An increase in household income will always lead to a rise in nominal interest rates due to higher demand for money, regardless of changes in the supply of loanable funds.

When household income increases, people tend to spend and invest more money, which raises the demand for money. This higher demand for money can put upward pressure on nominal interest rates because lenders want to charge more for borrowing money when more people want to borrow. However, it's important to remember that interest rates are also influenced by the supply of loanable funds. If the supply of money available for loans increases at the same time, it can offset the rise in interest rates. For example, if a local bank receives more deposits from increased savings, it might lower interest rates even if income is rising, showing that both demand and supply play a crucial role in determining interest rates.