On the Sensitivity of Impact Estimates for Fixed Ratio Assumptions

Abstract

Firms react to shortages in the supply of their inputs by looking for substitutes. We investigate the impact of finding such substitutes on estimates of the size of regional and national disaster impacts. To investigate this issue, we use the German multiregional supply-use table (MRSUT) for 2007, together with data on the direct impacts of the 2013 heavy floods of the German Elbe and the Danube rivers. Our analysis starts with a non-linear programming model that allows for maximum substitution possibilities. In that case there are little to no indirect damages in the directly affected regions, whereas negative indirect impacts of a magnitude of 5–7% and of up to 34% of the direct impact occur in other German regions and abroad, respectively. Adding the increasingly less plausible fixed ratios commonly used in standard Type I and extended Type II multiregional input-output and MRSUT models, results in (1) substantial increases in the magnitude of negative indirect impacts and (2) a significant shift in the intra-regional versus interregional and international distribution of these impacts. Our conclusion is that both demand-driven and supply-driven input-output and supply-use models tend to grossly overstate the indirect damages of negative supply shocks, which are part and parcel of most disasters.

Appendix

1.1 A.1 Impact of Adding Supply-Driven Fixed IO and SU Ratios to the NLP Model

1.1.1 A.1.1 A Supply-Driven Multiregional Supply-Use Model

The secondary question we investigate here, is whether adding the fixed ratios assumed in the supply-driven IO model produces a different outcome compared to adding the ratios of the demand-driven IO model, as discussed in the main text. First and foremost, it needs to be reiterated that the original quantity interpretation of the supply-driven IO model (Ghosh 1958) is generally considered extremely implausible (Oosterhaven 1988, 2012; Dietzenbacher 1997; DeMesnard 2009). In sum: the single homogeneous input assumption of this model implies that cars may drive without gasoline and factories may work without labour. Nevertheless, we discuss it here because, especially, natural disasters primarily constitute a shock to the supply-side of the economy, and because the name of this model suggests that it might be suited to simulate the quantity impacts of supply shocks (see Crowther and Haimes 2005, for at least one disaster application).²

As our base model is calibrated on a use-regionalized MRSU table (labelled as purchase only by Oosterhaven 1984, who describes a whole family of MRSUTs), we first need to formulate a supply-driven MRSU model that fits these detailed data (see Table 9.1). DeMesnard (2009) already formulated a supply-driven SU model for a closed economy when he discussed the unfitness of the commodity technology assumption while constructing a demand-driven SU model. Here, we will extend his SU model to fit to a use-regionalized MRSUT. It will be the mathematical mirror of the existing demand-driven MRSU model based on a use-regionalized MRSUT (Oosterhaven 1984). For briefness sake, we put the model directly in matrix notation.

First, any change in the supply of exogenous primary inputs w´ or endogenous intermediate inputs i´U of any regional industry leads to an equally large change in its total input x´:

$$ {\mathbf{x}}^{\acute{\mkern6mu}}={\mathbf{i}}^{\acute{\mkern6mu}}\mathbf{U}+{\mathbf{w}}^{\acute{\mkern6mu}} $$

(9.17)

where the vectors and matrices follow the layout of Table 9.1. In Eq. (9.17) all inputs are treated as perfect substitute for one another, just as the demand-driven model assumes that all outputs are perfect substitutes for one another.

Second, any change in total inputs x´ leads to an equally large change in the total supply of products by that industry Vi, while the latter are produced in a fixed product mix:

$$ \mathbf{V}=\hat{\mathbf{x}}\ \mathbf{M} $$

(9.18)

where ${m}_{ip}^r\in \mathbf{M}$ is calculated from the base-year MRSUT as ${v}_{ip}^{r, ex}/{x}_i^{r, ex}$. Note that Eq. (9.18) may technically be only realistic in case of some chemical industries. For other industries it must be based on the wish to service all purchasers proportionally, irrespective of their demand, which consequently is assumed to be perfectly elastic, just as the demand-driven model assumes supply to be perfectly elastic (Oosterhaven 1996, 2012).

Third, any change in the regional supply of any product i´V leads to an equally large change in the total supply g of that product:

$$ {\mathbf{g}}^{\acute{\mkern6mu}}={\mathbf{i}}^{\acute{\mkern6mu}}\mathbf{V} $$

(9.19)

Fourth, any change in the total regional supply of any product leads to a proportional increase (i.e., with fixed allocation coefficients) in the use of that product by all industries U and the use of that product by all final demand categories Y. Here a distinction between technical allocation coefficients B and spatial allocation coefficients T^g clarifies the multi-regional nature of the extension of the closed single-region SU model:

$$ \mathbf{U}=\hat{\mathbf{g}}\ \mathbf{B}\otimes {\mathbf{T}}^{\boldsymbol{g}}\ \mathrm{and}\ \mathbf{Y}=\hat{\mathbf{g}}\ {\mathbf{B}}^{\boldsymbol{y}}\otimes {\mathbf{T}}^{\boldsymbol{g}\boldsymbol{y}} $$

