Background

The Great Depression of the 1930's started in the United States but spread around the world. It is now generally accepted that ist origin was in the mismanagement of the money supply in the U.S. by the Federal Reserve System (the Fed). The Fed unknowingly allowed the money supply to decrease. That decrease in the money supply resulted in a general decrease in prices.This defation meant that the effective interest rate, the nominal interest rate minus the rate of inflation, went to record high levels of about ten percent even though the nominal interest rate was only about one percent. The record high real interest rate discouraged businesses from investing in new equipment. Such investment purchases are a major component of the demand for national production. The collapse of investment purchases resulted a significant decrease in production and people were put out of work. The unemploymet rate rose from 3 percent in 1929 to 25 percent in 1933. Once businesses had excess productive capacity the had no reason to purchase equipment to increase their productive capacity even if the real interest interest decreased. Thus the economic depression continued.

But an interesting aspect of the depression is that it spread to other countries with seemingly independent economies. But they were not truly independent. They engaged in trade and their exports were a significant component of the demand for their production. When production went down in the U.S. it decreased its imports from other countries. Their production decreased and their excess capacity discouraged their investment purcases further decreasing their productions. That led to diminished imports from te U.S. and the rest of the countries. Thus production in the U.S. further diminished leading to a further sequence of production drops. Each successive sequence of production drops is smaller so the system reaches at least temporary equilibrium. What is given below is an analysis that determines the level of the equilibria.

Analysis

Consider a system of n countries or regions. Construct an array of the form a_ij where A_ij stands for the purchases of the output of country i by buyers in country j. Let y_i the production level of country i and let Y the column vector of those production levels. Let A be the n×n matrix of the a_ij. For i≠j a_ij is the imports by country j from country i. Equivalently stated, for i≠j a_ij is the exports of country i to country j.

Equilibrium involves

Y = A1

where 1 is a column vector of 1's.

But the a_ij.s may be further refined. Generally

a_ij = b_ij + c_ijy_j. . .

In matrix form

A = B + CY

Thus equilibrium requires

Y = B1 + CY

This is equivalent to

(I − C)Y = B1

where I is the n×n identity matrix.

Thus the equlibrium outputs are given by

Y = (I − C)⁻¹B1

The Dynamic Version of the Model

Let Y_t be the column vector of outputs at time period t. Then

Y_t = B1 + CY_t−1

The equilibrium outputs Y are the outputs such that Y_t=Y_t−1. Such outputs satisfy the system of equations

Y = B1 + CY
and hence
have the solution
Y = (I − C)⁻¹B1