantity and price aggregator functions 
nonparametrically.  The monetary aggregation theory was extended to risk by Barnett (1995) and 
Barnett, Hinich, and Yue (2000).  In producing the Divisia index approximations to the theory’s 
aggregator functions under risk, Barnett, Liu, and Jensen (1997) and Barnett and Liu (2000) 
showed that a risk adjustment term should be added to the certainty-equivalent user cost in a 
consumption-based capital asset pricing model (CCAPM). The risk adjustment depends upon the 
covariance between the rates of return on monetary assets and the growth rate of consumption. 
Using the components of the usual Federal Reserve System monetary aggregates, Barnett, Liu, 
and Jensen (1997) showed, however, that the CCAPM risk adjustment is slight and the gain from 
replacing the unadjusted Divisia index with the extended index is usually small.  An overview of 
the relevant literature is provided in Barnett and Serletis (2000).  As in the equity premium 
puzzle literature, the small risk adjustment is caused by the low contemporaneous covariance, 
under intertemporal separability, between the rate of return on the asset and consumption growth. 
We also extend the model in Barnett, Liu, and Jensen (1997) to include multiple risky 
non-monetary assets, which are the assets that provide no liquidity services, other than their rates 
of return, and we show that any non-monetary asset can be used as the benchmark asset, when its 
rate of return is correctly adjusted. 
 
2. Consumer’s optimization problem 
 
  We assume that the representative consumer has an intertemporally non-separable 
general utility function,  , defined over current and past consumption and a 
vector of L current-period monetary assets, 
( , , , , ) ttt - 1 t - n Uc c c " m
1, 2, , (, , , ) tt tL mm m t ′ = " m . The consumer’s holdings of non-monetary assets are  1, 2, , ( , ,..., ) tt t K t kk k ′ = k .  To ensure the existence of a monetary aggregate, 
we further assume that there exists a linearly homogenous aggregator function,  , such that 
U can be written in the form 
() M ⋅
  1 ( , ,, ,) ( ( ) , ,, , ttt t n t tt t n Uc c c V M c c c −− −− 1 ) = " mm "
) s n
, t
1 t Y + +
. (2.1) 
Given initial wealth,  , the consumer seeks to maximize her expected lifetime utility 
function, 
t W
  , (2.2)  1
0
(, , , ,
s
tt s t s t s t
s
EUc c c β
∞
+++ − + −
= ∑ " m
subject to the following budget constraints, 
   (2.3) 
** *
,
11
LK
tt t t i t t j
ij
Wp c p m p k
==
=+ + ∑∑
**
tt t t pc pA =+ 
and 
  , (2.4) 
**
1, 1 , , 1 ,
11
LK
ti t t i t j t t j t
ij
WR p m R p k ++ +
==
=+ ∑∑ 
where  (0,1) β ∈  is the consumer’s subjective discount factor,   is the true cost-of-living index, 
and 
*
t p
,
11
LK
ti t
ij
, j t A mk
==
=+ ∑∑  is the real value of the asset portfolio.  Non-monetary asset j provides 
gross rate of return,  . Monetary asset i, having quantity  , has a gross rate of return  ,1 jt R + 
, it m ,1 it R + .  
The consumer’s income from any other sources, received at the beginning of period  , is  1 t + 1 t Y + .  
The consumer also is subject to the transversality condition  
 
* lim 0
s
tt s s pA β + →∞ = . (2.5) 
The first order conditions can be obtained as 
 
**
1, 1 1 tt t j t t t E Rp p λβ λ ++ + ⎡ ⎤ = ⎣ ⎦
  (2.6) 
and 
 
**
,1 , 1 ti t t t t i t t t Um E Rp p λβ λ ++ + 1 ⎡ ⎤ ∂∂= − ⎣ ⎦, (2.7) 
where   and  1, (, , , ) tt t t t UU c c c −− = " m n 1 ()
n
ttt t t t t n t E Uc U c U c λββ ++ = ∂∂ + ∂ ∂ + +∂ ∂ " .  
 
