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If economic growth is indeed highly desirable (see this question), why must this growth be exponential? With finite resources, exponential growth might hit limits rapidly (or be impossible?). Why not express growth in linear rather than exponential terms?

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-1: This question is too broad. It mixes optimal growth, possibility of infinite growth and how to express growth mathematically all together in one question. –  FooBar Dec 4 '14 at 21:51

4 Answers 4

up vote 7 down vote accepted

Growth as is meant here "must" be nothing in particular. It is a specific metric, the percentage change in yearly GNP/GDP, and it is what it is.
In Blanchard and Fischer 's "Lectures on Macroeconomics", in the introductory chapter 1, page 2, Figure 1.1, the logarithm of USA GNP 1874-1986 is graphed: and it is impressively linear , bar a disturbance around World-War II (a dive before it that was roughly equally compensated immediately after). But this means that

$$\ln Y \approx at \Rightarrow Y \approx e^{at}$$

(for the US Economy, $a \approx 0.030\;\; \text{to} \;\;0.037$ for the period).

It is the data that told us that "growth was exponential" during this period.
(Note that "exponential growth" usually includes the concept of constant growth rate, while in informal language, "exponential" may also refer to exploding paths, paths with increasing growth rate).
And so economic models were deemed relevant if they could replicate to a respectable degree the observed data.

The question "can this go on forever?" is an altogether different issue, starting with the meaning of the word "forever".

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Because linear functions don't match the data.

You can't express a series $$[1,2,4,9,16]$$

as $$f(x)=x+y$$

for any possible $y$.

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Because we use today's capital stock to produce tomorrow's output, some fraction of which is invested, so you should expect something like $dK/dt=\alpha f(K)$ where $f$ is increasing in $K$.

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  • growth makes most sense as a percentage. looking at absolute numbers does have value but percentage growth allows for some pretty good comparisons.

  • You seem to think exponential growth means infinite growth. It is a pretty logical assumption to make, but I believe it takes these models and uses them in a way they were not meant to be used. Economists seldom care about making predictions 200 years in the future. Exponential growth is quite bad at forecasting that far ahead in anything, in shorter time scales it isn't too bad (Source needed).

I'll try and make it clearer:

Consider a basic model of GDP growth. Suppose GDP is growing at 1% per year ($r=1.01$) and initially is at \$1,000,000. Let $Y_t$ denote the populations size $t$ years after the initial population of $Y_0 = \$1,000,000$. If one asks what the GDP will be in 50 years there are two options.

At 1% per year growth, the dynamic equation would be \begin{gather*} Y_{t+1} - P_t = 0.01 \, Y_t \end{gather*} and the corresponding iteration equation is \begin{gather*} Y_{t+1} = 1.01 \, Y_t \end{gather*} Starting with the initial condition, $Y_0 = 1,000,000$, we could calculate $P_1 = 1.01 \times 1,000,000 = 1,010,000$, $P_2 = 1.01 \times 1,010,000 = 1,020,100$ and so on for 50 iterations.

This is equivalent to:

\begin{gather*} Y_t = 1.01^t \left( 1,000,000 \right) \end{gather*} so that we immediately have a formula for the population after 50 years: \begin{gather*} Y_{50} = 1.01^{50} \left( 1,000,000 \right) = 1,644,631. \end{gather*}

A point I am trying to make here is that exponential growth is really just the size of something as a function of itself in a different state or time frame. If you want exponential growth over a longer timeframe, it makes sense to extend the model.

What if $r$ was endogenous to the model? As Y gets larger, r gets smaller. Still growing exponentially, and the size of the economy in $t+1$ is still dependent on the size of the economy in $t$.

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i feel this answer contains a lot of the intuition needed but is quite messy. I'll try and fix it up. Also, add a few sources. My main point is, exponential growth is a good way to view an economy in the short run, longer run models do not necessarily require this. –  Jamzy May 27 at 0:29

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