Household income is absolutely not the right metric to use here, because it'll always be proportional to the cost of the house out of necessity.
For example, if the cost of a house goes up relative to individual income, then more people in the family need to start working more hours, and more people live with roommates.
Household income stays proportionally the same, always, but individual income shows you how much people are struggling.
No, it's not the right metric. Which is why people don't use it.
Imagine you make $160K and buy the nicest house you can afford with that income.
Then you get married, and your spouse makes $100K. Your household income has increased to $260K, which means you can afford an even nicer house.
Your per capita income has decreased to $130K. By your logic, you can't afford a nicer house. In fact, with a second income you might no longer be able to afford your current house. That's nonsense.
When multiple people live in a house they all have the opportunity to contribute to paying for it. Some may contribute a lot, some (like children) may contribute nothing. The house you can afford depends on the total amount everyone contributes, aka household income.
if the cost of a house goes up
This doesn't make sense. The cost of a house is fixed when you buy it. It won't ever go up while you live there.
When everyone needs a separate individual house that they refuse to share, then per capita income will be relevant.
But in the real world, people buy houses because they want to share them with their family. On average, 2.5 people live in a single house. And the median household income is $63K in Michigan.
So if you really want to look at per capita income, that means there is $87.5K (2.5 x $35K) available to buy a house, which is easily enough to afford a $300K home.
You insist on using per capita income for some reason. That means that even in a single earner family, the income is considered to be divided equally among the family members.
If you are a single earner making $160K in a family with two adults and two kids, then when using per capita income you consider all four family members as making $40K each. That's the definition of per capita.
And that's why it's nonsense to say that per capita income of $40K is not enough to afford a house. A family of five with per capita income of $40K would be in the top 10%.
Household income is total family income, regardless of whether there is one earner or more than one. And it's not tautological. If your household income is very low, then your family may not be able to afford to buy a house.
A household includes all the people who occupy a housing unit (such as a house or apartment) as their usual place of residence.
A household includes the related family members and all the unrelated people, if any, such as lodgers, foster children, wards, or employees who share the housing unit. A person living alone in a housing unit, or a group of unrelated people sharing a housing unit such as partners or roomers, is also counted as a household.