Applied input-output volume models









>

= < ,

which can be decomposed as follows:

p = pA + p^b<c^a> + c^r, p^b = pB + c^b,

and c^r denotes the value added coefficients other than amortization and c^b the additional cost coefficients belonging to sectoral investments but not included into the commodity block of the table (e.g., taxes, when material cost is measured at base prices). p^b can be interpreted as the vector of price indexes of capital goods invested into various sectors, which are supposed to modify the value of amortization too (p^b<c^a>).

By appropriate substitutions we can again eliminate p^b and the second equation to an equivalent condensed form:

p = p(A + B<c^a>) + c^b<c^a> + c^r = pA^d + c^b<c^a> + c^r,

Partial closure thus extends the scope of the endogenously treated phenomena in the conventional input-output multiplier analyses. By the latter we mean the analysis which rests on a given input-output coefficient matrix and its Leontief inverse, and on a simple matrix-vector multiplication. For example, in last case on the following formulas:

∆x = (I−A^d)⁻¹∆y^e p = (c^b<c^a> + c^r)(I−A^d)⁻¹.

We could also see that the decomposed (structural) schemes are much more transparent then the augmented or the condensed (reduced) multiplier forms. Their only advantage was the computational convenience (a simple matrix-vector multiplication), which mattered in the early years of input-output analysis, but are no longer required.

2.1.4. Applied input-output volume models

We provide two examples to illustrate how one can formulate somewhat more complex input-output models in their structural form, based on the knowledge of coefficients gained from statistical input-output tables and potentially from other statistical sources. The various balance and functional equations of the input-output model will reappear in the computable general equilibrium models as well, thus, this illustration paves the way for the better understanding the latter models as well.

Table 2.4: The assumed structure and content of the input-output table

absolute values coefficients

X^h y^ch y^gh Y^vh z y^kh x A^h s^ch s^gh B^h s^z b^kh X^m y^cm y^gm Y^vm 0 y^km m A^m s^cm s^gm B^m 0 b^km t^am tcm tgm t^vm 0 tkm tm τ^am τcm τgm τ^vm 0 τkm

t^a tc tg t^v -tz tk t τ^a τc τg τ^v -τz τk

d w t^w

π

c^a c^w τ^w c^π

t^x τ^x

∑ x ^{yc yg} y^v z yk 1 1 1 1 1 1 (We advise the reader to come back to that table when he or she feels lost in the midst of the – at the first glance – somewhat complicated notations.)

Table 2.4 lists the data, both the absolute values and the coefficients (the absolute values divided by the appropriate column sum) in an input-output table format. The first two blocks row wise describe the sectoral product balances, i.e., distribution the sectoral products available from home production (x, index h) or import (m, index m), where the notations are as follows:

X^h, X^m, A^h, a^m intermediate use and their input coefficients y^ch, y^cm, s^ch, s^cm personal consumption and their coefficients y^gh, y^gm, s^gh, s^gm public consumption and their coefficients Y^vh, Y^vm, B^h, B^m sectoral investment and their coefficients y^kh, y^km, b^kh, b^km change in stocks and their coefficients

Following the two blocks one can find two rows. The first contains the import tariffs (t^am, tcm, tgm …; τ^am, τcm, τgm …), because we assume here that the volume of import is measured at their world market prices, converted to local currency. Therefore, the total value of imports, m

= m1 measures the amount of foreign currency needed for their purchase, converted to local currency. The second row contains net indirect taxes and subsidies (t^a, tc, tg …; τ^a, τc, τg …).

The volume of exports (z) is assumed to show the revenue of the producers, which may be higher than the actual price paid by the foreign buyers. Their difference, the export subsidy (tz) appears also in this block (with negative sign). By this correction the column total exports, z becomes equal to the foreign currency earned by them, converted to local currency. The difference of m and z measures thus the foreign trade deficit (de).

Finally, the remaining block contains the value added items and their coefficients:

d and c^a amortization

w, t^w and c^w, τ^w wages and wage surcharges π and c^π net operating surplus (profit)

t^x, τ^x production taxes/subsidies and their coefficients Based on the above data we can formulate alternative structural models that can be used for comparative static analysis. For example, we can split up final consumption in such a way that makes it easy to introduce exogenous or endogenous variables into an input-output model:

y^dh = y^h0 + s^vh·ycv + s^gh·yg + By^v, y^dm = y^m0 + s^vm·ycv + s^gm· yg + B^my^v, where the new symbols are as follows:

y^h0, y^m0 exogenously fixed part of final demand (cf. committed consumption) ycv level of variable (personal) consumption

s^vh, s^vm unit coefficients of (variable) personal consumption (if different from s^ch, s^cm) yg level of public consumption

y^v level of sectoral investment

We can split up (gross) investment into replacement and net investment (y^v = y^rv + y^nv) and the former can be made endogenous by means of the following definitional equation:

y^rv = <r^a><k>x,

where r^a<k> = c^a, k is the vector of capital coefficients per unit of output and r^a is the vector of the rates of amortization, as before.

