his file is created for educational purpose not for sale. Creator of this presentation is highly appreciated the Authors:Christopher D. Carroll, Jody Overland and David N. Weil Source: Journal of Economic Growth, Vol. 2, No. 4 (Dec., 1997), pp. 339-367 Published by: Springer

1. Comparison Utility in a
Growth Model
Authors: Christopher D. Carroll, Jody Overland and David N. Weil
Source: Journal of Economic Growth, Vol. 2, No. 4 (Dec., 1997), pp. 339-367
Published by: Springer
Presented By Mr. Warawut Runakham,
National Institute of Development Administration, NIDA
DE 7300 Macroeconomics Theory 1

2. Coverage of
presentation
1. Introduction:
I. Comparison Utility
II. Inward Looking Utility
III. Outward Looking Utility
2. Model
3. Results
4. Conclusion
5. Q&A
DE 7300 Macroeconomics Theory 2

3. Introduction
• Most modern growth theories are agents solving
intertemporal optimization problems. Decisions
about consumption determine savings, which in
turn becomes capital for future production
• Less attention has been paid to the utility function
that agents are assumed to be maximizing;
generally, it has been taken to be time separable,
and usually of the constant relative-risk-aversion.
DE 7300 Macroeconomics Theory 3
Growth
Consumption
and Saving
Capital
Accumulation
Expansion in
future
production
Utility
Function

4. Introduction
• In this article authors examine the implications for
growth of an alternative form of preference that is
used to explain puzzle problems in both pricing and
consumption literatures called in this case the
“comparison utility”
• Using comparison utility might change individual
optimization and the “Dynamic” of economy
populated by such individual
DE 7300 Macroeconomics Theory 4
Growth
Consumption
and Saving
Capital
Accumulation
Expansion in
future
production
Standard
Utility
Alternative
form of
Utility
Utility
Function

5. Standard Vs. Comparison Utility ?
Standard Utility
• Generally, utility comprise with
various forms: linear, Leontief,
Stone- Geary, Addictive utility
functions, etc.
DE 7300 Macroeconomics Theory 5
Comparison Utility
• Alternative form of preference, call this
general class of utility functions
“comparison utility”, which believed
that utility not only dependent on
consumption also the consumption
compared to other people’s consumption
and standard level of society
• called this standard level of consumption
of society as “ the reference stock”

6. Models
• In this article authors consider two specific versions of the comparison utility via
“Endogenous growth Model” with neo-classical production function
DE 7300 Macroeconomics Theory 6
Inward- looking Utility
1. Individual care current consumption
comparing to their past consumption
2. each household's reference stock is
simply an exponential moving
average of its own past consumption
Outward-looking Utility
1. Individual care their own
consumption comparing to other
people’s consumption or reference
stock
2. Consumption is dependent on
standard level of consumption of
community or in the class to which a
person belongs

7. Set up
Household's optimization problem
• Economies are populated by identical, infinitely live dynasties
• Households' utility depends on the comparison of their consumption at each point in time to a
reference stock
• instantaneous utility function for household i is
DE 7300 Macroeconomics Theory 7












1
1
),( i
i
ii
z
c
zcU
where
• Z is the reference stock
• 𝛾 is parameter indexes the importance of comparison utility if
• 0 ≤ 𝛾 ≤ 1
• 𝛾 =0, then only absolute level of consumption is important
• 𝛾 =1, then consumption relative to the reference stock is important
• 𝜎 is the coefficient of relative risk aversion
• 𝜎 ≥
1
1−𝛾
condition guarantees utility function is concave

8. Set up
• Production Function
• Capital depreciates at rate 𝛿 ≥ 0
• Household i stock of capital 𝑘𝑖
• Household maximize the discounted, infinite stream of utility
DE 7300 Macroeconomics Theory 8
Aky 
iii ckAk  )( 
dtezcU ii


0
),(max 

9. Set up
Reference Stock: Inward – looking case,
• Household based their reference stock on their own past consumption and take into account the effect
of their current actions on the future reference stock when making consumption decision (z is
exogenous)
• There is no interaction between households, then reference stock of household i is
Reference Stock: Outward – looking case,
• Reference stock for household i is given by
DE 7300 Macroeconomics Theory 9
)( ii zcz   where
• c is the average level of consumption for all households
)( iii zcz   where
• 𝜌 > 0 is the relative weight of consumption at different times
• The larger 𝜌 ; the more important is consumption in the recent past
• Ci is the consumption of individual

