Quantifying the relative importance of export industries in a small open economy during the great depression of the 1930s: an input–output approach

Abstract

This paper provides a quantification of the relative importance of export industries in a small open economy using new data provided by input–output tables describing the Finnish economy in 1928. The Finnish analysis of the Great Depression of the 1930s has been particularly focused on the importance of foreign trade. Despite the lack of quantified evidence, it is commonly accepted that the export industries had a major role in the economic development. The basic input–output framework is extended into a production–consumption model to produce a more elaborate model that provides a quantification of changes in final demand of some key industries in the economy. Results suggest that even though the role of export industries was dominant, domestic market industries and private consumption also had a significant role in the depression.

Appendices

Appendix 1: Reliability analysis of data

Here we suggest that the 95% confidence interval for total output and GDP can be calculated by applying the Lindeberg-Feller variant of the central limit theorem,¹⁷ according to which for independent variables, x _i, x ₂, …, x _n , whose expectation and variance are E(x _i ) = μ _i and Var(x _i ) = σ ² _i , respectively, the distribution of the standardised sum

$$ \frac{{\sum\limits_{i = 1}^{n} {(x_{i} - \mu_{i} )} }}{S},{\text{ where }}S = \sqrt {\sum\limits_{i = 1}^{n} {\sigma_{i}^{2} } } , $$

(1)

is close to the standard normal distribution for large n.¹⁸ When this normal approximation holds, the 95% confidence interval for the unknown $\sum\nolimits_{i = 1}^{n} {\mu_{i} } ,$ is

$$ \left( {\sum\limits_{i = 1}^{n} {x_{i} } - 1.96 \times S,\sum\limits_{i = 1}^{n} {x_{i} + 1.96 \times S} } \right). $$

(2)

In the case of the supply and use tables constructed, n = 30 since the final use table of domestic products presents output at the level of 30 product groups. This is large enough to justify the normal approximation. What we need to estimate is $\sum\nolimits_{i = 1}^{30} {\mu_{i} } ,$ the total output, which is the sum of unknown product-group outputs. For each product group i, there are two observations forμ _i : supply and use. It is assumed that errors in these observations are random and independent with zero means, but not necessarily normal.

The point estimate, x _i in (1) for the true product-group output μ _i is the mean of supply and use in the corresponding product group i. The point estimate for $\sum\nolimits_{i = 1}^{30} {\mu_{i} }$ is the sum of the product-group means, that is, $\sum\nolimits_{i = 1}^{30} {x_{i} } .$ From (2) it is possible to see that for the 95% confidence interval for$\sum\nolimits_{i = 1}^{30} {\mu_{i} },$ an estimate $\hat{S}$ is needed for the unknown S. For this, variance estimates s ² _i are required for unknown variances σ ² _i . These are obtained by applying the standard formula using information on supply and use by product group.

$$ s_{i}^{2} = \frac{{\left( {x_{c} - \frac{{x_{c} + (z_{c} + y_{c} + e_{c} )^{2} }}{2}} \right) + \left( {(z_{c} + y_{c} + e{\kern 1pt}_{c} ) - \frac{{x_{c} + (z_{c} + y_{c} + e_{c} )^{2} }}{2}} \right)}}{2 - 1} $$

(3)

Now the 95% confidence interval for the unknown total output $\sum\nolimits_{i = 1}^{30} {\mu_{i}}$ is obtained by plugging the calculated values $\sum\nolimits_{i = 1}^{30} {x_{i} }$ and $\hat {S}=\sqrt{s_{1}^{2}+s_{2}^{2}+\ldots+s_{30}^{2}}$ into (2).

Appendix 2: Basic input–output equations

The structure of description is based on Miller and Blair (1985).

2.1 Static open input–output model

Mathematically, the basic input-output assumptions are as follows. Denote the observed monetary flows from industry i to industry j by z _ij . Thus, if the economy is divided into n industries, and if the total output of industry i is denoted by X _i and the total final demand for industry i’s output by Y _i , the equation can be written as

$$ X_{i} = z_{i1} + z_{i2} + \ldots + z_{ii} + \ldots + z_{in} + Y_{i} . $$

(1)