Next we may assume that the level of variable personal consumption changes in proportion to wages, where the propensity to consume (ϕ) is a potential exogenous variable:

ycv = ϕ^{· cwx.}

Summing up, we have so far defined the following sets of equations:

x = A^hx + y^h0 + s^vh·y_cv + s^gh·y_g + B^hy^rv + B^hy^nv + z (2.1-6) m = A^mx + y^m0 + ϕ^·svm·ycv + s^gm·yg + B^my^rv + B^my^nv (2.1-7)

y^rv = <r^a><k>x (2.1-8)

ycv = ϕ^{· cwx (2.1-9)}

We have altogether 3n + 1 number of equations. If we chose x, m, y^rv and ycv as unknown (endogenous) variables, the number of which is also 3n + 1, we would arrive at a well defined model, which could be solved once the values of the parameters and potential exogenous variables (first of all z, yg, ϕ^{, ynv, yh0 and ym0}) are given. We could thus run comparative static simulations to test the likely effect of their changes on the endogenous variables.

Eliminating the unknowns other than x we reduce the set of equations to n equations, containing only the sectoral levels of output as variables:

x = (A^h + B^h<r^a><k> + ϕ^{·svh◦cw)x + yh0 + sgh·y}g + B^hy^nv + z, the solution of which is

x = (I − A^h − B^h<r^a><k> − ϕ^{·svh◦cw)−1(yh0 + sgh·y}g + B^hy^nv + z), where the coefficient matrix

(A^h + B^h<r^a><k> + ϕ^·svh◦cw)

is nothing but the coefficients of a partially closed input-output table. The structural form given by equations (2.1-6) - (2.1-9) is, however, a more transparent presentation of the model, than its reduced form.

The special advantage of the structural form is that it makes easy to redefine the model. We may for example introduce further variables and add further equations to the above core model.

For example, the definition of the balance of trade

d_e = p^wmu – p^wez, (2.1-10)

where p^wm and p^we are the exogenously given price indexes of imports and exports. As long as we consider d_e an endogenous variable this equation would be just an epilogue added to the rest of the equations, since d_e does not appear in them.

One can also define the demand for labour and capital

l_d = lx (2.1-11)

k_d = kx (2.1-12)

or the level of total exports and imports

z = 1z – t_z (2.1-13)

m = m1. (2.1-14)

As long as we do not change the rest of the model, the value of the new variables (d_e, l_d, k_d, z, m) can be simply calculated after we have solved already the model given by equations (2.1-6) - (2.1-9), since none of the new variables appears in them. We can, nevertheless, revise the model with respect to variables considered to be endogenous or exogenous. We might want, for example fix the value of k, interpreting it as capital constraint, at the cost of freeing some variable that was exogenous so far (e.g. ϕ ^{or y}g). We might want to fix the structure of exports (z = z·s^z) and make its level depend on the volume of imports (z = m·r^z).

All these changes would imply another model structure and require another type of solution. One more reason why one might prefer structural to reduced form. Consider, for example, a variant of the model consisting of (2.1-6a), (2.1-7) - (2.1-9), (2.1-10a), where

x = A^hx + y^h0 + s^vh·y_cv + s^gh·y_g + B^hy^rv + B^hy^nv + z·s^z (2.1-6a)

p^wmu − z·p^wes^z = d_e. (2.1-10a)

In this variant of the model we handle foreign trade (exports and imports) in a similar way as in the analysis based on an input-output model of type D2. One could define a variant analogous with an input-output model of type D1 by the set of equations (2.1-6b), (2.1-7) - (2.1-9), (2.1-10a) and (2.1-14), in which d_e becomes endogenous variable too, moving in proportion to the level of imports and exports, leaving thus their observed proportion (s_de) unchanged.

x = A^hx + y^h0 + s^vh·y_cv + s^gh·y_g + B^hy^rv + B^hy^nv + m·r^z, (2.1-6b)

s_de·m = d_e. (2.1-10b)

The possibilities to form models for economic policy analysis on the basis of statistical input-output tables is wide enough, nevertheless, constrained especially by the rigidity of the linear forms. For example, they do not allow for making some of the coefficients dependent on price changes. The price models based on the input-output tables are developed with no reference to the volume side either. Let us turn now our attention to applied input-output price models.

In document CGE Modelling: A training material (Pldal 54-58)