Question: Prove the law of diminishing marginal return.
Hints: Use calculus.
Solution:
All things are scarce, i.e. they must be finite. A finite growth must have finite area (total quantity) under curve like the growth and decay of an organism as below:
The definite integral of the growth curve above is finite as fuck.
To achieve finite growth, thr curve will be eventually concave. Here we use the method of contrapositive to show the concavity of absolute returns: Assume the total product is always convex (the second derivative being always greater than 0), behaving like exponential growth, then we can infer that absolute returns will look like logistic growth instead because scarcity guarantees that there should be a upper bound (i.e. a horizontal asymptote) that no quantity can exceed.
Then the exponential growth is impossible or the life span of everything will be infinity. Q.E.D.
Question: Prove the law of demand.
Solution: Here we use the proof by contrapositive as well. Assume the opposite is true, i.e. the demand curve behaves the supply curve such that as price increaes, the quantity demanded will increase.
You will say that this is the reality of the diamond market, and this shows your profound lack of precalculus knowledge on indifference curves such that you know the fuck about income effect. But no worry, we won't deploy algebraic manipulation of the rectangular hyperbolas to complete the proof.
Remember that income is not infinite. As price (opportunity cost) increases, your deductible income will decrease such that the marginal quantity of the same product you can buy must decrease. Then to parametrize the demand curve we will use y=c-mx instead of y=mx+c where all y, c, m, x are positive. Note that m is the slope of the demand curve that can vary while c must be a finite constant. That's why demand curve is always negatively sloped by the axiom of scarcity again.
Hence, the proposition that demand curve is positively sloped is false. Q.E.D.
Some stupid asses will claim that my rate of change of income is increasing such that the rate of change of my deducible income can be increasing. Then oh what the fuck, you assume that the rate of change of price is constant? You neglect the effect of inflation?
Now consider the demand curve again:
y=c-mx where y is price and c is your income. When we consider rate of change, we differentiate y with respect to time such that:
y'= c' -mx' -m'x
Now m and x must be positive, remember that in economics, quantity is the function of price, so if you think y is a function of x, you can go fuck yourself. Rearrange the above derivative, we have:
y' + mx' +m'x = c'
Then it is a first order ordinary differential equation you know. No worry, we keep on rarranging the terms:
x = \frac{c' - y' - mx'}{m'}
and
x' = \frac{c' - y' - m'x}{m}
This implies that when your income grows less proportionately than the price, the quantity of x must decrease leading to negative value of x'. The additional factor m'x can accelerates the negative magnitude of x'.
So dude, this is exactly what imcome effect means.
Now, think about it:
Is c' < y'?
This is true when the general price level always increases. Notice that c' <0 , i.e. your deductible income is always decreasing over time when you are buying everything owning to the nature of consumption, unless you need to buy nothing. Do not talk about saving in microeconomics.
Also, y' > 0 because transaction cost covering depreciations must be greater than zero and will grow over time. So the law of demand that x'< 0 is established.
In physics, we have the law of conservation of energy in mechanics. In chemistry, we have the law of conservation of charges in redox reactions. The manipulation is similar to the law of diminishing marginal return.
So it is false to think that a microbe can multiply infinitely all the time. In the short run, they grow exponentially. Soon, they use up all essential resources for survival (for example, global glucose sources for respiration) and no additional population can even survive. They will eventually all die for the rest of time irreversibly. This phenomenon is referred to logistic decay in biology.
Mathematically, if they resurrect when all things other than the population of their own being constant, this will violate the conservation of energy. That's why all organisms must have finite population no matter how the living habitat is.
So, can we earn infinite profits?
Economic analysis requires basic knowledge in accounting, not mathematics alone. For example, A pays $10000 for B when B eats A's shit, then B pays $10000 for A when A eats B's shit reciprocally. The net change is $0, but in national income accounting, the GDP made is already $20000, and this will keep on rocketing if such a eat-shit economics repeats infinitely.
Also, the law of diminishing marginal return will fail if there are 2 or more variable costs in a production, for example, technology level and exploitation of new crude oil sources.
But in physics, unless we are in open system, the conservation of energy will still hold water. For example, if we use up all seafoods for consumption without letting the edible ocean organisms reproduce with sufficient time, we will end up global malnutrition of essential amino acids and the whole human civilization will bound to vanish. The key is, according to the third law of thermodynamics:
\partial G = \partial H - T \partial S
Humans are unable to create any seafood artificially without astronomical energy supply.
So you see the importance of sustainable development and equity then?