(9.20)

where ⊗ indicates a cell-by-cell multiplication. The technical allocation ratios (i.e., technical output or sales coefficients) ${b}_{pj}^{r\ast}\in \mathbf{B}$ are calculated from the base-year MRSUT as ${b}_{pj}^{r\ast}={u}_{pj}^{r\ast, ex}/{g}_p^{r, ex}$, with the ${b}_{py}^{r\ast}\in {\mathbf{B}}^{\boldsymbol{y}}$calculated analogously. The trade destination ratios${t}_{pj}^{r\ast}\in {\mathbf{T}}^{\boldsymbol{g}}$ are calculated as ${t}_{pj}^{rs}={u}_{pj}^{rs, ex}/{u}_{pj}^{r\ast, ex}$, with the ${t}_{py}^{rs}\in {\mathbf{T}}^{\boldsymbol{gy}}$calculated analogously. Note again the importance of the assumption of a perfectly elastic demand in all markets, as opposed to the assumption of a perfectly elastic supply in the demand-driven IO model.

Appropriate sequential substitution leads to, respectively, the following base equation and subsequent solution for total industry input:

$$ {\mathbf{x}}^{\acute{\mkern6mu}}={\mathbf{x}}^{\acute{\mkern6mu}}\ \mathbf{M}\ \mathbf{B}\otimes {\mathbf{T}}^{\boldsymbol{g}}+{\mathbf{w}}^{\acute{\mkern6mu}}\Rightarrow {\mathbf{x}}^{\acute{\mkern6mu}}={\mathbf{w}}^{\acute{\mkern6mu}}{\left(\mathbf{I}-\mathbf{M}\ \mathbf{B}\otimes {\mathbf{T}}^{\boldsymbol{g}}\right)}^{-1} $$

(9.21)

In Eq. (9.21) both coefficient matrices M and B ⊗ T^g may be rectangular, but their product M B ⊗ T^g is square and has an industry-by-industry dimension. G = (I − M B ⊗ T^g)⁻¹ represents the multi-regional generalization of the Ghosh-inverse.

The solution for total product supply may, then, be calculated simply by means of:

$$ {\mathbf{g}}^{\acute{\mkern6mu}}={\mathbf{x}}^{\acute{\mkern6mu}}\mathbf{M} $$

(9.22)

1.1.2 A.1.2 The Impact of Adding Supply-Driven Fixed Ratios to the Base Model

The above supply-driven MRSU model, specifies the fixed ratio assumptions that we will sequentially add to the base model (9.1)–(9.5).

First, the fixed product mix ratios by regional industry:

$$ {v}_{ip}^r={x}_i^r{m}_{ip}^r,\forall i,p,r. $$

(9.23)

where ${m}_{ip}^r$ = share of product p in the output of regional industry i, with ${\sum}_p{m}_{ip}^r=1.$

Second, the fixed industry and final demand allocation ratios for regional product supply, now written out in full:

$$ {\sum}^s{u}_{pj}^{rs}={g}_p^r{b}_{pj}^{r\ast},{\sum}^s{y}_p^{rs}={g}_p^r{h}_p^{r\ast},{\sum}^s{f}_p^{rs}={g}_p^r{d}_p^{r\ast}\ \mathrm{and}\ {e}_p^r={g}_p^r{k}_p^r,\forall p,j,r $$

(9.24)

where ${b}_{pi}^{r\ast},{h}_p^{r\ast}\ \mathrm{and}\ {d}_p^{r\ast}$denote the technical allocation coefficients, i.e., sales regardless of their spatial destination per unit of regional supply as calculated from the rows of the MRSUT. The ${k}_p^r$ denote foreign export allocation coefficients, which do not need to be added separately as ${\sum}_i{b}_{pi}^{r\ast}+{h}_p^{r\ast}+{d}_p^{r\ast}+{k}_p^r=1,\forall p,r$, holds because of Eq. (9.2) in the main text.

Third, the cell-specific fixed intermediate and final output trade destination ratios, now again written out in full:

$$ {t}_{pi}^{rs}={u}_{pi}^{rs}/{u}_{pi}^{r\ast},{t}_{py}^{rs}={y}_p^{rs}/{y}_p^{r\ast},{t}_{pf}^{rs}={f}_p^{rs}/{f}_p^{r\ast},\forall r,s,p,i $$

(9.25)

where ${t}_{pi}^{rs},{t}_{py}^{rs}\ \mathrm{and}\ {t}_{pf}^{rs}$ represent the use of product p from region r per unit of total use of product p by i, y and f in region s. These shares are calculated from the rows of the MRSUT, with ${\sum}^s{t}_{pi}^{rs}=1$ by definition. The column-specific version of (9.25), which we do not use, as we have detailed cell-specific MRSUT information, would assume that the trade destination ratios for all different products p from region r are equal (cf. the FI multiregional SUT in Oosterhaven 1984).

