Theoretical Economics Letters, 2013, 3, 279-282 Published Online October 2013 (
Elucidating General Equilibrium Multiplier Effects:
A Differential Perspective
M. Alejandro Cardenete1, Ferran Sancho2
1Department of Economics, Universidad Loyola Andalucía, Sevilla, Spain
2Department of Economics, Universitat Autònoma de Barcelona, Barcelona, Spain
Received August 20, 2013; revised September 11, 2013; accepted September 18, 2013
Copyright © 2013 M. Alejandro Cardenete, Ferran Sancho. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Expenditure multipliers are routinely used to evaluate the effectiveness of government spending. When interested in
disaggregated effects, interindustry models provide the necessary tools to be able to look at very detailed sectorial re-
sults. These models are theoretically simple and empirically operational, which makes them easily implementable and
therefore popular with policy makers. They miss, however, quite a bit of the interaction that takes place at the micro
level. On the one hand, they ignore the role exerted by supply constraints in primary factors; on the other hand, they
look at the world as though it is fully linear. We overcome these limitations by using an opposite Walrasian general
equilibrium model to compute marginal multipliers. By using differential calculus, we also offer some insights regard-
ing the “under-the-hood” circuits of influence.
Keywords: Marginal Multipliers; General Equilibrium; Linear vs. Nonlinear Models
1. Introduction
Let us begin by considering the general setup of an eco-
nomy described by m endogenous variables and k exter-
nal exogenous variables, (say, policy instruments) affect-
ing the equilibrium state. In this economy, multipliers
connect exogenous injections xi (I = 1,2 ··· k) with en-
dogenous responses ej (j = 1,2 ··· m). If the vector func-
tion :mk m
(, )ex R represents the equilibrium state
, then it is possible to use differential calcu-
lus [1] to study the equilibrium dependence of endoge-
nous variables e with respect to exogenous one x. In this
case, we would have
,,deFex edeFex Xdx  .
Solving now for de would yield
(, )
deFexe Fexxdx
where M is a (m × k) matrix whose generic element
ej/xi = mji(e, x) is an estimate of the (marginal) multi-
plier effect exerted by injection xi upon endogenous
variable ej. Notice that in principle the multiplier matrix,
M itself may be variable since it depends on the particu-
lar equilibrium state e induced by instruments x and the
characteristics of the economy, as embodied in F. Since
the vector function F is not usually directly observable,
neither are its derivatives nor is matrix M, hence there is
the need for relying on some type of approximation. One
such approximation is to linearize the economy; another
is to use numerical methods for the evaluation of nonlin-
ear equilibrium relationships.
2. Linear Multipliers
We now consider a linear economy of the interindustry
type. For this type of economies we have n endogenous
(m = n) and n exogenous variables (k = n). The endoge-
nous variables correspond to total output in each of the n
sectors, whereas the exogenous variables describe final
demand. This includes discretional government demand
for the goods and services of each of the n sectors. We
will use the n-vector q to denote output (i.e. e = q now)
and keep x for the exogenous variables. The equilibrium
state for this linear economy is represented, again using
the vector function F, by q = F(q, x). Thanks to the line-
arity assumption this can be seen to adopt the form q =
Aq + x, where A is a n×n nonnegative, productive and
homothetic technical coefficient matrix. From the equi-
opyright © 2013 SciRes. TEL
librium state we can quickly write the reduced form
linking output with instruments as q = (I A)1x = Mx,
with M representing the multiplier matrix of the linear
economy. Because of the assumptions on A, the matrix
M is constant. Its entries are independent of the equilib-
rium state. Taking derivatives, it is quite simple now to
relate changes in output with external changes in instru-
  IA Mx
Multipliers are given directly by the cells in matrix M,
i.e. qj/xi = mji. All that is needed to compute (linear)
multipliers is the coefficients matrix A. Since this matrix
is readily available from official statistics, this explains
the popularity of linear models in policy oriented em-
pirical economics. Even more, linear models are so sim-
ple that we need not worry about prices. Prices, in fact,
can be seen to be completely independent from quantities
in linear interindustry models. Notice that if quantities
are not affected by prices, neither are multipliers. End of
the story, all needed multiplier information is contained
in matrix M. But we know that the actual story is bit
more complicated than that since, in general, prices and
quantities are mutually dependent.
