Investment, consumption and growth

The role of investment

• Production can be divided into investment, which is everything dedicated to producing, and consumption.
• Consumption is the slice of the pie that people take, and investment is the slice of the pie that is used for production. The sum of investment and consumption is existing production; it's the whole pie.
• A common idea, which is also reflected in some manuals, is that to increase economic growth, the investment rate, the percentage of production that is dedicated to investment, must be increased.
• It's a "common sense" idea: if the investment serves to produce, the greater the part dedicated to production, "common sense" says that the greater the production will be.
• But we must be careful, because in science "common sense" very often leads to errors. The usual "common sense" idea was that the Earth was motionless.
• Increasing investment reduces current consumption, the slice of the pie that people get. The common idea is that increasing the investment rate increases economic growth, allowing for a larger pie in the future and more slices for everyone.
• Again, "common sense": we must reduce the piece of the pie that people take in the present so that the piece they take in the future is bigger.
• But let's remember that the Earth was immobile according to "common sense".
• There are several errors in this common idea. Here we will focus on just one: the belief that increasing the global investment rate leads to greater economic growth. To test this, we will construct two models, one exponential and the other logistic, to challenge this common notion.
• Another mistake is assuming that people's share of the pie is guaranteed. But we won't examine that now.
• We will also show that, at an international or regional level, a "small" economy (a country, a region) with a higher investment rate ends up displacing those with a lower rate, even if its growth factor is slightly lower and even if it starts from a smaller size.

Exponential model. The role of investment

Optional. Exponential model

Item_​ = s X _t

The investment at time t is equal to the investment rate multiplied by production at time t�

C _t = X _t – Item_​

Consumption at time t equals production at time t minus the investment at time t�

X _t+1 = k Item_​

Production at time t+1 is equal to growth factor due to investment at time t

Investment at time _t

s Investment rate (which we assume to be a given constant)

X _t Production at time t

X _t+1 Production at time t+1

k Growth factor (1 plus the growth rate)

C _t Consumption at time t

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• The first equation tells us that the investment is a given percentage of the pie.
• The second one says that the pie is divided between investment and consumption.
• The third is that the pie in the next step is generated from the investment, multiplied by the growth factor, following the exponential equation.
• If we know s, ky X _t we can calculate I _t , C _t and X _t+1

Optional. Exponential model

• The first equation tells us that the investment is a given percentage of the pie.
• The second one says that the pie is divided between investment and consumption.
• The third is that the pie in the next step is generated from the investment, multiplied by the growth factor.

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• Let's assume that the investment rate s is 70%, that the growth factor k is 3, and that the output X _t is 0.1.
• From the first equation, since the investment rate s is 70%, the investment will be 70% of the production X _t , which is 0.1. Then the investment will be I _t = s X _t = 70% · 0.1 = 0.07.
• From the second equation, consumption will be production minus investment. Therefore, _C _{_t} = X _{_t} – I_t = 0.1 – 0.07 = 0.03.
• From the third equation, the production in the next step will be the growth factor k times the investment I _t , then X _t+1 = k I _t = 3 · 0.07 = 0.21.
• Since we have the production at the next time, we can repeat the procedure to calculate the investment and consumption at the next time.
• From the first equation, since the investment rate s is 70%, the investment will be 70% of the production at the next time X _t+1 , which is 0.21. Then the investment will be I _t+1 = s X _t+1 = 70% · 0.21 = 0.147.
• From the second equation, consumption will be production minus investment. Then C _t+1 = X _t+1 – I _t+1 = 0.21 – 0.147 = 0.063.
• From the third equation, the production in the next step will be the growth factor k times the investment I _t+1 , then X _t+2 = k I _t+1 = 3 · 0.147 = 0.441.
• Etc.

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Optional. Exponential model

We propose two exponential models, one with an investment rate s1 and another with an investment rate s2, in order to compare them.

The growth factor and initial production are the same in both.

We observe that in the exponential model, an increase in the investment rate increases growth. And that increasing growth can increase consumption in the future (unless the investment rate is 100%, obviously).

If economies are growing exponentially, the usual idea holds true: increasing the investment rate allows for increased consumption in the long run.

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By double-clicking on the spreadsheet, you can modify the investment rates or the growth factor in the yellow cells to see how the respective production and consumption changes.

Exponential model. The role of investment

• If the global economy followed exponential patterns, increasing the investment rate would increase growth, increasing the size of the pie.
• If we also assume (without justification) that people can take a constant percentage of the pie, and that the investment rate remains constant, their long-term consumption will also grow (unless the investment rate reaches 100%, obviously).
• “Common sense” in an exponential world is justified at least in the part that indicates that increasing the investment rate increases growth.
• Other parts of the "common sense" reasoning still need to be justified, such as the idea that people could always get a given percentage of the pie.

