MODELING INPUT-OUTPUT IMPACTS WITH SUBSTITUTIONS IN THE HOUSEHOLD SECTOR: A NUMERICAL EXAMPLE

Transcription

1 MDELNG NPUT-UTPUT MPACTS WTH SUBSTTUTNS N THE HUSEHLD SECTR: A NUMERCAL EXAMPLE Peter Gordon Ralph & Goldy Lewis Hall 321 School of Policy, Planning, and Development University of Southern California Los Angeles, CA pgordon@usc.edu Phone: (213) JiYoung Park Von Kleinsmid Center 382 School of Policy, Planning, and Development University of Southern California Los Angeles, CA jiyoungp@usc.edu Phone: (213) Harry W. Richardson Ralph & Goldy Lewis Hall 212 School of Policy, Planning, and Development University of Southern California Los Angeles, CA pgordon@usc.edu Phone: (213)

2 Abstract Although there have been many elaborations of the basic input-output approach, including multi-regional models, dynamic models, models with variable coefficients, supply-side models, etc., these approaches all have the same limitation. The fixed-coefficients production function assumptions ignore substitutions in response to price changes that can be expected to accompany most shocks skipping over the heart and soul of market economics. This research note suggests a simple approach to estimating new technical coefficients matrices after a shock so that short-term substitution effects can be studied. Given a reduction in income (as reflected in the value added row), households are likely to make substitutions, reducing their final demand by less than the application of the base-year - coefficients would indicate. f this can be observed, the application of RAS procedures can generate an appropriately modified A matrix. Due to some well known limits in applying the traditional RAS approach, we reformatted it and suggest a new economic model that can link coefficient adjustments to degrees of a priori substitutability and complementarity. Based on this resolution, we look forward to detailed studies of specific coefficients and how they evolve over the short term. JEL Classification: C67, D57, R19 Key words: nput-output, RAS, substitution effects. 2

3 . ssues The application of input-output (-) models to economic impact studies is widespread and long established. A Google-scholar search finds 677,000 - hits, and 87,500 when the phrase is qualified with the words economic impact. n most cases, the applications have changed little in over 50 years. The typical what if scenario is: f the vector of final demands (multiplicand) is perturbed in a particular way, what happens to the vector of total outputs in light of the Leontief inverse (multiplier)? f course, there are many elaborations of the basic approach, including multi-regional models, dynamic models, models with variable coefficients, supply-side models, etc. (Hewings, 1985; Miller and Blair, 1985). All these approaches have the same limitation. The fixed-coefficients production function assumptions ignore substitutions in response to price changes skipping over the heart and soul of market economics. The defense against this criticism is that the analysis is short-term with applications before agents have been able to discover the relevant substitution options. The point of this note is simple. f the analyst can observe simultaneous changes in the final demand column and the value added row, he can use this information to study the economic adjustments reflected in the an impacted technical coefficients matrix; there are new multiplier values that reflect real world substitutions, for example, ones made by the household sector. Facing a decline in final demand, firms may cut their wage bills asymmetrically; this may result in changes in household spending behavior that changes the multiplier impact.. Numerical Example Given a reduction in income (as reflected in the value added row), households will attempt to make substitutions, reducing their final demand by less than the application of the base-year - 3

4 coefficients would indicate. f this can be observed, the application of RAS procedures can generate an appropriately modified A matrix. Consider the following example, built upon Miller and Blair s (1985, p. 15) well known and highly simplified 2x2 example: Table 1. Example of 2x2 flow matrix x 1 x 2 F X x x V X Source: Miller and Blair (1985), p.15 where x = industry sector i. i F = total final demand column vector V = total value added row vector X = total output column vector. X = total input (outlay) row vector. 150 First, denote the 2x2 interindustry flow matrix as Z ( 200 we use the open model. Then, the matrix A is calculated as 500 ). Given the 2x2 flow matrix, 100 = ˆ A Z ( X ) (1) where, the hat on the vector X indicates a diagonal matrix transformed from X and hence Xˆ = Xˆ, i.e. from the given matrix 1 0 ˆ ( X ) 1000 = ˆ ( X ) Xˆ and Xˆ = = 4

5 From the example A and hence the Leontief inverse matrix, ( A) = Also, based on X, the matrix B is calculated as = Xˆ B ( ) 1 Z (2) Hence, we derive the so-called Goshian inverse matrix, ( B) Assume the observed exogenous losses of the value added vector, Δ V = ( ), then t (ΔX ) ( ) = ΔV ( B) = (3) where the superscript t represents the transpose of the matrix. From the losses of total output via the Goshian inverse matrix, a newly derived total input column vector ( NX ) is easily calibrated as NX = X + ΔX (4)