3. Risk-adjusted user cost of monetary assets 
 
3.1. The theory 
 
  We define the contemporaneous real user-cost price of the services of monetary asset ito 
be the ratio of the marginal utility of the monetary asset and the marginal utility of consumption, 
so that 
 
()
,
1 , .
tt
it it
tt t n
tt t
UU
m
it UU U n
t t cc c E
π
, m
λ ββ
++
∂∂
∂
∂∂ ∂
∂∂ ∂
==
++ + "
∂
 (3.1) 
  2We denote the vector of L monetary asset user costs by  1, 2, , (, , , ) tt tL t π ππ ′ = " π .  With the user 
costs defined above, we can show that the solution value of the exact monetary aggregate, 
, can be tracked accurately in continuous time by the generalized Divisia index, as proved 
in the perfect certainty special case by Barnett (1980). 
 
( t M m )
Proposition 1.  Let 
,,
,, 1
,
it it
L
lt lt l
m
it m s
π
π
=
=
∑
 be the user-cost-evaluated expenditure share. Under the weak-
separability assumption, (2.1), we have for any linearly homogenous monetary aggregator 
function,  , that  () M ⋅
  , (3.2)  ,
1
log log
L
ti t
i
dM s dm
=
=∑ , i t
where Mt = M(mt). 
 
  The exact price aggregate dual,  ( ) t τ Π =Π π , to the monetary quantity aggregator 
function, Mt = M(mt), is easily computed from factor reversal, 
1
()()
L
tt i t
i
i t M m Ππ
=
=∑ π m , so that 
                                                          
1 ()
()
L
it it
i
t
t
m
M
π
Π
= =
∑
π
m
.            (3.3)  
In continuous time, the user cost price dual can be tracked without error by the Divisia user cost 
price index 
                                                         ,
1
log log
L
ti t
i
ds d , i t Π π
=
=∑ .                                       (3.4) 
To get a more convenient expression for the user cost,  , it π , we define the pricing kernel 
to be 
  11 tt Q t β λλ ++ = . (3.5) 
Let 
**
,1 ,1 1 jt jt t t rR p p ++ =   +  be the real gross rate of return on non-monetary asset,  , j t k , and let 
**
,1 ,1 it it t t t rR p p ++ = +  be the real gross rate of return on monetary asset, mi,t.  We can prove the 
following. 
 
Proposition 2.  Given the real rate of return,  ,1 it r + , on a monetary asset and the real rate of 
return,  ,1 j t r +  , on an arbitrary non-monetary asset, the risk-adjusted real user-cost price of the 
services of the monetary asset can be obtained as 
 
 
,, 1 , ,
,
,1
(1 ) (1 ) it t jt jt t it
it
tj t
1 E rE
Er
r ω ω
π
+ +
+
+− +
=


, (3.6) 
where 
  ,1 (, it t t it Cov Q r, 1 ) ω + + = −  (3.7) 
and 
  ,1 (, jt t t jt Cov Q r , 1 ) ω + + = −  . (3.8) 
  3             
Corollary 1.  Under uncertainty we can choose any non-monetary asset as the “benchmark” 
asset, when computing the risk-adjusted user-cost prices of the services of monetary assets. 
 
Notice that Proposition 2 doesn’t require existence of a risk-free non-monetary asset (in 
real-terms).  To see the intuition of Proposition 2, assume that one of the non-monetary assets is 
risk-free with gross real interest rate of 
f
t r at time t. Further, as proved by Barnett (1978), the 
certainty-equivalent user cost,  ,
e
it π , of a monetary asset   is  , it m
 
,1
,
f
tt i t e
it f
t
rE r
r
π
+ −
= . (3.9) 
We can prove the following: 
 
,1
,,
f
tt i t e
it it it it f
t
rE r
r
, , π ωπω
+ −
=+ = +
, 1 )
, (3.10) 
where  . Therefore,  ,1 (, it t t it Cov Q r ω ++ =− , it π  could be larger or smaller than the certainty-
equivalent user cost,  ,
e
it π , depending on the sign of the covariance between   and  .  
 
