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While I was studying microeconomics, a question arose: I know what the indifference curve for one “good” good and one “bad” good looks like. But if both goods are bad, is the indifference curve concave? Because I know that concave indifference curves mean an extreme bundle is preferred. With two bads, I would not want to have a balanced bundle. Does that also mean that the assumption of monotonicity is violated?
I’m thankful for every answer.
With two bads, I would not want to have a balanced bundle.
This sounds like a personal preference. Personally I would rather be a little thirsty AND a little cold than very thirsty OR very cold.
On to the mathematical question:
But if both goods are bad, is the indifference curve concave?
Look at these indifference curves:
Can you tell if $I_1 \prec I_2$ or if $I_1 \succ I_2$ without further information? You cannot. If you assume monotonicity, you can, but that is not implicit.
A similar exercise: draw some indifference curves for $U(x,y) = xy$, then do the same for $\hat{U}(x,y) = -xy$. Notice that the two "maps" look the same, hence the curves have the same concavity/convexity; but according to $U$, $x,y$ are goods, while according to $\hat{U}$, $x,y$ are bads.
The definition of monotonicity is:
$\forall i, x_i’ > x_i \implies (x_1’,\dots,x_n’) \succ (x_1,\dots,x_n).$
What this means is that if you take a consumption vector (bundle) and increase the consumption of all the goods, the new bundle has to be better.
If the “goods” were bads, increasing consumption of them would lead to a worse bundle. This means that monotonicity is indeed violated if the “goods” are bads.
Being monotone means that the goods are always indeed, goods, not bads.
Bonus: The first definition I gave is also referred to as “weak monotonicity”.
Strong monotonicity as its name implies, is a property that implies weak monotonicity but not backwards. Its definition is:
$\forall i, x_i’ \geq x_i$ and $\exists j : x_j’ > x_j \implies (x_1’,\dots,x_n’) \succ (x_1,\dots,x_n).$
This means that increasing consumption of just one of the goods is enough to achieve a strictly better bundle. Many of the commonly used preferences concerning goods that are actually “goods” satisfy this property.
The most familiar exception being the Leontief or Perfect Complements preferences/utility function in which increasing consumption of both goods achieves a strictly better bundle (monotone) but increasing consumption of only one good at the corners/kinks of the indifference curves will leave you at an indifferent bundle (same utility level), hence not strictly monotone.
