# Are the indifference curves for bads concave?

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?

• Before considering the shapes of indifference curves, we need to be clear as to how the axes, and particuarly the scales of the axes, are defined. That's fairly straightforward for the standard textbook case of two private "good" goods: the axis for a good represents the quantity of the good consumed (or perhaps purchased) in a period. It's much less clear for "bad" goods. Except perhaps out of ignorance, no one is going to choose to purchase a bad. They may "consume" a bad, but probably in circumstances in which a public bad is "bundled" with a private good, eg they want to travel .... Feb 8 at 19:05
• ... from A to B but can't do so without experiencing the air pollution on the way, or want to achieve some challenge such as climbing a mountain but can't do so without suffering cold and taking a risk of accident. It isn't clear how an axis for such a bad would be defined. Feb 8 at 19:09

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.

• This is very helpful! Feb 7 at 23:12