,1 it r + 1 t Q +
3.2.  Approximation to the theory 
 
All of the consumption-based asset pricing models require us to make explicit 
assumptions about investors’ utility functions. An alternative approach, which is commonly 
practiced in finance, is to approximate  1 t Q +  by some simple function of observable 
macroeconomic factors that are believed to be closely related to investors’ marginal utility 
growth. For example, the well-known CAPM [Sharpe (1964) and Lintner (1965)] approximates 
 by a linear function of the rate of return on the market portfolio.  We show that there also 
exists a similar CAPM-type relationship among user costs of risky monetary assets, under the 
assumption that   is a linear function of the rate of return on a well-diversified wealth 
portfolio  
1 t Q +
1 t Q +
Specifically, define  ,1 A t r +  to be the share-weighted rate of return on the consumer’s asset 
portfolio, including both the monetary assets,   ( , it m 1 , , iL = " ), and the non-monetary assets,  , j t k  
( ). Then the traditional CAPM approximation to  1, , j = " K 1 t Q +  mentioned above is of the form 
, where  , and   can be time dependent.  Econometric methodology for possible 
extensions of the specification can be found in Barnett and Binner (2004). 
1 tt t A Qa b r + =− , 1 t + t a t b
Let  , it φ  and  , j t ϕ  denote the share of   and  , it m , j t k , respectively, in the portfolio’s stock 
value, so that 
, it φ  = 
,
,,
11
it
LK
lt jt
lj
m
mk
==
+ ∑ ∑
 = 
, it
t
m
A
          (3.11) 
and  
  4, j t ϕ  = 
,
,,
11
jt
LK
lt it
li
k
mk
==
+ ∑ ∑
=
, jt
t
k
A
.         (3.12) 
Then, by construction,  , where   ,1 , ,1 , ,1
11
LK
At it it jt jt
ij
rr φϕ ++
==
=+ ∑∑  r +
                     ,,
11
1
LK
it jt
ij
φϕ
==
+ = ∑∑ .        (3.13) 
 
Let  , where  ,, ,,
11
LK
At it it it jt
ij
Πφ π ϕ
==
=+ ∑∑  , π , j t π   is the user cost of non-monetary asset j.  We 
define  , A t Π  to be the user cost of the consumer’s asset wealth portfolio. But the user cost,  , j t π  , 
of every non-monetary asset is simply 0.  Hence equivalently  ,,
1
L
At it it
i
, Π φπ
=
=∑ .  The reason is 
that consumers do not pay a price, in terms of foregone interest, for the monetary services of 
non-monetary assets, since they provide no monetary services and provide only their investment 
rate of return.  We can show that our definition of  , A t Π  is consistent with Fisher’s factor reversal 
test, as follows.   
 
Result: The pair ( t A, , A t Π ) satisfies factor reversal, defined by: 
,, ,
11
LK
, , A t t it it jt jt
ij
A m Πππ
==
=+ ∑∑  k
)
.           (3.14) 
 
 
Observe that the wealth portfolio is different from the monetary services aggregate, 
. The portfolio weights in the asset wealth stock are the market-value-based shares, while 
the growth rate weights in the monetary services flow aggregate are the user-cost-evaluated 
shares.  
( t M m
  Suppose one of the non-monetary assets is (locally) risk-free with gross real interest rate 
f
t r , and let 
f
t r =E ,1 j t r +   for all j.  It follows that the certainty equivalent user cost of the asset 
wealth portfolio is 
,1
f
tt A t
f
t
rE r e
t r Α, Π
+ − = .  We can prove the following proposition. 
 
