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Capital, Consumption and Growth


(c) 2009 by Ed Seykota





EcoNowMics models lay out all the model assumptions and show exactly how these assumptions interact to generate model behavior.


EcoNowMics is an "open platform" for discussion, extension, clarification and growth of economic understanding.


In this essay, I lay out a model for capital, consumption and growth.


While this model is relatively simple, its implications reach far into the fabric of social interactions. 

Definition of Capital


In this EcoNowMics model, capital means "tool" - any thing that (1) is useful in building something and (2) lasts a long time, say, over three years.


For example, a butcher knife is capital in that it helps produce steaks. With care, it can last for decades. A disposable plastic picnic knife is not capital. You don't use it to manufacture anything and it is generally good for a one-time use.  A picnic knife is a consumer item. The machine that makes the plastic picnic knife is capital.


A factory that produces parts of houses is capital.  The faucets, doorknobs, rugs, lamps and bathtubs the factory produces are consumption items. Although they may last a long time, they do not help manufacture anything.


Money, cash, currency, bank balances and debt are not capital. They do not directly help manufacture anything - although they may help account for it or provide a medium of exchange for it.


A truck that carries nuts and bolts to a factory is capital.  A truck that takes you fishing is a consumption item. Things may be capital or consumption items depending on their use. 


In our model, we represent the total stock of capital as a level. In EcoNowMics, Capital has the units of measure, tools.





Capital Stock is a Level


Unit of Measure: tools.


Capital stock is a level. In the next section we see how various rates flow into and out of this level and change this level.

Capital Flows


Capital stock is a level. It is the total number of tools in a factory, or depending on the model, in the economy.  Capital changes as tools flow in and out of the factory.


The main flow into a factory is the capital formation rate.  This is the number of tools per year that a factory acquires by purchasing them, or by manufacturing them.  The main flow of capital out of the factory is depreciation - the rate at which tools wear out. 


Capital Formation


Capital formation rate is the rate of formation of new capital.  While some factories produce consumer items, like t-shirts, other factories produce tools, like wrenches.  Tool makers make capital.  If your factory buys wrenches from a tool maker, then your capital formation rate might be 100 wrenches per week.  Capital formation has the units of measure, tools per unit of time.


In this model we use "tool" as the standard unit of measure - although we could choose any other unit of measure such as "machine" or "factory."  In this model we measure all types of capital in tools.  A wrench or a screwdriver might each be one tool.  A power drill might then be equivalent to 25 tools while a large milling machine might be equivalent to 1000 tools and a chemical plant might be 1,000,000 tools.


The essential property of capital is that it is useful in manufacturing other things and that it lasts at least three years.  Using "tools" as a unit of measure is consistent with the definition of capital.  Some economists measure capital in dollars.  While capital may have a dollar value, it does not consist of dollars. It consists of tools.  You can't weld two beams together with a dollar bill; you need a welding tool.


Capital Depreciation


Depreciation is the rate at which capital wears out. If a milling machine has a useful lifetime of, say 20 years, then its depreciation rate is 1/20 milling machine per year or 5% per year.  If a milling machine is equivalent to 1000 tools, then the depreciation rate is 1000/20 = 50 tools/year. If a factory has 1000 wrenches you can envision 50 of them wearing out each year and leaving the factory.


In EcoNowMics, we represent capital formation and depreciation as rates.  Formation rate flows into capital (capital stock means the stock of capital) while depreciation flows out of capital.  In some circumstances formation rate might be negative, such as if a factory sells off some trucks to people to use as pleasure vehicles.




Capital Flows


We see in the chart above how the capital formation rate (tools/year) and the capital depreciation rate (tools/year) change the capital stock (tools).  In the next section we examine the dynamics of depreciation.



Depreciation is the rate at which tools wear out.  In EcoNowMics, we compute depreciation as the capital stock over the capital life time.



Capital Stock and Depreciation Rate


If a tool has a life time of, say, five years,

then we might say that on average,

one fifth of it wears out each year.


Depreciation Rate = Capital Stock / Life Time.


Units: tools/year = tools / years



For example, a factory has 100 tools and the average life time of the tools is 20 years.  The depreciation rate is 100 tools / 20 years = 5 tools per year. On average, every year, 5 tools wear out and flow out of the capital stock by depreciation.


Thus, if the factory starts the year with 100 tools and if there is no capital formation then at the end of the year, the factory winds up with 95 tools.





Depreciation Dynamics


At the start of the run Capital Stock is 100 tools

and Depreciation Rate is 5 tools per year.


