If households had a limit on their debt (say 5x-4x) relative to their income the severity of both booms and busts would decrease. In "booms" many consumers feel safe in taking on way more debt than they can effectively pay off. This unsustainable buying on credit turns the whole economy into a bubble. When the bubble deflates people loose their jobs and have no way to pay back their debts which makes the situation much worse. If people wouldn't be able to take on all that debt in the beginning there wouldn't have been such a big "boom" but later the "bust" wouldn't have been as bad. The main idea is to get away from booms and busts and to the "grow line" (the ultimate growth in the economy).
You cannot prove that limiting debt has a direct relation to effectively paying debt off. Mandatory classes in economics and debt management would be far more effective than having a 3rd party decide how much debt you can take on. I could argue that the 4 major credit bureaus already limit possible debt by deciding credit limits. This does not directly relate the action of paying off debt. Based on your model, a person making $100K a year would not be allowed to purchase a house over $400-$500k even though monthly payments would be well within a payable amount.
You cannot give a fixed (5x or 4x) factor based on to many variable in ones income and expense. We already have a system that does that. Credit bureaus already do this by providing a credit score but this is based off the likeliness of the debt being paid back vs a direct limit based on household income.