Proposition 3.  If one of the non-monetary assets is (locally) risk-free with gross real interest 
rate 
f
t r , and if  , where  1 tt t A t Qa b r ++ =− , 1,1 A t r +  is the gross real rate of return on the consumer’s 
wealth portfolio, then the user cost of any monetary asset i is given by 
 
  , (3.15)  ,, ,, , (
e
it it it At At ππ β Π Π −= − )
e
 
  5where  , it π  and  , A t Π are the user costs of asset iand of  the asset wealth portfolio, respectively, 
and 
,1
,
f
tt i t
f
t
rE r e
it r π
+ − =  and 
,1
,
f
tt A t
f
t
rE r e
At r Π
+ − =  are the certainty-equivalent user costs of asset iand the 
asset wealth portfolio, respectively.  The “beta” of asset   in equation (3.15)) is given by  i
 
 
,1 ,1
,
,1
(,
()
tA t i t
it
tA t
Cov r r
Var r
β
++
+
=
)
. (3.16) 
 
Proposition 3 is very similar to the standard CAPM formula for asset returns. In CAPM 
theory, the expected excess rate of return,  ,1
f
ti t t Er r + − , on an individual asset is determined by its 
covariance with the excess rate of return on the market portfolio,  ,1
f
tMt t Er r + − , in accordance 
with 
 
  ,1 , ,1 ()
f f
ti t t i t tMt t Er r Er r β + −= − + , (3.17) 
 
where 
,1 ,1
,1
(,
, ()
)
f f
ti t t M t t
f
tM t t
Cov r r r r
it Var r r β
++
+
−−
− = .  
This result implies that asset i’s risk premium depends on its market portfolio risk 
exposure, which is measured by the beta of this asset.  
 
 4. Concluding remarks 
 
  Simple sum monetary aggregates treat monetary assets with different rates of return as 
perfect substitutes. Barnett (1978, 1980) showed that the Divisia index, with user cost prices, is a 
more appropriate measure for monetary services, and derived the formula for the user cost of 
monetary asset services in the absence of uncertainty. Barnett, Liu, and Jensen (1997) extended 
the Divisia monetary quantity index to the case of uncertain returns and risk aversion. For risky 
monetary assets, however, the magnitude of the risk adjustment to the certainty equivalent user 
cost is unclear. Using a standard time-separable power utility function, Barnett, Liu, and Jensen 
(1997) showed that the difference between the unadjusted Divisia index and the index extended 
for risk is usually small. However, this result could be a consequence of the same problem that 
causes the equity premium puzzle in the asset pricing literature. The consumption-based asset 
pricing model with more general utility functions, most notably those that are intertemporally 
non-separable, can reproduce the large and time-varying risk premium observed in the data 
[Campbell and Cochrane (1999)].  We believe that similarly extended asset pricing models will 
provide larger and more accurate CCAPM adjustment to the user costs of monetary assets than 
those found in Barnett, Liu, and Jensen (1997). The current paper extends the basic result in 
Barnett, Liu, and Jensen (1997) in that manner.  The proofs and more discussion will be 
available in the full paper, now in working paper form, and to appear in vol. 1, no. 1. of the 
Annals of Finance as Barnett and Wu (2004).   
  
 
  6References 
 
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A. Barnett and A. Serletis (eds.) (2000), The Theory of Monetary Aggregation, North-Holland, 
Amsterdam, Chapter 1, 6-10. 
 
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aggregation theory”, Journal of Econometrics 14, 11-48.  Reprinted in W. A. Barnett and A. 
Serletis (eds.) (2000), The Theory of Monetary Aggregation, North-Holland, Amsterdam, 
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Barnett, W. A. (1995), “Exact aggregation under risk,” in W. A. Barnett, M. Salles, H. Moulin, 
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12, 245-273.  
 
Barnett, W. A. and A. Serletis (eds) (2000), The Theory of Monetary Aggregation, North-
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vol 1, no. 1, forthcoming. 
 
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  8

Intertemporally non-separable monetary-asset risk adjustment and aggregation

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Intertemporally non-separable monetary-asset risk adjustment and aggregation

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