3. Applied General Equilibrium Mul tip li ers
In a standard general equilibrium model the interactions
of supply and demand determine, at the same time, prices
and quantities. We use now a general equilibrium frame-
work to elucidate multipliers and compare them to their
linear counterpart. Endogenous variables include now n
output levels q and n prices p, that is, , so in
total we have 2n endogenous variables. Let us consider
again that the government decides how much to buy of
each of the n goods and services; the government’s de-
mand levels are denoted by the vector x representing
policy instruments. The structural function F represent-
ing the equilibrium state would now be of the type
:nn n
which in turn can be split in two func-
tions n n
R and 2
nn n
R determin-
ing quantities and prices, respectively. The complete
general equilibrium state is represented by (q, p) = F(q, p,
x), or using the fact that
FF, it can also be seen
We perform comparative statics on the equilibrium
state represented in Equation (2) considering an exoge-
nous change dx in the instruments x. We would obtain
qq qp qx
pq pp px
dqMdq Mdp Mdx
dpM dqM dpMdx
 (3)
where we use, in Equation (3), the notational convention
Fqpxp , and so
on. Solving for dp in the second expression in Equation
(3) and substituting the result in the first equation would
qq qp qx
qqqp pppq px
qq qppppq
qx qppppx
dqMdq Mdp Mdx
dq MIMMdq Mdx
 
 
 
Solving now for dq in Equation (4) we finally obtain
qq qppppq
qx qppppx
MIM Mqpxdx
 
 
stands for the general quantity mul-
tiplier matrix1. We now proceed to relate the linear mul-
tiplier matrix in Equation (1) with the general multi-
plier matrix
,,xqpM derived in Equation (5).
Recall first that in linear models quantities and prices
are independently determined. Under this assumption the
partial derivative matrices Mpq and Mqp would be such
that Mpq = Mqp = 0 and then Equation (5) can be easily
verified to reduce to
qq qx
dqIMM dx
 (6)
The simplified expression that appears in Equation (6)
is of course the differential version of the classical linear
multiplier expression picked up in Equation (1) above,
with Mqq = A and Mqxdx = Δx. The chains of interactions,
however, can be seen to be quite more complex in Equa-
tion (5) than in Equation (1), in accordance with the
higher complexity of nonlinear models vis-a-vis linear
Figure 1 below depicts the way the model’s intercon-
nections work. Facing an external disturbance in x, the
system first reacts with changes in prices and quantities
through matrices Mpx and Mqx. Price effects repeatedly
self multiply through the loop Mpp along the cost struc-
ture which, in turn, are affected by cross effects Mqp from
quantities to prices. Similarly, the initial effect of the
disturbance on quantities gets itself multiplied by the
chain reaction that moves directly from quantities to
quantities, i.e. Mqq, and indirectly from quantities to
looped prices and back to quantities via the combined
1A similar derivation, that we omit, would produce a general price
multiplier matrix.
Copyright © 2013 SciRes. TEL
Figure 1. Example of the circuits of influence.
effect measured by
qppp qp
Notice again the role exerted by cross effects, in this
case from prices to quantities through Mqp. The dashed
arrows show the only influence that would remain in the
typical linear model where prices and quantities are in-
dependently determined. The linear model effects would
be restricted to matrices Mqx and Mqq.
