DSE 2010 Question 27: Keynesian cross and the multiplier

Consider a closed-economy Keynesian model: prices fixed, output determined by aggregate demand, investment fixed at $I^{*}$, no government. Two household groups $A$ and $B$ share total income $Y$ as $Y^{A}=\lambda Y$ and $Y^{B}=(1-\lambda)Y$, $\lambda\in(0,1)$. Consumption is $C^{A}=c+c_{A}Y^{A}$ and $C^{B}=c+c_{B}Y^{B}$ with $0<c_{A}<c_{B}<1$. The investment multiplier is