Repunit-style numbers whose product equals their sum

Find two real numbers $x$ and $y$, each written using only the digit $1$ in decimal (a decimal point is allowed), such that $xy=x+y$. What is the common value of $xy$?

$12.1$. The condition rearranges to $x=\tfrac{y}{y-1}$. Taking $y=11$ gives $x=\tfrac{11}{10}=1.1$, and indeed $11\times1.1=11+1.1=12.1$.