The above equation thus describes the distribution of industry i’s output. It is the sum of all of its inter-industry sales as well as of sales to final demand. Each industry has a similar equation, reflecting the sales of its output. Thus,

$$ \begin{array}{l} \begin{aligned} {X_{1} = z_{11} + z_{12} + \ldots + z_{1i} + \ldots + z_{1n} + Y_{1} } \end{aligned} \hfill \\ \begin{aligned} {X_{2} = z_{21} + z_{22} + \ldots + z_{2i} + \ldots + z_{2n} + Y_{2} } \end{aligned} \hfill \\ \vdots \begin{aligned} \hfill \\ X_{i} = z_{i1} + z_{i2} + \ldots + z_{ii} + \ldots + z_{in} + Y_{i} \hfill \\ \end{aligned} \hfill \\ \vdots \begin{aligned} \hfill \\ X_{n} = z_{n1} + z_{n2} + \ldots + z_{ni} + \ldots + z_{nn} + Y_{n} \hfill \\ \end{aligned} \hfill \\ \end{array}. $$

(2)

The input–output theory assumes that for one unit of every industry’s output a fixed amount of input of each kind is required. Thus, the inter-industry flows from industry i to industry j depend entirely on the total output of industry j for that same period of time. These fixed input relationships between industries are obtained by dividing the entries in the column by the total output of the consuming industry. In its mathematical form, the computation of the input coefficient matrix is thus

$$ a_{ij} = \frac{{z_{ij} }}{{X_{j} }}, $$

(3)

where z _ij stands for the flow of input from industry i to industry j, and X _j is the total output of industry j. The corresponding output coefficients (b _ij ) describe the proportion of industry i’s total output sold to industry j. In its mathematical form

$$ b_{ij} = \frac{{z_{ij} }}{{X_{i} }}. $$

(4)

Now Eq. 2 can be rewritten as

$$ \begin{array}{l} \begin{aligned} {X_{1} = a_{11} X_{1} + a_{12} X_{2} + \ldots + a_{1i} X_{i} + \ldots + a_{1n} X_{n} + Y_{1} } \end{aligned} \\ \begin{aligned} {X_{2} = a_{21} X_{1} + a_{22} X_{2} + \ldots + a_{2i} X_{i} + \ldots + a_{2n} X_{n} + Y_{2} } \end{aligned} \\ \vdots \begin{aligned} \\ X_{i} = a_{i1} X_{1} + a_{i2} X_{2} + \ldots + a_{ii} X_{i} + \ldots + a_{in} X_{n} + Y_{i} \\ \end{aligned} \\ \vdots \begin{aligned} \\ X_{n} = a_{n1} X_{1} + a_{n2} X_{2} + \ldots + a_{ni} X_{i} + \ldots + a_{nn} X_{n} + Y_{n} \\ \end{aligned} \\ \end{array} . $$

(5)

Equation (5) can be solved in terms of X by moving all the X terms to the left-hand side. Thus,

$$ \begin{array}{l} \begin{aligned} {X_{1} - a_{11} X_{1} - a_{12} X_{2} - \ldots - a_{1i} X_{i} - \ldots - a_{1n} X_{n} = Y_{1} } \end{aligned} \\ \begin{aligned} {X_{2} - a_{21} X_{1} - a_{22} X_{2} - \ldots - a_{2i} X_{i} - \ldots - a_{2n} X_{n} = Y_{2} } \end{aligned} \\ \vdots \begin{aligned} \\ X_{i} - a_{i1} X_{1} - a_{i2} X_{2} - \ldots - a_{ii} X_{i} - \ldots - a_{in} X_{n} = Y_{i} \\ \end{aligned} \\ \vdots \begin{aligned} \\ X_{n} - a_{n1} X_{1} - a_{n2} X_{2} - \ldots - a_{ni} X_{i} - \ldots - a_{nn} X_{n} = Y_{n} \\ \end{aligned} \\ \end{array} . $$

(6)

For reasons of simplicity, the above equations are often presented in matrix form. Thus,

$$ A = \left[ {\begin{array}{llllllll} {a_{11} } & {a_{12} } & \cdots & {a_{1i} } & \cdots & {a_{1n} } \\ {a_{21} } & {a_{22} } & \cdots & {a_{2i} } & \cdots & {a_{2n} } \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {a_{n1} } & {a_{n2} } & \cdots & {a_{ni} } & \cdots & {a_{nn} } \\ \end{array} } \right], \quad X = \left[ {\begin{array}{l} {X_{1} } \\ {X_{2} } \\ \vdots \\ {X_{n} } \\ \end{array} } \right],Y = \left[ {\begin{array}{l} {Y_{1} } \\ {Y_{2} } \\ \vdots \\ {Y_{n} } \\\end{array} } \right] $$

