Now we can easily combine the economic-base and interindustry concepts to yield a quick feeling for multi-regional models. While we stop with this introduction, extensions to include transactions between industries in different regions are immediately obvious. In fact, a considerable literature has developed on interregional interindustry models and efforts have even been made to relate such models to "other worlds" such as the environment delineating the interactions of the human economy with the highly interactive elements of our land, sea, and air "economies."
In the following summary models, I have left implicit the definitions and identities which we labored over before. The common glossary is as follows:
Y: | income |
E: | income-related expenditures |
E_{i}: | income-related expenditures in region i |
E_{ij}: | income-related expenditures in region i by residents of region j |
A: | autonomous expenditures |
e: | marginal propensity to spend, dE(Y)/dY=e |
X: | exports |
M: | imports |
m: | marginal propensity to import, dM(Y)/dY=m |
_{i}: |
subscript for region (A subscript of 0 is for "autonomous value." ) |
The one-region models are the Keynesian and economic-base models. Their distinguishing characteristic is that the value of income is determined by some autonomous activity out there. It may be investor behavior, as in the Keynesian model, or it may be purchases by consumers outside the region, as in the economic-base model.
Assumptions:
E = E(Y) = eY
A = A_{0}
Equilibrium condition:
Y = E + A
Solution:
Y = eY + A_{0 }Y = [1/(1 - e)] A_{0}
Multiplier:
dY/dA = 1/(1-e)
Assumptions:
E_{1} = E_{1}(Y_{1}) = e_{1}Y_{1 }X_{1} = X_{0 }M_{1} = M_{1}(Y_{1}) = m_{1}Y_{1 }A_{1} = A_{0}
Equilibrium condition:
Y_{1} = E_{1} + A_{1} + X_{1} - M_{1}
Solution:
Y_{1} = e_{1}Y_{1} + A_{0} + X_{0} - m_{1}Y_{1 }Y_{1} = (A_{0} + X_{0})* 1/(1 - (e_{1} - m_{1}))
Multiplier:
dY_{1}/dX_{0} = dY_{1}/dA_{0} = 1/(1-(e_{1} - m_{1}))
What has been missing all along has been interaction between regions. Obviously this takes place in the real world; otherwise, there would be no regions. Exports by one region have to be imports into another economy.
The system can more easily be presented by laying out the system in the same form as above but stopping before the algebra requires the theory of determinants, which has either been forgotten or ignored by most of us. We will limit the statement to region one alone in this section.
Assumptions:
E_{1} = E_{1}(Y_{1}) = e_{1}Y_{1 }X_{1} = M_{2} = M_{2}(Y_{2}) = m_{2}Y_{2 }M_{1} = M_{1}(Y_{1}) = m_{1}Y_{1 }A_{1} = A_{0}
Equilibrium condition:
Y_{1} = E_{1} + X_{1} - M_{1} + A_{1}
Solution:
Y_{1} = e_{1}Y_{1} - m_{1}Y_{1} + m_{2}Y_{2} + A_{0 }dY_{1} = e_{1}dY_{ 1}- m_{1}dY_{ 1} + m_{2}(dY_{2}/dY_{1}) dY_{1 }+ dA_{0 }dY_{1} = e_{1}dY_{ 1}- m_{1}dY_{ 1} + m_{2}(dY_{2}/dY_{1})dY_{1}(dM_{1}/dX_{2}) + dA_{0 }dY_{1} = e_{1}dY_{ 1}- m_{1}dY_{ 1} + m_{2} m_{1} (dY_{2}/dX_{2}) dY_{1 } + dA_{0}
But this line of attack becomes unnecessarily complicated. We can't easily reduce the interactive term without knowing the multiplier for the other region (dY_{2}/dX_{2})!
Multiplier:
dY_{1}/d A_{0} = 1/(1-(e_{1} - m_{1} +m_{2} m_{1} (dY_{2}/dX_{2}))
dY_{1}/d A_{0} = 1/(1-(e_{1} - m_{1} + interregional effect))
Let us take a more familiar tack. (This approach originated with Alan Metzler (Metzler 1950); Harry Richardson provides the restatement on which the above is based (Richardson 1969)
This model can be more systematically expressed with the matrix algebra developed for input- output analysis. Matrix algebra permits us to immediately extend our logic from two regions as above to n regions without producing page after page of cumbersome calculations. Nevertheless, we will work with a two-region system for simplicity of discussion. We proceed as follows:
Assumptions:
As above, but now for both regions 1 and 2.
E_{i} = E_{i}(Y_{i}) = e_{i}Y_{i
}X_{i} = M_{j} = M_{j}(Y_{j}) = m_{j}Y_{j
}
Equilibrium conditions:
E_{1} + X_{1} - M_{1} + A_{1} = Y_{1
}E_{2} + X_{2} - M_{2} + A_{2} = Y_{2
}
Rewrite to align purchases from the two regions:
E_{1} - M_{1} + X_{1} + A_{1} = Y_{1 } X_{2} + E_{2} - M_{2} + A_{2} = Y_{2 }
Now, observing the following definitions, we can rewrite the equation system preparatory to matrix manipulation:
E_{11} = E_{1} - M_{1} = e_{1}Y_{1} - m_{1}Y_{1} = e_{11}Y_{1 } E_{12} = M_{2} = M_{2}(Y_{2}) = m_{2}Y_{2} = e_{12}Y_{2 } E_{21} = X_{2} = M_{1}(Y_{1}) = m_{1}Y_{1} = e_{21}Y_{1 } E_{22} = E_{2} - M_{2} = e_{2}Y_{2} - m_{2}Y_{2} = e_{22}Y_{2 }
Note that the e's now show double subscripts showing purchases from region i by region j. So the system becomes
E_{11} + E_{12} + A_{1} = Y_{1
} E_{21} + E_{22} + A_{2} = Y_{2
}or
e_{11}Y_{1} + e_{12}Y_{2} + A_{1} = Y_{1
} e_{21}Y_{1} + e_{22}Y_{2} + A_{2} = Y_{2
}
which translates into matrices as
or
eY + A = Y
where e is the matrix of e_{ij}. This solves exactly as would an input-output system:
Y - eY = A
(I - e)Y = A
(I - e)^{-1}(I - e)Y = (I - e)^{ -1}A
Y = (I - e)^{ -1}A
Now, to give the elements of this inverse a designation, set
rewrite the solution as
Y_{1} = R_{11}A_{1} + R_{12}A_{2 }Y_{2} = R_{21}A_{1} + R_{22}A_{2 }
The interregional multipliers are thus
dY_{i}/dA_{j} = R_{ij }
for each ith row and jth column.
Now, it is a simple matter to expand the system to treat n regions. The process is identical to that used in the two-region system but with larger matrices!
The obvious extension is to expand the cell entries to become interindustry transactions matrices. Thus the E_{12} entries would become _{kl}E_{ij}, showing purchases from industry k in region i by industry l in region j; where i=j, the regional transactions matrix for region i is inserted, and where i is not equal j, a matrix of imports is inserted.
The most commonly used multi-regional interindustry model of the United States was assembled by Professor Karen Polenske at MIT. Her model includes transactions generally for 50 industries in each of 44 states or sets of states.(Polenske 1980)
Economic-ecologic models append matrices depicting interaction between sectors in the natural environment and between those sectors and the industrial ones. The concept is similar to interregional interindustry analysis but implementation becomes a harrowing experience. The dominant example of this was a complex model of the Massachusetts Bay area. (Isard et al. 1972)
Both of these extensions are discussed in detail in the comprehensive input-output reference by Ronald Miller and Peter Blair. (Miller and Blair 1985)