Thursday, August 1, 2013

Modeling the Delayed Effects of Investment

In this post,  I will suggest a model of the delayed effects of investment.  A convenient point of entry is the excellent post by Nick Edmonds found at .

Edmonds correctly arrives at the equation


ΔL + ΔB + α0

1.      Y




( 1 - α1 )


where Y is income,  ∆L + ∆B is the change in bank lending plus the change in lending by bonds, aₒ is base consumer spending and a₁ is an adjustment factor relating all the terms.  This simple equation has the effect of hiding the long term effects of bank and bond investing.

We can separate and examine the long and short term effects of bank lending plus lending by bonds by defining lending as money that is borrowed and would result in spending that would count toward national income as measured by GDP.  This is a severe limitation that we enforce by placing a negative limit on equation 1.   To add this limit to equation 1, we would rewrite it to read

ΔL + ΔB + α0

2.      Y


-------------------     Limit ∆L + ∆B > zero.


( 1 - α1 )


In other words, bank and bond lending cannot be negative or below zero.  If they were negative, they would represent a transaction that is not counted as income but, instead, would be an investment transaction which is not counted as part of GDP.

We will avoid the need for limits by defining our terms carefully.  Here, we depart from the definitions of Edmonds, except for the Y term which is commonly used to identify income.

We will narrowly define investment loans from bonds and bank loans as being new loans and bonds, as contrasted to refunding or roll-over loans or bonds.  New loans and bonds are anticipated to be spent on new plant, labor, material or other item that would be counted as adding to the national GDP calculation.  We label new bank loans NBL and new bond issues NBI

We will assume that every income reporting period is expected to have a different consumer spending base.  We label the annual consumer spending base as ACSB

Up to this point, we have identified two sources of consumer income, borrowing (NBL + NBI) and base spending (ACSB).  We will follow Edmonds by ignoring the effects of Government until a later posting.  If we also ignore the delayed effects of new loan spending,  we can write

3.        Y = NBL + NBI + ACSB.

From equation 3, we can see that national consumer income is at least new investment spending plus base consumer spending.

Now we will examine the delayed effects of new money.  New investment spending moves money from the hands of savers into the hands of workers and materials providers.  This movement into new hands is likely to result in additional spending at some unforeseeable time in the future.  New hands holding new money will change the base consumer spending pattern Y both in the current time period and future time periods. To model this change in the most recent time period, equation 3 will be supplemented with the term a₂ times the value of new loans and new bonds (a₂ * ( NBL +NBI), where a₂  is the expansion factor for new money .  We will write

4.       Y = a₂*(NBL + NBI)  +  NBL + NBI + ACSB .

From equation 4, we can see that national consumer income is at least the factor a₂ times new investment spending,  plus new investment spending,  plus base consumer spending.

Investment in the next income period may not contain any new investment income but it is highly likely that investment income from the previous period will carry over into subsequent periods.  If equation 4 is used to predict an income for a future Y₁ with no future investment, we could write

5.       Y₁ = a₃*(NBL + NBI) + ACSB

where a₃ is the revised constant for the no-investment period.  NBL, NBI, and ACSB are equal to identical terms from the previous year’s calculation.

More can be done with this model as represented by equation 4.  Government can be added to show the effect of changing the base money supply.  ACSB can be traced back in time in an attempt to establish a base consumer spending level or find the annual value of a₂.  It is expected that these and additional subjects will be covered in future postings.