Agricultural Trade Multipliers: Methodology

Base Year Open Model Estimation

The following procedure can be used to estimate employment, output, and/or income related to exports of agricultural commodities when an Input/Output (I/O) transaction table is available.

Income Generation

Since income (or gross domestic product) measures, in an aggregated form, the sum of value added in various I/O sectors, then

(1)

                n
Output = ∑ X
               j=1

                n
Income = ∑ Vj
               j=1

where Vj is value added in sector j. Under an I/O structure, value added is a fixed proportion of output, so that income can be written in a matrix form as:

(2)

Output = X = (I-A)^-1 F

Income = Y = vX = v (I-A)^-1 F

Where X	= an n x 1 vector of sector outputs
(I-A)-1	= an n x n I/O total requirements matrix
F	= an n x 1 vector of final demand for agricultural exports
Y	= an n x 1 vector of income originating from each sector of the economy due to agricultural exports
v	= an n x n diagonal matrix of value added per dollar of sector output coefficients

Employment Generation

Using the above notations, employment in each sector of I/O industries is derived as:

(3)

E = L (I-A)^-1 F

Where (I-A)-1 and F are as previously defined
L	= an n x n diagonal matrix of civilian employment coefficients per dollar of sector output
E	= an n x 1 vector of sector employment needs related to the level of agricultural exports defined in vector F

Nonbase Year Estimation

To estimate output, income, and employment multipliers related to exports for years beyond the published I/O tables, one must work with less information because current year (I-A)^-1, v, and L are unavailable. Yet, there are observable changes that can be incorporated into the analysis, such as changes in labor productivity and in the sectoral composition of final demand. Changes in the composition of final demand may also require changes in industry output requirements, which, in turn, change interindustry demand. Likewise, increases in labor productivity imply that the same output can be produced with a smaller workforce or that more output can be produced with the same size workforce.

Changes in the yearly commodity composition of agricultural exports are available from the Foreign Agricultural Trade of the United States (FATUS) calendar year tables.

Nonbase year income is estimated through a modification of equation (2).

(4)

Y = qT

Where T	= v(I-A)-1 F'
q	= an n x n diagonal matrix of output originating price deflators
F'	= an n x 1 vector of current year exports

 

Nonbase year employment is estimated through a modification of equation (3).

Labor productivity changes in farming and in nonfarm sectors are available from USDA and the U.S. Department of Labor, respectively. Therefore, equation (3) is modified to incorporate the effect of productivity change in the generation of employment.

(5)

E = pW

Where p	= an n x n diagonal matrix showing the ratio of base year labor productivity to current year productivity
W	= L(I-A)-1 F'

 

Base Year Partially Closed Model Estimation

First, the Miyazawa process of income formation for the base year (2002) is derived as:

(6)

Where A	= a matrix of technical production coefficients (n x n)
V	= a matrix of household income payment coefficients by sector (1 x n)
C	= the coefficients of household consumption (n x 1)
M	= 435 square block matrix of 434 intermediate industry sectors and one household
n	= 434

Alternatively, the Miyazawa model can be expressed mathematically as follows:

X = AX + CVX + F

Where X	= an n x 1 vector of sector outputs
A	= a matrix of technical production coefficients (n x n)
C	= the coefficients of household consumption (n x 1)
V	= a matrix of household income payment coefficients by sector (1 x n)
F	= an n x 1 vector of final demand minus personal consumption

The solution to the Miyazawa model can be stated in the following way:

(7)

X = (I – A – CV)^-1 x F

or

X = B(I-CVB)^-1 x F

Where B=(I-A)-1	is the standard Leontief inverse
(I-CVB)-1	is the Miyazawa "subjoined inverse" or (I-M)-1 as described in equation (6)

Sectoral output associated with agricultural exports for the base year is derived as:

(8)

X' = (I-M)^-1 x F'

Where (I-M)-1	is as previously defined
X'	= an n' x 1 vector of sector outputs
F'	= an n' x 1 vector of agricultural exports
n'	= 435 (see below)

Under an I/O structure, value added is a fixed proportion of output, so that income can be written in a matrix form as:

(9)

Income = v x X' = v x (I-M)^-1 x F'

Where X',	(I-M)-1, and F' are as previously defined
v	= an n' x n' diagonal matrix of 'other' value added [value added not included in the endogenized household rows, per dollar of sector output] coefficients

Using the previous notation, employment in each sector can be derived as:

(10)

E = L x (I-M)^-1 x F'

Where (I-M)-1	and F' are as previously defined
L	= an n' x n' diagonal matrix of civilian employment coefficients per dollar of sector output
E	= an n' x 1 vector of sector employment needs, e j' s for meeting the total output required to satisfy activities related agricultural exports

Estimates of household expenditures are derived from the benchmark 2002 Input-Output Personal Consumption Expenditures data. Some of the data about household incomes are from unpublished sources at the Bureau of Economic Analysis, National Income and Product Accounts and incorporated into the endogenous household value-added row. As in steps (4) and (5) of the open model, we apply sectoral price deflators and labor productivity indices to make the 'constant dollar' measures of final demands (exports), in years other than in the base year (2002).

Further detail:

Export output multipliers derived from this partially closed model are the sum of the entire column of that export-producing sector within the (I-M)^-1 matrix. That is:

              n'
Output = ∑ X where n' = 435
             j=1

Some researchers in this field of economics feel the proper derivation of output multipliers from a partially closed model is:

              n
Output = ∑ X where n = 434     (using this model as an example)
             j=1

This is exclusive of the n + 1 row of the household sector. The household effects on output are presented as a separate multiplier or subset of the total. We present the first example.