6 Then, the final demand losses in next period can be obtained from the following equation, using the proportion of the final demand to total output in the previous period. NF = Pˆ X NX (5) where the proportionate column vector, P = ˆ 0.35 X ( X ) F = From equation (5), the final demand losses between two periods are easily calculated as Δ NF = NF - F (6) 342 The observed exogenous losses of final demand, or Δ F implied by the observed ΔV from equations (3) to (6), that is, are assumed to be smaller than those Δ NF, because households experiencing income declines will seek substitutions that mitigate negative consumption impacts. For example, they may buy more chicken and less beef. Hence, Δ F is assumed to be and the observed final demand in the next period is easily calculated as F(1) = F + ΔF

7 30. (7) 1450 The diagonal matrix of R according to the traditional RAS approach can be obtained from the ratio of F(1) to NF as R = [Fˆ (1)] ( NF) (8) Similar to the procedures shown in equations (3) to (7), the diagonal matrix of S can be obtained. First, based on the observed exogenous losses of final demand, and NV are obtained as follows: Δ F, the row vectors ΔX,, NX NV = NX PˆX =[ X + ΔX ] PˆX = [ X + {( A) ΔF} ] t PˆX t = ( ) = [( ) + ( ) ] = ( ) = ( ). (9) 7

8 where the proportionate row vector P X = V ˆ ( X ) = ( 1400) = ( ) (10) Because the given value added in the next period V (1) = V + Δ V = + ( ) ( ) = ( ), the diagonal matrix S from RAS can be obtained from the ratio of V (1) to NV as S = [Vˆ(1)] ( NV ) (11) Therefore, the new technical coefficients matrix (= substitution effects are obtained from RAS as follows: A(1) ) for the next period reflecting A(1) = RAS (12) To verify whether the new technical coefficients matrix reflects substitution effects, we compare results based on the same final demand losses. f the final demand losses are the same 5 as the given vector ΔF then the new total input losses, ΔX (1), will be 250 t (ΔX (1)) = ( A(1)) ΔF 8

9 (13) From the results of the total inputs reductions shown in equation (9), ΔX = ( ). Therefore, the total output impact of Σ X Δ X,, is calculated as X Σ = ΔX u 2 = (14) where t u N is the N-element unit row vector and superscript t is the transpose of u, i.e. u (1) X Similarly, Σ = X ( 1) u = (15) Δ 2 The output impact of the observed Δ F multiplied by the modified multiplier A(1) is smaller, vs This is as expected. Substitutions result in multiplier modifications that lead to lesser total output impacts. This confirms standard economic principles: substitutions dampen impacts. The recognition that Δ V can differ from the ΔF that the base-year coefficients would predict and that these differences can be estimated suggests that higherquality and more plausible findings and a more acceptable version of the - model that avoids the standard zero- adjustment consequences are within our reach.. Remarks However, in actual applications, the two vectors Δ F and ΔV are not easy to observe for all industry sectors after a disaster. n that case, we cannot apply the very simple approach 9

10 suggested above because the traditional RAS application is based on the availability of observed vectors ΔF and ΔV, and hence our suggested approach only amounts to an ideal model. We have developed two steps to resolve this problem. The first step is based on the idea that we do not need to gather all data for all industries in an actual event, because only a limited number of industry-specific changes of final demand and value added are directly affected by a disaster, such as, say, a port closure. ur impact analysis focuses on estimating indirect impacts due to the targeted direct losses. That is, other indirectly related industry sectors can be assumed to experience little or no changes in final demand and value added. The updated technical coefficients matrix just after an event is created under the assumption that untargeted industries should be indifferent to the changes between ex- and ante-events, and then, other new coefficients matrices would be updated by period. The second remedy is to develop an advanced approach different from the difference approach of traditional RAS. Although we might gather data on the two vector, including only changes in targeted sectors and no changes in other sectors, it is still hard to apply the traditional version of the RAS approach, because the denominators, NF and NV in equations (8) and (11) respectively, would include many zeros and hence could not be run. Therefore, we modified the difference approaches of the traditional RAS application with a new proportional approach. The proportional approach constrains the changes in untargeted industry sectors to one, reflecting no changes in final demand and value added vectors, as discussed in the first suggestion. The starting values of the proportional approach, however, would not be applied to other periods except the first period, when a disaster occurs. The two new impacted vectors would be estimated after the second period. We realize that many economic agencies, e.g. Bureau of Economic Analysis (BEA) update coefficients every year based on the traditional RAS approach, but our suggested approach only focuses on direct and indirect impact analysis due to a specific event, by month or by quarter instead of by year. This research note simply suggests an approach to estimating new technical coefficients matrices after a disaster so that short-term effects can be studied. 10

11 While our suggestion starts from an ideal data set by which we can gather all the information we might want, we were able to reformat our approach by suggesting two additional discussions to resolve the problem. Based on this resolution, we look forward to detailed studies of specific coefficients and how they evolve over the short term. We plan to test hypotheses that link coefficient adjustments to degrees of a priori substitutability and complementarity. References Hewings GJD. Regional nput-utput Analysis. Beverly Hills, CA: Sage Publications, nc.; 1985 Miller RE, Blair PD. nput-utput Analysis: Foundations and Extensions. Englewood Cliffs, NJ: Prentice-Hall, nc.;