Note that, from a calculation point of view, it is not efficient to add both (9.24) and (9.25) to the base scenario (9.1)–(9.5). It is more efficient to combine them, which gives:

$$ {u}_{pj}^{rs}={g}_p^r{b}_{pj}^{rs},{y}_p^{rs}={g}_p^r{h}_p^{rs}\ \mathrm{and}\ {f}_p^{rs}={g}_p^r{d}_p^{rs},\forall p,j,r,s $$

(9.26)

and to then add (9.26), with its fixed interregional allocation coefficients, to the base scenario instead.

Tables 9.6, 9.7, 9.8, 9.9 describe the impact of this sequential adding of fixed ratios to the base model. The first rows, again, show the outcomes of the base model as defined by the Eqs. (9.1)–(9.5), while the second to fourth rows show the outcomes for the sequential adding of fixed product mix ratios by industry, fixed technical allocation ratios and, finally, fixed trade destination ratios. As opposed to Tables 9.1, 9.2, 9.3, 9.4 in the main text, which include the impacts on foreign exports, Tables 9.6, 9.7, 9.8, 9.9 include the impacts on foreign imports. The reason is that adding input ratios in the main text fixes the structure of the columns of the MRSUT, leaving exports relatively unconstrained, whereas adding output ratios in the Appendix fixes the structure of the rows of the MRSUT, leaving imports relatively unconstrained.

Table 9.6 Indirect impacts in permilles of direct gross output impact of Danube floods, while adding fixed ratios to the base model

Table 9.7 Indirect impacts in permilles of direct gross output impact of Elbe floods in Sachsen, while adding fixed ratios to the base model

Table 9.8 Indirect impacts in permilles of direct gross output impact of Elbe floods in Sachsen-Anhalt, while adding fixed ratios to the base model

Table 9.9 Indirect impacts in permilles of direct gross output impact of Elbe floods in Thüringen, while adding fixed ratios to the base model

As to the impact of adding fixed product mix ratios per regional industry, it can be observed that the intra-regional indirect impact in all four regions increase by at least 11% (Bayern), whereas the interregional impacts change less and show a mixed behaviour. On the one hand, the interregional impacts in Bayern and Thüringen increase slightly by about 2%, while, on the other hand, a slight decrease 0.6% and 2% can be observed for Sachsen and Sachsen-Anhalt, respectively. The drop of imports from foreign countries changes uniformly across the four regions, whereby the largest drop can be observed in Bayern (1.5%) and the lowest in Sachsen (0.3%).

When fixed technical allocation coefficients are added on top of the fixed product mix ratios, the change in the indirect impacts is more uniform across the four regions. It can be observed that the intra-regional impacts increase substantially and are at least about 2.5 times (Sachsen and Sachsen-Anhalt) up to 10 times (Thüringen) larger than before. At the same time, the indirect impacts in all of Germany decrease significantly by about 49% for Bayern to about 75% for Sachsen. Separating industries that experience a positive indirect impact from those with a negative impact (second and third column), shows that this is due to an decrease in the negative indirect impacts combined with a substantial increase in the positive indirect impacts in the rest of Germany. In contrast, the drop of imports again increases uniformly, but is much larger compared to the case where only fixed product mix ratios by industry are imposed. As before, the largest changes apply to Bayern (27%) and the lowest to Sachsen (11%).

Adding fixed spatial allocation coefficients, finally, leads to a substantial increase in the indirect impacts, both, intra-regionally and interregionally. The only exception is Sachsen-Anhalt, where the intra-regional impacts decrease slightly. In the other three regions, the intra-regional impacts become about 3 (Bayern) to 7 (Sachsen) times larger compared to the case where only fixed technical allocation ratios are added. Regarding the interregional indirect impacts on the rest of Germany our outcomes show that positive indirect impacts vanish almost completely across all regions, while, at the same time, negative indirect impacts become 2.8 (Sachsen) to 3.4 (Thüringen) times larger than before. As in the cases before, adding fixed spatial allocation ratios again leads to a further increase in the drop of imports from foreign countries across all four regions and again this further increase is larger than before. However, the rank-order of regions changes, as the by far largest increase can now be observed for Thüringen (48%) followed by Bayern and Sachsen-Anhalt (both about 37%) and Sachsen (32%).

Comparing the indirect impacts across all of Germany shows that the total indirect impacts are relatively close to each other, ranging between about 13% to 18% of the direct impact. However, the extent to which these indirect impacts occur intra-regionally and interregionally is very different across the regions. The largest share of intra-regional impacts in nation-wide impacts of 20% can be observed for Bayern, whereas the lowest share of only 0.35% is observed for Sachsen-Anhalt.