4. Some Numerical Results
We show now some results of implementing these two
models, namely, a linear interindustry model first (see [2]
for a reference of linear models), and then a Walrasian
general equilibrium model (see [3,4] for examples and
details of empirical general equilibrium models and their
methodology). We use data from the Spanish economy [5]
for 2006 to calibrate both models. Calibration entails the
selection of parameters so as to reproduce the empirical
data as an equilibrium under both models—the linear and
Walrasian versions (see [6] for a step-by-step guide to
calibration). The Spanish data distinguishes 26 produc-
tive sectors. For each sector we show the multiplier ef-
fects under the two models in Table 1 below. For in-
stance, the linear multiplier value of 1.4620 indicates the
(positive) change in economy-wide production when a
unitary exogenous demand for Agriculture is injected
into the economy. The general equilibrium multiplier of
0.5341 tells a different story. Now overall production
would go down, once all general equilibrium adjustments
had taken place. Supply restrictions and interconnected
price and quantity effects explain the different sign. The
initial injection into Agriculture is not able to activate
any overall output increase. The need for more primary
factors to satisfy the extra demand for Agriculture re-
quires siphoning them from elsewhere in the economy,
triggering an economy-wide fall in production. The indi-
rect output substitution effects more than compensate the
direct output volume effect arising from the extra injec-
Unlike the general positive multiplier effects of linear
models, multipliers results can perfectly be negative in a
Table 1. Multiplier comparison.
Multiplier Estimates
Data Economic Sector Linear
Gen. Equilib.
1. Agriculture 1.4620 0.5431
2. Fisheries 1.0394 0.3305
3. Coal 1.0211 0.5179
4. Petroleum and Gas 1.0056 0.9825
5. Mining 1.0974 0.3106
6. Oil Refining 2.0322 1.3076
7. Electricity 3.1015 0.2129
8. Gas 1.2480 0.8665
9. Water 1.0640 0.5126
10. Foodstuffs 2.7998 0.4838
11. Clothing 1.3781 0.7127
12. Wood Products 1.5952 0.2936
13. Chemicals 1.8607 0.6111
14. Building Materials 2.0411 0.0953
15. Iron and Steel 2.2508 0.5351
16. Metal Products 2.0267 0.2699
17. Machinery 2.1037 0.6224
18. Vehicles 2.1284 0.8703
19. Other Transport. Equip. 1.3384 0.5763
20. Other Manufactures 2.2511 0.2646
21. Construction 6.2904 0.0082
22. Commerce 3.5528 1.7232
23. Transport. and Comm. 2.8001 03582
24. Other Services 2.2807 0.6793
25. Services for Sale 2.2151 1.1277
26. Public Services 2.2962 1-1607
Total multiplier effect 54.2805 3.7503
Walrasian general equilibrium model. Moreover, the
combined multiplier effects can be dramatically different
in size. Under the linear model, aggregate output effects
can be as high as 54.2805, meaning that an additional
unitary demand for each and every of the 26 sectors
would give rise, on average, to a multiplier effect of
54.2805/26 = 2.0877 new output units. The general equi-
librium model, however, reduces this estimate drastically,
with an average value of only 3.7503/26 = 0.1442 new
units of output.
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5. Conclusion
The interaction of demand and supply in goods, services
and productive factors in response to external, policy
oriented induced changes makes multiplier estimates be
substantially smaller, and even negatively valued, in a
general equilibrium model than in a linear model where
price effects are disregarded and resource constraints are
not binding. Expenditure policies designed upon optimis-
tically estimated linear multiplier values should therefore
be carefully reevaluated, and perhaps even abandoned. In
fact the use of the name “multiplier” could even be a
misnomer. Under general equilibrium, multipliers are not
systematically above 1, or even positive in sign for that
matter; hence output levels need not “multiply” over 1,
as the standard linear models conclude. The tradition is
however too strong to be changed, and we will still refer
to the effects of external injections in endogenous output
as “multipliers”, provided the modeling assumptions un-
der which they are estimated are explicitly laid out.
6. Acknowledgements
Support from research projects MINECO2009-11857 and
SGR2009-587 is gratefully acknowledged. The usual ca-
veat applies.
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