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Logistics model. The role of investment

Logistics model. The role of investment

• We don't live on an exponential planet. In reality, the environment always imposes limits, as the classics said.
• But we already saw that the classics underestimated the ability of technology to increase load capacity, the final steady state of logistical behavior.
• We also saw that we can model, using the logistics equation, the limitations imposed by the environment and the increase in carrying capacity due to technological advancement:
• Increasing the growth factor, the “intensive” technology,
• Or also by increasing the limiting constant, the “extensive” technology.
• To simplify, we will assume that "extensive" technology increases, so that the economy evolves with the logistics equation but with a constant of increasing limitation.
• We already did this on slide 24 of ppt 10.1.
• If we were to assume that the "intensive" technology also increases, and that the expansion factor k also increases, we would arrive at the same results.

Optional. Logistics model

Item_​ = s X _t

The investment at time t is equal to the investment rate multiplied by production at time t�

C _t = X _t – Item_​

Consumption at time t equals production at time t minus the investment at time t�

X _t+1 = k It _t ( 1 – I _t / P _t )

Production at time t+1 is equal to growth factor for the investment at time t by ( 1 less the investment at time t divided by the limiting constant at time t )�

Investment at time _t

s Investment rate (which we assume to be a given constant)

X _t Production at time t

X _t+1 Production at time t+1

k Growth factor (1 plus the growth rate)

P _t Limiting constant at time t

C _t Consumption at time t

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• The first equation tells us that the investment is a given percentage of the pie (just like before).
• The first is that the pie is divided between investment and consumption (just like before).
• The third is that the pie in the next step is generated from the investment, following the logistic equation (we have added the parentheses that limit growth).
• If we know s, k, P _t and X _t we can calculate I _t , C _t and X _t+1

Optional. Logistics model

• The first equation tells us that the investment is a given percentage of the pie (as before).
• The second one says that the pie is divided between investment and consumption (as before).
• The third is that the pie in the next step is generated from the investment, multiplied by the growth factor and by a parentheses that limits the growth; by 1 minus the investment divided by the limiting constant.

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• Let's assume that the investment rate s is 70%, that the growth factor k is 3, and that the output X _t is 0.1.
• From the first equation, since the investment rate s is 70%, the investment will be 70% of the production X _t , which is 0.1. Then the investment will be I _t = s X _t = 70% · 0.1 = 0.07.
• From the second equation, consumption will be production minus investment. Therefore, _C _{_t} = X _{_t} – I_t = 0.1 – 0.07 = 0.03.
• From the third equation, the output in the next step will be the growth factor k times the investment I _t and by the parentheses, then X _t+1 = k I _t (1 – I _t / P _t ) = 3 · 0.07 · (1 – 0.07 / 1) = 0.1953.
• Since we have the production at the next time, we can repeat the procedure to calculate the investment and consumption at the next time.
• From the first equation, since the investment rate s is 70%, the investment will be 70% of the production X _t+1 , which is 0.1953. Then the investment will be I _t+1 = s X _t+1 = 70% · 0.1953 = 0.13671.
• From the second equation, consumption will be production minus investment. Then C _t+1 = X _t+1 – I _t+1 = 0.1953 – 0.13671 = 0.05859.
• From the third equation, the production in the next step will be the growth factor k times the investment I _t+1 and by the parentheses, then X _t+2 = k I _t+1 (1 – I _t+1 / P _t+1 ) = 3 · 0.13671 · (1 – 0.13671 / 1.05) = 0.35673.
• Etc.

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Optional. Logistics model

We propose two parallel logistic models, in order to compare them, one with an investment rate s1 and another with an investment rate s2.

The growth factor, the limiting constants, and the initial production are the same in both.

We assume that the constraint constants increase over time to model extensive technological development. If desired, the constraint can be implemented with a different dynamic, or assumed to remain constant.

We observe that in the logistics model, an increase in the investment rate does not increase growth but does reduce consumption.

If economies are logistics-based, the usual approach doesn't work. Increasing the investment rate also reduces consumption in the long run.

By double-clicking on the spreadsheet, you can modify the investment rates, the growth factor ko, and the limitation constants in the yellow cells to see how the respective productions and consumptions change.

Logistics model; the role of investment

• If the investment rate is high and increases, the economy won't necessarily grow much more, because economic growth is constrained by environmental limitations. Increasing the investment rate only causes it to collide with these limits sooner (and more forcefully). The economy cannot grow beyond the limits imposed by environmental constraints, which are determined by technological advancements.
• But the higher the investment rate, the more consumption will decrease.
• To increase consumption, the investment rate should not be too high. Beyond a certain point, increased investment will not increase production and will instead decrease consumption in the long run.
• Only if the investment rate is very low will it affect growth and therefore consumption.
• “Common sense” in a logistics world is wrong for the economies of the planet as a whole.
• This would be true for all the world's economies if the environment didn't impose limitations, but that's not the case in a logistical world where resources are indeed limited. Nevertheless, we will see that a "small" economy can grow by increasing its investment rate at the expense of the growth of others.
• (We will study this in another topic, but a high rate of investment, by colliding more forcefully with the limits of the environment, also increases the amplitude of the economic cycle, implying that economic crises are more severe.)

