Representation of foreign trade in the I-O tables

So far we have been dealing with closed economies and did not bother with exports, imports and the balance of trade. We can do that in an abstract model, but in applied models one can not ignore them. We will use the following notation:

• index of home origin h, index of foreign origin (import) m,

thus, for example, the intermediate use of the sectoral commodities (X) can be split up into domestically produced (X^h) and imported part (X^m). We will use similarly the A, A^h and A^m input coefficient matrices.

• index of domestic use d, index of terms related to export or foreign use e,

for example, total final use can be split up into domestic use and exports: y = y^d + z.

Distinguishing final use with respect to origin as well we can write: y^hd and y^md, where y^h = y^hd + z.

Total use of imported goods in production and their input coefficients can thus be defined as x^m = 1X^m and a^m = 1A^m. Total import of various sectoral commodities, on the other hand, will be denoted by vector m, where m = X^m1 + y^md, and their sum is m = 1m. If we combine the two sources, domestic production and imports, we will denote their total by xˆ = x + m.

The above examples illustrate the notation to be used, but their meaning will become clearer from the position they will hold in the various I-O tables which will be presented bellow. We will present four possible ways in which foreign trade can be incorporated into the scheme of an input-output table, without violating its basic convention, whereby the sectoral row and column totals must be equal.

In the first arrangement (I/O table of type A) we treat imports as if all imported commodities were perfect substitutes for their domestic sectoral output. Therefore, domestic production and imports are presented together in the upper part of the input-output table. Thus, the total amounts of the distributed sectoral commodities equal to x + m = xˆ .

I/O table of type A

I-O table I-O coefficients

X y^d z xˆ Aˆ s^d s^z

h cˆ

(x) yd z (ŝ^h) 1 1

m ŝ^m

xˆ xˆ 1

In order to make the sectoral column sums to be equal to their row equivalent the amounts of imported commodities (m) had to be added to the domestic output (x) at the end of each column. At first glance this looks to be an extremely artificial and meaningless correction, especially the coefficients calculated from the above table. It can be shown, however, that this solution is not meaningless at all.

Let us write up first the commodity balance equations with the coefficients gained from I/O table of type A:

xˆ = Â xˆ + y^d + z,

by means of which we could estimate the likely effect of a change in final demand on the supply of sectoral commodities. This equation is equivalent the following two:

x = <ŝ^h>(Ax + y^d + z) = Â^hx + yˆ^hd + ˆz ^h m = <ŝ^m>(Ax + y^d + z) = Â^mx + yˆ^md + ˆz , ^m where A = X<x>⁻¹, Â^h = <ŝ^h>A, Â^m = <ŝ^m>A and so on.

From this transformation it turns out that the implicit assumption behind such calculation would be that share of domestic production (ŝ^h) and imports (ŝ^m) would remain the same after the changes take place, and this same composition would prevail in every area of use, including the exports as well. Notice that import would become endogenously determined in such a model.

As far as the pricing equation implied by the above table is concerned, i.e., equation pˆ = pˆÂ + pm⋅ŝ^m + cˆ ,

it is easy to see that it can also be rearranged in the following two sets of equations:

pˆ = p^h<ŝ^h> + p_m⋅ŝ^m p^h = p^hÂ^h + pm⋅â^m + c,

where p^h can be interpreted as the price index of the domestically produced sectoral commodities, pm the average or general price index of the imports, and â^m = 1Â^m = ŝ^mA.

Vector pˆ can thus be interpreted as an average users’ cost-price of the sectoral commodities.

The second arrangement (I/O table of type B) rests on the opposite assumption: imports are perfect complements to the domestic output of the same sectoral origin. The upper part of the table contains only the commodity balances of domestic production. The import used in production (x^m = 1X^m) or in final consumption (ym) are represented in a separate row.

I/O table of type B

I-O table I-O coefficients

X^h y^hd z x A^h s^hd s^z

x^m y_m 0 m a^m s_m 0

h c

x y_d z 1 1 1

Let us write up again the commodity balance equations with the coefficients gained from I/O table of type B:

x = A^hx + y^hd + z,

to which we can immediately add the following two equations:

m = a^mx + ym, m = A^mx + y^hm.

