3. The specific features of the GEM-E3 model
3.1. Household’s behaviour
Private consumption decisions are derived from an intertemporal model of the household sector with two stages. In a first stage the households decide each year on the allocation of their expected resources between present and future consumption of goods and leisure, by maximising over their entire life horizon an intertemporal utility function subject to an intertemporal budget constraint defining total available resources. It is assumed that at the end of his life they will have no savings left. The utility function has as arguments consumption of goods and leisure. The specification of the first stage problem is based on a Stone-Geary utility function. The discrete approximation of this problem can readily be solved5.
Figure 3.2: The consumption structure of the GEM-E3 model
T o ta l E x p e c te d In c o m e
L e is u re
L a b o u r S u p p ly
C o n s u m p tio n
S a v in g s
D u ra b le g o o d s N o n -d u ra b le g o o d s a n d se rv ic e s
• C a rs
• H e a tin g S y s te m s
• F o o d
• C lo th in g
• H o u sin g
• H o u sin g fu rn itu re a n d o p e ra tio n
• M e d ic a l c a re a n d h e a lth e x p e n se s
• P u rc h a s e d tra n s p o rt
• C o m m u n ic a tio n
• re c re a tio n , e n tre rta in m e n t e tc .
• O th e r s e rv ic e s
• F u e ls a n d p o w e r
• O p e ra tio n o f tra n s p o rts C o n s u m p tio n o f n o n -d u ra b le s
lin k e d to th e u s e o f d u ra b le s
D isp o sab le in c o m e
5 For a detailed presentation of the derivation of the demand functions using optimal control see C. Lluch (1973). A similar formulation can also be found in Jorgenson et. al (1977).
In the second stage households allocate their total consumption expenditure between expenditure on non-durable consumption categories (food, culture etc.) and services from durable goods (cars, heating systems, and electric appliances). In GEM-E3 the above general scheme is implemented with the structure as given in Figure 3.2.
Households, modelled through one representative consumer for each EU country, allocate in each period their total expected income between consumption of goods (both durables and non-durables) and services, leisure and savings in the first stage.
The Stone-Geary utility function, yielding a LES demand system is based on a Cobb–
Douglas utility function and the maximisation problem is written6:
Max
(
1)
t(
ln(
t)
ln(
t) )
t
U =
∑
+stp − ⋅ BH HCDTOTV −CH +BL LJV −CL where HCDTOTV represents the consumption of goods,LJV the consumption of leisure,
stp the subjective discount rate of the households, or social time preference, CH and CL the committed amount of consumption and leisure,
BH and BL the cost shares of consumption and leisure.
The expenditure choice is subject to the following budget constraint, which states that all available disposable income will be spent either now or some time in the future:
( ) ( )
( ) ( )
1 1
t
t t t t t
t
t
t t t t t
t
r HCDTOT PCI CH PLJ LJV PLJ CL
r YTR PLJ LTOT PCI CH PLJ CL
−
−
+ ⋅ − ⋅ + ⋅ − ⋅
= + ⋅ + ⋅ − ⋅ − ⋅
∑
∑
where r is the nominal discount rate (parameter),
PLJ LTOT⋅ is the value of the available time resources,
YTR is the total income of the households from sources other than wages (transfers).
The household behaviour is assumed to be formed as a sequential decision tree: based on assumptions about the future, the household decides the amount of leisure, by which they define their labour supply. Computing the Lagrangian of the above problem the first order conditions are obtained. These consist of the budget constraint, plus the two derived demand functions:
(
1+stp)
−t⋅BH/(
HCDTOTVt−CH)
− λ ⋅ +(
1 r)
−t⋅PCIt =0(
1+stp)
−t⋅BL/(
LJVt −CL)
− λ ⋅ +(
1 r)
−t⋅PLJt =0the value of the Lagrange multiplier λ can be derived by summing up these equations over time, and substituting them into the budget constraint.
6 Equations without numbering are not included in the model text, as they are only intermediate steps used for the derivation of other formulas.
Expressing now the above equations for the current time period (t = 0) and using the value of the multiplier, the two demand functions to be used in the model are obtained:
( )
stp BH
HCDTOTV CH YDISP PLJ LJV Obl
rr PCI
= + + ⋅ − (1)
( )
stp BL
LJV CL YDISP PLJ LJV Obl
rr PLJ
= + + ⋅ −
⋅ (2)
where Obl=PCI CH⋅ +PLJ CL⋅ is the value of committed consumption and rr the real discount rate.
