M. A. CARDENETE, F. SANCHO

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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-

ments

1

qx

IA Mx

2

(1)

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

,eqp

2

:nn n

RR

2

:

qn

which in turn can be split in two func-

tions n n

R

R and 2

:

nn n

R

qp

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

as

,,

,,

p

q

pFqpx

qFqpx

(2)

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

,,

q

qq

Fqpxq

,

,,

q

qp

Fqpxp , and so

on. Solving for dp in the second expression in Equation

(3) and substituting the result in the first equation would

yield

1

1

1

qq qp qx

qqqp pppq px

qx

qq qppppq

qx qppppx

dqMdq Mdp Mdx

dq MIMMdq Mdx

Mdx

MMIM Mdq

MMIM Mdx

(4)

Solving now for dq in Equation (4) we finally obtain

1

1

1,,

qq qppppq

qx qppppx

dqIMMI MM

MIM Mqpxdx

M

(5)

where

,,qpxM

M

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

1

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

ones.

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.

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