After one year, Capital Stock is about 95 tools.


As Capital Stock falls, so does Depreciation Rate

since there is less capital to depreciate.


Capital Stock and Depreciation Rate

both fall asymptotically, toward zero.


The depreciation loop is an example

of a simple first order negative feedback loop.


For more on simple first order negative loops see the EcoNowMics article, The Ways of Delays.


In the next section, we examine some of the dynamics of capital formation.

Capital Formation Rate


Capital Formation Rate is the rate at which factories acquire new capital. In this model, we do not distinguish between the production of capital and the employment of that capital.  Note: In the K-Wave model (later in this series) we acknowledge a delay between capital production and employment and show how it can contribute to the cyclical dynamics of the K-Wave. In this model, however, as soon as a factory produces a tool, that tool goes to work as capital in that factory or in another factory.  In this model, we consolidate all factories and have one consolidating Capital Stock.


Say a factory produces 5 tools per year and another factory acquires these tools and puts them to work.  The Capital Formation Rate is, then, 5 tools per year.  Let's also say that the factories are in business many years and already have 100 tools with an average life time of 20 years.  So the Depreciation Rate is 5 tools/year and this exactly balances the Capital Formation Rate.  Capital Stock neither grows or declines.  The factory is in "steady state."


In this section we see how the Capital Stock responds to changes in the Capital Formation Rate.




If The Formation Rate Balances the Depreciation Rate


Capital Stock remains constant

and the factory remains in steady state.



If you are familiar with the article, The Ways of Delays, you may recognize the structure above as a "lag" or "delay" or "tracker" and expect that the depreciation rate tracks formation rate.


In the simulation run below, we start with a factory in steady state with 100 tools, and formation and depreciation rates both at 5 tools per year.


At year five, we quadruple the formation rate to 20 tools per year.  Since the depreciation rate is only 5 tools / year, we have a net inflow of capital and capital stock begins rising.  Eventually capital rises four-fold, at which point the depreciation rate rises to matche the formation rate.


The depreciation rate "tracks" the formation rate.  At the new equilibrium, depreciation rate = capital stock / life time. 


Note: since, at steady state,

depreciation rate = formation rate, we also have

capital stock = formation rate x life time. 


For more on trackers, see The Ways of Delays.




The Factory Initially has Capital of 100 Tools


At year 5, the formation rate (green) rises to 20 tools/year.

Capital rises and brings depreciation rate with it.

Depreciation rate "tracks" formation rate

rising asymptotically toward 20 tools/year.


For more on "tracking" see The Ways of  Delays.


Next, we turn to the magic of capital, namely: positive feedback.

The Magic of Capital - Positive Feedback


The magic of capital is that you can use capital to produce more capital. This increases capital stock so capital production rate can go even higher - all in an exponentially growing positive feedback loop.


Other symptoms of this process are high employment, increasing prosperity and a higher standard of living for all.  We develop these themes in a later paper.


Capital formation is particularly vital when factories are free to evolve and respond to economic conditions by inventing new kinds of tools. 


This process of capital growth and innovation is the essence of capitalism.  You can measure the health and vitality of a capitalist system by the growth rate of capital. 


In the United States, we are currently experiencing a net decrease in manufacturing and a decline in capital stock. We no longer manufacture many automobiles or electronic devices in the US. The capital formation rate is less than the depreciation rate. The United States is no longer a capitalistic society.  It is a consumer-istic society.


For an example of capitalism at work, say each tool can produce two more tools in one year.  (The tool production time is six months.)  Of course, not all tools produce tools.  Many tools and factories that employ these tools produce consumer goods. Say that on average the economy as a whole devotes 2.5% of its production capacity to tool-making.   Say the economy starts out with a million tools.




Model Structure


Note: For more details, see "Model Definitions and Values"

at the end of this paper.


Production Capacity

= 1 million tools X 2 tools per tool per year

= 2 million tools / year.


Capital Formation Rate

= production capacity per year X 2.5%

= 50,000 tools per year.


Depreciation Rate

= 1,000,000 tools / 20 years

= 50,000 tools per year.


The Net Flow of Capital

= 50,000 tools / year - 50,000 tools / year

= zero.



Our simulation model verifies that the system is in steady state equilibrium.



For Investment Fraction = 2.5%


capital formation rate balances depreciation rate

and the system settles into no-growth steady state.



So far, this might not seem like much magic.  To see how the magic really works, we have to have a capital formation rate high enough to cover depreciation plus a little extra for growth.


