Let us assume a basic economy. Economists in Austerios are interested in comparing the levels of investment and unemployment. They already have some basic information.
They know the relationship between u', the rate of change of unemployment with respect to x, the level of consumption with respect to time.
du/dx = (8/15)x
I < (4/5) (x)^2
This last equation tells us, in Austerios, the level of investment must be strictly less than four-fifths the level of consumption-squared in the economy.
We must integrate to find an equation relating u to x.
Solving, we find:
u = (4/15) (x)^2 + constant
I'm going to assume away the constant here. It is most important when comparing base levels of consumption and unemployment, and I'm not concerned with the extreme case. If both u and x are sufficiently large compared to the constant, we get
u = (4/15) (x)^2
Now, we can find our relationship between the unemployment level and investment.
I < (4/5)[(15/4)u]
So, it must be the case that...
I <3 u
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