10. Set up
Inward-looking model
• Current Value Hamiltonian (CVH)
• There are two state variables 𝑘𝑖, 𝑧𝑖 that are
affected by household decision
• There are two associated co-state
variables Ψ, λ
DE 7300 Macroeconomics Theory 10
Outward-looking model
• Current Value Hamiltonian (CVH)
• z is taken as exogenous
• There is one state variable 𝑘𝑖 that is affected
by household decision
• There is one associated co-state variable 𝜋
• An equilibrium path will be one in which the
household’s expectation of average
consumption {𝑐} = household’s own planned
consumption path {𝑐𝑖}
))((),( iiii ckAzcUH   )(])[(),( zcckAzcUH  

11. Outward-looking model
• The necessary conditions for optimum
• The transversally condition
0


c
i
U
c
H
 )( A
k
H
i




0 


i
i
k
k
Solution to the Household’s Problem
• The reference stock affects household
decisions via its effect on the marginal utility
of consumption
• Take differentiate necessary condition wrt.
time
• , then
• Since, then rewrite,
DE 7300 Macroeconomics Theory 11
))((),( iiii ckAzcUH  
 cU
A
U
U
c
c
 


icziccc zUcUU  
A
U
zUcU
c
iczicc






12. Solution
• since all households are identical , the
economywide average level of 𝑐 and 𝑧 are
equal to the values for any single household.
• Authors drop the subscripts and combine
with equation of motion for reference stock
gives relation between consumption growth
and c/z along the optimal path:
DE 7300 Macroeconomics Theory 12
• Substituting
• then, growth rate of consumption along the
optimal path in terms of parameters and the
growth rate of the reference stock (as
exogenous)
ic
cz
ic
cc
zU
U
and
cU
U )1( 









z
z
A
c
c i
i
i 
)1(
1














 1)1(
1
z
c
z
z
A
c
c i



13. Solution
• In steady state, the values of c/z and 𝑐/𝑐 can
be recalculated as
DE 7300 Macroeconomics Theory 13





)1(
A
c
c















c
cA
z
c 



1
1
)1(
1
1
Figure 1. Phase diagram in (c/z, 𝑐/𝑐) space

14. Solution to the Household’s Problem
Inward-looking model
• Current Value Hamiltonian (CVH)
• reference stock corresponds to an average of
agent's own past levels of consumption can
be interpreted as a model of habit formation,
• the reference stock can be intuitively
designated the habit stock.
• Since all household are identical and
household with inward-looking utility have
no effect on each other, the authors drop the
subscripts, then the CVH is
DE 7300 Macroeconomics Theory 14
)(])[(),( iiiiii zcckAzcUH   )(])[(),( zcckAzcUH  
• Current Value Hamiltonian (CVH) with no
subscripts
• The necessary and sufficient conditions for
optimization are the first-order condition
0


cU
c
H
zU
z
H



  )(
 )( A
k
H




(B1)
(B2)
(B3)

15. Solution
Inward-looking model
• Transversally condition
DE 7300 Macroeconomics Theory 15
0)( t
• Differentiating with respect to time gives
• Combine with (B2), gives
• Combine (B9) and (B1) gives
• Combine (B10) and (B3),
0)()(lim 

tkte t
t

0)()(lim 

tzte t
t

0)( t
   cccc UzUc
(B4)
(B5)
(B6)
(B7)
(B8)
]][ zc UU    (B9)
zcc UUU   ))(( (B10)
zcc UUUA   )()(  (B11)

16. Solution
• Differentiating (B11) wrt. time, dividing by (B11) and combining with (B3) gives
• Substituting following expression into (B12),
DE 7300 Macroeconomics Theory 16
(B12)






























































z
c
A
c
c
z
z
A
z
c
z
z
z
c
z
z
z
z
c
c
z
z
c
c
c
c



)()1()2(
)1()1)1(()1(
)1)1()(1()1(2)1(
22





















 11
z
c
z
c
z
z
and
z
c
z
z



17. Solution to the Household’s Problem
• then simplifying, gives
• This equation gives the relationship between consumption, reference stock, and the first –
and second-time derivatives of consumption along an optimal path.
• Setting either 𝛾 = 0, so that reference stock is irrelevant to utility or 𝜌 = 0, so that the reference
stock is unchanging, causes (B13) to collapse to the FOC for consumption growth
DE 7300 Macroeconomics Theory 17
(B13)
0))(()(
)2(1)1(2)1(
)22)1(111)1((1









































A
z
c
A
z
c
A
z
c
c
c
c
c
A
z
c
z
c






18. Solution
• Construct a phrase diagram for inward-looking model
DE 7300 Macroeconomics Theory 18

19. Inward vs. Outward Looking
• Construct a phrase diagram for both models
DE 7300 Macroeconomics Theory 19

20. Comparison of Dynamics in
Outward and Inward looking Economies
DE 7300 Macroeconomics Theory 20

21. Comparison of Dynamics in
Outward and Inward looking Economies
DE 7300 Macroeconomics Theory 21