(7)

and let I denote the n X n identity matrix. Equation 6 then can be rewritten simply as

$$ {X} - {AX} = {Y}, $$

(8)

where matrix A stands for the input-output coefficients of the intermediate use of domestic products, X is the vector of output and Y is the net final demand. In the basic matrix,

$$ \begin{array}{*{20}c} {X - AX = Y} \\ {(I - A)X = Y} \\ {X = (I - A)^{ - 1} Y} \\ \end{array} , $$

(9)

where (I - A)^–1is the so-called Leontief inverse.²⁰ This basic model is also called the static open input–output model, and it forms the basis of most input–output analysis.

2.2 Production–consumption model

In order to arrive at the production–consumption model, the exogenous household sector is moved to the technically interrelated input–output coefficient table. This is also known as closing the model in relation to households, and was introduced by Leontief in his early work.

In practice, the household row and column are added to the coefficient table at the bottom and to the right respectively. The flow of money to households (wages) is shown in the last row ((n + 1)st row), representing at the same time the labour input required for the industry to produce its products. The money flow from the households is shown in the last column of the coefficient table ((n + 1)st column), representing the purchase of goods by the household sector. In its mathematical form the flow Eq. 1 now becomes

$$ X_{i} = z_{i1} + z_{i2} + \ldots + z_{ii} + z_{in} + z_{i,n + 1} + Y_{i} * $$

(10)

where the last term represents the remaining final demand for sector i’s output after the closing of the model in terms of households. In addition, the total value of households’ sales of labour services to other industries is presented as a new equation

$$ X_{n + 1} = z_{n + 1,1} + z_{n + 1,2} + \ldots + z_{n + 1,i} + \ldots + z_{n + 1,n} + z_{n + 1,n + 1} + Y_{n + 1} *. $$

(11)

The element z _{n+1, n+1} represents the household purchase of labour services (such as paid household work), while the last term on the right includes payments to government employees, for example. The household input coefficients are calculated in the same way as in the open model. The value of industry j’s purchase of labour is divided by the total output of the industry j. The “household industry’s” coefficients are obtained by dividing the purchase of industry i’s product by the total household consumption. Thus, the ith equation in (5) becomes simply

$$ X_{i} = a_{i1} X_{1} + a_{i2} X_{2} + \ldots + a_{ii} X_{i} + \ldots + a_{in} X_{n} + a_{i,n + 1} X_{n + 1} + Y_{i} * $$

(12)

and the new equation relating household output to the total output of industries becomes

$$ X_{n + 1} = a_{n + 1,1} X_{1} + a_{n + 1,2} X_{2} + \ldots + a_{n + 1,i} X_{i} + \ldots + a_{n + 1,n} X_{n} + a_{n + 1,n + 1} X_{n + 1} + Y_{n + 1} * $$

(13)

Similarly, the ith term in Eq. 6 becomes

$$ - a_{i1} X_{1} - a_{i2} X_{2} - \ldots + (1 - a_{ii} )X_{i} - \ldots - a_{in} X_{n} - a_{i,n + 1} X_{n + 1} = Y_{i} * $$

(14)

and for the household sector, rewriting Eq. 12 gives

$$ - a_{n + 1,1} X_{1} - a_{n + 1,2} X_{2} - \ldots - a_{n + 1,i} X_{i} - \ldots - a_{n + 1,n} X_{n} + (1 - a_{n + 1,n + 1} )X_{n + 1} = Y_{n + 1} * $$

(15)

Finally, if the (n + 1 X n + 1) coefficient matrix is nonsingular, the unique solution can again be found by using an inverse matrix. Thus, if we denote the new coefficient matrix A _new, the (n + 1)-element column vector of gross output X _new, and the (n + 1)-element vector of final demand, including that for the output of households Y _new, we can rewrite the solution of the open model simply as

$$ X_{\rm new} = (I - A_{\rm new} )^{ - 1} Y_{\rm new} . $$

(16)

Thus, with the above model it is possible to estimate the impact of a change in the final demand on the economy as a whole as well as on individual industries. It further takes into account the relationship between industries and households in terms of income and consumption. The impact (ΔX) is then written simply as the difference between the original output and the new calculated output

$$ \Updelta X = X - X_{\rm new} . $$

(17)