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The role of investment among economies competing for resources

The role of investment among economies competing for resources

• We will study the evolution of logistics economies that each operate with different investment rates, but which are limited by the planet's overall resources.

Optional. Logistics model; the role of investment in competing economies

I _{i t} = Yeah_​ X _{i t}

For each economy i, the investment at time t is equal to the investment rate multiplied by production at time t�

C _{i t} = X _{i t} – I _{i t}

For each economy i, consumption at time t equals production at time t minus the investment at time t�

X _i,t+1 = k _i I _i,t ( 1 – ∑ I _j,t / P _t )

For each economy i, the production at time t+1 is equal to growth factor for the investment at time t by ( 1 less the sum of investments at time t of all economies divided by the limiting constant at time t )

• The first equation tells us that for each economy, investment is a given percentage of the pie (just like before, but with an equation for each economy).
• The second is that for each economy the pie is divided between investment and consumption (the same as before, but with an equation for each economy).
• The third point is that for each economy, the pie in the next step is generated from its investment, following the logistic equation (we have added the parentheses that limit growth). Here, the availability of resources limits the total investment of all economies.
• If we know s _i , k _i , P _t and X _i,t we can calculate I _i,t , C _i,t and X _i,t+1 for each economy.

Optional. Economies competing in a logistics-driven world

We propose three economies that compete for the same resources with different investment rates.

We assume that the (common) constraint constant increases over time to model extensive technological development. If desired, the constraint can be implemented with a different dynamic, or it can be assumed to remain constant.

Here, the economy with a higher investment rate ends up displacing those with a lower rate, even if its growth factor is slightly lower and even if it starts from a smaller size.

By double-clicking on the spreadsheet, you can modify the investment rates, growth factors, and limitation constants in the yellow cells to see how the respective productions and consumptions change.

To compare the case with two economies, set the initial production of the third to 0.

Logistics model; the role of investment in “small” economies competing for resources

• The overall size of the world's economies is determined by the limits imposed by the environment; and these limits by technology.
• A relatively “small” economy, a company, a region or a country, is therefore limited by the resources of the planet as a whole, for which it competes with the rest of the economies.
• A "small" economy can grow by consuming resources, to the detriment of the growth of the rest of the world's economies.
• For a "small economy", increasing the investment rate does allow for greater growth.

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Investment, consumption and growth

The role of investment

• Across all the world's economies:
• If the world were exponential, increasing the investment rate would produce greater growth, a larger pie, and although the percentage of the pie for consumption is reduced, in the long term it could be larger.
• But if the world is logistically driven, increasing the investment rate wouldn't necessarily lead to greater growth, as growth would be subject to environmental constraints. The pie wouldn't be significantly larger, and the portion allocated to consumption would be smaller (if the investment rate isn't already very low).
• The world's economies operate within logistical constraints; they cannot exceed the limits of their environment. Only technological advancement can shift the carrying capacity (the final steady state) upwards. It is technological advancement that allows the pie to grow. Therefore, across all economies, increasing the investment rate will not lead to greater growth, but rather to a reduction in consumption.
• In a “small” economy:
• But a relatively "small" economy—a company, a region, or a country—is limited by the total resources of the planet as a whole, in competition with other economies. A "small" economy can grow by consuming resources at the expense of the growth of the rest of the world's economies. In this case, increasing the investment rate does allow for greater growth in the "small" economy, though not necessarily in the overall growth of the world's economies.

The role of investment

• There is therefore a tendency to increase the investment rate in "small" economies, even though this causes a reduction in the growth of the rest of the economies and also a reduction in global consumption.
• Reducing the global investment rate would not significantly decrease growth and would, in fact, increase consumption. However, this policy clashes with the individual interests of each country, which compete for global resources.
• Adam Smith argued that the pursuit of individual profit led to general benefit. But here we see a contradiction between the interests of each "small" economy and what is best for the world's economies as a whole. For each individual "small" economy, increasing the investment rate allows it to grow more and outcompete other economies. Conversely, for the world as a whole, it would be beneficial to reduce the investment rate.
• Adam Smith was wrong. Individual benefit does not always correspond to general benefit.
• Attributing something that is true for one element of a set, or even for each element of a set taken one by one, to all elements of a set is called a “fallacy of composition”.
• For example, if traffic laws are not applied to a single driver, A, that driver benefits (since they can choose to obey them or not). The same applies to a single driver, B. The same applies to any single driver. Each driver benefits if traffic laws are not applied to them alone. However, if traffic laws are not applied to all drivers simultaneously, the result is chaos in which everyone loses.
• In short, reducing the overall investment rate would not significantly reduce economic growth and would increase consumption.
• John Stuart Mill advocated halting economic and population growth before reaching the inevitable steady state, thus reducing the strain on resources and enabling people to live in a stable state free from poverty. To some extent, our perspective is inspired by his approach, even if it doesn't entirely align with it.

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