Given the fact that the model is calibrated to a base year data set in which households have a positive savings rate, the computed stp is less than rr. The savings rate computed from the above is not fixed but rather depends on such factors as the social time preference, the real interest rate and the relative shares of consumption and leisure in total potential disposable income.
In the second stage, total consumption is further decomposed into demand for specific consumption goods. For this allocation an integrated model of consumer demand for non durables and durables, developed by Conrad and Schröder (1991) is implemented. The rationale behind the distinction between durables and non durables is the assumption that the households obtain utility from consuming a non-durable goods or services and from using durable goods. So for the latter the consumer has to decide on the desired stock of the durable good based not only on the relative purchase cost of the durable, but also on the cost of those goods that are needed in connection with the durable (as for example fuels for cars or for heating systems).
The consumer problem can be written as
( ) i ( fixj ) j
i j
ND i DG
Max Uc=
∏ q
−γ
β∏
SDG −γ
βsubject to the constraint
( u fix )
i i j j j j
ND DG
HCDTOTV PC⋅ =
∑
p q +∑
p SDG + p I ,where Uc is the level of utility, PC is the consumption price, SDG is the stock of durables, γ is the minimum obliged consumption and β is the elasticity in private expenditure by category, non-durable goods and services are denoted by the index ND while durables by the index DG.
Under this specification, one can derive the following LES expenditure system for non durables:
( )
( , , ) ND ND DG DG ( ND) ND
ND DG ND ND
HCNDTOT =E U p SDG = PC ⋅ γ +Uc⋅ SDG− γ −β ⋅ PC β
∑ ∏ ∏
β ,which gives the (minimum) expenditure on non durables given the stock of durables and the utility level U. We obtain the derived demand functions for the non-durable goods by differentiating the expenditure function (Shephard's lemma):
HCFVND ND PCND HCNDTOT PC
where HCNDTOT is equal to E, the total expenditure on non durables.
The cost of using a durable is obtained by differentiating the above expenditure function with respect to the stock of each of the durables. This quantity represents the amount of non-durables that the consumer is willing to forsake for one extra unit of the particular durable:
∂
The cost of operating the durables, that is, consumption of linked non durables is included in the user's cost of the durable
(
PDUR)
:PDURDG = PCDG(rr + δDG) + TXPROP, DG(1 + rr) + ∑
LND λLND,DGPC LND,DG (3)
where δDG is the replacement rate for durable goods, TX is the property tax for the durables,
LND is the set defining all linked non-durable goods and λ is the consumption of non durables per unit of durable.
The last part of the equation links non-durable goods to the use of durables, Energy being the main linked non-durable good. Consumption of energy does not affect the expenditure of durables through the change in preferences but rather through the additional burden in the user cost.
The demand for linked non-durable goods, coupled with the use of the durable is then:
( )
, ,
ND DG ND ND DG
DG
LLNDC =
∑
λ ⋅ θ SDG (5)where λDG measures the proportion of the consumption of the linked non-durable good that is used along with the durable so as to provide positive service flow, θND DG, represents the minimum consumption of the non-durable that is needed for a positive service flow to be created. If there is no need for non-durable good the θND DG, in the first equation of the linked non-durables becomes zero. Therefore, we get:
HCFVND = CHND +
Total household’s expenditure is then the sum of consumption (for linked non-durables) plus investment in durables plus consumption in non-durables used with durables.
DG ND
Assuming a rate of replacement δ, this investment is equal to:
[ ]
(1 ) 1
DG DG DG
HCFV =SDG − − δ ⋅SDG − (8)
The demand for consumption categories is then transformed into demand for products through a consumption transformation matrix with fixed coefficients:
,
This equation determines the final consumption expenditure of the households. The consumption transformation matrix is also used to compute the consumption price as the weighted average of the consumers’ prices of products in private consumption (PH):
,
Production functions in GEM-E3 appear in the form of nested, constant return to scale CES functions. At the first level, production splits into two aggregates, one consisting of capital stock and the other of labour, materials, electricity and fuels. At the second level, the latter aggregate is further divided in their component parts. Figure 3.3 illustrates the nesting structure of the production functions.
The model considers 18 production sectors, each represented as a firm which decides on the supply of goods or services given their sales prices and the prices of production factors.
The stock of capital is fixed within each period, the supply curve of the produced goods exhibits, therefore, decreasing return to scale7.
The production function has the following form (for the1st nest):
7 This description applies only to the case, where capital is assumed immobile across sectors and countries.