Let's see what happens if we raise the investment fraction from 2.5% to 5%.



With Investment Fraction at 5%


capital grows from 1 million tools to about 12 million tools.


Now that seems like some magic.  At 10% investment fraction, the results are even more amazing.




With Investment Fraction at 10%


capital grows from 1 million tools to around 2 billion tools.



In this simulation you re-invest about 10% of what you produce and you wind up with a phenomenally booming economy - and you leave your kids with a standard of living that is about 2,000 times higher than your own. 


This looks pretty good in the simulation.  You might think that people would catch on to this and that, as a culture, we could invest 10% toward tooling-up.  However, in practice, we don't observe this happening.  What we see is factories closing, unemployment and production moving to other countries like China and India,


An investment fraction of close to zero percent produces a curve that is more consistent with our actual situation.




With Investment Fraction at Zero


capital stock depreciates asymptotically toward zero.


This curve is similar to the the curve

showing the decline in American manufacturing. 


One might naturally wonder how come a society would choose decay and decline over robust, vital growth.


To see how this can be, we extend the model to see what factors account for us having a low investment fraction.

Investment Fraction


The opposite of Capitalism is not Communism or Socialism; it is Consumer-ism.  The essential decision that creates the condition for capitalism is the decision to forego consumption in order to increase capital and production capacity.  In our model, this decision appears as the Fraction for Investment.


In this model, the Fraction for Investment competes with the Fraction for Costs (including labor), the Fraction for Consumption and the Fraction for Taxes.


If the Fraction for Investment is zero, capital stock depreciates asymptotically toward zero.  If the fraction is 100%, and we plow 100% of our production into new equipment - we find we are unable to pay for raw materials, wages or utilities and, again, we must shut down our factories.


Our model indicates an investment fraction of about 5% or 10% is enough to generate prosperity.


In this section, we continue to examine the central role of the Investment Fraction.  The following diagram shows the relationship of the Investment Fraction to other key fractions that operate to "divvy up" our total Production Capacity.




Model Showing The Four Fractions

of Total Production Capacity


1. Fraction for Consumption (Consumer Goods)

2. Fraction for Production Costs (To run the Factories)

3. Fraction for Taxes

4. Fraction for Investment



The way our society works, and the way the model works, is to set the Investment Fraction to what remains after we pay other costs.


If our Production Cost Fraction is 50%, our Consumption Fraction is 15% and our Tax Fraction is 32.5% we have 2.5% remaining for investment.


These numbers are somewhere in the ball park for the USA, circa 2009, and, as we know, a 2.5% investment fraction produces a zero-growth economy.


In this run, we try an experiment. We reduce Tax Fraction 5% from 32.5% to 27.5%.  This has an effect of increasing the investment fraction by 5%, from 2.5% to 7.5%.  The results, as you might imagine, are astounding.




Reducing the Tax Fraction


from 32.5% to 27.5% at year 5,

produces an initial decline in tax collections,

and then leads to a period of long-term growth

in capital stock, consumer goods production rate

and also in tax receipts.



The essential economic choice confronting a society is what fraction of production capacity to devote to re-investment.  The higher that rate, the more prosperous, vital and healthy the economy.


We see, however, that our society does not invest very much.  Indeed, we are currently shutting down factories.  To see how this can be, we have to look at the forces and factors that govern our policies for setting costs, consumption and taxes.


On this path we examine various popular economic theories such as:

1. Government spending is the same as investment.

2. You can spend your way out of a recession.

3. Government can stimulate business.


We extend the model to cover these items in the next paper.



Model Definitions and Values


Fraction for Investment = .025

The economy re-invests 2.5% of its production into new capital formation.


Capital Production Time = 1/2 year.

The time to produce one tool using one tool is one half of one year.


Capital Formation Rate (tools/year)

The capital formation rate is production rate * Fraction for Investment / Capital Production Time

=  (initially) 2,000,000 tools/year * 2.5%

= 50,000 tools per year.


Production Rate (tools/year)

Production rate = capital stock / production time

= (initially) 1,000,000 tools / .5 years

= 2,000,000 tools/year


Capital Life Time

The time for an average tool to wear out

= 20 years.


Depreciation Rate

The depreciation rate equals the Capital Stock / Capital Life time

= (initially) 1,000,000 tools / 20 years

= 50,000 tools / year.


The net formation rate is, therefore, initially,

= 50,000 tools/year - 50,000 tools/year

= zero