The first equation can be used to estimate the likely effect of a change in the final demand for domestically produced sectoral commodities on their output. The second and the third equations will show the repercussive effect of the above change on the imports, where the use of imports in production is endogenously determined via x.

The interpretation of the p^h = p^hA^h + p_m⋅a^m + c

price equation is straightforward. It can be used for tracing through the likely changes of the sectoral prices of domestically produced commodities resulting from an exogenous change in the cost of imports or some elements of value added.

A third version (I/O table of type C) is aimed at correcting the criticized problem of type A, that is, adding imports to domestic output to regain the equality of the last row and column in the table. Instead of that they propose to subtract imports from final demand, and make the sum of rows be equal to domestic production. In effect, instead of the exports the table contains only net exports (z − m).

I/O table of type C

I-O table I-O coefficients

X y^d z − m x A s^d

h 0 c 0

x yd 1 1

As a result both exports and imports will be exogenous variables in the input-output analysis based on the coefficients gained from I/O table of type C, as can be seen from the basic equations:

x = Ax + y^d + z − m, p = pA + c.

These equations are the straightforward extensions of the forms we got in the case of the closed model. It suffers, however, from the problem that imported goods are considered to be completely the same in all respect (quality, price) as their domestic counterparts. It provides no explanation for the size of foreign trade.

And finally, a fourth version (I/O table of type D) has been designed exactly with the purpose of making foreign trade, both exports and imports endogenous variables in the otherwise conventional input-output multiplier analysis. It is achieved by the introduction of an additional ‘extern trade’ sector, which purchases foreign currency by means of exports, which in turn finances imports. The total value of imports (m = x^m1 + ym) as a rule differs from the total value of exports (z = 1z). Their difference, the balance of trade (m − z = de) can be placed in different positions of the revised I-O table. We will show two possible arrangements here.

Both solutions are based on sound economic logic, both are meant to make up for the

The basic equation of the conventional input-output multiplier analysis, assuming change in final domestic demand, takes the following form in the case of D1:



These equations reveal the nature of this solution: the change in domestic final demand will not only directly effect domestic production, but indirectly too. It is assumed that export will change too, in order to restore the balance of trade which is upset by the change in imports. As a matter of fact, the change in the exports will be proportional to the change in the import, as can be seen from the first equation.

We can eliminate ∆m by substituting its value with the expression standing on the right hand side of the second equation. In this way we may derive the condensed form equivalent of the augmented input-output model equation:

∆x = (A^h + r^z◦a^m)∆x + r^z·∆ym + ∆y^hd = A^z∆x + r^z·∆ym + ∆y^hd.

where r^z◦a^m denotes the dyadic product of the two vectors. The matrix A^z = (A^h + r^z◦a^m) is now of the same size as matrix A^h, containing larger elements. Thus if the changes effects only the final demand for domestic goods, its impact on domestic output can be estimited by solving the equation

∆x = A^z∆x + ∆y^hd,

which looks exactly like the basic equation, but the input-output coefficient matrix contains now elements that transmit the indirect effect of the changing export as well.

This modification of the input-output system will influence the form and content of the price multiplier associated with it too. On the basis of the above augmented input-output coefficient matrix we get the following price form:

(

^{h e}

) (

^{h e}

)

^h_m ^z

⁽

^{, e}

⁾

which can be decomposed as follows:

p^h = p^hA^h + p_e·a^m + c, p_e = p^hr^z + c_e,

where c_e denotes the balance of trade coefficient, which belongs to the column of the exports in the value added part of the modified input-output table.

In the definition of the domestic price indexes, unlike in its open version, the cost of import (a^m) is revaluated by the price index p_e. The value of this latter index, given by the second equation, reflects changes in the domestic cost of earning foreign exchange via exports.

In document CGE Modelling: A training material (Pldal 48-53)